Development of a model for representing dynamic behaviour of synchronous motor variable speed drives

D. Kottick, MlEE B.Z. Kaplan

Indexing terms: Synchronous motors, Variablefrequency source, Three-phase sinusoidal oscillator, Permanent magnet motors

-~

Abstract: The paper introduces a compact method for representing synchronous motors con- nected to a source of variable frequency. The model is based on a three-phase sinusoidal oscil- lator that represents the components of the mag- netomotive force. The whole model is of seventh order and it accurately represents the dynamic behaviour of the motors. Even simulated dynamic behaviour, in which rapid frequency modulation of the voltage source is involved, agrees well with that obtained by experimenting with a laboratory system. One of the advantages of the model is its applicability for permanent magnet motors and permanent magnet brushless DC motors.

1 Introduction

The paper deals with the development and testing of a method for representing a synchronous motor supplied by a variable frequency source. The model is not entirely physical since it does not attempt to represent the pro- cesses in the machine by differential equations represent- ing each of the system items individually. The model represents the overall performance and not each of the components separately. A comprehensive simulation of the model is attained in spite of the fact that the motor is represented in a compact manner with the aid of a rela- tively simple three-phase oscillator. The principle of representing synchronous machines by modified oscil- lator equations for relatively simple cases has been intro- duced earlier [l-41. We have developed models for representing voltage regulated and unregulated gener- ators and motors. The unregulated synchronous gener- ator model [2] is similar to the synchronous motor model; they differ only in the direction of the stator cur- rents. The simulation capability of the unregulated gener- ator model to demonstrate relatively complicated transient phenomena has encouraged us to extend the

5Q IEE, 1995 Paper 1805B (Pt), first received 24th June and in revlsed form 5th December 1994 D. Kottick is with Israel Electric Corp. Ltd, Research & Development Div., P.O. Box IO, Haifa 31000, Israel B.Z. Kaplan is with Ben-Gunon University of the Negev, Dept. of Elec- trical & Comptuter Engineering, P.O. Box 653, Beer-Sheva 84105, Israel B.Z. Kaplan is the incumbent of the Abrahams-Curiel Chair in Electronic Instrumentation

IEE Proc.-Electr. Power Appl., Vol. 142, N o . 4 , July 1995

method and to test the performance of synchronous motors during complicated transient regimes.

In the present work the dynamic behaviour of a synchronous motor connected to a source of variable frequency is investigated. The advantage of the present model is that it is capable of simulating the waveforms of the variables at the line frequency, and not merely their average values. Hence, the model is especially suitable for simulating the dynamic behaviour of relatively small motors whose dynamic phenomena are of a rapid nature.

In recent years synchronous motors have been used in variable speed drive systems [ S , 61. Small permanent magnet motors are manufactured and used in control systems. A compact method for modelling instantaneous transient dynamics is therefore needed [7]. Another type of motor that is rapidly being developed is the D C brush- less motor [8, 91. Although the DC brushless motor is supplied by a direct voltage source, its operation is some- what similar to the operation of a synchronous motor. The rotating rotor of the DC brushless motor triggers (it is usually done with the aid of optical sensors) amplifiers that supply the stator windings in the correct order. Thus the operation of a brushless DC motor is similar to that of an oscillator. A brushless DC machine can be regarded as a sort of a synchronous machine possessing electronic feedback which forces the electronic part to oscillate con- tinuously and forces the machine to rotate indefinitely. The synchronous motor representation developed earlier [2, 41 evolved around a three-phase oscillator model. Hence, it appears that the present work possesses a certain relevance to recent efforts investigating the dynamics and control capabilities of brushless DC motors [SI. Furthermore, brushless DC motors are usually constructed with a permanent magnet rotor. The latter manifests another connection between the present work and brushless DC motors.

2 Synchronous motor model

A synchronous machine can operate either as a synchro- nous generator or as a synchronous motor. Synchronous generators are employed as electrical power sources. Syn- chronous motors are employed when relatively large reactive power is required. In these cases synchronous motors are superior to induction motors because their

This is an enlarged version of a paper that was presented at the 12th IMACS World Congress - Paris 1988.

269

reactive power can be controlled by the DC excitation current. Another area, rapidly developing, is that of adjustable speed drives of relative agility, that use smaller synchronous motors. The model that is developed in this Section is especially suitable for relatively small motors, since it is capable of simulating rapid dynamic pheno- mena with small time constants, which are typical of small motors.

