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Journal of ELECTRICAL ENGINEERING, VOL. 58, NO. 2, 2007, 7984

PERMANENT MAGNETS SYNCHRONOUSMOTOR CONTROL THEORY

Marek Stulrajter Valeria Hrabovcova Marek Franko

The paper presents different approaches of PMSM control strategies. Scalar control and vector control were chosen aspossible methods for AC motors control. Theoretical background briefly describes the properties of these control techniques.Several advantages and disadvantages are highlighted and confirmed by simulations.

K e y w o r d s: AC motor, PMSM, vector control, scalar control

1 INTRODUCTION

One way of controlling AC motors for variable speedapplications is through the open loop scalar control,which represents the most popular control strategy ofsquirrel cage AC motors. Nowadays it is used in appli-cations, where information about the angular speed neednot be known. It is suitable for a wide range of drives asit ensures robustness at the cost of reduced dynamic per-formance. Typical applications are pump and fan drivesand low-cost drives. The main idea of this method is vari-ation of the supply voltage frequency abstractedly fromthe shaft response (position, angular speed). The mag-nitude of the supply voltage is changed according to the

frequency in a constant ratio. Then the motor is in thecondition, where the magnetic flux represents the nominalvalue and the motor is neither overexcited nor underex-cited. The most advantage of this simple method is run-

ning in a sensorless mode because the control algorithmdoes not need information about the angular speed or ac-tual rotor position. On the contrary, the big disadvantagesare the speed dependence on the external load torque,mainly for IM, and reduced dynamic performances.

In the last years the dynamic performances are de-manded as high as possible. AC machines have found verypretentious applications in automotive industry, where

dynamic responses are important. Vector control (FieldOriented Control) of AC machines, as a novel approach inelectrical drives, provides very good performance in com-parison with the scalar control. Vector control eliminatesalmost all the disadvantages of constant V/f control. Themain idea of this method is based on controlling the mag-nitudes and angles of the space vectors. Different kinds ofvector controls cover a wide range of industrial applica-tions. When high performance, regarding torque dynam-ics is required, a technique which controls the torque in

a direct manner is an option that must be taken into ac-count. This type of control is known as Direct TorqueControl.

2 SCALAR CONTROL V/F OF PMSM

Constant volt per hertz control in an open loop is usedmore often in the squirrel cage IM applications. Using thistechnique for synchronous motors with permanent mag-nets offers a big advantage of sensorless control. Informa-tion about the angular speed can be estimated indirectlyfrom the frequency of the supply voltage. The angularspeed calculated from the supply voltage frequency ac-cording to (1) can be considered as the value of the rotor

angular speed if the external load torque is not higherthan the breakdown torque.

The mechanical synchronous angular speed s is pro-portional to the frequency fs of the supply voltage

s =2fs

p, (1)

where p is the number of pole pairs. The RMS value ofthe induced voltage of AC motors is given as

Ef =

2fsNskw . (2)

By neglecting the stator resistive voltage drop and assum-ing steady state conditions, the stator voltage is identicalto the induced one and the expression of magnetic fluxcan be written as

=Vsph

2fsNskw= c

Vsphfs

. (3)

To maintain the stator flux constant at its nominal valuein the base speed range, the voltage-to-frequency ratio is

University of Zilina, Faculty of Electrical Engineering, Department of Power Electrical Systems, Velky diel, 010 26 Zilina, Slovakia.

E-mail: [email protected], [email protected]

EVPU a.s., Trencianska 19, 018 51 Nova Dubnica. E-mail: [email protected]

ISSN 1335-3632 c2007 FEI STU

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80 M. Stulrajter V. Hrabovcova M. Franko: PERMANENT MAGNETS SYNCHRONOUS MOTOR CONTROL THEORY

kept constant, hence the name V/f control. If the ratiois different from the nominal one, the motor will becomeoverexcited or underexcited. The first case happens whenthe frequency value is lower than the nominal one andthe voltage is kept constant or if the voltage is higherthan that of the constant ratio V /f. This condition is

called overexcitation, which means that the magnetizingflux is higher than its nominal value. An increase of themagnetizing flux leads to a rise of the magnetizing cur-rent. In this case the hysteresis and eddy current lossesare not negligible. The second case represents underex-citation. The motor becomes underexcited because thevoltage is kept constant and the value of stator frequencyis higher than the nominal one. Scalar control of the syn-chronous motor can also be demonstrated via the torqueequation of SM, similar to that of an induction motor.The electromagnetic torque of the synchronous motor,when the stator resistance Rs is not negligible, is given

asTe = m

s

VsphEfZd

sin

L E

2

fRs

Zd

, (4)

where Vsph is the stator terminal phase voltage, Ef isthe induced voltage generated by the field current exci-

tation, Rs is the stator resistance, = arcsinRsZd

, L is

the load angle, which is negative, if it is motoring opera-tion, and positive if it is generating one. The synchronousreactance is Xd = 2fsLd and synchronous impedance

Zd =

R2s + X2

d . The second part of (4) in the brack-ets is cancelled, if Rs is neglected and Zd = Xd . Theinduced voltage of PMSM Ef is generated by the flux of

permanent magnets, hence Ef EPM .

