Dynamic Modeling of Three Phase and Single Phase Induction Motors Ping Zhou
Scott Stanton
Zoltan J. Cendes
Ansoft Corporation 4 Station Square, Pittsburgh, PA 15219 Abstracl: A general time-stepping approach is presented to solve coupled fieldl-circuit motion problems for the analysis of three phase and single phase induction motors. The circuit equations are used to describe Ithe winding connections, to include rotor end-ring effects and possible (externalimpedances, and to support voltage source with arbitrary waveforms. In addition, a moving surface method is proposed to deal with rotor movement without any remeshing.
Keywords: Three phase induction motors, single phase induction motors, linite element method, time-stepping, motion.
I. INTRODUCTION Induction motor designers are today being challenged to improve motor performances without substantially increasing manufacturing costs due to the increased competition in world markets and the pressure of increased cost of electrical energy. The solution to this challenge is greatly simplified by using reliable, accurate and practical simulation methods. With a good simulation, any desired performance characteristic and corresponding design configurations can be determined without fabricating an actual prototype. With the rapid growth of compul.er capacities and the development of numerical techniques, the finite element method has been recognized as a practical and accurate approach to aid in the induction motor design[ 1-31. When steady-state performance is the area of interest, a balanced three phase induction motor with a sinusoidal source supply can be analyzed economically using the complex magnetodynamic finite element model with the use of effective reluctivity[4,5]. However, if nonsinusoidal quantities and the teeth-slot effects caused by the rotation of rotor need to be considered, then a time stepping approach associated with the motion of the rotor is necessary. The external circuit equations and electromechanical equations are further coupled to account for voltage source supply, driving circuit and mechanical transientrb]. In this paper, a general time-stepping approach is presented to solve coupled electrical-magnetic-circuit-motion problems. The circuit equations are used to describe the winding connections, to include end ring effects and possible external impedances, and to support actual voltage sources with arbitrary waveforms. In addition, a moving surface method is proposed to deal with the rotor movement without any remeshing during the entire solution process. This modeling approach has been implemented in the Maxwell 2D Field Simulator. This allows the user to simulate and analyze the dynamic and transient behavior of various motor types under all operating conditions. Two application examples are presented: one is a 4-pole threephase induction motor, and the other one is a 2-pole singlephase capacitor-start capacitor-run induction motor.
The mathematical model describing the operation of electrical machines is a boundary value problem because of the distributed nature of electromagnetic fields within the machines. It is also an initial value problem because of the time dependence of the field and the source. In addition, the circuit equations should be used to describe the power supply and the connection of windings. In the model, two types of conductors are considered: solid conductors in which eddy current can be induced, and stranded conductors (windings) without eddy currents. For an electromagnetic problem involving motion, it is advantageous to write the field equations for rotor and stator in their own coordinate systems so as to avoid the rotational speed term appearing explicitly in the formulation and to preserve the symmetry of the final matrix of equations. Thus, the timedependent magnetic diffusion equations is: V X ~ V X A=
J , -at~ + + ~ v v + v ~ H ,
where A and V are magnetic vector potential and electric scalar potential, respectively, H , is the coercivity and represents the contribution from permanent magnet, J , is the source current density. For a two dimensional problem, the vectors have only one component in z-direction. In this case, the scalar potential V has a constant value on the cross section of a conductor, and the gradient of the scalar potential can be expressed by the voltage difference, V , , across a conductor between the far and near ends. Stranded conductors are normally connected to produce windings. To represent a voltage fed device, circuit equations must be coupled with the field equations. By applying Kirchoff's law, one can write the following circuit equation to relate the terminal voltage U, of a winding to its terminal current lf
N1 di dfSf.a _f_jjdA.dQ+R.if dt +L.zf +U, = U,
(2)
where N f is the total conductor number in this winding, a is the number of parallel branches in the winding, df is the polarity index (+1 or -1) representing forward and return paths; S denotes the total area of the cross section of this coil group. and L are equivalent resistance and inductance respectively, since they can also include external impedance connected with the winding in addition to the DC resistance and end-turn leakage inductance of the winding itself. U, is the voltage drop across a capacitor. In order to avoid integration due to the inclusion of capacitance, it is necessary to introduce another differential equation to relate capacitor voltage U, to the current Zf
R
Solid conductors, as used for the rotor bars of an induction motor, are large enough to model skin effects with finite elements. Furthermore, these conductors may be connected at both ends by means of end rings. In order to take end effects
Manuscript received September 30, 1998. Ping Zhou,
[email protected]. Scott Stanton,
[email protected]
Zollan J. Cendes,
[email protected]
0-7803-5293-9/99$10.00 0 1999 IEEE
11. MATHEMATICAL MODEL
556
into account, every portion of the ring located between two bars may be considered as an external impedance. By taking the squirrel cage as a polyphase circuit[4], one can obtain the following circuit equations
and
where re and 1, are the resistance and the inductance of a portion, between two bars, of the ring respectively, [ C ]is the connection bar matrix, i and J , are current and current density in the bars respectivety. The equation of motion is
Ja + ho = T , , + Tapp
Fig. 1. Flux plot at 420 rpm
I
I
where a is the angular acceleration, w is the angular velocity, J is the moment of inertia, h is the coefficient of friction, T,,, is the electromagnetic torque and Tapp is the external applied mechanical torque. External applied torque may either be load torque or may be an accelerating driving torque in the same direction as the electromagnetic torque. The sign of both the electromagnetic torque and the external applied torque is determined by the torque direction, where the anti-clockwise direction is positive and the clockwise direction is negative. At each time step, the electromagnetic torque is computed using the method of virtual work. Solving the equation of motion allows us to obtain the rotor angular acceleration, the angular velocity and the displacement. This angular displacement is in turn used to move the rotor to a new position. The rotor movement is taken care of by the moving surface method. The basic idea is to create two independent meshes sharing a common moving slip surface, which allows the stationary and moving parts to move freely with respect to each other. Essentially, the moving surface can be imagined to be split into two surfaces. One side is attached to the stationary mesh; the other is attached to the moving mesh. After any specified angular displacement of moving part, the two independent meshes are then coupled together by the finite element shape functions. Thus, part of the mesh is free to move to any specified angle without any remeshing.
