Editing Gyrator (section)

== Relationship to the ideal transformer ==

[[File:tellegen-gyrator-cascaded.svg|thumb|left|Cascaded gyrators]]
An ideal gyrator is similar to an ideal transformer in being a linear, lossless, passive, memoryless two-port device. However, whereas a transformer couples the voltage on port&nbsp;1 to the voltage on port&nbsp;2, and the current on port&nbsp;1 to the current on port&nbsp;2, the gyrator cross-couples voltage to current and current to voltage. [[Cascade connection|Cascading]] two gyrators achieves a voltage-to-voltage coupling identical to that of an ideal transformer.<ref name="tellegen1948" />

Cascaded gyrators of gyration resistance <math>R_1</math> and <math>R_2</math> are equivalent to a transformer of turns ratio <math>R_1 : R_2</math>. Cascading a transformer and a gyrator, or equivalently cascading three gyrators produces a single gyrator of gyration resistance <math>R_1 R_3/R_2</math>.

From the point of view of network theory, transformers are redundant when gyrators are available. Anything that can be built from resistors, capacitors, inductors, transformers and gyrators, can also be built using just resistors, gyrators and inductors (or capacitors).
<!-- force a break as there may be too little text to stop the picture intruding into the next section -->{{-}}

===Magnetic circuit analogy===
{{main|Gyrator–capacitor model}}

In the two-gyrator [[equivalent circuit]] for a transformer, described above, the gyrators may be identified with the transformer windings, and the loop connecting the gyrators with the transformer magnetic core. The electric current around the loop then corresponds to the rate-of-change of magnetic flux through the core, and the [[electromotive force]] (EMF) in the loop due to each gyrator corresponds to the [[magnetomotive force]] (MMF) in the core due to each winding.

The gyration resistances are in the same ratio as the winding turn-counts, but collectively of no particular magnitude. So, choosing an arbitrary conversion factor of <math>r</math> ohms per turn, a loop EMF <math>V</math> is related to a core MMF <math>\mathcal{F}</math> by

: <math>V = r \mathcal{F},</math>

and the loop current <math>I</math> is related to the core flux-rate <math>\dot{\Phi}</math> by

: <math>I = \frac{1}{r} \frac{\partial}{\partial t} \Phi.</math>

The core of a real, non-ideal, transformer has finite [[permeance]] <math>\mathcal{P}</math> (non-zero [[reluctance]] <math>\mathcal{R}</math>), such that the flux and total MMF satisfy

: <math>\Phi = \frac{\mathcal{F}}{\mathcal{R}} = \mathcal{P} \mathcal{F},</math>

which means that in the gyrator loop

: <math>I = \frac{\mathcal{P}}{r^2} \frac{\partial}{\partial t} V</math>

corresponding to the introduction of a series capacitor

: <math>C = \frac{1}{r^2} \mathcal{P}</math>

in the loop. This is Buntenbach's capacitance–permeance analogy, or the [[gyrator–capacitor model]] of magnetic circuits.