Transient Simulation of Magnetic Circuits Using the Permeance-Capacitance Analogy

Home , Magnetic circuit, Magnetic reluctance, Permeance

Jost Allmeling Wolfgang Hammer John Schonberger¨ Plexim GmbH Plexim GmbH TridonicAtco Schweiz AG Technoparkstrasse 1 Technoparkstrasse 1 Obere Allmeind 2 8005 Zurich, Switzerland 8005 Zurich, Switzerland 8755 Ennenda, Switzerland Email: [email protected] Email: [email protected] Email: [email protected]

Abstract—When modeling magnetic components, the R1 Lσ1 Lσ2 R2 permeance-capacitance analogy avoids the drawbacks of traditional equivalent circuits models. The magnetic circuit structure is easily derived from the core geometry, and the Ideal Transformer Lm Rfe energy relationship between electrical and magnetic domain N1:N2 is preserved. Non-linear core materials can be modeled with variable permeances, enabling the implementation of arbitrary saturation and hysteresis functions. Frequency-dependent losses can be realized with resistors in the magnetic circuit. Fig. 1. Transformer implementation with coupled inductors The magnetic domain has been implemented in the simulation software PLECS. To avoid numerical integration errors, Kirch- hoff’s current law must be applied to both the magnetic flux and circuit, in which inductances represent magnetic flux paths the flux-rate when solving the circuit equations. and losses incur at resistors. Magnetic coupling between I.INTRODUCTION windings is realized either with mutual inductances or with ideal transformers. Inductors and transformers are key components in modern power electronic circuits. Compared to other passive com- Using coupled inductors, magnetic components can be ponents they are difficult to model due to the non-linear implemented in any circuit simulator since only electrical behavior of magnetic core materials and the complex structure components are required. This approach is most commonly of components with coupled windings. used for representing standard magnetic components such as This paper compares different approaches to model mag- transformers. Fig. 1 shows an example for a two-winding L L netic components by means of equivalent circuits with lumped transformer, where σ1 and σ2 represent the leakage induc- L R elements. It highlights the advantages of the permeance- tances, m the non-linear magnetization inductance and fe capacitance analogy over the traditional coupled-inductor the iron losses. The copper resistances of the windings are R R model and the reluctance-resistance analogy. modeled with 1 and 2. Using the permeance-capacitance analogy it is shown how However, the equivalent circuit bears little resemblance to variable permeances are employed to model saturation of the the physical structure of the magnetic component. For exam- core material. The saturable core model can be extended to ple, parallel flux paths in the magnetic structure are modeled model frequency-depending losses and hysteresis. with series inductances in the equivalent circuit. For non-trivial Finally, the implementation of the magnetic domain in the magnetic components such as multiple-winding transformers commercial simulation software PLECS is described. The or integrated magnetic components, the equivalent circuit can system equations must be set up by applying Kirchhoff’s be difficult to derive and understand. In addition, equivalent current law to both flux-rate and flux in order to avoid circuits based on inductors are impossible to derive for non- numerical integration errors. planar magnetic components [4].

II.EQUIVALENTCIRCUITSFORMAGNETICCOMPONENTS B. Reluctance-resistance analogy To model complex magnetic structures with equivalent cir- The traditional approach to model magnetic structures cuits, three different approaches exist: Coupled-inductors, the with equivalent electrical circuits is the reluctance-resistance resistance-reluctance analogy and the capacitance-permeance analogy. The magnetomotive force (mmf) F is regarded as analogy. analogous to voltage and the magnetic ﬂux Φ as analogous to current. As a consequence, magnetic reluctance of the ﬂux A. Coupled inductors path R corresponds to electrical resistance: In the coupled inductor approach, the magnetic component F R = (1) is modeled directly in the electrical domain as an equivalent Φ L: N

A R e+ m+ v V A Ф

Electrical Magnetic R A

e- i K V F m- K: 1/N Fig. 3. Gyrator symbol and implementation Fig. 2. Implementation of magnetic interface

