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Helsinki University of Technology Department of Electrical andCommunications Engineering Laboratory of Electromechanics
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Helsinki University of Technology Department of Electrical and
Communications Engineering Laboratory of Electromechanics
Teknillinen korkeakoulu sähkö- ja tietoliikennetekniikan osasto sähkömekaniikan laboratorio
Espoo 2000 Raportti 63
INDUCTION MOTOR MODELLING FORVECTOR CONTROL PURPOSES
Mircea Popescu
Helsinki University of Technology
Department of Electrical and Communications Engineering
Laboratory of Electromechanics
Teknillinen korkeakoulu
Sähkö- ja tietoliikennetekniikan osasto
Sähkömekaniikan laboratorio
2
Popescu M., Induction Motor Modelling for Vector Control Purposes, Helsinki University ofTechnology, Laboratory of Electromechanics, Report, Espoo 2000, 144 p.
Keywords: Induction motor, vector control, d-q models, continuos time domain, discrete timedomain, linearization
Abstract
Widely used in many industrial applications, the induction motors represent the starting point when
an electrical drive system has to be designed. In modern control theory, the induction motor isdescribed by different mathematical models, according to the employed control method. In the
symmetrical three-phase version or in the unsymmetrical two-phase version, this electrical motortype can be associated with vector control strategy. Through this control method, the inductionmotor operation can be analysed in a similar way to a DC motor. The goal of this research is to
summarize the existing models and to develop new models, in order to obtain a unified approach onmodelling of the induction machines for vector control purposes. Starting from vector control
principles, the work suggests the d-q axes unified approach for all types of the induction motors.However, the space vector analysis is presented as a strong tool in modelling of the symmetricalinduction machines. When an electrical motor is viewed as a mathematical system, with inputs and
outputs, it can be analysed and described in multiple ways, considering different reference framesand state-space variables. All the mathematical possible models are illustrated in this report. Thesuggestions for what model is suitable for what application, are defined as well. As the practical
implementation of the vector control strategies require digital signal processors (DSP), from thecontinuos time domain models are derived the discrete time domain models. The discrete models
permit the implementation of the mathematical model of the induction motors, in order to obtainhigh efficiency sensorless drives. The stability of these various models is analysed.
Distribution:
Helsinki University of Technology
Laboratory of Electromechanics
P.O. Box 3000
FIN-02015 HUT
Tel: +358-9-451-2384
Fax: +358-9-451-2991
E-mail: [email protected]
© Mircea Popescu
ISBN 951-22-5219-8
ISSN 1456-6001
Picaset Oy
Helsinki 2000
4
Contens
Abstract ………………………………………………….……………………………….…
Preface …………………………………………………………………………………..…..
List of principal symbols …………………………………….………………………..…….
1. Vector control of induction motors - Overview……………………………..…………...1.1 Introduction …………………………………………………………………1.2 Algorithm of Vector Control ………………………………………………..
1.3 Field Orientation Control ……………………………………………………1.4 Direct Torque Control ………………………………………………………
2. Continous-time domain linear models of the three-phase induction machine …………..2.1 Introduction …………………………………………………………………
2.2 Voltage and flux linkage equations …………………………………………2.3 Space vector equations for three-phase induction machines ………………..
2.4 Vectorial equations system in a common reference frame …………….……2.5 Induction machine equations with stator referred rotor variables …………..2.6 Instantaneous electromagnetic torque ………………………………………
2.7 General equations of the induction machine in different reference frames ….2.7.1. Per unit system ………………………………………………………….2.7.2. Stationary reference frame equations. Block diagram …………………..
2.7.3. Rotor reference frame. Block diagram ………………………………….2.7.4. Synchronous reference frame. Block diagram ………………………….
2.8. D-Q Axes models of the three-phase induction machine …………………...2.8.1. Models with currents space vectors as state-space variables …………...2.8.2. Models with fluxes linkages space vectors as state-space variables ……
2.8.3. Models with mixed currents -flux space vectors as state-space variables2.9. Vector control strategies for three-phase induction machine ……………….
2.9.1. Stator flux field orientation (SFO) ………………………………………2.9.2. Rotor flux field orientation (RFO) ………………………………………2.9.3. Air-gap flux field orientation (AFO) ……………………………………
2.9.4. Stator current orientation (SCO) ………………………………………..2.9.5. Rotor current orientation (RCO) ………………………………………...
3. Continuos-time domain models of the single-phase induction machine ……………3.1. Introduction …………………………………………………………………
3.2. Voltage and flux-current equations of the single-phase induction machine ..3.3. Analysis of the single-phase induction machine in stationary reference
frame ………………………………………………………………………...3.4. Analysis of the steady-state operation for the symmetrical single-phase
induction machine …………………………………………………………..
3.5. Analysis of the unsymmetrical single-phase induction machine ……………3.6. Continuos linear models for single-phase induction machine ………………
3.6.1. Linear Γ model of the symmetrical single-phase induction machine .
2
3
6
99
11
1314
1717
1721
242627
292931
3234
414345
4751
525456
5860
6565
65
68
70
7172
73
5
3.6.2. Linear inverse Γ model of the symmetrical single-phase induction
machine …………………………………………………………..3.6.3. Universal model of the symmetrical single-phase induction machine
3.6.4. Linear Γ model of the unsymmetrical single-phase inductionmachine ……………………………………………………………..
3.6.5. Linear inverse Γ model of the unsymmetrical single-phase inductionmachine ……………………………………………………………...
3.6.6. Universal model of the unsymmetrical single-phase induction
machine ……………………………………………………………..3.7. D-Q axes models of the single-phase induction machine …………………..
3.7.1. Models with currents space vectors as state-space variables ……….3.7.2. Models with fluxes linkages space vectors as state-space variables ..3.7.3. Models with mixed currents-fluxes space vectors as state-space
variables ……………………………………………………………..3.8. Vector control strategies for single-phase induction machine ………………
3.8.1. Stator flux field orientation (SFO) ………………………………….3.8.2. Rotor flux field orientation (RFO) ………………………………….3.8.3. Air-gap flux field orientation (AFO) ………………………………..
3.8.4. Stator current orientation (SCO) ……………………………………3.8.5. Rotor current orientation (RCO) ……………………………………
4. Mathematical discrete models for the three-phase induction machine ……………..4.1. Introduction …………………………………………………..……………..
4.2. Bilinear transformation method (Tustin) ……………………………………4.3. Forward-differences method (Euler) ………………………………………..
4.4. Backward-differences method ………………………………………………4.5. Z-domain transfer functions …………………………………………………4.6. Stability analysis …………………………………………………………….
5. Mathematical discrete models for the single-phase induction machine …………….5.1. Introduction ………………………………………………………………….
5.2. Bilinear transformation method (Tustin) ……………………………………5.3. Forward-differences method (Euler) ………………………………………..
5.4. Backward-differences method ………………………………………………5.5. Z-domain transfer functions …………………………………………………
6. Linearisation of the induction machine mathematical models ……………………...6.1. Introduction …………………………………………………………………
6.2. Three-phase induction machine …………………………………………….6.3. Single-phase induction machine ……………………………………………
References …………………………………………………………………………………..
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838487
89
949597
99101
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108110113
115117
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123125
127129131
135
135136138
141
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List of principal symbols:
Scalar variables are denoted by plane letters. Vector variables are denoted by underlined letters.
Boldface symbols are used for matrix variables.
, , , , ,A B C D E F state-space coefficient matrix
ija coefficients for state-space variables
mB viscous friction coefficient
H relative inertia constantH transfer function matrix
(a,b,c)sI instantaneous stator phase current for the three-phase induction machine
(a,b,c)rI instantaneous rotor phase current for the three-phase induction machine
(d,q)sIɶ complex stator phase currents in d-q co-ordinates for steady-state analysis
(a,b)sIɶ complex stator phase currents in physical co-ordinates for steady-state
analysis
(d,q)rIɶ complex rotor phase currents in d-q co-ordinates for steady-state analysis
(a,b)rIɶ complex rotor phase currents in physical co-ordinates for steady-state
analysis
sI stator current space vector
rI rotor current space vector
(a,b,c)si instantaneous stator phase currents for the three-phase induction machine
(a,b,c)ri instantaneous rotor phase currents for the three-phase induction machine
(a,b)si instantaneous stator phase currents for the single-phase induction machine
(a,b)ri instantaneous rotor phase currents for the single-phase induction machine
(A,B,C)si phase currents for a three-phase system
(X,Y)si phase currents for an orthogonal two-phase system
(d,q)si instantaneous stator phase currents in d-q co-ordinates
(d,q)r'i instantaneous referred rotor phase currents in d-q co-ordinates
(d,q)R'i Γ models instantaneous referred rotor phase currents in d-q co-ordinates
s( , )R Ii stator phase currents in complex co-ordinates and per unit system
r ( , )' R Ii rotor phase currents in complex co-ordinates and per unit system
si stator current space vector in per unit system
r'i referred rotor current space vector in per unit system
J inertia constantj complex operator
p,iK proportional, respectively integrative constant for PI controllerss,r
s,rK transformation matrix from abc co-ordinates to d-q co-ordinates
k turns ratio for the unsymmetrical single-phase machine
(a,b,c)(s,r)L self-inductance for stator phase, respectively rotor
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l(s,r)L leakage inductance for symmetrical stator phase, respectively rotor
l(m,a)L leakage inductance for unsymmetrical stator phase: main, respectively
auxiliary
Ml magnetisation inductance in per unit system
sl total stator inductance in per unit system
r'l total rotor inductance in per unit system
(a,b,c),(s,r)M mutual inductance for stator phase, respectively rotor
s,rN turns number for stator phase, respectively rotor
P number of polesp derivative operator
sR per unit stator phase resistance for the three-phase induction machine
sr stator phase resistance for the symmetrical induction machine
mr main stator phase resistance for the unsymmetrical single-phase machine
ar auxiliary stator phase resistance for the unsymmetrical single-phase machine
rr rotor phase resistance for the symmetrical induction machine
r'r referred rotor phase resistance for the symmetrical induction machine
s Laplace operator
ms critical slip for the induction machine
T sampling period
eT instantaneous electromagnetic torque
LT load torque
t time
(a,b,c)sU instantaneous stator phase voltage for the three-phase induction machine
(a,b,c)rU instantaneous rotor phase voltage for the three-phase induction machine
s ( , )R IU stator phase voltage in complex co-ordinates and per unit system
°(d,q)sU complex stator phase voltages in d-q co-ordinates for steady-state analysis
°(a,b)sU complex stator phase voltages in physical co-ordinates for steady-state
analysis
sU stator voltage space vector
rU rotor voltage space vector
(d,q)su instantaneous stator phase voltages in d-q co-ordinates
(d,q)r'u instantaneous rotor phase voltages in d-q co-ordinates
su stator voltage space vector in per unit system
r'u referred rotor voltage space vector in per unit system
cW magnetic coenergy
bX base variable value for per unit system
x space vector variable
mx magnetisation reactance
lsx stator phase leakage reactance for the symmetrical induction machine
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l(m,a)x stator phase leakage reactance for the unsymmetrical induction machine:
main, respectively auxiliary
lr'x rotor leakage reactance
Lx Γ models equivalent leakage reactance for symmetrical induction machine
L(d,q)x Γ models equivalent leakage reactance for unsymmetrical induction machine
Mx Γ models equivalent magnetisation reactance for symmetrical induction
machine
M(d,q)x Γ models equivalent magnetisation reactance for unsymmetrical induction
machinez discrete co-ordinate
α space vector operator
γ Γ models turns ratio for symmetrical induction machine
(d,q)γ Γ models turns ratio for unsymmetrical induction machine
,R I∆ real, respectively imaginary component of a matrix determinant
(d,q)sψ stator flux linkage in d-q co-ordinates and per unit system
(d,q)mψ magnetisating flux linkage in d-q co-ordinates and per unit system
(d,q)rψ rotor flux linkage in d-q co-ordinates and per unit system
s( , )R Iψ stator flux linkage in complex co-ordinates and per unit system
r( , )' R Iψ rotor flux linkage in complex co-ordinates and per unit system
(d,q)sλ stator flux linkage in d-q co-ordinates and flux linkage units per second.
(a,b,c)sλ stator flux linkage fluxes in abs or ab co-ordinates
(d,q)r'λ referred rotor linkage fluxes in d-q co-ordinates and flux linkage units per
second
(d,q)R'λ Γ models referred rotor linkage fluxes in d-q co-ordinates and flux linkage
units per second
(a,b,c)rλ rotor flux linkage in abs or ab co-ordinates
r'λ referred rotor flux linkage space vector for three-phase induction machine
sλ stator flux linkage space vector for three-phase induction machine
γλ arbitrarily flux linkage space vector for three-phase induction machine
mℜ magnetic reluctance
s,rσ leakage factor for stator phase, respectively rotor phase
rθ periferical displacement between stator and rotor space vectors
kθ periferical displacement between stator and arbitrary space vectors
Ω relative angular frequency in per unit systemω angular frequency of the supply system
bω base angular frequency of the supply system
nω rated angular frequency of the induction machine in electrical degrees
rω rotor angular frequency in electrical degrees
9
1. VECTOR CONTROL OF INDUCTION MOTORS - OVERVIEW
1.1.Introduction
The electrical DC drive systems are still used in a wide range of industrial applications,although they are less reliable than the AC drives. Their advantage consists in simple and precisecommand and control structures.
The AC drives, sometimes more expensive but far more reliable, (Rajashekara et al. 1996)require complex modern control techniques. The design of a control system is realised in two
important steps:1. The drive system has to be converted into a mathematical model, in order to accomplish the
analysis and the evaluation of the system.
2. The imposed response of the drive system is obtained through an optimal regulator, whenexternal perturbations are present.
The induction motors are relatively cheap and rugged machines because their construction isrealised without slip rings or commutators. These advantages have determined an importantdevelopment of the electrical drives, with induction machine as the execution element, for all
related aspects: starting, braking, speed reversal, speed change, etc. The dynamic operation of theinduction machine drive system has an important role on the overall performance of the system of
which it is a part.There are two fundamental directions for the induction motor control:
• Analogue: direct measurement of the machine parameters (mainly the rotor speed), which are
compared to the reference signals through closed control loops;• Digital: estimation of the machine parameters in the sensorless control schemes (without
measuring the rotor speed), with the following implementation methodologies:• Slip frequency calculation method;
• Speed estimation using state equation;
• Estimation based on slot space harmonic voltages;
• Flux estimation and flux vector control;
• Direct control of torque and flux;
• Observer-based speed sensorless control;
• Model reference adaptive systems;
• Kalman filtering techniques;
• Sensorless control with parameter adaptation;
• Neural network based sensorless control;
• Fuzzy-logic based sensorless control.
Another classification of the control techniques for the induction machine is made by Holtz
(1998) from the point of view of the controlled signal:a) Scalar control:a.1 Voltage/frequency (or v/f) control;
a.2 Stator current control and slip frequency control. These techniques are mainly implementedthrough direct measurement of the machine parameters.
b) Vector control:b.1 Field orientation control (FOC): b.1.1. Indirect method; b.1.2. Direct method;b.2 Direct torque and stator flux vector control. These techniques are realised both in analogue
version (direct measurements) and digital version (estimation techniques)The development of accurate system models is fundamental to each stage in the design, analysis
and control of all electrical machines. The level of precision required of these models dependsentirely on the design stage under consideration. In particular, the mathematical description used in
10
machine design requires very fine tolerance levels as stated by Nabae et al. (1980) and Murata et al(1990). However, in the development of suitable models for control purposes, it is possible to make
certain assumptions that considerably simplify the resulting machine model. Nonetheless, thesemodels must incorporate the essential elements of both the electromagnetic and the mechanical
systems for both steady state and transient operating conditions (Nowotny and Lipo - 1996).Additionally, since modern electric machines are invariably fed from switching power conversionstages, the developed motor models should be valid for arbitrary applied voltage and current
waveforms. This work presents suitable models for use in digital current control of the inductionmotors. In addition, the limits of the validity of these models are summarised and, in some cases,
the models are extended to account for some non-idealities of the machine.Usually, the following assumptions are made (Lorenz et al. 1994):
• No magnetic saturation, i.e. machine inductance is not affected by current level.
• No saliency effects i.e. machine inductance are not functions of position.
• Negligible spatial mmf harmonics i.e. stator windings are arranged to produce sinusoidal mmf
distributions.
• The effects of the stator slots may be neglected.
• There is no fringing of the magnetic circuit.
• The magnetic field intensity is constant and radially directed across the air-gap.
• Eddy current and hysteresis effects are negligible.
The modern control theory for an electrical drive system requires the existence of a real-time,stabile, and precise mathematical model for each component of the system. The analysis and thedesign of the numerical command for such systems depend on the hardware and software resources.
If in communication techniques the real-time response of the system is not always compulsory, inindustrial processes the real-time response of the drive systems is essential.
The soft numerical command for the electrical drive systems is far more flexible to implementthan the hardware version. For the latter, lately there is an intense research effort for implementingASIC (application specific integrated circuit). The numerical command of the electrical drive
systems is a challenging task mainly due to the DSP (digital signal processing) technology. Now itis possible to realise linear and non-linear techniques for implementing continuos and discrete
mathematical models of the entire element of an electrical drive system, including the electricalmachine (Xu and Nowotny - 1990, 1992).
For the AC drives there are several solutions for implementing the command and the control of
the system. A quick summary of the existing technologies already out there in the field is givenbelow:
DC DrivesInitially the DC drives were used for variable speed control because they could easily achieve a
good torque and speed response with high accuracy. Field orientation of the motor is achieved using
a mechanical commutator with brushes. In DC, torque is controlled using the armature current andfield current. The main drawback of this technique is the reduced reliability of the DC motor - the
fact that brushes and commutators wear down and need regular servicing; that DC motors can becostly to purchase; and that they require encoders for positional feedback.AC Drives
The evolution of AC variable speed drive technology has been partly driven by the desire toemulate the performance of the DC drive, such as fast torque response and speed accuracy, while
using robust, cheap to purchase and relatively maintenance-free AC motors (Kelemen and Imecs -1987).AC Drives, frequency controlled using PWM
With this technique, sometimes known as scalar control, the field orientation of the motor is notused. Instead, the frequency and the voltage are the main control variables and are applied to thestator windings. The status of the rotor is ignored, meaning that no speed or position signal is fed
11
back. The drive is therefore regarded as an open-loop drive. This type of drive is suitable forapplications such as pumps and fans, which do not require high levels of accuracy or precision.
AC Drives, flux vector control using PWMHere, field orientation is achieved by mathematical modelling using microprocessors and
feedback of rotor speed and angular position relative to the stator field by means of an encoder (Vas- 1990). This results in a drive with greater stability and capable of fast torque response andaccurate speed control. But the drawback is the need for the encoder, which reduces drive system
reliability and adds cost. The controlling variables in a DC drive for torque are armature current andfield current, and armature voltage for torque. AC drives using the PWM principle; however, use
voltage and frequency as the controlling variables and these are controlled by a device called amodulator. A modulator adds considerable delay in the responsiveness of a motor to changes intorque and speed. Furthermore, with flux vector AC drives, a tacho-generator or position encoder is
invariably needed to obtain any real degree of accuracy. Such devices are costly and compromisethe simplicity of the AC induction motor.
AC Drives, sensorless flux vectorThe flux vector controlled drive with encoder feedback does offer very high levels of
performance across a wide power range and should not be confused with sensorless vector - or open
loop vector - drives, which offer performance only slightly superior to that of a standard inverterusing scalar control (Rajashekara et al. - 1996).
1.2.Algorithm of vector control
The induction motors are very common because they are inexpensive and robust, finding use ineverything from industrial applications such as pumps, fans, and blowers to home appliances.Traditionally, induction motors have been run at a single speed, which was determined by the
frequency of the main voltage and the number of poles in the motor. Controlling the speed of aninduction motor is far more difficult than controlling the speed of a DC motor since there is no
linear relationship between the motor current and the resulting torque as there is for a DC motor.The technique called vector control can be used to vary the speed of an induction motor over a
wide range. It was initially developed by Blaschke (1971-1973). In the vector control scheme, a
complex current is synthesised from two quadrature components, one of which is responsible forthe flux level in the motor, and another which controls the torque production in the motor.
Essentially, the control problem is reformulated to resemble the control of a DC motor. Vectorcontrol offers a number of benefits including speed control over a wide range, precise speedregulation, fast dynamic response, and operation above base speed.
The vector control algorithm is based on two fundamental ideas. The first is the flux and torqueproducing currents. An induction motor can be modelled most simply (and controlled most simply)
using two quadrature currents rather than the familiar three phase currents actually applied to themotor. These two currents called direct (Id) and quadrature (Iq) are responsible for producing fluxand torque respectively in the motor. By definition, the Iq current is in phase with the stator flux,
and Id is at right angles. Of course, the actual voltages applied to the motor and the resultingcurrents are in the familiar three-phase system. The move between a stationary reference frame and
a reference frame, which is rotating synchronous with the stator flux, becomes then the problem.This leads to the second fundamental idea behind vector control.
The second fundamental idea is that of reference frames. The idea of a reference frame is to
transform a quantity that is sinusoidal in one reference frame, to a constant value in a referenceframe, which is rotating at the same frequency. Once a sinusoidal quantity is transformed to a
constant value by careful choice of reference frame, it becomes possible to control that quantitywith traditional proportional integral (PI) controllers.
12
Vector transformsThe Park and Clarke vector transforms are one of the keys to vector control of induction motors.
I) Clarke transformThe forward Clarke (1943) transform does a magnitude invariant translation from a three phase
system into two orthogonal components. If the neutral - ground connection is neglected, thevariables in a three-phase system (A, B, and C) sum is equal to zero, and there is a redundantinformation. Therefore, the system can be reduced to two variables, called X and Y. The Clarke
transform is given by:
As
Xs
Bs
Ys
Cs
( )( ) 1 cos( ) cos(2 )2
( )( ) 0 sin( ) sin(2 )3
( )
i ti t
i ti t
i t
γ γ
γ γ
= ⋅ ⋅
(1)
where:
2
3
πγ =
Using the relation:
As Bs Cs( ) ( ) ( ) 0i t i t i t+ + = (2)
and the fact that:2 4 1
cos c o s3 3 2
π π = = −
(3)
Thus, the Clarke transform can be simplified to:
( )
As
X s
As CsB s
( )( )
1( ) ( )( )
3
i ti t
i t i ti t
= ⋅ −
(4)
The Clarke transform can also be understood using a vector diagram as shown in Fig. 1.1. In thefigure, A, B, and C are the axes of a three phase system, each offset 120° from the other. X and Y
are the axes of a two variable system where X is chosen to be coincident with A. To perform theClarke transform of a three variable system (iA, iB, iC), iX is equal to iA and iY is the scaledprojection of iB and iC onto the Y axis. The scaling is necessary to preserve the signal magnitudes
through the transform.
BY
i
C
A, X
0i
X
iB
iC
iY
s
iA
=
Fig. 1.1. Clarke Transform Vector Diagram
The Clarke transform preserves the magnitude, and realise a quadrature between the current
components.
13
II) Park transformThe Park (1929) transform is a vector rotation, which rotates a vector (defined by its quadrature
components) through a specified angle. The Park transform function implements the following setof equations:
x x
y y
Out ( ) In ( )cos( ) sin( )
Out ( ) In ( )sin( ) cos( )
t t
t t
θ θ
θ θ
− = ⋅
(5)
where θ is the angle to rotate the vector through. A reverse vector rotation can be accomplished
simply by changing the sign on the sin (θ) input value. The vector rotation is illustrated by Fig. 1.2.
Some references (Vas -1990, Nowotny and Lipo - 1996) describe the Park transform as acombination of the Clark and Park transforms presented here. Breaking into a three-variable-to-twotransform (i.e. the Clarke transform) and a vector rotation is done for efficiency of calculation: with
separate Park and Clarke transforms, only two trigonometric calculations are required as opposed to6 in the traditional Park transform.
QY
i
D
X
0i
X
iQ i
D
iY
θ
s
Fig.1.2. Park transform vector diagram
1. 3. Field orientation control (FOC)
Vector control techniques have made possible the application of induction motors for high-
performance applications where traditionally only DC drives were applied (Holtz - 1995). Thevector control scheme enables the control of the induction motor in the same way as separately
excitation DC motors. As in the DC motor, torque control of induction motor is achieved bycontrolling the torque current component and flux current component independently. The basicschemes of indirect and direct methods of vector control are shown in Figs. 1.3 –1.5. The direct
vector control method depends on the generation of unit vector signals from the stator or air-gapflux signals. The air-gap signals can be measured directly or estimated from the stator voltage and
current signals. The stator flux components can be directly computed from stator quantities. In thesesystems, rotor speed is not required for obtaining rotor field angle information. In the indirect vectorcontrol method, the rotor field angle and thus the unit vectors are indirectly obtained by summation
of the rotor speed and slip frequency.
Flux
Statorθs
Fig. 1.3. Position of the rotor flux vector
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FOC Inverter IM
Speed
sensor
Slip frequency
calculation
Fluxcommand
Torquecommand
+ +
Voltage Current
ωωrsr
1
s
θs
Fig. 1.4. Indirect vector control method
Flux
command
Torque
command
FOC Inverter
Voltage
CurrentIM
Speedsensor
Flux vectormeasurementor estimation
θs
Fig. 1.5. Direct vector control method
Fundamental requirements for the FOC are the knowledge of two currents (if the inductionmotor is star connected) and the rotor flux position. Knowledge of the rotor flux position is the coreof the FOC. In fact if there is an error in this variable the rotor flux is not aligned with d-axis and
the current components are incorrectly estimated. In the induction machine the rotor speed is notequal to the rotor flux speed (there is a slip speed; as such, a special method to calculate the rotor
flux position (angle) is needed. The basic method is the use of the current model.Thanks to FOC it becomes possible to control, directly and separately, the torque and flux of the
induction motors. Field oriented controlled induction machines obtain every DC machine
advantage: instantaneous control of the separate quantities allowing accurate transient and steady-state management.
1. 4. Direct torque control
The most modern technique is direct torque and stator flux vector control method (DTC). It hasbeen realised in an industrial way by ABB, by using the theoretical background proposed byBlashke and Depenbrock during 1971-1985. This solution is based both on field oriented control
(FOC) as well as on the direct self-control theory.Starting with a few basics in a variable speed drive the basic function is to control the flow of
energy from the mains to a process via the shaft of a motor. Two physical quantities describe thestate of the shaft: torque and speed. Controlling the flow of energy depends on controlling these
15
quantities. In practice either one of them is controlled and we speak of "torque control" or "speedcontrol". When a variable speed drive operates in torque control mode the speed is determined by
the load. Torque is a function of the actual current and actual flux in the machine. Likewise whenoperated in speed control the torque is determined by the load.
Variable speed drives are used in all industries to control precisely the speed of electric motorsdriving loads ranging from pumps and fans to complex drives on paper machines rolling millscranes and similar drives.
The idea is that motor flux and torque are used as primary control variables which is contrary tothe way in which traditional AC drives control input frequency and voltage, but is in principle
similar to what is done with a DC drive, where it is much more straightforward to achieve. Incontrast, traditional PWM and flux vector drives use output voltage and output frequency as theprimary control variables but these need to be pulse width modulated before being applied to the
motor. This modulator stage adds to the signal processing time and therefore limits the level oftorque and speed response time possible from the PWM drive.
In contrast, by controlling motor torque directly, DTC provides dynamic speed accuracyequivalent to closed loop AC and DC systems and torque response times that are 10 times faster. Itis also claimed that the DTC does not generate noise like that produced by conventional PWM AC
drives. And the wider spectrum of noise means that amplitudes are lower which helps to controlEMI and RFI emissions. The basic structure of direct torque and stator flux vector control is
presented in Fig. 1.6.
Flux
command
Torque
command
Flux
controller
Torque
controller
Voltage
vector
selection
Inverter
Voltage
Current
IM
Speed
sensor
Flux vector
measurement
or estimation
Fig. 1.6. Basic structure of direct torque and flux vector control
In DTC field orientation is achieved without feedback using advanced motor theory to calculate
the motor torque directly and stator flux without using a modulator or a requirement for atachogenerator or position encoder to feed back the speed or position of the motor shaft. Both
parameters are obtained instead from the motor itself. DTC's configuration also relies on two keydevelopments - the latest high-speed signal processing technology and a highly advanced motormodel precisely simulating the actual motor within the controller. A DSP (digital signal processor)
is used together with ASIC hardware to determine the switching logic of the inverter.The motor model is programmed with information about the motor, which enables it to
determine parameters including stator resistance, mutual inductance saturation coefficients andmotor inertia. The model also encompasses temperature compensation, which is essential for goodstatic speed accuracy without encoder.
16
In normal operation, measurements of the two motor phase currents and the drive DC linkvoltage, together with information about the switching state of the inverter are fed into the motor
model . The motor model then outputs control signals, which are accurate estimates of the actualmotor torque and actual stator flux. All control signals are transmitted via optical links for high
speed. In this way, the semiconductor switching devices of the inverter are supplied with anoptimum switching pattern for reaching or maintaining an accurate motor torque.
Also, both shaft speed and electrical frequency are calculated within the motor model. There is
no need to feedback any shaft speed or position with tachometers or encoders to meet the demandsof 95% of industrial applications. However, there will always be some special applications where
even greater speed accuracy will be needed and when the use of an encoder improves the accuracyof speed control in DTC. But even then, the encoder does not need to be as costly or as accurate asthe one used in traditional flux vector drives, as DTC only has to know the error in speed, not the
rotor position.The drive will have a torque response time typically better than 5ms. This compares with
response for both flux vector PWM drives and DC drives fitted with encoders. The newersensorless flux vector drives now being launched by other drives manufacturers have a torqueresponse measured in hundreds of milliseconds.
DTC also provides exceptional torque control linearity. For the first time with an open loop ACdrive, torque control can be obtained at low frequencies, including zero speed, where the nominal
torque step can be increased in less than 1ms. The dynamic speed accuracy of DTC drives is betterthan any open loop AC drives and comparable to DC drives, which use feedback.
DTC brings other special functions, not previously available with AC drives, including
automatic starting in all motor electromagnetic and mechanical states. There is no need foradditional parameter adjustments, such as torque boost or starting mode selection, such as flyingstart. DTC control automatically adapts itself to the required condition. In addition, based on exact
and rapid control of the drive intermediate DC link voltage, DTC can withstand sudden loadtransients caused by the process, without any overvoltage or overcurrent trip.
17
2. CONTINUOUS-TIME DOMAIN LINEAR MODELS OF THETHREE-PHASE INDUCTION MACHINE
2.1. Introduction
Until the last decades the three-phase induction machine was mainly used in constant speeddrives due to the control system performance, not to the operating principle of the machine.
Nowadays, this situation is completely changed. With the technical progress in power electronicsand microelectronics, the three-phase induction machine control becomes very flexible and highlyefficient. Since 1983, the year when the digital signal processor (DSP) appeared, the control theory
for this type of machine was permanently improved.New mathematical models have to be implemented for the three-phase induction machine in
order to analyse its operation both dynamically and in steady-state.
2.2. Voltage and flux linkage equations
The first mathematical model for the dynamic analysis of the induction machine was based on
the two real axis reference frame, developed initially by Park (1929) for the synchronous machine.Using the symmetric configuration of the induction machine, Kovacs and Racz (1959) haveelaborated the space complex vector theory, and obtained a model for the steady-state analysis of
the machine. Both theories are used for modelling the three-phase induction machine. Thefollowing assumptions are made when a complete equations system is written to describe the
continuous-time linear model of the induction machine (Krause et al. 1995):• Geometrical and electrical machine configuration is symmetrical;
• Space harmonics of the stator and rotor magnetic flux are negligible;
• Infinitely permeable iron;
• Stator and rotor windings are sinusoidally distributed in space and replaced by an equivalent
concentrated winding;
• Saliency effects, the slotting effects are neglected;
• Magnetic saturation, anisotropy effect, core loss and skin effect are negligible;
• Windings resistance and reactance do not vary with the temperature;
• Currents and voltages are sinusoidal terms.
• End and fringing effects are neglected
All these assumptions do not alter in a serious way the final result for a wide range of induction
machines.
18
as
csbs
cr
ar
br
Uas
Ias
Ucs
Ics
Ubs
Ibs
Ucr I crU
br
I
br
ar
ar
U I
θ
Fig. 2.1. The real model of the three-phase induction machinewith three stator windings and three rotor windings
For the machine stator in Fig.2.1 if we choose the stator reference frame, the voltage equations areas follows:
asas s as
bsbs s bs
cscs s cs
dU r I
dt
dU r I
dt
dU r I
dt
λ
λ
λ
= +
= +
= +
(6-8)
wherea s bs cs, ,U U U are the instantaneous stator voltages,
as bs c s, ,I I I are the instantaneous stator
currents, s as bs csr r r r= = = is the stator winding resistance and as bs cs, ,λ λ λ are the total magnetic
fluxes for the three stator windings.The flux-current relations are determined after detailing the total flux of a stator winding. For the
other two windings, there are valid similar relations:
as asas bsas csas aras bras crasλ λ λ λ λ λ λ= + + + + + (9)
where the flux components are:
asasλ the magnetic flux produced by stator phase current as in the stator phase winding as
bsasλ the magnetic flux produced by stator phase current bs in the stator phase winding as
csasλ the magnetic flux produced by stator phase current cs in the stator phase winding as
arasλ the magnetic flux produced by rotor phase current ar in the stator phase winding as
brasλ the magnetic flux produced by rotor phase current br in the stator phase winding as
crasλ the magnetic flux produced by rotor phase current cr in the stator phase winding as
These components are computed with the expressions:
asas as as
bsas bsas bs
csas csas cs
L I
M I
M I
λ
λ
λ
=
=
=
aras aras ar
bras bras br
cras cras cr
M I
M I
M I
λ
λ
λ
=
=
=
19
Self-inductance Las has two components, one created by the linkage magnetic flux Lmas and thesecond created by the leakage magnetic flux Llas:
lasmasas LLL += (10)
The mutual inductance, which is considered to be equal due to the machine symmetry, can also besplit in two components. However, the leakage flux created component in the mutual inductance
can be neglected. It results that:
asbs bsas ascs csas mas
1
2M M M M L= = = = − (11)
The mutual inductance within the stator and rotor windings varies with the relative space position
between them. The stator flux created by current from rotor phase ar in stator phase as depend onthe angle value θ:
2 w 2
aras asar mas mas
s w t
1cos cos
s
w kM M L L
w k kθ θ= = ⋅ ⋅ = ⋅ ⋅ (12)
where kt represents the turn ratio multiplied by the winding factor ratio. In a similar way the
relations for the others mutual inductances can be written:
bras asbr mas
t
cras ascr mas
t
1 2cos
3
1 4cos
3
M M Lk
M M Lk
πθ
πθ
= = ⋅ ⋅ +
= = ⋅ ⋅ +
(13-
14)
Due to the symmetrical configuration of the induction machine, we can deduce the total magneticflux for the stator phase winding as expressed as follows:
( )as mas las as mas bs mas cs mas ar
t
mas br mas cr
t t
1 1 1cos
2 2
1 2 1 4cos cos
3 3
L L I L I L I L Ik
L I L Ik k
λ θ
π πθ θ
= + ⋅ − ⋅ − ⋅ + ⋅ +
⋅ + + ⋅ +
(15)
For the rotor windings, by using a rotor reference frame, it can be developed a similar equationsystem to the stator case:
ar
ar r ar
barbr r br
cr
r crcr
dU r I
dt
dU r I
dt
dU r I
dt
λ
λ
λ
= +
= +
= +
(16-18)
where: ar br cr, ,U U U are the instantaneous rotor voltages, ar br cr, ,I I I are the instantaneous rotor
currents, r ar br crr r r r= = = is the rotor winding resistance and ar br cr, ,λ λ λ are the total magnetic
fluxes for the three rotor windings.The total rotor magnetic flux for the winding ar is described by:
ar arar brar crar asar bsar csar
ar ar brar br crar cr asar as bsar bs csar csL I M I M I M I M I M I
λ λ λ λ λ λ λ= + + + + + =
= + + + + + (19)
In this case the mutual inductance is:
20
brar arbr mar
crar arcr mar
1
2
1
2
M M L
M M L
= = −
= = −
(20-21)
asar aras t mar
bsar arbs t mar
csar arcs t mar
cos
2cos
3
4cos
3
M M k L
M M k L
M M k L
θ
πθ
πθ
= = ⋅
= = ⋅ +
= = ⋅ +
(22-24)
Due to the symmetrical windings and motor configuration, one can write the following relation:
mas t mar m
t
1L k L L
k⋅ = ⋅ = (25)
Through a similar algorithm as that one for the stator and rotor phase as, respectively ar, it is
possible to obtain another four equations: two for the stator phases bs, and cs, and two for the rotorphases br and cr. All six final equations can be grouped in a matrix form as follows:
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]
s
s s s
r
r r r
s s s s r
r r r r s
dU r I
dt
dU r I
dt
L I M I
L I M I
λ
λ
λ
λ
= +
= +
= ⋅ + ⋅
= ⋅ + ⋅
(26-29)
where: [ ] [ ] [ ] [ ] [ ] [ ]s r s r s r, , , , ,U U I I λ λ represent the transpose matrix for the stator and rotor
voltage, current, respectively flux vectors. As an example are given the flux matrix:
[ ] [ ]T
s as bs csλ λ λ λ= [ ] [ ]T
r ar br crλ λ λ λ= (30-31)
So we get:
[ ]
mas las mas mas s
s mas mas las mas mas s
mas mas mas las s
1 1 1 11
2 2 2 2
1 1 1 11
2 2 2 2
1 1 1 11
2 2 2 2
L L L L
L L L L L L
L L L L
σ
σ
σ
+ − − + − − = − + − = ⋅ − + − − − + − − +
(32)
[ ]
mar lar mar mar r
r mar mar lar mar mar r
mar mar mar lar r
1 1 1 11
2 2 2 2
1 1 1 11
2 2 2 2
1 1 1 11
2 2 2 2
L L L L
L L L L L L
L L L L
σ
σ
σ
+ − − + − − = − + − = ⋅ − + − − − + − − +
(33)
21
[ ]s mas
t
2 4cos cos cos
3 3
1 4 2cos cos cos
3 3
2 4cos cos cos
3 3
M Lk
π πθ θ θ
π πθ θ θ
π πθ θ θ
+ +
= ⋅ ⋅ + +
+ +
(34)
[ ]r t mar
4 2cos cos cos
3 3
2 4cos cos cos
3 3
4 2cos cos cos
3 3
M k L
π πθ θ θ
π πθ θ θ
π πθ θ θ
+ +
= ⋅ ⋅ + +
+ +
(35)
Note:
1) [ ] [ ]T
s rM M= the mutual stator inductance matrix equals the transpose matrix of the mutual rotor
inductance;
2) las as lar ar
s r
mas mas mar mar
1; 1L L L L
L L L Lσ σ= = − = = − are the stator, respectively rotor leakage factors.
