t

50 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS December 1984

Torque and speed harmonic analysisof a PWMCSI-fed induction motor drive

SYNOPSIS

P Pillay* BEng, MScEng, GradSA1EEEJ Odendal* PrEng, MSc(Eng), FSA1EE, M1EE, M1EEE

RG Harley* PrEng, MScEng, PhD, MSA1EE, M1EE, M1EEE

This paper analyses the transient and steady state behaviour of a pulse width modulated current-source-inverter fed induction motor drive. It pays par-ticular attention to the relationship between the harmonic components of the current, torque and speed, in order to provide strategies to minimize certaintorque and speed harmonics. It shows that pulse width modulation slows down the dynamic response time of the motor.

Indexing terms: PWM inverter, induction motor, analysis, current source inverter.

Listof principal symbolsCSI current source inverterdB decibelsFFT Fast Fourier TransformH inertia constant = Jw/2 J Wb!2IAL stator current of line AIBL stator current of line BICL stator current of line CIDc direct currentidl stator d-axis currentid2 rotor d-axis currentiql stator q-axis currentiq2 rotor q-axis currentJ inertia of motorLll stator self inductanceL22 rotor self inductanceLm mutual inductance[M] matrix M[M]T transpose of a matrix M[M]-I inverse of a matrix Mp derivative operator d/dtp. u. per unitPWM pulse width modulationR, stator phase resistanceR2 rotor phase resistanceTe electrical torqueTL load torquevdl stator d-axis voltagevd2 rotor d-axis voltageVql stator q-axis voltagevq2 rotor q-axis voltageCt angle of pulse width modulationA 21T/3rad.Wb base speed in rade/sWi fundamental frequency of inverter rade/sw, rotor speed in rade/se arbitrary angle = wt

I Introduction

There is an increasing need for variable speed drives,and with the advent of power semi-conductors the vari-

* The authors are with the Department of Electrical Engineering.University of Natal, King George V Avenue, Durban, 4001.

able frequency fed induction motor drive is becoming aserious competitor for the conventional dc motor drive.Frequency conversion usually takes place by first rectify-ing the fixed ac mains and then inverting to a new vari-able frequency. The dc link between rectifier and in-verter can be operated with the link voltage heldconstant, or with the link current held constant; the lat-ter method is illustrated in Fig 1and is usually referred toas a constant link current or current source inverter(CSI) .

w~IDC

B

CONTROL CONTROL

CI RCUITRY CI RCUITRY

Fig I Current source inverter fed induction motor

This paper investigates the behaviour of an inductionmotor supplied from a CSI which switches the dc linkcurrent sequentially through the stator phase windings.The stator phase current therefore consists of a series ofblocks or pulses per half cycle which represent a funda-mental sine wave plus numerous harmonics. As a conse-quence, the induced rotor current also contains numer-ous harmonics. All these current harmonics give rise tothe two types of parasitic torques,(!) namely those whichhave constant values and those which pulsate as a func-tion of time; both these types are described below:

(a) A steady parasitic torque is produced when a cur-rent harmonic in the rotor interacts with an air gap fluxharmonic of the same order. It has a non-zero constantaverage value and can either add to or subtract from thesteady torque (called the fundamental torque) producedby the fundamental current and fundamental air gap fluxto yield a net steady electrical torque. In a three phasebridge rectifier or inverter, all even harmonics of current

f;j.

1i

December /984 THE TRANSACfIONS OF THE SA INSTITUTE OF ELECfRICAL ENGINEERS 51

anu those divisible by three are absent on the ac side.hence only the 5th, 7th, 11th, 13th etc are present. Theconstant parasitic torque produced by the 5th harmonicrotor current interacting with the 5th harmonic air gapflux is called the 5th torque harmonic (although its valueis constant) and it acts in the opposite direction to thefundamental torque; on the other hand, the constantparasitic 7th torque harmonic acts in the same directionas the fundamental torque. All the group (a) parasitictorques have an odd order, with the 5th, 11th, 17th. 23rdetc components aiding the fundamental and the 7th,13th, 19th, 25th etc components opposing the funda-mental. The net effect of all these steady aiding and op-posing parasitic torque components is a slight reductionin the value of the net steady state torque of the motor.The effect of the group (a) parasitic torques is thereforeneglected for the purpose of this paper.

(b) A parasitic torque which pulsates as a function oftime around a zero value (also referred to as a torqueharmonic) is produced when a harmonic rotor currentinteracts with a harmonic air gap flux (including the fun-damental) of a different order. The average value of sucha pulsating parasitic (PP) torque is therefore zero. Theinteraction with the fundamental air gap flux producesthe largest pulsating parasitic (PP) torque componentand is therefore more important to consider than theother PP torque components produced by interactionwith the higher order harmonic fluxes.

