Evaluation of the Functional Quantities of a Polyphase Induction Motor Supplied by SCR voltage Control*

Copyright © (FAC Control in Power Electronics and Electrical Drives , Lausanne , Switzerland , (983

EVALUATION OF THE FUNCTIONAL QUANTITIES OF A POLYPHASE INDUCTION MOTOR SUPPLIED BY SCR VOLTAGE CONTROL* D. Casini and M. Lemmi Gigli Istituto di ElettrotecnI'ca, Universita di Pisa, Pisa, Italy

Abstract. The functional behaviour of a polyphase induction motor, supplied

by an electronic device, is investi~ated, A mathematical model is proposed which differs from the conventional ones. This model implies a theoretically infinite number of the phases. The results emphasize the limit features obtainable by increasing the number of the phases. Keywords. Electric drives; variable-speed gear;on-off control; a.c. motors;

polyphase induction motors. INTRODUCTION in the electronic device becomes. At the same time, the power components can be designed for a smaller power.

It is well known that the normally construc ted three-phase asynchronous motor presents some inconveniences when it is supplied by an electronic device. A specific constructi ve technique should be more ap~ropriate for util i zi ng the motor. Thi s is connected to the waveforms of voltage and current which are greatly affected by harmonics when there is an electronic supply.

However, the choice of the number of the phases involves not only constructive sides, but also functional characteristics. The aim of this paper is to investigate the effects resulting from the choice of a very high phase number. The phase number is assumed to be theoretically infinite. In such a way, the maximum advantages which could be obtained by increasing the phase number are emphasized.

If this criterion is followed, two different models would have to be worked out,in order to be employed, one with a traditional supply, the other with an electronic supply. In this case, the choice of phase number is free in the design of the motor supplied by an electronic device. Two or more phases can be adopted, or each slot can be separately supplied with respect to the adjioning one.

Only simulation procedures are shown in this paper. Some very simple wave-forms of supply voltages are employed in the simulation examples. No methods to obtain these voltages a re taken into account. A further analysis in this direction can be made at the moment of the practical application of the study in real cases with a fi nite phase number. It is foreseable that the wave-forms which can be obtained in practice are different from the sim~lified ones, which are shown in the examples. The same method of approach can also be used for the former.

As regards the electromagnetic design of the motor, there are no technical difficulties nor increased costS,on condition that the slot number is not greater than the usual one. I.e., the phase number cannot be greater than the usual slot number on a polar arc. The greater the phase number, the greater the number of components required

Only steady-state conditions are taken into account in this paper. The rotor is assumed to have a usual squirrel cage winding. Any dissymmetry is not considered in the elec-

~

This paper was carried out in the Pisa research unit of the CNR Research Group on Electrical Machines. CPE-J I o

283

D. Casini and M. Lemmi Gigli

284

trical quantities nor in the mechanical structure. No saturation of magnetic circuit is taken into account in this paper; but it cou 1d be introduced wi thout any diffi culty in an applicative study. The leakage fluxes are represented by means of constant inductive elements. Moreover it is assumed that the voltage drop in the electronic devices is the sum of two terms, one, independent from the current, and therefore calculable as a fixed voltage drop, the other, in proportion to the current, and thus calculable as an additional armature resistance. THE METHOD OF APPROACH When the phase number is theoretically infinite,it is not possible to utilize the usual methods of approach to the motor study. A different model (Lemmi Gigli, 1981,1982 , 1983) is introduced. Each stator or rotor phase is considered as an elementary conduc tor, disconnected from the adjoining ones. A cyclical supply is taken into account for the stator phases; the rotor phases are short-circuited. In each phase of both stator and rotor, the wave-forms of the functional quantities are shifted by an elementary time interval, in respect to the ones of the adjoining phases. This model has to be considered accurate on ly if the highest order of the harmonics,required for the representation of the phase currents, is low with respect to the phase number. Otherwise, the model gives approximate results. On the other hand, though,so me more common models are not accurate.For example, the d-q model represents two-phase windings and squirrel-cage windings with the same equations, while their behaviour is dif ferent as regards the effects of the current harmonics. In following the explained model, the linear densities of current must be introduced: As and Ar , for the representation of the stator and rotor currents,respectively. In both the stator and the rotor, the current linear density depends on the time, t, as well as on the air-gap abscissa, x. In steady-state conditions, the following occurs: aA

s

--=

at

aA

s

w- -

ax

( 1)