3) The matrix system determined above represents the flux-current equations set for the three-phase
induction machine in a reference frame attached separately to each armature.
2.3. Space vector equations for three-phase induction machines
Considering the assumptions made in the previous paragraph, the space vector notation and
concepts introduced by Racz and Kovacs (1959) are particularly useful. In this approach, allvariables are represented by polar vectors indicating the magnitude and angular position for therotating sinusoidal distribution.
A three-phase variable system can be uniquely described through a space vector, which is acomplex term and time dependent x(t) and a real homopolar component x0(t):
( )
( )
2
a b c
0 a b c
2( ) 1
3
1( )
3
x t x x x
x t x x x
α α= ⋅ ⋅ + ⋅ + ⋅
= ⋅ + +
(36)
where:
2
3
2
1
2
3
2
1
3
42
3
2
je
je
j
j
−−==
+−==
⋅
⋅
π
π
α
α
and xa, xb, xc are the phase variables.
The real axis direction coincides with that one of phase a. Usually, the neutral connection for athree-phase system is open, so that the homopolar component equals zero. The phase variables canbe easily obtained from the space vector notations:
[ ] 2
a b c( ) ( ) ( ) Re 1 ( )TT
x t x t x t x tα α = ⋅
22
The phase voltage for the induction machine can be expressed with the help of the space vectortransformation:
( )
( ) ( )
2
s as bs cs
2 2
as bs cs s as bs cs
21
3
2 21 1
3 3
U U U U
dI I I r
dt
α α
α α λ α λ α λ
= ⋅ ⋅ + ⋅ + ⋅ =
= ⋅ ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ + ⋅
(37)
or in a condensed form:
sss s
dU r I
dt
λ= + (38)
where ss s, ,U I λ are space vectors for stator voltage, current and flux. Similarly, we get the rotor
equation:
r
rr r
dU r I
dt
λ= + (39)
where rr r, ,U I λ are space vectors for rotor voltage, current and flux, respectively.
Since the machine is considered magnetically linear, the stator and flux linkage will be determined
as follows, making the notation:
[ ] [ ]21
3
2
3
2αα⋅=Α⋅
we get:
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]s rs s s
2 2 2
3 3 3L I M IΑ λ Α Α⋅ ⋅ = ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅
where:
[ ] [ ] [ ]
s
2
s mas s mas s
s
1 11
2 2
1 1 31 1
2 2 2
1 11
2 2
L L L
σ
Α α α σ σ Α
σ
+ − −
⋅ = ⋅ ⋅ − + − = ⋅ + ⋅
− − +
[ ] [ ] [ ]2
s m m
2 4cos cos cos
3 3
4 2 31 cos cos cos
3 3 2
2 4cos cos cos
3 3
jM L L e θ
π πθ θ θ
π πΑ α α θ θ θ Α
π πθ θ θ
+ +
⋅ = ⋅ ⋅ + + = ⋅ ⋅ ⋅
+ +
The condensed stator flux-current equation results from the last three equations:
s rs mas s m
3 3
2 2
jL I L I e
θλ σ
= ⋅ + ⋅ + ⋅ ⋅ ⋅
(40)
where: s rs , ,I Iλ are space vectors notations for stator flux linkage, current and rotor current.
Similarly, the rotor flux linkage-current equation is deductible:
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]r r sr r
2 2 2
3 3 3L I M IΑ λ Α Α⋅ ⋅ = ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ (41)
where:
23
[ ] [ ] [ ]
r
2
r m r mar r
r
1 11
2 2
1 1 31 1
2 2 2
1 11
2 2
L L L
σ
Α α α σ σ Α
σ
+ − −
⋅ = ⋅ ⋅ − + − = ⋅ + ⋅ − − +
[ ] [ ] [ ]2
r m m
4 2cos cos cos
3 3
2 4 31 cos cos cos
3 3 2
4 2cos cos cos
3 3
jM L L e θ
π πθ θ θ
π πΑ α α θ θ θ Α
π πθ θ θ
−
+ +
⋅ = ⋅ ⋅ + + = ⋅ ⋅ ⋅ + +
Finally, we get the condensed form of the rotor flux linkage-current equation:
r sr mar r m
3 3
2 2
jL I L I e
θλ σ − = ⋅ + ⋅ + ⋅ ⋅ ⋅
(42)
where r sr , ,I Iλ are the space vector notations for rotor flux linkage, current and stator current
respectively.For an easier manipulation of the equations we make the notations:
s mas s
r mar r
mas t mar
t
3
2
3
2
3 1 3
2 2
L L
L L
M L k Lk
σ
σ
= ⋅ +
= ⋅ +
= ⋅ ⋅ = ⋅ ⋅
The general set of voltage and flux linkage equations, written in space vector notations, is:
sss s
rrr r
s rs s
r sr r
j
j
dU r I
dt
dU r I
dt
L I M e I
L I M e I
θ
θ
λ
λ
λ
λ −
= +
= +
= + ⋅ ⋅
= + ⋅ ⋅
(43-46)
Some important conclusions have to be drawn regarding this mode of describing the machine
equations, through space vector notations:• The 12 scalar equations system written in natural reference frame is transformed in a 4 vector
equations system. This form is equivalent to substituting the real induction machine equippedwith three-phase windings on stator and rotor with a fictive machine equipped with single phasewinding on stator and rotor;
• An inconvenience of the developed system is that the stator equations system is written in stator
reference frame, and the rotor equations system is written in rotor reference frame, making the
analysis of the machine difficult;• The mutual inductance depends on the relative rotor position.
In Fig. 2.2 is illustrated the new fictive model of the induction machine from the space vector pointof view theory.
24
as
csbs
cr
ar
br
Us
Is
U
Ir
r
θ
Fig. 2.2.Model of the three-phase induction machinewith fictive one stator winding and one rotor winding
(space vector notations)
2.4. Vectorial equations system in a common reference frame
The analysis of an induction machine drive system has to be made when the stator and rotorvariables are represented in a common reference frame. When using the same space vector
notations, we can define an arbitrary reference frame, which rotates with the angular velocity ωk ,
and according to Fig. 3, the following relation is valid:( )
( ) ( ) kk j t
x t x t eθ−= ⋅ (47)
where: θk(t) is the time variable relative angle between the new reference frame and the stationary
reference frame initially considered.; xk(t) represents the space vector for the new reference frame.
The reverse transformation relation is:)()()( tjk
ketxtx θ⋅= (48)
The homopolar component being a scalar variable, is independent from the chosen reference frame.
Re0
Rek
x
ω
θ
k
k
Fig. 2.3. Transformation into an arbitrary reference frame
In an arbitrary common reference frame, the vectorial voltage and flux linkage equations become:
25
( ) ( )( )
( )
s rss rs
s ss s s s
s r
s s rs s
s rs
ss s
kk
k k
k k k k k
k
jk kjj
k kj j
k k
k k kj j j j jk k
k k
kj
k
d L I e M e Id L I M e IU r I U e r I e
dt dt
d dd I d Ir I e L e j e I M e j e I
dt dt dt dt
d L I M Ie r I j L
dt
θ θ θθθ
θ θ
θ θ θ θ θ
θ
θ θ
ω
− + ⋅ + ⋅+ ⋅= + = ⋅ = ⋅ + =
= ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ =
+= ⋅ + + ⋅ ⋅ ( ) s
s r ss sk
k
k k k kj
k
dI M I e r I j
dt
θ λω λ
+ = ⋅ + + ⋅ ⋅
(49)
( ) ( )( )( )
r srr sr ( ) ( )
r rr r r r
( ) ( ) ( ) ( ) ( )srr r sr r
( ) ( )
kk
k k
k k k k k
k kjjj
k kj j
kkk k kj j j j jk k
d L I e M e Id L I M e IU r I U e r I e
dt dt
d ddIdIr I e L e j e I M e j e I
dt dt dt dt
θ θθ θθ
θ θ θ θ
θ θ θ θ θ θ θ θ θ θθ θ θ θ
−−−
− −
− − − − −
⋅ + ⋅+ ⋅= + = ⋅ = ⋅ + =
− −= ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅
( ) ( )r sr( ) ( ) rr r s rr r r s( ) ( )k k
k kk
k k k k kj j
k k
d L I M I de r I j L I MI e r I j
dt dt
θ θ θ θ λω ω ω ω λ− −
=
+ = ⋅ + + ⋅ − ⋅ + = ⋅ + + ⋅ − ⋅ (50)
where ωω ,k are the angular velocity of the arbitrary reference frame, respectively the angular
velocity of the induction machine. Finally, if we consider P the number of poles for the inductionmachine, the electrical angular velocity of the rotor is ω)2/(P and the vectorial equations of the
induction machine written in a common arbitrary reference frame which rotates with angularvelocity
kω are:
s
ss s s
r
rr r r
s rs s
r sr r
( )
k
k k k
k
k
k k k
k
k k k
k k k
dU r I j
dt
dU r I j
dt
L I M I
L I M I
λω λ
λω ω λ
λ
λ
= + + ⋅ ⋅
= + + ⋅ − ⋅
= +
= +
(51-54)
The above equations system represent the mathematical model of the induction machine with stator
and rotor equipped with one fictive windings each in a common arbitrary reference frame. It has tobe highlighted that the mutual inductance does not depend on the relative rotor position (Kovacs -1984).
Us
Is
U Ir r
θ
kk
k
k k
S
R
ω
θ
λ
λ
sk
rk
k
k
ω
Fig. 2.4.Model of the three-phase induction machinewith fictive one stator winding and one rotor winding
(space vector notations) represented in a commonarbitrary reference frame
26
2.5. Induction machine equations with stator referred rotor variables
A complete unified equations system for the induction machine is obtained when both stator and
rotor variables are expressed in a common reference frame and the rotor variables are also statorreferred. Using the turns ratio and the winding factor ratio we obtain:
rr
t
r t r
r t r
2
r t r
1'
'
'
'
I Ik
U k U
k
r k r
λ λ
= ⋅
= ⋅
= ⋅
= ⋅
(55-58)
Now the stator and rotor flux linkage equations can be written as:
( )s r s ss mas s mas s mas t r
t
3 3 1'
2 2
k k k k k kL I L I L I k M I I
kλ σ σ
= ⋅ + ⋅ + ⋅ ⋅ = ⋅ ⋅ + ⋅ ⋅ +
(59)
( )2
r s s rr mar r mar r mar t r t
t t
3 3 1 1' '
2 2
k k k k kkL I L I L k I k M I Ik k
λ σ σ = ⋅ + ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ +
(60)
If the following notations are introduced:
ls s masL Lσ= ⋅ the stator leakage inductance2
lr t r mar'L k Lσ= ⋅ ⋅ the rotor leakage inductance
2
M t mas t mar
3 3
2 2L k M L k L= ⋅ = ⋅ = ⋅ ⋅ the main magnetisation inductance
M s r
k k kI I I= + the magnetisation current space vector
MM M
k kL Iλ = ⋅ the magnetisation flux
sls ls
k kL Iλ = ⋅ the stator leakage flux
lr lr r' ' 'k kL Iλ = ⋅ the rotor leakage flux
The flux linkage equations get the following form:
s Ms ls M ls M
k k k k kL I L Iλ λ λ= ⋅ + ⋅ = + (61)
( ) ( )Mr lr r M lr M
t t
1 1' ' ' '
k kk k kL I L Ik k
λ λ λ= ⋅ ⋅ + ⋅ = ⋅ + (62)
or in referred variables we obtain:
s ss ls M M r s M r( ) ' 'k k kk kL L I L I L I L Iλ = + ⋅ + ⋅ = ⋅ + ⋅ (63)
s sr lr M r M r r M' ( ' ) ' ' 'k kk k kL L I L I L I L Iλ = + ⋅ + ⋅ = ⋅ + ⋅ (64)
where Ls and L’r are the total stator, respectively rotor self-inductance.
The referred rotor voltage is given now by the relation:
( ) ( )lr M
r r r lr M
'' ' ' '
2
kk
kk k k
k
d PU r I j
dt
λ λω ω λ λ
+ = + + ⋅ − ⋅ +
(65)
Finally we can write a complete equations system, in a common arbitrary reference frame, withrotor variables referred to the stator, which define the induction machine. The result is a
mathematical model of the three-phase induction machine equipped with two fictive windings,rotating with angular velocity ωk. The simplified representation of this system is given in Fig. 2.5.
27
s
ss s s
rr r r r
ss s M r
sr r r M
'' ' ' ( ) '
2
'
' ' '
kk k k
k
kk k k
k
k k k
kk k
dU r I j
dt
d PU r I j
dt
L I L I
L I L I
λω λ
λω ω λ
λ
λ
= + + ⋅ ⋅
= + + ⋅ − ⋅
= +
= +
(66-69)
Us
Is
U
I
r
r
θ
k
k
k
k
k
S
R
ω
θ
λ
λ
sk
rk
k
k
ω
Fig. 2.5. Model of the three-phase induction machine withfictive one stator winding and one rotor winding (space vectornotations) represented in a common arbitrary reference frame,
and referred rotor variables
2.6. Instanteous electromagnetic torque
The input power for a three-phase induction machine with wounded rotor is:
* *s ri s r
1 1( ) Re Re ' '
2 2
k kk kP t m U I m U I= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ (70)
where m is the phase number (3), s r, 'k k
U U are the stator and rotor voltage space vectors, * *
s r, 'k kI I
are the conjugate stator and rotor current space vectors.
Using a detailed relation for the space vectors with rotor variables referred to the stator, we get:
i as as bs bs cs cs ar ar br br cr cr( ) ' ' ' ' ' 'P t U I U I U I U I U I U I= + + + + + (71)
From the complete equations system of the induction machine, the following expressions can beobtained for the input power of the machine:
* * * * * *s rs s s si s s r r r r r r
'3Re ' ' ' ' ' '
2 2
k kk k k k k k k k k k
k k
d d PP r I I I j I r I I I j I
dt dt
λ λω λ ω ω λ
= + + ⋅ + + + ⋅ −
(72)
* ** *2 2 s rs si s s r r r s r r
'3Re ' ' ' ' '
2 2
k kk k kk k k
k k
d d PP r I r I I I j I I
dt dt
λ λω λ ω ω λ
= + + + + ⋅ + − (73)
The first two terms from the power equation represent the Joule effect loss, the following two terms
represent the electromagnetic power due to the time variation of the magnetic energy, and the lastterm stands for the mechanical power available at the machine shaft, if the hysterezis loss, eddy
current loss and the stray losses are neglected. The mechanical power will be:
28
( ) ( )
** *
s s sm s M r r r M r
* *
s sM r M r
3Re ' ' '
2 2
3 3Re ' Im '
2 2 4
k k kk k k
k k
k kk k
PP j L I L I I L I L I I
P Pj L I I L I I
ω ω ω
ω ω
= ⋅ + ⋅ + − ⋅ + ⋅ =
= − ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅
(74)
Using the flux linkage relations, three expressions are deductible for the induction machinemechanical power:
** *
s sm r r s M r
**M Msr s r
r
3 3 3Im ' ' Im Im '
4 4 4
3 3Im ' Im '
4 ' 4
k k kk k k
k kk k
P P PP I I L I I
L LP PI
L
ω λ ω λ ω
ω λ ω λ λ∆
= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =
= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ (75)
where:2
s r M'L L L∆ = −which gives four expressions for the instantaneous electromagnetic torque if the mechanical power
is divided by the rotor angular velocity:
** *
s se r r s M r
**M Msr s r
r
3 3 3Im ' ' Im Im '
4 4 4
3 3Im ' Im '
4 ' 4
k k kk k k
k kk k
P P PT I I L I I
L LP PI
L
λ λ
λ λ λ∆
= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ =
= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅
(76)
The torque and the rotor speed a related by the mechanical equation:
e L
2 dT J T
P dt
ω = ⋅ +
(77)
where: J is the inertia of the rotor and in some cases the connected load. The first term on the right-hand side of the equation is the inertial torque. The load torque TL is positive for a torque load on
the shaft of the induction machine.Considering the voltage, flux linkage and mechanical equations, the complete relations system of
the induction machine can be written:
sss s s
rr r r r
ss s M r
sr r r M
** *
s se r r s M r
*Msr L
r
'' ' ' ( ) '
2
'
' ' '
3 3 3Im ' ' Im Im '
4 4 4
3 2Im '
4 '
kk k k
k
kk k k
k
k k k
kk k
k k kk k k
kk
dU r I j
dt
d PU r I j
dt
L I L I
L I L I
P P PT I I L I I
LP dI J T
L P dt
λω λ
λω ω λ
λ
λ
λ λ
ωλ
= + + ⋅ ⋅
= + + ⋅ − ⋅
= +
= +
= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ =
= ⋅ ⋅ ⋅ = ⋅ +
(78-82)
The above set of equations is valid for the following conditions:• All the equations are written in a common reference frame, which rotates with the arbitrary
angular velocity ωk.
• The rotor variables are referred to the stator;
• The stator and rotor variables are described by the space vector notations;
• The machine parameters are constant in a common reference frame for both armatures;
• The mechanical equation is written in real domain, and is independent of the chosen reference
frame.
29
2.7. General equations of the induction machine in different reference frames
When the mathematical model of the induction machine is established, several reference framescan be employed depending on application and the chosen strategy control. There are several main
reference frames: stationary fixed to the stator, synchronously fixed to the rotor, and revolving withan angular velocity equal to: the air-gap flux, the rotor flux, the stator voltage, the rotor currentspace vectors. The transformation from one reference frame to another is made by keeping constant
the value of the m.m.f. as all the reference frames are energetically equivalent. For the inductionmachine, the phasorial diagram of the main space vectors is illustrated in Fig. 2.6. The significance
of the index is as follows:s – stationary reference frame linked to phase as of the stator windingsr – rotating reference frame linked to the rotor shaft
m – air-gap flux reference framess – stator total synchronous reference frame
rs – rotor flux synchronous reference framek – arbitrary synchronous reference frameThe angles depicted in Fig. 2.6 are in electrical degrees.
ds
qs
L iM s
L i'M r
dss
dm
drs
dr
dkλ
λ
λ
λ
θ
θθ
θ
s
m
r
r
ks
rk
k
s
r
s θsm
Fig. 2.6. Definitions of the electrical angles between different reference frames
2.7.1. Per unit system
It is often convenient to express machine parameters and variables as per unit quantities. Themathematical model in per unit representation of the induction machine has some main features(Krause et al. 1995):
• All parameters have maximum value equal to unit;
• Two similar systems can be compared easier;
• The digital control is readily implemented;
When using the per unit system for writing the machine equations, the following observations have
to be made:a) The base torque is not the rated one. As the rated power output generally occurs at a speed
(rated speed) slightly less than synchronous, the base torque Tb will be less than rated torque bythe ratio of rated speed to synchronous speed of the machine.
b) In per unit system, the inductance value is equal to the reactance value.
30
c) If the flux linkage per second is used in the induction machine equations, it is per unitised bydividing by base voltage.
The following values of the machine parameters are used as base variables:
b sn,max s(rms),n
b sn.max s(rms),n
b
b
b
b b b s(rms),n s(rms)n
sb sn n
b sb sn n
bb
sb
bb
b
sn.max sn,maxbb
b sn
2
b
b
2
2
33
2
2
2 2 4
3
4
1
2
U U U
I I I
UZ
I
S U I U I
f
fP P P
U
LI
U IS PT
JH
S
ω ω π
πω ω ω
λω
λ
ω ω
ω
= = ⋅
= = ⋅
=
= =
= = ⋅
= ⋅ = ⋅ = ⋅
=
=
= = ⋅
= ⋅
-phase voltage, maximum value
-phase current, maximum value
-base impedance
-base power
-base stator variables angular velocity
-base rotor variables angular velocity
-base stator flux
-base inductance
-base torque
-inertia constant
The stator voltage equation in per unit system becomes:
ss s b s s
b b b b b
s
s s s ssn
1
1
k k k k
k
k
k k k
k
U r I I d j
U U I dt U U
du R i j
dt
λ ω λ
ψΩ ψ
ω
⋅= + ⋅ +
= + ⋅ + ⋅
(83-84)
Similarly we obtain the rotor voltage equation:
r
r r r r
sn
'1' ' ' ( ) '
k
k k k
k
du R i j
dt
ψΩ Ω ψ
ω= + ⋅ + ⋅ − ⋅ (85)
where the relative angular velocity is: b
ωΩ
ω=
In the previous per unit expressions we should note that the multiplying coefficient of the time fluxderivative is necessary, as the time is not referred. The relative time, defined as follows, can be
used:
relative sb sn
sn
2t
t t tT
ω ω π= = = ⋅ (86)
where Tsn represent the stator voltage supply period.
Finally the complete equations set in per unit system for the induction machine is:
31
s
s s s ssb
r
r r r r
sb
s s M rs
r r r M s
*
s Ls
1
'1' ' ' ( ) '
'
' ' '
1 3Im
2 4
k
k k k
k
k
k k k
k
k k k
kk k
k k
du R i j
dt
du R i j
dt
x i x i
x i x i
d Pi T
dt H
ψΩ ψ
ω
ψΩ Ω ψ
ω
ψ
ψ
Ωψ
= + ⋅ + ⋅
= + ⋅ + ⋅ − ⋅
= +
= +
= ⋅ ⋅ −
(87-91)
2.7.2. Stationary reference frame equations. Block diagram
If the stator voltages are unbalanced or discontinuous and the rotor-applied voltages are balanced
or zero, the most appropriate choice for the reference frame is the one fixed to the stator. Thisstationary reference frame was first employed by Stanley (1938). In a stationary reference frame,
fixed to the stator, the arbitrary angular velocity is zero, (ωk = 0) and the induction machine
equations system becomes:
s
s s s
sb
r
r r r r
sb
s s M rs
r r r M s
*
s Ls
1
'1' ' ' '
'
' ' '
1 3Im
2 4
k
k k
k
k k k
k k k
kk k
k k
du R i
dt
du R i j
dt
x i x i
x i x i
d Pi T
dt H
ψ
ω
ψΩ ψ
ω
ψ
ψ
Ωψ
= + ⋅
= + ⋅ − ⋅ ⋅
= +
= +
= ⋅ ⋅ −
(92-96)
The total leakage factor is described by relation 2
Mt
s r
1x
x xσ = − and if we note: 2
s r Mx x xδ = − the
following relations are deductible:
r M
s rs
s M
r r s
'
' '
k k k
kk k
x xi
x xi
ψ ψδ δ
ψ ψδ δ
= −
= −
(97-98)
A new equation system can be written in the two-axis co-ordinates system:
32
( )
qs
qs s qs
sb
ds
ds s ds
sb
qr
qr r qr dr
sb
dr
dr r dr qr
sb
qs s qs M qr
ds s ds M dr
qr r qr M qs
dr r dr M ds
ds qs q s d s
1
1
'1' ' ' '
'1' ' ' '
'
'
' ' '
' ' '
1 3
2 4
du R i
dt
du R i
dt
du R i
dt
du R i
dt
x i x i
x i x i
x i x i
x i x i
d Pi i
dt H
ψ
ω
ψ
ω
ψΩ ψ
ω
ψΩ ψ
ω
ψ
ψ
ψ
ψ
Ωψ ψ
= + ⋅
= + ⋅
= + ⋅ − ⋅
= + ⋅ + ⋅
= +
= +
= +
= +
= ⋅ − −L
T
(99-107)
If the induction machine is equipped with cage rotor, then the rotor voltage is zero. A complete
block diagram for the induction machine in stationary reference frame using as inputs stator voltageand currents and load torque, and as output the speed, is illustrated in Fig. 2.7.
ias
ibs
ics
uas
ubs
ucs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
ids
iqs
uds
uqs
Rs
Rs
+
+
-
+-
+
ω
ω
ψ
ψ
sn
sn
ds
qs
*
* +
+
--
1/s
1/s
1/sΩ
1/(2H)
Fig. 2.7. Block diagram of the induction machine in stationary reference frame
2.7.3. Rotor reference frame equations. Block diagram
The choice of the reference frame for the dynamic analysis of the induction machine, especiallywhen the rotor circuits are unbalanced, is more convenient to be fixed to the rotor frame. This
reference frame is in fact the Park’s transformation, initially developed for synchronous machineand than applied to the induction machine by Brereton (Fitzgerald et al -1990). The method ofreferring the machine variables to a rotor reference frame is most useful for field oriented control
33
systems. Transformation from the arbitrary frame to the rotor frame is made by substituting
kΩ Ω= , or from the stationary frame with the matrix relations:
dr dsr r
qr qsr r
ds drr r
qs qrr r
cos sin
sin cos
cos sin
sin cos
x x
x x
x x
x x
θ θ
θ θ
θ θ
θ θ
= ⋅ −
− = ⋅
(108-109)
where rθ is the electrical angle between the magnetic axis of the stator flux and the magnetic axis
of the rotor flux.
The complete equations set results as follows:
ss s s s
sb
r
r r r
sb
s s M rs
r r r M s
*
r r L
1
'1' ' '
'
' ' '
1 3Im ' '
2 4
k
k k k
k
k k
k k k
kk k
k k
du R i j
dt
du R i
dt
x i x i
x i x i
d Pi T
dt H
ψΩ ψ
ω
ψ
ω
ψ
ψ
Ωψ
= + ⋅ + ⋅ ⋅
= + ⋅
= +
= +
= ⋅ ⋅ −
(110-114)
Or expressed in orthogonal two-axis co-ordinate system:
( )
qs
qs s qs ds
sb
ds
ds s ds qs
sb
qr
qr r qr
sb
drdr r dr
sb
qs s qs M qr
ds s ds M dr
qr r qr M qs
dr r dr M ds
qr dr dr qr
1
1
'1' ' '
'1' ' '
'
'
' ' '
' ' '
1 3' ' ' '
2 4
du R i
dt
du R i
dt
du R i
dt
du R i
dt
x i x i
x i x i
x i x i
x i x i
d Pi i
dt H
ψΩψ
ω
ψΩψ
ω
ψ
ω
ψ
ω
ψ
ψ
ψ
ψ
Ωψ ψ
= + ⋅ +
= + ⋅ −
= + ⋅
= + ⋅
= +
= +
= +
= +
= ⋅ − −L
T
(115-123)
The block diagram of the induction machine model in rotor reference frame is presented in Fig.2.8. The input variables are stator currents and load torque, and as output variable is chosen the
angular velocity of the rotor. Also, the rotor voltages are considered zero (case of squirrel-cagerotor). It has to be observed that two algebraic loops appear in this diagram. This inconvenient leads
to a limited use of this reference frame to the wound rotor induction machine, when the blockdiagram is similar to that from Fig. 2.8, where inputs will be rotor voltages and currents.
34
ias
ibs
ics
TL
+ 1/3
2
+
-
+
+3-1/2
ids
iqs
x +
+
-
+-
+
ψ
ψ
dr
qr
*
* +
+
--
1/sΩ
1/(2H)
M
xM
1/x'r
1/x'r
i'
i'
1/s
1/s
dr
qr
-R'
-R'
r
r
ω
ω
sn
sn
3 ---> 2
e-j θ
rs
Fig.2.8. Block diagram of the induction machine in rotor reference frame (cage rotor case)
iar
ibr
icr
uar
ubr
ucr
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
idr
iqr
udr
uqr
Rr
R
+
+
-
+-
+
ω
ω
ψ
ψ
sn
sn
dr
qr
*
* +
-
+-
1/s
1/s
1/sΩ
1/(2H)
r
Fig. 2.9. Block diagram of the induction machine in rotor reference frame (wound rotor case)
2.7.4. Synchronous reference frame equations. Block diagram
The synchronously rotating reference frame, with angular velocity equal to that one of thepower supply system, is particularly convenient when incorporating the dynamic characteristics ofan induction machine into a digital computer program used to study the transient and dynamic
stability of power systems. The synchronously rotating reference frame may also be useful invariable frequency applications if we may assume that the stator voltages are a sinusoidal balanced
set. It was systematically developed by Kovacs (1984) and Krause et al (1995), or Lorenz et al.(1994).
35
When a two-axis co-ordinates reference frame is employed, it has to be fixed to different variablesof the machine. The main configurations are the synchronously reference frame fixed to: stator flux,
air-gap flux and rotor flux.I) Stator flux fixed synchronous reference frame.
In order to link the d-axis of the synchronous reference frame to the stator flux space vector, the q-component of this flux vector is defined equal to zero:
s s
dss
s
qs 0
ψ ψ
ψ
=
= (124)
The following equations set is obtainable:s s sqs qs dss s
sdss s
ds dss
sb
sqrs s s
qr qr drr s
sb
sdrs s s
dr dr qrr s
sb
s sqs qrs M
s s sds ds drs M
s s sqr qr qsr M
s s sdr dr dr M
1
1 '' ' ' ( ) '
1 '' ' ' ( ) '
0 '
'
' ' '
' ' '
u R i
du R i
dt
du R i
dt
du R i
dt
x i x i
x i x i
x i x i
x i x i
Ω ψ
ψ
ω
ψΩ Ω ψ
ω
ψΩ Ω ψ
ω
ψ
ψ
ψ
= +
= + ⋅
= + ⋅ + − ⋅
= + ⋅ − − ⋅
= +
= +
= +
= + s
s sds qs L
1 3
2 4
d Pi T
dt H
Ωψ
= ⋅ −
(125-133)
The block diagram from Fig. 2.10 presents the induction machine model in synchronous referenceframe linked to the stator flux, with inputs stator voltages and currents, and load torque. The outputsare synchronous and rotor angular velocity.
ias
ibs
i cs
uas
ubs
ucs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
i ds
iqs
uds
uqs
Rs +-
+
ω
ψ
sn
ds
*
+
+
-
1/s
1/sΩ
1/(2H)
Rs +
-
+
/D
N
Ωs
3 ----> 2
e-j θ
sss
3 ----> 2
θsss-j
e
Fig. 2.10. Block diagram of the induction machine in synchronous reference frame(linked to stator flux)
36
II) Rotor flux fixed synchronous reference frame.
In order to link the d-axis of the synchronous reference frame to the rotor flux space vector, the q-component of this flux vector is defined equal to zero:
r r
r dr
r
qr
' '
' 0
ψ ψ
ψ
=
= (134)
The induction machine characteristics can be analysed with the equation system:rqsr r r
qs qs dss s
sb
rdsr r r
ds ds qss s
sb
r r rqr qr drr s
rdrr r
dr drr
sb
r r rqs qs qrs M
r r rds ds drs M
r rqr qsr M
r r rdr dr dsr M
1
1
' 0 ' ' ( ) '
1 '' 0 ' '
'
'
0 ' '
' ' '
du R i
dt
du R i
dt
u R i
du R i
dt
x i x i
x i x i
x i x i
x i x i
d
d
ψΩ ψ
ω
ψΩ ψ
ω
Ω Ω ψ
ψ
ω
ψ
ψ
ψ
Ω
= + ⋅ +
= + ⋅ −
= = + − ⋅
= = + ⋅
= +
= +
= +
= +
r r r rMdr qr dr qsL L
r
1 3 1 3' ' '
2 4 2 4 '
LP Pi T i T
t H H Lψ ψ
= ⋅ − − = ⋅ ⋅ −
(135-143)
The block diagram from Fig. 2.11 presents the induction machine model in synchronousreference frame linked to the rotor flux, with inputs stator currents, and load torque. The outputs are
synchronous and rotor angular velocity. The rotor voltages are considered equal to zero (case of thecage rotor).
ias
ibs
i cs
TL
+ 1/3
2
+
-
+
+3
-1/2
i ds
iqs
+
+
ω
ψ
sn
dr
*
+
+
-
1/s
1/sΩ
1/(2H)
/D
N
Ωs
xM R'r /x'r
+
xM
R'r
/x'r
- Ω
xM
x'r
R'r
/x'r
3 ---> 2
e-j θsrs
Fig. 2.11. Block diagram of the induction machine in synchronous reference frame(linked to rotor flux)
37
III) Air-gap flux fixed synchronous reference frame.In order to link the d-axis of the synchronous reference frame to the air-gap flux space vector, the q-
component of this flux vector is defined equal to zero:m m
dmm
m
qm 0
ψ ψ
ψ
=
= (144)
The system is described by the relations:m
qsm m mqs qs dss s
sb
mdsm m m
ds ds qss s
sb
mqrm m m
qr qr drr s
sb
mdrm m m
dr dr qrr s
sb
m mqs qrM
m m mdm ds drM
m m mds dss M
1
1
1 '' 0 ' ' ( ) '
1 '' 0 ' ' ( ) '
0 ( ' )
( ' )
'
du R i
dt
du R i
dt
du R i
dt
du R i
dt
x i i
x i i
x i x i
ψΩ ψ
ω
ψΩ ψ
ω
ψΩ Ω ψ
ω
ψΩ Ω ψ
ω
ψ
ψ
= + ⋅ +
= + ⋅ −
= = + ⋅ + − ⋅
= = + ⋅ − − ⋅
= +
= +
= + m mdr dm dsls
m m m mqs qs qr qss M ls
m m m mqr qr qs qrr M lr
m m m m mdr dr ds dm drr M lr
m mdm qs L
'
' ' ' ' '
' ' ' ' '
1 3
2 4
x i
x i x i x i
x i x i x i
x i x i x i
d Pi T
dt H
ψ
ψ
ψ
ψ ψ
Ωψ
= +
= + =
= + =
= + = +
= ⋅ −
(145-155)
Fig. 2.12 illustrates the induction machine model in synchronous reference frame linked to the air-
gap flux, with inputs stator voltages and currents, synchronous angular velocity and load torque.The output is rotor angular velocity.
ias
ibs
ics
uas
ubs
ucs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
ids
iqs
uds
uqs
Rs +-
+
ωψ
sn
dm
*
+
+
-
1/s
1/sΩ
1/(2H)
Ωs
xls *
-+
xls
+
-
3 ----> 2
3 ----> 2
e-j
e-j
θ
θ
sm
sm
Fig. 2.12. Block diagram of the induction machine in synchronous reference frame(linked to air-gap flux)
38
IV) Stator current space vector fixed synchronous reference frame.In order to link the d-axis of the synchronous reference frame to the stator current space vector, the
q-component of this space vector is defined equal to zero:sc sc
s ds
sc
qs 0
i i
i
=
= (156)
This reference frame realises a special control of the machine torque and speed, by using only one
current component and the stator voltage. De-coupling circuits are needed for an independentcontrol of the speed and torque. The complete set of equations which describes the dynamic and
steady-state operation of the induction machine is.sc
qssc scqs dss
sb
scdssc sc sc
ds ds qss s
sb
scqrsc sc sc
qr qr drr s
sb
scdrsc sc sc
dr dr qrr s
sb
sc sc scds ds drs M
sc scqs qrM
scqr
1
1
1 '' 0 ' ' ( ) '
1 '' 0 ' ' ( ) '
'
'
'
du
dt
du R i
dt
du R i
dt
du R i
dt
x i x i
x i
x
ψΩ ψ
ω
ψΩ ψ
ω
ψΩ Ω ψ
ω
ψΩ Ω ψ
ω
ψ
ψ
ψ
= ⋅ +
= + ⋅ −
= = + ⋅ + − ⋅
= = + ⋅ − − ⋅
= +
=
= scqrr
sc sc scdr dr dsr M
sc scqs sd L
' '
' ' '
1 3
2 4
i
x i x i
d Pi T
dt H
ψ
Ωψ
= +
= ⋅ − ⋅ −
(157-165)
The block diagram from Fig. 2.13, uses as inputs the stator voltage space vector, the d-axis stator
current component, the load torque, and the synchronous angular velocity. As output is consideredthe rotor angular velocity (speed).
ias
ibs
ics
uas
ubs
ucs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
ids
iqs
uds
uqs
Rs +-
+
ω
ψ
sn
ds
*
+
-
1/s
1/sΩ
1/(2H)
Ωs
ωsn
1/s
*
*
++ +
-
ψqs
++
-
3 ---> 2
3 ----> 2
e-j
e-j
θ
θ
ssc
ssc
Fig. 2.13. Block diagram of the induction machine in synchronous reference frame
(linked to stator current space vector)
39
V) Rotor current space vector fixed synchronous reference frame.In order to link the d-axis of the synchronous reference frame to the rotor current space vector, the
q-component of this space vector is defined equal to zero:rc rc
r dr
rc
qr
' '
' 0
i i
i
=
= (166)
Applying the above conditions, a complete equations system for the induction machine expressed in
this particularly reference frame can be deduced:rc
qsrc rcqs dss
sb
rcdsrc rc rc
ds ds qss s
sb
rcqrrc rc
qr drs
sb
rcdrsc rc rc
dr dr qrr s
sb
rc rc rcds ds drs M
rc rcqs qss
rc rcqr qsM
rcd
1
1
1 '' 0 ( ) '
1 '' 0 ' ' ( ) '
'
'
'
du
dt
du R i
dt
du
dt
du R i
dt
x i x i
x i
x i
ψΩ ψ
ω
ψΩ ψ
ω
ψΩ Ω ψ
ω
ψΩ Ω ψ
ω
ψ
ψ
ψ
ψ
= ⋅ +
= + ⋅ −
= = ⋅ + − ⋅
= = + ⋅ − − ⋅
= +
=
=rc rc
r dr dsr M
rc rc rc rcqr dr qs drL M L
' '
1 3 1 3' ' '
2 4 2 4
x i x i
d P Pi T x i i T
dt H H
Ωψ
= +
= ⋅ ⋅ − = ⋅ ⋅ ⋅ −
(167-175)
In Fig. 2.14, the block diagram that describes the induction machine model has as inputs the d-axisstator voltage component and current space vector, the load torque, and the synchronous angularvelocity. As output is considered the rotor angular velocity (speed).
ias
i bs
ics
uas
ubs
ucs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
ids
iqs
uds
uqs
Rs +-
+
ω
ψ
sn
ds
+
-
1/s
1/sΩ
1/(2H)
Ωs
xs
*
+
ψqs
xs
+
+
-
*3P/4+
3 ---> 2
3 ----> 2
e-j
e-j
θ
θ
s rc
src
Fig. 2.14. Block diagram of the induction machine in synchronous reference frame
(linked to rotor current space vector)
40
VI) Magnetising current space vector fixed synchronous reference frame.In order to link the d-axis of the synchronous reference frame to the magnetising current space
vector, the q-component of this space vector is defined equal to zero:mc mcmc mc
m dm s r
mc
qm
'
0
i i i i
i
= = +
= (176)
Considering the above conditions for a synchronous reference frame, the complete set of equation
for the three-phase induction machine can be obtained:m c
qsmc mc m cqs qs dss s
sb
m cdsmc mc mc
ds ds qss s
sb
mcqrmc mc mc
qr qr drr s
sb
mcdrmc mc mc
dr dr qrr s
sb
mc mc m cqm qs qr
mc m cds dss M
1
1
1 '' 0 ' ' ( ) '
1 '' 0 ' ' ( ) '
0 '
du R i
dt
du R i
dt
du R i
dt
du R i
dt
i i i
x i x
ψΩ ψ
ω
ψΩ ψ
ω
ψΩ Ω ψ
ω
ψΩ Ω ψ
ω
ψ
= + ⋅ +
= + ⋅ −
= = + ⋅ + − ⋅
= = + ⋅ − − ⋅
= = +
= + mcdr
mc m c mc mcqs qs qr qss M ls
mc m c mc mcqr qs qr qrM r lr
mc mc mcdr dr dsr M
mc mcdm qsM L
'
'
' ' ' ' '
' ' '
1 3
2 4
i
x i x i x i
x i x i x i
x i x i
d Px i i T
dt H
ψ
ψ
ψ
Ω
= + =
= + =
= +
= ⋅ ⋅ ⋅ − (177-185)
A block diagram for the mathematical model of the induction machine in this reference frame ispresented in Fig. 2.15. As inputs are chosen the stator current space vector, the d-axis component of
the stator voltage space vector, the synchronous angular velocity and the load torque. The output isthe machine (rotor) angular velocity. The similarity to the synchronous reference frame linked tothe air-gap flux space vector can be observed, as the magnetising current is directly proportional
with the air-gap flux, through the magnetising inductance.
ias
ibs
i cs
u as
ubs
u cs
TL
+
+
1/3
2
1/3
2
+
-
+
+
-
+
+
+
3-1/2
3-1/2
ids
iqs
uds
u qs
Rs +-
+
ω
ψsn
dm
*
+
+
-
1/s
1/sΩ
1/(2H)
Ω s*
++
x ls
+
ψqs
x ls
3P/4
uds
3 --> 2
eθj
3 --> 2
ej θ−
−sm
sm
Fig. 2.15. Block diagram of the induction machine in synchronous reference frame(linked to magnetising current space vector)
41
2.8. D-Q Axes models of the three-phase induction machine
The three-phase induction machine can be modelled by using different state-space variables andkeeping as inputs the stator voltages and the load torque, and as outputs the electromagnetic torque
and the rotor angular velocity. The possible set of currents and flux linkages per second spacevectors are defined as follows (Nowotny and Lipo - 1996):
s ds qs
r dr qr
m dm qm
s ds qs
r dr qr
m dm qm
' '
' ' '
i i j i
i i j i
i i j i
j
j
j
λ λ λ
λ λ λ
λ λ λ
= + ⋅
= + ⋅
= + ⋅
= + ⋅
= + ⋅
= + ⋅
(186-191)
[ ] [ ]T
s r m s r m' 'x i i i λ λ λ= (192)
The d-q axes are orthogonal and fixed to the stator, d axis coincides to the magnetic axis of the aswinding.
