transient theory of synchronous generator under unbalanced conditions

621.313.322.016.313 Monograph No. 85SUPPLY SECTION

TRANSIENT THEORY OF SYNCHRONOUS GENERATORS UNDERUNBALANCED CONDITIONS

By Y. K. CHING, B.Sc., and B. ADKINS, M.A., Member.{The paper was first received 29 th May, and in revised form 26th August, 1953. It was published as an

INSTITUTION MONOGRAPH, 15th December, 1953.)

SUMMARYThe paper contains a mathematical theory of the operation of a

synchronous generator after the closing of a switch at its terminals,under the assumption that the speed remains constant after the change.The theory is first formulated for the general case where the externalcircuit consists of any unbalanced impedances and source voltages.It is then applied to the three well-known conditions of unbalancedshort-circuit. The method of solution is fully explained for the line-to-line short-circuit, while only the equations and the results are quotedfor the line-to-neutral and the double-line-to-neutral short-circuits.

The solution is obtained by the method of the Laplace transforma-tion, but its application is more difficult than in ordinary circuit theoryor in the case of a generator under balanced conditions, because thedifferential equations, although linear, have variable coefficients.The solutions are obtained as infinite series, which can be summed,however, for the cases given.

For each of the three short-circuit conditions, comparative experi-mental and theoretical results are given for a small synchronousmachine. The theoretical curves were calculated from the mathe-matical expressions, using the measured constants of the machine.

(1) INTRODUCTIONWhen the external connections of a 3-phase synchronous

generator are balanced between the phases, the voltages or thecurrents are sinusoidal under any condition of steady operation,and each forms a balanced 3-phase system. Under transientconditions the quantities are still substantially sinusoidal in form,but the amplitude varies with time and the wave may be dis-placed away from the zero line. For example, the current aftera symmetrical short-circuit is of this type, and its value as afunction of time can be calculated by means of an operationalsolution of the equations.1

When the external connections are unbalanced, the voltagesand currents are not sinusoidal and may contain pronouncedharmonics. During steady operation each of them is periodicand can be expressed as the sum of a fundamental and an infiniteseries of harmonics.2 Under transient conditions the wave is ofa periodic type but is not sinusoidal, and the shape of the wave,as well as its magnitude, varies with the passing of time.

The analysis of a single-phase short-circuit on a synchronousgenerator was first given by Doherty and Nickle,3 who obtainedexpressions for the transient currents in the armature and fieldcircuits and verified them by means of oscillograph tests. Afull treatment of the three alternative types of short-circuit ofa 3-phase generator (line-to-line, line-to-neutral and double-line-to-neutral) is contained in the recent book by Concordia,4

who investigates the open-phase voltages and the torque as wellas the currents. The method used by both writers is to derivethe initial values of the components by approximate methodsand to estimate a time-constant appropriate to each component.

The present paper gives a more rigorous and general analysis

Correspondence on Monographs is invited for consideration with a view topublication.

Mr. "" 'Uni

Mr. Ad kins is Reader in Electrical Engineering, Imperial College of Science andTechnology, University of London.

of the synchronous machine under unbalanced conditions.Operational expressions are obtained for the currents andvoltages with any combination of external impedances. How-ever, the evaluation of the quantities as functions of time is notfeasible in the general case and can be carried out only forsimplified conditions. Rigorous mathematical solutions havebeen obtained for the three short-circuit conditions, subject tocertain assumptions about the relative magnitudes of the para-meters, and lead to expressions which agree with and supplementthose obtained by Doherty and Nickle, and Concordia.

The solution for the balanced short-circuit is relatively simple,because when the equations are expressed in terms of direct- andquadrature-axis quantities, they take the form of linear differentialequations with constant coefficients in which the applied voltagesare known. In the unbalanced problem, the solution is moredifficult. The analysis given starts with equations containing thephase quantities and transforms them into new equations usinga, j8 components. They are linear differential equations but thecoefficients are variable.

(2) GENERAL EQUATIONS OF THE 3-PHASE SYNCHRONOUSMACHINE

(2.1) Assumptions

The machine analysed is an "ideal synchronous machine" asdefined by Park,5 and illustrated in Fig. 1. It has a field winding/

uiivauuu.Mr. Ching was formerly at the Imperial College of Science and Technology,diversity of London, and is now at King's College, Hong Kong.

Fig. 1.—Diagram of a 3-phase synchronous machine,and direct- and quadrature-axis damper windings kd and kq.The three armature phases are labelled a, b and c.

The principal assumptions are that(a) The machine runs at constant speed.(b) There is no magnetic saturation.(c) There are negligible space harmonic effects.

Assumption (c) corresponds to Park's definition of a sinu-soidally-distributed armature winding. All the harmonic wind-ing factors for the armature are zero, and consequently thecurrents produce no space harmonics of magnetomotive force.In a salient-pole machine, space harmonics of flux exist because

166]

CHING AND ADKINS: TRANSIENT THEORY OF UNBALANCED SYNCHRONOUS GENERATORS 167

of the variable permeance, but they cannot induce any voltage inthe armature winding.

In the problem considered, the generator is assumed at firstto be running steadily on open-circuit with a known field voltageefQ. At zero time the machine is connected suddenly to anexternal system consisting of linear static impedance elementsand voltage sources. The solution determines the currents inthe armature, field and damper circuits, the terminal voltagesand the torque, as functions of time.

(2.2) Voltage Equations of the MachineWith constant speed co, the angle 6 between the pole axis and

the axis of phase a is given by0 = cot + A

where A is the angle at zero time.The equations relating the terminal voltages to the currents

in the six circuits are expressed by the matrix equation

M = [*JM 0)where [Zm] =

a b c

where

[vm] =

a

b

c

f

kd

kg

Vat + *a

Vbt + eb

vct + ec(4) [/]

a

b

c

f

kd

kq

ia

lb

*c

' /

hd

ikq

(5)

ea — E sin 6

(6)

kd kg

, _ dfa + La0j-(

+ ?La2 cos 20

mr d

+ ^ - ( 2 0 - T )

+ ^La2cos(20+^)

-TrAffl/COS 6

-jMaf COS 6

— jMaq sin 6

M d

M*°dT

+ J(La2cos ( 2 0 - 3 )

, T dra + LaO-7£

+ d

StLa2cos(2B+2l)

+ j(Lal cos 26

d „ , ft, 2TT\

JtM«fcos (6-T)

d., . fa 2TT\

Ms4+ ^La2cos(26+^j

MsOdi

+ j(La2 cos 26

J. T d

-j-AfqfCOS 6

-Mqfcos(6-T)

-Mqfcos(6+T)

_L r d

r'+L'Jt

Mfkdjt

-rMtf COS 6

d.., fa 2TT\

dJM<*C0SV-j)

A* d

MfkdJt

, T d

r/cd + Lkdjt

— jMaq Sin 6

d., . / o 2TT\

-d<M°«Sm{d-3)

d.. . /a , 2TT\-^tMagsm (0+-j)

• r d

rkq + Lkg^

f

kd

kg

and [vt] and [/,] contain the six voltages and currents at theterminals.

