Magnetic Modeling

Inductors and transformers are key components in modern power electronic circuits. Compared to other passive components they are rather difficult to model for the following reasons:

Magnetic components, especially transformers with multiple windings can have complex geometric structures. The flux in the magnetic core may be split into several paths with different magnetic properties. In addition to the core flux, each winding has its own leakage flux.
Core materials such as iron alloy and ferrite express a highly non-linear behavior. At high flux densities, the core material saturates leading to a greatly reduced inductor impedance. Moreover, hysteresis effects and eddy currents cause frequency-depending losses.

In PLECS, the user can build complex magnetic components in a special magnetic circuit domain. Primitives such as windings, cores and air gaps are provided in the Magnetics Library. The available core models include saturation and hysteresis. Frequency dependent losses can be modeled with magnetic resistances. Windings form the interface between the electrical and the magnetic domain.

Alternatively, less complex magnetic components such as saturable inductors and single-phase transformers can be modeled directly in the electrical domain.

Equivalent circuits for magnetic components

To model complex magnetic structures with equivalent circuits, three different approaches exist: Coupled-inductors, the resistance-reluctance analogy and the capacitance-permeance analogy.

Coupled inductors

In the coupled inductor approach, the magnetic component is modeled directly in the electrical domain as an equivalent circuit, in which inductances represent magnetic flux paths and losses incur at resistors. Magnetic coupling between windings is realized either with mutual inductances or with ideal transformers.

Using coupled inductors, magnetic components can be implemented in any circuit simulator since only electrical components are required. This approach is most commonly used for representing standard magnetic components such as transformers. Fig. 82 shows an example for a two-winding transformer, where \(L_{\sigma1}\) and \(L_{\sigma2}\) represent the leakage inductances, \(L_{\rm m}\) the non-linear magnetizing inductance and \(R_{\rm fe}\) the iron losses. The copper resistances of the windings are modeled with \(R_1\) and \(R_2\).

../_images/transformer_schema.svg — Fig. 82 Transformer implementation with coupled inductors

However, the equivalent circuit bears little resemblance to the physical structure of the magnetic component. For example, parallel flux paths in the magnetic structure are modeled with series inductances in the equivalent circuit. For non-trivial magnetic components such as multiple-winding transformers or integrated magnetic components, the equivalent circuit can be difficult to derive and understand. In addition, equivalent circuits based on inductors are impossible to derive for non-planar magnetic components.

Reluctance-resistance analogy

The traditional approach to model magnetic structures with equivalent electrical circuits is the reluctance-resistance analogy. The magnetomotive force (MMF) \(\mathcal{F}\) is regarded as analogous to voltage and the magnetic flux \(\Phi\) as analogous to current. As a consequence, magnetic reluctance of the flux path \(\mathcal{R}\) corresponds to electrical resistance:

\[\mathcal{R} = \frac{\mathcal{F}}{\Phi}\]

The magnetic circuit is simple to derive from the core geometry: Each section of the flux path is represented by a reluctance and each winding becomes an MMF source.

To link the external electrical circuit with the magnetic circuit, a magnetic interface is required. The magnetic interface represents a winding and establishes a relationship between flux and MMF in the magnetic circuit and voltage \(v\) and current \(i\) at the electrical ports:

\[v = N \frac{\mathrm{d}\Phi}{\mathrm{d}t}\]

\[i = \frac{\mathcal{F}}{N}\]

where \(N\) is the number of turns. If the magnetic interface is implemented with an integrator, it can be solved by an ODE solver for ordinary differential equations:

\[\Phi = \frac{1}{N} \int v\, \mathrm{d}t\]

The schematic in Fig. 83 outlines a possible implementation of the magnetic interface in PLECS.

../_images/mag_interface_schema.svg — Fig. 83 Implementation of magnetic interface

Although the reluctance-resistance duality may appear natural and is widely accepted, it is an awkward choice for multiple reasons:

Physically, energy is stored in the magnetic field of a volume unit. In a magnetic circuit model with lumped elements, the reluctances should therefore be storage components. However, with the traditional choice of mmf and flux as magnetic system variables, reluctances are modeled as resistors, i.e. components that would usually dissipate energy. It is also confusing that the magnetic interface is a storage component.
To model energy dissipation in the core material, inductors must be employed in the magnetic circuit, which is even less intuitive.
Magnetic circuits with non-linear reluctances generate differential-algebraic equations (DAE) resp. algebraic loops that cannot be solved with the ODE solvers offered in PLECS.
The use of magnetic interfaces results in very stiff system equations for closely coupled windings.

