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IEEE TRANSACTIONS ON MAGNETICS, VOL 33, NO 2, MARCH 1997
lectromechanical System Simulation with Models Generated from Finite Element Solutions Thomas E. McDermott, Ping Zhou, and John Gilmore Ansoft Corporation, Pittsburgh, PA Zoltan Cendes Camegie Mellon University, Pittsburgh, PA
Ab~tra~t-This paper shows an effective way of coupling finite element analysis to system simulation tools. By creating equ~valentcircuit models for electromechanical devices, and using electrical analogs for mechanical subsystems, it is possible to perform accurate parametric design studies. The method is ideally suited to such problems as energy delivery in a pulse transformer, and transient motion and current waveforms in an actuator. I. BACKGROUM)
Finite element analysis has typically been coupled to circuit simulators using time-stepped field solutions [ 1,2]. This approach can be very accurate, but it involves long simulation times that may discourage parametric studies, especially in system-level simulation. A behavioral model for an electromechanical device can be extracted from a set of parametric finite element solutions. Once the model has been developed, it can be used in a series of system simulations using SPICE [3] or Saber [4],varying the electrical circuit and mechanical load parameters without re-running the field solutions. Typically, the system simulations run much faster than the parametric or time-stepped field solution, which is the main advantage of this method. Another advantage is that the models can be more widely used and distributed, without the need for a finite element solver and detailed geometry and materials information. 11. STATE-SPACE APPROACH
We refer to these as “basis fhctions” for the device. To evaluate these functions analytically, one needs an expression for the stored magnetic coenergy, W’, in terms of I and x. Then the force and flux linkages can be calculated from
awf
f =-
ax (4)
The force and flux linkage appear in differential equations involving the position and coil current, respectively
d 2x f ( I , x ) = mdt2
(5)
dt v = V,,,
- rI
(7)
These three equations describe the dynamic behavior of the simple system in Figure 1, including the nonlinear conversion of magnetic and mechanical energy. If the coupling field is electric rather than magnetic, the basis functions become
As an alternative to time-stepped field solutions, the electromechanical energy conversion may be represented as a conservative black box, which may be nonlinear [5]. For the coupling system shown in Figure 1, one can write expressions for the coil flux linkages and mechanical force as f =f ( J , x ) (1) h =h(I,x) (2)
Manuscript received March 19, 1996;revised August 22,1996 Tom McDemott, e-mail
[email protected];Ping Zhou, e-mal ping@ansoft com; John Gilmore, e-mail
[email protected];Zoltan Cendes, email
[email protected]. Fig. 1. Magnetic field coupling system. 0018-9464/97$10,00 0 1997 IEEE
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resulting from 1 amp in coil ,j. When i=j, a self inductance is calculated, otherwise a mutual inductance is calculated. The matrix of inductance values will be symmetric about the diagonal. If there are permanent magnets in the device, another linear field solution is performed with the same localized permeabilities, but all the coil currents set to zero. The permanent magnet flux linking each coil is evaluated from weighted integrals of the magnetic vector potential over the coil conductor cross sections. Note that apparent stored energy is used to calculate the inductances, so that we: can write
And the sys,temdifferential equations are
d'x
f ( v , x )= m-
dt2
Losses can be modeled with external damping elements in Figure 1, and magnetic diffusion or charge relaxation can also he approximated with external circuit elements. The state-space technique is very powerful. Unfortunately, it is difficult to dierive the necessary expressions for the stored coenergy, except by making simplifying assumptions.
