# The Gyrator

*Pieter-Tjerk de Boer, PA3FWM web@pa3fwm.nl*

(This is an adapted version of part of an article that I wrote for the Dutch
amateur radio magazine *Electron*, July 2015.)

We all know the transformer, which converts an (AC) voltage into a larger or smaller AC voltage. The ratio between the primary and secondary voltages equals the ratio of the number of turns of the transformer's coils. At the same time, also the current is transformed, in the opposite ratio. This is summarised in the left half of the figure.

The Dutch engineer/scientist Bernard Tellegen in 1947 originated the idea
(see B.D.H. Tellegen: The gyrator, a new electric network element,
Philips Research Reports, 1948)
that in theory one could design a kind of transformer
in which the primary *voltage* is converted to a secondary *current*,
and vice versa; see at right in the figure.
Current and voltage thus exchange roles.
He called this thing a gyrator, because of a theoretical analogy with a flywheel (gyroscope).

What does such a gyrator do?
If on the secondary side we connect a simple resistor R,
on the primary side we'll also measure a resistor, of value s^{2}/R,
where s is the gyrator's voltage current conversion ratio.
So, the larger the resistor connected on the secondary side, the smaller the resistance
measured on the primary side.

If we connect a capacitor C on the secondary side, things get more interesting.
Recall that in a capacitor, the current *leads* (i.e., is ahead of) the voltage
(you first feed a current into it, before charge accumulates and a voltage develops across it).
Our gyrator exchanges voltage and current, so at the primary side, the current will
*lag* the voltage, which is the characteristic of an *inductor*.
In other words, a gyrator converts a capacitor into a coil!
To be precise, if we connect a capacitance C on the secondary side,
at the primary side one measures an inductance of L=s^{2}C.
The reverse is also true: if an inductor is connected to the secondary side,
one measures a capacitance on the primary side.

And this is the use of a gyrator, *if* we would have one.
Large capacitors with good properties can be made relatively easily,
while large inductors tend to be physically large and have large losses
and/or parasitic capacitance.
With a gyrator, one could use such a good capacitor as a replacement for a large coil.

While a long piece of copper wire and a chunk of iron give a transformer that's quite close to what an ideal transformer does, realizing a gyrator is not so easy. In his article, Tellegen considers a couple of practical realizations, among others based on electromechanical systems and on metal plates, coils and a suspension of metal particles, but none of those seem to have been very succesful. So his work is essentially limited to a theoretical construct.

A practical gyrator that radio amateurs have heard about in their license test
(although the literature rarely describes it as a gyrator), is the quarter-wave transformer.
A transmission line (e.g., coaxial cable) of a quarter wave long and with an impedance Z_{0}
transforms an impedance Z_{1} connected at one end into Z_{0}^{2}/Z_{1}
at the other end.
That's precisely the same formula which also describes the gyrator.
The obvious limitation is that such a transmission line only does this on the
frequency at which it is a quarter wave long (or an odd multiple of it),
so it is not a general gyrator.

## An electronic gyrator

Nowadays, the gyrator is mostly known as an electronic circuit which in a limited range of voltage, current and frequency works as a gyrator, and needs a power supply for that. Many gyrator circuits have been designed, and a simple one is illustrated below (based on P. Strict: Gyrator acts as electronic choke, Electronics World + Wireless World, Sept. 1993).First consider the situation for DC; then we can ignore the capacitor. The transistor ensures that there is about 0.6 V over R2, and if its currect gain is large enough, its base current can be neglected, so the current through R1 will be approximately the same as through R2; if R1=R2, there will be another 0.6 V across R1. For DC, the whole circuit thus behaves as a resistor with value R3, with a constant extra voltage drop of 1.2 V.

For AC, the situation is different. For high frequencies, the capacitor is almost a short circuit. So if we apply a high-frequency AC voltage superposed on a DC voltage, the voltage on the capacitor will be practically constant. The transistor, acting as an emitter follower, will then ensure that the voltage across R3 is constant too, and thus that the current through the circuit is constant as well (apart from a small AC current through R1 and C). That looks like an inductor: one can't have much of an AC current in it.

For intermediate frequencies, some calculations are needed, but the outcome is that the whole thing behaves as an inductor with value C R1 R3, in series with a resistor R3, and a fixed 1.2 V voltage drop.

We previously defined the gyrator as a thing that effectively interchanges
voltage and current, and that's indeed the case here.
The *voltage* across the capacitor determines, via the emitter follower
and the emitter resistor, the *current* through the circuit as a whole.
And the *current* through the capacitor is determined by the *voltage*
across the circuit as a whole, since this voltages charges and discharges
the capacitor through R1.

The lion's share of the current through the circuit flows through the transistor, but since it has a finite current gain, it needs to get sufficient base current. That is only possible if R1 is small enough. O.t.o.h., the smaller R1 is, the larger the current which at high frequencies can flow through R1 and C, "around" the simulated inductor. As a consequence, the circuit will only behave as an inductor in a limited range of frquencies, but that may be enough.

The impedance graph in the figure also shows that for very high frequencies, the impedance decreases again. This is caused by parasitic capacitances in the circuit, in particular the capacitance parallel to R1.