The symbols r, L and M denote respectively resistance,self-inductance and mutual inductance, and the suffixes indi-cate the windings to which they relate. All the quantities aremeasured as fractions of an arbitrary unit, i.e. they are "pre-unit values." The derivation of the impedance matrix is givenin Reference 2.

By a simple substitution the equation may be modified sothat the field current in eqn. (1) is not the actual current if, butthe current ij- by which it differs from the initial steady currentIj-Q — e/olry. After the substitution has been made, the appliedfield voltage becomes zero and the armature voltages are modified.

The equation then becomes

k,]=K]['] 0)

(2)

and vat, vbt, vct are the phase terminal voltages.steady phase voltage.

E is the maximum

(7)

(2.3) Operational Equations for Phase Currents

In Appendix 10.1, the Laplace transformation6 defined by

is performed on eqn. (1) assuming the currents to be initiallyzero. By eliminating the field and damper currents and trans-ferring terms like ia(p + y2a>), etc., to the voltage matrix, theoperational voltage equations of the three armature circuits areobtained in the form

- [ZOO] POO] (8)

168

where [v(j>)] =

CHING AND ADKINS: TRANSIENT THEORY OF

*Jp) + ea(p) - }p{e-mZD(p + ja>)[IJLp + j2co) 4

^i/07) + ^ C P ) ~ ip{he~J2XLD(p +joo)[ia(p +j2a>) -

«rt(p) + ec(p) - i ^ - y z x l ^ + yW)[f> + y2co) -

- hib(p

±hib(p

hhib(p

+ j2co) + W

+ j2co) + h

+ j2cS) + h

lc{p + y'2oj)] 4- conj}

lic(p 4- j2a))] + conj}

llc(P + 72co)] 4- conj}

(9)

ra + i^[^o + AsG? + y60)

| ^ [ L 0 -4- hLs{p 4- yo>) 4

•Jp[Z,0 + h2Ls(j> 4- ;co) -

+ conj]

- conj]

f- conj]

ip[Lo

ip[Lc

+ h%

[L0 + 'i

+ hLs

<P

ip

+ ju)-{

P + j«»)

- conj]

+ conj]

conj]

lp[Lo -

MA)4

ra + ip[i

\-hLs(p +

- h2Ls(p 4

o + ^ O 3

yco) 4-

- fa) 4

+ yw)

conj]

conj]

4- conj]

do)

and (ID

Td0~Lflrf

Mfk

In the voltage and impedance matrices (9) and (10), ry - f ^ - M A ^ + Mrf£ / -?MfoWL0 = La0 2 M

(15)

s0

^ M 2

(rf + pLf)(rkd

~ 2Mfkd)rkq

• (12)

Md =

The term "conj" is used to indicate the conjugate functionobtained by replacing j by —j. It should be noted that it isnot the same as the numerical conjugate when p is complex, h isthe complex number d2"/3.

The operational impedances Ld(p) and Tq(p) may be expressedin terms of six time-constants as follows:

(16)

The form of eqns. (13) and (14) is more convenient for evalu-ating Ld(p) and Lq(p), particularly when approximations are to bemade. When p = 0, each factor becomes unity. When p islarge, the figure of unity in each factor may be ignored.

(2.4) Operational Equations for a-f& Components of Current

To solve the equations it is advantageous to transform theminto new equations in terms of a, jS, 0 components instead ofthe phase values. The transformation has the effect of replacingthe 3-phase machine with currents ia, ib, ic, by an equivalent2-phase machine with currents ia and z'p, allowance being madealso for a zero-component of current /0 which flows equally inall three phases.

The transformation matrix [S] and its inverse [S]-1 are

a j8 0

[S]= b

Tdop)(l + T^

Tq'op

(13) and

(14)

where !T 0, 7^Q, Td and 7^' are found by factorizing the numeratorand denominator of the expression for Ld(p). In many practicalcases, TdQ and Td are much larger than Td0 and T'd, and thesefour time-constants then have approximately the values given in equation iseqn. (15). TqQ and Tq have the values given in eqns. (16).

1

1

~~2

1

~2

a

2

1

V32

V32

b

j

V3

1

1

1

1

c

- 1

-V3

1

• • (17)

• (18)

Applying this transformation to eqn. (8), the new voltage

(19)

SYNCHRONOUS GENERATORS UNDER UNBALANCED CONDITIONS

where [«(/>)]'= [S]~l[v(p)] =

a

P0

*«,(/>) H

«P,(P) -

h 4CP) - */>[« mLD(p

\-h(p)-*p[je-'*ZD(p

+ ycu)fa(/> + y'2a>) -

+ »,;(/>+;2a>)

- conj] — -J-/7[ye J2^LD(p -\- jco)l

+ conj] + ip[e~->lxLD(p -\- joi)lr

tip)

i(p+j2aj)~

(p+y2co)H

h conj]

- conj]

169

(20)

a

J80

ra + h

a

p[Zs(/7 + yaj) + conj]

Ls(p + ya>) + conj]

P[jLs{p + JOJ) + conj]

^[I^Cp+yw) + conj]

0

r.+pZ*

(21)

. • (22)

The quantities in the new voltage matrix are

and

(23)

• 7 3 ^ ~The phase currents are related to the a, ft, 0 components of

current by the formula

wcosA+/?sinA\

P cos A — o> sin A

(3) DERIVATION OF OPERATIONAL EXPRESSIONS FORTHE CURRENTS

(3.1) Armature Currents

The three simultaneous eqns. (27) suffice to determine fa(p),ijj(p) and I0(p). By eliminating I0(p) and l${p) by the methodoutlined in Appendix 10.2, an equation containing ljj>) onlycan be obtained.

= Up) + i. . . . (28)

where the coefficients are known functions of p derived in thecourse of the elimination.

Eqn. (28) is a recurrence equation containing fa(p ± j2oS) aswell as Ia(p). The next step is to solve eqn. (28) in order todetermine fa(p) as an explicit function of p. The expression forIJJJ), derived by the method explained in Appendix 10.3, isobtained in the form of an infinite series

(24)

The simplification obtained by means of the above transforma-tion arises from the fact that, as far as the machine is concerned,the zero components are independent of the a and ft components.

