280 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 2, MARCHIAPRIL 1989

Analysis and Application of Three-phase Induction Motor Voltage Controller with

Improved Transient Performance WERNER DELEROI, JOHAN B. WOUDSTRA, AND AZZA A. FAHIM, MEMBER, IEEE

Abstract-An improved transient performance of three-phase induc- tion machines with controlled thyristor triggering is presented. A newly developed dynamic function for the thyristor triggering angle is utilized. The triggering angle function influences the phase modulation of the machine-supplied voltage, such that the normal transient problems are avoided. Simulation of transients, based on the analytical solution of the machine differential equations, for all modes of operation is provided. The beneficial effect of the suggested function in smoothing the transient flux, currents, and torque is illustrated. Application of the function to different switching conditions is performed with actual laboratory tests.

I . INTRODUCTION

HE SWITCHING of three-phase induction machine un- T der different operation conditions is one of the processes frequently performed through speed control, soft starting, energy-saving, and in wind energy applications [ 11. It is well- known that the transient behavior associated with frequent switching of induction machines is characterized by high cur- rent peaks and pulsating torques [2]. Such performance is most undesirable for both electrical supply and mechanical gearing systems.

In this paper, a solution for the aforementioned problem is suggested. It utilizes a common form of voltage controller with back-to-back thyristors in each supply line, with a new time-varying function for the thyristor triggering angle. By the use of this special function, the building up of the rotating main flux is delayed for some time period and the normally existing pulsating torque is reduced. The number of stator phases connected to the supply can change through the whole operational time, according to the state of the thyristors. The analysis of the machine performance is provided in this paper for the various modes of connection.

For the general application of the suggested triggering func- tion, different switching conditions were taken into consider- ation, including the possibility of switching the machine from standstill or at any running speed. This is done for both gen- erator and motor operational modes.

Paper IPCSD 87-42, approved by the Electric Machines Committee of the IEEE Industry Applications Society for presentation at the 1987 Industry Ap- plications Society Annual Meeting, Atlanta, GA, October 19-23. Manuscript released for publication December 14, 1988.

W. Deleroi and J. B. Woudstra are with the Faculty of Electrical Engi- neering. Technical University of Delft, Mekelweg 4, 2628 CD Delft, The Netherlands.

A . A. Fahim is with the National Research Center, Electrical and Electronic Research Institute, El-Tahir st. Dokki, Cairo, Egypt.

IEEE Log Number 88253 1 1 .

I L1

Fig. 1. Basic circuit of voltage controller.

To verify the effectiveness of the suggested triggering func- tion, theoretical and experimental results are reported. The improvement achieved in transient performance is shown by comparing it to a direct switching process.

II. VOLTAGE CONTROLLER CIRCUIT ANALYSIS The basic circuit used for the controller is shown in Fig. 1 .

The circuit shows the commonly used back-to-back thyristors in each supply line. The assumptions in the analysis are as follows.

1) The three-phase supply is balanced sinusoidal voltage. 2) For the motor, the windings are assumed to be symmet-

rical and star-connected. Saturation and eddy currents are neglected.

3) For the controller, the six thyristors are assumed to be ideal and symmetrical. The forward voltage is com- pletely blocked from the instant the thyristor anode is positive to the instant at which it is fired, and conduction continues only till the phase current reaches zero.

4) For the switching process, the machine speed is assumed not to change at the instant of switching.

Thyristor Triggering Angles Taking the thyristor firing angle a to be a constant angle

from the point of zero crossing of each phase voltage, the

0093-9994/89/03OO-O280$01 .OO 0 1989 IEEE

DELEROI et al.: INDUCTION MOTOR VOLTAGE CONTROLLER

+ I 1 2 3

2 8 1

g a t e 2 pulses 2

gate- pulses

Tlme

0"

60 2 Aa

Fig. 2. Triggering pulses for a(t) = 010

12d'Aa

Fig. 3. Triggering pulses for a(t) = 010 k A a .

train of triggering pulses required for the whole three-phase system will be separated by an angle of 60" as shown in Fig. 2. To allow variation in the value of a with time, i.e., a( t ) = a. f Aa(t ) , the train of the triggering pulses required will be separated by 60" +Aa(t) as shown in Fig. 3. (Symbols are defined in the Nomenclature at the end of the paper.)