A central item in the suggested model is a three-phase sinusoidal oscillator system. The quasilimit cycles of the system are associated with three-phase variables x, y. z which are arranged symmetrically in the steady-state with equal amplitudes as in a balanced three-phase volt- ages system [IO, 1 I ] :

w i = - ( y - z ) + E [ l - p(xZ - yz ) ] x

v 3

w i = /7 (.y - y ) + E [ 1 - p(zZ - xy)]z

\ / J

where E and p are positive constants and o is a param- eter that determines the frequency. The dynamic behav- iour of the oscillator represented by eqn. 1 is such that the frequency is determined instantaneously by a change in the parameter w [IO, 1 I]. A variation of the frequency does not involve a change of amplitude when the oscil- lator is already in its steady state [IO, 111 . This means that the behaviour of the oscillator is very similar to that of an ideal voltage controlled oscillator (VCO) in three phases. The steady-state amplitude of the oscillations is determined by the value of l/Jp. The oscillator system is distinguished in possessing a steady-state dynamic behav- iour with no direct interaction between the 'amplitude process' and the 'frequency process'. As a result, a varia- tion of the parameter w (even a sudden variation) does not cause any perturbation of the amplitude [l-31. This is due to the fact that the sizes of the evaluated levels: xz - yz, y z - zx, z 2 - x y are kept continuously constant (they are invariants) when x, y, z are three equal ampli- tude sine waves arranged symmetrically in a way similar to the voltages of a balanced three-phase system. Fur- thermore, the constant sizes of the previously evaluated levels are not affected in any way by change in w, even during the time when the change takes place (provided, of course, that the amplitudes have already reached steady- state when the frequency change process starts). The property of the xz - vz, y 2 - zx, z 2 - xy values to be three equal-size invariants, when x, y, z is a balanced three-phase triplet, assists in explaining the purity (they are distortionless) of the sine waves generated by eqn. 1

An important issue in the suggested model is that of choosing the x, y, z signals of the three-phase oscillator as components of a rotating magnetomotive force which possesses a constant value. One way of generating such a constant magnetomotive force in a real system is by sup- plying the rotor from a current source. The assumption of a current source exciting the rotor enables one to neglect any interaction of mutual impedances acting between the stator coil system and the rotor. Another possibility is to consider a small machine that includes an assembly of permanent magnets in the rotor. Such small machines, whose permanent magnets are manufactured from modern materials, have been discussed recently [7 ,

[lo, 111.

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121. These modern materials possess large coercive force; their hysteresis curve in the region of operation reduces almost to a straight line whose steepness is equal to po (the permeability of free space), and it passes the negative point of the coercive force on the horizontal axis and the positive point of the saturation inductance on the vertical axis. Hence, the permanent magnet in the model can be replaced by a coil supplied by a current source. The coil form is that of the magnet enclosure and it should cir- cumscribe an empty space of the same form as the magnet. The space should be empty as a result of the p,-like characteristic mentioned previously. This means that the current source assumption is of value also for modern permanent magnet machines.

The stator winding fluxes are regarded as produced by the x, y, z components of the rotating magnetomotive force described previously, together with a magneto- motive force generated by the currents in the stator. Hence:

where N is the number of turns in the stator coil, 9 is the magnetic reluctance per phase, i,, i, and i, are the cur- rents supplied by the source into the motor phases. As a result, the motor terminal voltages can be written as

N i N 2 di, c, = ri, + - + - -

d d dt

N j N 2 di 3 .% dt c, = ri, + - + - Ni N 2 di

t'; = ri, + - + - 2 3 d dt (3)

where r is the resistance of each of the stator windings. The terminal voltages can also be expressed in terms of the three-phase supply voltages:

U, = A , cos (2nf,t - 4) OY = A , cos (27cf,t - 120" - 4) U; = A , COS ( 2 ~ f , t - 240" - 4) (4)

where A , represents the amplitude of the voltages sup- plied to the motor and f , is the supply frequency. The dynamic behaviour of the motor stator phase currents is obtained from eqns. 3 and 4:

di, r i dt 1 I N _ - _ - cos ( 2 K f , t - 4) - - i, - -

I = N 2 / d is the self-inductance of each of the phase wind- ings. The rotor instantaneous angular velocity is deter- mined by the electromechanical interaction:

dw T = T, + T, = J - dt + b(w - wo)

1EE Proc -Electr Power Appl . , Vol 142. N o . 4 . July 1995

where oo is the synchronous angular velocity, b is an effective viscous damping coefficient which includes mainly electromechanical effects, and Te is the electro- mechanically produced torque, given by

where = bN2/w is the mutual inductance between the phases. The changes in the flux equations result in corre- sponding changes in the current equations, which are

T, consists of the torque applied for performing external mechanical work. As a result, the overall model employed for treating the dynamic processes of a syn- chronous motor consists of eqns. 1 , 5 and 6, which can be expressed as a system of seven state equations:

change. In the other mode of motor performance the

frequency to another, frequency of supply is suddenly made to jump from one

. U ?' = - (z ~ x ) + E [ 1 ~ p( yz - zx)]4

7 = - (U - 4') + E L 1 - p(? - xy) ]z

.3

v'3 . w

di, A , r i d t - I cos (2nf1t - 240" - 4) - - iz - -

I N - -_

Eqn. 10 replaces eqn. 5 and the rest of the model equa- tions are unchanged.