(1')

Where w and Sw are the frequencies (rad / sec) of stator and rotor current respective ly. The electrical balance of each phase brings the following equations: 3A

1

s

~

(2)

aA = A +T __r_ r dr at

(2')

RI [v(x)+e(x)]= As + Tds s

R'

r

e (x)

r

where l/R~ and l/R~ are the 1 i near densities of d.c. conductance, for the stator and the rotor windings respectively; for the stator, l / Rs can be inclusive of the effect due to the electronic de vices. Tds and Tdr are the leakage timeconstants of the single conductor for the stator and rotor respectively. At each time, the e.m.fs.,e s and er' which appear in (2) and (2'), are due to both sta tor and rotor currents, according to: lDw 2R

e (x)

s

f

x [As(x)+Ar(x)]dx

(3)

X-1T

x

e (x)

r

=

LDw

2R

f

[As (x)+Ar(x)] dx

(3')

X-1T

where l , D and R are the constant values of axial 1ength,air-gap diameter and reluctance respectively. It would not be difficult to take into account a variable reluctance due to the magnetic saturation, but this is not done in this paper . The equations (1) to (3) allow the evaluation of the functional quantities of the motor when it is supplied by a power source through an electronic devi ce. Both impressed-voltage and impressed-current procedures can be taken into account. In fact, the equation system can be resolved if the stator voltage or the stator current are known. In this paper, only impressed-voltage procedures are taken into account, in such a way that: - the current can flow through the ma in system of electronic switches only if it has the same direction as the impressed volta ge and if the switch system is turned on; - the current can also flow through a reverse semiconductor system if it has reverse

Evaluation of a Polyphase Induction Motor

285

In these conditions, when the stator current flows, the v(x) voltage, which appears in (2), can be assumed to be

v

v(x) =

V s

±

(4 )

t>V

where V is a constant voltage supplied by the e~ectronic apparatus and t>V is the constant component of the voltage drop in the electronic devices. The sign applying to Vs depends on whether the main elec-

t Fig. 1 - Three typical wave-forms of applied voltage: the a) wave has to be employed at a higher frequency; the b) and c) waves can be alternately employed at a lover frequency.

±

t I

direction in respect to the impressed voltage.

Fig . 2

Fig. 3

t,x

t,x

Fi g. 4

a

Figs. 2,3,4 - Typical wave-forms of the functional quantities of the motor, when the wave forms shown in fig. 1 ( a, b, c, respectively) are employed for the impressed voltage: a) impressed voltage (only referring to the on-off times of the main electronic switches);b) flux density at load; c) stator current at no load; d) stator current at load; e) rotor current at load.

t,x

286

D. Casini and M. Lemmi Gigli

tronic switches are turned on or off: if they are turned on, the sign is connected to the electronic control; if they are turned off, the sign is opposite to that of the current linear density. If the stator current achieves the zero value, it can vary, according to (2) ,only if the main switches are turned on, or if 1EsI > Vs. Otherwise, it keeps the zero va lue, till one or both of these conditions are met. THE WAVE-FORM OF THE FUNCTIONAL QUANTITIES OF THE MOTOR It is generally possible to apply only a numerical analysis to the equation system explained in the previous paragraph. This was done in this study, with reference to the three typical wave-forms of stator voltage, which are shown in fig . 1. Figs. 2 to 4 show some functional quantities of the motor corresponding to the same three types of applied voltage.

nal quantities of motor along the air- gap, at a generic time. The power supplied by the electroni c apparatus is P s

.~v( x) As (x ) dx

The power losses in the stator and windings are:

( 5)

rotor

P cus

R' ~A2( x ) dx s· s

( 6)

P cur

R' ~A2(x) dx r· r

(6 ' )

respectively. The stator losses also inclu de a part of the losses in the electronic devices. For an accurate evaluation, two values of l / R~ would have to be distinguished, depending on whether the current flows through the main devices or through the reverse semiconductors. The rest of the losses in the electronic apparatus is expressed by: (7)

According to (1), these wave-forms can be read versus the time, for a pre-defined phase, as well as versus the air-gap ab scissa, at a predefined time: obviously, two different time scales must be used for the stator and rotor quantities.