Ef = EPM =2fs

2PMNskws = 2fsPM . (5a)

After neglecting the stator resistance and rewriting thereactance and angular speed as a function of frequency,it is possible to rewrite the maximal torque as

Tm =3p

2fs

VsphEPM2fsLd

=3p

2fs

Vsph2fsPM2fsLd

. (5b)

All constant values in (5b) can be replaced with con-

stant C and then the modified expression for the maximaltorque is given as

Tm = CVsph

fs, (6)

where C = 3pPM2Ld

because the flux of PM is predicted as

PM = const. Taking into account (6), the torque will beconstant in a wide speed range up to the nominal speed ifthe ratio of stator voltage and frequency is kept constant:

Vsphfs

= const. (7)

Expression (5b) and consequently expression (7) are validonly if Rs can be neglected in comparison with the syn-chronous reactance Xd . This is valid for big machines

around the rated frequency. Since Xd is proportional tothe stator frequency, resistance Rs cannot be neglected inthe range of low frequencies (less than 10,Hz). Therefore,keeping the constant ratio Vsph/fs is not enough duringthe full speed range. In the range of low frequencies, thedecreasing of the voltage must be slower. This can be

achieved by keeping the voltage at a constant value inthe region of the low frequencies, as it is shown in Fig. 1.

Such a control strategy can be represented by the blockdiagram illustrated in Fig. 2 and simulation results shownin Fig. 3. The motor parameters are in Table 1.

The proposed approach was verified by a simulationtest. Via simulation results (Fig. 3) it is possible to an-alyze the applied constant volt per hertz control for sur-face mounted PMSM. It can be seen that the motor doesnot work under nominal conditions of the magnetic fluxduring the whole operating cycle. During starting up itis possible to observe big overexcitation (Fig. 3b), which

would cause saturation of the magnetic circuit, an in-crease of the magnetizing current (see Fig. 3c, id currentcomponent) and, hence, motor losses. Motor operationwith the stator flux kept at the nominal value occursonly in no load steady state condition where the mag-netizing current is zero. The motor, loaded by nominaltorque TL = 0.38 Nm becomes underexcited, Fig. 3b. Inorder to show the waveforms of the magnetic flux moretransparently, the frame was chosen instead of dqframe, Fig. 3b. The frame in all figures represents theframe coupled with the stator, -axis being aligned withthe axis of a-phase.

In fact, V /f control represents the most common con-trol strategy for asynchronous motors, as it was men-tioned in the first chapter. In the case of PMSM, the mainflux is produced by the permanent magnets PM on therotor. The current through the stator coils produces a fluxcalled the armature reaction flux a , because the statorof SM represents the armature. Then the total stator fluxis given as a vector sum of both of these fluxes, Fig. 4.

The armature reaction flux a = LaIs , where La isdefined as the stator inductance, can be resolved to thedq frame as follows:

ad = LdId = LdIs cos ,aq = LqIq = LqIs sin .

(8a)

Then the total flux in dq frame is given as:

d = ad + PM = LdId + PM ,

q = aq = LqIq .(8b)

The mutual relationship between total flux s and otherfluxes illustrated in Fig. 4 can be described as follows:

2

s =

LqIs sin 2

+

LdIs cos + PM2

. (9)

Angle , called the torque angle, is the angle between thevector of stator current Is and the direct axis which is

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Journal of ELECTRICAL ENGINEERING VOL. 58, NO. 2, 2007 81

Fig. 1. Stator voltage Vsph versus stator frequency fs .

Fig. 2. Block diagram of Volt per Hertz control of AC machines.

Fig. 3. Scalar control V/f of PMSM a) Stator currents in -frame, b) Stator fluxes in -frame, c) Stator currents in dq -frame.

Fig. 4. Phasor diagram illustrating relation between stator fluxesand stator current.