For normal steady-state operation, with current continu- ously flowing in the three phases, the triggering angle a is ct = cp, where cp, is the load-dependent phase-shift angle. For a < cp, the thyristor conduction requirements are not satis- fied, because I p h < 0 and v t h > 0. So the constraint for the triggering angle a is (Y 2 cp.

111. THEORETICAL ANALYSIS It is important to note that, by implementing the thyris-

tor controller, the number of machine phases connected to the supply can change during operation time. The different operational modes of the machine are

a) symmetrical three-phase operation (M3); b) unsymmetrical two-phase operation (M2); c) disconnected mode (MO).

Changing of the mode of operation is observed when 1) two thyristor firing pulses are provided (every 60" kAa(t)); 2) one of the phase currents becomes zero. The sequence of changing of the mode of operation is shown in Fig. 4, where ( t l - t3) are the time instants of providing the two triggering pulses.

A. Three-phase Operation Mode Using the generalized theory of ac machines, [3], the three-

phase stator and rotor voltage differential equations described

Fig. 4. Sequence of changing mode of operation

in a synchronous rotating two-axis frame are [4]

+

where

Ls 0 Lm 0

0 Ls 0 Lm

L , 0 L , 0

0 Lm 0 Lr

xs = ~ ( L S I + Lm)wLs

Xr = w ( L , ~ + L + m)wL,.

The solution of (1) is obtained analytically by computing the system eigenvalues and eigenvectors [5], [6]. The total solu- tion for the transient stator and rotor currents are calculated by combining the particular and the homogeneous solutions as follows:

j(t> = jpart(t) -k j h o m ( t ) .

To allow continuity between the solutions of different modes of operation, the current vector computed in the rotating (d- q) reference frame is transformed to a fixed reference frame (a-b) by multiplying with the transformation coordinate ma- trix.

B. Two-Phase Operation Mode For the two-phase operational mode, (1) must be modified.

Since the two operating phases have the same current, their effect can be represented by an equivalent single winding.

Further, to simplify calculations by keeping the rotor equa- tions unchanged, the mutual inductance between the rotor and the equivalent single-phase winding (i.e, the main flux) is taken to be the same as that between the original windings.

2 8 2 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 2, MARCHiAPRIL 1989

It is also assumed that the system is power invariant. The values for the stator voltage, the current vectors, and the ma- chine parameters, in relation to the original values used in the symmetrical two-phase systems, are as follows [7]:

v:= v, I:= &I.

R,*= (2/3)R,

x,*l = (2/3)Xs1 where the asterisk indicates the two-phase system. It must be remembered that the rotor parameters and the mutual induc- tance are kept the same. For the resultant modified system of the equations, the solution for the currents is obtained in the same manner as for three-phase system, by computing the particular and the homogeneous solutions.

C. Disconnected Mode When the stator current in mode M2 drops to zero, the

machine is in the disconnected mode. In this mode, the rotor currents start with their values at the end of the previous mode and decay to zero with the rotor time constant (7,). The time- varying solution of the currents for the disconnected mode are as follows:

Isd(t) = 0.0

Is&) = 0.0

I r d ( t ) = e-r'Tr{Zrd(T2) cos((1 - s>t>

- Irq(T2) sin((1 - s>t>>)

Irq(t) = e-"'r{Zrd(T2) sin((1 - s>t>

+ Irq(T2) cos((1 - s>t>>l where

r, = X J R ,

O I t I 6 0 " + A c Y - T ~

and T2 is the instant of time when stator current drops to zero.

D. Continuity of the Current Solution To ensure the continuity of the solutions when the oper-

ational mode changes, the initial value of the currents at a certain mode is taken to be the same as the final value of the currents at the mode advanced to it.

IV. TRIGGERING ANGLE FUNCTION The main objective of this work is to provide a function for

the triggering angle a( t ) , which allows smooth switching for the three-phase induction machine during both motor and gen- erator operational modes. Since the aforementioned function is applied between two fixed operation points, the initial value cyo and the final value c y f , the machine performance must be studied first under operation with constant values of C Y .

