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Title Analysis of Three-Phase Induction Motors Controlled by Thyristors

Author(s) Fujii, Tomoo; Watanabe, Masao; Ishizaki, Takemitsu

Editor(s)

CitationBulletin of University of Osaka Prefecture. Series A, Engineering and nat

ural sciences. 1971, 20(1), p.109-119

Issue Date 1971-09-30

URL http://hdl.handle.net/10466/8171

Rights

109

Analysis of Three-Phase Induction Motors Controlled by Thyristors

Tomoo FuJII*, Masao WATANABE** and Takemitsu IsHizAKi*

(Received June 15, 1971)

This paper deals with an analytic means of the three-phase induction motor whose

primary voltage is controlled by symmetrical triggering of inverse parallel connected pairs

of thyristors series with stator windings. On this contro!, the motor exciting voltage

waveforms become segments of sinusoids and the phase currents are discrete ones. Op-

erational modes of this motor control may be divided into two modes; three-phase oper-

ation and single-phase operation.

Motor characteristics are obtained through the Fourier analysis of the voltage waves

on each modes. And the infiuences of the harmonic components of the supply voltage

on the motor characteristics are investigated.

1. Introduction

As the thyristors of high current and voltage ratings become available at decreasing

cost, it is finding increased use in the control of polyphase induction motors. There are

two main areas of application. One of these is the use of thyristor inverter circults to

form an effective variable-frequency supply to the motor')'2), and the other is the use of

inverse parallel connected pairs of thyristors series with the stator winding, (hereafter

referred to as the thyristor pair), to form an effective variable-･voltage supply to the

Motor.3)･-io)p

This paper presents the primary voltage control using thyristor pair, which is one

of the latter application that is usefu1 fbr !ow power induction motor controls. The

waveforms of the motor exciting voltage controlled by symmetrical triggering of thyristor

pair are segments of sinusoids. Therefore the motor exciting voltage is rich in higher

harmonics.

The influence of these harmonics can easily be investigated by knowing the rate of

each harmonic, which is obtained from an expanslon of voltage waveforms in a Fourier

.senes.

Two of the representative operational modes occur within this control as fbllows,

mode I: normal three-phase operation, and in this mode the motor speed can

be controlled smoothly by adjusting exciting voltage,

mode I: single-phase operation, and in this mode the motor per se has no

starting torque.

The boundary of these modes is the function of triggering angle, current conducting

* Department of EIectrical Engineering, College of Engineering.

** Graduate Student, Department of Electrical Engineering, College of Engineering.

110 T. FuJii, M. WATANABE and T. IsHizAKi

angle, and the power factor angle of the motor equivalent circuit per phase.

2. Abstract ef the Primary Voltage Control

Control of the primary voltage of the induction motor has been made

technology concerning semiconductors. The fundamental principle of

voltage control is that the generated torque which corresponds to the speed is

to square of the supply voltage with voltage variation.

os"U--o-'j

Vl

V2

Vl > V2

TL

through the

the primary

proportional

n2 nl speed ' Fig. 1. Schematic speed vs. torque characteristics of three-phase induction motor with voltage variation. Dotted line shows load torque.

As shown in Fig. 1, the generated torque exists in equilibrium with load torque TL

at the speed ni in the voltage L. But when the voltage is reduced to K, its equilibrium

state is reached at reduced speed n2. This is the basic principle of the variable speed

operation of induction motors controlled by the primary voltage control.

In this reserch, thyristor pair is used to give the primary voltage control.

3. 1lnalysis

A connection of main circuit to give symmetrical control of a star connected three-

phase induction motor is shown in Fig. 2.

The foregoing analysis are based on the equivalent circuit of the motor. Equivalent

circuit for a three-phase induction motor may be reduced to resistance-inductance series

circuit. When a sinusoidal voltage controlled by thyristor pair is impressed on this

circuit, ihe relation among triggering angle a, current conducting angle P and power

factor angle g is given in eq. (1).

sin (a+,B-g)) == sin (a-q)e-Rl(wL)P (1)

where

-, toL , R, L: equivalent circuit resistance and inductance, 9 = tan R

,

Analysis

to == 2nf,

of

Phase control can be

tinuous. Relations of a vs.