Fig. 5. Basic phasor relationships for PMSM in synchronously ro-tating frame using d -axis alignment with rotor magnet flux PM .

aligned with the vector of the permanent magnet flux on

the rotor. As it can be seen in Fig. 4, the magnitude of

the total stator flux depends on the location and magni-

tude of the stator current phasor. For example, the motor

excited by the nominal flux during no-load steady state

operation produces a small no-load stator current. Then

the armature reaction fluxes in dq -frame are negligible. In

this case the magnitudes of the total stator flux and the

flux produced by the permanent magnets are considered

to be same:

PM s . (10)

Applying V /f control to PMSM does not allow to controlthe stator current, which causes variation of the magni-

tude of the main stator flux according to equation (9).

Then, the total flux of SMPM is much higher than the

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Fig. 6. Different locations of the stator current vector. Fig. 7. Phasor diagram of vector controlled PM motor for maxi-mum torque per stator current.

Fig. 8. Typical cascaded control structure for PMSM drives using closed speed control loop with inner current loop.

nominal one, if angle is less than 90 degrees (Fig. 3b-start up and Fig. 5a), and on the contrary, if this angle

is more than 90 degrees, the total flux is lower than itsnominal value (Fig. 3b-loading and Fig. 5b).

In case the vector of the stator current is located in

a manner that its d-component is in the same direction

and the same sense as flux of PM, the total flux in d-axisis given as a sum of the permanent magnet flux PMand of the contribution of the armature reaction flux ind- axis ad = LdId . Flux s represents the total flux inthe machine and the induced voltage Es generated by thisflux is higher than the induced voltage EPM generated bythe magnet, Fig. 5a. This case corresponds to the scalar

control during starting up, see Fig. 3. On the contrary, if

the vector of the stator current in d-axis is in oppositesense to the flux of PM, the total flux in d-axis is givenas a difference between the permanent magnet flux PMand the contribution of the armature flux in d-axis ad .The induced voltage Es generated by the main flux s islower than the induced voltage generated by permanent

magnets EPM , Fig. 5b.

3 VECTOR CONTROL

The vector control of PMSM allows separate closedloop control of both the flux and torque, hence, achieving

a similar control structure to that of a separately excited

DC machine.

Electromagnetic torque expressed in dq componentsof the currents is given as

Te =3p

2

PMiq + idiq(Ld Lq)

, (11)

where Ld and Lq are the d- and q - axis synchronous in-ductances. Each of the two terms in equation (11) has auseful physical interpretation. The first magnet torqueterm is independent of id but is directly proportional tothe stator current component iq . In contrast, the sec-ond reluctance torque term is proportional to the idiqcurrent component product and to the difference of theinductance values (Ld - Lq ).

As equation (11) shows, the torque depends on the ro-tor type and its inductances Ld , Lq and on permanentmagnets mounted on the rotor. The non-salient PMSMhave surface mounted magnets on the rotor and the reluc-tance term disappears since Lq equals Ld . On the con-trary, the electromagnetic torque is more dominated bythe reluctance component, when permanent magnets areinterior mounted and rotors saliency causes a differencein Lq and Ld .

It follows from equation (11) that if a smooth air gapis assumed (Ld = Lq ) and if on the rotor only the per-manent magnets represent the source of excitation, the

electromagnetic torque can be expressed as

Te =3p

2PMiq . (12)

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Journal of ELECTRICAL ENGINEERING VOL. 58, NO. 2, 2007 83

Fig. 9. Vector control of PMSM, a) stator currents in -frame,b) stator fluxes in -frame, c) stator currents in dq -frame.

Table 1. The motor parameters

Rated Motor Voltage 15 VRated Current 8.4 ARated speed 3000 min1

Terminal Resistance R2ph 0.33 Terminal Inductance L2ph 0.30 mH

Pole Pairs 3Rated Torque 0.38 NmFrequency 150 HzFlux of PM 0.0096 WbRotor Inertia 0.08 kgcm2

As it can be seen from equation (11), an optimally effi-cient operation is achieved by stator current control whichensures that the stator current phasor contains only a

quadrature axis component iq , see Figs. 6 and 7. This isanalogous to the separately excited DC machine, wherethis is achieved by consecutive switching of the armature

coil through the commutator [3].

General expression of the torque can be written as

Te =3p

2PM|is| sin , (13)

where angle is the torque angle. The flux of permanent

magnet PM has been assumed to be constant. In Fig. 6it is shown how the current component Iq is changingwith the change of Is position, which results in a changein angle .

For a given value of stator current, the maximumtorque can be obtained with an angle of = 90 . Forthis condition, the electrical variables are displayed in thephasor diagram of Fiq. 7. This mode of operation givesthe maximum torque per ampere of stator current andtherefore a high efficiency.

