Analysis of partly symmetrical machines by means of unitary transformation

ANALYSIS

OF PARTLY SYMMETRICAL MACHINES OF UNITARY TRANSFORMATION *

BY MEANS

BY

D. W. C. SHEN’

AND

H. N. G. BROADBENT

’

SUMMARY

The instantaneous impedance matrices of machines having electric and magnetic circuits on but one side of the air gap symmetrically arranged are shown to be Hermitian and unitary. Unitary matrices containing variable base vectors are used to transform the operational equations relating voltages and currents in six circuits, viz., three stationary armature coils and three rotating field coils on salient pole. The transformed equation is compact, convenient and amenable to transient solutions. It is correlated to previous results obtained by Park’s transformation in a simple manner. An analysis is then made on asynchronous operation under balanced terminal The results include the well-known voltages with the main field short circuited. Georges phenomenon of induction motor with single phase rotor. PARTIALLY-SYMMETRICAL

MACHINES

Polyphase machines may be classified according to their magnetic as well as their electrical symmetry of construction; the problem may be readily grouped into two classes. In the case of completely symmetrical machines, the polyphase windings on both sides of the air-gap are symmetrically arranged and the magnetic permeance is independent of the relative positions of the stator and the rotor. The so-called symmetrical alternator or motor studied by Rudenberg (1) 2 is simply a wound-rotor induction machine running at synchronous speed and excited by direct current. The partially symmetrical machines, the circuits on but one side of the air-gap being symmetrical, may be subdivided into round-rotor and salient-pole machines if only steady-state balanced operation is to be considered. In transient analysis, however, the most general case would be a salient-pole machine with three separate rotor windings, two in the direction of the polar axis and one in the direction of the inter-polar axis. It may be noted that when unbalanced condition is to be discussed, only the completely symmetrical machine will be free from harmonics. INSTANTANEOUS

IMPEDANCE

MATRICES

The instantaneous self-impedance matrix of the stator of a partially symmetrical machine Z,, may be written as two parts: one being the * This work has the support of the Electricity Association of Australia, whose funds were made available by the Electrical Research Board of C.S.I.R.O. 1 Department of Electrical Engineering, The University of Adelaide, Adelaide, Australia. 2 The boldface numbers in parentheses refer to the references appended to this paper. 473

D. W. C. SHEN

474

AND H.

[J. F. I.

IV. G. BROADBENT

constant part is of the usual form for a completely tionary network.

symmetrical sta-

(1) where ya = resistance of stator per phase to neutral, L, = Lao + Mao = constant part of the self-inductance of stator per phase (Lao) plus the constant part of the mutual inductances between two phases (Mao), P = f

= the time differential operator.

The variable part being a second harmonic function of time characterizes the saliency feature of the machine and is simply

Ma2

L,z cos 28

Ma, cos 2(8+120) cos 2(e- 120)

%Z =p where

M,zcos2(e+12o) LaBcos 2(e-- 120) M,, cos 28

(2) 1

M,~cos2(e-l2o) Maz cos 28 La2 cos

2(e+120)

Laz = maximum value of the second harmonic of the self-inductance of armature per phase, Ma2 = maximum value of the second harmonic of the mutual inductance between two phases, 8 = it, v being the rotational angular velocity between the armature and the field windings. In terms of the following two matrices, +

;

z]

G=[;2

pl

.2r a,

=

;3

+

Ht)

YJ

2s a2

=

~~‘3

Z,z may be expressed as za2

=

p(Laz

+

;

p

(

Ma2)

L2dze

&2O(H

+6Ma2s-~2e)

+

p(La2

G

+

p

;

Ma2)

(L,2~-i~~

t-2je(H”

;

M.2d2’)

+

Ht*)

G*.

(3j

UK.,

I().jL.]