In order to complete the equations the external source voltagesand the external impedances must be introduced. These maybe expressed in terms of a, ft, 0 components by operationalmatrices of the following form:

W) = A0(j>)FaiP)

where

. (29)

1

+ Y%p) - 1

and

a

ft0

V<xs(P)

V$S(P)

VQS(P)

(25)

A_n{j>) = ( - *)

1

YJip-j2w)

- j2u>)

a

ft0

a

*.(*)

P^(P)

z^ip)

Z0&)

0

^BOCP)

^oo(^)

1 - - j2*J)Ka(p ~ J4to)

The complete equation is then

• • (26)

' • • • (27)

1 - . . .

7*(/>) = conjugate function of YJjj)

_ iKa(p)K^(p + j2co)Yl(p+j2u>)

(30)

A similar expression for l$(p) may be obtained by eliminatingia{p) and fo(jj) from eqn. (27). I0(p) can then be found bysubstituting ia(p) and I^(p) in eqn. (27).

Finally the values of the currents as functions of time are

170 CHING AND ADKINS: TRANSIENT THEORY OF

determined by the inverse transforms of the operationalexpressions.

(3.2) Field and Damper CurrentsIn deriving expressions for the field and damper currents it

is convenient to introduce the axis components id and iq of thearmature currents, defined as follows:

U *

iq =

cos

sin 0 +

+ ib cos (d ~ ~ ) + ic cos (0 + Y

(9 - 2fib sin /c sin

The transforms of these quantities are

UP) -= ib~JlUp +/«) + conj + he-Mb(p +y«)+ conj + h2€~Îc(p +jco) + conj ]

jto) + yfp(p + yo;)] + conj(31)

The voltage equations (100), (101) and (102) in Appendix 10.1can be written

+ conj +jh2e~Wc(p +yo>) + conj]

- ipO + ju)j\ - conj

kdkd (32)

0 = WaqPtqiP) + ('** + LkqP)lkq{p)

The following expressions are obtained by solving eqns. (32),

- Mfkd) pld(p)rf rkd

~ Vf

Lkd ~ 2M/kd).

rf + p(Lf — M/kd) pid(j>)kd-2Mfkd)Zd(p)

(33)

Hence if fa(/>) and i$(p) are known, f/(/?),b f d i'( i h f f h

Hence if fa(/>) and i$(p) are known, f/(/?), fÔ?) and ikjj>)can be found. i'f(p) is the transform of the superimposed fieldcurrent which is added to the original steady current 7y0.

(3.3) Electromagnetic TorqueThe torque developed by the machine is given by the following

expression due to Park:7

T - ifjdiq - tfiqid (36)

where tpd - f [Ldid + Mîj- + ikd)]

and iftq = f (Z,^ + JWaffifc^(37)

Alternatively the transforms if*d(p) and ifjq(j>) are given, interms of the armature currents and the original steady fieldcurrent 7y0, by

. . (38)

(4) APPLICATION OF THE THEORY TO THE SINGLE-PHASELINE-TO-LINE SHORT-CIRCUIT

(4.1) Armature Current(4.1.1) The Operational Equation.

The single-phase line-to-line short-circuit is the simplest caseof unsymmetrical transient operation of a synchronous machine.In this Section the transient currents and other quantities aredetermined as functions of time by solving the equations obtainedin Section 2, with certain simplifying assumptions. The short-circuit takes place between lines 6 and c at the instant when theopen phase a is at an angle A from the field axis, and as before,the speed is assumed to remain constant after the short-circuit.

Under this condition, ia(j>) = lo(p) = %,(p) — 0. Therecurrence equation for z'pQ?) follows directly from the secondequation of eqn. (19), which becomes

+ J'2co) - conj . (39)

and gives on rearranging, an equation of the same form aseqn. (28), but with jS instead of a,

+y'2eo)

(40)

wherep cos A — co sin A

ra + %P[LS(P +M+ conj]

pZD(p +jto)

-jco)(41)

(j> +JOJ) + conj]

The operational expression for fpO) is similar to eqn. (29),

= BQ{p)F^{p) + 2 [Bn(j>)F&(j> +j2naj) + conj]71—1

(42)

where

(34) °n\P) = \— ?<

(35) and Y^(p)=l-

(43)

Eqns. (42) and (43) are expressed in a simpler form than themore general eqns. (29) and (30) because, for the special caseunder consideration, F§(j>) is a real function of p . In obtainingthe conjugate quantity in eqn. (42), it is understood that Bn(p)is substituted from eqn. (43) before j is replaced by —y.

(4.1.2) General Method of Solution.To determineri§ as a function of time, the inverse transform

of the expression for itfjj) in eqn. (42) must be evaluated. Todo this, the poles of each term like Bn(p)F^,(j> + j2nco) must befound and the function split up into partial fractions, by thefollowing method. If p = pk is one of the values of p whichmakes fp(p) infinite, the corresponding partial fraction is

where Ck = [(p - pk)i^{p)]p,-Pk

and the inverse transform is

SYNCHRONOUS GENERATORS UNDER UNBALANCED CONDITIONS 171

The solution therefore consists of several infinite series ofexponential terms. The values of pk are imaginary or complexand occur in conjugate pairs, so that the final result consists ofan infinite number of real sinusoidal terms some of which havean exponential decrement factor.

Because of the complicated nature of fp(p), containing as itdoes the continued fractions Y§(p), it is necessary to makesome simplifying assumptions about the relative magnitudes ofthe parameters, as explained later. With these assumptions, itis found that the solution can be obtained in four parts,

(a) Steady-state component i$s

(b) Field-transient component i'%

(c) Sub-transient component /£

(d) Armature-transient component /pa

Pk = - xa ± J2nco

(4.1.3) Steady-State Component.There are evidently poles at p — ± j(2n + 1)&>, depending on

the factors (p + jco)(p — jco) in the denominator ofThese poles determine the steady-state component i$s.