Permeance-capacitance analogy

To avoid the drawbacks of the reluctance-resistance analogy the alternative permeance-capacitance analogy is most appropriate. Here, the MMF \(\mathcal{F}\) is again the across-quantity (analogous to voltage), while the rate-of-change of magnetic flux \(\dot{\Phi}\) is the through-quantity (analogous to current). With this choice of system variables, magnetic permeance \(\mathcal{P}\) corresponds to capacitance:

\[\dot{\Phi} = \mathcal{P} \, \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t}\]

Hence it is convenient to use permeance \(\mathcal{P}\) instead of the reciprocal reluctance \(\mathcal{R}\) to model flux path elements. Because permeance is modeled with storage components, the energy relationship between the actual and equivalent magnetic circuit is preserved. The permeance value of a volume element is given by:

\[\mathcal{P} = \frac{1}{\mathcal{R}} = \frac{\mu_0 \mu_\mathrm{r} A}{l}\]

where \(\mu_0 = 4 \pi \times 10^{-7}\,\mathrm{N}/\mathrm{A^2}\) is the magnetic constant, \(\mu_\mathrm{r}\) is the relative permeability of the material, \(A\) is the cross-sectional area and \(l\) the length of the flux path.

Magnetic resistors (analogous to electrical resistors) can be used in the magnetic circuit to model losses. They can be connected in series or in parallel to a permeance component, depending on the nature of the specific loss. The energy relationship is maintained as the power

\[P_\mathrm{loss} = \mathcal{F}\,\dot{\Phi}\]

converted into heat in a magnetic resistor corresponds to the power loss in the electrical circuit.

Windings form the interface between the electrical and the magnetic domain. A winding of \(N\) turns is described with the equations below. The left-hand side of the equations refers to the electrical domain, the right-hand side to the magnetic domain.

\[v = N \dot{\Phi}\]

\[i = \frac{\mathcal{F}}{N}\]

Because a winding converts through-quantities (\(\dot{\Phi}\) resp. \(i\)) in one domain into across-quantities (\(v\) resp. \(\mathcal{F}\)) in the other domain, it can be implemented with a gyrator, in which \(N\) is the gyrator resistance \(R\). Fig. 84 shows the symbol for a gyrator and a possible implementation in PLECS.

../_images/gyrator_symbol.svg — Fig. 84 Gyrator symbol and implementation

../_images/gyrator_schema.svg — Fig. 84 Gyrator symbol and implementation

In principle, the gyrator component could be used with regular capacitors to build magnetic circuits. However, neither the gyrator symbol nor the capacitor adequately resemble a winding respectively a flux path. Moreover, any direct connection between the electrical and magnetic domain made by mistake would lead to non-causal systems that are very difficult to debug. Therefore, dedicated magnetic components should be used when modeling magnetic circuits.

Magnetic Circuit Domain in PLECS

The magnetic domain provided in PLECS is based on the permeance-capacitance analogy. The magnetic library comprises windings, constant and variable permeances as well as magnetic resistors. By connecting them according to the physical structure the user can create equivalent circuits for arbitrary magnetic components. The two-winding transformer from Fig. 82 will look like the schematic in Fig. 85 when modeled in the magnetic domain.

../_images/transformer_magnetic_schema.svg — Fig. 85 Transformer implementation in the magnetic domain

\(\mathcal{P}_{\sigma1}\) and \(\mathcal{P}_{\sigma2}\) represent the permeances of the leakage flux path, \(\mathcal{P}_{\rm m}\) the non-linear permeance of the core, and \(G_{\rm fe}\) dissipates the iron losses. The winding resistances \(R_1\) and \(R_2\) are modeled in the electrical domain.

Modeling Non-Linear Magnetic Material

Non-linear magnetic material properties such as saturation and hysteresis can be modeled using the variable permeance component. The permeance is determined by the signal fed into the input of the component. The flux-rate through a variable permeance \(\mathcal{P}(t)\) is governed by the equation:

\[\dot{\Phi} = \frac{\mathrm{d}}{\mathrm{d} t} \left( \mathcal{P} \cdot \mathcal{F} \right) = \mathcal{P} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} + \frac{\mathrm{d}}{\mathrm{d} t} \mathcal{P} \cdot \mathcal{F}\]

Since \(\mathcal{F}\) is the state variable the equation must be solved for \(\frac{\mathrm{d}\mathcal{F}}{\mathrm{d}t}\). Therefore, the control signal must provide the values of both \(\mathcal{P}(t)\) and \(\frac{\mathrm{d}}{\mathrm{d} t} \mathcal{P}(t)\).

The control signals must also provide the flux \(\Phi(t)\) through the permeance. This enables the solver to enforce Kirchhoff’s current law for all branches \(k\) of a node:

\[\sum_{k=1}^n \Phi_k = 0\]

When specifying the characteristic of a non-linear permeance, we need to distinguish carefully between the total permeance \(\mathcal{P}_\mathrm{tot}(\mathcal{F}) = \Phi / \mathcal{F}\) and the differential permeance \(\mathcal{P}_\mathrm{diff}(\mathcal{F}) = \mathrm{d}\Phi / \mathrm{d} \mathcal{F}\).