The 5th and 7th current harmonics, when interactingwith the fundamental air gap flux, produce a 6th har-monic PP torque; in a similar way the lith and 13th cur-rent harmonics produce a 12th harmonic PP torque etc.The group (b) parasitic torques therefore pulsate at the6th, 12th, 18th, 24th etc harmonic frequencies. Howevertheir magnitudes are 1/6th, 1/12th, 1/18th, 1/24th etc re-spectively of the steady fundamental torque for a motorsupplied by rectangular blocks of current; in that casethe 6th harmonic PP torque is usually the largest andmost troublesome of all, especially at low speeds whenthe torque pulsation can be in the 30-100 Hz rangewhere shaft mechanical resonances often occur. Sus-tained operation at such resonant frequencies could re-sult in accelerated wear of gear teeth and/or reducedshaft life, thus leading to an early fatigue failure.

Various methods have been proposed (2)in an attemptto eliminate these group (b) parasitic pulsating torques:

The first method is to use a twelve pulse inverterwhich creates no 6th harmonic PP torque but only the12th and higher PP torque harmonics. However thismethod is expensive since twelve thyristors or otherswitching devices are required in the inverter instead ofonly six, in addition to a transformer with two phaseshifted secondary windings (delta and star).

The second method is to regulate the instantaneousvalue of link current(3.4)by using a feedback control sys-tem. Fig 2 (a) shows the assumed rectangular blocks ofstator current from a current source inverter, and typicalaccompanying parasitic torque pulsations appear in Fig2 (b). For the current in Fig 2 (a) to have flat tops, thelink current IDC in Fig 1should have no ripple. However,Lipo (3)suggests that the parasitic torque pulsation can

be reduced by regulating (the link current and therefore)the line current to contain some ripple as shown in Fig 2(c); Fig 2 (d) illustrates that the resulting torque pulsa-tions are reduced to impulses of extremely short dura-tion and in practice the rotor and load inertias effectivelydampen out the effects of these impulses upon thespeed.

The third method is to switch the inverter elements ina pulse width modulated (PWM) fashion and produce aPWM train of current pulses in an effort to reduce thecurrent harmonics and hence the torque harmonics.

Previous authors have considered these parasitic pul-sating torques as a steady state problem only, and noattention has been given to the transient behaviour of aPWM CSI -fed induction motor. Of the steadv state solu-tions, Lienau (5)proposes that for a three pulse PWMwave (shown in Fig 3), the width of the side pulse shouldbe a = 12°, but Chin and Tomita (2)state that ashould be7,5°. This paper calculates the transient response tolarge disturbances and then uses the Fast Fourier Trans-form (FFT) of the time domain torque and speedwaveforms to further investigate the relationship be-tween a and the amplitudes of their individual harmoniccomponents. It also investigates the interrelationshipbetween current, torque and speed harmonics.

(a) Une current when link current has no ripple

(b) Parasitic torque pulsations for motor line current waveform of (a)

IKt

I eSwtK

011

6"

11

2'511 11"6

(c) Une current when link current is regulated to have ripple

(d) Parasitic torque pulsations for motor line current waveform of (c)

Fig 2 Waveforms of current and torque

I. . .

11 11 511 11 711 311 1171" 21106" 2 6" h ""2 """6

52 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS December /984

0T[

6T[

2"

centre pulse side pulse

Fig 3 Pulse width modulated line current waveform

2 Theory

2.1 Analysisof the PWMcurrent waveform appliedto themotor

Fig 2 (a) shows 120° rectangular blocks of stator linecurrent from a typical current source inverter. WhenPWM is applied to the inverter the line currentwaveform changes to that shown in Fig 3 in which a canbe varied between 0° and 30°; for a = 0° the waveformdegenerates to the 120°blocks of Fig 2 (a).

The waveform of Fig 3 can be represented as the fol-lowing Fourier series (Appendix A):

x

iAL(Wt) = L [2I/mT][cos(n(30°- a» - cos(n300)n=1

- cos(n(150° - a» + cos(n(30° + a»- cos(n(150° + a» + cos(n1500)] sin(nwt) (1)

Expressions for iBLand in are obtained by replacing nwtin Eqn (1) by (nwt - J..)and (nwt + J..)respectively.