The transform of the steady-state component is given by

= 2 - T - T T T - f r + conJ • • (44)

where Cn = [(/> + j2n + 1 CL»)/pO )]/» =» —y<2«+i>«o • • (45)

Only two of the terms in l$(p) have a pole at/? — —j(2n + l)co,and consequently only these two terms need to be considered.Hence eqn. (45) reduces to

+ j2nco)

• (46)

Table 1

FORMULAE FOR FUNCTIONS IN fp(/>)

Function

KQ\J>)

K%\p)

YQ\P)

Y$(p)

KQ,(J>)

Y$(p)

Bo(p)

Approximate value when p ~ yw

2(JL'q — Lrf)

Ld(p — Jco) + 2- CP — y*") + •£</ "1" -^9

2[Lq(p — jco) — Zd(p — jco)]

"Edip — jco) + Zq(p — jco) + L'J + L'4

j [Eq(p — jco) — Ed(j> — jco)]2

/ \Ed(P — JCO) + Zq(j? — jco) + L'J + Lq] XX X [Zd(P — jco) + Zq(p — jco) + 2Lm\] J

L~d(P — jco) + Zq(p — jco) + 2Lm\

L"d(p — jco) + Zq(p — jco) + L'J + Lq

2[Lq(p — jco) — Zd(p — jco)]

L~d(p — jco) + Eq(j) — jco) + 7Lm\

( [L~d(P — jco) + Lq(p — jco) + L'J + Lq] X1 X X &d(j> - jco) + Eq(p - jco) + 2Lml] J4 (Ed(j> — jco) + Lm]\Lq(j> — jco) + Lm\]

2E 1 yeA10 Zd(p — jco) + Zq(p — jco) + L'J + L'4 P ~ J0*

2E 1 -yc-yx00 Ed(p — jco) + Zq(p — jco) + L'J + L'4 P ~ J0*

Approximate value when p ~ — /to

2[Lq(p + jco) — Ld(p + jco)]L~d(p + jco) + Tq(p + jco) + L'J + L'q'

2(L'q' - L'J)

L~d(P + jco) + Eqfjp + jco) + L'J + L'4

Ed(p + jco) + Eg(j) + jco) + 2Lmi

L~d(p + jco) + Zq{p + jco) + L'J + Lq

j [Lq{p + jco) — Zd(p + yco)]2

/ [Ld(P +JCO) + Zq{p + JCO) + L'J + L'q'] \X X [Zd(p + jco) + T^p + jco) + 2LmX] J

2[I«G> + jco) - Ld(p + jco)]L~d(p + jco) + Lq{p + jco) + 2Lm\

f [Ld(p + jco) + Zq(p + jco) + L'J + L'4] \1 X x lL~d(p + jco) + Lq(j} + jco) + 2Lmi] J4 [Ed(j> + jco) + Zm][Lq(j> + jco) + Lml]

2E 1 -ye-/Xco Zd(p + jco) + Eg(p + jco) + L'J + L'4 P+ }<»

2E 1 yWX

^ Ld(p + jco) + Lq(j> + jco) + L'J + Lq'P+ J10

Approximate value whenp ~ ± yvw (v > 2)

Lq'-LdL'J + L'4

Lq — L'JL'J + L'4

1 WL'J + VI4f)2

2 L'J + L'4

2 L'J + Lq'

- *<JL'4—"\/L'J _,•£—77-77-7—7777 -2JD\

VLd + VLq


(47)

In evaluating eqn. (46) it is assumed that ra is negligible com-pared with coLd, a)Lq, coLd or coLq, L"d and L'q', being given by

L'd - (T'dTd'lTdQTd'0)Ld

Lg = (TqITq0)Lq

and that 2co7^0, 2cur^, 2ojTd0, 2coTd, 2o)Tq0 and 2w7^' arelarge compared with unity. These assumptions are used inobtaining the formulae, given in Table 1, for the functions inI$(j>) when p has values near to ±jco or multiples of ±jco.Table 2 gives the values to which some of the functions reduce

Table 2

VALUE OF FUNCTIONS WHEN p — ±jo>

Function

K$(p)Yfa(p)

Bo(j>)

F$(j>)

F$(p + j2co)

Value ofP

-jco

~J(O

- j .

-jco

Approximate value of function

2(LQ-Ld)Ld + Lq + 2Lm\

1 (Ld + Lq + L'J + L'q'XLd + Lq + 2Lm\)4 (Ld + Lml){Lq + Lml)

2E 1 -yWXco Ld + Lq + L'J + Lq p + jco

2E 1 y'eACO Ld + Lq + L'J + Lq p + JCO

when p = ±jio. It may be noted that the continued fractionsYp,(p) and Y^ip) are evaluated by solving a quadratic equation.In the Tables the symbol Lml is introduced.

By using the values of the functions in Table 2, and in thelast column of Table 1, eqn. (46) simplifies to

s-i t __

w Ld -f Lml

where bx is a constant having the value shown in Table 1.

Substitution of eqn. (48) in eqn. (44) gives

E 1 » I f_&,v«;e-i(2H+i)x

(48)

1 oo r(—bt\n}€-jQn+\yk 1

-Lm, w=oL^+X2/i+ 1) J (49)

The inverse transform of eqn. (49) gives the explicit expressionfor the steady-state current (with 6 — cot + A)

2E(-6,)" sin (2H+1)0 . (50)

(4.1.4) Field-Transient and Sub-Transient Components

If the values of the functions in Table 1 are used to evaluatethe terms Bn(p)F$(p + j2nco) and Bn+l(p)F&[p + j2n + leu],when p is nearly equal to —j(2n + l)o>, the following result isobtained:

2nio) + Bn ^ l

' -j2n-

/>'+./"(51)

where />' = / » + y'2/io).Eqn. (51) shows that /p(^) has poles at values of p' deter-

mined by the factor <f>(p/) in the denominator, where

T'd'(p'

T'd0(p' +ycu)][l d + '•ml (52)

The corresponding factor containing Lq(p' + ya>) has can-celled out because of the approximations made. A moreaccurate solution would contain small terms depending on thequadrature-axis quantities.

<£(pO is the quotient of two quadratic expressions in / / , ofwhich the numerator determines the poles of fp(/?). The twolinear factors of the numerator may be found approximately byusing the fact that T'd0 and T'd are much greater than Td'Q and Td .

The poles which determine the field-transient component arefound by substituting [1 + T'd'^{p' +yco)]/[l + Td(j>' + jco)] = 1in eqn. (52). Then

(53)

(54)where

and

a =•"ml

L'd + Lml T,•ml x dO

>-l±LJd0

(55)

a! is a real quantity which is small compared with co andwhose value is independent of n.

Eqn. (53) shows that i$(p) has a pole at/? = — a! — j(2n + l)a>.The solution therefore has a decrement factor e~a''. The resultis usually expressed in terms of a line-to-line transient time-constant, 7rf(/_/), where

7" - 1 - L'd + V(LdLg') ^,

The corresponding component of fp(/?) can now be found,

I 1 \= -E-(

co \L'H + Lml

« r ( - bl)nj€

nio\_ac' + p + /+j(2n+ + conj• ] • (57)

The inverse transform of eqn. (57) gives the explicit expressionfor the field-transient component of current,

1 1

L'H -'ml ml'

71 = 0( - bi)n sin (2/i + 1)0 . (58)

The poles which determine the sub-transient component arefound by substituting [1 + T'd0(p' +jco) ]/[l -f T'd(p' + jo)] =T'dolTd in eqn. (52). Then


dQ( jco)

where

and

a =

" — d d T

(59)

(60)

(61)

Hence there is a pole at p = — a" — j(2n + l)w. Thecorresponding component of l$(p) can therefore be found, and is

ml

1 ^V, + LmJ

c o n j J (62)

and the exph'cit expression for the sub-transient component ofcurrent is

tt> \L'd + Lmi L'd + Lm

S ( - />!)" sin (2n + 1)0 . (63)

_ A is the line-to-line sub-transient time-constant.