ANALYSIS

BY UNT_~KY

In the above, the asterisk (*) denotes the conjugate means the transpose of the matrix. As for the armature-reaction matrix, it may be mutual-inductance matrix of the rotor with respect transpose of M,,, the mutual-inductance matrix respect to the rotor. Thus, JJ*, cos e PM,, = p Mz,cos0 [ &fgasin e

47.5

TILWSFOKMIATION

MI, cos (0 i14zacos (e -iIds, sin (6 -

120) 120) 120)

and the subscript, t, noted that Mar, the to the stator, is the of the stator with

M1,cos (e + 120) Mzn cos (e + 120) -MBa sin (0 + 120) I aMI, aMz, aMa,

- ja2Mh - ja2M20 c--j@ (4) - jaZMaa I

in which Ml,, Mza, MS0 are the maximum values of mutual inductances between one armature phase and the following: the main field winding, the second winding on the direct axis, and the winding along the quadrature axis respectively. Let the resistances and self-inductances of the two direct-axis field windings and the one quadrature-axis winding be rl, r2, ra and L1, L2, La, respectively. Also let the mutual inductance between the two coincident-axis windings be M12; then the self-admittance matrix for the rotor is ~1 + LIP Yf, =

M,zp

I

0

M,zp 72

-I- L2p

0

0

-1

0

.

(9

Ya + L,p 1

By performing the operation, pMaf.Yrr.pMI,, the armature-reaction matrix can be decomposed into the following Hermitian and unitary matrices :

A(p) and B(p) together with their conjugates are operational functions (see Appendix). The instantaneous impedance matrix of the machine is 2 = Z,, - PMaf.Yff .PMfar and from Eqs. 1, 2, and 3, may be put into the

D. W. C. SHEN AND H. N. G. BROADBENT

476

[J. F. I.

following form :

11

3

+ Ya + p-L - 3A*(p) 3 + p(L02 - Ma’)

11

1 1

a2

a

a

1

a2

i a2

a

1

+28(H*

+

+

p(La2

-

6

Maz)

a

a2

62,B(H

+

H

1>

H,*)

6 + 3L2P

+ JJa2p 3

-

3B(p)

+ 3LazP

+ Ma2~ z

-

3B*(p)

Ej2sG

e_i2BG*

(7)

Denoting the emf. due to d-c. excitation by E,, which is equal to - PM,, . Ide, the voltage equation of a partially-symmetrical machine is

V, =

E, +2-I,

where

(8)

v, =

Owing to the varying exponential terms in the above operators, the voltage equation cannot be decomposed into linear differential equations with constant coefficients. However, from a study of the form of 2 in Eq. 7, it is readily seen that unitary transformations with variable base vectors may lead to the linearization of the differential matrix equations. UNITARY

TRANSFORMATION

WITH VARIABLE BASE VECTORS

It is interesting to note that the first three matrices occurring in Eq. 7 characterize the internal reaction of a completely symmetrical machine, while the remaining ones describe the feature of saliency. In fact, the main object of the above analysis is to consider the machine as a whole instead of treating its reactions as many reactance comFrom such a general impedance matrix, ponents from the beginning. whose coefficients are expressible in terms of resistances, inductances, and the differential operators, it is at once apparent which terms taken by themselves give rise to complete symmetry, which terms, if coexistent with the former, will result in practical symmetry, and finally which terms in the latter are due to the saliency of poles.

Dec., rg.jr.1

Let

us

ANAl,Ysrs

BY

UKITAIIY

477

TRANSFORMATION

introduce the following unitary matrices:

(9)

and transform the voltage equation (8). We get U*(V, -E,) = U~Z~U~“~U.1,.

(11)

lxt the components of the new voltage and current matrices bc:

U. (V, - E,) =

V” V/‘,

and

I V7l1 The transformed equation takes the following form: VfJ

(+lGpO

v, =

ZdP+jv)

I

-3B(P+ju)

. (12)

ra+L.(p-jv)-311*(p-jv:

VT,

From Eq. 12, it may be noted that if the stator current of the machine is constrained to be strictly sinusoidal of fundamental frequency, there will be a third-harmonic voltage drop in the internal impedance due to the inequality of La2 and Maz. Conversely, if the line-to-neutral voltage is constrained to be positive sequence of fundamental voltage, a zero-sequence third-harmonic current will appear. However, if the operation does not involve neutral, there could be no zero-sequence current and both the voltage and the current arc sinusoidal. By assuming La2 = Maa, which is approximately true in practice, Eq. 12 simplifies to 7.7 +

=

(Lo - 2Mdp 0

0

0

0

ra+L,(P+jv)-3A(P+jv) $L,&-jv)73B*(p-jv)

%&+jv)-3B(p+jv) r=+L,(P-jv)-3A*(P-j~)

io

IiI

i. . (13) i,

Equations 12 and 13 are linear differential equations, the operational solutions can be easily obtained and interpreted by Laplace or Heaviside method. We shall not, however, go into details in this paper.