As there are four voltage equations, it is necessary to consider two of the space vectors as state-variables in order to obtain a solution for the equations system. Let the selected pair of state-space
variables be denoted as 21 , xx . The set of six state-space variables will be expressed in terms of the
two selected state-space variables:
[ ] [ ] [ ]
11 12
21 22
T T31 32
s r m s r m 1 2
41 42
51 52
61 62
' '
a a
a a
a ax i i i x x
a a
a a
a a
λ λ λ
= = ⋅
(193)
Only fourteen out of the fifteen state-space possibilities represent valid mathematical models forthe three-phase induction machine (the pair of state-space variables that comprises airgap flux
linkage per second space vector and the magnetising current space vector cannot be selected asthese vectors have the same direction). There are three types of models:1. current state-space variables models;
2. flux linkages state-space variables models;3. mixed currents-flux linkages state-space variable models.
In conjunction with the mechanical equation:
( )r e L2
Pp T T
Jω = ⋅ − (194)
we obtain a complete version of the three-phase induction machine model, viewed as the key for a
motion control system.The starting point for the state-variable models is given by the voltage equations system written in
stationary reference frame:p= ⋅ + ⋅u L x R x (195)
where: x is the selected set of state-variables and represents also the output of the model, u is theinput vector (stator voltages), L is the coefficients matrix (it can be formed by reactance values, ornon-dimensional elements) for multiplying the time derivative of the state-variables, R is the
coefficients matrix (it can be formed by resistances and reactances values or non-dimensionalelements) for multiplying the state-variables and p stands for the differential operator (d/dt).
42
It results the general form of the state-variables system:
p = ⋅ + ⋅x A x B u (196)
where:1
1
−
−
= − ⋅
=
A L R
B LAll the following mathematical models permit a discretisation for the implementation of
controllers in the drive systems with three-phase induction machines. Classically, the current state-
space and the flux state-space models are the chosen options for the manufacturers of drive systems.The original vector orientation scheme was based on the alignment of the synchronous reference
frame to the rotor flux linkage space vector. Then the vector control strategy was extended byconsidering as well the stator and air-gap flux space vectors, as alignment of the reference frame.However, for a complete analysis of the vector control strategies, each of the six space vectors that
can be selected as state-space variables represents a possible basis for a reference frame. If themagnetising current space vector selection as alignment gives similar results to the air-gap flux
space vector case, the stator and rotor current space vectors are still to be further analysed as newoptions for vector control.
An easy to follow steps algorithm for implementing vector oriented control systems is obtained
as follows:1. A complete mathematical model of the three-phase induction machine is developed in stationary
reference frame, according to the chosen set of state-space variables;2. The rotor based variables are completely expressed in the new state-variable system;
3. The rotor angular velocity ωr term is substituted with (ω r - ω k) where ω k is the angular
velocity of the synchronous frame;4. The synchronous reference frame is selected linked to one of the space vectors, which means
that the q-axis component of the reference space vector is null;5. The torque equation is computed according to the selected flux or current space vector in the
synchronous reference frame.The block diagram of modelling an induction motor for vector control purpose is detailed in Fig.2.16.
ux=Ax+Bup [a ]
ije
-j
Te
(open loop)
Te
(closed loop)
stationary frame
x
synchronous frame
θ
Tl
+-
+
2JP
ωr
Fig. 2.16. Block diagram form of the induction machine mathematical model
Fig. 2.17 shows the general two-axis equivalent circuit for the induction machine, in an arbitraryreference frame. This equivalent circuit represents the starting point for determining the matrixequations, which will be derived further.
43
+
-
+
-
+
-
+
-
+ - - +
+ -- +
uqs
uds
u'qr
u'dr
i qs
ids
i'qr
i'dr
LM
LM
r s
rs
r'r
r'r
Lls
L'lr
Lls
L'lr
ω λ (ω −ω)λ
ω λ (ω −ω)λ
ds
qs
k
k
k
k
dr'
'qr
Fig. 2.17. Arbitrary reference-frame equivalent circuits for a three-phase,
symmetrical induction machine
Note: For compactness, the elements from Fig. 2.17. have to be transformed in terms of reactances,
while the flux linkage has to be expressed in flux linkage units per second, or volts.
2.8.1. Models with currents space vectors state-space variables
I) As complex state variables, the currents space vectorsT
qs ds qr dr, , ' , 'i i i i = x are usually
assumed. The stator current space vector is considered generally as the right choice, because it
corresponds to directly measurable quantities. This model is readily available from voltages andflux linkages per second equations, and it can be expressed in a matrix form as follows:
2
s r m r m r m r r
b b
2qs qss rm r m r r m r
ds b b
qrb m s s m s r s rr r
dr b b
s m m s s r s rr r
b b
' ' '
' ' '
' ' '
'
' '
r x x x r x x
D D D D
i ir xx x x x r
i iD D D Dp
i x r x x x r x x
i D D D D
x x x r x x x r
D D D D
ω ω
ω ω
ω ω
ω ω
ω ω ω
ω ω
ω ω
ω ω
− − ⋅ − ⋅
⋅ − ⋅ ⋅ = ⋅ ⋅ − ⋅
− ⋅ − ⋅ −
r m
qsr m
ds ds
qr qrsm
dr dr
sm
'0 0
'0 0
' '0 0
' '
0 0
x x
D Dux x
uD D
i uxx
D Di u
xx
D D
− − + ⋅ − −
(197)
where: 2
s r m'D x x x= −If the stator, rotor and total leakage factor definitions are used:
s ls m s m
r lr m r m
s r
(1 )
' ' (1 )
11
(1 )(1 )
x x x x
x x x x
σ
σ
σσ σ
= + = +
= + = +
= −+ +
we obtain:
2 2
m s r s r m( )1
D x xσ
σ σ σ σσ
= ⋅ + + = ⋅−
44
The electromagnetic torque can be computed as:
( )m
e qs dr ds qr
b
3' '
4
xPT i i i i
ω= ⋅ ⋅ − (198)
II) The stator and the magnetising space vector currents as state-space variablesT
qs ds qm dm, , ,i i i i = x represent another mathematical d-q axis model based on currents space
vectors. By selecting the magnetising current space vector among the state-space variables, it is
possible to include the saturation effect in modelling the induction motor. Also, the stator currentspace vector is a measurable quantity, and determines a precise and accurate option for
implementing controllers. The state matrix elements are all non-zero, which implies acomputational effort similar to the previously analysed model.
r r
s r r m m lr m r m r
b b
qs r r
m lr s r r m m r m r
b bds
qm r rb
r ls s lr ls lr ls r ls r
b bdm
r r
ls lr r ls s lr ls r ls r
b b
' ' ' ' '
' ' ' ' '1
' ' ' ' '
' ' ' ' '
r x r x x x x r x x
ix x r x r x x x x r
ip
i Dr x r x x x x r x x
i
x x r x r x x x x r
ω ω
ω ω
ω ω
ω ω
ω ωω
ω ω
ω ω
ω ω
− − ⋅ − ⋅
− ⋅ − − ⋅
= ⋅
− − ⋅ − ⋅
⋅ − − ⋅ −
qs
ds
qm
dm
r m qs
r m ds
lr ls qr
lr ls dr
' 0 0
0 ' 01
' 0 0 '
0 ' 0 '
i
i
i
i
x x u
x x u
x x uD
x x u
⋅ +
− − + ⋅ ⋅
(199)
The instantaneous electromagnetic torque may be expressed in terms of the state-space variables:
( )m
e q s d m d s q m
b
3
4
xPT i i i i
ω= ⋅ ⋅ − (200)
III) The third possible combination of current space vectors as state-space variables is obtained
by selecting magnetising and rotor currents T
qr dr qm dm' , ' , ,i i i i = x . When compared to the other
currents space vector models, the similar computational burden for obtaining the outputs of thesystem is obvious. The main difference between them is the presence of the global parameters (self-
reactances) in the stator and rotor current state-space variables model, while in the other two modelsan accurate determination of the leakage reactances or leakage factors is necessary. A complete
description of the state-variables in matrix notation is given below:
45
r r
s m r s s lr s m s m
b b
qr r r
s lr s m r s s m s m
b bdr
qm r rb
s lr r ls ls lr s lr ls m
b bdm
r r
ls lr s lr r ls ls m s lr
b b
' '
'' '
' 1
' ' ' '
' ' ' '
r x r x x x r x x x
ix x r x r x x x r x
ip
i Dr x r x x x r x x x
i
x x r x r x x x r x
ω ω
ω ω
ω ω
ω ω
ω ωω
ω ω
ω ω
ω ω
− −
− − − = ⋅
− −
− − − −
qr
dr
qm
dm
m s qs
m s ds
lr ls qr
lr ls dr
'
'
0 0
0 01
' 0 0 '
0 ' 0 '
i
i
i
i
x x u
x x u
x x uD
x x u
+
− − + ⋅
(201)
The instantaneous electromagnetic torque is expressed as follows:
( )m
e dr qm qr dm
b
3' '
4
xPT i i i i
ω= ⋅ ⋅ − (202)
2.8.2. Models with flux linkages as state-space varaibles
I) When flux linkages per second are selected as state-space variables, the models are lesscomputationally demanding than in the currents state-space variables version. As each flux contains
information about two currents space vectors components, the state matrix contains zero elements.
One option is to select the stator and rotor flux linkages space vectors T
qs ds qr dr, , ' , 'λ λ λ λ = x for
describing the mathematical model of the machine. For this system the matrix equations are asfollows:
s r s m
qs qs qss r s m
ds ds ds
r sr m rqr qr qrb
bdr dr dr
r sr m r
b
'0 0
'0 0
''' ' '0' ' '
''0
r x r x
D D
ur x r x
D D upr xr x u
D D u
r xr x
D D
λ λ
λ λωλ λωωλ λ
ω
ω
−
− ⋅ = ⋅ +−
−
(203)
The electromagnetic torque is determined with the equivalent relation:
( )m
e qs dr ds qr
b
3' '
4
xPT
Dλ λ λ λ
ω= ⋅ ⋅ −
⋅ (204)
By comparison with the current models, it can be observed that the computational burden isessentially lower. Due to this important feature, this model is the most suitable for discretisation inmotion control strategies.
II) An alternative to model the three-phase induction machine with flux space vectors as state-variable system, is the selection of air-gap flux space vector among the set of independent variables.
This choice presents the advantage of an easier saturation effect modelling, but the disadvantage ofan increased computational burden. The air-gap flux is a measurable quantity, and this advantageimposes it in many practical solutions for vector control schemes.
46
A first approach is given by the stator and air-gap flux space vectors selected as state-space
variablesT
qs ds qm dm, , ,λ λ λ λ = x .
s s
ls ls
s sqs
ls lsds
s lr m s m lr r s ls m lr sr m m lr r rqmb
ls b ls b ls mdm
s lr m ls m lr sm lr r r m r
b ls b ls m
0 0
0 0
' ' ' '' '1
' '' '1
r r
x x
r r
x xp
r x x r x x r x x x x xr x x x
x D D D x D D D x x
r x x x x x xx x r x
D x D D D x x
λ
λ
ω ωλω
ω ωλ
ω ω
ω ω
−
− =
− + − ⋅ − ⋅ ⋅ +
⋅ − + − ⋅ ⋅ +
qs
ds
qm
dm
s lr m r s
ls
qs
dsls mlr m
qr
drls mlr m
' '
1 0 0 0
0 1 0 0
'0 0
'
''0 0
r x x r x
x D D
u
ux xx x
uD D
ux xx x
D D
λ
λ
λ
λ
⋅ +
−
⋅
(205)The electromagnetic torque expression depends on stator leakage factor:
e qs dm ds qm
ls b
3 1
4
PT
xλ λ λ λ
ω = ⋅ ⋅ − (206)
III) The third option of selecting flux space vectors as state-space variables is similar to theprevious model. It comprises the air-gap (magnetising) flux and rotor flux space vectors
T
qm dm qr dr, , ' , 'λ λ λ λ = x in the set of independent variables. The same computational effort is
required, and the electromagnetic torque is determined in a suitable form for vector control if the
rotor leakage factor is known:
s r r m ls s m r m ls m ls r
lr lr b
qm s r r m ls m ls s m r m lsr
lr b lrdm
qr r rb
lr lr bdr
r r r
lr b lr
' ' '0
' '
' ' '0
' '
' ' '0
' ''
' '0
' '
r
r x r x x r x r x x x x
D x D D x D D
r x r x x x x r x r x x
D x D D D x Dp
r r
x x
r r
x x
ω
ω
λ ω
ωλ
λ ωω
ωλ
ω
ω
− + − ⋅
− + − ⋅ − =
−
− −
qm
dm
qr
dr
m lsm lr
qs
dsm lsm lr
qr
dr
'
'
'0 0
'0 0
'
0 0 1 0 '
0 0 0 1
x xx x
uD D
ux xx x
D D u
u
λ
λ
λ
λ
⋅ +
+ ⋅
(207)
The resulting relation for computing the instantaneous electromagnetic torque is as follows:
47
( )e qm dr dm qr
lr b
3 1' '
4 '
PT
xλ λ λ λ
ω= ⋅ ⋅ − (208)
2.8.3. Models with mixed currents – flux space vectors state-space variables
I) For obtaining an acceptable computational effort as well as measurable output quantities, the
general accepted solutions are the mixed currents-flux state-space variable models. If the statorvariables are chosen for modelling the three-phase induction machine system, then a mixed flux
linkages-currents state-space variables model T
qs ds qs ds, , ,i iλ λ = x is developed. This mathematical
model is selected when stator flux oriented strategy is implemented. The matrix equations and theelectromagnetic torque relation are:
s
qs qss
ds dss r r sr r r r
b bqs qsb
ds dss r r sr r r r
b b
qs
r
r
0 0 0
0 0 0
' '' '
' '' '
1 0 0 0
0 1 0 0
'0 1 0
'0 0 1
r
r
r x r xr xp
D D Di i
i ir x r xx r
D D D
u
ux
D
x
D
λ λ
λ λω ω
ω ωω
ω ω
ω ω
− − + − ⋅ −= ⋅ + + ⋅ − −
+ ⋅− −
ds
qr
dr
'
'
u
u
(209)
The instantaneous electromagnetic torque relation depends only on the magnitude of the outputvector components.
( )e ds qs qs ds
b
3 1
4
PT i iλ λ
ω= ⋅ ⋅ − (210)
II) Another important mixed flux linkages-currents state-space variables modelT
qr dr qr dr' , ' , ' , 'i iλ λ = x is that expressed in rotor quantities. This model represents the optimum
solution for rotor flux oriented control strategy in a drive system with an induction machine.However, as it is impossible to measure the rotor currents if the machine is equipped with cage
rotor, there are limitations in using this model for vector control strategies. The state-spacevariables system is detailed below:
rr
b
qr qrrr
bdr dr m
qr qrs s s r r srb
bdr dr m
s s s r r sr
b
0 ' 00 0 1 0
' ' 0 0 0 10 0 '' '
0 0 0' '' '
0' '
0 0 0' '
0
r
rp x
i ir x r x r x D
D D Di i x
Dx r r x r x
D D D
ω
ω
λ λω
ωλ λ
ωω
ω
ω
ω
−
− − = ⋅ + − +
⋅ − − +
− ⋅ −
qs
ds
qr
dr
'
'
u
u
u
u
⋅
(211)
48
The electromagnetic torque expression depends only on the magnitude of the output vectorcomponents:
( )e qr dr dr qr
b
3 1' ' ' '
4
PT i iλ λ
ω= ⋅ − (212)
III) The most used alternative model for rotor flux oriented control strategies is that which
comprises the stator current and rotor flux linkage T
qr dr qs ds' , ' , ,i iλ λ = x as state-space variables. It
contains the advantages of measurable output quantities (stator currents) and acceptable
computational burden. The matrix equation and the electromagnetic torque relation are presentedbelow:
2 2
s r r m r m m r
r r b
2 2qs qss r r m m r r m
ds dsr b r
qr qrb r m r r
dr drr r b
r m r r
r b r
' ' '0
' '
' ' '0
' '
' '' '0
' '' '
' '0
' '
r x r x r x x
x D x D D
i ir x r x x r x
i ip x D D x D
r x r
x x
r x r
x x
ω
ω
ω
ω
λ λω ω
λ λω
ω
ω
+− − ⋅
+
− ⋅ = ⋅ −
− −
r m
qs
dsr m
qr
dr
'0 0
'0 0
'
0 0 1 0 '
0 0 0 1
x x
uD D
ux x
D D u
u
+
−
−+ ⋅
(213)
and
( )me qs dr ds qr
r b
3' '
4 '
xPT i i
xλ λ
ω= ⋅ ⋅ − (214)
IV) A theoretical mathematical model is that with mixed rotor currents space vector and stator
flux linkages space vector as state-space variables T
qs ds qr dr, , ' , 'i iλ λ = x . It should be noted that
the advantage of this model is that only the stator winding parameter is necessary and as such the
influence of rotor parameters is minimised. This model can be used for an unconventional statorflux oriented control with rotor current components producing the torque and the flux. The
expressions for implementing this model are:
49
2 2
s m r s s mr m r
s b s bsm
2 2qr qrs m r s s mr m
dr dr smb s s
qs qsb s m s
ds dss s
s m s
s s
'
0 0' ''
' '0 0
0 0 1 0 0
0 0
r x r x r x x
x D x D D xx
i ir x r x r xx D D
i i xxp x D D x D
D Dr x r
x x
r x r
x x
ω ω
ω ω
ω
ω
λ λω
λ λ
+− ⋅
+
− − − = ⋅ + −
−
qs
ds
qr
dr
'
0 '
0 1 0 0
u
u
u
u
⋅
(215)
and
( )m
e qs dr ds qr
s
3' '
4 b
xPT i i
xλ λ
ω= ⋅ ⋅ − (216)
V) The mixed stator current space vector and air-gap flux space vector T
qm dm qs ds, , ,i iλ λ = x as
state-space variables belongs to one of the most complex types of models. It preserves information
regarding both stator and rotor parameters. This mathematical model is the suitable choice for theair-gap flux orientation control strategy. The greatest advantage of this model is that by using Hall
sensors or tapped stator windings, all the output vector components are measurable. Due to itsversatility, this model is widely used in controllers implementation, especially for medium speedapplications. It permits also the simulation or modelling of the saturation effect. By comparison to
the previous mixed models, the state-space matrix contains only non-zero elements, which leads togreater computational effort.
qs s r r m m lr r b r r r b
ds m lr r b s r r m r r b r
qm m r m s lr ls m lr r b ls r ls r r bb
dm ls m lr r b m r m s lr ls
' ' ' ( / ) ' ' ( / )
' ( / ) ' ' ' ( / ) '1
( ' ' ) ' ( / ) ' ' ( / )
' ( / ) ( ' ' )
i r x r x x x r x
i x x r x r x x rp
x r x r x x x x x r x xD
x x x x r x r x x
ω ω ω ω
ω ω ω ω
λ ω ω ω ωω
λ ω ω
− − − − − = ⋅ − − −
− − −
qs
ds
qm
r r b ls r dm
r m qs
r m ds
lr m ls m qr
lr m ls m dr
' ( / ) '
' 0 0
0 ' 01
' 0 0 '
0 ' 0 '
i
i
x x r
x x u
x x u
x x x x uD
x x x x u
λ
ω ω λ
⋅ +
−
− − + ⋅ ⋅
(217)The electromagnetic torque is computed as:
( )e qs dm ds qm
b
3 1
4
PT i iλ λ
ω= ⋅ ⋅ − (218)
VI) Another theoretical model, similar in form to the precedent one, is the mixed rotor current
space vector and air-gap flux space vector as state-space variables T
qm dm qr dr, , ' , 'i iλ λ = x Its main
shortcoming is the presence of the unmeasurable rotor currents among the state-space variables.The matrix equation of the system is given below:
50
qr s m r s s lr r b s s r b
dr s lr r b s m r s s r b s
qm s m lr r m ls m ls lr r b s lr m ls r bb
dm m ls lr r b s m lr r m ls m ls
' ' ' ( / ) ( / )
' ' ( / ) ' ( / )1
' ' ' ( / ) ' ( / )
' ( / ) ' ' (
i r x r x x x r x
i x x r x r x x rp
r x x r x x x x x r x x xD
x x x r x x r x x x x
ω ω ω ω
ω ω ω ω
λ ω ω ω ωω
λ ω ω
− − − − − − = ⋅ − −
− − −
qr
dr
qm
r b s lr dm
m s qs
m s ds
m lr m ls qr
m lr m ls dr
'
'
/ ) '
0 0
0 01
' 0 0 '
0 ' 0 '
i
i
r x
x x u
x x u
x x x x uD
x x x x u
λ
ω ω λ
⋅ +
−
− − + ⋅ ⋅
(219)
The electromagnetic torque relation becomes:
( )e dr qm qr dm
b
3 1
4
PT i iλ λ
ω= ⋅ ⋅ − (220)
VII) If the magnetising current space vector is selected as state-space variable together with one
of the flux linkages space vectors (i.e. stator flux linkages)T
qm dm qs ds, , ,i i λ λ = x , the computation
of the state matrix elements gives several null results. This choice for a set of state-space variablespresents only theoretical importance, as the output vector components are unmeasurable. Also the
computational demand of the model does not make it a practical option for vector controlimplementation.
s lr m r s ls r ls s lrr lr r
ls b ls b
qm qms m lr r s ls r ls s lrr lr r
b ls b lsdm dm
qs s m sb
ls lsds
s m s
ls ls
' ' ' ' '
' ' ' ''
0 0
0 0
r x x r x x r x r x x
x D x D D
i ir x x r x x r x r xx
x D D x Di ip
r x r
x x
r x r
x x
ω ω
ω ω
ω ω
ω ω
λ λω
λ
− − − ⋅
− −
− ⋅ = ⋅
−
−
qs
ds
lslr
qs
dslslr
qr
dr
'0 0
'0 0
'
1 0 0 0 '
0 1 0 0
xx
uD D
uxx
D D u
u
λ
+
+ ⋅
(221)
The electromagnetic torque relation:
( )me dm qs qm ds
b ls
3
4
xPT i i
xλ λ
ω= ⋅ ⋅ − (222)
VIII) One other model analysed for theoretical reasons, is the mixed current-flux linkages spacevector model which realises a the connection between the magnetising current space-vector and the
rotor flux linkage space vector, selected as state-space variables T
qm dm qr dr, , ' , 'i i λ λ = x As the
output vector components cannot be measured, this model, like the previous one, is prohibitive forimplementation in vector control strategies. However, from computational effort, the state matrixcontains the same number of zero elements (four) and the instantaneous electromagnetic torque is
determined in a suitable form comparable to the classical vector control implementations.
51
r ls m s r lr s lr r ls ls r
lr lr b
qm r ls m s r lr ls s lr r lsr
lr b lrdm
qr r m r rb
lr lr bdr
r m r r
lr b lr
' ' ' ' '0
' '
' ' ' ' '0
' '
' ' '0
' ''
' '0
' '
r x x r x x r x r x x
x D x D D
i r x x r x x x r x r x
x D D x Dip
r x r
x x
r x r
x x
ω
ω
ω
ω
λ ωω
ωλ
ω
ω
− − ⋅
− −
− ⋅ =
−
− −
qm
dm
qr
dr
lslr
qs
dslslr
qr
dr
'
'
'0 0
'0 0
'
0 0 1 0 '
0 0 0 1
i
i
xx
uD D
uxx
D D u
u
λ
λ
⋅ +
+ ⋅
(223)
The electromagnetic torque is determined as follows:
( )me qm dr dm qr
b lr
3' '
4 '
xPT i i
xλ λ
ω= ⋅ ⋅ − (224)
2.9. Vector control startegies for three-phase induction machine
The aim of vector control is usually to decouple the stator current is into its flux producing andtorque producing components (ids, iqs respectively) in order to obtain a decoupled control of the flux
and the electromagnetic torque. For this reason a special reference frame is selected fixed todifferent space vector variables. The reference frame has to be synchronous, as all the space vectors
have the same angular velocity given by the supply voltage frequency (Vas - 1990, 1992), (Slemon- 1994), (Kelemen and Imecs - 1987).
Generally, the term of vector control is associated with field orientation control. This means that
the special synchronous reference frame is linked to one of the flux linkages space vectors. Theoriginal field orientation sheme, developed more than twenty five years ago, was based on the
alignement of the reference frame to the rotor flux. After 1985, (DeDoncker and Nowotny - 1988),(Erdman and Hoft - 1990) this control strategy was extented to the air-gap flux and to the stator fluxalignement of the synchronuous reference frame. However, for vector control schemes there are two
more possibilities, related to the currents space vectors: stator and rotor currents. All of thesecontrol strategies are investigated in this chapter, according to the modelling point of view of the
induction machine. An easy to follow steps algorithm for implementing vector oriented controlsystems is obtained as follows (Lai - 1999):1. A complete mathematical model of the three-phase induction machine is developed in stationary
reference frame, according to the chosen set of state-space variables;2. The rotor based variables are completely expressed in the new state-variable system;
3. The rotor angular velocity ωr term is substituted with (ω r - ω k) where ω k is the angular
velocity of the synchronous frame;
4. The synchronous reference frame is selected linked to one of the space vectors, which meansthat the q-axis component of the reference space vector is null;
5. The torque equation is computed according to the selected flux or current space vector in the
synchronous reference frame.The transformation of reference frames for the induction motor vector control can be summarised as
shown in Fig. 2.18.
52
Drive
ControllerInverse 2/3 Clark/ Park
Transformation
3/2 Clark/ Park
Transformation
Induction
motor
Stationary reference frameSynchronuous reference frame
Two-axis co-ordinate systemThree-phase system
Fig. 2.18. Block diagram of transformation of frames and co-ordinate systemsfor induction motor vector control
2.9.1. Stator flux field orientation (SFO)
For this vector control strategy, the set of state-space variables formed by stator flux linkage and
current space vectors T
qs ds qs ds, , ,i iλ λ = x is selected. The mathematical model is given in the
chapter dedicated to d-q models of the three-phase induction machine.The rotor based variables expressed in state-space variables terms are:
( )
( )
( )
( )
s s s
qr qs s qs
m
s s s
dr ds s ds
m
s s s
qr r qs qs
m
s s s
dr r ds ds
m
1'
1'
1' '
1' '
i x ix
i x ix
x D ix
x D ix
λ
λ
λ λ
λ λ
= −
= −
= − ⋅
= − ⋅
(225-228)
To obtain the machine equations in the synchronuous stator flux reference frame, one has to
eliminate the rotor flux from the rotor voltage equations and then force the q component of thestator flux to be zero. The stator voltages equations remain unchanged. If a cage rotor is considered,
the resultant equations are as follows:
( )s s s ssr s qs r ds ds r qs
b b b
s s s ss s
r s ds qs r r ds r qs
b b b b
' ' ' 0
' ' ' '
p pr x D i x D i x
p pr x D i D i r x x
ωλ λ
ω ω ω
ω ωλ λ
ω ω ω ω
+ − − ⋅ − =
+ − ⋅ = + −
(229-230)
where the definition used for the slip speed is: s e rω ω ω= − .
If the special reference frame is fixed to the stator flux linkage vector, the q-component of thisflux vector is defined equal to zero:
s
qs
s s
s ds
0λ
λ λ
=
= (231)
From stator flux linkage equations, the q-current components are given by:
53
s s
qs qs
s ss
qr qs
m
i i
xi i
x
=
= − (232)
The electromagnetic torque relation and slip speed can be derived in stator field orientation controlas:
( )
s s
e ds qs
b
s
b r s qs
s s s
r ds ds
3 1
4
'
'
PT i
r x D p i
x D i
λω
ωω
λ
= ⋅
+ ⋅=
− ⋅
(233-234)
The second dynamic equation of the machine, shows that there is a coupling between the stator
current components. Consequently, any change in the torque producing component s
qsi without
changing s
dsi accordingly will cause a transient in the stator flux. A decoupler is necessary to
overcome this disadvantage. Therefore the command current of the d-axis component of the statorcurrent can be calculated as follows:
s s si
ds p ds dq
s
s qss
dq
b r s'
Ki K i
p
D ii
r x D p
∆λ
ω
ω
= + +
⋅=
+ ⋅
(235-236)
where Kp and Ki are proportional, respectively integral coefficients of the flux controller. This
controller can be PI type or soft computing technique type (fuzzy, neural-network) as demonstratedby Xu and Nowotny (1988).
Steady-state perfomance of a stator flux oriented systemLetting the derivative operator p = 0, one can obtain the steady-state voltage equations of the
induction machine. After several manipulations of the system equations, the d current componentsare given by:
2
s s sb r s s r m sr
ds qs qs2
s m m b rm
2
s ss r m s
dr qs
m b r
' ''
'
'
'
r x x x xxi i i
x x rx
x x xi i
x r
ω ω
ω ω
ω
ω
−= ⋅ +
−= −
(237)
it yields the slip speed equation:2 s 2 2 s 2 2 s
s qs s b r m s b r s qs' ( ' ) 0i D r x r x iω ω ω λ ω− + = (238)
The solutions of the above equations have to be real, for a given stator flux linkage. This means that
the determinant of the second order equation satisfy the condition:
( ) ( ) ( )2 222 s 2 s
m b r s b r s qs' 4 ' 0x r D r x i∆ ω λ ω= − ≥ (239)
The maximum values for the q component of the stator current, the slip speed and theelectromagnetic torque (pull-out torque) are:
54
( )
( )
( )( )
2 s
s m s
qsmax
s
r ss max
22 s
m s
e maxb s
2
'
3
4 2
xi
Dx
r x
D
xPT
Dx
λ
ω
λ
ω
=
=
= ⋅
(240-242)
If the angular slip velocity is larger than ( )s maxω static instability will arise. This maximum
value depends only on machine parameters and not on stator flux level. Neveretheless, the pull-outtorque depends on the square of the flux magnitude. The pull-out torque determines the limit for the
system stability operation zone, when the stator flux oriented control is employed. It is possible tolimit the torque command not to exceed the pull out torque for a given stator flux. Some importantconclusions cand be drawn for this vector control strategy:
• The electromagnetic torque and stator flux producing components of the stator current are not
decoupled;
• A parameter dependent decoupling network has to be included;
• Torque and flux control does not require speed feed back;
• For operation at low speed it is difficult to estimate the stator flux;
• It is a good alternative for medium performance drive.
2.9.2. Rotor flux field orientation (RFO)
For this vector control strategy, the set of state-space variables formed by rotor flux linkage and
stator current space vectors T
qr dr qs ds' , ' , ,i iλ λ = x is selected. If the induction motor is equipped
with wound rotor, as the rotor currents are also measurable, the rotor current space vector can be
also selected as state-space variables together with the rotor flux linkage space vectorT
qr dr qr dr' , ' , ' , 'i iλ λ = x . The mathematical model is given in the chapter dedicated to d-q models
of the three-phase induction machine. The first case will be analysed, for the second one the
algoritm is similar.The stator flux linkage and rotor current space vectors components expressed as functions in
terms of state-space variables are:
( )
( )
( )
( )
r r r
qs m qr qs
r
r r r
ds m dr ds
r
r r r
qr qr m qs
r
r r r
dr dr m ds
r
1'
'
1'
'
1' '
'
1' '
'
x D ix
x D ix
i x ix
i x ix
λ λ
λ λ
λ
λ
= + ⋅
= + ⋅
= −
= −
(243-246)
The voltage equations re-written in terms of the state-space variables become:
55
r r r r rs s
qs s qs ds m qr m dr
r b b r b b
r r r r rs sds s ds qs m dr m qr
r b b r b b
r r r rsr rqr m qs qr dr
r r b b
r rr
dr m ds
r
' '' '
' '' '
' '' 0 ' '
' '
'' 0
'
D p D pu r i i x x
x x
D p D pu r i i x x
x x
r r pu x i
x x
ru x i
x
ω ωλ λ
ω ω ω ω
ω ωλ λ
ω ω ω ω
ωλ λ
ω ω
= + ⋅ + ⋅ + +
= + ⋅ − ⋅ + −
= = − + + +
= = − r rsr
dr qr
r b b
'' '
'
r p
x
ωλ λ
ω ω
+ + −
(247-250)
If the special reference frame is fixed to the rotor flux linkage vector, the q-component of thisflux vector is defined equal to zero:
r
qr
r r
r dr
' 0
' '
λ
λ λ
=
= (251)
From stator flux linkage equations, the q-current components are given by:r r
qs qs
r rm
qr qs
r
'
i i
xi i
x
=
= − (252)
The flux producing component of the stator current is determined as follows:
rr
dr
r br
ds
m
'1 '
'
x p
ri
x
λω
+ ⋅
= (253)
The above relation shows that there is no need of a current decoupler in rotor field orientationscheme. Both stator current components (torque and flux producing) can be controlled
independently.
Steady-state perfomance of a rotor flux oriented systemLetting the derivative operator p =0, one can obtain the steady-state voltage equations of the
induction machine. After several manipulations of the system equations, the d current components
are given by:rrqsr dr b r
ds
m s r
r
dr
' '
'
0
iri
x x
i
λ ω
ω= = ⋅
=
(254)
It yields the slip speed equation:
r rm
s dr b r qs
r
' ' 0'
xr i
xω λ ω− ⋅ = (255)
which for a given rotor flux has always real solution. Thus the resulting current controlled slipspeed and the electromagnetic torque are:
( )( )
r
m r b qs
s r
r dr
22
r r rm m r
e dr qs qs2
r b sr
'
' '
'3 3'
4 ' 4 '
x r i
x
x x rP PT i i
x x
ωω
λ
λω ω
=
= ⋅ = ⋅ ⋅
(256-257)
The most important features of rotor field oriented (RFO) vector control are:
56
• As it is the original vector control strategy developed by Blashke (1971), the RFO represents the
most popular approach in vector control strategies;
• It provides complete decoupling of the torque and flux producing components of the stator
current;
• There are possible direct vector control strategy (the usage of sensors or model to provide
feedback of the flux magnitude and orientation) and indirect vector control strategy (the usage
of assumed slip frequency relationship to achieve field orientation);• The direct vector control strategy is the optimum choice for medium and high-speed
applications;• The indirect vector control strategy is the optimum choice for low-speed applications;
2.9.3. Air-gap flux field orientation (AFO)
For this vector control strategy, the set of state-space variables formed by air-gap flux linkage
and stator current space vectors T
qm dm qs ds, , ,i iλ λ = x is selected. If the induction motor is equipped
with wound rotor, as the rotor currents are also measurable, the rotor current space vector can be
also selected as state-space variables together with the air-gap flux linkage space vectorT
qm dm qr dr, , ' , 'i iλ λ = x . The mathematical model is given in the chapter dedicated to d-q models of
the three-phase induction machine. As both cases are similar to analyse, only the first case will bedetailed.
