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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:
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.
To model complex magnetic structures with equivalent circuits, three different approaches exist: Coupled-inductors, the resistance-reluctance analogy and the capacitance-permeance analogy.
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.
The figure below shows an example for a two-winding transformer, where and
represent the leakage inductances,
the non-linear magnetization
inductance and
the iron losses. The copper resistances of the windings are
modeled with
and
.
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.
The traditional approach to model magnetic structures with equivalent electrical
circuits is the reluctance-resistance analogy. The magnetomotive force (MMF) is
regarded as analogous to voltage and the magnetic flux
as analogous to current.
As a consequence, magnetic reluctance of the flux path
corresponds to electrical
resistance:
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 and
current
at the electrical ports:
where 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:
The schematic below outlines a possible implementation of the magnetic interface in PLECS.
Implementation of magnetic interface
Although the reluctance-resistance duality may appear natural and is widely accepted, it is an awkward choice for multiple reasons:
To avoid the drawbacks of the reluctance-resistance analogy the alternative
permeance-capacitance analogy is most appropriate. Here, the MMF is again the
across-quantity (analogous to voltage), while the rate-of-change of magnetic
flux
is the through-quantity (analogous to current). With this choice
of system variables, magnetic permeance
corresponds to capacitance:
Hence it is convenient to use permeance instead of the reciprocal reluctance
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:
where is the magnetic constant,
is the relative
permeability of the material,
is the cross-sectional area and
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
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 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.
Because a winding converts through-quantities ( resp.
) in one domain into
across-quantities (
resp.
) in the other domain, it can be implemented with a
gyrator, in which
is the gyrator resistance
. The figure below shows the
symbol for a gyrator and a possible implementation in PLECS.
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.
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 above will look like the schematic below when modeled in the magnetic domain.
Transformer implementation in the magnetic domain
and
represent the permeances of the leakage flux path,
the
non-linear permeance of the core, and
dissipates the iron losses. The winding
resistances
and
are modeled in the electrical domain.
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 is governed by the equation:
Since is the state variable the equation must be solved for
. Therefore, the
control signal must provide the values of both
and
.
The control signals must also provide the flux through the permeance. This
enables the solver to enforce Kirchhoff's current law for all branches
of a node:
When specifying the characteristic of a non-linear permeance, we need to
distinguish carefully between the total permeance and the
differential permeance
.
If the total permeance is known the flux-rate
through a time-varying
permeance is calculated as:
In this case, the control signal for the variable permeance component is:
In most cases, however, the differential permeance is provided to
characterize magnetic saturation and hysteresis. With
the control signal is
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:
The coth function has three degrees of freedom which are set by the coefficients
,
and
. These coefficients can by found e.g. using a least-squares
fitting procedure. Calculating the derivate of
with respect to
yields
With the relationships and
the control signal
for the
variable permeance is easily derived from the equation above.