QTZ. . . (64)

where

(4.1.5) Armature-Transient Component.A further set of poles can be found by evaluating the terms

of /3G?) for p 2i ± j2noj. Table 3 gives approximate values ofthe functions when p ~ 0. In making these approximations, thefield and damper resistances are neglected and the armatureresistance is assumed to be small compared with the reactancesat normal frequency.

Table 3APPROXIMATE VALUES OF FUNCTIONS

Function

Hip)

$ >

2?o(p)

F$ip)

Approximate value when p — 0

\ P(^«' - ^d')J r^ -|~ iPK^d ' ^Q )

\ j 1 PWL'q' — -\/Ld')2

J 4 ra + \p{L'd' + L'q)

ra + ip(Ld' + L'q')ra + pLm\

E sin A 1co ra + MLd' + Lq')

If the values of the functions in Table 3 and in the last columnof Table 1 are used to evaluate Bn(p)F$(p + j2nw) when p isnearly equal to —j2nco, the following result is obtained:

£"sinA(—»'=0

ra+P'L(65)

Eqn. (65) shows that i$(j)) has a pole at a value of p deter-mined by the factor (ra + p'Lml) in the denominator. Thusthere is a pole at p = — a — jincu, where

(66)

The corresponding time-constant is the line-to-line armature-transient time-constant, Ta{i_ty where

(67)

The component of fp(p) can now be found as follows:

1(x)(68)haiP) = —j— j — + 2 ( , i ••,— H- conj) .

a>Lml [_oca + p n=i\afl + p +j2ncu / JThe inverse transform gives the explicit expression for the

armature-transient component of current.

(4.1.6) Resultant Armature Current.The armature current after a line-to-line short-circuit has

been obtained as the sum of four components given by eqns. (50),(58), (63) and (69). The operational expression for ltfj>) maybe written in the form

h(J>) = - i

( -r-d[p+J(fr

jEsinAf 1

l)co] +LmlP

(— b\_<xa+p ' n^i\oca+p+j2iuo

The complete expression for /p(/) is

2E

where

+ conj

+ conj I

(70)

- 6!)« cos 2/iwl (71)

(L'df

w '-d "f -^ml. . . (72)

The phase currents are given by

ml

The solution thus contains four harmonic series as explainedin Section 4.1.2. It may be noted that in each series the mag-nitudes of the harmonics are in geometric progression with thesame ratio —bx [except for the zero-frequency term in /pa(/)],and that each series has a decrement factor which is the samefor all harmonics.

(4.2) Field and Damper CurrentsThe transforms of the field and damper currents are given by

eqns. (33), (34) and (35) in terms of the axis quantities ld(j>) andii), which are related to /aO) and l^{p) by eqns. (31). Using


the value of l^j>) given by eqn. (71), and putting fa(/>) = 0in eqns. (31), the following expressions are obtained:

2£ sin Ae-'

1)0

( - ^ ^ cos 2«0 . (79)

x S ?

?(1 +/>j)sinA

+ conj

w = -

'ml

E{\ ~ bj)

1

Since v$t = vOt = 0, eqn. (79) gives also the actual terminalvoltage va across the open phase.

(4.4) Torque

To determine the torque, id, iq, tf/d and i//g must be found.(1V\ U a n d iq

a r e derived as the inverse transforms of eqns. (73)* " ' " K } and (74)

conj

-d(P ~\ J2na>) • { - L ,ml

+ conj."I

- (_ biye-Jto+V*

++ conj

. . . . (74)

Substituting these values in eqns. (33), (34) and (35), andevaluating the inverse transforms, the following explicit expres-sions for the field and damper currents are obtained:

+

+ :n=0

. (80)

to(Ld' +

:-'/7"a(/-/)2(-61)"COS(2/J + l)0 . (81)"/ x^/nl n=0

ipd and ipq are obtained by substituting eqns. (73) and (74)in eqns. (38), and evaluating the inverse transforms,

./. r* _ 3 LmE

^•f 1" ^kd

L a ' -^ml[— ^ I ) " " 1 cos 2«0

)« sin ,i . .- M

fkd

S (~ î)""1 cos

sin A( - 6^" sin

n - 0, ] } . .

. . . . (82)

_ 3 (1 -

3 ( 1 - 6+ 2 :

(76)

-'/r«(/-/)2 ( - b1ycos(2n + 1)0

. . . . (83)

Substituting eqns. (80), (81), (82) and (83) in eqn. (36) andmaking use of the following products of harmonic series:

oo oo

2 (~ î)"-1 sin 2«0 x 2 (~ i)""00

= \ 2 "(— î)""1 sin2(n +

ml

( - h)n-1 sin 2nd

-'ml 71-0. . (77)

( - 61)«sin(2« + 1)0 x 2 ( - 6i)wcos(2« + 1)00 «=0

/ i i

(4.3) Open-Phase Voltage «, «,From eqn. (19), with Up) = 0, the a-component of the ^(-bl)

nsm(2n+1)9 x^(-b1)"-^terminal voltage is given by , "

( ^ ) ( - * ' ) ° - ' s i n ( 2 "

- ( _1

+ conj] . (78) . ( _ ftj)n

Substituting the values of ea(p) and fpCp) from eqns. (23) "-° "-1

and (70) and evaluating the inverse transform, the explicit _ i y> (n — —^—\—b )"-lcos(2n+ 1)9expression for the open-phase voltage is obtained, «=oV 1 — bJ l

. (84)


the explicit expression for the short-circuit torque is obtained,

E2 2 sin A

E2[ 1 + b.d + Lml

7 1 - 1»(— ^ I ) " " 1 sin 2nd (85)

(4.5) Alternative Expressions for Currents and Voltage

By using the summation formulae

a_ [B + V(AB)] sin d

M'-O

n 0

(A-B)cos2d

\A + B+{A-B) cos 20

2 (2« + 1)6" sin (2n + 1)0

(AB)j

and

2 2(n + 1)6" cos 2(n +n '0

where

b =

(A-B)cos20]'

A-B+(A + B)cos2d- B) cos 20] 2

+

. (86)

reactances. It has, in effect, a uniform air-gap and no damperwindings. The same machine was used for each of the threeshort-circuit conditions.