478

D. W. C. SHEN

ANI)

I-i.

REFERENCE

N.

BROADBENT

G.

[J. F. I.

FRAMES

The reference frames of machines have recently been discussed by several machine experts. Kron (2) has classified the reference frames of a synchronous machine. Ku (3) in his recent paper has emphasized the advantages of rotating axes for the transient analysis of both rotating and stationary networks. Rotating reference axes are necessary for balanced polyphase machines whose axes may be assumed rotating with the revolving field, thereby reducing the analysis to that of direct current machines. It is important in hunting studies. Park’s (4) two reaction theory recently discussed by Adkins (5) and Vowels (6) employs rotating axes in the transient study of synchronous machines. It simplifies the problems by transforming the machine from slip ring to two quadrature brush sets, and the resultant direct and quadrature currents become direct currents. In a previous paper (7) the author pointed out that the normalized Clarke’s co-ordinates and Park’s two-reaction co-ordinates may be obtained from the phase axes by orthogonal transformations, while the stationary and the rotating symmetrical co-ordinates are obtained by In fact, the normalized Clarke’s co-ordinates unitary transformations. are related to the stationary symmetrical co-ordinates in the same way as the two-reaction co-ordinates are relate\. to the rotating symmetrical co-ordinates. To correlate Eq. 13 with Park’s result, we may introduce the following simple transformation (8) :

I ::/I Ii-1: Vd

_j

TJ4

and transform (13) to

-l

v8

L

v,

r vdl

(1.9

where

D,(p) Qd(p)

= =

Q,(p)=

(14)

(La - %Lz)v + S-P

(La + %& ya +

- WvP

(La - 3Laz)P - SOP2

Dec., 1952.J

ANALYSTS

BY

UNITARY

479

TRANSFORMATION

and a = Mla’(~z + L2P) -

2M1aMzaM12p + M2a2(71 + LIP)

(71 + L~p)(rz + L2p) -

M,22p2

M&l2 A= Y3bw

For steady state synchronous operation, P = 0, v = w, the above operational two reaction impedance functions reduce to Dd(O) = Ya,

Q&N

Q*(O) = Ya the direct axis synchronous reactance

= w(La + 3La2) = wLd = Xd,

D,(O) = - w(L, -

wL, = X,, the quadrature axis synchro-

$L,2) = -

PERFORMANCE

nous reactance.

CHARACTERISTIC

ON BALANCED

VOLTAGES

Equations 12 and 13 give all the information both transient and steady state of the machine under any terminal conditions. We shall investigate, however, the performance of a machine under balanced voltage. Let the voltage applied to the motor be a balanced positive sequence given by (16) then

(17) where s is the slip under asynchronous operations. Substituting (17) into (13) and solving for & and i,, we get i = 4\i4V~i+~[~,+~,&+jr) -3fl(p+jv)]-I

i,=

~~VC-~~“~[L’.&~+~V)

-3B(p+jv)J

(18)

D(P) ~y~~-~~~‘[~~+~~(p-j~)--3il*(p-j~~p~

~\ITVE-+~[L’&--~V)

-3B*(p-jv)]

(19)

wherein L’a2

=

3La2

and D(P) =

Ya+La(p+jv) -3A(p+jv) Lft12(P-jv) -3B”(P---jv)

L’az(P+.iv) -3B(p++jv) -fa+L,(p+jv) -3A’(p--jv)

*

(20)

480

D. W. C. SHEN

AND

H. N. G. BROADBENT

[J. F. 1.

To obtain the steady-state value, it is only necessary to change p in the operators to jsw or -jsw as the case may be. Then we get

-

++[-(SW

- Y)L’,z - 3B*j(sw - V)].

