Gyrator–capacitor model

The model makes permeance elements analogous to electrical capacitance (see magnetic capacitance section) rather than electrical resistance (see magnetic reluctance).

Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

[3][4] The gyrator–capacitor model is an example of a group of analogies that preserve energy flow across energy domains by making power conjugate pairs of variables in the various domains analogous.

It fills the same role as the impedance analogy for the mechanical domain.

Symbols for elements and variables that are part of the model magnetic circuit may be written with a subscript of M. For example,

This is because transducers between the electrical and magnetic domains in this model are usually represented by gyrators.

[1]: 100 Transducers that are not based on magnetic induction may not be represented by a gyrator.

For instance, a Hall effect sensor is modelled by a transformer.

The resistance–reluctance model uses the same equivalence between magnetic voltage and magnetomotive force.

, is an alternate name for the time rate of change of flux,

[4]: 37  The magnetic current flowing through an element of cross section,

The resistance–reluctance model uses a different equivalence, taking magnetic current to be an alternate name for flux,

Magnetic capacitance is an alternate name for permeance, (SI unit: H).

Permeance of an element is an extensive property defined as the magnetic flux,

, through the cross sectional surface of the element divided by the magnetomotive force,

where: For phasor analysis, the magnetic permeability[5] and the permeance are complex values.

(SI unit: F) is the analogy to inductance in an electrical circuit.

[citation needed] The notion of magnetic inductance is employed in analysis and computation of circuit behavior in the gyrator–capacitor model in a way analogous to inductance in electrical circuits.

This example shows a three-phase transformer modeled by the gyrator-capacitor approach.

The magnetic circuit is split into seven reluctance or permeance elements.

The value of each capacitor in farads is the same as the inductance of the associated permeance in henrys.

Magnetic flux in each permeance element in webers is numerically equal to the charge in the associate capacitance in coulombs.

The gyrator-capacitor approach can accommodate leakage inductance and air gaps in the magnetic circuit.

Gaps and leakage flux have a permeance which can be added to the equivalent circuit as capacitors.

The permeance of the gap is computed in the same way as the substantive elements, except a relative permeability of unity is used.

The permeance of the leakage flux may be difficult to compute due to complex geometry.

CPL and CSL represent the primary and secondary leakage inductance respectively.

Magnetic complex impedance (SI unit: S) is determined by:

The magnetic reactance of an undeveloped circuit with the inductance and the capacitance which are connected in series, is equal:

On a complex plane, the magnetic reactance appears as the side of the resistance triangle for circuit of an alternating current.

A simple transformer and its gyrator-capacitor model. R is the reluctance of the physical magnetic circuit.
Definition of Gyrator as used by Hamill in the gyrator-capacitor approach paper.
Permeance of a rectangular prism element
Circuit equivalence between a magnetic inductance and an electric capacitance.
Three phase transformer with windings and permeance elements.
Schematic using gyrator-capacitor model for transformer windings and capacitors for permeance elements
Transformer with gap and leakage flux.
Gyrator-capacitor model of a transformer with a gap and leakage flux.
Circuit equivalence between a magnetic impedance and an electric admittance.