For operation as a synchronous machine the appropriaterating would be about 110 volts, 16 amp, 3 kVA. The constantsof the machine are given below both in actual values and inper-unit values relative to this rating.

xd = xq = x'q' — 6 0 ohms =1-51 per-unit.x'd — xd = 0-54 ohm = 0-136 per-unit.

x0 = 0-90 ohm = 0-227 per-unit.ra = 0-15 ohm = 0 038 per-unit.

T'd0 = 0073 sec.

Fig. 2 shows a set of oscillograms taken after a line-to-lineshort-circuit (phase 6 to phase c) from 25% normal voltage.Fig. 3 shows calculated curves of the same quantities drawn tothe same scale and corresponding to the same instant of switching.The numerical expressions from which the calculations weremade are also given in Fig. 3.

(5) LINE-TO-NEUTRAL SHORT-CIRCUIT(5.1) Operational Equation for the Armature Current

With phase a short-circuited, the following relations hold forthe following terminal voltages and currents:

= 0

the quantities ib, va and if can be expressed as

. . . . (88)

The equation for ia(p) is obtained by adding the first and thirdequations of the matrix equation (19). In order to make theresult more general, it is assumed that there is some externalresistance in the neutral circuit, so that the total resistance is r0.The difference between r0 and ra can then also allow for thefact that in practice there is a difference, not allowed for in the

lb = - 'c = ~ ^TT-— Xd) C 0 S

;[>i(0 sin 0 — e~'/r«(/-/) sin A)]

va =

and

- ** COS 20[x'd' + xq' + (x'q' - jf^) cos 20]2 w (x'q' - x'd') cos

+ Lkd-

where the x's are the corresponding L's multiplied by o>.

(4.6) Comparison with Experimental Results

In order to verify the results of the theory, a series of tests wasmade on a 7^-h.p. 110-volt 1 500-r.p.m. star-connected inductionmotor. The machine was operated as a synchronous machineby exciting two rotor phases and leaving the third rotor phaseopen. The machine represents a relatively simplified case of thesynchronous machine, but it is one which gives rise to pro-nounced harmonics because of the widely differing sub-transient

1cq + (x'q' - xd')cos2d

sin A sin 0CQ ~ xd) c o s 20

(87)

theory, between the zero- and positive-sequence resistances.equation is

+ L conjj]}/«

The

. . . . (89)

Eqn. (88) can readily be put in the form of eqn. (28), and thesame method of solution used as for the line-to-line short-circuit.

176 CHING AND ADEEMS: TRANSIENT THEORY OF

Fig. 2.—Oscillograms of current and voltage after a line-to-line short-circuit.

S 100

S o \i2TT

ex ort - 5 0

-.£-100a of-250

5-500>-75O

YY

Fig. 3.—Calculated values of current and voltage after a line-to-line short-circuit.

/ ' (f 3 3 8 ) E - " o - O 2 . 9 + Q-298) +TT

-298)+ O-866s-'/o-«76]

t'ac = 2r<» (aftc the short-circuit)

146sinG-161sin'eV (0702CF20 + cos 26)2

63-8 + 76-3cos26298) + ( 1 .


MWlWPSL ,

• Fig. 4.—Oscillograms of current and voltage after a line-to-neutral short-circuit.

« <&I 100

• • ^ Q

50a

I °} ~5°' " - 1 0 0„ 300% 200> 100>2 °

-100

w 100"5 0> -100

A

r*"•200

- 3 0 0V v v

Fig. 5.—Calculated values of current and voltage after a line-to-neutral short-circuit.. , / I \ , 15-3 cos 0

// = 15.5e-'/0-0288 + 23-6(i-36-cos26 ~ 1-08J(0-607s-'/°-288 + 0-393) 420 *2

ia = ~ 1-36 - cos26[(0'607e~</0'0288 + °"393)COS ° + O-766e-*/o-«58]

39-6sin6-18-lsin*6-123cos6+ 150cos*G ./„„,„t* ~ (1 -36-cos 26)2 (0-607£-'/0-0288 + O-393)

48-7 - 13-4 sin 26 - 66-3 cos 26 , /„„ ,„(1 -36-cos 26)2 E

39-6sin6-18-ls in*8 +(1 - 36 —

48-7 + 13-4 sin 2 6 - 66-3 cos 20r_.fT (1-36 -cos26)2


(5.2) Expressions for Currents and VoltagesThe values of armature current ia, field current i'f, and open-phase voltages vbt and vct, after a line-to-neutral short-circuit at the

instant when phase a is at an angle A from the field axis, are as follows:

[g2{t) COS 9 - e-"ra(/-n) COS A]x'J + x'q' + x 0 - (x'q' - x'd') cos 20

vbt =

y/3xo[(2x'q' - x'd' + jx0) sin 9 - {xq' - x'^ sin3 0] - 2(xq' + $x0) [{xq' - 2x'd' - jx0) cos 0 - (xq' - x Q cos3 9][xd + xq + x0 - (xq - xd') cos 20p

-»> cos AV3^o sin 0 - W~

[V +

2fl]- (xq - xd) COS Id]

V3xo[(2xq' - x'd' + jx0) sin g - (^ ' - x Q sin3 6] + 2(xq' + ^ [ ( ^ - 2xd' - ^ 0 ) cos 9 - (x'q' - xjQ cos3 6][xd + ^^' + x0 - (xq - xd) cos 29]2

3 o sin 6 + ( < - y - ^ - ( ^ + xj' + x0) cos 20]-* cos A - xd) cos

m2

1[x'q' - x'd' \x'd' + x'q' + x0 - {x'q

r - x'd') cos 2Q 2xm2

cos A cos 9- (^^' - *dO COS

.(90)

where

cd±_

wm2

Xd -V X w2

. Xd \ Xmx</ ^ xm2

/-«)

m2 Xo

(91)

= 0O) = 0 (92)

Using eqns. (92) in conjunction with eqn. (19), two simul-taneous equations for ijji) and f^(p) are obtained,

f Zs{p + joj) + conj]}fa(»

+

+

conj]faCp)

+ conj]}fp(/>)

+ y2o>)

(5.3) Comparison with Experimental ResultsFig. 4 shows a set of oscillograms taken after a line-to-neutral

short-circuit (phase a) from 25 % normal voltage on the machinedescribed in Section 4.6. Fig. 5 shows the correspondingcalculated curves and the numerical expressions from which thecalculations were made.

