EE 410/510: Electromechanical Systems Chapter 5

EE 410/510: Electromechanical SystemsChapter 5

• Chapter 5. Induction Machines• Fundamental Analysis and Control of Induction u da e ta a ys s a d Co t o o duct o

Motors

• Two phase induction motors

• Lagrange Eqns. (optional)

• Torque speed characteristics and controlTorque speed characteristics and control

• Three phase induction motors in machine variables

• Simulation and Analysis of Induction Motors in MATLAB

Note: We will be skipping multiple sections of this chapter in attempt to provide a clear introduction to the material and allow us to move onto other equally important topics

All figures taken from primary textbook unless otherwise cited.5/21/2010 1

AC Motors: Induction MachinesAC Motors: Induction Machines

• Three main torque and energy conservation mechanisms for electromagnetic motion devicesdevices

– Induction: Electromagnetic torque is the result of time varying electromagnetic fields present due to time varying voltage or motion or the rotor w.r.t. the stator

– Syncrhonous: Torque results because of the interaction of a time varying field generated in the stator windings and a stationary field established by the windings or magnets in the motor

– Variable reluctance: Torque produced to minimize the reluctance of the electromagnetic system. Thus the torque is created in attempt to align the minimum reluctance path of the rotor with the time varying rotating air gap.

Two Phase Induction MotorTwo Phase Induction Motor

For a squirrel – cage motor:

Note: Each of the two sets of inductive circuits are orthogonal. This means that the mutual inductance between lss and lrr is zero.


circuit resistance:

M i fl

circuit resistance:

Magnetic flux through each inductor

Mutual inductance between coupled circuits

Mutual inductance of circuits that are orthogonal in space


C li t f th t ti t i i d tCyclic nature of the rotating system in r on inductance

Magnetic flux through each inductor can then be written asMagnetic flux through each inductor can then be written as


One can then write the equation for magnetic flux in matrix form:

Where each of the inductance terms is composed of a 2x2 matrix:

N is the number of turns in the inductor, and R is the reluctance




AC Motors: Induction MachinesAC Motors: Induction MachinesAn alternate derivation using the relations previously described is provided below

AC Motors: Induction MachinesAC Motors: Induction MachinesThese equations can be rewritten in terms of differentials of current vs. time


AC Motors: Induction MachinesAC Motors: Induction MachinesNow that we have differential equations for current, we can write the mech. ODE

The mechanical rotational velocity and rotation angle is equal to two times electrical angular component divided by the number of magnetic poles in the systemthe number of magnetic poles in the system

AC Motors: Induction MachinesAC Motors: Induction MachinesThese equations can be rewritten in terms of differentials of current vs. time

AC Motors: Induction MachinesAC Motors: Induction MachinesNext we need to develop the equation for the electromagnetic torque in the system

where

The self inductance terms as well as the leakage inductances are not functions of angular displacement. Thus only the mutual inductance term, Lsr , provides electromagnetic torque.

recalling

AC Motors: Induction MachinesAC Motors: Induction MachinesThus one can now write the mechanical equations of motion for a two phase AC motor

Complete set of ODE’s governing a two phase motor

AC Motors: Induction MachinesAC Motors: Induction MachinesGoverning ODEs written in matrix form. Note that these are HIGHLY nonlinear

Remember that for squirrel cage motors (very common devices) that

2 Phase Induction Motor2 Phase Induction Motor

Torque Speed Characteristics and Control f d iof Induction Motors

• As we have observed, the electromagnetic torque generated by induction motors is a f i f b h h d ll di lfunction of both the stator and rotor currents, as well as rotor displacement

• Control of induction motors is achieved by changing the frequency and magnitude of theControl of induction motors is achieved by changing the frequency and magnitude of the voltages supplied to the phase windings.

• Remember to use the voltage rating of the stator windings as the maximum allowable applied voltage, or risk device failure due to resistive heating of the stator windings

O d fi th h l l it f i d ti hi• One defines the synchronous angular velocity of induction machines as

• Note that the electrical angular velocity of an induction machine will always be less than or equal to the synchronous angular velocity. q y g y

– Equal values for electronic and synchronous angular velocities are only achieved under zero load and zero friction conditions


• One can generate a steady state response in terms of electrical angular velocity vs. the l i delectromagnetic torque generated

• Using this approach, industrial standards have been developed to classify induction machines into four distinct classes: A,B,C,D

• Each class is defined by its “slip” which provides an efficiency ratio for the electrical to y p p ymechanical angular velocity


• For steady state operation (and neglecting friction)

• Acceleration of the motor to steady state from zero requires Testart > TL0• The motor then accelerates until rc is reached at the maximum electromechanical torque

• The torque is then decreased back to TL as the speed of the motor increases to r

• Note that e requires the torque on the system to go to zero implying no load or friction forces and that this value is slightly higher than forces, and that this value is slightly higher than r


• Most industrial motors are either type A or type B which have a normal starting torque and a l lilow slip.

