Steady-state analysis of a self-controlled synchronous motor

ELSE VIER

Electric Power Systems Research 36 (1996) 211-216

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Steady-state analysis of a self-controlled synchronous motor S.P. Srivastava, V.K. Verma, Bhim Singh Department of Electrical Engineering, University of Roorkee, Roorkee 247 667, India Received 9 October 1995; accepted 28 November 1995

Abstract This paper deals with a simple mathematical model for steady-state analysis of a self-controlled synchronous motor in terms of d,q variables. The flux linkage of the damper winding is considered constant under steady-state conditions. With the input voltage to the synchronous motor and the machine equations in d,q form, torque-current and speed-current relations are developed. The computed results are given and discussed in detail for a 3 kVA, three-phase, 400 V, four-pole, inverter fed synchronous motor and it is shown that this system has the same characteristics as a conventional DC motor. Keywords: Self-controlled synchronous motors; Commutatorless motors

I. Introduction The inverter fed self-controlled synchronous motor (commutatorless DC motor) has become popular in the last decade due to its high power rating, variable-drive quality and ease of control as in conventional DC motor drives. It has a major advantage in the commutation of the inverter thyristors by the machine terminal voltage when the synchronous motor operates under overexcited conditions and also in the triggering frequency being decided by the speed of the motor. The motor always maintains the synchronous motor quality. Due to the self-commutations of the inverter, the hardware requirement is reduced drastically. This drive exhibits similar steady-state and transient characteristics to DC motor drives. The steady-state analysis of these motors has been reported in the literature [1-3]. In the analysis, it is assumed that a quasi-square current is fed to the synchronous motor. Fourier analysis of the current waveform has been used to obtain the steady-state behaviour of the motor. The commutation process of the inverter, feeding an overexcited synchronous motor, has also been reported [4,5] along with numerical solution of the machine equations in state variable form using phase variables [4,5]. These methods require large computing times. A generalized analysis in terms of machine winding inductances and other basic parameters has been given by Takeda et al. [6] and the effects of parameter variation have been studied. 0378-7796/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0378-7796(95)01034-3

In this work, a simple and effective mathematical model is developed for the steady-state analysis of a DC commutatorless motor (self-controlled synchronous motor), using terminal voltage commutation, in terms of d,q variables. The flux linkage of the damper winding is considered constant under steady-state conditions. The computer results are given and discussed in detail for a 3 kVA, three-phase, 400 V, four-pole, inverter fed synchronous motor.

2. Principles of operation An inverter fed self-controlled synchronous motor is treated as a commutatorless DC motor. The inverter and its triggering arrangement replaces the commutator of a conventional DC motor. Similar to a DC motor, the average DC voltage applied to the inverter and the DC link currents are responsible for the operation of the motor in this case. The machine can operate as both a series and a shunt DC motor. Fig. 1 shows a schematic diagram for a commutatorless DC motor. The triggering pulses for the inverter thyristors are derived from the induced e.m.f, at the motor terminals. The frequency of the input current is determined by the speed of the synchronous motor with an overexcited field. A phase controlled converter is used as a variable DC supply. The voltage output varies according to the requirement of the load. Two converters are connected through a large inductance to make the DC current smooth.

212

S.P. Srivastava et al./' Electric Power Systems Research 36 (1996) 211 216

(iii) during commutation, the current varies linearly in the short-circuited phases; (iv) the inverter operates in the 120 ° mode; (v) harmonics in the stator currents can be neglected; (vi) the field current is ripple free. The matrix used for conversion of the phase variables to d,q variables is

RsXd.c. 3-,0

50Hz 400V ACSupply

'-.

±0

Rlf,spled

>

I

0c soo.,.

~

E:I= I coso -sin 0

cos,o 12o, - s i n ( O - 120)

-

sin(0 + 120)

Fig. 1. Schematic diagram o f the D C commutator]ess motor.