A . Operation with Constant Triggering Angle CY

The constraint in this paper for the value of CYO is that the first stator current peak does not exceed the rated value. Note

a-

Fig. 5 . First peaks of stator current versus switching angle 01

TABLE I

Machine Parameters (per unit)

R, = 0.042

X , = 1.500

R , = 0.040 X,, = 0.085 X,, = 0.113

1 1.0

I - 1,

08

06

04

Oi

C w t -

Fig. 6. Half-cycle of stator current waveform for different values of a.

that this restriction on the first peak does not guarantee that the following peaks of the current will not exceed this limita- tion. However, the overall performance is improved, as will be shown. By plotting the relationship between the values of the first peaks obtained for different switching processes with different values of (YO (Fig. 5), the theoretical calculations with the motor parameters of Table I show that, for Zpeak = 1.0 pu, the range of CYO lies between 110" and 120". This range appears to be acceptable for most motor parameters. On the other hand, the final value of CY (i.e., ay) is defined up to the constraint that af 2 p. For c y f = cp, continuous current flow will be provided at the end of the transient; while for af > p, periods of current interruption will be recognized in the current waveform which could be utilized for motor speed control. In Fig. 6 a half-cycle of the steady-state current

DELEROI el al.: INDUCTION MOTOR VOLTAGE CONTROLLER 283

I

6 4 2 0 -2 -4 -6 1%) - 5

Fig, 7. Range of C I ~ values for motor and generator operation

waveform is shown for different values of a and for s = 4 percent. For a > cp, three parts of the three-phase mode and two parts of the single-phase mode are recognized. The rest of the time interval has a zero current. By increasing the firing angle, the parts of single-phase mode are increased, the limit where a = 95" and the whole region is covered with the single-phase mode.

If af is chosen to be greater than cp, there are two cases to be studied. These are the motor and generator operational modes. For motor operation, with symmetrical phase voltage and af L 150", the conduction requirements are not satisfied. This is due to a change in polarity of the line voltage across the operating thyristors. In the generator region, the stability requirements have to be satisfied for the chosen value of af. To check the stability, the final stator current should behave in a steady-state manner.

Accordingly, in Fig. 7 the chosen region of af for motor operation is between the value of a = cp and the upper limit (a-b), where a = 150", while in the generator region, the upper limit (6-c) is defined from the stability requirements. It is important to mention that, if af exceeds the upper limit in the generator mode, the stator current will exhibit an unduly high peak before decaying to zero. To avoid such a case, af should not exceed cp at the corresponding operating slip.

As a further calculation step, before defining the time- varying switching function, the torque-speed characteristics of the motor under investigation are provided for different constant values of a (Fig. 8). Comparing the normal torque- speed characteristics of the induction motor, it is observed that, for constant values of the firing angle, the breakdown torque peak has disappeared. The conclusion is that it is possi- ble to start (s = 1) an induction motor with a constant torque. It is necessary to use more than one time-varying switching function. Switching while the machine already rotates with a speed corresponding to a slip between -sn I s I s, can be done with one switching function, because the breakdown torque peak is passed in the torque-speed characteristic. B. Operation with Time- Variable Switching Function

To define the time-varying function a( t ) , it is preferable to consider the possibility of switching while the machine rotates

0.6 0.4 0.2 c s

o b ' ' ' ' ' ' i Fig. 8. Torque-speed characteristic for constant values of a.

w t - Fig. 9. i versus time for different values of triggering frequency fa.

(without excitation current) with a speed corresponding to a slip between -sn I s I sn. The constraint utilized for smoothing the switching process is chosen as

Is I I , = 1 pu.

As a first trial, a straight line relation for a( t ) between the two endpoints a0 and af is employed. However, the results showed that smoothness of the transient current is not sat- isfied. By allowing variation of a in small steps such that for every new value of a the corresponding transient current does not exceed the permissible value, it was found that the variation of CY with time behaves in a cosine function form.

From the preceding, the triggering function can be written as

a( t ) = a + b cos(ct)

where a , b , and c are constants. To calculate a, b , and c, the two defined values of a0 and af and the rate of change of the function fa are utilized. To determine fa (frequency of the triggering function), the time variation of I,,, (transient current peaks) is plotted for different trial values of fa and at s = -4 percent (see Fig. 9). It was found that, for the motor under investigation, the frequency fa = 3.5 Hz provides the most smooth Imax variation (curve b). Since the value of fa is dependent on the value of the switching slip, the relation between fa and the slip is defined. The main idea of defining fa@) is to relate two cases of switching with different values of slip. Knowing the variation of the power angle cp with the slip, and with the assumption that af = cp, the relationship af(s) is defined (Fig. 10). Using a previously chosen value of fa,, which satisfies smooth switching at s = sl, a new

2 84 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 2, MARCH/APRIL 1989

value of triggering frequency fo2 for switching at s = s2 is determined: (fol/fa2) = ( A B / A C ) , where AB and AC are as shown in Fig. 10.