Also in this case two values of 6V could the be distinguished according to where current flows.

The wave-form of flu x density is greatly affected by harmonics, but rotates, unchanged when the time varies, along the air-gap at the synchronous speed . The torque is constant when the time varies , as is the current supplied by the electronic apparatus. All this is independent of the wave-form chosen for the impressed voltage. These results are, obviously,limit results, which have to be reshuffled when real conditions are taken into account with a realistic phase number.

The equations referred to in the two previ ous paragraphs were resolved in order to simulate two control procedures for the motor supply,when the frequency of the impressed voltage varies. In these procedures, the two wave-forms called b) and c) in fig. 1 are alternately employed. In both cases, the mode of shutting the voltage wa ve-form is chosen so that the flu x density maintains the same maximum value at all frequencies and, for each frequency, at each slip .

When the wave-forms of the functional quan tities shown in figs. 3 to 4 are known,it is also possible to evaluate the power supplied by the electronic apparatus, the power losses in the electronic devices,in the stator and rotor windings, the torque, etc. The iron losses are not considered can in this model, and their evaluation be made separately,on the basis of the wave-form of the flu x density . All these quantities are constant as the time varies, and its evaluation can be made by integrating the distribution of functio-

If, in a semi-period, there is only a single time interval in which the main electronic switches are turned on (case b in fig. 1), the control regulates the extent of that interval. If there are several in tervals in which the main electronic switches are turned on (case c in fig. 1) the control maintains as constant the lenght of the switching - on time intervals and regul ates the 1enght of the switchi ng - off time intervals. These control features are also idealstic: with a realistic phase number, they could only be approximate.

SOME EXAMPLES

Evaluation of a Polyphase Induction Motor

This all applies only situations of low speed. At higher speeds the simple square wave is employed in both the procedures. Figs. 5 to 7 summarize the results of the si mulation. The best results are achieved by employing the most complex waveform (the one called c in fig. 1). This is connected to the higher mean value of the flux densi ty, which can be obtained, under the same maximum value of the flux density.

t

CONCLUSIONS

If the phase number of the motor is brought there to a theoretically infinite value, are several advantages in the behaviour. The wave-form of the flux density is not sinusoidal, but rotates at a synchronous speed; the toraue is exactly constant, as is the current supplied by the electronic apparatus. All this occurs, which ever supply method is chosen, so that the control procedu -

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re can be optimized without any negative effect on the torque and on the supply cur rent. Some simulations show interesting features that can be achieved within a large speed range. It is obvious that these results can never be obtained with a finite number of phases. However, it is logical to think that the greater the phase number, the nearer the behaviour becomes to that described. In this paper no investigations are made in order to quantify this approximation, because this paper is only a preliminarstudy adressed to show an alternative way to the PWM procedures . Therefore it seems too early to make any comparisons with the PWM procedures; on the other hand, there are several PWM procedures with different effects on the motor behaviour,and a comparison with an "optimized PWM procedure" is not possible.

REFERENCES Ferraris, P., M. Lazzari, F. Villata (1981) The phase number choice as a method for altering the quality Factors of a polyphase inverter-fed induction motor drive. Symposiwn on nectrical Machines jbr Special Purposes. Bologna Italia , Settembre 1981. pp. 341-350.

Lemmi Gigli, M. (1981). A polyphase motor with armature voltage control .Symposiwn on Electr ical Machines for Special Purposes . Bologna Italia . pp. 371-378. Lemmi Gigli, M. (1982). The effects of the magnetic saturation in a polyphase motor with armature voltage control. ICEM Budapest, Settembre 1982 . pp. 587-590. Lemmi Gigli, M. (1983). A polyphase motor with armature voltage control including a current limitation. Symposiwn on nect rical Dri ves f or ground Transportation . Po sitano, Maggio 1983 .

Nelson, R.H. and P.C. Krause (1974). Induction Machine analysis for arbitrary displacement between multiple winding sets. I EEE Trans on PAS- 93 . pp . 841-848.