If the total permeance \(\mathcal{P}_\mathrm{tot}(\mathcal{F})\) is known, the flux-rate \(\dot{\Phi}\) through a time-varying permeance is calculated as:

\[\begin{split}\dot{\Phi} & = \frac{\mathrm{d} \Phi}{\mathrm{d}t} \\ & = \frac{\mathrm{d}}{\mathrm{d} t} \left( \mathcal{P}_\mathrm{tot} \cdot \mathcal{F} \right) \\ & = \mathcal{P}_{\mathrm{tot}} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} + \frac{\mathrm{d} \mathcal{P}_{\mathrm{tot}}}{\mathrm{d} t} \cdot \mathcal{F} \\ & = \mathcal{P}_{\mathrm{tot}} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} + \frac{\mathrm{d} \mathcal{P}_{\mathrm{tot}}}{\mathrm{d} \mathcal{F}} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} \cdot \mathcal{F}\\ & = \left( \mathcal{P}_{\mathrm{tot}} + \frac{\mathrm{d} \mathcal{P}_{\mathrm{tot}}}{\mathrm{d} \mathcal{F}} \cdot \mathcal{F} \right) \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t}\end{split}\]

In this case, the control signal for the variable permeance component is:

\[\begin{split}\left[ \begin{array}{c} \mathcal{P}(t) \\ \frac{\mathrm{d}}{\mathrm{d} t} \mathcal{P}(t) \\ \Phi(t) \end{array} \right] = \left[ \begin{array}{c} \mathcal{P}_\mathrm{tot} + \frac{\mathrm{d}}{\mathrm{d} \mathcal{F}} \mathcal{P}_{\mathrm{tot}} \cdot \mathcal{F} \\ 0 \\ \mathcal{P}_\mathrm{tot} \cdot \mathcal{F} \end{array} \right]\end{split}\]

In most cases, however, the differential permeance \(\mathcal{P}_\mathrm{diff}(\mathcal{F})\) is provided to characterize magnetic saturation and hysteresis. With:

\[\begin{split}\dot{\Phi} & = \frac{\mathrm{d} \Phi}{\mathrm{d}t} \\ & = \frac{\mathrm{d} \Phi}{\mathrm{d} \mathcal{F}} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} \\ & = \mathcal{P}_\mathrm{diff} \cdot \frac{\mathrm{d} \mathcal{F}}{\mathrm{d} t} \quad ,\end{split}\]

the control signal is:

\[\begin{split}\left[ \begin{array}{c} \mathcal{P}(t) \\ \frac{\mathrm{d}}{\mathrm{d} t} \mathcal{P}(t) \\ \Phi(t) \end{array} \right] = \left[ \begin{array}{c} \mathcal{P}_\mathrm{diff} \\ 0 \\ \mathcal{P}_\mathrm{tot} \cdot \mathcal{F} \end{array} \right]\end{split}\]

Saturation Curves for Soft-Magnetic Material

Curve fitting techniques can be employed to model the properties of ferromagnetic material. As an example, a saturation curve adapted from the modified Langevian equation for bulk magnetization without interdomain coupling is used, which is referred to as the \(\coth\) function:

\[B = B_\mathrm{sat} \left( \coth \frac{3H}{a}-\frac{a}{3H} \right) + \mu_\mathrm{sat} H\]

The \(\coth\) function has three degrees of freedom which are set by the coefficients \(B_\mathrm{sat}\), \(a\) and \(\mu_\mathrm{sat}\). These coefficients can by found e.g. using a least-squares fitting procedure. Calculating the derivate of \(B\) with respect to \(H\) yields:

\[\frac{\mathrm{d} B}{\mathrm{d} H} = B_\mathrm{sat} \left( \frac{\tanh^{2} \left( H/a \right) -1}{a\,\tanh^{2} \left( H/a \right)} - \frac{a}{H^2} \right) + \mu_\mathrm{sat}\]

With the relationships \(\Phi = B \cdot A\) and \(\mathcal{F} = H \cdot l\) the control signal \(\mathcal{P}_\mathrm{diff}\) for the variable permeance is easily derived from the equation above.

References

S. El-Hamamsy and E. Chang, “Magnetics modeling for computer-aided design of power electronics circuits,” in Power Electronics Specialists Conference, vol. 2, pp. 635-645, 1989.

R. W. Buntenbach, “Improved circuit models for inductors wound on dissipative magnetic cores,” in Proc. 2nd Asilomar Conf. Circuits Syst., Pacific Grove, CA, Oct. 1968, pp. 229-236 (IEEE Publ. No. 68C64-ASIL).

R. W. Buntenbach, “Analogs between magnetic and electrical circuits,” in Electron. Products, vol. 12, pp. 108-113, 1969.

D. Hamill, “Lumped equivalent circuits of magnetic components: the gyrator-capacitor approach,” in IEEE Transactions on Power Electronics, vol. 8, pp. 97-103, 1993.

D. Hamill, “Gyrator-capacitor modeling: A better way of understanding magnetic components,” in APEC Conference Proceedings pp. 326-332, 1994.