Dynamic Modeling of Universal Motors - IEEE Xplore

Dynamic Modeling of Universal Motors Ping Zhou

John R. Brauer Scott Stanton Zoltan J. Cendes

Ansoft Corporation, 4 Station Square, Pittsburgh, PA 15219 USA

Roderick N. Ebben Milwaukee Electric Tool Corporation, 13135 W. Lisbon road, Brookfield, WI 53005 USA Abstract: A general time-steppingapproach is presented to solve coupled field-circuit-motionproblems for universal motors. Circuit equations are used to describe the dynamic connections of field and armature coils, to include the end leakage inductance and the voltage drop across brushes, and to support a voltage source with arbitrary waveforms. In addition, the effects of the coils undergoing commutation have been considered by having them removed temporarily from the main winding and excited by a controlled current source. I. INTRODUCTION There is extensive literature on the application of the finite element method to the analysis of electrical machines. Many commercial software packages are also available, which can be used by engineers without specialist knowledge of field computation. However, no attempt has been made to deal with the modeling of universal motors with the only exception in Ref [ 11, despite the fact that universal motors dominate the mainssupplied domestic appliance and portable machine-tool markets. This is due to the advantages of wide variable-speed range, high starting torque and attractive powedweight ratio. The complexity of universal motor modeling stems from the existence of many unique facts in operating the machines. First, when a universal motor is operated from an AC source, the presence of both transformer induced voltage and rotational induced voltage makes the commonly used snapshot approach no longer applicable. Secondly, the winding configuration is constantly changing. Each coil in the armature circuit, whenever passing over a stationary brush, will transit from one group of the armature coils, after being momentarily removed from the main circuit of the winding due to being short-circuited by the brush, to the other group. Thirdly, all the conductors in both armature and stator are connected together and they are fed by a voltage source, rather than current source. In addition, in order to obtain the control of speed and torque, the voltage source may be adjustable by power electronics. Finally, accounting for the effects of those short-circuited coils undergoing commutation further complicates the problem. In this paper, a general time-stepping approach is presented to solve coupled field-circuit-motion problems. The circuit equations are used to describe the dynamic connections of field winding and two groups of armature coils, to include the end leakage inductance and the voltage drop across brushes, and to support a voltage source with arbitrary waveforms. The effects of saturation, slotting, rotor movement and both time and space harmonics are taken into account. In addition, the effects of the coils undergoing commutation have been considered by having them removed temporarily from the main circuit and excited by a controlled current source. This allows one to obtain steady state and transient behavior of universal motors under any operating conditions.

11. MODELING APPROACH

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+ h~ = T e n , + Tap,, (4) where A and V are magnetic vector potential and electric scalar potential, respectively, H , is the coercivity and represents the contribution from permanent magnets, J,y is the source current density. For a 2D problem, the vectors have only one cornponent in z-direction. One equation in the form of (2) is required for each winding to relate the terminal voltage u , with ~ its terminal current if, where U, is the voltage drop across a capacitor, Nf is the total conductor number in the winding, U is the number of parallel branches, df is the polarity index (+I or -1) representing forward and return paths and Sf denotes the total area of the cross section of the winding. R , L and C are resistance, inductance and capacitance respectively. They can also include external impedance in addition to the resistance and end leakage inductance of winding itself. Finally, Eqn. (4) is used to describe the motion of the rotor when mechanical transient is involved, where w is the angular velocity, J is the moment of inertia, h is the coefficient of friction, Ten, is the electromagnetic torque computed by the virtual work method based on the field solutions and Tu,, is the external applied mechanical torque. This electromecianical coupling model can be applied to investigate such transient behavior of a universal motor as starting or after a load condition change. In the case of running at a specified constant steady-state speed, the Eqn. (4)is excluded. In both transient and steady-state cases, a moving surface method is used to deal with the arbitrary rotor movement without the need of remeshing[2]. While the above mathematical model is fairly general, the commutation issue has not been explored when the model is to be applied to the analysis of a universal motor. First, the winding configuration keeps changing due to the transition of a coil undergoing commutation from one group of armature coils to the other group. In addition, during the time when the brushes are simultaneously in contact with two adjacent commutator segments, the coils connected to these segments are short-circuited and thus have to be temporarily removed from the main circuits. Furthermore, the current in the coil undergoing commutation must reverse at the end of the short commutation interval. When the coil is about to leave the brush, if the current reversal is not complete, the current jumps to its final value and may cause sparking. The quantitative analysis of the current reversal and the securing of good commutation is beyond the scope of this paper because it is more an empirical art than quantitative science due to the electrical behavior of the carbon-copper contact film whose resistance is a function of current density, current direction, temperature, brush material, moisture, and atmospheric pressure. However, it is desirable to be able to account for the impacts of the commutation process on the motor performance in terms of a user-specified current reversal pattern. In order to model the incessant change of the winding connection, it is essential for each coil on the armature to be able

To calculate time-dependent magnetic fields including movement, the following system of coupled differential equations has to be solved: V X V V ~ =A JS-oaA + o V V + V x H c (1) at 0-7803-5293-9/99$10.00 0 1999 IEEE 419

to dynamically locate the group to which it belongs between the two groups of armature coils at each time step. This can be achieved by making the polarity index df for each coil in the Eqn. (2) as a function of position as shown in Fig. l(a). Whenever a coiil on the rotor passes over a stationary brush, the change of its polarity can be used to indicate the transition from one coil group to the other.