The sequence of switching of the inverter thyristors decides the direction of rotation. The electromagnetic events are repeated every re/3 radians. In each re/3 angle of rotation there are two modes of operation, the commutation and the conduction mode. As shown later in Figs. 4 and 5, in the conduction mode the input current will flow through the two phases of the armature winding of the synchronous machine (say a and c) and in the commutation mode all three phase windings (a, b and c) of the armature will carry the current. Again in the conduction mode, the next two phase windings (a and b) will carry the current. In the commutation mode, the two phase windings of the armature, in which the commutation is to take place (c and b), are treated as a short-circuit (see Fig. 4). The damper winding may be treated as an energy storing device and acts as a flywheel. Under steady-state conditions, when the speed is constant, the flux linkage to the damper winding may be treated as constant for simplification. The motor will operate at an average speed due to the average torque developed.

(1) The voltage equations of the motor in d,q variable form in the steady-state condition [7] are Vd = riao -- cO~,q

(2)

Vq = riqo + ~oCd

(3)

where r is the resistance of the phase winding of the armature and ~ba and ~/q are the flux linkages along the d- and q-axis, respectively. The flux linkage to the d-axis damper winding [7] is (Jdd = Xmdid + Xdaiaa

where lad is the damper winding current. With assumptions (ii) and (vi), since under steady-state conditions the damper winding flux linkage is constant, the damper current flows in such a way as to cancel the flux due to higher harmonic components of the armature current. The damper flux is determined only by the fundamental component of the armature current [8]. The steady-state d-axis flux linkage ~'d can be expressed as g'd = (Xd -- Xd)id0 + Xmdlr

(4)

Similarly, ~tq = (Xq -- x;)iqO 3. Mathematical

(5)

model

From Fig. 2, the flux linkages along the d,q axes are In the commutatorless motor drive consisting of an inverter fed overexcited synchronous motor, the inverter and the synchronous motor need to be analysed to study the performance. The circuit conditions in the conduction mode are such that two phase windings are connected in series with the DC source and in the commutation mode all three phases are connected to the same source with two phase circuits short-circuited. Since the electromagnetic events repeat every re/3 radians, one conduction and one commutation mode are sufficient to evaluate the steady-state performance of the drive system. The following assumptions are made to simplify the analysis: (i) the DC link inductance is large enough to ensure smooth DC current; (ii) the damper winding resistance can be neglected;

~/d = @g COS 5,

@q = ~/g sin 6

(6)

d-axis~ I

q-axis

J %=< (d-axis armatur ¢ reaclion )

Fig. 2. Phasor diagram for an overexcited synchronous motor.

S.P. Srivastava et al./Electric Power Systems Research 36 (1996) 211-216

~/g = (~//d 2 3t- ~/q2)1/2,

tan • = ~/q/~//d

(7)

213

ea

As the harmonics are neglected, the steady-state current through the phase winding will be Iac. F r o m Fig. 2, the steady-state currents along the two axes m a y be written as

,"

ido = - Idc sin(~ + ~b) iq0 =-

(8)

Idc cos(~ + ~b)

,

I ",, "~

I I /b"

zi01 ix,S\,

and

\

\

\ 0

(9) --0

and the voltages along the two axes, Vd and vq, in terms of the applied voltage V to the m o t o r are VO= - - V s i n 0 e

i

vq=VCos~

.;&_

(10)

I -- -- -- No load case

where

Load case V=

(/)d 2 -~- /)q2)1/2 and

tan 0~ = /)d//)q

(11) ia=idc

is the angle by which the voltage shifts from no load to the loaded condition. Since the input power to the inverter is to be taken by the synchronous motor, the power balance across the inverter can be written as Eala~ = VI cos ~b = Vdiao +

Uq/q0

(12)

V and I are the machine voltage and current in p.u. and Va, Vq, ido and iq0 are the axis voltages and currents. Since the power factor cos q5 can be expressed as cos ~b = 0.5[cos(7 - u) + cos

7]