(6) DOUBLE-LINE-TO-NEUTRAL SHORT-CIRCUIT(6.1) Operational Equations for the Armature Currents

With phases b and c short-circuited, the following relationshold for the terminal voltages and currents:

conj

(93)

] + conj

. . . . (94)

By means of the method of elimination explained in Appen-dix 10.2, separate equations of the form of eqn. (28) are derived.From this point the solution follows the same lines as in Section 4.r0 is introduced in the same way as in Section 5.1.

(6.2) Expressions for Currents and VoltagesThe values of armature currents ib and ic, open-phase voltage

vat, and field current if, after a double-line-to-neutral short-circuit at the instant when phase a is at an angle A from the fieldaxis, are as follows:

SYNCHRONOUS GENERATORS UNDER UNBALANCED CONDITIONS

<\/3x'q' cos 6 — {xq + 2x0) sin 0

179

x'd'x'q' + xo(xd + xq) + x o ( 4 ' - *d') cos 20

t " *'1f? ) + xo(xq ~ xd) cos

xdxq + xo(x'd' + x'q') + xQ(xq' - x^O cos 20

a' cos 0 + (x ' ' + 2x0) sin 02 63V 'x^'x ' ' + xo(xy + x'O + xo(x" - x'd') cos

-ECOsX€-'lT<xUl-n] - x'd"> c o s - W - x'd) s i n 2d

xdxq + xo(xd + x ') + xo(.xq ~ x'^ cos

V 3 „ c. A _/ / 7 Mn x'd' + x^ + 4x0 - (xqf - 4 Q cos 26

—^—•" S i n A t ' (3U' — "J ;—r;—; -r ; -4

^ - x'd') sin77———; ;—r;—; -rjr ; .—77 77r ^ 7 ;

xd xq + X Q ^ + x g ) + XQCX^ - xd) cos 20[2xo(xq' - 2x'd

r) - x'd'x'q'] sin 0 - ^ ( ^ - xfi sin3 0

K * i ' + ' + *;o + oK' - *io cos 20],/T cos A sin 20

3xoxdxq(xq ~ '^ + XQ{X<' + xfl + xo(x- - 4 0 cos 20P

-4k — M,lJkd

rv"v"

[xdxq1

j) cos 20]2

— 4 0

y + fcd — 2Mfkd xd + xc Lf x'd + <

^•W - Mfkd

Lf xd + xe

Lf+ Lkd - 2Mffkd V K ' ( 4 ' + 2x0)] -

Vt^'^'C^y + 2xo)(x ' + 2x0)]dXq

- ' r f O i

*q)

- M,

(4 ' — 4 ' ) cos 20-afkd

XdXq - Xd) COS 20

co)(x- + 2x0)]M cos A cos 0 + ( < + 2A:0)e-'/%//-n) sin A sin 0)q J

(95)

where

y. _xdxq

lm3

- VK'(4 ' + 2*0)]

4' +

xd 4" x^ x^ + x,y" 4- v

e JtlTd(U-n) +

d0

1d{ll-ri) " -T~

xm3 + 2*0

(6.3) Comparison with Experimental Results

Fig. 6 shows a set of oscillograms taken after a double-line-to-neutral short-circuit (phases b and c) from 25% normal voltageon the machine described in Section 4.6. Fig. 7 shows thecorresponding calculated curves and the numerical expressionsfrom which the calculations were made.

(7) CONCLUSIONThe theory explained in the paper provides a rigorous basis

for the investigation of unbalanced conditions in synchronous(96) machines. The results agree with those given in References 3

and 4 and include several additional formulae not hithertopublished.

Doherty and Nickle gave sets of oscillograms and calculationsfor the line-to-line short-circuit similar to those in Figs. 2 and 3.However, their theoretical formulae, like the tests, only appliedto the simplified case of a machine without a damper winding.

Concordia considerably extended the theoretical results butgave no tests. His book contains complete expressions for thearmature currents for each of the three types of short-circuit,but only gives initial values of field current and open-phasevoltages. Thus the present paper, apart from the greater rigour

180 CHING AND ADHNS: TRANSffiNT THEORY OF

Fig. 6.—Oscillograms of current and voltage after a double-line-to-neutral short-circuit.

Fig. 7.—Calculated values of current and voltage after a double-line-to-neutral short-circuit.

if « 50- le-'/o-oi2i + ^ — g ^ ^ 2 6 ~ 6 O)(O-835s~' / o 0 1 2 1 + 0-165) -

67-2 cos 6 - 5 0 - 5 sin 9 .,

175 cos 6s"</o-°22 2Olsinee~'/o-°278•86 + cos 29

13-3 + 10-5 cos 29 + 6-4 sin

1-86 + cos 29

„ „»., 22-7 - 12-2 cos 2 9 - 2 0 - 1 sin 29

/. 1-86 +

19 sin 9 - 243 sin1 8"Ur86Hrcos"26)2~(

+ O.,65) -

1-86

13-3 4- 10-5 cos 29 - 6 - 4 sin 281-86 + cos 28

1-86 + cos 26

22-7 - 12-2 cos 26 + 20-1 sin 261-86 + cos 26

. _ / > . 0 W .

68-1 sin 2 8 ^ 0 • ™ (28-6 + 53-1 cos 28)E-(/°-°"8° °121 + 0-165> + (1-86 +cos 28)2 + (1-86 + cos 26)2


of its method, supplements the results already available, and givesexperimental confirmation for the two more difficult types ofshort-circuit. Moreover it provides a general method by meansof which more complicated practical conditions can beinvestigated.

(8) ACKNOWLEDGMENTSMuch of the paper is based upon a thesis presented to London

University by one of the authors (Mr. Ching).The authors wish to thank the Imperial College of Science

and Technology and Professor Willis Jackson for providing thefacilities for carrying out the work recorded in the paper, andalso many colleagues for their advice and assistance.

(9) REFERENCES(1) ADKTNS, B.: "Transient Theory of Synchronous Generators

connected to Power Systems," Proceedings I.E.E., 1951,98, Part II, p. 510.

(2) CHING, Y. K., and HUMPHREY DAVIES, M. W.: "Harmonicsin Synchronous Machines under Steady Conditions,"Proceedings I.E.E. (to be published).

(3) DOHERTY, R. E., and NICKLE, C. A.: "SynchronousMachines—IV," Transactions of the American I.E.E., 1928,47, p. 457.

(4) CONCORDIA, C.: "Synchronous Machines" (John Wiley andSons, New York, 1951).

(5) PARK, R. H.: "Definition of an Ideal Synchronous Machineand Formula for the Armature Flux Linkages," GeneralElectric Renew, 1928, 31, p. 332.

(6) CARSLAW, H. S., and JAEGER, J. C.: "Operational Methodsin Applied Mathematics" (Oxford University Press, 1939).