(22)

D (.isw) The above two expressions can be conveniently ing forms:

put into the follow(23)

If the operation does not involve neutral, the line current can be calculated from

Hence the current in line a is of the form, i.z = 1, cos (wt + 41) + I
1)wt + 42)

(26)

where @V I-=dm

GiV 108_-l)U= ~ GzF

(27)

Equation 26 shows that the stator currents consist of two frequencies and the current of each frequency forms a balanced system. The average torque may be calculated as the algebraic sum of the torques produced by each balanced system as if the other were absent. For rotor speed between standstill and half synchronous value, the torque of currents having (2s - 1) times impressed frequency is negative and should be subtracted from that produced by currents of imThe exact shape of the resultant curve depends on pressed frequency. the constants of the machine. However, the well-known Georges

Dec.,

ANALYSIS BY UNITARY TRANSFORMATION

19jZ.l

481

phenomenon (9) of having a dip of the resultant torque near halfsynchronous speed is present in every case. SPECIAL CASES

Let us consider the simpler case of machines having main windings only. The operators in Eq. 13 reduce to the following:

JfZP(P + $)

A”(p - j,) = C”($; 1

A(p + jV) = 4(71 + LIP) A(p

+ j,>

B(P -I- $1

=

A”(p

- jv)

= B”(p

;) 1

-

jv)

(28)

and the denominator becomes

aP)

= [% + UP + _i4lCr. + L(P - j4lhl + -LPI - [ra + L(p - _i~)lWP(P + _i4 - [rcz + -L(P + jz~)]tM~p(P -

jv>. (29)

Therefore, the impedances, i, and iz, in Eqs. 23 and 24 are given below:

(a)

Synchronous Speed, s = 0 For synchronous operation, 2,

=

g2

=

Ya’+ Xa” - X20’?= Ya2 + xlix, ye - jX, yn - 2,

(32)

Yn2+ xa2 - x202= ya2 + x,x, .iXzo _iX2,

(33)

where Xd = X, + Xan is the direct-axis synchronous reactance, and X, = X, - Xzo is the quadrature-axis synchronous reactance. The two components of stator current are of the same frequency ; hence the two corresponding admittances, I’1 and Pz, may be added in series if the displacement angle is constrained to be zero. Thus, the impedance per phase is 1

i= FI

+

Ya2+ XdX, Ii2

=

7a

-

jX,

for ~2 is smaller than either X,? or XdX,.

2L$kr.+jXd 4

(34)

D. W. C. SHEN

482

AND

H. N. G. BROADBENT

[J.F. I.

Similarly with angle constrained to be 90°, the impedance per phase is 1

i=

Yo2 + x+X,

=2-y,

+ jX,. d

(35)

The above reveals, incidentally, the principle of the well known slip tests. Thus, if the machine is run at small slip, the ratio of the maximum voltage to the minimum current gives the direct-axis synchronous reactance, Eq. 34, and that of the minimum voltage to the maximum current, Eq. 35, gives the quadrature-axis synchronous reactance. For round-rotor machines, La2 = 0, 2, = co, the impedance per phase will have the value, 2?, = -ra -I- jX,, where X, = Xd = X, is the synchronous reactance independent of the displacement angle. (b) Half-Synchronous

Speed, s = 3

At half-synchronous speed, the current is of impressed frequency only. The internal impedance of the machine becomes simply

i~=r.+jX~+

$Xm2

(36)

2r1+ jX1

which does not depend upon the saliency of the machine and therefore is of the same form as that of a polyphase induction motor operated at half speed; however, in the latter case, the rotor reaction is

B-G2 2~1 + _iX, instead, being three times that of the present case. Assuming that the rotor resistance is negligible, Eq. 36 can be conveniently written as

2j;l=yo+jXa

(1-g >=

ya +

j7X,

(37)

1

$Xm2 may be interpreted as the equivalent coefficient of where 7 = 1 - Xkyl coupling between the stator and the rotor. The conductance and susceptance parts are g=

respectively.

7.

Ya2+ r2xlz2

and

b =

7xa Ya2+ 72xa2 ’

Dec.,

ANALYSIS BY UMTAICY TKANSFOIWATIOK

1c)jL.l

(c) Standstill,

1k3

s = 1

At standstill, the current is again of impressed frequency and the two admittances, Y, and Iiz, may be considered to be in parallel; their equivalent value is simply Y1 + Yz.. (d) Infinite

Speed, s =

00

Since infinite The second term of Eq. 26 is of infinite frequency. It is slip cannot be obtained physically, this is, of course, fictitious. interesting, however, to investigate the forms of i, and i, for these In order to show the relationship of i, and i, limiting conditions. with their values at synchronous or half-synchronous speeds, let us introduce the coefficient of coupling, u, defined by Q -_ I - 3 x,2 2x,x, and consider the simple case of cylindrical rotor only. we find

Neglecting Xza,

2

(38)

2, =

an d

g2=j

Ya2 + x,2 (

7-l

(a>2 7

)

2

(39)

u

Since the stator current in this case is not a simple sinusoidal wave of fundamental frequency, the equivalent admittance for infinite slip should be obtained by combining the two terms in the usual manner. Thus, 2

(40)

y = j+y2;li_.

