## Waveguide Transformers and Gyrators

The ideal transformer, depicted in Fig. C.37 a, is a lossless two-port electric circuit element which scales up voltage by a constant [110,35]. In other words, the voltage at port 2 is always times the voltage at port 1. Since power is voltage times current, the current at port 2 must be times the current at port 1 in order for the transformer to be lossless. The scaling constant is called the turns ratio because transformers are built by coiling wire around two sides of a magnetically permeable torus, and the number of winds around the port 2 side divided by the winding count on the port 1 side gives the voltage stepping constant .

In the case of mechanical circuits, the two-port transformer relations appear as

where and denote force and velocity, respectively. We now convert these transformer describing equations to the wave variable formulation. Let and denote the wave impedances on the port 1 and port 2 sides, respectively, and define velocity as positive into the transformer. Then

Similarly,

We see that choosing

eliminates the scattering terms and gives the simple relations

The corresponding wave flow diagram is shown in Fig. C.37 b.

Thus, a transformer with a voltage gain corresponds to simply changing the wave impedance from to , where . Note that the transformer implements a change in wave impedance without scattering as occurs in physical impedance steps (§C.8).

### Gyrators

Another way to define the ideal waveguide transformer is to ask for a two-port element that joins two waveguide sections of differing wave impedance in such a way that signal power is preserved and no scattering occurs. From Ohm's Law for traveling waves (Eq.(6.6)), and from the definition of power wavesC.7.5), we see that to bridge an impedance discontinuity between and with no power change and no scattering requires the relations

Therefore, the junction equations for a transformer can be chosen as

 (C.125)

where

 (C.126)

Choosing the negative square root for gives a gyrator [35]. Gyrators are often used in electronic circuits to replace inductors with capacitors. The gyrator can be interpreted as a transformer in cascade with a dualizer [433]. A dualizer converts one from wave variable type (such as force) to the other (such as velocity) in the waveguide.

The dualizer is readily derived from Ohm's Law for traveling waves:

In this case, velocity waves in section are converted to force waves in section , and vice versa (all at wave impedance ). The wave impedance can be changed as well by cascading a transformer with the dualizer, which changes to (where we assume ). Finally, the velocity waves in section can be scaled to equal their corresponding force waves by introducing a transformer on the left, which then coincides Eq.(C.126) (but with a minus sign in the second equation).

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