The nonlinear stabilisation terms of the oscillator model (eqn. 1) can sometimes impose phenomena similar to those that are usually represented by mutual induc- tances. This was demonstrated in References 1 and 3 which dealt with the representation of voltage-regulated synchronous generators. In the latter cases the stabilisa- tion terms of the oscillator included the following com- ponents: P$ - c y r z , 1): - C = L > ~ , v i ~ ~ , t > , . If the voltage in one of the phases changes (for example. as a result of unbalanced loading) the change would influence the other phases, and the currents in all three phases are eventually affected. In the case of a synchronous gener- ator, the voltages are produced by the treated machine, while in the case of motor operation of the machine, the terminal voltages are supplied by the electric source and hence they cannot be simulated as being affected by the model nonlinearities. In the present variable speed syn- chronous motor model the stabilisation terms of the imbedded oscillator consist of: xz - yz, y 2 - zx, z 2 - xy. In this case, a change in the voltage of one of the phases does not cause any change in the other two phases. Hence, in this case the introduction of mutual induc- tances is also helpful.

3 Comparison of simulation and experimental results

best expiessed 6 matrix form: I

rdixi I + M r-

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(10)

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The frequency of the supply in the first mode of tests

(1 1)

can be written as follows:

fi = 50 + B sin (2nFt)

where B = 1 Hz,f, is the varied supply frequency in eqn. 4 and F is a relatively lower frequency of modulation.

L 10K -

O g K j

modulation frequency F = 6 Hz). Fig. 4 demonstrates related experimental results when the frequency of modu- lation is 6 Hz. The phenomenon represented by the latter results is similar to that of a second-order system [16].

modulat ion frequency. H z 0

Fig. 1 Block diagram of the variable frequency power supply. The system consists mainly of a low power three-phase V C O and an ampliJica- tion system based on operational ampl8ers

Table 1 presents experimental and simulation results. In the first column the modulation frequency ( F ) is pre- sented. The second and third columns present the experi- mental and simulation rotor mechanical movement variations, respectively. The results in Table 1 indicate that when the modulation frequency is low the motor is capable of following the changes almost instantaneously. For higher modulation frequencies the mechanical rotor movement has an overreaction, and for even higher modulation frequencies the rotor is not able to follow the rapid supply changes.

Table 1 : Experiments and simulations responses of the rotor movement to a frequency modulated supply at various modulating frequencies

Modulation Simulation Experimental frequency, F mechanical mechanical (Hz) movement movement

W z ) (Hz)

1.3 1 1 2.5 1.05 1.05 3.5 1.15 1.1 4 1.1 1.15 6 0.8 0.8 8 0.4 0.5

Fig. 2 shows graphically the effect of input frequency modulation on rotor mechanical movement. Amplitude results are shown in Fig. 2a, and the corresponding phase responses are compared in Fig. 2b. It can be concluded that higher modulation frequency results in a larger phase difference between the input frequency variation and the corresponding rotor movement. Fig. 3 shows the detailed simulation results related to one of the points in Fig. 2 (a point related to the case where the supply

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1 0 2 4 6 6 10

modulation frequency, H z b

The response of the rotor movement to a frequency modulated Fig. 2 supply at various modulation frequencies D amplitude response b phase response A experiment 0 simulation

mechanical frequency N 551 53 I r g L9

5 51 C

2 L7

032 0 6 4 0 9 6 1 2 8 time,s

Fig. 3 at 6 H r in Fig. 2

Detailed output of the simulation work which yielded the result

Fig. 4 Detailed experimental response to a frequency modulated source when thefrequency ofmodulation is 6 Hz upper trace ~ the Instantaneous frequency of the input, namely. f, = 50 + I nn (2n6t) lower trace - the related frequency of the rotor movement

The performance of the motor when the supply fre- quency jumps suddenly from one frequency to another

I E E Proc.-Electr. Power Appl., Vol. 142, No . 4 , July 1995

was also investigated. Figs. 5 and 6 show the rotor mechanical frequency when the input frequency was suddenly changed from 50 to 35 Hz in the simulation and in the experiment, respectively.