The stator and rotor flux linkages and rotor current space vectors components expressed asfunctions in terms of state-space variables are:
( )( )
( )
( )
m m m
qs qm s m qs
m m m
ds dm s m ds
m m mrqr qm r m qs
m
m m mrqs qm r m ds
m
'' '
''
x x i
x x i
xx x i
x
xx x i
x
λ λ
λ λ
λ λ
λ λ
= + −
= + −
= − −
= − −
(258-261)
r m m
qr qm qs
m
r m m
dr dm ds
m
1'
1'
i ix
i ix
λ
λ
= −
= −
(262-263)
The voltage equations of the machine expressed in state-variables terms, are readily deductible
now:
( )
( )
m m m m ms
qs s s m qs qm dm s m ds
b b b
m m m m ms
ds s s m ds dm qm s m qs
b b b
m m m msr r rqr r r m qs qm dm r m ds
b m m b b m
( ) ( )
( ) ( )
' ' '' 0 ' ( ' ) ( ' )
p pu r x x i x x i
p pu r x x i x x i
r x xp pu r x x i x x i
x x x
ωλ λ
ω ω ω
ωλ λ
ω ω ω
ωλ λ
ω ω ω
= + − + + + −
= + − + − + −
= = − + − + + ⋅ + − −
m
m m m m msr r rdr r r m ds dm qm r m qs
b m m b b m
' ' '' 0 ' ( ' ) ( ' )
r x xp pu r x x i x x i
x x x
ωλ λ
ω ω ω
= = − + − + + ⋅ − − −
(264-267)
57
If the special reference frame is fixed to the air-gap flux linkage vector, the q-component of thisflux vector is defined equal to zero:
m
qm
m m
m dm
0λ
λ λ
=
= (268)
From air-gap flux linkage equations, the q-current components are given by:m m
qs qs
m m
qr qs'
i i
i i
=
= − (269)
The flux producing component of the stator current is determined as follows:
m m msr r
r r m ds dm r m qs
b m m b b
' '' ( ' ) ( ' )
r xp pr x x i x x i
x x
ωλ
ω ω ω
+ − = + ⋅ + −
(270)
One can note that the d-axis component of stator current for AFO is not only controllled by the d-axis air-gap flux componennt, but also by the q-axis (torque producing) component of the stator
current. It is necessary to decouple the stator current components, in order to achieve a linearcontrol. For this reason, the command of current of the d-axis component is computed as follows:
m m mi
ds p dm dq
m
s r m qsm
dq
b r r m
( ' )
' ( ' )
Ki K i
p
x x ii
r x x p
∆λ
ω
ω
= + +
− ⋅=
+ − ⋅
(271-272)
The electromagnetic torque relation and slip speed can be derived in air-gap field orientationcontrol as:
[ ]
m m
e dm qs
b
m
b r r m qs
sm mr
dm r m ds
m
3 1
4
' ( ' )
'( ' )
PT i
r x x p i
xx x i
x
λω
ωω
λ
= ⋅
+ − ⋅=
− − ⋅
(273-274)
Steady-state perfomance of an air-gap flux oriented systemLetting the derivative operator p =0, the steady-state voltage equations of the induction machine
are readily determined. After several manipulations of the system equations, the d currentcomponents are given by:
m m mb sr r
ds qs r m qs
m s m b
m
m r qsm s
dr
r b
' '( ' )
( ' )
'
r xi i x x i
x x
x x ii
r
ω ω
ω ω
ω
ω
= ⋅ + − ⋅
−= ⋅
(275)
It yields the slip speed equation:2
2 m m mr ms qs s dm r b q s
r b
( ' )' 0
'
x xi r i
rω ω λ ω
ω
−⋅ ⋅ − + = (276)
The solutions of the above equations have to be real, for a given stator flux linkage. This means that
the determinant of the second order equation satisfy the condition:
( ) ( ) ( )2 22m m
dm r m qs4 ' 0x x i∆ λ= − − ⋅ ≥ (277)
The maximum values for the q component of the stator current, the slip speed and theelectromagnetic torque (pull-out torque) are:
58
( )
( )
( )( )
mm dm
qs maxr m
rs max
r m
2m
dm
e maxr m
2( ' )
'
'
3
8 '
ix x
r
x x
PT
x x
λ
ω
λ
=−
=−
= ⋅−
(278-280)
When the angular slip velocity is larger than ( )s maxω instability will occur. One should note that
the maximum (pull-out) angular slip velocity depends only on the rotor parameters and does not
depends on the air-gap flux. However, the maximum (pull-out) torque is proportional to the squareof the air-gap flux magnitude and thus a small increment of air-gap flux will detetmine a significantincrement of the electromagnetic torque.
Some important conclusions cand be drawn for the air-gap field orientation (AFO) controlstrategy:
• The electromagnetic torque and air-gap flux producing components of the stator current are not
decoupled;
• A parameter dependent decoupling network has to be included;
• Torque and flux control does not require speed feed back;
• No sophisticated parameter estimation methods or model based observers is required;
• It is a good alternative for low and medium performance drives as the air-gap flux can be
measured directly;
2.9.4. Stator current orientation (SCO)
If the field orientation control is well established with multiple practical solutions for different
industrial applications, an unconventional method of controlling the speed and torque for theinduction machine, is given by the selection of currents space vectors as linking basis for thesynchronuous reference frame. The set of state-space variables is identical with the stator flux field
orientation control (SFO): the stator flux and current space vectorsT
qs ds qs ds, , ,i iλ λ = x . The
mathematical model is identical to that used in SFO case.The rotor based variables expressed in state-space variables terms are:
( )
( )
( )
( )
sc sc sc
qr qs s qs
m
sc sc sc
dr ds s ds
m
sc sc sc
qr r qs qs
m
sc sc sc
dr r ds ds
m
1'
1'
1' '
1' '
i x ix
i x ix
x D ix
x D ix
λ
λ
λ λ
λ λ
= −
= −
= − ⋅
= − ⋅
(281-284)
To obtain the machine equations in the synchronuous stator current reference frame, one has to
eliminate the rotor flux from the rotor voltage equations and then force the q component of thestator current to be zero. The stator voltages equations remain unchanged. If a cage rotor isconsidered, the resultant equations are as follows:
59
( )sc sc sc scs
r s qs r ds ds r r qs
b b b
sc sc sc scs s
r s ds qs r r ds r qs
b b b b
' ' ' ' 0
' ' ' ' 0
p pr x D i x D i r x
p pr x D i D i r x x
ωλ λ
ω ω ω
ω ωλ λ
ω ω ω ω
+ − − ⋅ − + =
+ − ⋅ − + + =
(285-286)
If the special reference frame is fixed to the stator current space vector, the q-component of thiscurrent vector is defined equal to zero:
sc
qs
s sc
s ds
0i
i i
=
= (287)
Different from the SFO case, we have to express the q-flux components, by considering the fluxlinkages equations:
sc sc
qs qs
sc scr
qr qs
m
''
x
x
λ λ
λ λ
=
= (288)
The electromagnetic torque relation and slip speed can be derived in stator current orientation
control as:
( )
sc sc
e qs ds
b
sc
b r s r qs
s s sc
ds r ds
3 1
4
' '
'
PT i
r x x p
D i x
λω
ω λω
λ
= − ⋅
+ ⋅=
⋅ −
(289-290)
From the second dynamic equation of the machine, a relation between the d-axis component of the
stator current and stator flux linkage space vector components can be deduced:
sc sc scsr s ds r r ds r qs
b b b
' ' ' ' 0p p
r x D i r x xω
λ λω ω ω
+ − + + =
(291)
The previous dynamic equation of the machine, shows that there is a coupling between the stator
flux linkage components. Consequently, any change in the torque producing component sc
dsλ
without changing sc
qsλ accordingly will cause a trensient in the stator flux. A decoupler is necessary
to overcome this disadvantage. Therefore the command current of the d-axis component of thestator current can be calculated as follows:
sc sc sci
ds p ds dq
scs
r qs
sc b
dq
r r
b
'
' '
KK i
p
x
pr x
λ ∆ λ
ωλ
ωλ
ω
= + +
=
+
(291-292)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. This
controller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomance of a stator current oriented systemLetting the derivative operator p =0, one can obtain the steady-state voltage equations of the
induction machine. After several manipulations of the system equations, the d current components
are given by:
60
sc sc scs s r br
ds qs qs2 2
r b sm m
sc scbr
dr qs
m s
''
'
'
x rx D
r x x
r
x
ω ωλ λ λ
ω ω
ωλ λ
ω
= − ⋅ ⋅ ⋅ − ⋅ ⋅
= − ⋅
(293)
it yields the slip speed equation:2
sc sc scs sr
qs ds qs2
b r m b r m
' 1 1 10
' '
xi
r x r x
ω ωλ λ
ω ω
⋅ + ⋅ + =
(294)
The solutions of the above equations have to be real, for a given stator current. This means that the
determinant of the second order equation satisfy the condition:22 2 scsc
qsds r
2r r m
'4 0
' '
i x
r r x
λ∆
= − ≥
(295)
The maximum values for the q component of the stator flux linkage, the slip speed and theelectromagnetic torque (pull-out torque) are:
( )
( )
( )( )
2 scsc m dsqs
maxr
rs max
r
22 scm ds
e maxb r
2 '
'
'
3
4 2 '
x i
x
r
x
x iPT
x
λ
ω
ω
=
=
= ⋅
(296-299)
If the angular slip velocity is larger than ( )s maxω static instability will arise. This maximum value
depends only on machine parameters and not on stator current or flux level. Nevertheless, the pull-
out torque depends on the square of the stator current magnitude. The pull-out torque determinesthe limit for the system stability operation zone, when the stator current space vector oriented
control is employed. It is possible to limit the torque command not to exceed the pull out torque fora given stator current. Some important conclusions can be drawn for the stator current orientation(SCO) vector control strategy:
• The electromagnetic torque and stator current related components of the stator flux linkage are
not decoupled;
• A parameter dependent decoupling network has to be included;
• Torque and flux control does not require speed feed back;
• An operation at low speed is easy to realise as there is no need to estimate the stator flux;
• It is a good alternative for wide range speed drives.
2.9.5. Rotor current orientation (RCO)
Another unconventional vector control strategy is the one wich comprises a synchronuousreference frame linked to the rotor current space vector. The rotor current orientation control
strategy (RCO) is analysed for a cage rotor induction motor. The selected set of state-spacevariables is given by two options if we consider the criteria of direct measurable quantities:
I) The stator flux linkage and rotor current space vectors: T
qs ds qr dr, , ' , 'i iλ λ = x ;
When the stator flux and and the rotor current are selected as state-space variables, one can derivethe rotor flux and stator current functions in terms of state variables as follows:
61
( )
( )
rc rc rc
qs qs m qr
s
rc rc rc
ds ds m dr
s
rc rc rcm
qr qs qr
s s
rc rc rcmdr ds dr
s s
1'
1'
' '
' '
i x ix
i x ix
x Di
x x
x Di
x x
λ
λ
λ λ
λ λ
= −
= −
= +
= +
(300-304)
To obtain the machine equations in the synchronuous rotor current reference frame, one has toeliminate the rotor flux from the rotor voltage equations and then force the q component of the rotorcurrent to be zero. The stator voltages equations are re-written also. The resultant equations have
the following form:
rc rc rc rcs m s s
qs qr qs ds
s s b b
rc rc rc rcs m s s
ds dr ds qs
s s b b
rc rc rc rcs sm mr qr dr qs ds
s b s b s b s b
rc sr dr
s b s
'
'
0 ' ' '
0 ' '
r x r pu i
x x
r x r pu i
x x
x xD p D pr i i
x x x x
D p Dr i
x x
ωλ λ
ω ω
ωλ λ
ω ω
ω ωλ λ
ω ω ω ω
ω
ω ω
= − + + +
= − + + −
= + ⋅ + ⋅ + ⋅ + ⋅
= + ⋅ − ⋅
rc rc rcsm mqr ds qs
b s b s b
'x xp
ix x
ωλ λ
ω ω+ ⋅ − ⋅
(305-308)
If the special reference frame is fixed to the rotor current space vector, the q-component of this
current vector is defined equal to zero:rc
qr
rc rc
r dr
' 0
'
i
i i
=
= (309)
Different from the RFO case, we have to express the q-flux components, by considering the fluxlinkages equations:
rc rc
qs qs
rc scs
qr qs
m
'x
x
λ λ
λ λ
=
= (310)
The electromagnetic torque relation and slip speed can be derived in stator current orientation
control as:
rc rcm
e qs dr
s b
rc
m qs
s rc rc
dr m ds
3'
4
'
xPT i
x
x p
D i x
λω
λω
λ
= ⋅
− ⋅=
⋅ +
(311-312)
From the second dynamic equation of the machine, a relation between the d-axis component of the
rotor current and stator flux linkage space vector components can be deduced:
rc rc rcsm m
r dr ds qs
s b s b s b
0 ' 'x xD p p
r ix x x
ωλ λ
ω ω ω
= + ⋅ + ⋅ − ⋅
(313)
A coupling between the stator flux linkage components appears, as can be deduced from theprevious dynamic equation of the machine. Consequently, any change in the torque producing
62
component rc
dsλ without changing rc
qsλ accordingly, will cause a transient in the stator flux. The
command current of the d-axis component of the stator current can be calculated in order toovercome this disadvantage:
rc rc rci
ds p dr dq
rc sc
dq s qs
'
1
KK i
p
p
λ ∆ λ
λ ω λ
= + +
= ⋅
(314-315)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. Thiscontroller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomance of a rotor current oriented systemAfter several manipulations of the system equations if we let the derivative operator p = 0, the d
current components are obtained from the steady-state voltage equations of the induction machine:
rc rcs
ds qs
s r b
rc
dr
'
' 0
D
x r
ωλ λ
ω
λ
= − ⋅
=
(316)
It yields the slip angular velocity equation:
rc rcsm
dr qs2
s bs r
0'
x DDi
x x r
ωλ
ω⋅ − ⋅ = (317)
which for a given rotor flux has always real solution. Thus the slip speed and the electromagnetictorque are:
rc
s r b dr
s rc
m qs
rc rcm
e qs dr
s b
' '
3'
4
x r i
x
xPT i
x
ωω
λ
λω
=
= ⋅
(318-319)
The most important features of rotor current oriented (RCO) vector control with mixed flux andcurrent state-space variables are:• The electromagnetic torque and rotor current related components of the stator flux linkage are
not decoupled;• The decoupling network that has to be included is parameter independent and easy to
implement;• Torque and flux control require speed feed back;
• There is no stability limit as there is no pull-out slip speed or electromagnetic torque,
• An operation at low speed is easy to realise as there is no need to estimate the stator flux;
• It is a good alternative for wide range speed drives.
II) The stator and rotor currents T
qs ds qr dr, , ' , 'i i i i = x . When the stator and the rotor current are
selected as state-space variables, one can derive the stator and rotor flux as functions in terms ofstate variables from the classical flux linkage equations.
To obtain the machine equations in the synchronuous rotor current reference frame, one has toeliminate the rotor flux from the rotor voltage equations and then force the q component of the rotorcurrent to be zero. The stator voltages equations are re-written also. The resultant equations have
the following form:
63
rc rc rc rc rcs s
qs s s qs m qr s ds m dr
b b b b
rc rc rc rc rcs s
ds s qs m qr s s ds m dr
b b b b
rc rc rc rcs sm qs r r qr m ds r dr
b b b b
rcsm qs
b
' '
' '
0 ' ' ' ' '
0
p pu r x i x i x i x i
p pu x i x i r x i x i
p px i r x i x i x i
x i
ω ω
ω ω ω ω
ω ω
ω ω ω ω
ω ω
ω ω ω ω
ω
ω
= + + + +
= − − + + +
= + + + +
= − rc rc rcsr qr m ds r r dr
b b b
' ' ' ' 'p p
x i x i r x iω
ω ω ω
− + + +
(320-323)
If the special reference frame is fixed to the rotor current space vector, the q-component of this
current vector is defined equal to zero:rc
qr
rc rc
r dr
' 0
'
i
i i
=
= (324)
Different from the (I) case of RCO control strategy, for the (II) case we have to express the stator
flux linkage q-components, by considering the flux linkages equations:rc rc
qs s qs
rc sc rcm
qr m qs qs
s
'
x i
xx i
x
λ
λ λ
=
= = (325)
From the second dynamic equation of the machine, a relation between the d-axis component of therotor current and stator current space vector components can be deduced:
rc rc rcsm ds m qs r r dr
b b b
' ' 'p p
x i x i r x iω
ω ω ω
= − +
(326)
The electromagnetic torque relation and slip speed can be derived in stator current orientationcontrol as:
rc rcm
e qs dr
b
rc
m qs
s rc rc
m ds r dr
3'
4
' '
xPT i i
x pi
x i x i
ω
ω
= ⋅ ⋅
⋅= −
+
(327-328)
A coupling between the d-axis and q-axis stator current components appears, as can be deducedfrom the previous dynamic equation of the machine. Consequently, any change in the torque
producing component rc
dsi without changing rc
qsi accordingly, will cause a transient in the stator flux.
The command current of the d-axis component of the stator current can be calculated in order to
overcome this disadvantage:
rc rc r ci
ds p dr dq
rc sc
dq s q s
'
1
Ki K i i
p
i ip
∆
ω
= + +
= ⋅
(329-330)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. Thiscontroller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomance of a rotor current oriented systemAfter several manipulations of the system equations if we let the derivative operator p = 0, the d
flux components are obtained from the steady-state voltage equations of the induction machine:
64
rc rcs
ds qs
r b
rc
dr
'
' 0
Di
r
ωλ
ω
λ
= − ⋅
=
(331)
It yields the slip angular velocity equation:
rc rcsm
dr qs
s bs r
0'
x DDi i
x x r
ω
ω⋅ − ⋅ = (332)
which for a given rotor flux has always real solution. Thus the slip speed and the electromagnetictorque are:
rc
r b dr
s rc
m qs
rc rcm
e qs dr
b
' '
3'
4
r i
x i
xPT i i
ωω
ω
=
= ⋅
(333-334)
The rotor current oriented (RCO) vector control with currents state-space variables presents the
same features as the mixed flux-current state variables model. However the current model is easierto be implemented as it provides directly the estimated values for the two-axis co-ordinate stator
current. Thus the command for a current PWM inverter is readily obtainable. Also, as feedbackmeasurements, the stator currents sensors are more reliable and cheaper than the flux sensors.
65
3. CONTINUOS-TIME DOMAIN LINEAR MODELS OF THESINGLE- PHASE INDUCTION MACHINE
3.1. Introduction
The induction machine is used in a wide range of applications as an electrical to mechanicalenergy converter. The single-phase induction machine is the most used converter in home
appliances.The analysis of the induction machine is essentially the same for a three-phase, two-phase or
single-phase machine. An accurate mathematical model for the induction machine is necessary tobe determined in vector control operation. This model has to be suitable for the analysis of both thesteady-state operation and the dynamic operation of the system. If the double-revolving field
(Veinott - 1959), or the symmetrical component theories (Fortescue - 1918) permit a detailedsimulation and modelling of the single-phase induction machine in steady-state operation, for the
dynamic analysis of the machine a different mathematical approach has to be found.. Starting fromthe reference frame theory, with voltages, currents and fluxes referred to a two-axis quadrature co-ordinates system, a general model for the single-phase induction machine is developed according to
Krause (1965) and Krause et al (1995). Different from the three-phase induction machine, wheretwo approaches are valid (d-q axes and space vector theory), the single-phase induction machine is
completely described only by a two-axis quadrature axis.
3.2. Voltage and flux-current equations of the single-phase induction machine
In Fig. 3.1 is illustrated a single-phase induction machine. The following assumptions have been
made:a) electrically orthogonal stator windings with sinusoidal distribution;b) only the fundamental-space-harmonic-component of the air-gap flux distribution will be
considered;c) magnetic-saturation effects, core loss and stray load losses are negligible;
d) magnetic-diffusion (i.e. deep-bar) effects in the rotor will be ignored. This assumption istypically valid in small induction machine. It is further justified by the fact that under mostoperating conditions, the single-phase induction motor will be operating at low slips and hence
the rotor currents will be at frequencies sufficiently low that magnetic-diffusion effects areinsignificant.
e) temperature effect on windings resistance and reactance value is negligible;f) the lamination magnetic permeability is considered infinite;g) in steady-state operation, the voltages and currents are sinusoidal.
66
bs axisbr axis
as axis
ar axis
u as
ubs
rs
r s
Ls
Ls
ias
ibs
rr
Lr
Lr
rr
iar
ibr
bs’
as’
as
br’ar
ar’
bs
brθ r
ω rΦr
Φs
TL
Te
-
+
+-
statorrotor
Fig.3.1. The real model of the symmetrical single-phase induction machine withsquirrel-cage rotor represented by two identical windings
The machine rotor is described by two identical magnetically orthogonal windings. Consideringan arbitrarily reference frame, the spatial position of the stator winding is characterised by the
electrical angle Φ s and the spatial position of the rotor winding is characterised by the electrical
angle Φ r. The angular speed of the rotor is ω r and the displacement between stator and rotor
windings is θ r. These angles are linked through the relation:
s r rΦ Φ θ= + (335)
The voltage equations related to the machine from Fig.3.1, can be expressed as follows:
as s
bs s
2cos
2cos( )
u V t
u V t
ω
ω ϕ
=
= + (336-337)
asas s as
bs
bs s bs
arr ar
br
r br
0
0
du r i
dt
du r i
dt
dr i
dt
dr i
dt
λ
λ
λ
λ
= +
= +
= +
= +
(338-342)
As the machine is magnetically linear, the fluxes are easily determined from the currents andinductance values. Particularly, it can be written:
67
as asas as asbs bs asar ar asbr br
bs bsas as bsbs bs bsar ar bsbr br
as aras as arbs bs arar ar arbr br
br bras as brbs bs brar ar brbr br
L i M i M i M i
M i L i M i M i
M i M i L i M i
M i M i M i L i
λ
λ
λ
λ
= + + +
= + + +
= + + +
= + + +
(343-346)
The mutual inductances given in the above relations are defined by the subscripts. Applying thereciprocity principle, the following identities are valid:
asbs bsas asar arasM M M M= =
For further development of the model, the terms from the previously equations are grouped in amatrix form:
abs s abs sr abr
T
abr sr abs r abr( )
λ
λ
= +
= +
L i L i
L i L i (347-348)
The indexes denote stator fluxes (as, bs), respectively rotor fluxes (ar, br). The self-inductance can
be written by including a leakage inductance caused by the leakage flux and a magnetisation(mutual) inductance caused by the magnetic flux which links the stator and rotor core:
ss ls ms
rr lr mr
L L L
L L L
= +
= + (349-350)
where the first term stands for the leakage inductance, and the second one for the mutual inductance
between stator and rotor windings. The mutual inductance can be determined:2
s(r)
ss(rr)
m
NL =
ℜ (351)
where: Ns(r) is the turns number and ℜm the magnetic reluctance depending on the air-gap value and
on the magnetic core dimensions.As the magnetic axis of the stator, respectively rotor windings are orthogonal, it results in null
mutual inductance between the stator and rotor windings. This is one of the main differences fromthe three-phase induction machine where the stator (rotor) windings are placed at 1200 electricalspace angle, which determines mutual linkage among the same armature windings.
Due to the relative movement between stator and rotor windings a magnetic linkage will appear.The following defining relations for the mutual inductance can be expressed:
asar sr r
asbr sr r
bsar sr r
bsbs sr r
s r
sr
cos
sin
sin
cos
m
M L
M L
M L
L L
N NL
θ
θ
θ
θ
=
= −
=
=
=ℜ
(352)
The electromagnetic torque of the single-phase induction machine can be determined from the
generalised forces law:
c r
e r
r
( , )( , )
2
W iPT i
θθ
θ
∂= ⋅
∂ (353)
where Wc represents the conenergy which is equal to the magnetic energy of the linkage field asfollows:
2 2 2 2as bs ar brf ss ss rr rr ms a s ar r
m s a s br r ms bs ar r ms bs br r
1( ' ' ' ' ) ' cos
2
' sin ' sin ' cos
W L i L i L i L i L i i
L i i L i i L i i
θ
θ θ θ
= + + + + −
− + + (354)
68
In the above relation, the superscript (’) stands for referred rotor windings to the stator according tothe expressions:
r
abr abr
s
sabr abr
r
2
s
rr rr
r
'
'
'
Ni i
N
N
N
NL L
N
λ λ
=
=
=
(356-357)
The electromagnetic torque relation can be simplified and presented in the following form:
( ) ( )e ms as ar bs br r as br bs ar r' ' sin ' ' cos2
PT L i i i i i i i iθ θ = − + + − (358)
This relation shows that there are two components of the instantaneous electromagnetic torque:
an average component with constant value for a given value of the rotor speed and a pulsatingcomponent with a frequency double of the currents frequency (ω e). The pulsating torque
component determines an important magnetic noise for the single-phase induction machinecompared to the three-phase induction machine. The mechanical equation that links the torque and
the rotor speed is:
r
e m r L
2 2dT J B T
P dt P
ωω= ⋅ + + (359)
where J is the rotor inertia, B is the viscous friction coefficient associated to the rotational system ofthe machine and with the mechanical load, and P represents the number of poles for the analysedmachine.
3.3. Analysis of the single-phase induction machine in stationary reference frame
In order to eliminate the time dependence of the voltage and flux equations terms, a variablestransformation into a new reference frame is necessary. This transformation is given by the
following relations (Krause et al - 1995):ssqds abs=f K f (360)srqdr abr' '=f K f (361)
ss
cos sin
sin cos
θ θ
θ θ
= −
K (362)
r rsr
r r
cos( ) sin( )
sin( ) cos( )
θ θ θ θ
θ θ θ θ
− − = − − −
K (363)
where f can be fluxes, voltages, currents in new co-ordinates (d-q) or in classical co-ordinates (a-b).
Index (s) stands for statoric terms and index (r) stands for the rotoric terms. One should note that thetwo transformation matrixes depend on the generalised co-ordinate θ which expresses the
periferical displacement of the chosen reference frame.Reference frames linked to stator, rotor or arbitrarily can be chosen for the polyphase induction
machine. As for the single-phase induction machine, the stator windings are not identical, the only
transformation that maintains the windings parameters (resistance, inductance) unchanged is thestationary reference frame transformation (θ =0).
The voltage and linkage flux equations are expressed for the previous assumptions made at thebeginning of this chapter.
69
Note: For compactness, the flux linkages will be further described by flux linkage units persecond or volts, and the inductance equivalent circuit elements will be transformed in reactance
elements.
qs s qs qs
b
dq s ds ds
b
rr qr qr dr
b b
rr dr dr qr
b b
0 ' ' ' '
0 ' ' ' '
pu r i
pu r i
pr i
pr i
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + −
= + +
(364-367)
qs s qs m qs qr
ds s ds m ds dr
qr r qr m qs qr
dr r dr m ds dr
( ' )
( ' )
' ' ' ( ' )
' ' ' ( ' )
x i x i i
x i x i i
x i x i i
x i x i i
λ
λ
λ
λ
= + +
= + +
= + +
= + +
(368-371)
m
e qs dr ds qr
b
( ' ' )2
xPT i i i i
ω= ⋅ ⋅ − (372)
In the reference frame systems theory we have to note that the parameters do not depend on therelative position between the stator and the rotor. The time derivative (d/dt) in the above equations
is denoted as p, and ω b andω r represent the angular base speed given by the supply frequency,
respectively the electrical rotor speed. The qr terms are referred to the qs winding and the dr terms
are referred to the ds winding. It is important to observe that stator windings of the single-phaseinduction machine physically represent the d-q co-ordinates windings, different from the three-
phase induction machine where the two-axis co-ordinate equivalent windings are fictive.The dynamic analysis of the symmetrical single-phase induction machine can be accomplished
by using the equivalent circuit from Fig. 3.2. The only modification that has to be done in order to
obtain the real value of the terminal quantities (voltage, current) is the multiplication by (-1) factorfor the auxiliary (ds) winding parameters.
rs xs x’r (ωr/ωb)λ’dr
+ + -
iqs xm iqr r’r
uqs
_
rs xs x’r (ωr/ωb)λ’QR
+ - +
ids xm i’dr r’r
uds
_
Fig. 3.2. Equivalent circuit of the symmetrical single-phase induction machineconform to reference frame system theory
70
3.4. Analysis of the steady-state operation for the symmetrical single-phase induction machine
In a wide range of applications the single-phase induction machine is equipped with a cage rotor.During the steady-state operation, the stator parameters are variable with the stator voltage
frequency ω , and the rotor parameters are variable with the slip frequency ω - ω r. From the
stationary reference frame equations, by letting the differential operator p be replaced by the
complex operator jω , the following matrix relation is established (Krause and Thomas - 1965):
°
°
s ss m
b b
qs qss ss m
b bds ds
r r qrm m r rr rr
b b b b dr
r r rm m rr r rr
b b b b
0 0
0 0
0' ' '
0
' ' '
r j x j x
U Ir j x j x
IU
Ij x x r j x x
I
x j x x r j x
ω ω
ω ω
ω ω
ω ω
ω ωω ω
ω ω ω ω
ω ω ω ω
ω ω ω ω
+
+
= ⋅ − + −
+
ɶ
ɶ
ɶ
ɶ
(373)
where
ss s m
rr r m' '
x x x
x x x
= +
= + (374-375)
A symmetrical two-phase system is defined by the identities:
QS DS
QR DR
F jF
F jF
= −
= −
ɶ ɶ
ɶ ɶ (376-377)
where~F represents a complex variable with current or voltage significance. The four equations
from the above matrix relation are interdependent. If it is used, the variable slip s is defined as:
rs
ω ω
ω
−= (378)
and also through the inverse transformation from the stationary reference frame (d-q) to thephysical one (a-b), the relations for the steady-state operation analysis of the symmetrical single-phase induction machine can be written:°
as s s as m as ar
rar m as ar
( ) ( )
'0 ' ( )r
U r jx I jx I I
rjx I jx I I
s
= + + +
= + + +
ɶ ɶ ɶ
ɶ ɶ ɶ (379-380)
The above equations suggest the equivalent circuit from Fig. 3.3.
rs
jxs
jx’r
r’/s
jxm
Ias
I’ar
Uas
+
-
Fig. 3.3. Equivalent circuit of the symmetrical single-phase induction machine
for the steady-state operation analysis
71
With slight modifications (xm becomes 3/2xm) it can be observed that the equivalent circuit fromFig. 3.3 can be used also for the three-phase induction machine performance in steady-state
operation.The electromagnetic torque equation is:
22
m b r as*me as ar 2 2
b r rr
2( /2)( / )( ' / )2 Re
2 ( ' / ) '
P x r s IxPT jI I
r s x
ω
ω
= = +
ɶɶ ɶ (381)
°2
2asm b r
e 2 2 2
s r m ss rr r ss s rr
2( /2)( / )( ' )
( ' ( ' )) ( ' )
P x r s UT
r r s x x x r x sr x
ω=
+ − + + (382)
It is important to highlight that the positive values of the torque are obtainable when the slip s ispositive (motor operation) and the negative values when the slip s is negative (generator operation).
By setting the torque/slip derivative equal to zero, the relation for the critical slip can be obtained:
m r
2 2
s ss
2 2 2 2
m ss rr s rr
'
( ' ) '
s r G
r xG
x x x r x
=
+= ±
− +
(383-384)
The positive value corresponds to motor operation, and the negative one to generator operation.
If at start-up (s = 1) the torque is directly dependent to the variation of rotor resistance, as themagnetisation reactance value is considered to be much higher than the stator or rotor resistancevalue, the maximum torque value is not dependent to the rotor resistance value:
°2
2asm
e,max 22 2
s m ss rr ss s rr
2( /2)( / )
( ' ) ( ' )
P x G UT
r G x x x x Gr x
ω=
+ − + +
(385)
3.5. Analysis of the unsymmetrical single-phase induction machine
A new equations set and a new equivalent circuit for the unsymmetrical single-phase induction
machine can be obtained by eliminating from the initial assumptions the one referring to identicalstator windings (Krause - 1965):
Voltage equations:
qs m qs qs
b
ds a ds ds
b
r
r qr qr qr
b b
2 r
r dr dr dr
b b
10 ' ' ' '
0 ' ' ' '
pu r i
pu r i
pr i
k
pk r i k
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + − ⋅
= + + ⋅
(386-389)
Flux equations:
qs lm qs m qs qr
2
ds la ds m ds dr
qr sr qr m qs qr
2 2
dr sr dr m ds dr
( ' )
( ' )
' ' ' ( ' )
' ' ' ( ' )
x i x i i
x i k x i i
x i x i i
k x i k x i i
λ
λ
λ
λ
= + +
= + +
= + +
= + +
(390-393)
The electromagnetic torque equation is:
72
m
e qs dr ds qr
b
( ' ' )2
xPT k i i i i
ω= ⋅ ⋅ ⋅ − (394)
The mathematical model presented in Fig. 3.4 permits the analysis of steady state as well asdynamic operation of the unsymmetrical single-phase induction machine. This model is alsosuitable for implementing non-linearities effects such as: core loss or saturation of the main or
leakage inductance.
rm xlm x’sr (1/k) (ωr/ω
b)λ’dr
+ + -
iqs i’qr r’r
uqs
_
ra xla k2 x’sr k (ωr/ωb)λ’qr
+ - +
ids i’dr k 2 r’r
uds
_
xm
k2 xm
Fig. 3.4. Equivalent circuit of the unsymmetrical single-phase induction machine
3.6. Linear models for single-phase induction motors
The analysis of the induction machine-drives is generally made using a conventional linear
mathematical machine model, either in the form of self and mutual inductances, or the familiar T-form of equivalent circuit. This type of model has been developed in the previous paragraph,
dedicated to symmetrical and single-phase induction machines.When the vector control of these machines is implemented, such kind of models is adequate for
many situations, especially for air-gap flux orientation control strategy. However, they are more
complex than necessary for the analysis of most linear machines.If for the polyphased induction machine two approaches are valid related to vector control
analysis, i.e. the space vector notation and the two-axis co-ordinates reference frame, the single-phase induction machine can be completely described only by the latter approach in stationaryreference frame. As for the polyphased induction machines, the single-phase version can be
controlled with a scheme which maintains correct angular relationship between the stator currentvector and one flux vector (stator, air-gap or rotor flux), by either direct or indirect methods. All the
field-oriented methods suffer from specific theoretical and practical problems. The indirect methodsare highly dependent on the machine rotor parameters (varying with load and temperature), andhave good speed control performance only if precise shat encoders are used to calculate the
electrical frequency. The direct methods present minor dependency on rotor parameters, but are notable to measure the selected flux vector at low speed, i.e. zero frequency. This is why only
estimations of the selected flux vector can make possible the total control of the speed in directmethods. For estimation, usually the terminal quantities are measured and a mathematical modelwith related parameters is used. Thus, the correct estimation of the machine parameters is essential
for all types of field oriented control schemes.
73
The original field orientation scheme, developed for the three-phase induction machine, wasbased on the alignment of the rotor flux linkage because the torque and the rotor flux are related to
each other in a straightforward manner, without any de-coupling circuit. To calculate the rotor flux,the stator and rotor leakage inductance and the main inductance are necessary. If some other
equivalent circuits are implemented, it can be obtained a Γ model like that one illustrated by
Slemon (1989), where the stator and rotor leakage inductance are viewed as a total measurable
inductance.
3.6.1. Linear Γ model of the symmetrical single-phase induction machine
The classical T-form circuit model can be transformed into simpler models with no loss of
information or accuracy. Since rotor variables can be seen from the stator reference frame asreferred variables depending on the transformation turn ratio, we can choose a value such that the
magnetisation inductance is equal to the total stator inductance. This would give the following setof rotor variables related to those from the T-form equivalent circuit:
dR
qR
dr qr dr qr
dR
qR
'
' 1' ' ' '
'
' 1
i ii
i
γ
λ γ
λλ λ
γ
γ
= ⋅
(395)
where m s m/( )x x xγ = + . The above relation combined with the voltage equations gives the
equivalent d-q modified model for the symmetrical single-phase induction machine presented in
Fig. 3.5. This configuration has been denoted as the Γ form model due to its inductance structure.
The parameters of this equivalent circuit are related to those of the T-form d-q mathematical modelthrough the relations:
M m s m
2 2L r m s m s r
2R r
( ) ( )
' '
x x x x
x x x x x x x
r r
γ
γ γ γ
γ
= ⋅ = +
= + ⋅ − + = ⋅ + ⋅
= ⋅
(396)
rs x
L (ω
r/ω
b)λ’
dR
+ + -
iqs
xM
i’qR
r’R
uqs
_
rs x
L (ω
r/ω
b)λ’
qR
+ - +
ids
xM
i’dR
r’R
uds
_
Fig. 3.5. Equivalent linear d-q Γ form circuit for the
symmetrical single-phase induction machine
74
These parameters can be derived directly from the usual no load and standstill measurements on themachine.