2.2 Two axis analysisof the current source inverter fedinductionmotor

The following analysis of the CSI-fed induction motoris based on the well known (7)two-axis theory. An ideal-ised symmetric motor is assumed, with a balanced sinu-soidal airgap mmf and a linear magnetic circuit. Iron andmechanical losses, stray load losses and mechanicaldamping are all neglected. All motor resistances and in-ductances are independent of frequency, which limitsthe usefulness of these models to wound rotor and singlecage rotors with shallow bars.

The two-axis voltage equations of a voltage fed induc-tion motor are expressed below in terms of currents anda reference frame rotating in synchronism with the fun-damental component of the stator current:

[v] = [R][i] + [L]P[i] + w;[F][i]+sw;[G][i]

where [v]= [Vdh Vqh VdZ, Vq2V

[i] = [idhiqh id2,iq2V

s = (w; - W,)/Wi

The other matrices in Eqn (2) appear in Appendix B.In a short circuited rotor circuit VdZ,Vq2are zero.

However, in the case of a current source inverter idh

iqlare independent predefined variables (obtained froma Park's transform of iAL,iBb ied and differential equa-tions are only required for the rotor currents id2, iq2 suchthat:(6)

[V2] = [R2][i2] + [L2]P[i2]+ [Lm]p[id+ Wi[G1][id + SWi[G2][i2] (6)

[V2] = [Vd2,vdT[id = [idh iqlV[i2] = [idZ,iq2V

The other matrices in Eqn (6) appear in Appendix B.Eqn (6) can be rearranged to yield the following differ-ential equations for the rotor currents:

p[i2] = -[B]{ {[R2] + sw;[G2]}[i2]+ sw;[Gd[id + [Lm]p[id}

where

(7)

where [B] = [L2]-I. Eqn (7) is nonlinear and is inte-grated numericallystep-by-stepto yieldvalues for idZ,iq2which are used together with idh iqlto compute the elec-trical torque T, from:

T, = wbLmUdZiql - iq2 idl)/3 (8)

The mechanical motion is described by:

pw, = (T, - Td/J (9)

(2)

(3)

(4)

(5)

For a sinusoidal line current to the motor, idh iql areconstant quantities in a synchronously rotating refer-ence frame. However, in the case of an inverter fedmotor, where the line current consists of a series of har-monics, idh iql are functions of time (see Appendix D)and are defined by the Park transform operating on eachharmonic component and summating the result. Ex-pressions for pidh piql are required in Eqn (6) and arefound by differentiating the summated series express-ions for idl, iqlas shown in Appendix D.

A computer program was developed to predict the dy-namic behaviour of the PWM CSI-fed induction motordrive when it is subjected to a variety of severe disturb-ances such as a start-up, a change in load after runningup, or a change in values of link current IDeor inverteroutput frequency Wi'The independent variables are idl'iql, pidl, piql and their accuracydepends on the number nof harmonics used in the Fourier series. The value of nhas to be infinity in order to represent the wave exactly,

-

December 1984 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS 53

but it has been shown elsewhere (6)that thirty-one cur-rent harmonics (n = 31) usually yield sufficient accuracywhile also keeping computation time down.

The program starts by finding the magnitude of eachof the thirty-one harmonic current components. It thencalculates idl, iql, pidl, piq, and uses these values to findid2,iq2,Teand w, at each step of the integration process.

3 ResultsThis section uses the above techniques to evaluate the

predicted response of a PWM CSI-fed 2 kW inductionmotor of which the parameters appear in Appendix E.Throughout these investigations the PWM waves arerepresented by thirty-one current harmonics (n = 31),and the inertia constant H = 0,8 see unless otherwisestated.

3.1Start-up of a PWMCSI-fedinductionmotorFig 4 shows the electrical torque transients for a no-

load start-up when the motor is supplied from a 10HzPWM current wave for modulation angles a of 0°. 7,5°,12°and 20°. An increase in a reduces the duration of thecentre block of current in Fig 3 and hence the magnitudeof the fundamental components of the current wave (Fig7) and of the torque wave, and hence the value of the netaccelerating torque. As a result the larger values of aincrease the start-up time as shown by the speed curvesof Fig 5 which also illustrate the speed pulsations. Thepulsating parasitic torques do not contribute to the start-up time since their average values are zero. This analysis

1.5~~~~~~:~ __D~~RE~~_n-- .n

=: iIL I

1.1111;

!

--- -1

LIJ !I5 -'5

1~-1.11_-11.11111

n + + - ~ n- 4-.511 1.11111 1.511 2.1111

TIME (a) SEC.