They are obtained by numerical solutions of eq

Three-Rhase induction Motors Controlled by 11hyristors 111

f: line frequency.

tttt ,,, . ,. inq.uctio.nmQtor Va ' ' '''' ' ''' '' -' r..-. ".--L i TU ' ' ' ･.v ･- -,- 1

, Vc

Vb

Fig. 2. Main circuit of "thyristor-pair"

-induction motor drive.

done for gfE{Ia:f{:z, in this region the line current is discon-

P with various power factor angle g are shown in Fig. 3.9)

.(1). .

7

5 tr･16

2z13

6v"-yn, ftq

vr/2

T/3

vrf6

o

A 'D

'ko£e'?"."o

n

qfie1tSreJr'-1?J

"sre"'9)6r--J."X".･"..qpq}'r"/vJe

'tcrJ-

'

c

/

'

'

NK---Nx-'x"'-N"xx

Nx

..QN<kN.<O"S"Elj,l}g,N

GXxxXXxNxXN<Y

~xxxxNxx

'Hxilii'l',lix/g,,,,xssxx

''x

.

xxYo rr/6

Fig. 3.

z13 vr12 2z/3 a (rad .)

Relations of a vs. B(B' )with various ep.

5 7/6 rr

F

112 T. FuJii, M. WATANABE and T. IsHlzAKI

3-1. 0perational Mode and Exciting Voltage Waveform.9)'iO)

Typical line-to-neutral stator winding voltage waveforms on Fig. 2 are shown in

Figs. 4N7. Fig. 4 and Fig. 5 are theoretical waveforms and Fig. 6 and Fig. 7 are oscil-

lograms corresponds to Fig. 4 and Fig. 5.

In Fig. 6 and Fig. 7, line current waveforms are also shown.

As shown in Fig. 4, voltage waveforms are series of segments of sinusoidal line-to-

neutral voltage V, line-to-line voltage V 23 VeJ'(ndf6) and V23 Ve-j'(ti6) .

These voltage waveforms will maintain for -2-z<p<n and q<a:E{ 2 z, as shown

33by the area ABCD in Fig. 3.

N<Vle

xXx

･×

v

owt -

sh112 vab. /

-

itli;Lt)llllli,,

l. i

112 veg

/

l/1l

bv-

xx

o lr･･a

!xlN ix ll

to

y/

1

1/y

/ xx

ke

o

Kva'

IX lx IX lxx IX Nwt+ "l6 X rl;

Nxva

lt2 hb l Xx lt2 ve,

xx x N

o

N--

a+rt13 1 at2nl3 n

, / fi- .

Nxx....;-T/

tep=o

//

N

(a)

xlva

jxx

i /1

wt re

a=rc!6,

"l3

/h

XsNX ']ii,tab )

. //

'2#t3

,4

!l

.

,

,

Vb r( l XXN

l

il?YC:i

i] i a+2,ttt3"

ti

o

/1

1

l

NJ

/

alTt3tr+P-a

N N

a+B-2f13

N/l

Fig. 4.

[/'">/×<i []y/

(b) a=T/2, ep=z/3

Theoritical waveforms ofexciting voltage in mode

(a): resistive load,

(b): inductive load.

, a+fi-"13

dl N a+"t3 N.-- -H

stator

I.

!

o a-ri3 1 l, 1

J

a.

I /<va I

I i

cr == 11z118,

.

2vl3

/ ",

1x

!

v.

.

i (a)

,

s'r.l6 alti3

riLl'"l

li

i

ep -o l

1

7rJ6

/ //Xn

,

'l

vc

~(,Fx,/vf.

fN

,

tsIN

i1'1ill

xxy4/XNl/t

ll2V"b.lIXXAi><Nit

1t2vc#

sxlt1l1

,' .l I

'

/"t- Xrrt32at3I'

ra+2et3

t[

o･a-rl]'1 ,1,1a

xx

l

xX</llv

,l

i,(i<liv

･fi

1ll[

a+rt3

v

/<xx

,>

<T

Fig. 5.

(b) a=11n/18, ¢==z/3

Theoritical waveforms ofexciting voltage in mode

(a): resistive load,

(b): inductive load.

stator

]･

x

Analysis qf 71hree-Phase induction Motors Controlled by Tlhyristors 113

'v

i

o

o

-aF6-fa) a - n16, P=O

Fig. 6.