62 01

55 61

L 3001 1 I I 1 I I I I I

0 003 006 009 012 t1rne.s

Fig. 5 motor input frequency way changedfrom 50 H z to 35 H z

Simulation result of the mechanical rotor frequency when the

Fig. 6 motor input frequency was changedfrom 50 H z to 35 H z

Experimental result o f the mechanical rotor frequency when the

The simulation results and the experimental results yield similar responses; namely, the response demon- strates a mechanical movement similar to that of a second-order system. The rotor exhibits decaying angular oscillations with respect to a spatial vector which rotates at the newly established synchronous speed. The param- eters -5, the damping ratio, and w d , the damped natural frequency of the approximately deduced second-order system, are similar both in experiments and simulations (5 = 0.67 in both the simulation and experiment, the damped natural frequency in the simulation was wd = 36 rad/s and wd = 35 rad/s in the experiment). They also agree with the results of other authors [16].

4 Conclusions

The investigations demonstrate that the method of mod- elling synchronous motors suggested in the paper pro- vides reliable and relatively precise results. The emphasis has been on the transient behaviour of synchronous motors when connected to an AC three-phase supply whose frequency is changed in various possible regimes of variation against time. One of the advantages of this method is that, unlike some other methods, it is capable of representing short-term phenomena of durations which can even be shorter than the supply voltage period, and not merely average variations. For example, models that are commonly used by electric utilities for representing transient behaviour provide only average results 117, IS]. The suggested model (because of its compactness), is of order that is not significantly higher than the order of other models. Its order is even smaller than that of certain models that supply merely average information. The new model should assist not only in the

IEE Proc - t lectr . Power Appl., Vol. 142, N o . 4 , July 1995

overall design of variable speed control systems but may also be utilised for the internal programming of micro- computers which are installed in systems in order to par- ticipate in the online control of the motors. It is worth mentioning that the present model preserves the well known [16] second-order behaviour (with, however, some additional nonlinear flavour) of the rotor dynamics.

One of the advantages of the model is in obtaining all the phase variables by employing the same procedures. We do not rely on trigonometric transformations to obtain some of the variables. This contributes to the uni- formity of the model as a whole, which in turn helps in simplifying the implementation of the model in various computers. Furthermore, trignommetric operations are usually either time consuming or resource consuming when represented by digital computers. As a result, the present model possesses a time saving property, which is helpful for an online digital operation. Another disadvan- tage of the trigonometric procedure is the fact that it is correct only for fully balanced circumstances. The electri- cal variables are assumed there to be symmetrical even during transients. The method suggested in the present paper does not possess such a limitation: it can allow a certain independence of the variables of each of the phases. This is despite the distinguished compactness of our model.

Another advantage of the present model, explained in Section 2, is due to its simple adaptability for rep- resenting permanent magnet machines. Many permanent magnet synchronous machines are built in a way that causes their back EMFs to be sinusoidal [19]. This situ- ation is easily catered for by the present model. This can be concluded from eqn. 3 which represents the back EMF variations against time (when the currents are assumed to be zero). The represented back EMFs are indeed sinusoidal, since the x, y, z values provided by the oscillator of eqn. 1 are also sinusoidal. A large propor- tion, however, of commercial and experimental brushless DC motors possess a trapezoidal back EMF [19]. It would be helpful if the presently suggested method could also cater for the trapezoidal case. It would then mean that an oscillator similar to that of eqn. 1 could be devised, in which the x, y, z waveforms are trapezoidal integrals. Three-phase oscillator models similar to the one represented by eqn. 1, whose x, y, z waveforms are not sinusoidal but trapezoidal, square or triangular are discussed in References 20-22. Even more generalised versions of the oscillators can also be constructed. Far- reaching generalisations were comprehensively discussed in the two-phase case [23, 241, and it can be extended to three phases. Hence, an oscillator similar to that of eqn. 1 which is capable of generating appropriate x, y, z wave- forms for representing permanent magnet machines with trapezoidal back EMFs, can be provided.

One of the justifications for proposing this method lies in its advantage for representing asymmetrical flux patterns during transients. This was easily demonstrated when utilising the model for representing generators [ 1-31, Simulations of unbalanced load transients were easily performed with the aid of the model. It seems very probable that the same advantage is also theoretically valid when dealing with motors. However, inducing the relevant transients by running the motor against an externally pulsating mechanical load is difficult to achieve experimentally. Furthermore, rapidly pulsating large mechanical torques occur relatively rarely in practi- cal situations where brushless DC motors are employed as components in control systems. In addition, the

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occurrence of an asymmetrical fault in the electrical supply side of the motor when it is a part of a robust control system is also rare. It is, therefore, believed that the suggested model can cater for asymmetrical faults and transients also in the presently treated motor case. However, this facility in the present case is far less impor- tant than its counterpart in the synchronous generator case [1/3].

5 References

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274 IEE Proc.-Electr. Power Appl., Vol. 142, N o . 4 , July 1995