(14)

as

--la¢sin(~o--U/2)

iqo = Ia¢ cos(~ + 7 -- u/2) = Ia¢ cos(70 -- u/2)

(15) (16)

where 70 = 7 + e. The D C link voltage relation under steady-state conditions is Vdc= RIa~ + Eo

ic =ldc

Fig. 3. W a v e f o r m s of induced e.m.f, at load and no-load conditions.

of the drive are obtained. The waveforms of the current and induced e.m.f, are shown in Fig. 3. The broken line shows the no-load waveshape and the solid line the loaded condition waveshape. The two modes, namely, commutation and conduction, are analysed to determine the average voltage expression. 3. I. Commutation mode

As the harmonics are neglected, the d- and q-axis currents under steady-state conditions can be expressed /dO = - - I d c s i n ( e + 7 - - u / 2 ) =

ib=ld¢

(13)

where u is the overlap angle, if u is small, cos q~ = cos(~ - u/2)

ib=ld¢"~

X

(17)

Before c o m m u t a t i o n starts, conduction is assumed through the a and b phases and commutation will take place from phase b to phase c. The circuit condition is shown in Fig. 4. The instantaneous induced e.m.f. during commutation is Vl = va - (Vb + Vc)/2

(21)

Using Eq. (1), the instantaneous induced e.m.f, in d,q variables is vl = (3/2)[va cos 0 - Vq sin 0]

At the instant when commutation starts, the phase currents are as follows:

The input/output voltage relation through the inverter can be expressed as

ia =

Ea = V c o s 7 + E x

Using Eq. (1), the currents in d,q variables are

(18)

Ex is the voltage drop due to overlap and can be expressed as Ex = (rcl6)~x'~I,l~ = 0.5 V[cos(~ -- u) -- cos 7]

(19)

so

Ed = 0.5 V[cos(7 -- u) + cos 7]

(20)

Using the above relations and the circuit conditions in the different modes, the performance characteristics

(22)

- - ib =

Id~,

i¢ = 0

(23)

id = - (2/x/~)Idc sin(0 - r#3)

(24)

/q = - (2/X/~)Id¢ COS(0 -- r#3)

(25)

Using Eqs. (2), (3), (6), (7), (15), (16), (22) and (23), the instantaneous induced e.m.f, can be expressed as v, = (3/2)rldc - (3/2)COOg sin(6 + 0)

(26)

where c~ is the angle of shift due to the armature reaction.

214

S.P. Srivastava et al./'Electric Power Systems Research 36 ( 1 9 9 6 ) 2 1 1 - 2 1 6

n2 Ed=3(

70 + u f

~/2 -- ;'o + m3 UI d0 -~-

f

/-)260t

(33)

~/2 }'0 ~,'2 -- ;'O -- It By Eqs. (17), (26), (31) and (33) the average input voltage to the DC link is expressed as Vac = / d c R ~ q + ( 3 ~ / = ) C O ¢ g

cos(u/2) cos(70 - c5- u/2)

where Req = R q-

2r - (3r/2rc)u

The overlap angle u is very small so

Fig. 4. Circuit in the commutation period.

R~q = R + 2r and Vdc = IdcReq 4- ( 3 ~ / a ' ) ~ @ g COS(@o-- g -- U/2)

From Eq. (34) the average speed of the motor, co, in electrical rad/s can be expressed as

b/- c

rdc -- IdcReq co = ( 3 x / 3 / a ' ) @ g COS(7 o -- 6 - -

(35)

u/2)

As the average torque developed by a motor is T, the power balance equation will be

< ldl;

Fig. 5. Circuit in the conduction interval.