The control structures for a wide variety of PMSMdrives have the same characteristics. The most popular

control technique is the cascaded one using classical tech-niques to achieve torque, speed and position control inPMSM motion control system.

Speed control can be conveniently achieved in thePMSM drive by closing a speed feedback loop around theinner torque/current loop as illustrated in Fig. 8. Thetorque request is generated by the speed controller in de-pendence on the speed error. By keeping Id current frozenat zero, the phasor of stator current will be placed just inthe quadrature axis (see Fig. 7) and the maximal drivingtorque will be achieved. This can be obtained by settingthe demanded current Id to zero.

As it can be seen from Fig. 9, vector control providesmuch better performance in comparison with scalar con-trol. The motor is in a condition, where the magnetizingflux is kept in its nominal value during the whole oper-ation cycle as it can be observed in Figs. 9b,c. The d-component of stator current is kept at zero all the timeand the magnitudes of the total stator flux and the fluxof the permanent magnet are equal, Fig. 9b. Using sucha control strategy avoids the motor to become under- orover-excited.

4 COMPARISON OF PROPOSED

PMSM CONTROL STRATEGIES

By means of Fig. 10 the dynamic performances of bothcontrol strategies can be compared. Vector control gener-ally provides a possibility to choose different starting-uppaths. In order to compare the vector control with thescalar control in open loop, start-up with uniform accel-eration was used. This can be also seen in Fig. 9a,c, wherethe currents reach the same magnitude during the wholestarting-up. Referring to Fig. 10, at t = 0.3 s the mo-tor should reach the nominal angular speed. As it can beseen from the detailed picture, by using the scalar control

a certain overshoot occurs with settling time T sc1. In thecase of vector control only a negligible overshot occurs.At time t = 0.4 s, the motor gets loaded by the nominalload torque and again certain oscillation on the rotor can

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Fig. 10. Angular speed of PMSM controlled by Vector control andScalar control strategies.

be observed. Comparing the settling time of scalar con-trol T sc2 with the settling time of vector control T vc2shows that vector control provides better dynamic per-

formances. This is caused by the absence of the speedfeedback in the scalar control strategy.

It can be seen from the details in Fig. 10, that the useof angular feedback and of the speed controller in vectorcontrol strategy (see Fig. 8) avoid any overshoots.

5 CONCLUSIONS

Nowadays, many research teams try to develop thesensorless control of PMSM. In the paper there was shownthat it is possible to use a scalar control which brings

an advantage of sensorless control. On the other hand,the fact that the scalar control in an open loop doesnot provide a possibility to control the currents causes

different behaviours of PMSM during various operationcycles like the start-up and loading.

The vector control as the most common method ofSM control was used to compare the performances ofthe proposed scalar control approach. The paper confirmsnot only better dynamics performances of vector control(Fig. 10) but also shows that AC motors with PM must becontrolled by the vector control to avoid oversaturation ofthe magnetic circuit, increasing the magnetizing currentand, hence, losses.

Acknoweledgments

This work has been supported by projects VEGANo. 1/2052/05, VEGA No. 1/3086/06

and GACR 102/06/1320.

References

[1] HRABOVCOVA, V.RAFAJDUS, P.FRANKO, M.HU-

DAK, P. : Measurements and Modeling of Electrical Machines,

EDIS publisher of University of Zilina, 2004. (in Slovak)

[2] FRANKO, M.HRABOVCOVA, V.HUDAK, P.: Measure-ment and Simulation of Permanent Magnet Synchronous Ma-chines, XI. International Symposium on Electric Machinery inPrague, ISEM 2003, 1012 September 2003.

[3] BOSE, B. K.: Power Electronics and Variable Frequency Drives.Technology and Applications, Institute of Electrical and Elec-tronics Engineers, Inc, New York, 1997.

[4] VAS, P. : Sensorless Vector and Direct Torque Control, Pub-lished in the United States by OxfordUniversity, 1998.

Received 17 June 2006

Valeria Hrabovcova is professor of electrical engineering

at the Faculty of Electrical Engineering, University of Zilina.Her main working area is investigation of electrical machineperformances.

Marek Stulrajter was born in 1980. In 2003 he graduated(MSc) from the Department of Power Electrical system, Fac-ulty of Electrical Engeneering, University. In 2006 he gainedhis (PhD) at the same department. Now he is with the samefaculty as a researcher.

Marek Franko was born in 1977. In 2001 he graduated(MSc) from the Department of Power Electrical Systems, Fac-

ulty of Electrical Engineering, University of Zilina. Now he iswith the Electrotechnical Research and Design Company inNova Dubnica.

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