The ratio of power to the square of the voltage will give the conductance part of the admittance. Hence, R=

Ya 2

and

b = dY2 - g2.

The value of the susceptance will be approximately the same as Y since g is usually small compared with b.

D. W. C. SHEN

484

H. N. G.

AND

[J. I;. I.

BROADBENT

CONCLUSION

It can readily be inferred from the preceding treatment that the unitary transformation of Eq. 18 is just as useful and usable in studying the performance of partly symmetrical machines as Park’s transformation. Moreover, it can throw additional light on the subject when Park’s transformation fails to show. For example, under steady state balanced operation, Eq. 13 becomes simply

(Ya+

jx,>r-

jx,J*

= & - p

where X, = wL,, Xaz = wLfaz, 1 is the armature current, f* its conjugate. If B is taken as the axis of reference, and the angle between 2 and 1 be a, (13~) becomes [(r.

+ Xaz sin 2a) + j(X,

- Xaz cos 2a)]f

= IJ - P.

(13b)

Note that the term Xa2 sin 2a! corresponds to an additional resistance, and when sin 2a! is negative, we have the effect of negative resistance. This seems to be a novel feature of the saliency effect. Acknowledgment The authors are indebted to Professor E. 0. Willoughby for help in Grateful thanks are due to Professor Y. H. preparing the manuscript. Ku of M.I.T. for his valuable suggestions and criticism. REFERENCES (1)

R. R~~DENBERG,“Kunzschlaszstrom Elektrische

Sctialtvorgange,

(2) G. KRON, “Classification

bein Betrieb von Groszkraftwerken,”

Berlin,

1925 ;

Springer, Berlin, 1926.

of the Reference

Frames of a Synchronous

Machine,”

Trans.

A. I. E. E., Vol. 69, Pt. 11, p. 720 (1950). (3) Y. H. Ku, “Transient Analysis of Rotating Machines and Stationary Networks by Means of Rotating Reference Frames,” Trans. A. I. E. E. (Reprint Tl-176) (1951). (4) R. H. PARK, “Two-reaction

Theory

of Synchronous

48, No. 2, p. 716 (1929). (5) B. ADKINS, “Transient Theory of Synchronous

Machines,”

Generators

Trans. A. I. E. E., Vol.

Connected

to Power Systems,”

Journal I. E. E., Pt. 11, Aug. 1951. (6) R. E. VOWELS, “Transient 33, April, 1951. (7) D. W. C. SHEN, “Generalized

Analysis of Synchronous Co-ordinates

Machines,”

in Substitutive

I. E. E. Monograph

Networks,”

No.

Phil. Mug., Vol. 39,

p. 890, Nov. 1948. Impedance Matrices of n-Phase Partially Symmetrical (8) D. W. C. SHEN, “Operational Machines,” Aust. J. Sci. Res., Ser. A, Vol. 4, NO. 4, p. 544 (1951). Motors with (9) H. L. GARBARINO AND E. T. B. GROSS, “The Georges Phenomenon-Induction Unbalanced Rotor Impedances,” Trans. A. I. E. E., Vol. 69, Pt. 11, p. 1569 (1950).

Ik.,

19j.?.]

ANALYSIS

BY

UNITARY

TRANSFOI:MATION

48s

APPENDIX Rotor Reaction Since M,f is the transpose of M,,, The rotor reaction of the machine is pM,,.Yff.pM,,. hence by Eqs. 4 and 5, after carrying out the multiplication, we get the four matrices given in Eq. 6. The operational functions, A(p) and B(p) are

where the symbol, p + be replaced by p - ju.

(p - jv), meaus that all p’s on the right of the above equation are to A*(p) and R*(p) are the corresponding conjugate functions.