ALPHA-12 DEGREES

;:) 1.5,__-+ +--' " + 1a: 1.IIJII 1

.51

111.11

LIJ I5 -.5~@5 I... I

-1. IIL +-n ---+--- + ~11.1111 .511 1.1111 1.511 2.11111

TIME (C) SEC.

therefore proves that the dynamic behaviour of the in-duction motor is affected by the value of the modulationangle. The link current could have been increased tocompensate for a drop in fundamental current, but this isimpossible if the link current already equals rated cur-rent, and would moreover require complex closed-loopfeedback linking the modulation angle back to the recti-fier's current controller.

3.2 Changeof loadFigs 4 and 5 further show the torque and speed curves

of the induction motor when a load of 0,2 p. u. is applied1,25 s after switching on. Pulse width modulating thecurrent wave clearly forces the motor to be more slug-gish and hence impairs its dynamic performance.

3.3 Determination of the optimum modulationangleThe optimum value of the modulation angle a is de-

fined as that value of a which ensures a minimum valuefor a harmonic component of the parasitic pulsatingmotor torque or speed. Due to the nonlinearities in theequations, the speed harmonics cannot be expressed asexplicit functions of a. However, it is possible to select aseries of values for a and analyse by Fast Fourier Trans-form (FFT), the time domain waveform of torque orspeed either during start-up or at final steady state speedafter start-up, depending on the particular region of in-terest. This investigation considers the following valuesof modulation angle a:

0° 5° 7,5° 10° 12° 15° 20° 30°

1. 51i1r-- ~~~H"--7:.5... DE~REES- -. TI

Ii-

;:)

a:1.1I1L,

..:.

.5'-'T

I!. II

-1.lIraL .111.111111 - '. 5111

TIME

.- -. ..1.1111

(b)

~ t-1. 5111 2.1111

SEC.

1.5'1-- ALPHA-2" pEGREES TIj

;:)

a: I.IIIi1..

;i-I

T

.5~'

11.11 +

I

-1.IIel.111.1111

. 0-1.1111

(d)

-, --- 41.511 2.1111

SEC.':50

TIME

Fig 4 Predicted start-up torque for different values of modulation angle

54 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS December 1984

SPEED

1. 25

:J

Q..

1. 121121

.75

.5121

0WWQ..(f)

.25

121.121121

121.121121 .5121

TIME

~10 a. 0°2, a = 7. SO3, a = 12°4, a = 20°

1. 121121 1. 5121 2.121121

Fig 5 Predicted start.up speed for different values of modulation angle

SEC

ALPHA-111 DEGREES-+-

;ja:

~-I

111.111

-.5....zUJ

~ -1.111~u

-1.5ii.1II1II

1. 5,

5111.111111

TIME

ALPHA-12 DEGREES

1111111.111111

(a)

15111.111111

mi 11 i-SEC.

2111111.111111

--+-

;ja:

1.111

.5

111.111

-.5....zUJa: -1.111a:~u

-1.51ii.1II1II 15111.111111

mill i-SEC.

2111111.1111115111.111111

TIME

1111111.111111

(C)

;j

a:

1.5

j1.111

.5

I

ALPHA-7.S DEGREES.

t

i-T-I

111.111

It

I

~ -.51~ -1.1111

a-I.5~1II1II111.111111 TIME

DEGREESt-

.1111111.111111

(b)

.15111.111111

mi 11 i-SEC.

j2111111.111111

;j

a:

111.111

-.5....ZUJ

~ -1.111~u

-1. 5ii.IIIIII 15111.111111

mill i-SEC.5111.111111

TIME

Fig 6 Steady pulsating stator current for different modulation angles

1111111.111111

(d)2111111.111111

December /984 THE TRANSACfIONS OF THE SA INSTITUTE OF ELECfRICAL ENGINEERS 55

a = 0° corresponds to 120°rectangular blocks of currentand is hereafter referred to as the base case for purposesof comparison.

Fig 6 shows only the four line current waveforms for aequal to 0°, 7,5°, 12°and 20° since the waveforms for theother four values of a are similar in appearance. How-ever, Fig 7 shows the Fast Fourier Transforms (FFT) ofpeak-to-peak values of the individual harmonics presentin the current waveforms for all eight values of a.

Fig 7 (a), which applies to the base case, clearly showsthat the magnitude of each current harmonic is inverselyproportional to its harmonic order, eg the 5th currentharmonic has a magnitude of l!5th that of the fundamen-tal (10 Hz), while the 7th current harmonic has a magni-tude of l/7th that of the fundamental, etc. However, thisproperty is only true of the base case where no PWM is

.applied. When the current ispulsewidthmodulated, themagnitudes of the current harmonics do not bear anyfixed relation to each other as illustrated in the rest of Fig7. For a = 5°for example, the 11th and 13th current har-monics are almost zero, but for a= 10°,the 29th and 31stcurrent harmonics are almost zero.