To) = EdIac

=

(V~c

--

Ro~Idc)Ia~

so

3.2. Conduct&n interval

T = (3xfg/~r)g, glac cos(7o - ~ - u/2) During conduction, the current flows from the a to c phase winding. The circuit condition is shown in Fig. 5. At the instant when conduction starts, the phase currents are i, = - ic = Idc,

(34)

ib = 0

(27)

(36)

If P is the number of pole pairs and COrnthe mechanical speed in rad/s, V d c - - ldcRo. corn (3x/f3/g)P~g cos(7o - ~ - u/2)

and the average torque developed by the motor is Using Eq. (1), the currents in d,q variables are

T = (3x~/gr)P~gIac cos(70 - 6 - u/2) id = (2/Xf3)Idc sin(0 + re/3)

(28)

iq = (2/xfS)Idc cos(0 + re/3)

(29)

and the instantaneous induced e.m.f, is v2 = va - vc

(30)

By Eq. (1), the e.m.f, in d,q variable form is

v2 = - xfS[vd sin(0 - 2,r/3) + Uq C O S ( 0

--

21r/3)]

(31)

By Eqs. (2), (3), (6), (7), (29) and (3l), the expression for the instantaneous induced e.m.f, is

v2 = 2rld~ + X~CO~'g COS(0 + ~ + ~r/3)

(32)

where 6 is again the angle of shift due to the armature reaction. The DC input voltage to the link can be expressed as

Vac= Rldc + Eo The average value to the induced e.m.f, is as follows:

For a given value of 7 (the actual leading angle of commutation) and overlap angle u (which should be small), the torque-current and speed-current characteristics can be evaluated using the above expressions at different DC link voltages and field excitations. The procedure is explained below. The inverter input voltage is evaluated by Eq. (17) and the machine voltage V is computed using Eq. (20). With the phasor diagram shown in Fig. 2, the excitation voltage Er can be evaluated in terms of actual current and machine parameters. This gives the idea of angle ~ also, which is the angle between the machine voltage V and the quadrature axis. Now 7o (leading angle of commutation at no load) is determined by knowing the values of and 7. With these data the steady-state axis currents can be determined by Eqs. (15) and (16) and the flux linkage Og, along with the shift angle c5, can be determined by Eqs. (6) and (7). By substituting these values in Eqs. (35) and (36) and varying the current, the speed current and torque-current characteristics can

S.P. Srivastava et al./Electric Power Systems Research 36 (1996) 211 216

215

1.6

1

1.2

R U.

1.2

n. v i,i

a:0.8

--g.

g0 0.t Q Z ~t

t-I hi

-

0.4

~

~

ILl O.

,

tn 0.4 --

LU

;X,, 0.C

0.0

, 0.4

,

Vd c = 1.0 R U . ( c o n s L )

)

0.8 1.2 CURRENT(RU.)--'--~

0.0! 0.0

1.6

I

0.6

I

I

0.8

,

TOROUE (R U.)

Fig. 6. Torque-current and speed-current characteristics at Vdc = 1.0 and 0.8 p.u.

1 . 6 ~

I

1.2 >

,

I

1.6

Fig. 8. Effect of excitation on torque speed characteristics.

shown in Fig. 8 for a fixed input voltage. Since the field excitation is constant in Figs. 6 and 7, a shunt motor characteristic is obtained. The characteristics obtained have the same drooping nature as the DC shunt motor characteristics under armature voltage control. The speed drop is mainly due to the resistance of the DC link and armature. Fig. 8 shows the field excitation control of the drive. Beyond a certain value of torque, due to the demagnetizing effect of the armature reaction, the motor becomes unstable.

1.2

go.8-

If = 1.8 P. U.(consL)

5. Conclusions o.0

0.0

,

I

0.4

i

I

I

0.8 TORQUE (P. U.)

I

1.2

I

I

1.6

Fig. 7. Torque speed characteristics at Vac = 1.0 and 0.8 p.u.

be obtained. With these two results, the final t o r q u e speed characteristics are computed, a s shown in Figs. 6 and 7. The computer results are given for a 3 kVA, 400V, Y-connected, four-pole synchronous motor. Other parameters are: R = 0 . 0 0 8 , Xmd=0.715, 7 = 0.965, U = 0.107, r = 0.086, xd = 0.76 and Xq = 0.33 (all in p.u.). For a fixed excitation and voltage the current varied.