Using the two preceding calculated points ( fal, sl ) and (fa2,s2), a straight line relation for variation of fa@) is ob- tained (Fig. 11). From this relation, the triggering function frequency can be determined at the desired value of switching slip as shown in Fig. 12.

v. EVALUATION OF THE CONTROLLER

To confirm the effectiveness of the controller with the pro- posed triggering function, computational and experimental re- sults are provided. The transient performance of a three-phase induction machine with the parameters of Table I was com- puted. To allow comparison, the performances of both the di- rect switching process and the controller implementation are provided.

The time variations of the stator current, torque, and the speed, for switching processes carried out at slip s = 0, are shown in Figs. 13 and 14. For the direct switching process, the maximum transient current approaches a value of 5 pu, while for controlled switching, the maximum amplitude is around 1 pu. Also, from torque variation, the direct switching process shows a considerable torque peak with a nonzero average. For the same case, by using the controller, the amplitude of the transient pulsating torque is quite improved, with an average tending to zero. Further, the frequency of the pulsating torque is in the range of 300 Hz, which makes the torque pulsation

For more detailed explanation of the transient behavior, the vector trajectory of the air-gap flux during the transient is pro- vided in Fig. 15, where every half-cycle a is marked. For the direct switching process (the continuous-line plot), the field buildup is remarkably fast. After the first half-cycle, the flux exceeds the steady-state value and a dc flux component is rec-

t i

18-

16.

"-

12-

.

- unrecognized by the load. I N

ognized. This peak value of flux corresponds to the maximum current peak of 5 pu, which is shown in Fig. 12. The final steady-state value of the flux is almost approached within two cycles (4a). Comparing the same situation with the controlled switching (the dotted-line plot), the flux buildup is quite grad- ual, and the steady-state value is achieved within a reasonable number of cycles, which makes the variation in the flux con- siderably smoother.

To emphasize the advantage of the suggested switching function, Fig. 16 shows a switching process in the genera- tor operational mode at s = -0.04. By using the switching function, the average transient torque is gradually decreased from zero to the desired steady-state value, while the current varies smoothly between the range of 1.0-0.8 pu.

To justify the range of accuracy achieved in practical ap- plication by using the suggested controlled switching, the fol- lowing experimental results are reported. In Fig. 17, the time variation of the switching function a(t) and the corresponding phase current is provided for the same case (s = 0) as in Fig. 12.

From the current values of both results, it can be seen that the experimental and theoretical results are in good agree- ment. Fig. 18 shows the experimental results for the controlled

J ' , ; , : , , 0"

0 -4 -8 -12 - s (%)

Fig. 10. Final triggering angle CY, versus slip.

0 - A -8 -12 (%) S-

Fig. 11. Triggering frequency fa versus slip.

0 2n Ln 6n 8n 10n 1211 1411 16n 18n wt -

Fig. 12. Time variation of i for different slip values.

>controller a wlthout b with

0 0 2n 4n 6n Br 1011 1211 1411 16r 181-1 2 0 ~

ut - Fig. 13. Stator current for switching at s = 0.0. (a) Without controller.

(b) With controller.

DELEROI et al.: INDUCTION MOTOR VOLTAGE CONTROLLER

S. 0

I " \ / \

a= wothout ~= w,th >controller

980

L i n 4n 6n 8n 10n 1211 1Ln 16n l8n 20n 2in 26n

wt - Fig. 14. Torque and speed variation for switching at s = 0.0. (a) Without

controller. (b) With controller.

without controller 7

Fig. 15. Air-gap flux vector trajectory.

! ! I s=-004

w t - Fig. 16. Torque and stator current variation for controlled switching at s =

-0.04.

switching at s = 4 percent. For a controlled switching process at s = 1.0, the experimental results (Fig. 19) show the time variations of both the stator current and the mechanical speed. Noting these results, it is clear that transient speed measure- ment is possible with the standard equipments when the con- troller is employed, since the variation in the speed is quite

t a

2 8 5

n

I 0 ' 80 120 (ms) 160 40

t-

40 80 120 (ms) 160 t -

Fig. 17. (a) Trigger function. (b) Corresponding phase current. "'..- a,

500

0 t-

Fig. 18. Experimental results for triggering angle and stator current for controlled switching at s = 0.04.