The following sections evaluate the effects which dif-ferent values of a have on the torque and speed pulsa-tions.

Torque pulsationsThe 6th harmonic component of the pulsating torque

. is usually the most troublesome one (produced by the5th and 7th harmonic currents) and some designersselect an a of 12° since this has zero 5th harmoniccurrent.

Fig 8 shows the eighth pulsating torque waveforms(for the eight values of a) after the motor has reached afinal steady speed. The corresponding eight FFTs whichshow the peak-to-peak magnitudes of the individual tor-que harmonics look similar to Fig 7 and are thereforesummarized in Table 1. Note that the fundamental com-ponent of the current waveform in Fig 6 (a) has a peakvalue of 1,047 of the magnitude of the de link currentwhich in this case is 1p.u. However a peak magnitude of1,414 p.u. is needed for the fundamental component ofcurrent to produce rated torque. Since torque is propor-tional to the square of the current, the magnitude of thefundamental component of torque when the de link cur-rent is Ip.u. is (1,047/1,414)2 = 0,548p.u. Hence thepeak-to-peak value of the 6th torque harmonic mustequal 2(0,548/6) = 0,183 p. u. which is verified in Table 1for a = 0°, while that of the 12th torque harmonic is2(0,548/12) = 0,092 p.u. which is also verified in Table 1.

Fig 8 (a) shows the steady pulsating torque of the basecase (no PWM)<8)after start-up with a large 6th har-monic content at 60 Hz for a fundamental frequency of10 Hz. Column 1of Table 1confirms that the amplitudesof the various frequency components present in the tor-que pulsations are inversely proportional to their order.

Fig 8 (b) shows the torque pulsation produced whenPWM is applied with a = 5°. During the period markedas Tin this diagram, the torque pulsation due to the cen-tral block of current has an amplitude which is less thanthat of the base case (Fig 8 (a)) and shorter in duration.There is however an additional torque pulsation oneither side of T in Fig 8 (b) due to the side pulses of the

PWM current wave.As a is increased to 7.so and even larger (Figs 8 (c) to

(h)), the pulsating torque produced by the central cur-rent block steadily reduces in magnitude and durationwhile that due to the side pulses increases. Eventually ata = 30°, the central block of pulsating torque disappearsaltogether, and only the side pulses remain. The FFTresults in Table 1 show that in contrast to the base case,the amplitudes of the harmonics present in the PWMproduced pulsating torques are not inversely propor-tional to their harmonic order, and in fact do not bearany fixed relation to each other, just as the current har-monics in Fig 7 (b) to (h) did not bear any relation toeach other.

Table I Peak-to-peak amplitudes (p,u,) of various torque harmon-ics for different modulation angles when the inertia constant H =0,8s.

However, there is a relation between each torque har-monic and certain current harmonics, eg for a = 5°(Table 1) the 12th torque harmonic is zero since the 11thand 13th current harmonics (Fig 7 (b)) are approxima-tely zero. Similarly the 30th torque harmonic is zero fora = 10°(the 29th and 31st current harmonicsare zero).Large current harmonics therefore produce large torqueharmonics. If it is required to have an extremely low 12thtorque harmonic (regardless of the magnitudes of theother harmonics) a should be 5°. Similarly the smallest6th and 30th torque harmonics are produced by a = 10°,the smallest 18th by a = 20° and the smallest 24th by a =30°. These various harmonic components of torque allact together on the rotating shaft to produce a speedwaveform, and it is often the severity of a particularspeed harmonic which might be of concern to a designerin which case his analysis to find the best value of ashould include an evaluation of the harmonic compo-nents present in the pulsating speed.

Speed pulsationsThe harmonic components of the base case (a = 0)

speed pulsations which are summarized in Table 2, haverelative amplitudes which follow the relative amplitudesof base-case harmonic torques in Table 1; however it isquite clear that this relationship does not extend to thePWM produced results in the rest of Table 2. For exam-ple, for a = 7,5° in T~ble 1, the 6th harmonic of torque issmaller than the 18th or 24th harmonics but the 6th har-monic of speed is the largest of all the harmonics in Table2; similarly for a = 15°the 6th torque harmonic is smallerthan the 12th, but the 6th speed harmonic is larger thanthe 12th. The usual (5)choice ofthe 'optimal' value of atobe 12°,givesa lower6th torque harmonic than a = 7,5°,but the latter has a lower 6th speed harmonic. Howeverthe results indicate that of all the values of switching

0' 5' 7,5' 10' 12' 15' 20' 30'

harmonic no.