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A. DC Source Supply The main interest here is two important steady state characteristics: speed-current curves and speed-torque curves. For this investigation, three different voltages 60,90 and 120 V are examined. The analysis is considerably facilitated if the speed is input as a step function of time as presented in Fig. 2(c), where speed varies from 6,000 to 15,000 rpm with each speed spanned 0.01s that is enough for the current and torque to reach steady state. Figures 2(a) and 2(b) show the computed current and torque curves when the motor operates at the prescribed terminal voltage of 90 volt. The average values are obtained by

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180

(a) without considering commutation process

B. df

to 6.664 ohm, which corresponds to the running temperature from 35 to 70 O C . The end winding leakage inductance is 0.00238H. The moment of inertia of the rotor is 6.27E-5 k g - m* and the friction coefficient is set to 3.18E-6 N-m-s. The commutation interval is 15 degrees since the width of brushes equals one commutator segment. Without losing generality, the linear commutation process is assumed.

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(b) considering commutation process Fig. 1 Polarity index as a function of position

In reality, however, the transition of the coil group and the current reversal can not be accomplished instantaneously due to the finite dimension of brushes and the existence of inductance in the coils undergoing commutation. To this end, the definition of polarity index is extended to include the commutation process as shown in Fig. l(b), where T is the commutation interval representing the actual brush size. The transition curves can describe different commutation processes. For example, the solid line in the Fig. l(b) represents the linear commutation and the dashed line is the case of delayed commutation. At every time step, once the polarity index of a coil, d y ) , is detected to be between the commutation interval T ,the program will take two actions. First, this coil is temporarily removed from the main circuits since this coil is short-circuited by the brush and no induced voltage will contribute to the left hand of circuit equation (2). Second, the excitation of the coil undergoing commutation is replaced by a controlled current source and determined by where i?" is the terminal current of main circuit at current time step and a is the number of parallel branches. Thus, this scheme can be conveniently used to simulate different commutation processes in terms of the user specified polarity function. In addition, the voltage drop across a brush can be modeled by a nonlinear resistor as a function of position and current. 111. APPLICATION EXAMPLE The proposed method is applied to a typical small electrical tool motor with two stator poles and twelve rotor slots[ 11. In the case of dc source supply, the total winding resistance is assumed to vary from 5.789 to 6.227 ohm with the applied terminal voltage and running speed, which corresponds to the operating temperature from 30 to 50 O C . This is because a higher thermal load is associated with a higher applied voltage and a lower operating speed. When the motor operates with an ac source:,additional heat sources have to be taken into account due to the fact that core loss occurs in the stator with alternating flux. In this case, the resistance is assumed to vary from 5.898

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(c) s'peed aS a step function of time Fig. 2 Current and torque curves at different constant speed

averaging the values on the curves over the last 0.002s of each speed interval. Figures 3 and 4 are the computed and measured current-speed and torque-speed curves. The friction and windage torque loss has been deducted from the computed torque[ 11. B. AC Source Supply The computations were carried out under a sinusoidal voltage source of rms value 120 V and 60 Hz. Similarly, let the

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neous torque varies at twice the supply frequency. These computed results of the rms current and time-average torque agree well with measurements[3]. Finally, Fig. 9 shows the transient behavior of speed during starting and load changing. The motor starts with a load torque of 0.24 N-m and then at t = 0.15s the load torque is changed to 0.138 N-m. Similarly, various transient responses can also be easily computed when the terminal voltage suddenly changes ~~~

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speed be a step function varying from 6,000 to 21,500 rpm. The time interval associated with each speed is chosen to span 0.05s, which is considered to be enough for the current and torque to reach steady state. Figures 5 and 6 are the computed current and torque waveforms at 2 1,500rpm. The current waveform is well matched with the measured one[3]. It can be seen that the instantaneous torque varies at twice the supply frequency. The slot effects can also be clearly observed from both figures. The computation of rms value from the current curve can be carried out over the last cycle ( 1/60 sec) of each speed interval. Fig. 7 is the obtained curve of rms current vs. speed. The time -average torque vs. speed as shown in Fig. 8 is computed over the last portion ( 1 / 120 sec) of each speed because the instantawinding current vs hmc I

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Fig. 9 Speed responses during starting and load changing

IV. CONCLUSION In this paper, a general approach based on the coupling of field equations, circuit equations and motion equations is presented for the dynamic analysis of universal motors. It can be employed to provide a virtual prototype for a new design with complete physical accuracy. V. REFERENCES

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[ l ] R.N. Ebben, J.R. Brauer, G.C. Lizalek and Z.J. Cendes, "Performance curves of a DC motor predicted using parametric finite element analysis", Digests of IEEE Con$ on Electromagnetic Field Computation, June 1998. [2] P. Zhou, S . Stanton and Z. J. Cendes, "Dynamic modeling of three phase and single phase induction motors", A companion paper to be presented at IEDMDC-99, Seattle, May, 1999. [3] R.N. Ebben, J.R. Brauer, Z.J. Cendes and N.A. Demerdash, "Prediction of performance characteristics of a universal motor using parametric finite element analysis", A companion paper to be presented at IEDMDC-99, Seattle, May, 1999.

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Fig. 6 Computed torque waveform at 120V ac supply and 21,500 rpm

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