This Γ model represents an appropriate solution for the analysis of scalar control and vector
control (stator flux oriented) single-phase induction machine (Slemon - 1994). The following
equations describe the complete operation of the system:Stator and rotor voltage:
qs s qs qs
b
ds s ds ds
b
r
R qR qR dR
b b
r
R dR dR qR
b b
0 ' ' ' '
0 ' ' ' '
pu r i
pu r i
pr i
pr i
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + −
= + +
(397-400)
Stator and rotor flux linkage:
qs M qM M qs qR
ds M dM M ds dR
qR L qR M qM M qs L M qR
dR L dR M dM M ds L M dR
( ' )
( ' )
' ' ( ) '
' ' ( ) '
x i x i i
x i x i i
x i x i x i x x i
x i x i x i x x i
λ
λ
λ
λ
= = +
= = +
= + = + +
= + = + +
(401-404)
Electromagnetic torque:
( ) ( )M
e qs dR ds qR qs ds ds qs
b
' '2 2
xP PT i i i i i iλ λ
ω= ⋅ ⋅ − = ⋅ − (405)
In the steady-state (p = 0) the torque expression becomes:
( )( )
2 2
qs ds R b
e 22
R L
'2
2 '
r sPT
r sx
λ λ ω+ = ⋅ +
(406)
where the slip s is given by: b r b( ) /s ω ω ω= −
3.6.2. Linear inverse Γ model of the symmetrical single-phase induction machine
If the arbitrarily turns ratio of referring the rotor parameters to the stator is chosen such that themagnetisation inductance is equal to the total rotor inductance, a new set of variables is obtainable:
dR
qR
dr qr dr qr
dR
qR
'
'' '
'' 1' ' ' '
'' '
'' 1
'
i ii
i
γ
λ γ
λλ λ
γ
γ
= ⋅
(407)
where: m m r'' /( )x x xγ = + . Using the voltage equations given for T-form equivalent circuit, it will
result a new configuration denoted as inverse Γ form model shown in Fig.3.6 in which:
75
2
s m r m m
L s r
r m
2
M m r m
2
R r
( )( )' '
( )
' ' ' ( )
' ' ' '
x x x x xx x x
x x
x x x x
r r
γ
γ γ
γ
+ + −= + ⋅ =
+
= ⋅ = ⋅ +
= ⋅
(408)
The inverse Γ form mathematical model of the single-phase induction machine is particularly
appropriate to analyse the vector-controlled machine in rotor field oriented systems.
rs x’L (ωr/ωb)λ’’dR
+ + -
iqs x’M i’’qR r’’R
uqs
_
rs x’L (ωr/ωb)λ’’qR
+ - +
ids x’M i’’dR r’’R
uds
_
Fig. 3.6. Equivalent linear d-q inverse Γ form circuit for the symmetrical
single-phase induction machine
The following equations can be written to describe the operation of the machine:Stator and rotor voltage:
qs s qs qs
b
ds s ds ds
b
r
R qR qR dR
b b
r
R dR dR qR
b b
0 ' ' ' ' '' ''
0 '' '' '' ''
pu r i
pu r i
pr i
pr i
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + −
= + +
(409-412)
Stator and rotor flux linkage:
qs L qs M qs qR M L qs M qR
ds L ds M ds dR M L ds M dR
qR M qM M qs qR
dR M dM M ds dR
' ' ( '' ) ( ' ' ) ' '
' ' ( ' ' ) ( ' ' ) ' '
'' ' ' ' ( ' ' )
' ' ' ' ' ( ' ' )
x i x i i x x i x i
x i x i i x x i x i
x i x i i
x i x i i
λ
λ
λ
λ
= + + = + +
= + + = + +
= = +
= = +
(413-416)
Electromagnetic torque:
( ) ( ) ( )M
e qs dR ds qR qs dR ds qR dR qR qR dR
b
'' ' ' ' '' '' '' ' ' '' ''
2 2 2
xP P PT i i i i i i i iλ λ λ λ
ω
= ⋅ − = ⋅ − = ⋅ −
(417)
In steady-state operation the torque becomes:
76
( )2 2
dR qR b
e
R
' ' ''2
2 ''
sPT
r
λ λ ω+ ⋅ = ⋅ ⋅ (418)
where s denotes the slip for the analysed machine.
3.6.3. Universal model of the symmetrical single-phase induction machine
The modern theory of the vector controlled induction machine demonstrates that severalreference flux vectors can be chosen for an independent control of the torque and flux in the
machine (DeDoncker et al - 1995). A unified theory simplifies and unifies the calculations of themachine parameters and makes a change of the reference vector flux more flexible. The idea is tolink the stationary reference frame (the only valid option for single-phase induction machine) to an
arbitrary flux vector γλ which can be derived from the flux linkage equations given at the previous
analysed models (T-form, Γ-form, inverse Γ-form) by means of a turn ratio transformation.
An arbitrary turn ratio γ is used to multiply the real rotor vector flux r dr qrjλ λ λ= + ⋅ and
defining the flux linkage equations as:
( )( )
ds s m ds m dr m ds m ds
qs s m qs m qr m qs m qs
x x i x i x i x i
x x i x i x i x i
λ γ γ
λ γ γ
= + ⋅ + − ⋅ + ⋅
= + ⋅ + − ⋅ + ⋅ (419-420)
( )( )
dR dr m ds r m dr m dr m dr
qR qr m qs r m qr m qr m qr
' '
' '
x i x x i x i x i
x i x x i x i x i
λ γ λ γ γ
λ γ λ γ γ
= ⋅ = ⋅ + ⋅ + ⋅ − +
= ⋅ = ⋅ + ⋅ + ⋅ − + (421-423)
The transformed rotor current d(q)R'i equals:
qrdrdR qR
''' ; '
iii i
γ γ= = (423)
The flux linkage equations reduce to:
( )( ) ( ) ( )( )( )( ) ( ) ( )( )
( )( ) ( ) ( )( )( )( ) ( )
ds s m ds m ds dR s m ds d
qs s m qs m qs qR s m qs q
dR r m dR m ds dR d r m dR
qR r m qR m qs qR
1 ' 1 '
1 ' 1 '
' 1 ' ' ' 1 '
' 1 ' '
x x i x i i x x i
x x i x i i x x i
x x i x i i x x i
x x i x i i
γ
γ
γ
λ γ γ γ λ
λ γ γ γ λ
λ γ γ γ γ λ γ γ γ
λ γ γ γ γ λ
= + − ⋅ ⋅ + ⋅ ⋅ + = + − ⋅ ⋅ +
= + − ⋅ ⋅ + ⋅ ⋅ + = + − ⋅ ⋅ +
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + = + ⋅ ⋅ + − ⋅ ⋅
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + = ( )( )q r m qR' 1 'x x iγ γ γ γ+ ⋅ ⋅ + − ⋅ ⋅
(424-427)This flux linkage can be represented by the equivalent circuit given in Fig. 3.7.
77
xs
+ (1- γ )xm
ids
γ
γ
(1−γ)
γ(γ +(γ−1)
γ(γ +(γ−1)x +s
xm
x xr m
)
x x )r m
xm
xm
iqs
iqs
+ i'qR
i'qR
i'dR
ids
+ i'dR
Fig. 3.7. Flux linkage equivalent circuit with arbitrary turn ratio
The rotor quantities are now calculated with the equations:
r
R dR dR qR
b b
r
R qR qR dR
b b
2
R r
0 ' ' ' '
0 ' ' ' '
'
pr i
pr i
r r
ωλ λ
ω ω
ωλ λ
ω ω
γ
= + +
= + −
= ⋅
(428-430)
Through variation of the arbitrary turn ratio “γ” the appropriate reference vector flux for controlling
the machine can be selected. Table 3.I summarise the choices that have to be made:
TABLE 3.I . Specific choices of the turn ratio “γ”
Turn ratio Mathematical model λd(q)γ Flux selected
m
m r
x
x xγ =
+Inverse Γ-form λd(q)r Rotor flux
1=γ T-form λd(q)m Air-gap flux
m s
m
x x
xγ
+=
Γ-form λd(q)s Stator flux
The universal mathematical model given in Fig. 3.7 permits a choice between different flux vectors
by selecting only one parameter “γ” , denoted the arbitrary turn ratio. All other basic machine
parameters remain unaffected.
3.6.4. Linear Γ model of the unsymmetrical single-phase induction machine
Usually the single-phase induction machine is equipped with non-identical sinusoidallydistributed windings arranged in space quadrature. From the T-form equivalent circuit developed
78
for this type of machine in previous paragraphs, it can be developed a linear Γ-form equivalent
circuit as shown in Fig. 3.8.Since rotor variables can be seen from the stator reference frame as referred variables depending
on the transformation turn ratio, we can choose a value such that the magnetisation inductance is
equal to the total stator inductance on each of the two-axis co-ordinates. This would give thefollowing set of rotor variables related to those from the T-form equivalent circuit:
d
dR q
qR
dr qr dr qr
ddR
qR
q
'
' 1' ' ' '
'
' 1
i ii
i
γ
λ γ
λλ λ
γ
γ
= ⋅
(431)
where
q m lm m
2 2
d m la m
/( )
/( )
x x x
k x x k x
γ
γ
= +
= + (432-433)
For an identical distribution of the windings, we have with a good approximation 2
la lmx k x= , and
this gives equal turn ratio for each axis ( d qγ γ= ).
The above relations combined with the voltage equations give the equivalent d-q modified model
for the unsymmetrical single-phase induction machine. The parameters of the Γ-form model are
related to those of the T-form d-q mathematical model through the relations:
Mq q m lm m
2 2
Lq r m q lm m lm q r q
2
R q r
2 2 2Md d m sa m Mq
2 22 2 2 2Ld r m d la m la d r q Lq
( ) ( )
' '
( ) ( )
x x x x
x x x x x x x
r r
x k x x k x k x
x k x x x k x x k x k x
γ
γ γ γ
γ
γ
γ γ γ
= ⋅ = +
= + ⋅ − + = ⋅ + ⋅
= ⋅
= ⋅ = + ≅
= + ⋅ − + = ⋅ + ⋅ ≅
(434-438)
r sm xLq (ωr/kωb)λ’dR
+ + -
iq s xM q i’qR r’R
uqs
_
r sa xLd (kωr/ωb)λ’qR
+ - +
ids xM d i’dR k2 r’R
uds
_
Fig. 3.8. Equivalent linear d-q Γ form circuit for the
unsymmetrical single-phase induction machine
It results an equations system as follows:Stator and rotor voltage:
79
qs sm qs qs
b
ds sa ds ds
b
rR qR qR dR
b b
2 r
R dR dR qR
b b
10 ' ' ' '
0 ' ' ' '
pu r i
pu r i
pr i
k
pk r i k
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + − ⋅
= + + ⋅
(439-442)
Flux linkage:
( )( ) ( )
( )( ) ( )
qs Mq qs qR
2
ds Md ds dR Mq ds dR
qR Mq qs Mq Lq qR
2
dR Md ds Md Ld dR Mq ds Mq Lq dR
'
' '
' '
' ' '
x i i
x i i k x i i
x i x x i
x i x x i k x i x x i
λ
λ
λ
λ
= ⋅ +
= ⋅ + = ⋅ +
= ⋅ + + ⋅
= ⋅ + + ⋅ = ⋅ ⋅ + + ⋅
(443-447)
Electromagnetic torque:
( )Mq
e qs dR ds qR qs ds ds qs
b
1' '
2 2
xP PT k i i i i i k i
kλ λ
ω
= ⋅ ⋅ − = ⋅ ⋅ − ⋅
(448)
This Γ-form model represents an appropriate solution for the analysis of scalar control and vector
control (stator flux oriented) of the unsymmetrical single-phase induction machine drives.
3.6.5. Linear inverse Γ model of the unsymmetrical single-phase induction machine
Another possible approach on analysing the unsymmetrical single-phase induction machine is
the one when an arbitrary turn ratio is used such that the magnetisation inductance in each axis isequal to the correspondent total rotor inductance. The following set of variables is readilyobtainable:
d
dR q
qR
dr qr dr qr
ddR
qR
q
'
'' '
'' 1' ' ' '
'''
'' 1
'
i ii
i
γ
λ γ
λλ λ
γ
γ
= ⋅
(449)
where:2
m m
q d q2
r m r m
' ' '( )
x k x
x x k x xγ γ γ= = =
+ + (450)
The linear inverse Γ-form model of the unsymmetrical single-phase induction machine is
presented in Fig. 3.9. This configuration is deductible from the T-form model using the aboveexpressions and new equivalent machine parameters defined as follows:
80
( )
Mq q m
2 2
Md d m Mq
Lq lm q r
2 2
Ld la d r Lq
2'
R q(d) r
' '
' ' '
' '
' ' '
' '
x x
x k x k x
x x x
x x k x k x
r r
γ
γ
γ
γ
γ
= ⋅
= ⋅ =
= + ⋅
= + ⋅ ≅
= ⋅
(451-455)
rsm
x’Lq
(ωr/kω
b)λ’’
dR
+ + -
iqs
x’Mq
i’’qR
r’’R
uqs
---
rsa
x’Ld
(kωr/ω
b)λ’’
qR
+ - +
ids
x’Md
i’’dR
k2 r’’R
uds
---
Fig.3.9. Equivalent linear d-q inverse Γ form circuit for the
unsymmetrical single-phase induction machine
The complete equations system is detailed below:Stator and rotor voltage:
qs sm qs qs
b
ds sa ds ds
b
r
R qR qR dR
b b
2 rR dR dR qR
b b
10 ' ' ' '
0 ' ' ' '
pu r i
pu r i
pr i
k
pk r i k
λω
λω
ωλ λ
ω ω
ωλ λ
ω ω
= +
= +
= + − ⋅
= + + ⋅
(456-459)
Flux linkage:
( )
qs Lq qs Mq qs qR Mq Lq qs Mq qR
ds Ld ds Md ds dR Md Ld ds Md dR
2
Mq Lq ds Mq dR
qR Mq qM Mq qs qR
2
dR Md dM Md ds dR Mq ds dR
' ' ( ' ) ( ' ' ) ' '
' ' ( ' ) ( ' ' ) ' '
( ' ' ) ' '
' ' ' ' ( ' )
' ' ' ' ( ' ) ' ( '
x i x i i x x i x i
x i x i i x x i x i
k x x i x i
x i x i i
x i x i i k x i i
λ
λ
λ
λ
= + + = + +
= + + = + + =
= ⋅ + +
= = +
= = + = + )
(460-463)
Electromagnetic torque:
81
( )Mq
e qs dR ds qR qs dR ds qR
b
dR qR qR dR
' 1' ' ' '
2 2
1' ' ' '
2
xP PT k i i i i i k i
k
Pk i i
k
λ λω
λ λ
= ⋅ ⋅ ⋅ − = ⋅ ⋅ − ⋅ =
= ⋅ ⋅ − ⋅
(464)
The circuit model from Fig. 3.9, is particularly suitable for analysis and understanding the vector
control systems with single-phase induction machine when the rotor flux vector is chosen asreference.
3.6.6. Universal model of the unsymmetrical single-phase induction machine
All the developed mathematical models for the unsymmetrical induction machine can begenerically presented with a universal model that preserves the characteristics for each of theequivalent form circuits.
The basic idea is to calculate in the stationary reference frame (the only valid option for single-phase induction machine that maintains constant parameters) an arbitrary flux vector. The fluxvector is determined from the flux linkage equations given in various forms of equivalent circuits
(T-form, Γ-form, inverse Γ-form) by using an arbitrary turn ratio transformation.
This turn ratio, that characterises the two-axis reference frame, has two values: γq and γd. By
multiplying the real rotor vector flux r dr qrjλ λ λ= + ⋅ with the turn ratio we can define the flux
linkage equations as:
( )
( )qs lm m qs m qr q m qs q m qs
2 2 2
ds la m ds m dr d m ds d m ds
'
'
x x i x i x i x i
x k x i x i k x i k x i
λ γ γ
λ γ γ
= + ⋅ + − ⋅ + ⋅
= + ⋅ + − ⋅ + ⋅ (465-466)
( )( )
2 2 2 2
dR d dr d m ds d r m dr m dr m dr
qR q qr q m qs q r m qr m qr m qr
' ' ' ' '
' ' ' ' '
k x i k x x i k x i k x i
x i x x i x i x i
λ γ λ γ γ
λ γ λ γ γ
= ⋅ = ⋅ + ⋅ + ⋅ − +
= ⋅ = ⋅ + ⋅ + ⋅ − + (467-468)
The transformed rotor current Rqdi )(' equals:
qrdr
dR qR
d q
''' ; '
iii i
γ γ= = (469)
The flux linkage equations reduce to:
( )( ) ( )
( )( )( )( ) ( )( )( )
( )( ) ( )( )( )
d
q
d
2 2
ds la d m ds d m ds dR
2
la d m ds d
qs lm q m qs q m qs qR
lm q m qs q
2 2
dR d d r d m dR d m ds dR
2
d d d r d m dR
qR
1 '
1 '
1 '
1 '
' 1 ' '
' 1 '
'
x k x i k x i i
x k x i
x x i x i i
x x i
k x x i k x i i
k x x i
γ
γ
γ
λ γ γ
γ λ
λ γ γ
γ λ
λ γ γ γ γ
λ γ γ γ
λ
= + − ⋅ ⋅ + ⋅ ⋅ + =
= + − ⋅ ⋅ +
= + − ⋅ ⋅ + ⋅ ⋅ + =
= + − ⋅ ⋅ +
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + =
= + ⋅ ⋅ + − ⋅ ⋅
( )( ) ( )( )( )q
q q r q m qR q m qs qR
q q q r q m qR
1 ' '
' 1 '
x x i x i i
x x iγ
γ γ γ γ
λ γ γ γ
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + =
= + ⋅ ⋅ + − ⋅ ⋅
(470-473)
This flux linkage can be represented by the equivalent circuit given in Fig. 3.10. The rotorquantities are now calculated with the equations:
82
rR dR dR qR
b b
2 r
R qR qR dR
b b
2
R q(d) r
0 ' ' ' '
10 ' ' ' '
'
pr i k
pk r i
k
r r
ωλ λ
ω ω
ωλ λ
ω ω
γ
= + + ⋅
= + − ⋅
= ⋅
(474-476)
Through variation of the arbitrary turn ratios “γq” and “γd” there are several options to select the
appropriate reference vector flux for controlling the single-phase induction machine. Table 3.II
summarises the available choices:
xsm
+ (1- γ )xm
ids
γ
γ
(1−γ )
γ (γ +(γ−1)
γ (γ + (γ − 1) x +sa
xm
x xr m
)
x x )r m
xm
xm
iqs
iqs
+ i'qR
i'qR
i'dR
ids
+ i'dR
q
d
q
dk
2
k2
q q q
d d dk
2
Fig. 3.10. Flux linkage equivalent circuit with arbitrary turn ratiofor unsymmetrical single-phase induction machine
TABLE 3.II
Specific choices of the turn ratio “γq” and “γd”
Turn ratio Mathematical
modelλd(q)γ Flux selected
mq
m r
2
m
d q2
m r( )
x
x x
k x
k x x
γ
γ γ
=+
= =+
Inverse Γ-form λd(q)r Rotor flux
q d 1γ γ= = T-form λd(q)m Air-gap flux
m lmq
m
2
m la
d q2
m
x x
x
k x x
k x
γ
γ γ
+=
+= ≅
Γ-form λd(q)s Stator flux
83
The universal mathematical model given in Fig. 3.10 permits a choice between different flux
vectors by selecting only two parameters “γq” and “γd” that have practically equal values, and
denote the arbitrary turn ratio for each axis of the system. As in the symmetrical single-phaseinduction machine case, all other basic machine parameters remain unaffected.
3.7. D-Q axes models of the single-phase induction machine
The single-phase induction machine can be modelled by using different state-space variables,keeping as inputs the stator voltages and the load torque, and as outputs the electromagnetic torque
and rotor angular velocity. Let us define the possible set of currents and flux linkages per secondspace vectors as follows:
s ds qs
r dr qr
m dm qm
s ds qs
r dr qr
m dm qm
' '
' ' '
i i j i
i i j i
i i j i
j
j
j
λ λ λ
λ λ λ
λ λ λ
= + ⋅
= + ⋅
= + ⋅
= + ⋅
= + ⋅
= + ⋅
(477-482)
[ ] [ ]T
s r m s r m' 'x i i i λ λ λ= (483)
Two-space vectors have to be selected from the six available vectors. Let the selected pair of
state-space variables be denoted as 21 , xx . It is possible to express the set of six state-space
variables in terms of the two selected state-space variables as follows:
[ ] [ ] [ ]
11 12
21 22
T T31 32
s r m s r m 1 2
41 42
51 52
61 62
' '
a a
a a
a ax i i i x x
a a
a a
a a
λ λ λ
= = ⋅
(484)
where the coefficients aij, depend on the chosen set of state-space variables.Out of fifteen possible state-space versions obtained from combining two space vectors from the
total of six, only fourteen represent valid mathematical models for the single-phase inductionmachine. Please note that the pair of state-space variables that comprise the airgap flux linkagespace vector and the magnetising current space vector cannot be selected as these vectors have the
same direction. These models can be classified in three types: currents space vectors, flux linkagesspace vectors and mixed currents-flux linkages space vectors models.
In conjunction with the mechanical equation:
( )r e L2
Pp T T
Jω = ⋅ − (485)
we obtain a complete version of the single-phase induction machine model, viewed as the key for a
motion control system.For an unitary approach on all the possible models, and considering that the magnetising
reactance has generally much greater value than the leakage reactance, the following approximation
is made without altering the final results:2 2 2
la m lm m s( )x k x k x x k x+ ≅ + = (486)
84
The starting point for the state-variable models is given by voltage equations system written instationary reference frame:
p= ⋅ + ⋅u L x R x (487)
where: x is the selected set of state-variables and represent also the output of the model, u is the
input vector (stator voltages), L is the coefficients matrix (it can be formed by reactance values, ornon-dimensional elements) for multiplying the time derivative of the state-variables, R is the
coefficients matrix (it can be formed by resistances and reactances values or non-dimensionalelements) for multiplying the state-variables and p stand for the differential operator (d/dt). It resultsthe general form of the state-variables system:
p = ⋅ + ⋅x A x B u (488)
where:1
1
−
−
= − ⋅
=
A L R
B L (489-490)
All the following mathematical models can theoretically be discretised and used for
implementation of controllers in the drive systems with single-phase induction machines. However,practically only few of them are suitable for this purpose as the computational effort, and implicitly
the cost of the hardware, is a decisive factor to be considered.When the vector control strategy has to be chosen, each of the six space vectors that can be
selected as state-space variables represents a possible basis for a reference frame. The algorithm for
implementing field oriented control systems is obtained as follows:1. The rotor flux space vector is expressed as function of state-variables;
2. The rotor angular velocity ω r term is substituted with (ω r - ω k) where ω k is the angular
velocity of the synchronous frame;
3. The dynamic equations that relates the rotor angular velocity to rotor flux linkages and voltagesare deduced in the new state-variable system;
4. The synchronous reference frame is selected linked to one of the space vectors, which means
that the q-axis component of the reference space vector is null;5. The torque equation is computed according to the selected flux or current space vector in the
synchronous reference frame.
3.7.1. Models with currents space vectors as state-space variables
I) The most used model of the single-phase induction machine is the currents space vectors state-
space variables T
qs ds qr dr, , ' , 'i i i i = x . An important feature of this model is that the stator and
leakage reactance, arbitrarily considered equal in many implementations, are now included in the
stator and self-reactance values. As the stator currents are easily measurable quantities, a controllerbased on these state variables gives good performance accuracy. The model is readily available
from the equations of voltages and flux linkages per second, and it can be expressed in a matrixform as follows:
85
2
r m m r m r m r r
b b
2qs a rm r m r r m r
2ds b b
qrb s m s r s rm m r r
dr b b
s m m a s r s rr r
2
b b
' ' '
' ' '1
' ' '
'
' '1
x r x k x r x x k
D D D D
i r xx x x x r
i D k D D k Dp k
i x x x r x xx r k k
i D D D D
x x x r x x x r
D k D D k Dk
ω ω
ω ω
ω ω
ω ω
ω ω ω
ω ω
ω ω
ω ω
− − ⋅ − ⋅
⋅ − ⋅ ⋅ ⋅ = ⋅ − ⋅
− ⋅ ⋅ − ⋅ −
qs
ds
qr
dr
r m
qs
2 2
r m ds
qrsm
dr
2 2
m s
'
'
'0 0
1 10 0
'
'0 0
'
1 10 0
i
i
i
i
x x
D D
u
k x D k x D u
uxx
D D u
k x D k x D
⋅ +
−
− ⋅ −
−
(491)
where:2
s r m'D x x x= −The electromagnetic torque can be computed as:
( )m
e qs dr ds qr
b
' '2
xPT k i i i i
ω= ⋅ ⋅ ⋅ − (492)
As it can be observed, a shortcoming for this model is the full system matrix, with all 16 elements
non-zero values.
II) Another mathematical d-q axis model based on currents space vectors is the one whichcomprises the stator and the magnetising space vector currents as state-space variables
T
qs ds qm dm, , ,i i i i = x . The state matrix elements are all non-zero, which implies a computational
effort similar to the previously analysed model.
86
m r r m m lr r m r m r
1 1 b 1 1 b
2 2qs a r r mm lr r m r r r m
ds 2 b 2 2 b 2
qmb r lsr lm m lr lm lr r lm r r
dm 1 1 b 1 1 b
la lr r
2
' ' ' ' '
' '' ' '
'' ' ' '
'
rr x r x x x k x r x x k
A A A A
i r x k r xx x k x x k k r x
i A A A Ap
i r xr x r x x x k x x k
i A A A A
x x
A k
ω ω
ω ω
ω ω
ω ω
ω ω ω
ω ω
ω
ω
+− ⋅ − ⋅
+− ⋅ − ⋅
= −
− ⋅ − ⋅
⋅
qs
ds
qm
dm
r la a lr la r r lar
b 2 2 b 2
r m
1 1
q sr m
2 2 d s
ls q rlr
1 1 d r
lalr
2
2 2
' ' ' '
'0 0
'0 0
''0 0
'
'0 0
i
i
i
i
r x r x x x r x
A A k A
x x
A A
ux x
A A u
x ux
A A u
xx
A k A
ω
ω
⋅ +
− − ⋅ −
−
− + ⋅
(493)where:
1 lm r m lr
2 2
2 m lr la r
' '
' '
A x x x x D
A k x x x x k D
= + =
= + ≅The instantaneous electromagnetic torque may be expressed in terms of the state-space variables:
( )m
e qs dm ds qm
b2
xPT k i i i i
ω= ⋅ ⋅ ⋅ − (494)
III) The third possible combination of current space vectors as state-space variables is obtained
by selecting magnetising and rotor currents T
qr dr qm dm' , ' , ,i i i i = x . When compared to the other
current space vector models, the similar computing burden for obtaining the outputs of the system is
obvious. The main difference between them is the presence of the global parameters (self-reactances) in the first model, while in the other two models an accurate determination of the
leakage reactance is necessary. A complete description of the state-variables in matrix notation isgiven below:
87
m m r s s lr s mr m m r
1 1 b 1 1 b
2q r s lr a m r s s m a mr r
2 b 2 2 b 2d r
q m m lr r lm lm lr r m lr lm m rb
1 1 b 1 1 bd m
la lr a lrr
2 b
' '
' ' '
'
' ' ' '
' '
r x r x x x x xk r x k
A A A A
i x x r x k r x x x r xk k
A A A Aip
i r x r x x x k r x x x k
A A A Ai
x x r x
A k
ω ω
ω ω
ω ω
ω ω
ω ωω
ω ω
ω
ω
+− ⋅ ⋅
+ − ⋅ − − ⋅
= −
⋅ − ⋅
− ⋅
q r
d r
qm
dm
2
r la la m a lrr
2 2 b 2
sm
1 1
qssm
2 2 ds
q rlr lm
1 1 d r
lalr
22 2
'
'
' '
0 0
0 0
''0 0
'
'0 0
i
i
i
i
k r x x x r x
A A k A
xx
A A
uxx
A A u
ux x
A A u
xx
A k A
ω
ω
⋅ +
− − ⋅ −
−
− + ⋅
(495)where:
DkxxkxxA
DxxxxA
slrmla
slrmlm
22
2
1
'
'
≅+=
=+=
The instantaneous electromagnetic torque is expressed as follows:
( )m
e dr qm qr dm
b
' '2
xPT k i i i i
ω= ⋅ ⋅ ⋅ − (496)
3.7.2. Models with flux linkages as state-space variables
I) In this case the flux linkages space vectors, T
qs ds qr dr, , ' , 'λ λ λ λ = x are used for describing the
mathematical model of the machine. The matrix equations for this system are as follows:
m r m m
qs qs qsa r a m
2 2
ds ds ds
r sr m rqr qr qrb
bdr dr dr
r sr m r
b
'0 0
'0 0
''' ' '0
' ' '
''0
r x r x
D D
ur x r x
k D k D upr xr x u
D D k u
r xr x k
D D
λ λ
λ λ
ωλ λωωλ λ
ω
ω
−
− ⋅ = ⋅ +−
− −
(497)
where D has the same significance as in the previous cases.
The electromagnetic torque is determined with the equivalent relation:
( )m
e qs dr ds qr
b
' '2
xPT
k Dλ λ λ λ
ω= ⋅ ⋅ −
⋅ (498)
As opposed to the current models, it can be observed that the computational burden is substantially
lower. Due to this important feature, this model is the most suitable for discretisation in motioncontrol strategies.
88
II) An alternative to model the single-phase induction machine with flux space vectors as state-variable system, is the selection of air-gap flux space vector among the set of independent variables.
A first approach is given by the stator and air-gap flux space vectors selected as state-space
variablesT
qs ds qm dm, , ,λ λ λ λ = x .
m m
lm lm
a aqs
la lads
2
r s lr sm lr m r m lm m lr r m m lr lm m rqmb
lm 1 1 la 1 b lm 1 1 1 b la mdm
2 2
la m lr a lr mr r m
lm 2 b la 2
0 0
0 0
' '' ' ' '1
' ' '
r r
x x
r r
x xp
r x k x xr x x r x x x x k r x x x x
x A A x A x A A A k x x
x x x r k x xk k r x
x A x A A
λ
λ
ω ωλω
ω ωλ
ω
ω
−
− =
− + − ⋅ − ⋅ ⋅ +
⋅ − +
qs
ds
qm
dm
2 2
la m lr s a lr m r sr
2 2 b lm m la 2 2
qs
lr m lm m ds
1 1 qr
2drla mlr m
2 2
' ' '1
1 0 0 0
0 1 0 0
'0 0
'
''0 0
x x x x r k x x k r xk
A x x x A A
u
x x x x u
A A u
ux xk x x
A A
λ
λ
λ
λ
ω
ω
⋅ +
− ⋅ ⋅ + −
+ ⋅
(499)where:
1 lm m lr s
2 2
2 la m lr s
'
'
A x x x x D
A x x k x x k D
= + =
= + ≅The electromagnetic torque expression shows that this model is prohibitive for implementation for
modelling the single-phase induction machine in a vector control system and becomes:
e qs ds qs dm dm qm
b lm la la lm
1 1 1
2
P k kT
kx x x kxλ λ λ λ λ λ
ω
= ⋅ ⋅ ⋅ − + −
(500)
which can be simplified only if the approximation 2
la lmx k x= is reasonable to be made:
e qs dm dm qm
lm b
1
2
PT
kxλ λ λ λ
ω = ⋅ ⋅ − (501)
III) The third option of selecting flux space vectors as state-space variables is the one which
comprises the air-gap (magnetising) flux and rotor flux space vectors T
qm dm qr dr, , ' , 'λ λ λ λ = x . By
comparison to the previously analysed model, this one presumes the same computational effort, butthe electromagnetic torque is determined in a suitable form for vector control, without any
supplementary approximations.
89
m r r m lm m m r m lm m lm r
1 lr 1 1 lr 1 1 b
qm a r r m la m la a m r m lar
2 lr 2 2 b 2 lr 2dm
qr r r rb
lr lr bdr
r r r
lr b lr
' ' '0
' '
' ' '0
' '
' ' '0
' ''
' '0
' '
r x r x x r x r x x x x
A x A A x A A k
r x r x x x x r x r x xk
A x A A A x Ap
r r
x x k
r k r
x x
ω
ω
λ ω
ωλ
λ ωω
ωλ
ω
ω
− + − ⋅
− + − ⋅ − =
−
− −
qm
dm
qr
dr
m lr m lr
1 1 qs
dsm lam lr
2 2 qr
dr
'
'
' '0 0
'0 0
'
0 0 1 0 '
0 0 0 1
x x x x
A A u
ux xx x
A A u
u
λ
λ
λ
λ
⋅ +
+ ⋅
(502)where:
1 m lr lm r
2 2
2 m lr lm r
' '
' '
A x x x x D
A k x x x x k D
= + =
= + ≅The resulting relation for computing instantaneous electromagnetic torque is as follows:
( )e qm dr dm qr
lr b
1' '
2 '
PT
kxλ λ λ λ
ω= ⋅ ⋅ − (503)
3.7.3. Models with mixed currents – flux space vectors state-space variables
I) If the stator variables are chosen for modelling the single-phase induction machine system,
then a mixed flux linkages-currents state-space variables model T
qs ds qs ds, , ,i iλ λ = x is developed.
The matrix equations and the electromagnetic torque relation are:
m
aqs qs
m r r sr r r rds dsr
b bqs qsb
2rds dsm r r sr r r r
22 2
b b
0 0 0 1 0 0 0
0 0 0 0 1 0 0
' '' ' '0 0 0
'' '' '0 0 0
r
r
r x r xr x kp xD D k Di i D
xi ir x k r xx r
k DD k kk D k D
λ λ
ω ωλ λ
ω ωω
ω ω
ω ω
− − + − ⋅ −= ⋅ + + ⋅ − −
qs
ds
qr
dr
'
'
u
u
u
u
⋅
(504)
e ds qs qs ds
b
1 1
2
PT i k i
kλ λ
ω
= ⋅ −
(505)
II) Another important mixed flux linkages-currents state-space variables modelT
qr dr qr dr' , ' , ' , 'i iλ λ = x is that expressed in rotor quantities. This model can be easily used for the
rotor flux oriented control of the machine. We can describe the system as follows:
90
r
r
b
2rqr qr
r
bdr drm
s m r r sm rqr qrb
b mdr dr
22
s a a r r sr
4 2
b
0 0
0 0 0 0
' ' 0 0 0 00 0 '' '
0 0 0' '' '
0' '
0 0 0' '
0
rk
kk r
p x
x r x r xri i D
D D D xi i
k Dx r r x k r x
D k k D k D
ω
ω
ωλ λ
ωλ λ
ωω
ω
ω
ω
−
− − = ⋅ + − +
⋅ − − +
− ⋅ −
qs
ds
qr
dr
'
'
u
u
u
u
⋅
(506)The electromagnetic torque relation is given below:
e qr dr dr qr
b
1 1' ' ' '
2
PT k i i
kλ λ
ω
= ⋅ −
(507)
III) A similar model to the previous one is that with stator current and rotor flux linkageT
qr dr qs ds' , ' , ,i iλ λ = x as state-space variables. It represents an alternative model for rotor flux
oriented control strategies. The matrix equation and electromagnetic torque relation are presented
below:
2 2m r r m r m m r
r r b
2 2 2qs qsa r r m m r r m
2 2ds dsbr r
qr qrb r m r r
dr drr r b
2r m r r
r b r
' ' '0
' '
' ' '0
' '
' '' '0
' '' '
' '0
' '
r x r x r x x
x D x D D k
i ir x k r x x k r x
i iDk x D k x Dp
r x r
x x k
k r x k r
x x
ω
ω
ω
ω
λ λω ω
λ λω
ω
ω
+− − ⋅
+
− ⋅ = ⋅ −
− −
r m
qs
r m ds
2 2
qr
dr
'0 0
'0 0
'
0 0 1 0 '
0 0 0 1
x x
uD D
x x u
k D k D u
u
+
− − ⋅
(508)
and torque expression is:
m
e qs dr ds qr
r b
1' '
2 '
xPT i k i
x kλ λ
ω
= ⋅ ⋅ ⋅ − ⋅
(509)
IV) A rarely used mathematical model is that with mixed rotor currents space vector and stator
flux linkages space vector as state-space variables T
qs ds qr dr, , ' , 'i iλ λ = x . However, the advantage
of this model is that merely only stator winding parameters are necessary, so the influence of rotor
parameters is minimised. It can be used for an unconventional stator flux oriented control with rotorcurrent components producing the torque and the flux. The expressions for implementing this
model are:
91
2 2
m m r s r m m m r
s b s bsm
2 2 2qr qra m r s a mr m
2 4dr drb s s
qs qsb m m m
ds dss s
a m a
2
s s
'
0 0' ''
' '
0 0
0 0
r x r x k r x x
x D x D D k xx
i ir x k r x r xx D D
i ik Dp k x D k Dx
r x r
x x
r x r
x k x
ω ω
ω ω
ω
ω
λ λω
λ λ
+− ⋅
+
− − − = ⋅ + −
−
qs
s dsm
2 2
qr
dr
0 0'
1 0 0 0 '
0 1 0 0
u
x ux
k D k D u
u
⋅
(510)where the significance of D was already stated.
The main system output, the electromagnetic torque, can be computed as:
m
e qs dr ds qr
s b
1' '
2
xPT k i i
x kλ λ
ω
= ⋅ ⋅ ⋅ − ⋅
(511)
V) The mixed stator current space vector and air-gap flux space vector as state-space variablesT
qm dm qs ds, , ,i iλ λ = x belongs to one of the more complex model types. It preserves information
regarding both stator and rotor parameters. Different from the previous mixed models, the state-
space matrix contains only non-zero elements, which leads to greater computational effort.
m r r m m lr r r r r
1 1 b 1 1 b
2qs a r r mm lr r r r r
2 b 2 2 b 2ds
qm m r m m lr lm m lr r lm r lm r rb
1 1 b 1 1 bdm
la m lr mr
2 b
' ' ' ' '
' '' ' '
( ' ' ) ' ' '
'
r x r x x x k r x
A A A A k
i r x k r xx x k x k r
A A A Aip
x r x r x x x x k x r x x
A A A A k
x x x xk
A
ω ω
ω ω
ω ω
ω ω
λ ω ωω
ω ωλ
ω
ω
+− ⋅ − ⋅
+ ⋅ − ⋅
= −
− ⋅ − ⋅
− ⋅
qs
ds
qm
dm
2lr a la r la r la rr
2 2 b 2
r m
1 1
r m qs
2 2 ds
lr m lm m qr
1 1 dr
2
la mlr m
2 2
( ' ' ) ' '
'0 0
'0 0
' '0 0
'
'0 0
i
i
k x r x r x x x rk
A A A
x x
A A
x x u
A A u
x x x x u
A A u
x xk x x
A A
λ
λ
ω
ω
⋅ +
− + − ⋅ −
−
+ ⋅
(512)where:
1 lm r lr m
2 2
2 lr m la r
' '
' '
A x x x x D
A k x x x x k D
= + =
= + ≅The electromagnetic torque is computed as:
92
e qs dm ds qm
b
1 1
2
PT i k i
kλ λ
ω
= ⋅ ⋅ ⋅ − ⋅
(513)
VI) A similar model to the precedent one, is the mixed rotor current space vector and air-gap flux
space vector as state-space variables T
qm dm qr dr, , ' , 'i iλ λ = x . The matrix equation of the system is
given below:
m m r s s lr r m m r
1 1 b 1 1 b
2qr s lr a m r s s ar r
2
2 b 2 2 bdr 2
qm m r lm m lr lm m lr r lm m lm m rb
1 1 b 1 1 bdm
la m lr mr
2 b
' '
' ' '
'
( ' ' ) '
' (
r x r x x x k r x
A A A A k
i x x r x k r x x rk k
A A Ai k Ap
x r x r x x x x k x r x x
A A A A k
x x x xk
A
ω ω
ω ω
ω ω
ω ω
λ ω ωω
ω ωλ
ω
ω
+− ⋅ ⋅
+ − ⋅ − − ⋅ −
= − +
⋅ − ⋅
− ⋅
qs
ds
qm
dm
2 2
lr a la r la m la ar
2 2 b 2
sm
1 1
s qsm
2 2 ds
lr m lm m qr
1 1 dr
2
la mlr m
2 2
' ' )
0 0
0 0
' '0 0
'
'0 0
i
i
k x r k x r x x x rk
A A A
xx
A A
x ux
A A u
x x x x u
A A u
x xk x x
A A
λ
λ
ω
ω
⋅ +
− − ⋅
−
− + ⋅
(514)where:
1 lm m lr s
2 2
2 la m lr s
'A x x x x D
A x x k x x k D
= + =
= + ≅The electromagnetic torque relation becomes:
e dr qm qr dm
b
1 1' '
2
PT k i i
kλ λ
ω
= ⋅ ⋅ ⋅ − ⋅
(515)
VII) If the magnetising current space vector is selected as state-space variable together with one
of flux linkages space vectors T
qm dm qs ds, , ,i i λ λ = x , the state matrix computation gives several
null elements. Nevertheless, the torque expression is more complicated and the final relation can be
written in the same form as in previous cases only through some new assumptions,.
