Variable Magnetic Permeance

Purpose

Variable permeance controlled by external signal

Library

Magnetic / Components

Description

This component provides a magnetic flux path with a variable permeance. The component is used to model non-linear magnetic material properties such as saturation and hysteresis. 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:

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

Since \(\mathcal{F}\) is the state variable the equation above must be solved for \(\frac{\mathrm{d}\mathcal{F}}{\mathrm{d}t}\). The control signal must provide the values of \(\mathcal{P}(t)\), \(\frac{\mathrm{d}}{\mathrm{d} t} \mathcal{P}(t)\) and \(\Phi\) as a vector. It is the responsibility of the user to provide the appropriate signals.

Modeling non-linear material properties

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:

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

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

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

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

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

the control signal is:

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

Parameters

Initial MMF

Magneto-motive force at simulation start, in ampere-turns (\(\mathrm{A}\)).

Probe Signals

MMF

The magneto-motive force measured from the marked to the unmarked terminal, in ampere-turns (\(\mathrm{A}\)).

Flux

The magnetic flux flowing through the component, in webers (\(\mathrm{Wb}\)). A flux entering at the marked terminal is counted as positive.