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
[#!DesoerAndKuh!#,#!Belevitch!#]. 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).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University