93
m lr m r s ls lm 2 r r lm m lr lm lr r
ls 1 la 1 b ls 1 la 1 b
2qm la 1 a m lr r s la la lr r la a lrr
lm 2 b la 2 lm 2 la 2dm
qs m m mb
lm lmds
' ' ' ' '
( ' ' ) ' ' '
0 0
0
r
b
r x x r x x x A k r x r x x x k
x A x A x A x A
i x A k r x x r x x x x r x r x
x A k x A x A k x Aip
r x r
x x
ω ω
ω ω
ω ω
ω ω
λω
λ
− −⋅ − ⋅
− − − ⋅ ⋅
=
−
qm
dm
qs
ds
2
a m a
la la
lr lm
1 1 qs
la dslr
2
2 2 qr
dr
0
'0 0
'0 0
'
1 0 0 0 '
0 1 0 0
i
i
k r x r
x x
x x
A A u
x ux
A k A u
u
λ
λ
⋅ +
−
+ ⋅
(516)where:
1 m lm s lr
2 2
2 m la s lr
'
'
A x x x x D
A x x k x x k D
= + =
= + ≅
The electromagnetic torque relation, with the approximation: 2
la lmx k x≅ can be written as:
3
e m dm qs qm ds qs ds
b la lm lm la
mdm qs qm ds
b lm
1 1 1
2
1
2
P k kT x i i
x kx kx x
xPk i i
x k
λ λ λ λω
λ λω
= ⋅ ⋅ ⋅ ⋅ − ⋅ + − ≅
≅ ⋅ ⋅ ⋅ − ⋅
(517)
The presence of oscillating terms given by the product between flux linkages, makes this modelprohibitive for implementing in vector control strategies.
VIII) The last mixed current-flux linkages space vector model realises the connection between
the magnetising current space-vector and the rotor flux linkage space vectorT
qm dm qr dr, , ' , 'i i λ λ = x , selected as state-space variables. The state matrix contains the same
number of zero elements (four) and the instantaneous electromagnetic torque is determined in asuitable form for vector control implementation.
94
r lm m m r lr m lr r lm lm r
lr 1 lr 1 1 b
r la m a r lr la a lr la rrqm
2
lr 2 2 b lr 2dm
r m r rqrb
lr lr bdr
2
r m r r
lr b lr
' ' ' ' '0
' '
' ' ' ' '0
' '
' ''0
' ''
' '0
' '
r x x r x x r x r x x
x A x A A k
r x x r x x x r x x ri
x A A k k x Aip
r x r
x x k
k r x rk
x x
ω
ω
ω
ω
ωλω
ωλ
ω
ω
− −⋅
− − − ⋅
=
−
− −
qm
dm
qr
dr
lr lm
1 1 qs
la dslr
2
2 2 qr
dr
'
'
'0 0
'0 0
'
0 0 1 0 '
0 0 0 1
i
i
x x
A A u
x ux
A k A u
u
λ
λ
⋅ +
+ ⋅
(518)where:
DkxxxxkA
DxxxxA
rlalrm
rlslrm
22
2
1
''
''
≅+=
=+=
The electromagnetic torque is determined as follows:
m
e qm dr dm qr
b lr
1' '
2 '
xPT i k i
x kλ λ
ω
= ⋅ ⋅ ⋅ − ⋅
(519)
3.8. Vector control strategies for single-phase induction machine
The aim of vector control is usually to decouple the stator current is into its flux producing andtorque producing components (ids, iqs respectively) in order to obtain a decoupled control of the fluxand the electromagnetic torque. For this reason a special reference frame is selected fixed to
different space vector variables. The reference frame has to be synchronous, as all the space vectorshave the same angular velocity given by the supply voltage frequency.
As the stator windings of the single-phase induction motor are usually unsymmetrical, the vectorcontrol principles have to be implemented in a special way. The machine parameters differ fromaxis d to axis q. The waveform of the electromagnetic torque demonstrates the unbalance of the
system. Even for equal amplitude, orthogonal stator currents iqS and idS, the torque contains ACterms. A new torque relation has to be developed for each case of vector control strategy of single-
phase induction machine (Correa et al. - 1998, 1999).Generally, the considerations made for the three-phase induction machine when the vector
control sheme has to be developed, are valid. An easy to follow steps algorithm for implementing
vector oriented control systems is obtained as follows:1. A complete mathematical model of the single-phase induction machine is developed in
stationary reference frame, the only reference frame that maintains constant parameters,according to the chosen set of state-space variables;
2. A suitable torque expression is determined, in order to eliminate the influence of AC terms;
3. The rotor based variables are completely expressed in the new state-variable system;4. The rotor angular velocity ωr term is substituted with ωs = (ωr - ωk) where ωk and ωs are the
angular velocity of the synchronous frame, respectively the slip angular velocity;
95
5. The synchronous reference frame is selected linked to one of the space vectors, which meansthat the q-axis component of the reference space vector is null;
6. The torque equation is computed according to the selected flux or current space vector in thesynchronous reference frame.
The transformation of reference frames for the induction motor vector control can be summarised asshown in Fig. 3.11.
Drive
ControllerInverse 2/2 Clark/ Park
Transformation
2/2 Clark/ Park
Transformation
Induction
motor
Stationary reference frameSynchronuous reference frame
Two-axis co-ordinate systemSingle-phase system
Fig. 3.11. Block diagram of transformation of frames and co-ordinate systemsfor single-phase induction motor vector control
3.8.1. Stator field orientation (SFO)
For this vector control strategy, the set of state-space variables formed by stator flux linkage and
current space vectors T
qs ds qs ds, , ,i iλ λ = x is selected. The mathematical model is given in the
chapter dedicated to d-q models of the single-phase induction machine.
The rotor based variables expressed in state-space variables terms are:
( )
( )
( )
s s s
qr qs s qs
m
s s
s ds s dsdr 2
mm
s s s
qr r qs qs
m
s s 2 s
dr r ds ds
m
1'
'
1' '
1' '
i x ix
x ii
xk x
x D ix
x k D ix
λ
λ
λ λ
λ λ
= −
= −
= − ⋅
= − ⋅
(520-523)
To obtain the machine equations in the synchronuous stator flux reference frame, one has toeliminate the rotor flux from the rotor voltage equations and then force the q component of thestator flux to be zero. The stator voltages equations remain unchanged. If a cage rotor is considered,
the resultant equations are as follows:
s s s ss
r s qs r ds ds r r qs
b b b
2 s s s ss s
r s ds qs r r ds r qs
b b b b
1' ' ' ' 0
' ' ' '
p pr x D i x k D i r x
k
p pk r x D i k D i r x k x
ωλ λ
ω ω ω
ω ωλ λ
ω ω ω ω
+ − − ⋅ ⋅ − + =
+ − ⋅ = + −
(524-525)
96
where the definition used for the slip speed is: s e rω ω ω= − . The electromagnetic torque expression
has to be re-written:
s s s s
e ds qs qs ds
b
1 1
2
PT i k i
kλ λ
ω
= ⋅ −
(526)
If we make the notation:s 2 s
qs qs1i k i= the above torque relation may be expressed as for the
symmetrical induction machine:
( )s s s s
e ds qs1 qs ds
b2
P kT i iλ λ
ω= ⋅ − (527)
If the special reference frame is fixed to the stator flux linkage vector, the q-component of this flux
vector is defined equal to zero:s
qs
s s
ds s
0λ
λ λ
=
= (528)
From stator flux linkage equations, the q-current components are given by:s s
qs1 qs1
2s ssqr qs1
m
'
i i
k xi i
x
=
= − (529)
The electromagnetic torque relation and slip speed can be derived in stator field orientation control
as:
( )
s s
e d s q s 1
b
2 s
b r s qs1
ss s
r ds ds
2
'
1'
P kT i
k r x D p i
x k D ik
λω
ωω
λ
= ⋅
+ ⋅=
− ⋅ ⋅
(530-531)
The second dynamic equation of the machine, shows that there is a coupling between the stator
current components. Consequently, any change in the torque producing component s
qs1i without
changing s
dsi accordingly will cause a transient in the stator flux. A decoupler is necessary to
overcome this disadvantage. Therefore the command current of the d-axis component of the statorcurrent can be calculated as follows:
s s si
ds p ds dq
s
s qs1s
dq
b r s'
Ki K i
p
k D ii
r x D p
∆λ
ω
ω
= + +
⋅=
+ ⋅
(532-533)
where Kp and Ki are proportional, respectively integral coefficients of the flux controller. Thiscontroller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomance of a stator flux oriented system
Letting the derivative operator p = 0, one can obtain the steady-state voltage equations of theinduction machine. After several manipulation of the system equations, the d current componentsare given by:
97
s
s ss ds
ds qs1 2
r s b s
s
s sds s
dr ds2
mm
'
'
kDi i
r x k x
xi i
xk x
ω λ
ω
λ
= ⋅ +
= −
(534-535)
it yields the slip speed equation:2 22 2
s s 2 ss s mqs1 s r s q s 1
b r s b s
' 0'
xk Di k r x i
r x kx
ω ωλ
ω ω
− ⋅ + =
(536)
The solutions of the above equations have to be real, for a given stator flux linkage. This means that
the determinant of the second order equation satisfy the condition:
( )2
2 s2
4 2 sm s
qs1
s
4 0x
k D ikx
λ∆
= − ≥
The maximum values for the q component of the stator current, the slip speed and the
electromagnetic torque (pull-out torque) are:
( )
( )
( )( )
2 ss m sqs1 2max
s
r ss max
22 s
m s
e 2maxb s
2
'
22
xi
k D x
r x
kD
xPT
D xk
λ
ω
λ
ω
=
=
= ⋅
(537-539)
The problems related to static stability of the single-phase induction machine when this vectorcontrol strategy is applied, are similar to the polyphase machine case. It has to pointed out that thecritical values for current q-component, slip angular velocity and electromagnetic torque depend
also on the turns ratio k of the unsymmetrical stator windings. If we detail this consideration, it canbe observed that having a motor equipped with a main stator winding with fixed parameters
(resistance, reactance) the pull out torque and critical rotor speed are such that (Popescu -2000):• A maximum torque value and minimum rotor speed are obtained if the turns ratio k < 1, which
is the case of the usual split-phase motors configuration;• The medium values are obtained if the turns ratio k = 1, which is the case of the symmetrical
motors configuration;
• A minimum torque value and maximum rotor speed are obtained if the turns ratio k > 1, which
is the case of the capacitor run motors configuration;
Some important conclusions can be drawn for this vector control strategy, besides those onesvalid for the polyphase motor:
• The estimated equivalent value for the torque component of the stator current has to be
corrected by dividing to the square of the turns ratio value;
• There is no need for changing the number of co-ordinates from the real machine system to the
control system.
3.8.2. Rotor flux orientation (RFO)
For this vector control strategy, the set of state-space variables formed by rotor flux linkage and
stator current space vectors T
qr dr qs ds' , ' , ,i iλ λ = x is selected. The mathematical model is given in
the chapter dedicated to d-q models of the single-phase induction machine.
The stator flux linkage and rotor current space vectors components expressed as functions interms of state-space variables are:
98
( )
( )
( )
r r r
qs m qr qs
r
r r 2 r
ds m dr ds
r
r r r
qr qr m qs
r
r
r rdrdr m ds2
r
1'
'
1'
'
1' '
'
'1'
'
x D ix
x k D ix
i x ix
i x ix k
λ λ
λ λ
λ
λ
= + ⋅
= + ⋅
= −
= −
(540-543)
The rotor voltage equations re-written in terms of the state-space variables become:
r r r rsr r
qr m qs qr dr
r r b b
r 2 r r rsr rdr m ds dr qr
r r b b
' ' 1' 0 ' '
' '
' '' 0 ' '
' '
r r pu x i
x x k
r r pu k x i k
x x
ωλ λ
ω ω
ωλ λ
ω ω
= = − + + + ⋅
= = − + + −
(544-545)
The electromagnetic torque expression has to be re-written:
r r r rm
e dr qs qr ds
r b
1' '
2 '
xPT i k i
x kλ λ
ω
= ⋅ −
(546)
If we make the notation:r 2 r
qs qs1i k i= the above torque relation may be expressed as for the
symmetrical induction machine:
( )r r r sm
e dr qs1 qr ds
r b
' '2 '
xP kT i i
xλ λ
ω= ⋅ − (547)
If the special reference frame is fixed to the rotor flux linkage vector, the q-component of this
flux vector is defined equal to zero:r
qr
r r
dr r
' 0
' '
λ
λ λ
=
= (548)
From rotor flux linkage equations, the q-current components are given by:r rqs1 qs1
2r rmqr qs1
r
''
i i
k xi i
x
=
= − ⋅ (549)
The flux producing component of the stator current is determined as follows:
rr
dr
r br
ds 2
m
'1 '
'
x p
ri
k x
λω
+ ⋅
= (550)
The above relation shows that there is no need of a current decoupler in rotor field orientation
scheme. Both stator current components (torque and flux producing) can be controlledindependently.
Steady-state perfomance of a rotor flux oriented systemLetting the derivative operator p =0, one can obtain the steady-state voltage equations of the
induction machine. After several manipulation of the system equations, the d current componentsare given by:
99
rrr qs1r dr b
ds 2
s rm
r
dr
''
'
' 0
k r ii
xk x
i
λ ω
ω
⋅= = ⋅
=
(551)
It yields the slip speed equation:
r
rs dr mb r qs12
r
'' 0
'
xk r i
xk
ω λω− ⋅ ⋅ = (552)
which for a given rotor flux has always real solution. Thus the resulting current controlled slipspeed and the electromagnetic torque are:
( ) ( )
3 r r
m r b q s 1 m r b qs
s r r
r dr r dr
2 24 2
2 2r r r rm m r m r
e dr qs1 qs1 qs
r b r s r s
' '
' ' ' '
' ''
2 ' 2 ' 2 '
k x r i k x r i
x x
x x k r x k rP P PT k i i i
x x x
ω ωω
λ λ
λω ω ω
⋅= =
= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅
(553-554)
By comparison with the three-phase induction motor, for the rotor field orientation control
strategy, three differences have been highlighted by Popescu and Navrapescu (2000):• The stady-state values for the slip angular velocity and torque are proportional with the turns
ratio value, respectively the square of turns ratio k which determines the option for k > 1, as thetorque response is the determinant factor in an electrical drive system;
• The estimated equivalent value for the torque component of the stator current has to be
corrected by dividing to the square of the turns ratio value;• There is no need for changing the number of co-ordinates from the real machine system to the
control system;• The problem of the stator windings asymmetrical configuration is overcome by using the value
of turns ratio k in computing the estimated value for the torque and flux producing componentsof the stator current.
3.8.3. Air-gap flux field orientation (AFO)
For this vector control strategy, the set of state-space variables formed by air-gap flux linkage
and stator current space vectors T
qm dm qs ds, , ,i iλ λ = x is selected. The mathematical model is given
in the chapter dedicated to d-q models of the single-phase induction machine.
The stator and rotor flux linkages and rotor current space vectors components expressed asfunctions in terms of state-space variables are:
( )( )
( )
( )
m m m
qs qm s m qs
m m 2 m
ds dm s m ds
m m mrqr qm r m qs
m
m m 2 mrdr qm r m ds
m
'' '
'' '
x x i
k x x i
xx x i
x
xk x x i
x
λ λ
λ λ
λ λ
λ λ
= + −
= + −
= − −
= − −
(555-558)
r m m
qr qm qs
m
r m m
dr dm ds2
m
1'
1'
i ix
i ik x
λ
λ
= −
= −
(559-560)
100
The voltage equations of the machine expressed in state-variables terms, are readily deductiblenow:
m m m m msr r r
qr r r m qs qm dm r m ds
b m m b b m
m 2 m m m msr r rdr r r m ds dm qm r m qs
b m m b b m
' ' '1' 0 ' ( ' ) ( ' )
' ' '' 0 ' ( ' ) ( ' )
r x xp pu r x x i k x x i
x x k x
r x xp pu k r x x i k x x i
x x x
ωλ λ
ω ω ω
ωλ λ
ω ω ω
= = − + − + + ⋅ + ⋅ − ⋅ −
= = − + − + + ⋅ − − −
(561-562)
The electromagnetic torque expression has to be re-written:
m m m m
e dm qs qm ds
b
1 1
2
PT i k i
kλ λ
ω
= ⋅ −
(563)
If we make the notation:m 2 m
qs qs1i k i= the above torque relation may be expressed as for the
symmetrical induction machine:
( )m m m m
e dm qs1 qm ds
b2
P kT i iλ λ
ω= ⋅ − (564)
If the special reference frame is fixed to the air-gap flux linkage vector, the q-component of thisflux vector is defined equal to zero:
m
qm
m m
dm m
0λ
λ λ
=
= (565)
From the air-gap flux linkage equations, the q-current components are given by:m m
qs1 qs1
m 2 m
qr qs1'
i i
i k i
=
= − ⋅ (566)
The flux producing component of the stator current is determined as follows:
2 m m 3 msr r
r r m ds dm r m qs1
b m m b b
' '' ( ' ) ( ' )
r xp pk r x x i k x x i
x x
ωλ
ω ω ω
+ − = + ⋅ + −
(567)
When an air-gap field orientation control is employed, it is necessary to decouple the stator
current components, in order to achieve a linear control. For this reason, the command of current ofthe d-axis component is computed as follows:
m m mi
ds p dm dq
m
s r m qs1m
dq
b r r m
( ' )
' ( ' )
Ki K i
p
k x x ii
r x x p
∆λ
ω
ω
= + +
⋅ − ⋅=
+ −
(568-569)
The electromagnetic torque relation and slip speed can be derived in air-gap field orientationcontrol as:
[ ]
m m
e dm qs
b
2 m
b r r m qs
sm mr
dm r m ds
m
2
' ( ' )
' 1( ' )
P kT i
r x x p k i
xk x x i
x k
λω
ωω
λ
= ⋅
+ − ⋅=
⋅ − ⋅ − ⋅
(570-571)
101
Steady-state perfomance of a air-gap flux oriented systemLetting the derivative operator p = 0, the steady-state voltage equations of the induction machine
are readily determined. After several manipulation of the system equations, the d currentcomponents are given by:
m m mb sr
ds r qs1 r m qs1
m s r b
m
m r qs1m s
dr
r b
'' ( ' )
'
( ' )'
'
xki r i x x i
x r
k x x ii
r
ω ω
ω ω
ω
ω
= ⋅ + − ⋅
−= ⋅
(572)
It yields the slip speed equation:2 m2
m ms s dmr m
qs1 r qs12
b r b
( ' )' 0
'
k x xi k r i
r k
ω ω λ
ω ω
−⋅ ⋅ − ⋅ + ⋅ =
(573)
The solutions of the above equations have to be real, for a given stator flux linkage. This means thatthe determinant of the second order equation satisfy the condition:
( ) ( )2
m222 mdm
r m qs124 ' 0k x x i
k
λ∆
= − − ⋅ ≥
(574)
The maximum values for the q component of the stator current, the slip speed and theelectromagnetic torque (pull-out torque) are:
( )
( )
( )( )( )
mm dmqs1 3max
r m
r
s maxr m
2m
dm
e 2maxb r m
2 ( ' )
'
'
1
2 2 '
ik x x
r
x x
PT
k x x
λ
ω
λ
ω
=−
=−
= ⋅ ⋅−
(575-577)
For the air-gap field orientation control strategy, there have to be pointed out several important
features:• The turns ratio k determines the effects on the critical value of the electromagnetic torque of the
single-phase machine: lower torque for k > 1, and higher torque for k < 1, when the main statorwinding parameters are kept constant;
• The pull-out value of the slip speed does not depend on the unsymetrical configuration of the
stator windings, and it is expressed with an identical relation to that deduced for the three-phaseinduction machine;
• The estimated equivalent value for the torque component of the stator current has to be
corrected by dividing to the square of the turns ratio value;
• There is no need for changing the number of co-ordinates from the real machine system to the
control system;
• The problem of the stator windings asymmetrical configuration is overcome by using the value
of turns ratio k in computing the estimated value for the torque and flux producing components
of the stator current.
3.8.4. Stator current orientation control (SCO)
For a complete comparison to the vector control strategies applied to three-phase induction
motor, the current orientation control shemes have to be analysed as well. One option is if the set ofstate-space variables is identical with the stator flux field orientation control (SFO): the stator flux
102
and current space vectors T
, , ,qs ds qs dsi iλ λ = x . The mathematical model is identical to that used in
SFO case.The rotor based variables expressed in state-space variables terms are:
( )
( )
( )
( )
sc sc sc
qr qs s qs
m
sc sc 2 sc
dr ds s ds2
m
sc sc sc
qr r qs qs
m
sc sc 2 sc
dr r ds ds
m
1'
1'
1' '
1' '
i x ix
i k x ik x
x D ix
x k D ix
λ
λ
λ λ
λ λ
= −
= −
= − ⋅
= − ⋅
(578-581)
To obtain the machine equations in the synchronuous stator current reference frame, one has to
eliminate the rotor flux from the rotor voltage equations and then force the q component of thestator current to be zero. The stator voltages equations remain unchanged. If a cage rotor is
considered, the resultant equations are as follows:
sc sc sc scs
r s qs r ds ds r r qs
b b b
2 sc sc sc scs s
r s ds qs r r ds r qs
b b b b
1' ' ' ' 0
' ' ' '
p pr x D i x k D i r x
k
p pk r x D i k D i r x k x
ωλ λ
ω ω ω
ω ωλ λ
ω ω ω ω
+ − − ⋅ ⋅ − + =
+ − ⋅ = + −
(582-583)
The electromagnetic torque expression has to be re-written:
sc sc sc sc
e ds qs qs ds
b
1 1
2
PT i k i
kλ λ
ω
= ⋅ −
(584)
If we make the notation: sc sc
qs qs12
1
kλ λ= the above torque relation may be expressed as for the
symmetrical induction machine:
( )sc sc sc sc
e ds qs qs1ds
b
1
2
PT i i
kλ λ
ω= ⋅ − (585)
If the special reference frame is fixed to the stator current space vector, the q-component of thiscurrent vector is defined equal to zero:
sc
qs
s sc
s ds
0i
i i
=
= (586)
Different from the SFO case, we have to express the q-flux components, by considering the fluxlinkages equations:
sc sc
qs1 qs1
sc scrqr qs12
m
''
x
k x
λ λ
λ λ
=
= (587)
The electromagnetic torque relation and slip speed can be derived in stator current orientationcontrol as:
( )( )
sc sc
e qs1 ds
b
sc
b r s r qs1
s 2 s sc
ds r ds
1
2
' '
'
PT i
k
r x x p
k k D i x
λω
ω λω
λ
= − ⋅
+ ⋅=
⋅ −
(588-589)
103
The relation between the d-axis component of the stator current and stator flux linkage space vectorcomponents can be deduced as follows:
2 sc sc scsr s ds r r ds r qs1
b b b
1' ' ' ' 0
p pk r x D i r x x
k
ωλ λ
ω ω ω
+ − + + ⋅ =
(590)
Similarly to the SFO case, there is a coupling, but between the stator flux linkage components
used as control variables. Consequently, any change in the torque producing component sc
dsλ
without changing sc
qsλ accordingly, will cause a transient in the stator flux. A decoupler is necessary
to overcome this disadvantage:
sc sc sci
ds p ds dq
scs
r qs1
sc b
dq
r r
b
1'
' '
KK i
p
xk
pr x
λ ∆ λ
ωλ
ωλ
ω
= + +
⋅
=
+
(591-592)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. This
controller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomance of a stator current oriented system
Letting the derivative operator p = 0, one can obtain the steady-state voltage equations of theinduction machine:
sc sc scs s r br
ds qs1 qs12 2
r b sm m
sc scbr
dr qs1
m s
''
'
'
x rx D
kr x kx
r
kx
ω ωλ λ λ
ω ω
ωλ λ
ω
= − ⋅ ⋅ ⋅ − ⋅ ⋅
= − ⋅
(593)
it yields the slip speed equation:2 2
sc 2 sc scs sr r
qs1 ds qs12
b m r b m
' '10
'
x rk i
x k r k x
ω ωλ λ
ω ω
⋅ + ⋅ + = ⋅ ⋅
(594)
The solutions of the above equations have to be real, for a given stator current. This means that thedeterminant of the second order equation satisfy the condition:
( )22 sc
2 qs2 sc r
ds 2 2m m
'4 0
xk i
x k x
λ∆
= − ≥
(595)
The maximum values for the q component of the stator flux linkage, the slip speed and the
electromagnetic torque (pull-out torque) are:
( )
( )
( )( )
2 2 scsc m dsqs1
maxr
rs max
r
22 scm ds
e m a xb r
2 '
'
'
2 2 '
k x i
x
k r
x
k x iPT
x
λ
ω
ω
=
⋅=
⋅= ⋅
(596-598)
The analysis of this vector control strategy leads to the same conclusions as for the three-phaseinduction machine case, but with the differences imposed by the asymmetrical motor configuration
and the number of phases for the supply voltage:
104
• The turns ratio k determines different effects on the critical values of the rotor speed and
electromagnetic torque response of the single-phase machine: the electromagnetic torque critical
values reaches a maximum for k > 1, while the rotor speed reaches a maximum for k < 1, whenthe main stator winding parameters are kept constant;
• The estimated equivalent value for the torque component of the stator flux has to be corrected
by multypling with the square of the turns ratio value k;
• There is no need for changing the number of co-ordinates from the real machine system to the
control system;
3.8.5. Rotor current orientation control (RCO)
If the synchronuous reference frame is linked to the rotor current space vector it results anotherunconventional vector control strategy. The rotor current orientation control strategy (RCO) is
analysed for a cage rotor induction motor. There are two options in selecting the set of state-spacevariables, if we consider the criteria of direct measurable quantities:
I. The state-space variables are the stator flux linkage and rotor current space vectors:T
qs ds qr dr, , ' , 'i iλ λ = x . When the stator flux and and the rotor current are selected as state-space
variables, one can derive the rotor flux and stator current functions in terms of state variables asfollows:
( )rc rc rc
qs qs m qr
s
rc
rc rcds
ds m dr2
s
rc rc rcm
qr qs qr
s s
2rc rc rcm
dr ds dr
s s
1'
1'
' '
' '
i x ix
i x ix k
x Di
x x
x k Di
x x
λ
λ
λ λ
λ λ
= −
= −
= +
= +
(599-602)
To obtain the machine equations in the synchronuous rotor current reference frame, one has to
eliminate the rotor flux from the rotor voltage equations and then force the q component of the rotorcurrent to be zero:
rc rc rc rcs sm m
r qr dr qs ds
s b s b s b s b
2 rc rc rc rcs sm mr dr qr ds qs
s b s b s b s b
10 ' ' '
0 ' ' '
x xD p kD pr i i
x x x k x
x kxD p kD pk r i i
x x x x
ω ωλ λ
ω ω ω ω
ω ωλ λ
ω ω ω ω
= + ⋅ + ⋅ + ⋅ + ⋅ ⋅
= + ⋅ − ⋅ + ⋅ − ⋅
(603-604)
The electromagnetic torque expression has to be re-written:
rc rc rc rcm
e qs dr ds qr
b s
1' '
2
xPT k i i
x kλ λ
ω
= ⋅ −
(605)
If we make the notation: rc rc
qs qs12
1
kλ λ= the above torque relation may be expressed as for the
symmetrical induction machine:
( )rc rc rc rcm
e qs1 dr ds qr
b s
' '2
xPT i i
k xλ λ
ω= ⋅ − (606)
If the special reference frame is fixed to the rotor current space vector, the q-component of this
current vector is defined equal to zero:
105
rc
qr
rc rc
dr r
' 0
' '
i
i i
=
= (607)
Different from the RFO case, we have to express the q-flux components, by considering the flux
linkages equations:rc rc
qs1 qs1
rc scsqr qs12
m
'x
k x
λ λ
λ λ
=
= (608)
The electromagnetic torque relation and slip speed can be derived in stator current orientationcontrol as:
rc rcme qs1 dr
s b
rcm
qs12
src rc
dr m ds
'2
1'
xPT i
kx
xp
k
k D i xk
λω
λω
λ
= ⋅
− ⋅=
⋅ ⋅ +
(609-610)
A relation between the d-axis component of the rotor current and stator flux linkage space vectorcomponents can be deduced:
2 rc rc rcsm m
r dr ds qs1
s b s b s b
0 ' 'x xD p p
k r ix x kx
ωλ λ
ω ω ω
= + ⋅ + ⋅ − ⋅
(611)
Due to the coupling between the stator flux linkage components, any change in the torque
producing component rc
dsλ without changing rc
qsλ accordingly, will cause a trensient in the stator
flux. To overcome this disadvantage, the command current of the d-axis component of the statorcurrent is expressed as follows:
rc rc rci
ds p dr dq
sc
qs1rc
dq s
'
1
KK i
p
p k
λ ∆ λ
λλ ω
= + +
= ⋅ ⋅
(612-613)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. Thiscontroller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomanceAfter several manipulation of the system equations if we let the derivative operator p = 0, the d
current components are obtained from the steady-state voltage equations of the induction machine:
rc rcs
ds qs1
s r b
rc
dr
'
' 0
D
kx r
ωλ λ
ω
λ
= − ⋅
=
(614)
It yields the slip angular velocity equation:2
rc rcsmdr qs12
s bs r
0'
x Dk Di
x kx r
ωλ
ω⋅ − ⋅ = (615)
which for a given rotor flux has always real solution. Thus the slip speed and the electromagnetic
torque are:
106
3 rc rc
s r b dr s r b dr
s rc rc
m qs1 m qs
rc rc rc rcm m
e qs1 dr qs dr
s b s b
' ' ' '
' '2 2
k x r i kxr i
x x
x kxP PT i i
kx x
ω ωω
λ λ
λ λω ω
= =
= ⋅ = ⋅
(616-617)
The rotor current oriented (RCO) vector control with mixed flux and current state-spacevariables differs from the variant applied to the three-phase induction machine by the following:
• The transformation between the reference frames (stationary to synchronous and vice-versa) is
easier as there is no need of changing the number of variables;• The asymmetry of the stator windings is modeled only by using a supplementary parameter, the
turns ratio k;• In steady-state operation, the torque and slip speed response are influenced by the unsymmetrical
configuration of the motor: higher torque and slip speed for k < 1.
II. The state-space variables are stator and rotor currents T
qs ds qr dr, , ' , 'i i i i = x . When the stator
and the rotor current are selected as state-space variables, one can derive the stator and rotor flux asfunctions in terms of state variables from the classical flux linkage equations.
To obtain the machine equations in the synchronuous rotor current reference frame, one has to
eliminate the rotor flux from the rotor voltage equations and then force the q component of the rotorcurrent to be zero. The stator voltages equations are re-written also. The resultant equations have
the following form:
rc rc rc rcs s
m qs r r qr m ds r dr
b b b b
rc rc 2 rc 2 rcs sm qs r qr m ds r r dr
b b b b
0 ' ' ' ' '
0 ' ' ' ' '
p px i r x i kx i kx i
p pkx i kx i k x i k r x i
ω ω
ω ω ω ω
ω ω
ω ω ω ω
= + + + +
= − − + + +
(618-619)
The electromagnetic torque expression has the expression:
( )rc rc rc rcm
e qs dr ds qr
b
' '2
kxPT i i i i
ω= ⋅ − (620)
It can be observed that the above relation is readily available for implementing a vector control
strategy.When linking the special synchronuous reference frame to the rotor current space vector, the q-
component of this current vector is defined equal to zero:rc
qr
rc rc
dr r
' 0
' '
i
i i
=
= (621)
Different from the (I ) case of RCO control strategy, for the (II ) case we have to express the stator
flux linkage q-components, by considering the flux linkages equations:rc rc
qs s qs
rc sc rcm
qr m qs qs
s
'
x i
xx i
x
λ
λ λ
=
= = (622)
The electromagnetic torque relation and slip speed can be derived in this type of rotor currentorientation control as:
107
( )
rc rcm
e qs dr
b
rc
m qs
s rc rc
m ds r dr
'2
' '
kxPT i i
x pi
k x i x i
ω
ω
= ⋅ ⋅
⋅= −
+
(623-624)
From the second dynamic equation of the machine, a relation between the d-axis component of therotor current and stator current space vector components can be expressed as:
2 rc rc 2 rcsm ds m qs r r dr
b b b
' ' 'p p
k x i kx i k r x iω
ω ω ω
= − +
(625)
A coupling between the d-axis and q-axis stator current components appears, and consequently,
any change in the torque producing component rc
dsi without changing rc
qsi accordingly, will cause a
transient in the stator flux. To overcome this disadvantage the command current of the d-axiscomponent of the stator current has to be expressed in the following form:
rc rc rci
ds p dr dq
sc
qsrc
dq s
'
1
Ki K i i
p
ii
p k
∆
ω
= + +
= ⋅ ⋅
(626-627)
where Kp and Ki are proportional, respectively integral coefficients of the current controller. Thiscontroller can be PI type or soft computing technique type (fuzzy, neural-network).
Steady-state perfomanceBy letting the derivative operator p = 0, the d flux components are obtained from the steady-state
voltage equations of the induction machine:
rc rcs
ds qs
r b
rc
dr
'
' 0
kDi
r
ωλ
ω
λ
= − ⋅
=
(628)
It yields the slip angular velocity equation:2
rc rcsmdr qs
s s r b
' 0'
x kDk Di i
x x r
ω
ω⋅ − ⋅ = (629)
which for a given rotor flux has always real solution. Thus the slip speed and the electromagnetic
torque are:rc
r b dr
s rc
m qs
rc rcm
e qs dr
b
' '
'2
kr i
x i
kxPT i i
ωω
ω
=
= ⋅
(630-631)
A comparison with the rotor current orientation control detailed for the three-phase inductionmachine permits to highlight the same conclusions. However, there has to be pointed out the
necessary corrections:• The transformation between synchronous reference frame and the stationary reference frame is
made without changing the number of variables;• The influence of the unsymmetrical stator windings configuration can be evidenced by using the
turns ratio k, and it determines different effect over the rotor speed and torque response of themotor: higher electromagnetic torque and lower rotor speed for k > 1, respectively lower torqueand higher rotor speed for k < 1.
108
4. MATHEMATICAL DISCRETE MODELS FOR THETHREE-PHASE INDUCTION MACHINE
4.1. Introduction
Advances in very-large-scale integration (VLSI) technology made possible the real-time
modelling for many industrial applications. Real-time simulation is used increasingly in theautomatic control field. One important application is the advanced AC motor control, i.e. vector
control, where the immeasurable quantities, like the cage rotor parameters (flux, current), can beestimated by a simulator operating in parallel with the real motor. The simulation became animportant alternative against the measurement, as the latter is complex, noise sensitive, and
expensive. It has to be also mentioned that the approximate simulation always introduces someerror between the true dynamic behaviour and the modelled behaviour (Vainio et al - 1992).
The continuous and discrete models of the induction machine are equivalent if their timeresponse is similar for typical inputs (step or sinusoidal input). However, there are several discretemathematical models equivalent to the continuous model. Iron loss and saturation can be omitted if
the stator and rotor fluxes are limited to stay below the wide saturation border of the iron core,which is the case in most practical applications. For this equivalence, the approximation mode of
the response for two consecutive sampling rates is determinant. Usually, when a time continuoussystem is discretized, the designer has to chose the properties of the system that will be maintained:the zero and pole number, the response characteristics for pulse, step and linear input, the DC
amplify, the frequency response.In Table 4.I there are presented several possible discretization methods for the time continuous
systems.
TABLE 4.I
No Discretization method Continuous–discrete equivalence equation
1 Forward-difference method (Euler)1
11
−
−
⋅−
=zT
zs
2 Backward-difference method
T
zs
11 −−=
3 Bilinear transformation method(Tustin)
1
1
1
12−
−
+
−⋅=
z
z
Ts
4 Frequency prewarping method
2tan
2,
1
121
1 T
Tz
z
Ts D
A
⋅⋅=
+−
⋅=−
− ωω
5 Pulse invariance method [ ] )()(1
sGLZTzGD−⋅=
6 Step invariance method ( )
⋅−= −−
s
sGLZzzG D
)(1)( 11
7 Matched zero-pole method Zero/pole from s=-a is placed in discrete to z=e-aT .
Zero/pole from ±∞ is placed in discrete to z=-1
All of the above discretization methods are compared for the induction machine seen as a discretesystem. The assumptions valid for the continuous model are also valid for the discrete systems. The
results of the discretization are analysed from the point of view of computation complexity and theresponse stability for a step input signal. The analysis of the different models is essential for the
design of electrical drive system with numerical command. A special attention is given to torqueestimation from the data acquisition process (voltage or current quantities).