6th 0,184 0,136 0,080 0,007 0,060 0,159 0,300 0,348

12th 0,090 0,000 0,088 0,172 0,222 0,250 0.158 0.039

18th 0,059 0.059 0.141 0,172 0,149 0,058 0,047 0,092

24th 0,044 0,089 0,132 0,087 0,018 0,040 0,076 0,026

30th 0,036 0,088 0,084 0,002 0,032 0,032 0.061 0,072

'I

56 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS December /984

angles considered, a value of (}'= 10°producesthelowest6th harmonic component of both speed and torque pul-sations.

These relationships between corresponding harmon-ics of the parasitic pulsating torque and speed are theresult of the complex manner in which the torque har-monics produce the speed harmonics. Unlike the re-

lation between current and torque harmonics (large cur-rent harmonics produce large torque harmonics), largetorque harmonics do not necessarily produce largespeed harmonics. The reason for this is that the rotorinertia smooths out the effects of the higher order torqueharmonics and hence does not respond to them as well asit does to the lower order harmonics.

ALPHA-S'

,;.:

2321 29 Jl )1 37

'"a;;,I--'Q.%<

0

-I n 100FREQ.

ALPHA-7. 5 0

200 300 /400-j -

0 jOOFREU.

ALPHA- 10'

200 00 400(a) HZ. (b) HZ.

,;.:

,;.:

'"

~

' '"gig 11 13 17 19

5 5 III 1 7 A ~ ~ A . 29 31~ 0 ~ 0< <

-1'0 100 200 300 40~ -10 100 -2Cio 300 4'00FREQ. ( C ) HZ. FREQ. ( d) HZ.

,;1 , ,,_.,,' I ; . "",.,,'

i 3 I .: 3 IIj'

Hil

"'~ I"

1111 11a 1 a:J :JI- - 1719 I- "

! 0.' "', " ~ -: III . n 1\ 1\ .""" ,,"-10 100 200 JU(, 4(;u () 100 200 300 400

FREQ. (e) HZ. FREQ. ( f) HZ.

~ ufl~,-". t ~ 't"'.w

'~Il !

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FREQ. (9) HZ. FREQ. (h) HZ.

Fig 7 Fast Fourier Transforms of steady pulsating stator current for different modulation angles

ALPHA-O

4,,;.:

Ji

'I ':JI-

U 7 I,IJ 1719-'

0 ...J\/\2J 25 2 J 1

Q. . .%<

December 1984 THE TRANSACTIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS 57

If the design criterion is to minimize the 6th speed har-monic then (Yshould be 10°. This is usually the most im-portant speed harmonic to minimize. since for 5 to 50 Hzsupply frequency operation. this pulsation lies in the 30to 300 Hz range. where shaft mechanical resonancescould occur. However this conclusion rests on an analy-sis which so far has been carried out on only one value ofinertia. The following section therefore investigateswhether the value of (Y= 10°dependsonthevalueofH.Torqueand speedpulsations with reduced inertia

The results in Tables 1 and 2 were repeated but withthe inertia constant H reduced to OA s: a comparison ofthese results with those in Table 1 when H was 0.8 sshowed that the relative torque harmonics were identi-cal as expected since they do not depend on H.

However. due to the lower inertia. the speed pulsa-tions for H = OA s were found to be larger than those forH = 0,8 s in Table 1. What is more important though, isthat the relative magnitudes of the speed harmonics for agiven value of (Ywere the same, regardless of the inertiaconstant; as an example. for (Y= 5°and H = 0,8 s (Table2), the 12th speed harmonic is the lowest and the 6th isthe highest with the others approximately equal in mag-nitude, but this is also true for (Y= 5°and H = 0,4 s. Simi-lar comparisons for other values of (Yin Table 2 lead tothe same conclusion, namely that the optimum value of (Ycan be chosen independently of the inertia constant H.In other words an (Y= 10°ensures the lowest value of 6thharmonic speed, irrespective of the value of the motorand load inertia.

Table 3 summarises the optimum values of the modu-lation angle in order to minimize certain harmonic com-ponents present in the torque and speed pulsation re-sults presented above.