Fig. 4.

v

i

o

o

N a l- B --ti

(b) a =z/2, 9- a!3

Oscillograms of voltage and current waveformslcorresponding to (a): resistive load, (b): inductive load.

v

o

i o

v

i

o

o

ba Vi.J frat6 rl (a). a-11nl18,g)-O (b) a-11nl18, op-vr13 Fig. 7. 0scillograms of voltage and current waveforms corresponding to Fig. 5. (a): resistive load, (b): inductive load.

In this region, motor stator winding has a period impressed by the three-phase line

voltage simultaneously. Therefore the motor can operate as an ordinary three-phase

induction motor.

In Fig. 5, voltage waveforms are series of segments of sinusoidal line-to-line voltage

V23 J7EJ'(¢i6) and V2-t V-i'(ti6), and line-to-neutral voltage V disappears. This means

that the motor stator winding is excited only by line-to-line voltage at any instant, and

the motor operates on the condition of the single-phase operation and per se has no

2nstarting torque. This single-phase operation occurs fbr Of{gfiE{gl}-z, -g-･SlaE{;z as

shown by the area BEC in Fig. 3.

Because of the line-to-line voltage leads the line voltage by the time-phase angle

-ii-, the triggering angle referred to the line-to-line voltage becomes a+ :-. ･

Therefore the stator winding voltage will be zero at a=ire and also conduction

6angle must be read corresponding to this leading triggering angle. These relations are

shown in dotted line in Fig. 3. Here we define the fbrmer operational mode to mode I,

and the latter one to mode I.

114 T. FuJii, M. WATANABE and T. IsmzAKi

3-2. Analysis of Exciting Voltage Waveforms

In order to investigate the performance of an induction motor whose exciting voltage

is nonsinusoidal waveform, it is one of available methods to know the rate of harmonics

by Fourler analysis. The Fourier series for exciting voltage wavefbrm in mode I, Fig.

4, is as fo11ows,

vi(t) == -il- V-2- l7(VAi2 + Bi2 sin (tot+ ri)+Z k, ! 1 VCi2+ Di2 sin (kwt + Si)l

(2)

where

Ai = t((2P--il-n) cos a+sin a-sin (2B+a)]

BJ == t((2P- ; a) sin a+cos a-cos (2P+a)]

c, = -i- [sin (a+p)(-k cos (kp-gkz)+k cos (k3-5n)+2k cos kfl]

+cos (a+ fi)(sin (kP- ; kz)-sin (kP- i z)-2 sin k31

+ sin a(-2k-k cos -:-z+k cos -;-kz)

+ cos a(sin 5z - sin -il-kza)]

Di = -}-[sin (a+P)(k sin (kP- i kz)-k sin (kfi-5z)-2k sin kfll

+ cos (a + fi) (sin (kfi - -Z- kn) - sin (kP - -l?･ z) - 2 sin kP)

+ sin a(k sin -fl- n -k sin ;kz)

+ cos a(2 + cos e n - cosl kz)]

., DI -, BI , 6i=tan , rl == tan C, AI

and in mode II, Fig. 5, is as fbllows,

v!(t) = X/2't. V(VAn2-+Bll2 sin (tot+r")+Zk, 1--1 VCll2+Dff2 sin (ktot+s il )l

(3)

where

' All = 3(P' cosa-sin P' cos (a+pt+-Ii-)]

Bl = 3 (P' sin a+sin fi' sin (a+3'+ g )]

Analysis of tTlhree-Rhase induction Atfotors Controlled by 71hyristors 115

Cl = vX-3-[sin (a+B')(-2 sin (kP'+-il-k) cos -(i7,k-2V-Ii-k cos

(kpt + -Z- k) cos -(li' k)

+ cos (a + P')(2VZi- sin (kP' + -[i- k) dos -Z-k- 2k cos

(kP' + -(i- k) cos -ii- k]

, + sin a(sin {i- k+V 3 k+V-gU k cosg k)

+ cos a(V-I sin {i- k + k + k cos -ll- k)]

D fi = v'-Ii-[sin (a+ P')(-2 cos (kP'+ -ii-k) cos -(i-k -2VJI k sin

(kP' + -il- k) cos -(i-k]

+ cos (a + P')(2V-li- cos (feP' +-Zk) cos -Ii-k-2k sin (kP' + -(i-k) cos -li-k]

+ sin a(1 + cos --"i k - V '3- k sin -li- k)

+ cos a(- VL3- - V li- cos {i-k-k sin -li-k)]

", Dl -, Bff ,6fi == tan ･ rll =: tan Cg A fi

conducting angle P' for line-to-line voltage rerfers to dotted line in Fig. 3,

and k == 6n+1, 6n +3, 6n +5, (n == O, 1, 2, 3, ･･･).