109
The synchronous reference frame is used for determining the discrete model of the inductionmachine. The following parameters can be defined:
s s rkω ω ω ω ω= − =
For an easy to follow analysis, the index k will be omitted in the machine equations. The generalform equations used to determine the discrete mathematical induction machine in arbitrary
synchronuos reference frames and per unit system are:
ss s s s s
n
( )1( ) ( ) ( )
d tU t r i t j t
dt
ψω ψ
ω= + + (632)
r
r r r r
n
' ( )10 ' ' ( ) ' ( )
d tr i t j t
dt
ψω ψ
ω= + + (633)
s s M rs( ) ( ) ' ( )t l i t l i tψ γ= ⋅ + ⋅ ⋅ (634)
2
r r M sr( ) ' ' ( ) ( )t l i t l i tψ γ γ= ⋅ ⋅ + ⋅ ⋅ (635)
2rr r r rr' ; ' ; '
ii r rψ γ ψ γ
γ= = ⋅ = ⋅ (636)
( )e l
M
1dt t
dt T
ω= ⋅ − (637)
Depending on the synchronuous reference frame the following relations are valid:a) Stator reference frame:
s
M
*
e ss
3Im
2
l
l
t P i
γ
ψ
=
= ⋅
(638)
b) Rotor reference frame:
M
r
*Me sr
r
'
3Im
2
l
l
lt P i
l
γ
ψ
=
= ⋅
(639)
c) Airgap reference frame:
*
e sm
M s rm
1
3Im
2
( ' )
t P i
l i i
γ
ψ
ψ
=
= ⋅
= +
(640)
If the determinant of the flux system equations is noted with δ and with d its inverse, the voltage
equations are re-written as follows:
ss s r M r ss s
n
( )1( ) ( ' ' ) ( )
d tU t r d l l j t
dt
ψγ ψ ψ ω ψ
ω= ⋅ ⋅ ⋅ ⋅ − ⋅ + + ⋅ ⋅ (641)
r
r s r M s r r
n
' ( )10 ' ( ' ) ' ( )
d tr d l l j t
dt
ψψ γ ψ ω ψ
ω= ⋅ ⋅ ⋅ − ⋅ ⋅ + + ⋅ ⋅ (642)
For zero initial conditions, if the Laplace transformation is applied, it results the matrix system:
s r s s M sss
r M r s rn r r
'
' ' 0' '
r l d j r l d Us
r l d r l d j
ψψ γ ω
γ ωω ψ ψ
− ⋅ − ⋅ = ⋅ + ⋅ − −
(643)
110
or in synthetisized form:( ) ( ) ( )s s s s⋅ = ⋅ + ⋅Y A Y B U (644)
One can note that by comparing with the state variable system described by the equations:
( ) ( ) ( )
( ) ( ) ( )
s s s s
s s s
⋅ = ⋅ + ⋅
= ⋅ + ⋅
X A X B U
Y C X D U (645)
the state variable vector is identical to the output vector (fluxes vector in this case). The input vectoris the voltage vector. Through identification we obtain:
s r s s M
n
r M r s r
n
'
' '
r l d j r l d
r l d r l d j
γ ωω
γ ω
ω
− ⋅ − = ⋅ ⋅ − − =
A
B
(646)
4.2. Bilinear transformation method (Tustin)
The relation (3) from the Table I is known as bilinear transformation or Tustin and it makes theconnection between continuous to discrete domain. This method gives better results than thetrapezoidal approximation, or other discretization method illustrated in Table 4.II, by considering
first four order of integration operators. The describing system relation becomes:2 1
( ) ( ) ( )1
zz z z
T z
−⋅ ⋅ = ⋅ + ⋅
+Y A Y B U (647)
where T is the signal sampling period for continuous domain, and f is the sampling frequency.
TABLE 4.II
Method
s
12
1
s 3
1
s 4
1
sTustin
1
1
2 −
+⋅
z
zT2
22
)1(
)1(
4 −
+⋅
z
zT3
33
)1(
)1(
8 −
+⋅
z
zT4
44
)1(
)1(
16 −
+⋅
z
zT
Boxer-
Thaler 1
1
2 −
+⋅
z
zT
2
22
)1(
110
12 −
++⋅
z
zzT3
23
)1(2 −
+⋅
z
zzT4
234
)1(
4
6 −
++⋅
z
zzzT
Madwed
1
1
2 −
+⋅
z
zT
2
22
)1(
14
6 −
++⋅
z
zzT3
233
)1(
11111
24 −
+++⋅
z
zzzT4
2344
)1(
16626
120 −
+++⋅
z
zzzT
2( 1) ( ) ( 1) ( ) ( 1) ( )z z z z z z
T⋅ − ⋅ = + ⋅ ⋅ + + ⋅ ⋅Y A Y B U
1 1
1
1
2 2( ) ( ) ( ) ( ) ( 1) ( )
2 2( ) ( ) ( 1) ( )
2 2 2( ) ( ) ( 1) ( )
2 2( )
z z z z z z z zT T
z z z z zT T
z z z z zT T T
z zT T
− −
−−
⋅ − ⋅ = ⋅ ⋅ + ⋅ + + ⋅ ⋅
⋅ − ⋅ ⋅ = ⋅ + ⋅ + + ⋅ ⋅
⋅ = ⋅ − ⋅ ⋅ + ⋅ + ⋅ − ⋅ ⋅ + ⋅
= ⋅ − ⋅ ⋅ + ⋅ ⋅
Y Y A Y A Y B U
I A Y I A Y B U
Y I A I A Y I A B U
Y I A I A Y
1
1
1 1
2( ) (1 ) ( )
1 1( ) ( ) (1 ) ( )
z z zT
z z z z z∆ ∆
−−
− −
+ ⋅ − ⋅ ⋅ + ⋅
= ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅
I A B U
Y C Y D U
(648)
where ∆ is the matrix determinant: 2
T
⋅ − I A :
111
[ ]
2 2 2
n s r n r s n s r M r s
n r n s r s r s
(2 ' ) (2 ' ) ( ' )
(2 ' ) (2 ' )
R Ij f r l d f r l d r r l d
j f r l d f r l d
∆ ∆ ∆ ω γ ω ω γ ω ω
ω ω ω γ ω ω
= + ⋅ = + ⋅ ⋅ ⋅ + ⋅ − ⋅ ⋅ + +
+ ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ + ⋅ (649)
[ ]
2 2 211 n s r n r s n s r M r s
n r n s r s n r s
12 n s M
21 n r M
2 2 222 n s r n r s n s r M r s
n r
(2 ' ) (2 ' ) ( ' )
(2 ' ) (2 ' )
4
4 '
(2 ) (2 ' ) ( )
(2
c f r l d f r l d r r l d
j f r l d f r l d
c f r l d
c f r l d
c f r l d f r l d r r l d
j
ω γ ω ω γ ω ω
ω ω ω γ ω ω
ω γ
ω
ω γ ω ω γ ω ω
ω ω
= − ⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ + +
+ ⋅ ⋅ ⋅ − ⋅ − ⋅ +
= ⋅
=
= + ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ + +
+ ⋅ ⋅ − ⋅[ ]n s r s n r s
2
11 n n r s r n
2
21 r M n
) (2 ' )
(2 ' )
'
f r l d f r l d
d f r l d j
d r l d
ω γ ω ω
ω ω ω ω
ω
+ ⋅ + ⋅ −
= ⋅ + + ⋅
=
(650)
Observation: d12, and d22 are not calculated as the induction machine is considered to have cagerotor.
[ ]
[ ]
11 12 r 11 s ss s
r 21 22 r 21 s ss
1 1( ) ( 1) ' ( 1) ( 1) ( )
1 1' ( ) ( 1) ' ( 1) ( 1) ( )
n c n c n d U n U n
n c n c n d U n U n
ψ ψ ψ∆ ∆
ψ ψ ψ∆ ∆
= ⋅ ⋅ − + ⋅ − + ⋅ ⋅ − + = ⋅ ⋅ − + ⋅ − + ⋅ ⋅ − +
(651)
where n and n – 1 are two consecutive sampling periods in discrete domain.
The induction machine discrete model is presented in Fig. 4.1. For this model the stator voltagerepresents the input vector and the fluxes are the expressed as output vector.
s
z-1
+
+
+
d11
c11
d12
∆
∆
1/
1/
z-1
z-1
c12
c21
c22
s
r
U
Ψ
Ψ'
Fig. 4.1. The induction machine discrete model
The blocks with thicker lines denote complex multipliers. The following matrix relation isobtained from the separation of the real and the imaginary parts of the flux equations.
s s s s
s s s s
2 2 2 2r r
r r
( ) ( 1) ( 1) ( )
( ) ( 1) ( 1) ( )1 1
' ( ) ' ( 1) 0
' ( ) ' ( 1) 0
R R R R
I I I I
R RR I R I
I I
n n U n U n
n n U n U n
n n
n n
ψ ψ
ψ ψ
ψ ψ∆ ∆ ∆ ∆
ψ ψ
− − + − − + = ⋅ ⋅ + ⋅ ⋅ −+ +
−
E F (652)
where:
112
IR
RR
RIIR
RIRR
RIIR
IIRR
IR
RR
IR
RR
RIIR
IIRR
dff
dff
ddff
ddff
ccee
ccee
cee
cee
cee
cee
ccee
ccee
∆=−=
∆==
∆−∆=−=
∆+∆==
∆−∆=−=
∆+∆==
∆=−=
∆==
∆=−=
∆==
∆−∆=−=
∆+∆==
214132
214231
11112112
11112211
22224334
22224433
214132
214231
122314
122413
11112112
11112211
(653)
The complete discrete model of the induction machine, based on the bilinear method is given in Fig.
4.2. The electromagnetic torque expressed in relative units is considering different reference framesfor the induction machine:a) Stator reference frame
( )
( )
*
e s s s s s s s s ss
s r s M r s r s M r M s r s r
3 3 3Im Im ( ) ( )
2 2 2
3 3( ' ' ) ( ' ' ) ( ' ' )
2 2
R I R I R I I R
R I I I R R I R R I
t P i P j i j i P i i
P l d l d l d l d P l d
ψ ψ ψ ψ ψ
ψ γ ψ ψ ψ γ ψ ψ ψ ψ ψ ψ
= ⋅ = − ⋅ + ⋅ = ⋅ − ⋅ =
= ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ ⋅ − ⋅
(654)
b) Rotor reference frame:
( )
( )
( )
*
e s r r s s r s r sr
2
r r s M r r r s M r r s r s r
M s r s r
3 3 3Im Im ( ' ' ) ( ) ' '
2 2 2
3 3' ( ' ' ) ' ( ' ' ) ' ( ' ' )
2 2
3' '
2
R I R I R I I R
R I I I R R I R R I
I R R I
t P i P j i j i P i i
P l d l d l d l d P l d
P l d
γ ψ γ ψ ψ γ ψ ψ
γ ψ γ ψ ψ ψ γ ψ ψ γ ψ ψ ψ ψ
ψ ψ ψ ψ
= ⋅ ⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅ = ⋅ ⋅ ⋅ − ⋅ =
= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ ⋅ ⋅ − ⋅ =
= ⋅ ⋅ ⋅ − ⋅
(655)c) Airgap reference frame:
( )
( )
( ) ( )
*
e s m m s s m s m sm
m r s M r m r s M r
M s r M r s M r s M r
M s r M
3 3 3Im Im ( ) ( )
2 2 2
3( ' ' ) ( ' ' )
2
( ' ) ' ( ) ' '3
2 ( ' )
R I R I R I I R
R I I I R R
R R I I
I
t P i P j i j i P i i
P l d l d l d l d
l d l l l l l d l dP
l d l l
ψ ψ ψ ψ ψ
ψ γ ψ ψ ψ γ ψ ψ
ψ γ γ ψ γ γ ψ ψ
ψ γ γ
= ⋅ = − ⋅ ⋅ + ⋅ = ⋅ − ⋅ =
= ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ − ⋅ =
⋅ ⋅ ⋅ − ⋅ + ⋅ − ⋅ ⋅ ⋅ − ⋅ −=
− ⋅ ⋅ ⋅ − ⋅ +( ) ( )( ) ( )
( ) ( )( )
r s M r s M r
s r M r s M r s M r
M M s r s r
s r M r s M r s M r
' ( ) ' '
( ' ) ' ( ) ' '3 3' '
2 2( ' ) ' ( ) ' '
I R R
R R I I
I R R I
I I R R
l l l d l d
l l l l l d l dP l d P l d
l l l l l d l d
ψ γ γ ψ ψ
ψ ψ ψ ψψ ψ ψ ψ
ψ ψ ψ ψ
= ⋅ − ⋅ ⋅ ⋅ − ⋅
⋅ − + − ⋅ ⋅ − ⋅ −= ⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅
− ⋅ − + − ⋅ ⋅ − ⋅ (656)
From the torque equations, written in stator, rotor and air-gap reference frames, one can note that
the same general relation can be implemented, though the equivalent rotor flux stands for differentsignificance, i.e. only in air-gap reference frame it has the real physical rotor flux.
113
As it can be observed by studying the block diagram of the discrete model of the inductionmachine, for a complete implementation there are necessary 31 multiplier blocks, 23 summing
blocks and 4 delay blocks. However, this model can be simplified further. The total number ofmultiplier blocks can vary from one implementation to another, due to the place of this operation in
the block diagram. Also the frequency response is variable according to the implementation version(Vainio et al - 1992).
+
+
+
+
+
+
z-1
z-1
UsR(n)
UsI(n)
f11
f12
f21
f22
f31
f32
f41
f42
e11
e12
1/
e13
e14
e21
e22
1/
e23
e24
e31
e32
e33
e34
1/
e41
e42
e43
e44
1/
z-1
z-1
z-1
z-1
*
*
+-
+
lM
d
Torque
∆
∆
∆
∆
Ψ
Ψ
Ψ
Ψ
sR(n)
rI(n)
sI(n)
rR(n)
t
'
'
Fig. 4.2. Complete discrete induction machine model based on
bilinear transformation method (Tustin)
4.3. Forward-differences method (Euler)
In this method the continuous-time derivative is approximated by the relation (1) from Table 4.I
or by a scaled difference of two successive samples:
T
nXnX
dt
dX )()1( −+= (657)
where T is the sampling period. The discrete-time model is in this case defined by the following
equations:
( ) s ss s r M r ss s
n
( 1) ( 1)1( ) ' ( ) ' ( ) ( )
n nU n r l d n l d n j n
T
ψ ψγ ψ ψ ω ψ
ω
+ − += ⋅ ⋅ ⋅ − ⋅ + ⋅ + ⋅ ⋅ (658)
( ) r rr s r M rs r
n
' ( 1) ( 1)10 ' ' ( ) ( ) ( )
n nr l d n l d n j n
T
ψ ψψ γ ψ ω ψ
ω
+ − += ⋅ ⋅ − ⋅ ⋅ + ⋅ + ⋅ ⋅ (659)
In matrix form it results:
114
s r s s M
n ss s
n n
r rr M r s r
n
1'
( 1) ( ) ( )
1 0' ( 1) ' ( )' '
r l d j r l dn nT U n
T Tn n
r l d r l d jT
γ ωψ ψω
ω ωψ ψ
γ ωω
− ⋅ ⋅ − ⋅ + = ⋅ ⋅ + ⋅ + ⋅ − ⋅ − ⋅
(660)
After separating the real and the imaginary components, it will result:
[ ]
[ ] [ ]
s s n s r s s s
n
n s M r r n s s
1( 1) ( 1) ( ' ) ( ) ( )
' ( ) ' ( ) ( ) ( )
R I R I
R I R I
n j n T r l d j n j nT
T r l d n j n T U n j U n
ψ ψ ω γ ω ψ ψω
ω γ ψ ψ ω
+ + ⋅ + = ⋅ − ⋅ + − ⋅ ⋅ + ⋅ +
+ ⋅ ⋅ ⋅ + ⋅ + ⋅ + ⋅
(661)
[ ]
[ ]
r r n r M s s
n r s r r r
n
' ( 1) ' ( 1) ' ( ) ( )
1( ' ) ' ( ) ' ( )
R I R I
R I
n j n T r l d n j n
T r l d j n j nT
ψ ψ ω γ ψ ψ
ω ω ψ ψω
+ + ⋅ + = ⋅ ⋅ ⋅ + ⋅ +
+ ⋅ − + − ⋅ ⋅ + ⋅ (662)
n s r n s n s Ms s
n s n s r n s Ms s
n r M n r s n rr r
n r M n n r sr
1 ' 0( 1) ( )
1 ' 0( 1) ( )
' 0 1 '' ( 1)
0 ' 1 '' ( 1)
R R
I I
R
I
T rl d T T rl dn n
T T rl d T rl dn n
T r l d T r l d Tn
T r l d T T r l dn
ω γ ω ω ωψ ψ
ω ω ω γ ωψ ψ
ω γ ω ω ωψ ψ
ω γ ω ω ωψ
− ⋅ ⋅ ⋅+ − ⋅ − ⋅ ⋅ ⋅+ = ⋅ ⋅ ⋅ − ⋅ ⋅+
⋅ ⋅ − ⋅ − ⋅+
s
s
n
r
( )
( )
( ) 0
( ) 0
R
I
R
I
U n
U nT
n
n
ω
ψ
+ ⋅
(663)or:
n( 1) ( ) ( )n n T n+ = ⋅ + ⋅A Uψ ψ ω (664)
For the current vector is valid the following relation:
( ) ( )n n= ⋅i B ψ (665)
or the matrix form equation:
r Ms s
r Ms s
M sr r
M sr r
' 0 0( ) ( )
0 ' 0( ) ( )
0 0' ( ) ' ( )
0 0' ( ) ' ( )
R R
I I
R R
I I
l d l di n n
l d l di n n
l d l di n n
l d l di n n
γ ψ
γ ψ
γ ψ
γ ψ
⋅ − ⋅ − = ⋅ − ⋅
− ⋅
(666)
For this case the complete discrete model of the induction machine is described in Fig. 4.3.
115
UsR(n)
UsI(n)
z-1
z-1
+
+
+
+
z-1
z-1
z-1
z-1
*
*
-
+
+
lM
d
Torque
t
a11
a12
a13
a21
a22
a24
a31
a33
a34
a41
a43
a44
ψ
ψ
ψ
ψ
ω
ω
n
n
T
T sR(n)
rI(n)
sI(n)
rR(n)
Fig. 4.3. Complete discrete induction machine model based onforward-differences method (Euler)
4.4. Backward-differences method
This simple method produces a stabile discrete system for a stabile time continuous system.Even some unstable continuous systems can be transformed to stabile version through this
discretization method. However, a higher sampling frequency has to be adopted in order to avoidthe frequency response distortions of the system. With relation (2) from Table 4.I the following
equations can be implemented:1
1
1
( )
z
T
T z T
−
−
−⋅ = ⋅ + ⋅
− ⋅ ⋅ = ⋅ + ⋅ ⋅
Y A Y B U
I A Y Y B U
(667)
n s r s n n s M
n r M n r s r n
1 '
' 1 '
T r l d j T Trl dT
T r l d Tr l d j T
ω γ ω ω ω
ω γ ω ω ω
+ ⋅ + − − ⋅ = − ⋅ + + I A (668)
[ ]
2 22 2n s r s n n r s r n n s r M
2 22 2
n s r n r s n s r s r M
n r n s r s n r s
(1 ' ) (1 ' ) '
(1 ' ) (1 ' ) ( ' )
(1 ' ) (1 ' )
T R Ij T r l d j T Tr l d j T T r r l d
T r l d Tr l d T r r l d
j T T r l d Tr l d
∆ ∆ ∆ ω γ ω ω ω ω ω ω γ
ω γ ω ω ω ω γ
ω ω ω γ ω ω
− = + = + ⋅ + ⋅ + + − ⋅ =
= + ⋅ ⋅ + − ⋅ − ⋅ +
+ ⋅ ⋅ + ⋅ + ⋅ +
I A
(669)
n r s r n n s M1
n r M n s r s n
1 '1( )
' 1 'T
Tr l d j T Trl dT
T r l d T r l d j T
ω ω ω ω
ω γ ω γ ω ω∆−
−
+ + − ⋅ = ⋅ ⋅ + ⋅ + I A
I A (670)
By substituting vectors Y and U with the flux vector, respectively stator voltage, it follows that:
116
ss s1 1
r r
ss
2 2 2 2
r
( ) ( 1) ( )( ) ( )
0' ( ) ' ( 1)
( 1) ( )1 1
0' ( 1)R I R I
n n U nT T T
n n
n U n
n
ψ ψ
ψ ψ
ψ
ψ∆ ∆ ∆ ∆
− −−
= − ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ = − −
= ⋅ ⋅ + ⋅ ⋅ −+ +
I A I A B
G H
(671)
When the real and the imaginary parts are separated, it results that:
s s s
s s s
2 2 2 2
r r
r r
( ) ( 1) ( )
( ) ( 1) ( )1 1
' ( ) ' ( 1) 0
' ( ) ' ( 1) 0
R R R
I I I
R RR I R I
I I
n n U n
n n U n
n n
n n
ψ ψ
ψ ψ
ψ ψ∆ ∆ ∆ ∆
ψ ψ
− − = ⋅ ⋅ + ⋅ ⋅ −+ +
−
L M (672)
11 22 11 11
12 21 11 11
13 24 12
14 23 12
31 42 21
32 41 21
33 44 22 22
34 43 22 22
11 22 11 11
12 12 11 11
31 42 21
32
R R I I
R I I R
R
I
R
I
R R I I
R I I R
R R I I
R I I R
R R
l l g g
l l g g
l l g
l l g
l l g
l l g
l l g g
l l g g
m m h h
m m h h
m m h
m
∆ ∆
∆ ∆
∆
∆
∆
∆
∆ ∆
∆ ∆
∆ ∆
∆ ∆
∆
= = +
= − = −
= =
= − =
= =
= − =
= = +
= − = +
= = +
= − = −
= =
41 21R Im h ∆= − =
(673)
The complete discrete mathematical model of the machine, obtained through the backward-difference method is presented in Fig. 4.4. A similar analysis was made for the step and pulseinvariance methods by Vainio et al (1992). The results permit some important conclusions to be
drawn:• The pulse invariance method and backward-difference method determine similar and very
resembling discrete mathematical model for the machine;• The step invariance method and the forward-difference method determine similar and very
resembling mathematical discrete model for the machine.Nevertheless, although the methods are very similar in results, the frequency characteristics of themodel differ from one method to another.
The computational burden for the three described methods is summarised in Table 4.III. As theDSP-ASIC implementation is an important cost issue, the designer has to choose a compromise
between the complexity and the accuracy of the model. When the comparison is made, one can notethe advantage of the forward-difference method (Euler) for less computing time. When accuracy isthe determining factor, the bilinear transformation method (Tustin) has to be chosen for the
implementation of the mathematical discrete model of the machine. It is also possible to apply ahybrid approach where the stator equation is discretized using the forward-difference (Euler)
method, whereas the rotor equation is converted using the bilinear transformation (Tustin) or vice-versa. As one might expect, the resulting response and computational complexity are between thoseof the complete methods. This could be exploited when implementing the discretized model with a
programmable signal processor.
117
TABLE 4.III
Method \ Real operation Additions Multiplications Delays
Forward-difference method (Euler) 11 17 6
Backward-difference method 21 27 4
Bilinear transformation method (Tustin) 23 31 4
z -1
z -1
z-1
z -1
l12
l11
l13
l14
l21
l22
l23
l24
l31
l32
l33
l34
l41
l42
l43
l44
+
+
++
+
+
+
+
+
+
+
+
+
*
*
lM
d
Torque
t
m11
m21
m31
m41
m12
m22
m32
m42
UsR(n)
UsI(n)
sR(n)
rI(n)
sI(n)
rR(n)
∆ ∗ Ψ
∆ ∗ Ψ
∆ ∗ Ψ
∆ ∗ Ψ
∆_
+
Fig. 4.4. Complete discrete induction machine model based onbackward-differences method
4.5. Z-domain transfer functions
The general form of a state-variable system using matrix notation in z-domain is:
[ ]
s s s
s s s
T
s 1
( 1) ( ) ( )
( ) ( ) ( )
( ) ( ), ..., ( )m
n n n
n n n
n x n x n
+ = ⋅ + ⋅
= ⋅ + ⋅
=
x A x B u
y C x D u
x
(674)
where xs(n) is the vector for the state variables, us (n) and ys(n) are the input, respectively the outputvectors, and A is the state matrix. The impulse response sequence in terms of the state-variabledescription is given by the relation:
1
, for 0( )
, for 0k
kk
k−
==
⋅ >
Dh
C A (675)
118
and the transfer function matrix:
1
0
( ) ( ) ( )k
k
z k z zψ
∞− −
=
= ⋅ = + ⋅ − ⋅∑H h D C I A B (676)
For the induction machine model case, there are four real outputs (the stator and rotor fluxes)
and two real inputs (stator voltage). Also, the state variables are the outputs directly. We canestablish the notation:
s
s
s s
r
r
( )
( )( ) ( )
( )
( )
R
I
R
I
n
nn n
n
n
ψ
ψ
ψ
ψ
= =
x y
The state matrix A, the input vector us and the coefficients matrix B, C, and D have differentelement according to the transformation method used for implementing the discrete mathematical
machine model.
I) Bilinear transformation method (Tustin):The state matrix A is calculated as follows:
11 11 11 12 12
11 11 11 12 12
2 2
21 21 22 22 22 22
21 21 22 22 22 22
1
R R I I R I R I R I
R I R R I I R I R I
R R R I R R I I R I I RR I
R I R R R I I R R R I I
c c c c c
c c c c c
c c c c c c
c c c c c c
∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
+ − + − = ⋅ + −+
− − + +
A (677)
where:
[ ]
2 2 2
n s r n r s n s r M r s
n r n s r s r s
(2 ' ) (2 ' ) ( ' )
(2 ' ) (2 ' )
R Ij f r l d f r l d r r l d
j f r l d f r l d
∆ ∆ ∆ ω γ ω ω γ ω ω
ω ω ω γ ω ω
= + ⋅ = + ⋅ ⋅ ⋅ + ⋅ − ⋅ ⋅ + +
+ ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ + ⋅ (678)
[ ]
2 2 2
11 n s r n r s n s r M r s
n r n s r s n r s
12 n s M
21 n r M
2 2 2
22 n s r n r s n s r M r s
n r
(2 ' ) (2 ' ) ( ' )
(2 ' ) (2 ' )
4
4 '
(2 ' ) (2 ' ) ( ' )
c f r l d f r l d r r l d
j f r l d f r l d
c f r l d
c f r l d
c f r l d f r l d r r l d
j
ω γ ω ω γ ω ω
ω ω ω γ ω ω
ω γ
ω
ω γ ω ω γ ω ω
ω ω
= − ⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ + +
+ ⋅ ⋅ ⋅ − ⋅ − ⋅ +
= ⋅
=
= + ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ + +
+ ⋅ ⋅ − ⋅[ ]n s r s n r s(2 ' ) (2 ' )f r l d f r l dω γ ω ω+ ⋅ + ⋅ −
(679)
The input vector is determined with the relation:
s s
s s
( 1) ( )
( 1) ( )
R R
s
I I
U n U n
U n U n
+ − = + −
u (680)
The coefficients matrix are:
11 11 11 11
11 11 11 11
2 221 21
21 21
1
R R I I R I I R
R I I R R R I I
R I R IR I
R I R I
d d d d
d d d d
d d
d d
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆∆ ∆
∆ ∆
+ − − + + = ⋅ +
−
B (681)
where:2
11 n n r s r n
2
21 r M n
(2 ' )
'
d f r l d j
d r l d
ω ω ω ω
ω
= ⋅ + + ⋅
=
119
By identification, we establish the other coefficients matrix:
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
= =
C D (682)
From the transfer function matrix relation, we obtain a 4 X 2 matrix giving the transfer functionfrom both inputs to the four outputs:
11 11 11 12 12
2 2 2 2 2 2 2 2
11 11 11 12 12
2 2 2 2 2 2 2 2
2 2
21 21 22 22 22
2 2 2 2 2 2
1( )
R R I I R I R I R I
R I R I R I R II
R I R R I I R I R I
R I R I R I R I
R R R I R R I I R IR I
R I R I R I
c c c c cz
c c c c cz
zc c c c c
z
ψ
∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
+− − − −
+ + + +
+− −
+ + + += ⋅
+ − ++− − −
+ + +
H
1
22
2 2
21 21 22 22 22 22
2 2 2 2 2 2 2 2
11 11 11 11
11 11 11 11
21 21
21 21
I R
R I
R I R R R I I R R R I I
R I R I R I R I
R R I I R I I R
R I I R R R I I
R I R I
R I R I
c
c c c c c cz
d d d d
d d d d
d d
d d
∆
∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆
∆ ∆
− ⋅
+ − + − − + + + +
+ − − + +⋅
−
(683)
II) Forward-difference method (Euler)The state matrix A is described by the relation:
n s r n s n s M
n s n s r n s M
n r M n r s n r
n r M n n r s
1 ' 0
1 ' 0
' 0 1 '
0 ' 1 '
T r l d T T r l d
T T r l d T r l d
T r l d T r l d T
T r l d T T r l d
ω γ ω ω ω
ω ω ω γ ω
ω γ ω ω ω
ω γ ω ω ω
− ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅
⋅ ⋅ − ⋅ − ⋅
A (684)
The input vector and the corespondent matrix coefficient are:
s
s
s
( )( )
( )
R
I
U nn
U n
=
u
n
n
0
0
0 0
0 0
T
T
ω
ω
=
B (685-686)
The other matrix C and D have the same value as for the bilinear transformation case. It results the
transfer function matrix as follows:1
n s r n s n s M n
n s n s r n s M n
n r M n r s n r
n r M n n r s
1 ' 0 0
1 ' 0 0( )
' 0 1 ' 0 0
0 ' 1 ' 0 0
z T rl d T T rl d T
T z T rl d T rl d Tz
T r l d z T r l d T
T r l d T z T r l d
ψ
ω γ ω ω ω ω
ω ω ω γ ω ω
ω γ ω ω ω
ω γ ω ω ω
−− + ⋅ ⋅ − − ⋅
⋅ − + ⋅ ⋅ − ⋅ = ⋅ − ⋅ ⋅ − + ⋅ − ⋅
− ⋅ ⋅ ⋅ − + ⋅
H
(687)
120
III) The backward-difference methodThe state matrix A and coefficients matrix B can be calculated with the expressions:
11 11 11 11 12 12
11 11 11 11 12 12
2 2
21 21 22 22 22 22
21 21 22 22 22 22
1
R R I I R I I R R I
R I I R R R I I I R
R I R R I I R I I RR I
I R R I I R R R I I
g g g g g g
g g g g g g
g g g g g g
g g g g g g
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
+ − − + + − = + −+
− − + +
A (688)
where:
( ) n r s r n n s M11 11 12
n r M n s r s n21 22 22
n r s r n r n n r s n s M n s M
n r M n
1 '
' 1 '
(1 ' ) ( (1 ' ))
'
R I
R I
R I
R I R I R I
R I
Tr ld j T Trl dg jg gj
T r l d T rl d j Tg g g
Tr ld T j T Tr l d Trl d j Trl d
T r l d j T
ω ωω ω∆ ∆
ω γ ω γ ωω
∆ ω ∆ωω ∆ωω ∆ ω ∆ω ∆ω
∆ω γ ∆ω γ
+ ++ = = − ⋅ = ⋅ + ⋅ ++
+ + + − + −=
⋅ − ⋅
G
r M n s r s n s n n s r' (1 ) ( (1 ' ))R I R Ir l d T rld T j T T rl d∆ ω γ ∆ωω ∆ωω ∆ ω γ
+ ⋅ + + − + ⋅
(689)
and:
11 11 11 11
11 11 11 11
2 221 21
21 21
1
R R I I R I I R
R I I R R R I I
R IR I
I R
h h h h
h h h h
h h
h h
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆∆ ∆
∆ ∆
+ − − + + = ⋅ +
−
B (690)
where:
n r s r n n s M11 11 12
n
n r M n s r s n21 22 22
1 '
' 1 '
R I
R I
Tr l d j T Trl dh jh hT
T r l d T rl d j Th h jh
ω ωω ωω
ω γ ω γ ωω
+ ++ = = ⋅ ⋅ + ⋅ ++
H (691)
As the input vector is s
s
( )
( )
R
s
I
U n
U n
=
u and the coefficients matrix C and D are calculated in a similar
way to the previous cases, it results the following transfer function matrix for two inputs (voltage)and four outputs (fluxes):
11 11 11 11 12 12
2 2 2 2 2 2 2 2
11 11 11 11 12 12
2 2 2 2 2 2 2 2
21 21 22 22 22
2 2 2 2 2 2
( )
R R I I R I I R R I
R I R I R I R I
R I I R R R I I I R
R I R I R I R I
R I R R I I R I
R I R I R I
g g g g g gz
g g g g g gz
zg g g g g
z
ψ
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
+ −− − − −
+ + + +
− + +− − −
+ + + +=
+ −− − − −
+ + +
H
1
22
2 2
21 21 22 22 22 22
2 2 2 2 2 2 2 2
11 11 11 11
11 11 11 11
2 221 21
21 21
1
I R
R I
I R R I I R R R I I
R I R I R I R I
R R I I R I I R
R I I R R R I I
R IR I
I R
g
g g g g g gz
h h h h
h h h h
h h
h h
∆
∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
∆ ∆∆ ∆
∆ ∆
− ⋅
+ − + + − − − + + + +
+ −− + +⋅ ⋅+
−
(692)
121
4.6. Stability analysis
If the resulting discrete-time system is unstable, it can be transformed in a stable one bydecreasing the sampling rate. It is possible to select the the sampling rate such that the discrete-time
system is always stable, assuming that the pole locations in the continuos s domain are known.However, this presumes an increased computing burden, as there are processed more sampling ratesin a time unit. Therefore it is important to analyse the value for the minimum sampling rate and to
determine an optimum value as stated by Franklin et al (1997).The z-domain transfer function is obtained from an analog prototype transfer function by using
the following substitutions according to the approximation method:
T
zs
1−= forward-difference method (Euler)
Tz
zs
⋅−
=1 backward-difference method
1
12
+−
⋅=z
z
Ts biliniar transformation method (Tustin)
Thus the interdependence between an s-domain pole and the corresponding z-domain pole is:
Tsz polepole 1 += forward-difference method (Euler)
Tsz
pole
pole1
1
−= backward-difference method
Ts
Tsz
pole
pole
pole2
2
−
+= biliniar transformation method (Tustin)
For a stable s-domain pole (i.e., α < 0):
βα js +=pole
A z-domain pole is stable if it is located inside the unit circle, i.e., its modulus is less than the unit
(one in relative units):2 2
pole
2 2 2 2
2 2
1 (1 ) ( ) 1
21 2 1
z T T
T T T T
α β
αα α β
α β
< ⇒ + + <
+ + + < ⇒ < −+
forward-difference method (Euler) (693)
pole 2 2
2 2 2 2
2 2
11 1
(1 ) ( )
21 2 1
zT T
T T T T
α β
αα α β
α β
< ⇒ <
− + − + + > ⇒ > +
backward-difference method (694)
2 2
pole 2 2
(2 ) ( )1 1
(2 ) ( )
0
T Tz
T T
T
α β
α β
+ +< ⇒ <
− + >
biliniar transformation method (Tustin) (695)
Expressed in terms of the sampling frequency (fsample =1/T), the above conditions are:
α
βα
2
22
sample
+−>f forward-difference method (Euler) (696)
α
βα
2
22
sample
+<f backward-difference method (697)
0sample >f biliniar transformation method (Tustin) (698)
122
These expressions prove that the sampling rate can always be selected such that the discrete-timesystem is stable if the original continuos-time system is stable as well.
The poles position may be variable due to three important factors:• The simplified assumptions for linearity of the continuous-time model;
• The parameters variation due to environmental effects: temperature, humidity;
• The implementation of the system using fixed-point digital signal processors. The fixed point
determines round errors and scaling or quantification errors.So, even for poles placed inside the unit circle, there is a possibility of unstable operation of the
system. A pole location very near to the unit location can be problematic, mainly in environmentswith short word length of the DSP. Therefore it may be desirable to maximise the distance betweenthe critical pole and the unit circle and set the sampling rate accordingly.
The derivative of the modulus of the pole gives the relations:
2222
22pole
21
222
TTT
TT
dT
zd
βαα
βαα
+++
++= forward-difference method (Euler) (699)
32222
22pole
)21(
222
TTT
TT
dT
zd
βαα
βαα
++−
++−= backward-difference method (700)
[ ]
22
22
22222pole
)()2(
)()2(
4)(4)(4
TT
TT
TT
dT
zd
βα
βα
αβαβαα
+−++
++−+= biliniar transformation method (Tustin) (701)
By setting the derivative equal to zero it is obtained an expression for the optimum sampling
frequency:
β
βα 22
optsample,
+−=f forward-difference method (Euler) (702)
β
βα 22
optsample,
+=f backward-difference method (703)
0optsample, >f biliniar transformation method (Tustin) as the derivative is always positive. (704)
One should notice that the above sampling rate values do not optimise the resemblance between
time-domain or frequency-domain responses of the continuos-time and the discrete-time system.The stable methods (forward-differences, backward-differences and biliniar transformation) map
an s-domain point βα j+ into the z domain as follows:
a) Forward-difference method (Euler)
βα
βα
j
jmz
−−−
=)1(
(705)
where m=1 for the ”optimum” sampling rate; m=2 for the minimum sampling rate
b) Backward-difference method
αβ
αβ
)1( mj
jz
−−+
= (706)
where m=1 for the ”optimum” sampling rate; m=2 for the minimum sampling rate
c) Biliniar transformation method (Tustin)1=z the unit circle, (707)
i.e. the system is stable for any stable s-domain poles (α < 0)
123
5. MATHEMATICAL DISCRETE MODELS FORTHE SINGLE-PHASE INDUCTION MACHINE
5.1. Introduction
Discrete-time computational models have to be derived for an advanced motor control of
squirrel-cage type single-phase induction machine. The real-time vector control analysis for thesingle-phase induction machine can be realised using mathematical discrete models, in a similar
way to the three-phase induction machine. Some differences will appear due to the asymmetry ofthe single-phase induction machine stator configuration.