N
hi= L,,I, + C L ,+A, ~I~
(15)
j=l
dh, v. =I dt
I1 I. APPLICATION OFFINITEELEMENT SOLUTIONS
Instead of evaluating the basis functions from an analytical expression of the stored coenergy, finite element analysis may be used to evaluate the force, torque, flux linkage, or charge from field solutions. This procedure produces a taible of force and flux linkage output values, for example, versus current and position input values. The advantage of the finite element approach is that complicated geometries and nonlinear material properties can be included more easily than in the analytical approach. In the fiinite element method, force and torque are calculated using the method of virtual work. For example, the force on an object is calculated from the derivative of stored coenergy
Inductances in a nonlinear system are evaluated with localized permeabilities obtained from a field solution at the specified operating point. Then the material properties are frozen in each element, and a linear field solution is performed for each coil to be included in the inductance matrix, with one amp flowing in that coil and all other sources turnled off. The B and H solutions are saved, and all the self and mutual inductances are calculated from
(14) 1
V
where W,, is the apparent energy created by current I (which is set to 1 amp), Bi is the magnetic flux density resulting from 1 amp in coil i, and Hj is the magnetic field
where N is the number of coils and represents coil flux linkages created by permanent magnets, if there are any. This procedure varies from many other circuit models of nonlinear inductance, which use incremental inductance (dudi) linearized around the operating point. Reference [6] explores the difference between using apparent and incremental inductanoes in circuit simulation. Using a spreadsheet interface, a series of field solutions is performed to characterize the device behavior for systemlevel simulations. Each coil is an electrical port, with ampturns as input and apparent inductance as output. Each moving assembly is a mechanical port, with position or angle as the input, and force or torque as the output. The resulting finite element solutions provide piecewise linear basis functions for an equivalent circuit model. Motion is accounted for by referencing the derived lumped parameters to their corresponding coils, conductors, and masses; the electromagnetic field solution does not include any velocity terms. Iv. INTERFACE TO SYSTEM SIMULATIONSOFTWARE
To use the equivalent circuit models in SPICE, a piecewise linear dependent source component was added to Berkeley SPICE version 3F5. This component evaluates the lookup table produced by the finite element solutions. For example, given position and current, it would return flux and force. Mechanical subsystems can be represented using their electrical analogs in ,SPICE, summarized in Table I. In this analog, force or torque corresponds to current, and velocity corresponds to voltiig. A mass or rotating inertia is represented with a capacitor, a shaft or spring with an inductor, and viscou!; damping with a resistor. Mechanical hardstops can be simulated with controlled switches. Rotating mechanical loads can be represented with a
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nonlinear resistor, using a V-I characteristic to match the speed-torque curve. The new "B source" component in Berkeley SPICE 3F5 provides considerable modeling flexibility for nonlinear electrical and mechanical loads. Unlike previous versions of SPICE that provided only polynomial nonlinear dependent sources, the B source can include more general expressions. However, it is not a replacement for the multi-dimensional piecewise linear component developed here, because of the various hazards in fitting analytical expressions to numerical values. TABLEI MECHANICAL ANALOGS Rotational
Translational
Electrical
z [N-in]
F [NI
1 [AI
o [rads]
w
v [VI
J [kg-m2]
m [kg1
K [N-&rad] D [N-m-s/rad] Power = zo
K [N/m] D [N-s/ml Power = Fw
C[Fl 1K r1m 11R [1/a] Power = IV
e=lo+oa
x = Iw+xo
v, = i v + vno
[&SI
Using the electrical analogs, a force output from the piecewise linear source can be applied as a current to a capacitance representing the device mass. To provide the position input, it is necessary to integrate the voltage across the capacitor, which may be done with SPICE components. Given the coil flux linkages, a back emf is generated by differentiating the flux. Figure 2 shows the necessary SPICE components for a linear actuator corresponding to Figure 1. If the device is based on the electric rather than the magnetic field, Figure 3 shows the necessary SPICE components. Examples in the following section were run in both our modified SPICE, and the commercial Saber simulator.
I
integrator
Saber uses piecewise linear soluti SPICE uses the continuous Newton Raphson algorithm. Saber includes an extensive library of mechanical components. The differentiating circuits in Figures 2 and 3 are not required because Saber can solve the coupling equations directly. V. EXAMPLES
The first example is a linear actuat axisymmetric with a stroke of 0.0127 m, 12500 turns, and SS430 material in the armature and stop. Parameters of interest are the time to closure, and the waveshape of the coil current, which may be used for electronic sensing. Fig shows the laboratory test current waveshape, along simulation results using the finite-element derived model, and a simpler electromagnet model based on magnetic circuit analysis. The simpler model provides a reasonable estimate of the time to closure, but is not as good at simulating the total coil current waveshape. The finiteelement derived model required 121 separate field solutions, varying the core position and coil current as parameters. The second example is a pulse transformer, which is required to deliver a specified energy to a nonlinear load. This transformer has a C core, 0.023 m thick, with a turns ratio of 12700/140. A circuit model in the form of Figure 5 was derived from finite element solutions. T the magnetizing inductance, L,forty-one separate field solutions were performed, varying the primary coil current as a parameter with the secondary current set to zero. This simulates the laboratory open-circuit test. While the parametric field solutions for for the leakage inductances, it is much
I
mechanica I ana I og
Fig. 2. SPICE components for a magnetic field device.