4 ConclusionsThis paper has analysed the steady state and dynamic

behaviour of a PWM CSI-fed induction motor drive.The motor was represented by its two-axis equations(without omitting certain voltage terms) and the currentpulses were represented as a harmonic Fourier series.Harmonic components give rise to a pulsating torqueand speed of which the severity depends on the amountof pulse width modulation. The following conclusionscan be drawn from the results:

(a) When each half cycle of the stator line current con-.sists of 120°rectangular blocks, instead of a PWM trainof smaller blocks, the magnitudes of the current har-monics, the torque harmonics, as well as the speed har-monics are inversely proportional to their order. WhenPWM is applied this relation no longer holds.(b) There is a clear correlation between the currentand torque harmonics whether PWM is applied or not,the correlation being that the larger the relevant currenthar~onics, the larger the corresponding torque har-mOnIc.(c) There is no clear correlation between the corre-sponding harmonic components of torque and speedwhen PWM is used. For example, a particularly large 6thharmonic of torque does not necessarily produce a par-ticularly large 6th harmonic of speed, since the motor

inertia smooths out the effects of the higher order torqueharmonics. In other words, the variations in the speedcaused by the higher order torque harmonics of a givenmagnitude are smaller than those caused by the lowerorder torque harmonics of the same magnitude. How-ever, the smallest torque harmonic does produce thesmallest corresponding speed harmonic.(d) In a PWM CSI system it is therefore a fallacy toargue that the design object should be to minimize the5th current harmonic only by using (Y= 12° because thisdoes not result in a minimum 6th harmonic of speed ortorque. To minimise the 6th speed or torque harmonic.the modulation angle (Yshould be 10°since this value re-duces both the 5th and 7th current harmonics to such anextent that it produces a smaller torque harmonic thanthat produced when using a modulation angle of 12°;moreover, this angle does not depend on the value of theinertia. Chin and Tomita(2) state that (Yshould be 7,5°.but it is not contradictory to this paper since they definean integral that in effect summates all torque harmonicsup to the 24th and they thereafter attempt to minimizethis sum. This paper uses a completely different method(FFf) to find the magnitudes of the individual current,torque and speed harmonics; it investigates the re-lationship between them, and provides criteria for theminimization of anyone particular harmonic of torqueor speed.(e) The dynamic behaviour of the induction motor isaffected by the value of the modulation angle (Yof thePWM CSI. Due to the reduction of the fundamental cur-rent component, the motor becomes sluggish and oper-ates at a larger steady state slip as (Yis increased. How-ever at (Y= 10°, the dynamic behaviour is not muchslower.

5 AcknowledgementsThe authors acknowledge the assistance of D C Levy,

R C S Peplow, and H L Nattrass in the Digital ProcessesLaboratory of the Department of Electronic Engin-eering, University of Natal. They also acknowledge thehelpful suggestions and interest shown in this work by

Table 2 Peak-to-peak amplitudes (dB)of various speed harmonicsfor different modulation angles when the inertia constant H =0,8s.

Table 3 Optimum values of modulation angle Q'

::s:0" 5" 7,5" 10" 12" 15" 20" 30"

harmonic no.

6th -0,055 -0,058 -0,064 -0,083 -0,065 -0,057 -0,052 -0.050

12th -0,068 -0,105 -0,068 -0,062 -0,061 -0,059 -0.062 -0,076

18th -0,071 -0,075 -0,068 -0.066 -0,067 -0,074 -0,074 -0,070

24th 0,080 0,074 -0,072 -0,074 -0,085 -0,079 -0,074 -0,084

30th -0,084 -0,076 -0.078 -0,111 -0,084 -0,082 -0,079 - 0,078

Harmonic Angle for minimum Angle for minimumnumber torque harmonic speed harmonic

6h 10 1012th 5 5

18th 20 2024th 30 3030th 10 10

58 THE TRANSACfIONS OF THE SA INSTITUTE OF ELECfRICAL ENGINEERS

M A Lahoud of the Department of Electrical Engin-eering, University of Natal. They are also grateful forfinancial support received from the CSIR and the Uni-versity of Natal.

6 ReferencesI MURPHY, J M D: 'Thyristor control of ac motors', (Book), Perga-

mon Press. 1973. ISBN No 0-08-016943-0.

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Fig 8 Steady pulsating torque for different modulation angles

<c.

December 1984

2 CHIN. T H and TOMITA. H: 'The principles of eliminating pulsatingtorque in current source inverter induction motor systems'. IEEE.VoIIA-I7. No 2, March/April 1981. pp 160-166.

3 LIPo. T A: 'Analysis and control of torque pulsations in current fedinduction motor drives', IEEE IAS Anl/I/al Meeting, 1978. paperCH 1227-5/78, pp 89-96.

4 CHIN. T H and TOMITA, H: 'Elimination of torque pulsation by thecurrent instantaneous value control in squirrel cage inductionmotors fed with controlled current inverter'. Elec Eng il/ Japan,Vol 98. No 4. 1978. pp 105-1/2.