The voltage waveshape contains only odd harmonics in a three-phase system. And

the (6n+ 1)th harmonics are of positive sequence nature, the (6n+5)th harmonics have

negative sequence nature while the (6n+3)th harmonics have a zero sequence nature.

Calculated results of eq. (2) and eq. (3) are shown in Fig. 8.

' On this calculation, in the area of FGHF in Fig. 3, fl' is always equal to {i-, and

in this case VC"2+ D"2, eq. (3), is independent of a, i.e. the harmonic components

become constant with a variation.

3-3. Harmonic Circuit and Analysis of Characteristics

The equivalent circuit per phase fbr kth harmonic is shown in Fig. 9.

The diflierences between this circuit compared to the circuit at fundamental line

frequency are those needed to take account the harmonic frequency, i.e. for a time harmonic

of order k, as fo11ows,

(a) all reactances are evaluated at the harmonic frequency Zijl, whereA is the fun-

damental frequen¢y,

116

.1.0

Ats. ,GS･illi

O.8

O.6

O.4

O.2

.o

Fig. 8.

T. Fujil, M. ･WATANABE and T.,

k=1

cosg= O.5

t-N /il . sNN, tt/7

11 .tt-~ t'N N13 X x

t r ' l, 'vt

x

IsHIZAKI

(ep= vr!3)

s-.s x s N N N NN

--- N t' N ts

IJrk

O T13 n/2 2vr13' 5n/6 a (rad.).

Harmonic voltage components for inductive load.

Iik ri kri r'2 kx'2 t'2-k iOk

go bo lk f2 (1 ],Sk)

Fig. 9. Equivalent circuit fbr kth time harmonic.

(b) the slip is the harmonic slip sk, and sk is given as fo11ows,

if synchronous speed of fundamenta1 field = IVI,

synchronous speed of kth harmonic = kAl}

rotor speed = N

Ak-N fundamental slip, s= M

kth harmonic slip, sk .. kATk ± N

klVk

-k±(1-s) k

(4)

Analysis of 71hree･･1]Viase lvduction Mbtors Controlled by T7iyristors 117

where positive sign is for the (6n+5)th harmonics and negative sign ls for the

(6n+1) th harmonics, ' ･ kth harmonic secondary frequency, nk ;;sklof1 r ' ･

-{k±(1-s)}A (5) '' 1';when's'is's'm"'all 6rk>(1=s), b'kN-1 dndftla 1!2ij1';' '' i' ' ''''''''i '' (6)

' ... . t.t .. In order to apply analytical methods based on the principle of superposition, certain

simplifying assumptions are paade regarding the motor.

' (1) The rotor has not the structure of double squirrel-cage or deep-slot squirrel-N

cage. ･ (2) I7)le does not become over the rated voltage.

(3) Saturation, hysteresis and eddy currents are ignored.

Considering 21k, Z,k, and V)le as follows, ･ - '･

. 21k == ri+fax,, 4k =ri+1'slekx2, l9>le == g,-j' bke ' '' (7)

'each characteristic is given in eqs. (8).-v(13).

exiciting current, '' '''･'''''idk = ?le4,+,,21i+k 2il,4,･]?r,. (8)

' ' ' -- primary current, Lk !t'i' tik4,+,S,'z+,4+kz･iY,k4,v, iii (Ak-.iBk)Vk (9)

-t- secondary current, i'2k == Vlez,k+sk2Zlk2k+Yli k4kvk (10)

cos gk == vA,A, ll:B,, (il) porwer factor,

primary input power, Pik = 31 Vhll41 cos qk (12)

pk=3usle)2 rE (1-s,) (13) output power, Sk ' This is only the analysis for kth harrnonic. So, if the suppiy voltage contains

various harmonics, the primary current is obt.ained by taking the square root of the sum

of the squares of all harminic currents.

primary current, ii,i'= VXX I.i,,I2 (14) le Similary :

primaryvoltage,' l",l=V]XIP,I2 ' . ' (15) le

tt priipary input power, Pi -- ZPik . .(16) .re

output power, P=t ZPle ' ' (17) t. re

118 T. FuJ!1, M. WATANABE and T. IsHTzAKi

P･ eMciency. op =: (18) Pi

power factor, cosg == v,-:i-IPlirtl (i9)

Therefore by knowing the rate of the viotage for aech harmonic frQm Fig. 8, each

characteristic is obtained.

