Analysis of Transient Phenomena of Synchronous Machines by Means of Spiral Vector Method. By Sakae YAMAMURA, M. J. A. (Communicated Jan.

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1 No. 1] Proc. Japan Acad., 72, Ser. B (1996) 1 Analysis of Transient Phenomena of Synchronous Machines by Means of Spiral Vector Method By Sakae YAMAMURA, M. J. A. (Communicated Jan. 12, 1996) Abstract : The synchronous machine has a long history of development, analysis and usage. Now almost all large generators are synchronous generators, which are synonymous with AC generators. They have generally saliency of magnetic poles, which makes machine inductances timely variant and makes their analyses complicated and difficult. The conventional theories have not provided adequate analytical results of the machine and they still leave the machine analysis somewhat in confusion. The spiral vector method, which has been proposed by the author, will be applied in this paper to analysis of transient phenomena of the machine. It was reported in the recent proceedings.3~ But this time the new idea of "damped inductance" will be introduced to further the SV theory of the synchronous machine. Key words: Spiral vector method; synchronous machine; sadden short circuit; damped inductance. Introduction. The synchronous machine has more or less saliency. It is caused by non-uniform magnetic path or unsymmetrical windings of the machine. As the rotar rotates, its inductances vary timely with frequency higher than fundamental frequency, and its analysis becomes complicated and difficult. The Park's equations and the two-reaction theory have been resorted to for many years since the beginning of the century. Both are based on the d, q axis theory, and involve the cumbersome variable transformations. Neverthless the present state of analyses of the conventional theories is not adequate nor sufficient, in spite of wide-spread and important roles played by the synchronous machine. It is the case especially in transient phenomenon analyses. Recently the synchronous motor began to replace DC control motors. They are permanentmagnet-excited synchronous motors and are often called brushless DC motor. Due to the presence of the permanent magnets on the rotor, the magnetic paths become unsymmetrical and nonuniform and the motor has conspicuous saliency. Quite often the quadracture axis inductance Lq is much larger than the direct axis inductance Ld. It seems also that transient phenomenon of the synchronous control motor, which has important influence on its control performances, are not well investigated. In order to improve the present state of analyses, the author applied the spiral vector method to the analysis of the synchronous machine and obtained the good results, which were reported in the recent proceedings of the Japan Academy and books.l~-3) Most synchronous machines have damper winding. It is thought that the damper winding stabilizes operation of the synchronous machine. On the other hand the damper winding makes analyses of the synchronous machine more conplicated and makes measurement of individual inductances more difficult. In this paper the idea of "damped inductance" will be introduced, to remove difficulties in analyses and in measuring inductances. Circuit equations of synchronous machine. In circuit equations of the electrical machinery ramped circuit constants are coefficients of state variables of the machine. In the spiral vector method of analysis state variables are expressed in SVs, which are exponential time functions with a complex index. SV can express both steady and transient states of AC and DC, and can express most kinds of state variables, which appear in electrical engineering. And thus the SV method unifies the electric circuit theories, which are separated by the two different expressions of phasor and real values. It has other merits; the

2 2 S. YAMAMURA [Vol. 72(B), and Aad is that from the damper winding. Inserting eqs. [1]-[4], eq. [5] becomes Aga = 32~ L 2 I I i I et cos (w't+ Bpi ) + 32 L v~ 2 x Ii I e-at cos [(2w- w') t + 2S + Aaf + Aad [6] Expressing variables of this equation in SVs, we get Aga = 32 Li2I I i I e-at+,cw't) + 32 Lv2I ~I Ii et SV method makes symmetrical component method applicable to transient state analyses and makes phase segregation possible to the three phase circuits and machines.1~,2) Fig. 1 shows the schematic model of the three-phase synchronous machine with field and damper windings. Self- and mutual inductances of the three-phase armature winding are given by eqs. [1] and [2]. La = L + Lvcos(2wt + 2cpd) = L + Lvcos2B Lb =L+L cos 28-2 v 3 L c =L+Lcos v 26+2~r ~, B=wt+ ~] d [1 Mab =M+Mcos 28-2 ~r v 3 Mbc = M + Mvcos (2 6) Mca =M+Mcos v Tr. [2] In these equations the following relations hold. L=M, M=-L/2 [3] Symmetrical three phase amature currents are expressed in both real values and SVs as below. is = 2 I Ii I e-atcos(w't + Bpi) = real [\ I Ii I e-at+(w't+co1)] i= b V 2 I I I e-atcos w't + ~o1-2 7r ic = Fig. 1. Model of synchronous machine. = real [~ I Ii I 2 I I I e-atcos w't ~r = real [~ I Ii I e-ail+;(w't+w1+3n)1 [4] Magnetic flux linkage of phase a winding with the rotating magnetic field in the air gap is ~ga = Laia + Mabib + Mcaic + ~a f + dab. [5] Here ' a f is flux linkage from the field winding X ei(2wt+2~d)-;cw't+~1) + Aof + 'dad = 3 La i x Lve;2Bia* + Aa f + Aad. [7] Here * indicates conjugate. Let transient state field current be expressed by i ft = 2 I I f I e-2'tcos (w't + ~P ) = real['/ I I fl ] [8] This current gives rise to the following flux linkage in phase a winding. ' a f= Ml fcos6 2 I f e-a'tcos (co't + q f) = ~ 2 I I f I Mif et[ ` cos{(w+ w')t 2 + (cd + (pf)} + cos{(w - w')t+ (co c f)}i [9] Expressing this equation in SVs, we get of = 1~ 2 I I f I Mif 2 a-~ ft[a;(w+w') t+;cwd+~ot) + ei(ui-w')id] =Mif 2 [e' if + e' 2f*] [10] Now the circuit equation of phase a winding can be writen as below. Va - - Riga - l,j ia - PAga + P' of +YAad [11] Here li is leakage inductance of the armature winding. Inserting eqs. [7] and [10], eq. [11] becomes V1 = - R 1 i 1 - Lst'~i 1-32 L v~' (e j2bi * 1 + Mif e'bi + e'ei + 3M 2 idy ( e'bi) d '~ +e1. [12] 3.* H ere Aad = 2 Mid (e;eid) is flux linkage from damper winding and ei = jw A e1 is internal induced voltage. Circuit equation of the field winding is vf = Rfif + (lf + Lf)E'i t - 32 M1fY(e'Bi d *) +3MfdYid. 2 [13] Here Mfd i5 mutual inductance between field and