llrne (sec1
(a) torque response
111. THREEPHASE INDUCTION MOTOR EXAMPLE
t,ml5ecI I "
The first example is a 3-hp polyphase induction motor[3]. This motor has four poles, 36 stator slots and 44 rotor bars. Its wye-connected 3-phase stator winding is energized by 460 volts at 60 Hz. Taking advantage of motor symmetry and dynamic anti-cyclic boundary conditions, only one pole is modeled as shown in Fig. 1. The moment of inertia of the rotor is 0.02 1 k g - m2 and the friction coefficient is set to 0.0015 Nm-s. A special CAD tool has been developed to provide motor designers with a familiar, intuitive and easy-to-use environment, through which he or she can harness the full power of FEA without being confronted with finite element details. Using parameterized input, it automatically produces the geometry and all the necessary input data for the field computation. Table 1 and Table 2 show the computed average torques and rms values of phase current respectively together with the measurements[3]. The motor start up with the rated applied load of 12.1 N-m. Figure 2 shows the variation of the torque and speed of the rotor during the starting. There is no speed overshooting
I I
"""
I
L"
I
X
l
-
l
l
(b) speed response Fig. 2. Variation of torque and speed dunng starting
TABLE 1. Averaged torques and measurements
I Speed (RPM) I Computed(N.m) (Measured(N.m)(
I locked rotor pull up .~ rated load
I
0
47.1
49
1230
56.3 12.03
57.9 12.2
1747
1
I
I
TABLE 2. Comparison of Currents
locked rotor DUI1 UD
rated load
I
Speed (RPM) 0 1230 I741
Computed (A) 31.3 23.1 4.28
Measured (A) 39 24
I
4.5
since the motor starts with afull load. It can be recognized from figure 2(b) that the simulated rated speed is 1743 rpm, which is very close to the measured speed at1747 rpm.
557
Iv. SINGLE PHASE INDUCTION MOTOR EXAMPLE
winding current vs lime
The second example is a 3-hp, 2 pole, capacitor-start capacitor-run single phase induction motor as shown in Fig. 3(a). The stator has 20 large slots and 4 small slots. Two stator windings, namely the main and the auxiliary winding, are of concentric type. Each slot has a different number of turns for achievinlg best performance. One of the two capacitors is used only for starting and will be cut off by a centrifugal switch at 70 percent of synchronous speed, which can be modeled by a functionlalcapacitor varying with speed as shown in Fig. 3(b). The applied functional load is represented by input curve as shown in Fig. 3(c).The moment of rotor inertia is 0.0035 k g - m2 and the friction coefficient is 0.0015 N-m-s.
-------t
I
(a) geometry model’for field computation +””U
~
h
8
*
1
(a) torque
*si
-.
75 50
‘ \
25 0;
016
018
I;&&‘
i
4
k
*
*
;
m
A
w
LIYr I
i ”
i
l
i
(c) functional load (b) functional capacitance Fig 3. Single phase induction motor simulation setup A..
..........
..............
.. ... .
1,rnI%I ,
(b) speed Fig. 5 Torque and speed responses
..... ..
.
. ......... .. .. ... .. .
V. REFERENCES locked rotor main winding current (A) aux winding currtznt (A)
rated at 1745 rmp
computed
mesured
computed
measured
80.2
78.6
13.3
12.8
9.2
9.0
3.4
3.3
6.5
6.3
both windings during start-up. A sudden change of current in auxiliary winding at about t = 0.137s can be observed becauise of the switching off of the start capacitor. Figure 5 shows, the torque and speed responses during start-up. As expected, the torque pulsations are much more significant than those of the three phase induction motor. This double-statorfrequemcy torque pulsations are produced by the interaction of oppositely rotating flux and mmf waves which glide past each other at twice synchronous speed.
[l] S.C. Tandon, “Finite element analysis of induction machines”, IEEE Trans. on Magnetics, vol. 18, no. 6, November 1982, pp. 1722-1724. [2] S. Williamson and M.J.Robinson,” Calculation of cage induction motor equivalent circuit parameters using finite elements”, IEE proceedings-B, vol. 138, no. 5 , September 1991, pp. 262-276. [3] John Brauer, Hamid Sadeghi and Robert Oesterlei, “Polyphase induction motor performance computed directly by finite elements”, IEEE Trans. on Energy Conversion, vol. 12, 1998, inpress. [4] E. Vassent, G. Meunier and A. Foggia, “Simulation of induction machines using complex magnetodynamic finite element method coupled with the circuit equations”, IEEE, Trans. on Mugnetics, vol. 27, no. 6, September 1991, pp. 4246-4249. [ 5 ] Ping Zhou, John Gilmore, Zsolt Badics and Zoltan J. Cendes, “Finite element analysis of induction motors based on computing detailed equivalent circuit parameters”, IEEE Truns. on magnetics, vol. 34, no. 5, September, 1998, pp. 3499-3502. [6] T.W. Preston, A.B. J. Reece, P.S. Sangha, “Induction motor analysis by time-stepping techniques”, IEEE Trans. on Magnetics, vol. 24, no. 1, JanUW, 1988, pp. 471-474.
558