Calculated characteristics are shown in Fig. 10, where triggering angle a=:-l},

stator line-to-line voltge (rms), l7L=150 (I7), and equivalent circuit constants are as

fo11ows,

r, = O.827 (2), r,' =: O.784 (a), x, = x', = 1.505 (2)

g, = O.O02 (cr), b, = O.024 (cr).

Characteristics for sinusoidal operation with the same line-to-line voltage are given

in dotted line.

A*s.Vs8o.g

80

60

40

20

iO

8

6

4

t"'L

<--'N

2

o

' t lt

ll tt tt tt tllt ti tt t:tt..-1 tt '

7

--< ttttl tt

ll --t" ttt 1- tp ' xt /1 / tl// t//

/ --t-t

t"

-- tt --t-

t//

--

`"-"-....ecsS q n

A --- -..-.

sinusoiclal

nonsinusoidal

Fig. 10.

O O.Ol O.02 O.03 O.04 O.Q5 sCalculated characteristics of a three-phase induction motor.

4. Conclustion

The !osses of an induction motor with nonsinusoidal waveforme impressed upon its

terminals can be markedly different from its sinusoidal losses, depending upon the har-

monic content of the impressed waveform.ii)"2) The most noticeable consequence of time

harmonics is the increase in motor losses. The harmonic-produced steady-state torques

are usually small and can be neglected in comparision with the fundamental torque. As

Analysis of 7hree-Phase induction Motors Controlled by Thyristors 119

the harmonic slip sk, eq, (6), is nearly equal to unity for higher harmonics, the motor is

locked rotor condition, the harmonic components can scarcely produce torques.

The motor can smoothly be controlled only fbr mode I. And near the boundary of

the mode I and mode ll, the motor performance becomes unstable. In mode ll, the

motor is fbrced to drive in single-phase operation. Therefore it is desirable to eontrol

in the region of mode I.

From eq. (6), the kth harmonic secondary frequency, f>k=tA!11, this means that the

secondary frequency is very high. When the supply voltage is rich in harmonics, the cir-

cuit constants become nonlinear by the influence of skin effect.'3)"`) In particular, the

secondary resistance, which disturbs the characteristics of the motor, cannot be held

constant, but increases with frequency.

This effect will be presented in another paper.

References

1) Bradley, D.A., Clark, C.D., Davis, R.M., and Jones, D.A., PIEE 111, 1833 (1964).

2) K,Y.G. Li, PIEE 116, 1571 (1969).

3) P.L. Alger and Y.H. Ku, IEEE Trans., PAS-76, 1335 (1957).

4) G.C. Jain, IEEE Trans., PAS-83, 561 (1964). S) Shepherd, W. and J. Stanly, IEEE Internat'1 Conv. Rec., Pt 4, 155 (l964).

6) Shepherd, W. and J. Stanly, ibid., 135 (1964).

7) Shepherd, W., IEEE Trans., IGA-4, 304 (1968).

8) Derek A., Paice, IEEE Trans., PAS-87, 58S (1968).

9) "I'. Fu,jii and T. Ishizaki, Lecture in Annual Meeting of Kansai Branch of I.E.E of Japan

(1969).10) M. Watanabe, T. Fujii, and T. Ishizaki, Lecture in Annual Meeting of I.E.E ofJapan (1971).

11) Eugene A. Klingshirn, and Howard E. Jordan, IEEE Trans., PAS-87, 624 (1968).12) B.J. Chalmers, and B.R. Sarkar, PIEE 115, 1777 (1968).

13) M. Watanabe, T. Fuiii, and T. Ishizaki, Lecture in Annual Meeting of Kansai Branch of

I.E.E of Japan (1970).

14) T. Fzljii, M. Watanabe, and T. Ishizaki, Lecture in Annual Meeting of I.E.E of Japan (1970).