3 No. 1] Transient Phenomena of Synchronous Machine 3 damper windings. Circuit equation of the damper winding is 0=Rdid + (id +3Ld!'i 2 d Table I. Damped in ductances - 3 M 2 ldp(e'bi 1 *) + Mf~P~if. [14] Eqs. [12], [13] and [14] must be solved simultaneously. Their homogeneous equations are R1iIt + LpiI st =- 32 Lvp(ej2eiIt*) + Ml 2 f ~ ( ejeift + e'bi ft *) + 3 M e'ei *. 15 Ri +(l +L)pi +3M i f ft f f 1 t 2 fdl' dt = 32 Mlf(e'BiIt*). [16] Rdidt + (id +3Ld pidt +M-~i 2 f~' ft = 32 Mldp(e10 It*). [17] In order to facilitate the simultaneous solution of these homogeneous equations, we shall assume that Rd term is negligible in eq. [17], compared with reactance term. Thus we have 2dt = 3M 2 1d (e'b21t*)- M f d 2ft. [18] ld+23ld ld+3l 2 d Inserting this equation into eqs. [15] and [16], we get the following equations. R M2 1iIt + L- s -j- ld 3 ~ i It 2 vp (e ilt) 2 p (e ift) + M1f 2 _ 32 M1dMfd 3 p (Let* e if) [ 19 ] 2 R ift + (i1 + L- 3 M fd f f 2 3 p ft id + Z Ld = 3 (Mf_ 3 MfdMld )p(efbii t*) 20 These two equations indicate that damper current decreases inductances as shown in Table I. Let the inductances with prime be called damped i inductances. Making use of the damped inductances, eqs. [19] and [20] become R1iI + Lp' ii =- t st 2 v lit = Be + R f+ ( A +jw)(lf f) X AB-At+ie Bee-A ft + 32 lf+ MY Lf Ae-at+;e A f = l'+l' Rf [ 25 ] f f Inserting eqs. [23] and [25], eq. [21] becomes Rlilt + Lp' ii =- s t 2 v 3 Lp (ej2bae-at) + MY e;e -Aft + 3 MY Ae-At+;e 2 p Be 2 lf + L' f +(M '- 2Ml f [,o' e 3 L p (ej2biit*) + Ml f p(ejeift) + (M1; - MY p(ejeift*) [ ] Rfift + (l' f + Lf')pift = 32 Mlf'p(ejeiIt*). [22] Assuming the first approximation of i1 to be ift = Ae-Ate [23] insertion of this equations to eq. [22] gives Rfift + (l' f + L')pift f = 32 Mp lf] ' (Ae-~t). [24 The SV theorem gives the general solution of this equdtiuii as follows; if Be-~`t + 3 i p 2 fm+ylf Ae-~t-;e [26] The superposition theorem and SV theorems give the following solution process. Collecting terms with factor A exp. (- At), we get -At 3 ( R2Ae LS ( A )Ae 2 MY 2 j2 if Mif\