5 LIENAU. N. MULLER-HELLMANN. A and CHRISTOPHER.H: 'Power

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December 1984 THE TRANSACfIONS OF THE SA INSTITUTE OF ELECTRICAL ENGINEERS 59

converters for feeding asynchronous traction motors of single-phase ac vehicles'. Conf Rec 1977, 1nt Semiconductor Power Con-verter Conf pp 295-304.

6 HARLEY,R G, PILLAY,Pand OOENOAL, EJ 0: 'Analysingthe dynamicbehaviour of an induction motor fed from a current source in-verter'. Accepted for publication in Trans of SA lEE.

7 AOKINS, B and HARLEY,R G: 'The general theory of alternating cur-rent machines'. (Book), 1SBN No0412155605, Chapmafl & Hal/,1975.

8 SUBRAHMANYAN,V, YUVARAJAN,Sand RAMASWAMI,B: 'Analysisof commutations of a current source inverter feeding on inductionmotor load'. 1EEE Trans, Vol1A-16, No 3, May/lune 1980, pp332-341.

7 AppendicesA, Derivation of Fourier Seriesfor the PW M line current

The waveform for iALin Fig 3 is symmetrical about thewt axis and f(wt) = -f( -wt). Hence iAL can be rep-resented as a Fourier series(9)such that

iAL = BI sin( wt) + B2 sin(2wt) + + Bn sin(nwt) (A. 1)

where

Bn = [21/mr][cos(n(30° - £1')) - cos(n300)- cos(n(150° - £1'))+ cos(n(30° + £1'))- cos(n(150° - £1')+ cos(n(1500)] (A.2)

Hence

'"

iAL = L Bnsin(nwt)n=1'"

iBL= L Bn sin(n,(wt - 2n:/3))n=1'"

(A.3)

iCL= L En sin(n (wt + 2n:/3))n=1

B. Elements of matrices

=

[

RIO 0 0

]

0 RI 0 00 0 R2 00 0 0 R?

=

[

0 L,\ 0 L:

]

-Lll 0 -Lm 00 0 0 00 0 0 0

[L] =

[

Lll 0 Lm 0

]

0 L" 0 LmLm 0 Ln 0

[G] =

[

~ ~m ~ ~n

]

0 0 0 00 Lm 0 Ln-Lm 0 -Ln 0

[L2]= [Ln 0 ]0 LnI= [R2 0 ]0 R2

] = [0 Lm]-Lm 0

[G2]=[

0 L22]- Ln 0

[Lm]=[Lm 0

]0 Lm

C. Definition of Park's Transformation Matrix

[F Odql] = [P 11][[ Fabcd (C.1)

where

[P II]= v273

[

v172 v172 v172

]

cos( 0) cos( 0 - Je) cos( 0 + },)

sin(0) sin(0 - Je) sin(0 + Je)

[Fabel] = [PII]-I[Fodqd

(C.2)

(C.3)

[PII] -I = v273

[

v172.

COS(O) sin(O)

]v172 COS(O-Je) sin(O-Je) (C.4)v172 COS(0 + Je) sin(0 + Je)

In the synchronously rotating reference frame where theframe rotates at speed w, 0 = wt.

D. Derivation of the derivatives of the stator currents

From the orthogonal Park's transform (Appendix C)

idl = v273[iALcos( 0) + iBLcos( 0 - Je)+ iCLCOS(O+Je)] (D.1)

Hence

pidl = Y273 [piAL'cosO+ piBL'cos( 0 - Je)+ piCLCOS(0 + Je) - iALsinO- iBLsin(O- Je) - iCLsin(O- Je)]

iql = Y273 [iALsin( 0) + iBLsin( 0 - Je)+ iCLsin( 0 + Je)]

(D.2)

(D.3)

Hence

piql = Y273 [piAL.sinO + piBL.sin( 0 - Je)+ piCL.sin( 0 + Je)+ iALcosO+ iBLCOS(O- Je)+ iCLCOS(O+Je)]

E. Induction motor parametersStar connected statorBase timeBase speedBase powerBase stator voltageBase stator currentBase stator impedanceBase torqueNumber of polesPower factor at rated loadEfficiency at rated loadRated output powerStator resistance R,Rotor resistance R2Stator leakage reactanceXIRotor leakage reactance X2Magnetizing reactance XmStator leakage inductance L,Rotor leakage inductanceL2Magnetizing inductance Lm

(D.4)

1s1 rade/s3385,8VA220 V (phase)5,144 A (phase)42,77 ohm21,56Nm40,83377,9%2,2kW0,086 p.u.0,046p.u.O,077P'U.

]expressed

0,066p.u. at 50Hz1,57p.u.0,000 146p.u.0,00021 p. u.0,005 p.u.