" 0 :C 0 5 15 (sec) 10

t-

12001

n 600

300

(sec) w t -

Fig. 19. Experimental results for current and speed variation for controlled switching at s = 1.0.

VI. CONCLUSION

The switching of a three-phase induction machine is achieved without the transient problems associated with high currents and pulsating torque. Utilizing a dynamic function

smooth and no more abrupt variations are recognized. for the triggering angle of the voltage-controlled thyristors

286 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 2, MARCHIAPRIL 1989

proves to be a simple and effective way to improve transient performance. By employing the proper triggering function, the rate at which the main flux builds up is decreased, and transient torque is smoothed. Defining a value for the max- imum required transient current is possible by choosing the starting angle of the triggering function.

Simulation of the transient performance, based on the an- alytical solution of the machine differential equation under different modes of operation, is established. The simulation model results show that a smooth switching process for an induction machine, running at any speed, for both motor and generator operation, is achievable. Also, results from the prac- tical realization of the suggested function proved the validity of this technique. The authors will discuss in the future the problem of the switching process with a remainder flux.

NOMENCLATURE

CY Firing angle (rad). cp Current phase shift (rad). r , Rotor time constant ( S K I ) .

Triggering frequency (Hz) . Phase current (pu). Rated current (pu). Speed (r/min). Stator resistance (pu). Rotor resistance (pu). Slip. Rated slip. Thyristor voltage (pu).

X , Stator reactance (pu). X , Rotor reactance (pu). X,, Stator leakage reactance (pu). X,, Rotor leakage reactance (pu).

111

121

r31

141

REFERENCES

V. R. Stefanovic, “Present trends in variable speed ac drives,” in Conf. Rec. 1983 Int. Power Elec. Conf., pp. 3 3 8 4 9 . H. Rehaoulia and M. Poloujadoff, “Transient behavior of the resultant airgap field during run up of an induction machine,” IEEE Trans. Energy Conversion, vol. EC-I, no. 4, Dec. 1986. C. V. Jones, The Unified Theory of ElectricalMachines. London: Butterworth, 1967. B. Adkins and R. G. Harley, The General Theory of Alternating Current Machine, Application to Practical Problems. London: Chapman & Hall, 1975.

[5] W. Deleroi, “Ausgleichsvorgange der symmetrische Drehfeldmas- chine der Asynchronmaschine, beschrieben durch frie Ausgleich- swellen,” Arch. Elektrotech., vol. 65, pp. 1-10, 1982. D. W. Novotny and J. J . A. Melkebeek, “Dynamic response of volt- age driven induction machine,” Elec. Machines Power Syst., vol.

W. Deleroi J. B. Woudstra, and A. A. Fahim, “Analysis of induction machine operation with changing number of stator phases,” to be published.

[6]

10, pp. 149-179, 1985. 171

Werner Deleroi received the following degrees in electrical engineering from the Technical University of Braunschweig, Germany: M.Sc. (1963), Ph.D. (1968).

He was appointed to the Chair of Power Electron- ics at the University of Hamburg in 1975. The next appointment led him, in 1978, to his present field of activity as an electrical engineering Professor at the Technical University of Delft, The Netherlands.

Dr. Deleroi received an award for literature from the German Association of Electrical Engineers, Power Systems Division.

Johan B. Woudstra received the B A . degree from the Technical High School in Alkmaar (1978) and the M.Sc. degree in electrical engineering from the Technical University of Delft, The Netherlands (1986).

He is a member of the scientific staff of the Power Electronics and Electrical Machines Division of the Faculty of Electrical Engineering of the Technical University of Delft. His research work is on elec- trical drives control and power electronics.

Azza A. Fahim (M’88) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Cairo University in 1973, 1977, and 1984, respec- tively.

Since 1974 she has been with the National Re- search Center, Cairo, Egypt, where she is presently an Associate Research Scientist. She has been a graduate Visitor Scholar at Iowa State University and the University of Wisconsin-Madison from 1981 to 1984. She has also been a Postdoctoral Re- search Fellow at the Technical Universih, of Delft,

The Netherlands, from 1986 to 1987. She has conducted a numbe; of studies on tubular induction motors. Her current research interests include power electronic control and the transient analysis of electrical machines.