Three-phase Induction Motor Dynamic Mathematical Model

Three-phase Induction Motor Dynamic Mathematical Model Emesto Ruppert Filho and Ronald0 Martins de Souza DSCE/FEEC/UNICAMP C.P. 6101 - CEP 13081-970 Campinas-SP, Brasil E-mail: [email protected]

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Abstract The induction motor dynamic model, frequently used in motor dynamic studies, is constituted by four voltage differential equations and one mechanical differential equation being very known among the electrical machine researchers. The main goal of this paper is to present a more comprehensive three-phase induction motor dynamic mathematical model including the skin effect, the temperature influence on the parameters and allowing for the stator and rotor winding and stator and rotor core average temperature evaluation. This model is useful for any type of motor dynamic studies mainly those including fast motor speed changings, intermittent loading and in case of motors fed from non-sinusoidal voltages contributing to the energy conservation and power quality subjects. I. INTRODUCTION

The induction motor dynamic mathematical model, frequently used in motor dynamic studies like motor control, drive specifications, electrical protection, Starting high inertia loads, fast and large load changings, sucessive startings, locked rotor, etc., is one represented by the differential equations (1) to (5). It is a model written in terms Of the winding linkage fluxes per second (Ul-VOlts) referred to 8 SYnC~OnOUSlYrotating dq reference frame (angular speed o ).

if the motor is available, provided by the motor manufacturer or could be calculated using manufacturer technical catalog or bulletin data when the former attempts are not satisfied. The main goal of this paper is to present a more comprehensive three-phase induction motor dynamic mathematical model including the skin effect, the temperature influence on the parameters, the stator and rotor winding and stator and rotor core average temperature evaluation. 11. DYNAMIC MODEL INCLUDING TEMPERATURE

INFLUENCE AND SKTN EFFECT

To take into account the temperature influence on the winding electrical resistances, equations (6) and (7) are used. In those equations A0 sw and A0 are the stator and rotor winding temperature rises calculated by performing the Of the thermal differential equations "Prising the dynamic thermal mathmatical model (8) to (11) Where WjI,Wj2,Wcl and Wc2 are the Stator and the rotor winding

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and core losses, respectively, calculated at each integration step by (12) to (15), e, is the motor emf and r, the magnetic loss resistance.

(4)

This model was developed many years ago and it is presented in many books and papers among them it could be mentioned [l] to [3]. It is very familiar to the researchers dealing with electrical machine dynamic studies. The parameters appearing in the equations are the machine winding electrical resistances (rs ,rr ) , the leakage and magnetizing reactances (Xes,Xk ,X,), included in the linkage fluxes. Those parameters values could be measured

0~7803~3946-0/97/S10.00 0 1997 IEEE.

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wj2

=c,-

Wc2 +Gwc(O,

de, dt -0,)

-0,)

(10)

de, =C , -+G,0, dt

(11)

+G,C(Q,

2

Wjl = 3rs@sw )is

(12)

w,

rotor winding (c) and rotor core (d) for a motor starting followed by continuous operation with rated load during 160 minutes. It is also shown in the paper how the experimental job was performed using a data acquisition system.

3e2,

=-

rm

W,l = Wc2 = 0,5 W, (according [4])

(15) 60

The coefficients C, ,C, ,C, and C , are the thermal capacities of the stator and rotor windings and cores, respectively, and ,G and ,G are the conductances representing the heat transferred between stator winding and stator core and rotor winding and rotor core, respectively. It is possible to calculate C’s and G s using some statements on motor temperature rises normally presented in electrical machine standards [ 5 ] , [6]. To include the skin effect two different mathematical models are presented: one based on reference [7] that is a nonlinear model and other based on [SI that is a linear model. It is shown in the paper that linear model fits very well the steady-state torque x speed curve for motor NEMA C class so that, rotor winding resistance and rotor winding leakage reactance can be described as slip (s) functions as shown in the equations (16) to (19). rr (s) = rro + sArr