A simple mathematical model has been developed for the steady-state analysis of a self-controlled synchronous motor drive (i.e. shunt type commutatorless motor). The torque-speed characteristics of a 3 kVA, 400 V, four-pole, three-phase synchronous motor, operating as a commutatorless DC motor, were computed and plotted. The characteristics obtained are similar to those of a DC shunt motor with armature voltage control. The developed model can also be used for the DC series motor mode. The use of this motor drive avoids the limitations of conventional DC motors for industrial uses. The control philosophy in the closedloop condition will also be the same as that of the conventional DC motor.

4. Characteristics 6. Nomenclature

At different DC input voltages to the link inverter, the t o r q u e - c u r r e n t and speed-current characteristics for a fixed value of 7 were computed and the results are shown in Fig. 6. As the DC voltage is obtained by a controlled supply-side converter, the variable voltage is obtained by varying the firing angle of the converter. Using these two characteristics, the torque-speed characteristics were obtained and are shown in Fig. 7 for constant excitation. The effect of excitation is also

All quantities are in p.u. form. ea, eb, ec

G Ef i~,ib, i¢ id, iq idO, iq0

instantaneous induced e.m.fs DC input voltage to inverter excitation voltage instantaneous phase currents d- and q-axis currents steady-state d- and q-axis currents

216

idd I

P F

R Req

T Ua, Ub, Uc Ud, Uq

V vdc Xd, Xq /t rt Xd, Xq Xdd Xmd

Xs

S.P. Srivastava et al. /Electric Power Systems Research 36 (1996) 211-216

damper winding current instantaneous current during commutation machine current DC link current excitation c u r r e n t number of pole pairs phase resistance of machine winding link resistance resistance of equivalent circuit average torque developed by motor overlap angle instantaneous phase voltages d- and q-axis voltages machine voltage DC input voltage to link d- and q-axis reactances at rated frequency d- and q-axis subtransient reactances at rated frequency damper winding reactance d-axis magnetizing reactance at rated frequency mean value of d- and q-axis subtransient reactances at rated frequency

Greek letters c~ voltage angle shift from no-load to loaded condition 7 actual leading angle of commutation at load

70 cos 0 0d, 0q Odd 0g

~o ~Om

leading angle of commutation at no load angle of shift due to armature reaction power factor d- and q-axis flux linkages d-axis damper winding flux linkage resultant air-gap flux linkage at load electrical angular speed mechanical angular speed

References

[1] N. Sato, Induced voltage commutation type commutatorless motor, Electr. Eng. in Jpn., 91 (1971) 114.

[2] J. Rosa, Utilization and rating of machine commutated inverter fed synchronous motor drives, IEEE Trans. Ind. Appl., IA-15 (1979) 155. [3] F.C. Brockhurst, Performance equations for d.c. commutatorless motor using salient pole synchronous type machine, IEEE Trans. Ind. Appl., IA-16 (3) (1980) 362. [41 E.P. Cornell and D.W. Novotny, Commutation by armature induced voltage in self-controlled synchronous machine, IEEE Trans. Power Appar. Syst., PAS-74 (1974) 760. [5] A.C. Williamson, N.A.H. Issa and A.R.A.M. Mukky, Variable speed inverter fed synchronous motor employing natural commutation, Proc. Inst. Electr. Eng., 125 (1978) 113. [6] Y. Takeda, S. Morimoto and T. Hirsa, Generalised analysis for steady state characteristics of D.C. commutatorless motors, lEE Proc. B, 130 (6) (1983) 273. [7] C. Concordia, Synchronous Machines, Wiley, New York, 1951. [8] T. Kataoka and S. Nishikata, Transient performance analysis of self-controlled synchronous motor, IEEE Trans. Ind. Appl., IAÂ7(2) (1981) 152.