Saber offers some advantages over SPICE for this application:
integrator
mechanical analog
Fig. 3. SPICE components for an electric field device
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mm
capacitances will typiically be shorted out by ground connections. In three-phase transformers with delta windings or ungrounded neutrals, some of the Figure 5 capacitances will be paralleled with other winding terminal capacitances. Both the leakage inductances and the capacitances were assumed to be linear, so their calculation required only a few more field solutions. Table I1 shows simulated and measured energy delivery for two different core materials, using both finite-element derived models, and a simpler model based on magnetic circuit analysis.
t
TABLEI1 PULSE TRANSFORMER LOADENERGY DELIVERED
Fig. 4. Coil currents in a linear actuator.
obtain leakage inductances from a simulated bucking test [7]. Equal and opposite amp-turns are applied to each winding (pair-wise if there are more than two windings). In this case all of the stored field energy is due to leakage flux, and the leakage branch inductance for each winding is calculated from
Li= j A - J d v
(17)
vi
The capacitance elements in Figure 5 are derived from electrostatic field solutions using a 3D model. The capacitance matrix is set up with the core and the case or tank grounded, and each coil specified as a single conductor. From the capacitance matrix values C, C2, and C,, the equivalent circuit parameters are found from
CO l = Cl + C”‘ c,, = c, + c, c,, = -cm The total capacitance is usually split equally between each winding terminal as shown in Figure 5; some of these
I \ TO.~*CIO 0.5*C12 -
Fig. 5 . Transfonner equivalent circuit model.
0.5*C20rp‘ -
Model
MI9 Core
Composite Core
Magnetic Circuit E A Model Measurement
60.2 mJ 53.9 mJ 50.0 mJ (production req.)
42.4 mJ 34.5 mT 32.4 mJ
-
In both examples, all of the finite element solutions were completed in less than three hours on a Pentium 90 class machine. Subsequent isystem simulations were completed in a matter of seconds using either SPICE or Saber. Once the equivalent circuit model has been derived from the field solutions, there is no significant difference in system simulation time between the finite-element based model and the magnetic circuit model. REFERENCES Shen, G. L. Meunier, and J. L. Coulomb, “Solution of magnetic fields and electrical circuits combined problems”, IEEE Transactions on Magnetics, vol. 21, no. 6, pp. 2288-2291, 1985. J. R. Brauer and J. J. Ruehl, “Coupled nonlinear electromagnetic and structural finite element analsyis of an actuator excited by an electric circuit”, EEE Transactions on Magnetics, vol. 31, no. 3, pp. 1861-1864, 1995. L. W. Nagel, “SPICE2: a computer program to simulate semiconductor circuits”, Memo UCB,BRL M530, Electronics Research Laboratory, University of Califomia, Berkeley, CA, 1975. H A. Mantooth and M. Vlach, “Beyond SPICE with Saber and MAST”, Analogy white paper M[P-0161, Beaverton, Oregon. H. H. Woodson and J. IR. Melcher, Electromechanical Dvnamics, reprint, vol. 1. Malabar, Florida: Robert E. Krieger, 1990, pp. 60-79. T. W. Nehl and N. Demerdash, “Finite element state space modeling environments for electric motor drives”, IEEE Tutorid on Adjustable Speed Drives,92 EH0 362-4-PWR, 1992, pp. 109-126. D. A. Lowther and P. P. Silvester, Comuuter-Aided Design in Magnetics, Berlin: Springer-Verlag, 1986, pp 163-176.