The same general considerations regarding the discretisation process, valid for the three-phase
machine, apply.The starting point for converting the continuous-time model of the single-phase machine into a
discrete one is the system of voltage and flux linkage equations. However, by comparison with thethree-phase induction machine discretisation two main differences have to be highlighted:
• The reference system is stationary, fixed to the stator;
• The asymmetrical configuration of the stator windings permits only the analysis in two-axis co-
ordinates system. The space vector notation cannot be used.The complete set of equations for an unsymmetrical single-phase induction machine using theuniversal mathematical model, with flux linkage per second units and reactance elements, is as
follows:Stator voltage equations:
qs m qs qs
b
ds a ds ds
b
1
1
du r i
dt
du r i
dt
λω
λω
= + ⋅
= + ⋅
(708-709)
Flux linkages equations:
( )( ) ( )
( )( )( )( ) ( )( )( )
( )( ) ( )( )( )
d
q
d
2 2
ds la d m ds d m ds dR
2
la d m ds d
qs lm q m qs q m qs qR
lm q m qs q
2 2
dR d d r d m dR d m ds dR
2
d d d r d m dR
1 '
1 '
1 '
1 '
' ' 1 ' '
' ' 1 '
'
x k x i k x i i
x k x i
x x i x i i
x x i
k x x i k x i i
k x x i
γ
γ
γ
λ γ γ
γ λ
λ γ γ
γ λ
λ γ γ γ γ
λ γ γ γ
λ
= + − ⋅ ⋅ + ⋅ ⋅ + =
= + − ⋅ ⋅ +
= + − ⋅ ⋅ + ⋅ ⋅ + =
= + − ⋅ ⋅ +
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + =
= + ⋅ ⋅ + − ⋅ ⋅
( )( ) ( )( )( )q
qR q q r q m qR q m qs qR
q q q r q m qR
' 1 ' '
' 1 '
x x i x i i
x x iγ
γ γ γ γ
λ γ γ γ
= ⋅ ⋅ + − ⋅ ⋅ + ⋅ ⋅ + =
= + ⋅ ⋅ + − ⋅ ⋅
(710-713)
Rotor voltage equations (cage rotor case):
rR dR dR qR
b b
2 rR qR qR dR
b b
2
R q(d) r
0 ' ' ' '
10 ' ' ' '
' '
pr i k
pk r i
k
r r
ωλ λ
ω ω
ωλ λ
ω ω
γ
= + + ⋅
= + − ⋅
= ⋅
(714-716)
124
The transformed rotor current d(q)R'i equals:
qrdr
dR qR
d q
''' ; '
iii i
γ γ= = (717)
The electromagnetic torque equation is:
( )
me d qs dR q ds qR qR dR dR qR ds qs qs ds
b b b
mqs dR ds qR2
d qb s r m
1 1 1 1( ' ' ) ' ' ' '
2 2 2
1 1' '
2 '
xP P PT k i i i i k i i i k i
k k
xP
k x x x
γ γ λ λ λ λω ω ω
λ λ λ λγ γω
= ⋅ ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ − = ⋅ − =
= ⋅ ⋅ ⋅ − ⋅ ⋅ −
(718)If the flux linkages are chosen as independent variables, the currents can be deduced from theexpressions:
qs r qR m
q
qs
ds r dR m
d
ds 2
qR s qs m
q
qR
q
dR s ds m
d
dR 2
d
1' '
1' '
1'
'
1'
'
x x
iD
x x
ik D
x x
iD
x x
ik D
λ λγ
λ λγ
λ λγ
γ
λ λγ
γ
− ⋅
=
− ⋅
=
⋅ −
=
−
=
(719-722)
where:
s lm m
r lr m
2
s r m
' '
'
x x x
x x x
D x x x
= +
= +
= −Recapping from the continuous linear mathematical models for the unsymmetrical single-phase
induction machine, there are three specific choices of the turn ratio “γq” and “γd”:
a) The rotor flux is selected as reference (inverse Γ-form model):
m
q
m r
2
m
d q2
m r
'
( ' )
x
x x
k x
k x x
γ
γ γ
=+
= =+
(723)
b) The air-gap flux is selected as reference (T-form model):
q d 1γ γ= = (724)
c) The stator flux is selected as reference (Γ-form model):
m lm
q
m
2
m la
d q2
m
x x
x
k x x
k x
γ
γ γ
+=
+= ≅
(725)
If the inverse of D is noted with d, the voltage equations will be re-written in a new form:
125
qs m r qs m qR qs
b
ads r ds m dR ds2
b
rR s qR m qs qR dR2
b b
rR s dR m ds dR2
b b
1 1' '
1 1' '
1 1 1 10 ' ' ' '
1 1 10 ' ' '
du r x d x d
dt
r du x d x d
dtk
dr x d x d
dt k
dr x d x d k
dt
λ λ λγ ω
λ λ λγ ω
ωλ λ λ λ
γ ω ωγ
ωλ λ λ λ
γ ω ωγ
= ⋅ ⋅ − ⋅ ⋅ + ⋅
= ⋅ ⋅ − ⋅ ⋅ + ⋅
= ⋅ ⋅ − ⋅ + ⋅ − ⋅
= ⋅ ⋅ − ⋅ + ⋅ + ⋅
qR'
(726-729))
For null initial conditions, if the Laplace transformation is applied, it results a matrix system:
qs m r m m qs qs
2 2
ds a r a m ds ds
2
qR R m R s r b qRb
2
dR R m r b R s dR
' 0 / 0
0 ' / 0 /( )
' ' / 0 ' / /( ) ' 0
' 0 ' / ( / ) ' / ' 0
r x d r x d u
r x d k r x d k us
r x d r x d k
r x d k r x d
λ γ λ
λ γ λ
λ γ γ ω ω λω
λ γ ω ω γ λ
− − ⋅ ⋅ = ⋅ + −
−
(730)
or synthetically:
( ) ( ) ( )s s s s= ⋅ + ⋅Y A Y B U (731)
By comparing to the general form of a state variable system:
( ) ( ) ( )
( ) ( ) ( )
s s s s
s s s
⋅ = ⋅ + ⋅
= ⋅ + ⋅
X A X B U
Y C X D U (732)
It can be observed that the state variable vector is identical to the output vector (in this case the fluxlinkages per second vector). The inputs vector is the voltages vector (in this case only the statorvoltages, as the rotor is short-circuited). By identification, we obtain:
m r m m
2 2
a r a m
b 2
R m R s r b
2
R m r b R s
b
' 0 / 0
0 ' / 0 /( )
' / 0 ' / /( )
0 ' / ( / ) ' /
r x d r x d
r x d k r x d k
r x d r x d k
r x d k r x d
γ
γω
γ γ ω ω
γ ω ω γ
ω
− − ⋅ = ⋅ −
− =
A
B
(733-734)
5.2. Bilinear transformation method (Tustin)
Recapping from the mathematical discrete model of three-phase induction machine analysis, the
following relation is the bilinear transformation (Tustin) from continuous to discrete domain:
1
1
1
12−
−
+−
⋅=z
z
Ts
The matrix equation from continuous domain becomes in discrete:
2 1( ) ( ) ( )
1
zz z z
T z
−⋅ ⋅ = ⋅ + ⋅
+Y A Y B U (735)
which gives the relation for the inputs vector Y(z):
( ) ( ) ( ) ( )1 11 1( ) 2 2 ( ) 2 1 ( )z f f z z f z z
− −− −= ⋅ − ⋅ ⋅ + ⋅ ⋅ + ⋅ − ⋅ ⋅ + ⋅Y I A I A Y I A B U (736)
where:
126
b m r b m m
b a r b a m
2 2
b R m b R s r
2
b R m b R s
r 2
2 ' 0 / 0
'0 2 0
2 ' '0 2
' '0 2
f r x d r x d
r x d r x df
k k
f r x d r x df
k
r x d r x dk f
ω ω γ
ω ω
γ
ω ω ω
γ γ
ω ωω
γ γ
± ± ⋅ = ± ± ±
I A
∓
∓
∓ ∓∓
∓
(737)
The expression of the flux linkages per second in the discrete time domain is obtained when usingthe bilinear transformation:
qs qs qs qs
ds ds ds ds
qR qR
dR dR
( ) ( 1) ( 1) ( )
( ) ( 1) ( 1) ( )
' ( ) ' ( 1) 0
' ( ) ' ( 1) 0
n n u n u n
n n u n u n
n n
n n
λ λ
λ λ
λ λ
λ λ
− − + − − + = ⋅ + ⋅ −
−
C D (738)
( ) ( )
( )
1
1
b
2 2
2
f f
fω
−
−
= − ⋅ +
= ⋅ −
C A A
D A (739)
The above relations lead to the mathematical discrete model of the single-phase induction
machine from Fig. 5.1.The electromagnetic torque is computed using the stator and rotor flux linkages as independentvariables:
( )( )m m
e qs dR ds qR qs dR ds qR2d q bb s r m
1 1' ' ' '
2 2'
x x dP PT
kk x x xλ λ λ λ λ λ λ λ
γ γ ω γω
= ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ −
(740)
Considering the continuous time domain and using the Laplace transformation, the rotor angularvelocity value is readily available from the torque expression:
( )r e L
1 1
2
PT T
s Jω
= ⋅ ⋅ − (741)
The currents vector can be determined according to the following matrix equation:
qs r m qs
2 2
ds r m ds
2
qR m s qR
2 2 2
dR m s dR
' 0 / 0
0 ' / 0 /( )
' / 0 / 0 '
' 0 /( ) 0 /( ) '
i x d x d
i x d k x d k
i x d x d
i x d k x d k
γ λ
γ λ
γ γ λ
γ γ λ
− − = ⋅ −
−
(742)
or in a condensed form:( ) ( )n n= ⋅i F λ (743)
127
+
+
+
+
+
+
z-1
z-1
Uds(n)
Uqs(n)
d11
d12
d21
d22
d31
d32
d41
d42
c11
c12
c13
c14
c21
c22
c23
c24
c31
c32
c33
c34
c41
c42
c43
c44
z-1
z-1
z-1
z-1
*
*
+-
+
xM
d
Torque
λ
λ
λ
λ
ds(n)
qR(n)
qs(n)
dR(n)
T
'
'
ω γb
k/( )
e
P2_
Fig.5.1. Complete discrete single-phase induction machine model based on
bilinear transformation method (Tustin)
The analysis of the block diagram given in Fig. 1 shows that for a complete implementation of themodel 27 multipliers, 23 additions operations and 4 delay blocks are necessary. However, thismodel can be simplified depending on the type of implementation. The general structure of the
physical implementation depends on the placement of the blocks and the system frequency responsewill be modified accordingly.
5.3. Forward-differences method (Euler)
The system will become unstable if the sampling period for a discrete model is incorrectlychosen. The transformation from the continuous-time to the discrete-time domain can be made
using the Euler method, which by definition is:
T
nxnx
dt
dx )()1( −+= (744)
where T is the sampling period.
The voltage equations for the single-phase induction machine with cage rotor are transformed asfollows:
128
qs qs
qs m r qs m qR
b
a ds ds
ds r ds m dR2
b
qR qR rR s qR m qs dR2
b b
( 1) ( )1 1( ) ' ( ) ' ( )
( 1) ( )1 1( ) ' ( ) ' ( )
' ( 1) ' ( )1 1 1 10 ' ' ( ) ( ) '
n nu n r x d n x d n
T
r n nu n x d n x d n
Tk
n nr x d n x d n
T k
λ λλ λ
γ ω
λ λλ λ
γ ω
λ λ ωλ λ λ
γ ω ωγ
+ − = ⋅ ⋅ − ⋅ + ⋅
+ − = ⋅ ⋅ − ⋅ + ⋅
+ −
= ⋅ ⋅ − ⋅ + ⋅ − ⋅
dR dR r
R s dR m ds qR2
b b
( )
' ( 1) ' ( )1 1 10 ' ' ( ) ( ) ' ( )
n
n nr x d n x d n k n
T
λ λ ωλ λ λ
γ ω ωγ
+ −= ⋅ ⋅ − ⋅ + ⋅ + ⋅
(745-748)
which gives the matrix relation for the flux linkages per second in discrete representation:
qs m r b m m b qs
2 2
ds a r b a m b ds
2
qR R m b R s b r qR
2
dR R m b r R s b
( 1) ' 1 0 / 0 ( )
( 1) 0 ' / 1 0 /( ) ( )
' ( 1) ' / 0 ' / 1 / ' (
' ( 1) 0 ' / ' / 1
n r x d T r x d T n
n r x d T k r x d T k n
n r x d T r x d T T k
n r x d T k T r x d T
λ ω ω γ λ
λ ω ω γ λ
λ ω γ ω γ ω λ
λ ω γ ω ω γ
+ − + + − + = ⋅ + − +
+ − − +
qs
ds
b
dR
( )
( )
) 0
' ( ) 0
u n
u nT
n
n
ω
λ
+ ⋅
(749)
or in a condensed form:
b( 1) ( ) ( )n n T n+ = ⋅ + ⋅E uλ λ ω (750)
As the flux linkages per second vector is the independent variables vector, the currents aredetermined according to the following matrix equation:
qs r m qs
2 2
ds r m ds
2
qR m s qR
2 2 2
dR m s dR
' 0 / 0
0 ' / 0 /( )
' / 0 / 0 '
' 0 /( ) 0 /( ) '
i x d x d
i x d k x d k
i x d x d
i x d k x d k
γ λ
γ λ
γ γ λ
γ γ λ
− − = ⋅ −
−
(751)
or in a condensed form:( ) ( )n n= ⋅i F λ (752)
Similar to the bilinear transformation case, the electromagnetic torque is computed with the
relation:
( ) ( )m m
e qs dR ds qR qs dR ds qR2d q bb s r m
1 1' ' ' '
2 2'
x x dP PT
kk x x xλ λ λ λ λ λ λ λ
γ γ ω γω
= ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ −
(753)
For the forward-differences method (Euler), the mathematical discrete model of the single-phaseinduction machine is described by the block diagram in Fig. 5.2. This model is characterised by the
following number of blocks requested for implementation: 9 additions, 15 multipliers and 6 delayblocks. By comparison with the previous discretisation method, it is obviously the simpler
implementation structure for the forward-difference method. The computational burden is 1.8 timesgreater with the bilinear transformation than it is with the forward-difference method.
129
Uqs(n)
Uds(n)
z-1
z-1
+
+
+
+
z-1
z-1
z-1
z-1
*
*
-
+
+
(P/2) xM
d/( k
Torque
T
e11
e13
e22
e24
e31
e33
e34
e41
e43
e44
λ
λ
λ
λ
ω
ω
b
b
T
Tqs(n)
dR(n)
ds(n)
qR(n)
e
ω γ )b
Fig.5.2. Complete discrete single-phase induction machine model based onforward-difference method (Euler)
5.4. Backward-differences method
This simply to apply discretisation method allows the implementation of a stable discrete systemif the analogue version in continuous time domain is also stable. Using this method, even unstable
continuous models can be transformed in discrete stable systems. Nevertheless, it must be statedthat due to the distortions in the frequency response of the system, a lower sampling period has to
be used. The flux linkages per second vector can be obtained by applying the relation used for thesame transformation type from the three-phase induction machine to the single-phase inductionmachine:
( )1
11( ) ( ) ( ) ( ) ( ) ( )
zz z z T z z z T z
T
−−−
⋅ = ⋅ + ⋅ ⇒ − ⋅ ⋅ = ⋅ + ⋅ ⋅Y A Y B U I A Y Y B U (754)
or in a matrix form:
( ) ( )
qs qs qs
1 1ds ds ds
b
qR qR
dR dR
( ) ( 1) ( )
( ) ( 1) ( )
' ( ) ' ( 1) 0
' ( ) ' ( 1) 0
n n u n
n n u nT T T
n n
n n
λ λ
λ λω
λ λ
λ λ
− −
− − = − ⋅ ⋅ + − ⋅ ⋅ ⋅ −
−
I A I A (755)
or in a condensed form:( ) ( 1) ( )n n n= ⋅ − + ⋅G H uλ λ (756)
where:
130
( )
1
b m m
b m r
11 12 13 14b a r b a m
2 21 21 22 23 24
31 32 33 34b r m b r s r
2
41 42 43 44
b r m b r sr 2
1 ' 0 0
'0 1 0
' '0 1
' '0 1
Tr x dTr x d
g g g gTr x d Tr x d
g g g gk kT
g g g gTr x d Tr x d T
k g g g g
Tr x d Tr x dk T
ωω
γ
ω ω
γ
ω ω ω
γ γ
ω ωω
γ γ
−
−
+ −
+ − = − ⋅ = =
− + −
− +
G I A
(757)
and:
bTω= ⋅H G (758)
The matrix elements gij , i,j = 1, 2, 3, 4 expressions are not detailed here, due to the space
limitations.Like in the previous cases, the electromagnetic torque represents the output of the discrete
system:
( )( )m m
e qs dR ds qR qs dR ds qR2d q bb s r m
1 1' ' ' '
2 2'
x x dP PT
kk x x xλ λ λ λ λ λ λ λ
γ γ ω γω
= ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ −
(759)
The discrete mathematical model obtained through the backward transformation method isillustrated in Fig. 5.3. A comparative computational burden for different methods of discretisation isgiven in Table 5.I.
TABLE 5.I
Method \ Real operation Additions Multiplications Delays
Forward-difference method (Euler) 9 15 6
Backward-difference method 21 27 4
Bilinear transformation method (Tustin) 23 27 4
The following conclusions can be drawn regarding different implementation options of a discrete
mathematical model for the single-phase induction machine:• the pulse invariance method and backward-difference method determine similar discrete
mathematical model for the machine;• the step invariance method and the forward-difference method determine similar mathematical
discrete model for the machine.• the frequency characteristic of the model differs from one method to another.
• the forward-difference method (Euler) presents the minimum computing time.
• when accuracy is the determining factor, the bilinear transformation method (Tustin) has to be
chosen for the implementation of the mathematical discrete model of the machine.• it is also possible to apply a hybrid approach where the stator and rotor equations are discretised
using different transformation methods.
131
z -1
z -1
z-1
z -1
g12
g11
g13
g14
g21
g22
g23
g24
g31
g32
g33
g34
g41
g42
g43
g44
+
+
++
+
+
+
+
+
+
+
+
+
*
*
(P/2) xM
d
Torque
T
h11
h31
h21
h41
h12
h32
h22
h42
Uqs(n)
Uds(n)
qs(n)
qR(n)
ds(n)
dR(n)
λ
λ
λ
λ
_
+
e
ω γb
k/ ( )
Fig. 5.3. Complete discrete single-phase induction machine model based on
backward-difference method
5.5. Z-domain transfer functions
Recapping some general considerations of linear algebra, a state-variable system using matrixnotation in z-domain is described as:
[ ]
s s s
s s
T
s 1
( 1) ( ) ( )
( ) ( ) ( )
( ) ( ), ..., ( )
s
m
n n n
n n n
n x n x n
+ = ⋅ + ⋅
= ⋅ + ⋅
=
x A x B u
y C x D u
x
(760)
where xs(n) is the vector for the state variables, us (n) and ys(n) are the input and the output vectors
respectively, and A is the state matrix. The impulse response sequence in terms of the state-variabledescription is given by the relation:
1
, for 0( )
, for 0k
kk
k−
==
⋅ >
Dh
C A (761)
and the transfer function matrix:
1
0
( ) ( ) ( )k
k
z k z zλ
∞− −
=
= ⋅ = + ⋅ − ⋅∑H h D C I A B (762)
For the analysed system, i.e. single-phase induction machine, there are four real outputs (thestator and rotor flux linkages per second) and two real inputs (stator voltages). Also, the state
variables are the outputs directly. We can establish the notation:
132
qs
ds
s s
qR
dR
( )
( )( ) ( )
' ( )
' ( )
n
nn n
n
n
λ
λ
λ
λ
= =
x y (763)
The state matrix A, the input vector us and the coefficients matrix B, C, and D have differentelements according to the transformation method used for implementing the discrete mathematical
model for the single-phase induction machine.
I) Bilinear transformation method (Tustin):By comparing the already established relations with the general form of a state-variable description,for this transformation method we have the notation:
The inputs vector:
qs qs
ds ds
s
( ) ( 1)
( ) ( 1)( )
0
0
u n u n
u n u nn
+ − + − =
u (764)
The state matrix:1
b m r b m m
b a r b a m
2 2
b R m b R s r
2
b R m b R s
b 2
b m b m m
b a r b a m
2 2
b R b R
2 ' 0 / 0
'0 2 0
' '0 2
' '0 2
2 ' 0 / 0
'0 2 0
' '0 2
r
m
f r x d r x d
r x d r x df
k k
r x d r x df
k
r x d r x dk f
f r x d r x d
r x d r x df
k k
r x d rf
ω ω γ
ω ω
γ
ω ω ω
γ γ
ω ωω
γ γ
ω ω γ
ω ω
γ
ω ω
γ
−+ −
+ − = ⋅− − + − + +
− +
− +
+ −
A
11 12 13 14
21 22 23 24
s r
31 32 33 342
41 42 43 44
b R m b R sr 2
' '0 2
a a a a
a a a ax d
a a a ak
a a a ar x d r x d
k f
ω
γ
ω ωω
γ γ
=+
+ − −
(765)
The coefficients matrix is:1
b m r b m m
b a r b a m
11 12 13 142 2
21 22 23 24
b b R m b R s r
31 32 33 342
41 42 43 44
b R m b R s
r 2
2 ' 0 / 0
'0 2 0
' '0 2
' '0 2
f r x d r x d
r x d r x db b b bf
k kb b b b
r x d r x df b b b b
kb b b b
r x d r x dk f
ω ω γ
ω ω
γ
ω ω ω ω
γ γ
ω ωω
γ γ
−+ −
+ − = ⋅ =− − +
− + +
B
(766)The elements of the matrix A and B require too much space to be detailed here.
133
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
= =
C D (767)
We obtain the transfer function described by the 4 rows, 4 columns non-zero elements matrix:1
11 12 13 14 11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24 21 22 23 24
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
( ) ( ) ( ) ( )
( ) ( ) ( ) (( )
z a a a a b b b b h z h z h z h z
a z a a a b b b b h z h z h z hz
a a z a a b b b b
a a a z a b b b b
λ
−− − − −
− − − − = ⋅ = − − − − − − − −
H31 32 33 34
41 42 43 44
)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
z
h z h z h z h z
h z h z h z h z
(768)
where the elements hij of the transfer function matrix may be computed through linear algebraiccomputation.
The transfer function for the currents vector considered as output in relation to the voltagesvector as input can be determined as well:
m mr 11 31 r 12 32
r m r m21 41 22 422 2 2 2
s sm m31 11 32 122 2
s sm m41 21 42 222 2 2 2
' ( ) ( ) ' ( ) ( )
' '( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( )
i
x xx h z h z x h z h z
x x x xh z h z h z h z
k k k kz d
x xx xh z h z h z h z
x xx xh z h z h z h z
k k k k
γ γ
γ γ
γ γγ γ
γ γ γ γ
− −
− −= ⋅
− −
− −
H
(769)
The corresponding input vector will be:
1
s
2
u
u
=
u where:
( )( )
1
1 qs
1
2 ds
1
1
u u z
u u z
−
−
= ⋅ +
= ⋅ +
II) Forward-differences method (Euler)The state matrix A is described by the relation:
m r b m m b
2 2
a r b a m b
2
R m b R s b r
2
R m b r R s b
' 1 0 / 0
0 ' / 1 0 /( )
' / 0 ' / 1 /
0 ' / ' / 1
r x d T r x d T
r x d T k r x d T k
r x d T r x d T T k
r x d T k T r x d T
ω ω γ
ω ω γ
ω γ ω γ ω
ω γ ω ω γ
− + − + = − +
− − +
A (770)
The input vector and the corespondent matrix coefficients are:
qs
s
ds
( )( )
( )
u nn
u n
=
u
b
b
0
0
0 0
0 0
T
T
ω
ω
=
B (771-772)
The other matrix C and D have the same value as for the bilinear transformation case. The transfer
function matrix results as follows:
134
1
m r b m m b b
2 2
a r b a m b b
2
R m b R s b r
2R m b r R s b
' 1 0 / 0 0
0 ' / 1 0 /( ) 0( )
' / 0 ' / 1 / 0 0
0 ' / ' / 1 0 0
z r x d T r x d T T
z r x d T k r x d T k Tz
r x d T z r x d T T k
r x d T k T z r x d T
λ
ω ω γ ω
ω ω γ ω
ω γ ω γ ω
ω γ ω ω γ
−+ − −
+ − − = ⋅ − + − −
− + −
H
(773)A detailed expression for each element of the matrix Hλ (z) with 4 rows, 2 columns of non-zero
elements, is beyond the scope of this work. The transfer function valid for considering currentsvector as outputs, when voltages vector represents the inputs, is identical in symbolic form with thebilinear transformation method case.
III) Backward-differences method
Similar to the cases previously analysed, the state matrix A consists the main computational burdenwhen determining the transfer function matrix Hλ. By comparing the established relation for the
discrete domain model through this method, we get the following notation:
1
b m m
b m r
11 12 13 14b a r b a m
2 2
21 22 23 24
31 32 33 34b r m b r s r
2
41 42 43 44
b r m b r sr 2
1 ' 0 0
'0 1 0
' '0 1
' '0 1
Tr x dTr x d
a a a aTr x d Tr x d
a a a ak k
a a a aTr x d Tr x d T
k a a a a
Tr x d Tr x dk T
ωω
γ
ω ω
γ
ω ω ω
γ γ
ω ωω
γ γ
−
+ −
+ − = =
− + −
− +
A
(774)and the input vector us and the coefficients matrix B are:
qs
ds
s
( )
( )( )
0
0
u n
u nn
=
u bTω= ⋅B A (775-776)
The matrix C and D are identical to that ones previously determined.
We obtain the transfer function matrix Hλ as 4 rows, 4 columns matrix with non zero elements:1
11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24
b
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
( )
z a a a a a a a a
a z a a a a a a az T
a a z a a a a a a
a a a a a a a a
λ ω
−− − − −
− − − − = ⋅ ⋅ − − − − − − − −
H (777)
The complete expressions for computing the transfer function matrix elements require intense linearalgebra computation, and therefore it is recommended to determine these values according to the
concrete machine parameters. All the considerations made for the currents vector input case are alsovalid.
135
6. LINEARISATION OF THE INDUCTION MACHINE MATHEMATICALMODEL
6.1. Introduction
An important problem related to the modelling of the induction machine is the non-linearity of
the equations that describe its operation. This phenomenon appears in the voltage equations and theelectromagnetic torque relation as well, due to the products between the state variables. When a
control system with induction machine is designed, it is very useful to linearise the machineequations.
Basically, the linearised equations are obtainable in two ways (Krause et al - 1995), (Rogers -
1965). Firstly, the most used method is the Taylor series expansion of a particular variable (forexample voltage, current, flux linkage, or torque) around the steady-state operating point and then
by neglecting all second-order terms. Alternatively, it is possible to obtain the linearised equationsby expressing all the variables as the sum of their values in the operating point and their incrementalvalue, by neglecting the terms which comprise products of incremental values and by eliminating
the steady-state terms. This is called the small-signal form of the machine equations. The result is adifferential linearised equations set which describes the dynamic behaviour of the machine when
small displacements from the operation point are present. The induction machine can be analysed inthis way as a linear system, and it is possible to apply the basic linear system theory in order tocompute the eigenvalues and to establish transfer functions for use in the design of controls for
these machines.By definition, the initial displacements of the state-variables are considered to be zero. That is,
any machine variable can be written as:
0x x x= + ∆ (778)
where 0x is the value for the variable x in the fixed-operation point, and x∆ is an incremental value
from this value.The equations set for the state-variables are:
( ) ( ) ( ) ( )
( ) ( ) ( )
p t t t t
t t t
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅
x A x B u E z
y C x D u (779-780)
where x is the state-variables vector, u is the inputs vector, z is the perturbations vector, y is the
outputs vector, A the state system matrix, B the inputs matrix, C the outputs matrix, D the input-output matrix, E the perturbations matrix.
If u is set to zero the general solution of the homogenous or force-free linear differential
equation becomes:te= Ax K (781)
where K is a vector formed by an arbitrary set of initial conditions. The exponential teA representsthe unforced response of the system. It is defined as the state transition matrix. Small-signal
stability is assured if all elements of the transition matrix approach zero asymptotically as timeapproaches infinity. For the stability analysis, the characteristic equation of A, defined as follows, is
used:
( )det 0λ− =A I (782)
In the previous equation I is the identity matrix and λ are the roots of the characteristic equation,
referred as eigenvalues, characteristic roots or latent roots.The eigenvalues provide a simple means of predicting the behaviour of an induction machine at
any balanced operating conditions. For a real eigenvalue, the induction machine has an exponential
response, and signifies a movement away from the operating point. A real eigenvalue is positive
136
over the positive-slope region of the torque-speed curve, and becomes negative after maximumsteady-state torque. When the eigenvalues are complex they occur as conjugate pairs and signify a
mode of oscillation of the state variables. Negative real parts correspond to oscillations, whichdecrease exponentially with time, meaning a stable condition, while positive real parts correspond
to an exponential increase with time, an unstable condition.Usually, the starting point for the machine system analysis is from the voltage equations,
combined with the mechanical equation. So, if the input vector is formed by the stator and rotor
voltages plus the load torque, the matrix equation of the system is written as follows:
[ ]p p= ⋅ + ⋅ = + ⋅u L x R x L R x (783)
where [ ]p +L R is denoted as the motional impedance matrix of the machine.
It is possible to obtain the eigenvalues by substituting the differential operator p with the roots
symbol λ, thus the eigen-motional impedance matrix Zm can be formed, and the characteristic
equation will be:
[ ]det ( ) 0m λ =Z (784)
The design and analysis of controls associated with machines (i.e. vector control) require thetransfer function of the actual electrical machines, viewed as a system. Using the previous state-variable set of equations, and substituting the differential operator p with the Laplace operator s, the
input-output transfer function can be expressed as follows, considering no perturbations (z = 0):
[ ] 1( )( )
( )
ss s
s
−= = ⋅ − ⋅ +y
H C I A B Du
(785)
If the inputs vector is zero, the output-perturbation transfer function can be computed as:
[ ] 1( )( )
( )z
ss s
s
−= = ⋅ − ⋅y
H C I A Ez
(786)
6.2. Three-phase induction machine linearisation
In the concrete case of the symmetrical three-phase induction machine, the per unit version isselected for compactness, and the input vector u is chosen to be the voltages vector. Practically, it
contains two terms for the stator voltages, and two terms for the rotor voltages expressed in two-axis coordinate system. However, a single input variables or a linear combination of several inputvariables can also be selected. In this formulation, we can express u as:
iu= ⋅ ∆u G (787)
where G is a column matrix and ∆ui is an input variable such as a linear combination of several
input variables i.e. the amplitude of the terminal voltage.
The perturbations vector is usually given by the variation of the load torque:
LT= ∆z (788)
Recapping from the d-q modelling of the three-phase induction machine, there are fourteenpossible sets of state-variables: currents, flux linkages or mixed flux linkages and currents pairs. Let
the selected pair of incremental state-space variables be denoted as 1 2,x x∆ ∆ and detailed as:
1 1d 1q
2 2d 2q
x x j x
x x j x
∆ = ∆ + ∆
∆ = ∆ + ∆ (789)
The small-signal expression for electromagnetic torque can be deduced from the large-signal torqueexpression:
( )e 1q 2d 1d 2qT K x x x x= ⋅ − (790)
137
where K denotes a constant corresponding to the selected set of state-variables (i.e. mxK
D= for flux
linkages as state-variables).If we apply the small-displacement approach to the state-variables, it results:
( )e 1q0 2d 2d0 1q 1d0 2q 2q0 1dT K x x x x x x x x∆ = ⋅ ∆ + ∆ − ∆ − ∆ (791)
For a complete description of the linearised induction machine model, we have to include also themechanical equation, expressed in small-signal form:
( )r
e L
b
1
2p T T
H
ω
ω
∆= ⋅ ∆ − ∆ (791)
The starting point for the linearised state-variable models is given by voltage equations system
written in stationary reference frame:p= ⋅ + ⋅u L x R x (792)
where: x is the selected set of state-variables and represent also the output of the model, u is thepreviously detailed input vector, L is the coefficients matrix (it can be formed by reactance values,or non-dimensional elements) for multiplying the time derivative of the state-variables, R is the
coefficients matrix (it can be formed by resistances and reactances values or non-dimensionalelements) for multiplying the state-variables.
Finally, the elements for the three-phase induction machine mathematical model in linearisedform will be:
( ) ( ) ( ) ( )
( ) ( ) ( )
p t t t t
t t t
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅
x A x B u E z
y C x D u (793-794)
where:T
1q 1d 2q 2d r( )t x x x x ω = ∆ ∆ ∆ ∆ ∆ x (795)
T
qs ds qr dr L( )t u u u u T = ∆ ∆ ∆ ∆ ∆ u (796)
and:
1
12
21
21 2d0 2q0 1d0 1q0
T
12 dr0 1d0 2d0 qr0 1q0 2q0
(4,1)
;1(1,4) 0
2
0 0 ( , ) ( , ;
H
K x x x x
x x x xλ λ
−
−
= − ⋅
=
= = −
= ⋅ − −
= −
1A M S
B M
L 0R r
M S0l
l
r
(797-798)
5 5
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
= = = =
C I D 0 (799-800)
The perturbations vector can be extracted from the inputs vector as:
L( )t T= ∆z (801)
and:
138
T1
0 0 0 02H
= − E (802)
The eigenvalues can be determined from the characteristic equation. The other way of
determining the roots of this equation is by using the motional matrix impedance:
[ ] [ ]det ( ) det 0m λ λ= + =Z M S (803)
In Fig. 6.1 is illustrated the small-signal equivalent circuit of the three-phase induction machine instationary reference frame:
+
-
+
-
+
-
+
-
- +
+ -
uqs
uds
u'qr
u' dr
i qs
ids
i'qr
i'dr
LM
LM
r s
r s
r'r
r' rLls L'lr
Lls
L'lr
−ω ∆λ dr'
∆∆
∆
∆
∆
∆
∆
∆
0
ω ∆λ0 qr' ∆ω λ
qr0+ −
− +
− −
∆ω λ dr0− '
'
Fig. 6.1. Small-signal stationary reference-frame equivalent circuits for a
three-phase, symmetrical induction machine
6.3. Single-phase induction machine linearisation
In the concrete case of the unsymmetrical single-phase induction machine, the flux linkage persecond version is selected for compactness, and the input vector u is chosen to be the voltagesvector. Comparing with the symmetrical three-phase induction machine case, the same general
considerations are valid. The asymmetrical configuration of this induction machine type can beincluded readily in the state-variable equations form by using the turns' ratio value k.
The small-signal expression for electromagnetic torque can be deduced from the large-signaltorque expression in two forms, depending on the selected set of state-variables:
I) Flux linkages models or currents models:
( )e 1q 2d 1d 2qT K x x x x= ⋅ − (804)
II) Mixed flux linkages and currents models (x1 current space vector, x2 flux linkage space
vector):
e 1q 2d 1d 2q
1T K x x kx x
k
= ⋅ −
(805)
where K denotes a constant corresponding to the selected set of state-variables. For example:
m
b2
xPK
k Dω= ⋅ if the stator and rotor flux linkages are selected as state-variables.
If we apply the small-displacement approach to the state-variables, it results:
I) Flux linkages models or currents models:
139
( )e 1q0 2d 2d0 1q 1d0 2q 2q0 1dT K x x x x x x x x∆ = ⋅ ∆ + ∆ − ∆ − ∆ (806)
II) Mixed flux linkages and currents models (x1 current space vector, x2 flux linkage space
vector):
e 1q0 2d 2d0 1q 1d0 2q 2q0 1d
1 1T K x x x x kx x kx x
k k
∆ = ⋅ ∆ + ∆ − ∆ − ∆
(807)
For a complete description of the linearised induction machine model, we have to include also the
mechanical equation, expressed in small-signal form:
( )r
e L
b2
Pp T T
J
ω
ω
∆= ⋅ ∆ − ∆ (808)
Following are the final elements , which describe the linearised model of the single-phase inductionmachine:
( ) ( ) ( ) ( )
( ) ( ) ( )
p t t t t
t t t
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅
x A x B u E z
y C x D u (809-810)
where:T
1q 1d 2q 2d r( )t x x x x ω = ∆ ∆ ∆ ∆ ∆ x (811)
T
qs ds qr dr L( )t u u u u T = ∆ ∆ ∆ ∆ ∆ u (812)
1
12
21
21 2d0 2q0 1d0 1q0 21 2d0 2qo 1d0 1q0
T
12 dr0 1d0 2d0 qr0 1q0 2q0
(4,1)
;2(1,4) 0
1 1or
10 0 ( , ) ( , ) ;
J
P
K x x x x K x kx kx xk k
x x k x xk
λ λ
−
−
= − ⋅
=
= = −
= ⋅ − − = ⋅ − −
= −
1A M S
B M
L 0R r
M S0l
l l
r
(813-814)
5 5
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
= = = =
C I D 0 (815-816)
The perturbations vector can be extracted from the inputs vector as:
L( )t T= ∆z and
T
0 0 0 02
P
J
= − E (817-818)
The eigenvalues can be computed either from the characteristic equation or by using the motional
impedance matrix of the machine:
[ ] [ ]det ( ) det 0m λ λ= + =Z M S (819)
In Fig. 6.2 is illustrated the small-signal equivalent circuit of the single-phase induction machine instationary reference frame:
140
+
-
+
-
+
-
+
-
- +
+ -
uqs
uds
u'qr
u' dr
i qs
ids
i'qr
i'dr
LM
LM
r m
r a
r'r
r'r
Lla L'lr
Llm
L'lr
−ω ∆λ dr'
∆∆
∆
∆
∆
∆
∆
∆
0
ω ∆λ0 qr' ∆ω λ
qr0+ −
− +
− −
∆ω λ dr0− '
'
k2
k2 k k
k2
(1/k) (1/k)
Fig. 6.2. Small-signal stationary reference-frame equivalent circuits for asingle-phase, unsymmetrical induction machine
141
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