## 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 .

*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 waves (§C.7.5), we see that to bridge an impedance discontinuity between and with no power change and no scattering requires the relationswhere

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:

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The Digital Waveguide Oscillator

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