40

20

U -to

0

400

TWO

800

I S W TdOU

Figure 1 : Dynarmc motor torque x speed curves using linear and non-linearskin effect model

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JO

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(16)

Figure 2 : Steady-state torque x speed curves using hear and non-linear skin effect model

Parameters rm,Xem are referred to rated slip and rrl and Xerl to unit slip. 111. RESULTS

Experimental and simulation results presented are related to a three-phase, 3HP, 220V, 4 poles, NEMA C class motor. Figure 1 shows simulated dynamic motor torque x motor speed curves for linear and non-linear skin effect models while figure 2 shows the simulated steady-state motor torque x motor speed curves for linear and non-linear skin effect models. It can be seen that linear skin effect model produces a torque depression between starting and breakdown torques. Figure 3 and 4 shows the simulated temperature rise curves (continuous trace) and experimental temperature rise curves (dotted trace) in the stator winding (a), stator core (b),

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M

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1W

6)

120 1

Figure 3 : Simulated (curve 1) and experimental (curve 2) temperature rise a) stator-winding, b) stator-core.

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IV. ADDITIONAL COMMENTS

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Figure 4 : Simulated (curve 1) and experimental (curve 2) temperature rises c) rotor-winding, d) rotorare.

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Figure 5 presents dynamic torque x speed curves when temperature and skin effect are considered (curve 1) and not considered (curve 2) in the simulation showing the necessity to add those effects in the motor dynamic mathematical model.

The dynamic model presented in this paper has shown very accurated simulation results. Besides electrical and mechanical quantities it permits the average temperature rises in the stator and rotor windings and in stator and rotor cores evaluation. It is useful for any type of motor dynamic studies mainly those including fast motor speed changings like startings, brakings, stoppings and large load changings. It is addressed mainly to people who needs to spec@ motors for any drive or to perform transient studies in electrical energy systems or in electrical industrial systems and also, to motor designers who need to have a fast dynamic evaluation of the motor average temperature rises, while improving motor efficiency and power factor, during the design time, because those values optimization depend on the temperatures. The authors understand that this paper, this way, is also a contribution to the energy conservation and power quality subjects. Researches are going on to analyze, with this model, the intermittent motor loading according to different duties and also to analyze motor performance under non-sinusoidal voltage. ACKNOWLEDGEMENTS

The authors would like to thank to FAPESP for the financial support of this research. REFERENCES

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[11 P.C. Krause, 'Xnalysis ofElec?ricMachiner", McG~aw-Hill,1986. [2] P.K Kovics, "Transient Pl"ena in Electrical Machines", Ekevier, 1984. [3] T.A Lip0 and P.C. Krause, "Stability Analysis of a Rectifier-Inverter Induction Motor Drive", IHZE Trans. on Power Apparatus and Sysrems, vol. PAS-88, 1969, pp. 55-66. [4] D.S. Zhu, G. Champenois and C. GNszqnski., "Coupling of Electrical and Thermal Models of an Induction Motor for Performance Predictions", Proc. Znt. Conf on Elecmcal Machines, ICEM.90, Mass., USA, 1990, pp. 281-286. [5] NEMA Standard MG1,1978, patt 20. [6] IEC standard IEC 34-1, part 1,1983. [7] Kostenko, M. and Piotrovsky, L., ''Electrical Machines", MZR Publishers, Moscow, Russia, 1969. [SI M. Akbaba and S.Q.Fakhro, "New Model for Singleunit Representation of Induction Motor Loads, Including Skin Effect, for Power System Transient Stability Studies", IEE Proceedings-B, vol. 6, nQ139, November 1992.

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Figure 5 : Simulated dynamic torque x speed curves 1) consideringtemperature and linear skin effect, 2) rot consideringthase effects.

The complete paper will also present another results for sucessive startings and locked rotor operation.

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