(7) PARK, R. H.: "Two-reaction Theory of SynchronousMachines," Transactions of the American I.E.E., 1929,48, p. 716.

(10) APPENDICES

(10.1) Derivation of the Simultaneous Operational Equations

Assuming that the currents are initially zero, and using theformulae

where the symbol Vindicates the Laplace transformation definedin Section 2.3. The six eqns. (1) can be expressed in operationalform as follows:

I = (rfl + LOQPVJJ)) + MsOp!b(p) + M^picip)

l2/>{e--'2X[fa(P +j2a>) + hib(p +j2uS)

j2to)\ + conj} + \Mafp\e~Mj-e{p +/co) + conj]

x>nj] (97)

i = M^pijji) + (ra + LaQp)ib<j>) + M^pljj))

l2p{[he-JK[ia(P +y'2w) + Mb(j> +j2uS)

j2u))] + conj} + hMtfP [he~Jtffe(p +ya>) + conj]conj] (98)

conj]

(99)

vct{p) + ejj>) = M^PW + MsOplb(p) + (r

+ $La2p{h2e-JK[Ia(p +j2co) + h!b(p +J2a>)

+ h2lc(p +j2«))] + conj

0 = WtfP{e-J*[la(P + » + hh(P +Jw) + h%(P +J<»)] + c o nJ}

+ h2lc(p+jco)] +conj}+ (rkq + Lkqp)lkq(p) . (102)

where lfe(p) = l'f(jj) + lkd(p) and h ~ eJW\

From eqns. (100) and (101), lfe{p) may be determined,

conj}. . . . (103)

where Zd(p) has the value given in eqn. (12).The values of lfe(jp + joj) and ikq(p + jo>) and their con-

jugates are readily obtained from eqns. (102) and (103), andwhen substituted in eqns. (97), (93) and (99) give the rows of thematrix equation (8).

(10.2) Derivation of the Recurrence Equation

By eliminating tQ(p) from eqn. (27), two simultaneous equationsare obtained for la(p) and I$(p) in the following form:

fa(P) = Z«(p)«p) + Z^{p)l^p)+ j2cS)]-j2a>)] . (104)

+ J2a>) M ]j2GS)-jl9l(j>--j2u>)] . (105)

in which the new functions are deduced in the course of theelimination.

To eliminate f$(p) from eqns. (104) and (105), multiply eqn. (105)by j and add it to eqn. (104), and then replace p by p + j2a> inthe resulting equation,

[Z«(p +j2co) +jZ&£p +j2a>)] Up +j2uS)+ j [Z$ <J> +j2co) -jZ^(p +J2co)] f0O \ j2co)

(106)

Elimination of I&(p + j2cS) from eqns. (104) and (106) thenleads to eqn. (107). Similarly, eqn. (108) is obtained byderiving an equation conjugate to eqn. (106) and using it toeliminate l$(j> — j2cS) from eqn. (105),

. . . (107)

. . . . (108)

Finally eqn. (28) is obtained by eliminating /p(/?) from eqns.(107) and (108).

(10.3) Solution of the Recurrence EquationIf p is replaced by p ± j2<x> in eqn. (28) and the resulting

expressions are substituted in the original eqn. (28), a new

182 CHING AND ADK1NS: TRANSIENT THEORY OF UNBALANCED SYNCHRONOUS GENERATORS

The series is assumed to be convergent from the nature of theequation involving Fa(p), FJj? ± j2a>), !a(p) and ijj> ± jAoi) isobtained. By repeating this process, it can be concluded that physical problem, that iseqn. (28) has a solution of the form

+ y2«co) . . . (109)Lim An(p) = 0

71—>00

The proposition is proved by replacing p by p + j2nu) ineqn. (28) with n taking all integral values from — oo to + oo.The result is an infinite number of s'milar equations which canbe represented by the matrix equation

[y.] =[«*»][*«] • • • - O i o )

where [yn] is an infinite column matrix whose general element is

yn~ F*(P + J2fla>) • • • • (in)

[xm] is an infinite column matrix whose general element is

xm ~ Ja(/7 + j2moS) . . . . (112)

and [anm] is an infinite square matrix whose general element is

anm -- 1 when m — n — 0

? \-j2no)) when m — n = 1

\j2no)) when An — n — — 1

~ 0 when \m — n\ > 2

By Cramer's rule, when m — 0, xm becomes xQ and is given by

1 (Cofactor of an0) x yn (114)

where \anm\ is the determinant of the matrix [a^] . Substitutingeqns. (Ill) and (112) in eqn. (114) gives eqn. (109), and alsodefines the coefficient A0(p),

A0(p)-= (Cofactor of anQ) ~ \anm\ (115)

Consequently eqn. (109) is a solution of eqn. (28).Substituting eqn. (109) in eqn. (28) and equating coefficients

gives

An(p) + ^€"J2^KiX(p)An_l(p +y2co)

l(p-j2co)=0 . (116)

except for n — 0, when

A0<J>) + i +j2at)

Hence when n is large, eqn. (116) becomes approximately

ip + j2co) <* 0 . . (118)

Now replace n by n — 1 in eqn. (116) and p by p —j2co ineqn. (118) and eliminate An(p —j2a)).

- J2a>)]An_l{p)[1 - i+ $€-WKtt(p)An_2(j>+j2a>) = 0 . (119)

Repetition of this process indicates that the correct form ofeqn. (118), when n is a positive integer (not zero), is

Ye&)AJLp) + te-J*KJLp)Am-l(p+j2w)==0 . (120)

where YJji) is the continued fraction given in eqn. (30).This result may be proved as follows:Assume that eqn. (120) is correct, and replace n by n -f 1

and p by p — ;2co. Then

(113) Y(X(p -j2co)An+l(p -J2w) - j2uS)An(p) = 0

or

From the closed expression for Ya(j>) in eqn. (30)

Ya(p-j2a>)_lKJJ>-J2M) 4 1 - YJj>) ' ' ' ( 1 2 2 )

Substitute, eqn. (122) in eqn. (121) and multiply throughout by

- j2w) =1 (117)

Addition of eqns. (120) and (123) gives the original eqn. (116).By repeated use of eqn. (120) the expression for An(p) in

eqn. (30) can be deduced.A similar argument leads to the following relation for the

coefficients with negative suffixes:

Yl(p)A_n(p) + ^mK(pU_n+l(p-j2a>) = 0 . (124)

and to the expression for A_n(j>) in eqn. (30). The value ofA0(p) in eqn. (30) is obtained by putting n — 1 in eqns. (120)and (124) and substituting them in eqn. (117).

transient theory of synchronous generator under unbalanced conditions

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