• Type C motors have two rotors and thus require higher starting torques. Slip is generally greater in this class as well.

• Type D motors have a high rotor resistance and approximately 10 – 20 times the slip of types yp g pp y p ypA, B, and C.

• Two additional motor classes, E, and F, have

low starting torque, but high leakage inductances

l di t hi h li lG = generatorM tleading to high slip values M = motorB = breaking

2 vs. 3 Phase Induction Motor Torque2 vs. 3 Phase Induction Motor Torque

• We have derived the electromagnetic torque for a two‐phase motor as:

• One can guarantee balanced operation of two‐phase induction motors using either:

• The equation for a Three‐phase motor is:

or

q p

• Effective operation can be achieved using

Motor Control and OperationMotor Control and Operation

• Voltage control is achieved by changing the magnitude of the applied voltages circuits in the statorstator.

• However voltage control reduces the effective start torque of the system and prevents adequate control of type A, B, and C motors

• Thus frequency control is more widely used for standard motor operation. In frequency control, the voltage is stepped between zero and the phase voltage desired. The frequency (and effective duty cycle) of the system is controlled using the concept of slip and synchronous angular velocity, where the control frequency is given by =2f.


• One can further minimize losses by regulating the applied voltage as the frequency is changedchanged.

• As one can see in the figures below, the voltage is decreased linearly while reducing the frequency

• A constant volts per hertz control can is achieved by maintaining the following experimental relationship

• Or one can vary the system performance slightly by relating


• Control of both the voltage and the frequency provide a multi‐variable control scheme in whichwhich

to further improve performance optimization

Example 1: Torque Speed CharacteristicsExample 1: Torque Speed Characteristics

• Calculate the torque‐speed characteristic for a 4 pole induction motor.

24 5 hrs = 24.5 ohmXs = 10 ohmX’r = 40 ohmXm = 25 ohm

110 V

fmax = 60 Hzf1 = 20 Hzf2 = 40 Hzf3 = 60 Hz

f = 60 Hz

f = 40 Hzumax = 110 V 3

X = magnetizing reactance f = 20 Hz

f 40 Hz

Example 2: Analysis the Performance of a h d iTwo‐Phase Induction Motor

• Assume a motor operated at 115 Volts, and 60Hz has a 4 pole design. Use the differential equations previously derived to describe the dynamics of the motorequations previously derived to describe the dynamics of the motor.

• By adding the concept of torque‐speed characteristic performance, describe the following A and D class motors

• Using:

• We find that:

• Zero load and applied load conditions are examined using TL =0, and TL = 5 N‐m.

• Transient dynamics are developed using the following Simulink program based on our i l d l d i f ODE d d ib hpreviously developed series of ODEs used to describe the system.


• Control of both the voltage and the frequency provide a multi‐variable control scheme in whichwhich

to further improve performance optimization

3 Phase Induction Motor Equations3 Phase Induction Motor Equations

3 Phase Induction Motor Example3 Phase Induction Motor Example

SIMULINK DiscussionsSIMULINK Discussions

• 20‐30 minutes20 30 minutes

Pulse Width ModulationPulse Width Modulation• Pulse width modulation is the primary means for

induction motor control.

• Uses an inverter to supply fixed step voltages basedUses an inverter to supply fixed step voltages based over the current motor phases

• The additional applied voltage sums with the individual 3‐phase voltages to provide discrete Fourier transforms that provide inductor with the phasetransforms that provide inductor with the phase required for operation

• Positive or negative voltage outputs from the switch inverter allow for direct phase matched control of the f db k i h h b d i hfeedback current in the system, thereby reducing the time required to achieve the ideal operating condition

3-phase hard switch inverter

Hard vs. Soft Switch InverterHard vs. Soft Switch Inverter

• Hard switching provides digital l itype pulsing

• Soft witching using a capacitivly coupled diode yields rounds the top of the pulse to provide a smoother transfer function.

3-phase soft switch inverter

Six Step InvertersSix Step Inverters

• Square wave voltage source i f dinverters are often used

• Known as six step inverters

• Each switch is closed for 180o of the pulse and closed for the other phalf

• Each set of switches is offset by 60o to provide three phase operationoperation

Voltage Control

Thyristor Inversion: Current ControlThyristor Inversion: Current Control

• Current driven inverter with rectified l h i h ll d b hvoltage source that is then polled to both

positive and negative terms to provide three current values: ‐i, 0, I

• Each value is 60 degrees out of phase providing rotation dictating the rotation of the motor

• Application of current control from an internal voltage source helps modulate g pthe phase of the system to control the slip

• Result: the DC linked inductor smoothens the current out in the systemthe current out in the system

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EE 410/510: Electromechanical Systems Chapter 5