4 ~ r s f S - 4 S. YAMAMURA [Vol. 72(B), which gives x lr r if +L (- A)Ae-ate Ri [27] L 3 Mi M' [28] S 2 Mif 2 f l' f + i L' f Collecting terms with factor B exp. (- Aft + j6), in eq. [26], we get M1;(- Af + jw) -aft+;e iit - R i + (- Af+ jw) LS Be r -A t+o -L/Be I J [29] Collecting terms with factor A exp. (- At + j26) in eq. [26], we get l _ (A+j2w) it R 1 + (- Af + j2w)ls x- ---L 2 V + 34lf+Lf Mi;Mi f Ae-at+;2e j L' _ 32 L v + 4l 3 if if Ae-at+,2ẹ 30 S f'+l f [ ] The general transient solution (the complimentary solution) is given by adding transient terms of eqs. [23], [29] and [30], as follows; i it - Ae-at + Lr if Be-aft+;e _ 3 L + 3 MifMi; l Ae at;,2a [31 f f To improve approximation of transient solutions in eq. [25] and [31], these are iteratively inserted into eqs. [21] and [22] respectively. And we get the following results. LS ~ 2 v 4 l'+l'~ ] A = Ri L _ 3 S 2 Mif Mif 2 Mif lf' + L' f 3 Lv2 + _2 3 2 LS Rf Af = 3 M, iit = Ae-at + if+lf-21 LS if _ 3 L Lv M1fMlf LSlf + Lf f r+il, Sv 2 M if Be2 -aft+;e + LS L 2 v + 34lf+Lf Mif Mi f Ae-at+;2e [32] [33] [34] Fig. 2. Damped position ~d Adding steady state terms to eqs. [25] and [34], we obtain the general solutions. There are four transient current components in eq. [34]; decaying DC, decaying AC of fundamental frequency and decaying AC of double frequency. There is no sub-transient of fundamental frequency, which the coventional theory calls for, Measurement of inductances. When the machines has a damper winding, it is difficult to measure individual inductances. But the damped inductances introduced in this paper are easier to measure and they are sufficient to calculate performances of the machines. Measurement of damped synchronous inductance LS will be now explained. Let the machine rotor stand still and three phase AC voltages are impressed to terminals of the armature winding and let field current be kept zero. Then we have machine speed w = 0 and i f = 0 and eq. [12] becomes J j2cpd. V1 = - Riii - Lspii - 2 Lv p (d 21) and eq. [18] becomes synchronous inductance L' Mldp(id*e'~d) [35] 1 d2 = 3 Mid 3 e f~di i' [36] Inserting this, eq. [35] becomes v1 =-R1i1 -L i1-3lej2p~d i s~' 2 v l +-;- 9 Mid2 L'i id + 2Ld rotor =. - L' s + 32 Lvej2 ~d p ii. [37] Here Ls is damped synchronous inductance in Table I, With the machine rotor stand still, inductances at the machines winding terminals are measured for different values of cod by changing rotor position, Fig. 2 shows an example of such measurement, which gives LS = 0.014H andlv = 0.005H for the synchronous machine in Table II.

5 No. 1] Transient Phenomena of Synchronous Machine 5 Fig. 3. Sudden three-phase short circuit currents of the synchronous machine in Table II. Fig. 4. Sudden three-phase short circuit currents of the synchronous machine in Table II. Table II. Rating of a synchronous machine Damped mutual indictance Mif between armature and field windings is measured, with the rotor at standstill and armature winding terminals opened. Then eq. [12] becomes V1 '. Mif - 32 i MidMd 3 )P if = M1f'pif. [38 d + 2 Ld By impressing AC voltage to terminals of the field winding, i f and vl are measured to determine M1. Other inductances in Table I are not difficult to measure. Fig. 5. Transient component currents. Comparison between calculation and test. The synchronous machine in Table II was tested for sudden three-phase short circuit. The currents were also calculated with the analytical solutions derived in the preceeding section.

6 6 S. YAMAMURA [Vol. 72(B), Table III. Measured circuit constants Circuit constants were measured and the result is shown in Table III. Making use of the circuit constants, eqs. [25] and [34] were calculated for sudden three-phase short circuit currents. Arbitrary constants A and B were determined for the initial condition of it = 0 and i f = 0 at t = 0. Result of the calculation is shown in Fig. 3. The machine was also tested for sudden short circuit. Its oscillograph is shown in Fig. 4. The test result and the calculation result are in good agreement. Each component of the transcient currents was also calculated and is shown in Fig. 5, which reveals details of the structure of the short circuit phenomena. Conclusion. The idea of damped inductances was introduced to the SV method analysis of the synchronous machine, and it produced more accurate and detailed transient analysis of the synchronous machine. The conventional theories have not provided analysis of transient phenomena of the synchronous machine, but have provided only experimental formulas, which are based on xd, xd, Td, and Td determined experimentally. It is hoped that the SV analysis of the synchronous machine would contribute to better understanding and control of the machine and the electric power transmission network. 1) 2) 3) S. S. S. References Yamamura (1986) AC Motors for High Performance Applications (book). Marcel Dekker, New York, Brasil. Yamamura (1992) Spiral Vector Theory of AC Circuits and Machines (book). Oxford University Press, Oxford. Yamamura (1995) Proc. Japan Acad. 71B,