to current). With this choice of system variables, magnetic The magnetic circuit is simple to derive from the core geome- permeance P corresponds to capacitance: try: Each section of the flux path is represented by a reluctance dF Φ˙ = P (5) and each winding becomes an mmf source. dt To link the external electrical circuit with the magnetic Hence it is convenient to use permeance P instead of the circuit, a magnetic interface is required [1]. The magnetic reciprocal reluctance R to model flux path elements. Because interface represents a winding and establishes a relationship permeance is modeled with storage components, the energy between flux and mmf in the magnetic circuit and voltage v relationship between the actual and equivalent magnetic circuit and current i at the electrical ports: is preserved. The permeance value of a volume element is dΦ given by v = N (2) 1 µ µ A dt P = = 0 r (6) F R l i = (3) −7 2 N where µ0 = 4π × 10 N/A is the magnetic constant, µr N is the number of turns. If the magnetic interface is imple- is the relative permeability of the material, A is the cross- mented with an integrator it can be solved by an ODE solver: sectional area and l the length of the flux path. Dissipator components (analogous to electrical resistors) 1 Φ = v dt (4) can be used in the magnetic circuit to model losses. They N ˆ can be connected in series or in parallel to a permeance Fig. 2 outlines a possible implementation of the magnetic component, depending on the nature of the specific loss. The interface in a circuit simulation software. energy relationship is maintained as the power Although the reluctance-resistance duality may appear nat- P = F Φ˙ (7) ural and is widely accepted, it is an unfortunate choice for loss multiple reasons: converted into heat in a dissipator corresponds to the power • Physically, energy is stored in the magnetic field of a loss in the electrical circuit. volume unit. In a magnetic circuit model with lumped Windings form the interface between the electrical and the elements, the reluctances should therefore be storage magnetic domain. A winding of N turns is described with the components. However, with the traditional choice of mmf equations below. The left-hand side of the equations refers and flux as magnetic system variables, reluctances are to the electrical domain, the right-hand side to the magnetic modeled as resistors, i.e. components that would usually domain. dissipate energy. It is also confusing that the magnetic v = NΦ˙ (8) interface is a storage component. F • To model energy dissipation in the core material, induc- i = (9) N tors must be employed in the magnetic circuit, which is even less intuitive. Because a winding converts through-quantities (Φ˙ resp. i) in • Magnetic circuits with non-linear reluctances generate one domain into across-quantities (v resp. F ) in the other differential-algebraic equations resp. algebraic loops that domain, it can be implemented with a gyrator, in which N cannot be solved with ODE solvers. is the gyrator resistance R [5]. Fig. 3 shows the symbol for • The use of magnetic interfaces results in very stiff system a gyrator and its implementation in the simulation software equations for closely coupled windings. PLECS. The gyrator component could be used with regular capac- C. Permeance-capacitance analogy itors to build magnetic circuits. However, neither the gyrator To avoid the drawbacks of the reluctance-resistance anal- symbol nor the capacitor adequately resemble a winding ogy the authors are advocating the alternative permeance- respectively a flux path. Moreover, any direct connection capacitance analogy [2]–[5]. Here, the mmf F is again the between the electrical and magnetic domain made by mistake across-quantity (analogous to voltage), while the rate-of- would lead to non-causal systems that are very difficult to change of magnetic flux Φ˙ is the through-quantity (analogous debug. III.MAGNETICCIRCUITDOMAININ PLECS In most cases, however, the differential permeance Pdiff (F ) The permeance-capacitance analogy has been implemented is provided to characterize magnetic saturation and hysteresis. in PLECS by means of a special domain. The available With dΦ magnetic components include windings, constant and variable Φ˙ = permeances as well as dissipators. By connecting them accord- dt ing to the physical structure the user can create equivalent dΦ dF = · circuits for arbitrary magnetic components. The transformer dF dt from Fig. 1 will look like in Fig. 4 when modeled in the dF magnetic domain. = P · (13) diff dt the control signal is Pm Gfe " # " # P(t) Pdiff R1 R2 = (14) d dt P(t) 0 N1 Pσ3 Pσ4 N2 B. Saturation curves for soft-magnetic material Curve fitting techniques can be employed to model the Fig. 4. Transformer implementation in the magnetic domain properties of ferromagnetic material. Here, two functions for modeling the non-linear primary saturation curve in soft Pσ1 and Pσ2 represent the permeances of the leakage flux magnetic materials are presented. path, Pm the non-linear permeance of the core, and Gfe 1) coth fit: The first function, referred to as the coth fit, was dissipates the iron losses. The winding resistances R1 and R2 adapted from the Langevian equation for bulk magnetization are modeled in the electrical domain. without interdomain coupling [6]–[8] and is given as follows: 3H a A. Modeling non-linear magnetic material B = B coth − + µ H (15) sat a 3H sat Non-linear magnetic material properties such as saturation and hysteresis can be modeled using the variable permeance Calculating the derivate of B with respect to H yields: component. The permeance is determined by the signal fed dB tanh2 (H/a) − 1 a = Bsat − + µsat (16) into the input of the component. The flux-rate through a dH a tanh2 (H/a) H2 variable permeance P(t) is governed by the equation 2) atan fit: The second function, referred to as the atan fit, d dF d Φ˙ = (P· F ) = P· + P· F (10) has been proposed in [9]: dt dt dt 2 −1 πH The control signal must provide the values of both P(t) and B = Bsat tan + µsatH (17) d π 2a dt P(t). When specifying the characteristic of a non-linear per- dB Bsat = + µsat (18) meance, we need to distinguish carefully between the total dH π 2 a 1 + 2 H/a permeance Ptot(F ) = Φ/F and the differential permeance Both fitting functions have three degrees of freedom which Pdiff (F ) = dΦ/dF . are set by the coefficients µsat, Bsat and a. µsat is the fully If the total permeance Ptot(F ) is known the flux-rate Φ˙ through a time-varying permeance is calculated as saturated permeability, which is usually dΦ µsat = µ0 (19) Φ˙ = dt Bsat defines the knee of the saturation transition between d unsaturated and saturated permeability as illustrated in Fig. = (Ptot · F ) dt 5:

dF dP Bsat = (B − µsatH) (20) = P · + tot · F H→∞ tot dt dt The coefﬁcient a can be determined using the unsaturated

dF dPtot dF permeability µunsat at H = 0: = Ptot · + · · F dt dF dt a = Bsat/ (µunsat − µsat) (21) dPtot dF Fig. 5 illustrates the saturation characteristics for both = Ptot + · F · (11) dF dt fitting function. The saturation curves differ only around the In this case, the control signal for the variable permeance transition between unsaturated and saturated permeability. The component is coth fit expresses a slightly tighter transition. " # " d # With the relationships Φ = B · A and F = H · l the control P(t) Ptot + Ptot · F = dF (12) signal Pdiff for the variable permeance is easily derived from d dt P(t) 0 (16) and (18). B and iteratively minimized by tweaking Fk. For all variable µ permeances the flux Φk must be provided. Thus, the input sat coth fit signal of a saturable permeance becomes: B sat      d  P(t) Pdiff Ptot + dF Ptot · F  d P(t)  =  0  =  0   dt      atan fit Φ(t) Ptot · F Ptot · F (28)

IV. CONCLUSIONS µ unsat The permeance-capacitance analogy implemented in PLECS provides a powerful modeling domain for magnetic circuits. H The structure of the magnetic circuit can be derived easily from the core geometry. Any non-linear characteristic of the Fig. 5. Saturation characteristics of coth and atan fitting functions core material can be modeled using the variable permeance component. By applying Kirchhoff’s current law to both the flux-rate and the flux itself, the solver can integrate the C. Solving magnetic circuit equations magnetic circuit equations without accumulating numerical The system equations of the equivalent circuit are deter- errors. mined by Kirchhoff’s current and voltage laws. At each node, the sum of directed flux-rates in all branches n is zero: REFERENCES n [1] S. El-Hamamsy and E. Chang, “Magnetics modeling for computer-aided X ˙ design of power electronics circuits,” in Power Electronics Specialists Φk = 0 (22) Conference, vol. 2, pp. 635–645, 1989. k=1 [2] R. W. Buntenbach, “Improved circuit models for inductors wound on dissipative magnetic cores,” in Proc. 2nd Asilomar Conf. Circuits Syst., Around any closed circuit loop, the directed sum of the mmf Pacific Grove, CA, Oct. 1968, pp. 229–236 (IEEE Publ. No. 68C64- is zero: ASIL). n [3] R. W. Buntenbach, “Analogs between magnetic and electrical circuits,” X F = 0 (23) in Electron. Producs, vol. 12, pp. 108–113, 1969. k [4] D. Hamill, “Lumped equivalent circuits of magnetic components: the k=1 gyrator-capacitor approach,” in IEEE Transactions on Power Electronics, vol. 8, pp. 97–103, 1993. However, since Kirchhoff’s current law was not applied to the [5] D. Hamill, “Gyrator-capacitor modeling: A better way of understanding magnetic flux itself, Gauss’s law of magnetism magnetic components,” in APEC Conference Proceedings pp. 326–332, 1994. [6] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,” in IEEE B · dA = 0 (24) " Transactions on Magnetics, vol. 19, no. 5, pp. 21832185, 1983. ∂V [7] D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,” in Journal of Applied Physics, vol. 55, no. 6, pp. 21152120, 1984. would be violated, if [8] D. C. Jiles, J. B. Thoelke and M. K. Devine, “Numerical determination • the sum of all directed fluxes at simulation start was not of hysteresis parameters for the modelilng of magnetic properties using the theory of ferromagnetic hysteresis,” in IEEE Trans. on Magnetics, zero, or vol. 28, no. 1, pp. 27–35, 1992. • an error accumulated during the numerical integration. [9] C. Perez-Rojas, “Fitting saturation and hysteresis via arctangent functions,” in IEEE Power Engineering Review, vol. 20, no. 1, pp. 55–57, Therefore, the solver needs to enforce Kirchhoff’s current law 2000. for the flux n X Φk = 0 (25) k=1 explicitly. In case of linear systems with constant permeances Pk, this can simply be achieved by including n n X X Φk = Pk · Fk = 0 (26) k=1 k=1 in the system equations. To solve circuits with variable permeances Pk(t) the resid- ual flux Φres must be computed for every node n X Φres = Φk(Fk, t) = 0 (27) k=1