Variable Inductor

Purpose

Inductance controlled by signal

Library

Electrical / Passive Components

Description

This component models a variable inductor. The inductance is determined by the signal fed into the input of the component. The voltage across a variable inductance is determined by the equation:

\[v = L\cdot\frac{\mathrm{d}i}{\mathrm{d}t} + \frac{\mathrm{d}L}{\mathrm{d}t} \cdot i\]

Since \(i\) is the state variable the equation above must be solved for \(\frac{\mathrm{d}i}{\mathrm{d}t}\). The control signal must provide the values of both \(L\) and \(\frac{\mathrm{d}}{\mathrm{d}t}L\) in the following form: \(\left[ L_1\; L_2 \ldots L_n\; \frac{\mathrm{d}}{\mathrm{d}t}L_1\; \frac{\mathrm{d}}{\mathrm{d}t}L_2 \ldots \frac{\mathrm{d}}{\mathrm{d}t}L_n\right]\). It is the responsibility of the user to provide the appropriate signals for a particular purpose (see further below).

If the component has multiple phases, you can choose to include the inductive coupling of the phases. In this case the control signal vector must contain the elements of the inductivity matrix (row by row) followed by their derivatives with respect to time, e.g. for two coupled phases: \(\left[ L_{11}\; L_{12}\; L_{21}\; L_{22}\; \frac{\mathrm{d}}{\mathrm{d}t}L_{11}\; \frac{\mathrm{d}}{\mathrm{d}t}L_{12}\; \frac{\mathrm{d}}{\mathrm{d}t}L_{21}\; \frac{\mathrm{d}}{\mathrm{d}t}L_{22}\right]\). The control signal thus has a width of \(2\cdot n^2\), \(n\) being the number of phases.

Note

The momentary inductance may not be set to zero. In case of coupled inductors, the inductivity matrix may not be singular.

There are two common use cases for variable inductors, which are described in detail below: saturable inductors, in which the inductance is a function of the current and actuators, in which the inductance is a function of an external quantity, such as a solenoid with a movable core.

For a more complex example of a variable inductor that depends on both the inductor current and an external quantity see the Switched Reluctance Machine.

Saturable Inductor Modeling

When specifying the characteristic of a saturable inductor, you need to distinguish carefully between the total inductivity \(L_{\mathrm{tot}}(i) = \Psi/i\) and the differential inductivity \(L_{\mathrm{diff}}(i) = \mathrm{d}\Psi / \mathrm{d}i\). See also the piece-wise linear Saturable Inductor.

With the total inductivity \(L_{\mathrm{tot}}(i) = \Psi/i\) you have

\[\begin{split}\begin{aligned} v & = \frac{\mathrm{d}\Psi}{\mathrm{d}t}\\ & = \frac{\mathrm{d}}{\mathrm{d}t}\left(L_{\mathrm{tot}} \cdot i\right)\\ & = L_{\mathrm{tot}} \cdot \frac{\mathrm{d}i}{\mathrm{d}t} + \frac{dL_{\mathrm{tot}}}{\mathrm{d}t} \cdot i\\ & = L_{\mathrm{tot}} \cdot \frac{\mathrm{d}i}{\mathrm{d}t} + \frac{dL_{\mathrm{tot}}}{\mathrm{d}i} \cdot \frac{\mathrm{d}i}{\mathrm{d}t} \cdot i \\ & = \left(L_{\mathrm{tot}} + \frac{\mathrm{d}L_{\mathrm{tot}}}{\mathrm{d}i}\cdot i\right) \cdot \frac{\mathrm{d}i}{\mathrm{d}t} \quad , \end{aligned}\end{split}\]

which can be implemented as in Fig. 205.

../../_images/ltot_schema.svg — Fig. 205 Schematic for first equation

With the differential inductivity \(L_{\mathrm{diff}}(i) = \mathrm{d}\Psi / \mathrm{d}i\) you have

\[\begin{split}\begin{aligned} v & = \frac{\mathrm{d}\Psi}{\mathrm{d}t}\\ & = \frac{\mathrm{d}\Psi}{\mathrm{d}i} \cdot \frac{\mathrm{d}i}{\mathrm{d}t}\\ & = L_{\mathrm{diff}} \cdot \frac{\mathrm{d}i}{\mathrm{d}t} \quad , \end{aligned}\end{split}\]

which can be implemented as in Fig. 206.

../../_images/ldiff_schema.svg — Fig. 206 Schematic for second equation

Note that in both cases the \(\frac{\mathrm{d}}{\mathrm{d}t}L\)-input of the Variable Inductor is zero!

Actuator Modeling

In an actuator the inductivity is determined by an external quantity such as the position \(x\) of the movable core in a solenoid: \(L = L(x)\). Therefore you have

\[\begin{split}\begin{aligned} v & = L\cdot\frac{\mathrm{d}i}{\mathrm{d}t} + \frac{\mathrm{d}L}{\mathrm{d}t}\cdot i\\ & = L\cdot\frac{\mathrm{d}i}{\mathrm{d}t} + \frac{\mathrm{d}L}{\mathrm{d}x} \cdot \frac{\mathrm{d}x}{\mathrm{d}t} \cdot i \quad , \end{aligned}\end{split}\]

which can be implemented as in Fig. 207.

../../_images/l_actuator_schema.svg — Fig. 207 Schematic for third equation

Note that \(x\) is preferably calculated as the integral of \(\mathrm{d}x/\mathrm{d}t\) rather than calculating \(\mathrm{d}x/\mathrm{d}t\) as the derivative of \(x\).

Example Model

See the example model “Variable Inductor”.
Find it in PLECS under Help > PLECS Documentation > List of Example Models.

Parameters

Inductive coupling: Specifies whether the phases should be coupled inductively. This parameter determines how the elements of the control signal are interpreted. The default is off.
Initial current: The initial current through the inductor at simulation start, in amperes \((\mathrm{A})\). This parameter may either be a scalar or a vector corresponding to the implicit width of the component. The direction of a positive initial current is indicated by a small arrow in the component symbol. The default of the initial current is 0.

Probe Signals

Inductor current: The current flowing through the inductor, in amperes \((\mathrm{A})\). The direction of a positive current is indicated with a small arrow in the component symbol.
Inductor voltage: The voltage measured across the inductor, in volts \((\mathrm{V})\).