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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 $ R_{i-1}$ and $ R_i$ with no power change and no scattering requires the relations

$\displaystyle \frac{[f^{{+}}_i]^2}{R_i} = \frac{[f^{{+}}_{i-1}]^2}{R_{i-1}}, \qquad\qquad
\frac{[f^{{-}}_i]^2}{R_i} = \frac{[f^{{-}}_{i-1}]^2}{R_{i-1}}.
$

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

$\displaystyle f^{{+}}_i= g_i\, f^{{+}}_{i-1}\qquad\qquad f^{{-}}_{i-1}= g_i^{-1}\, f^{{-}}_i \protect$ (C.122)

where

$\displaystyle g_i \isdef \pm\sqrt{\frac{R_i}{R_{i-1}}} \protect$ (C.123)

Choosing the negative square root for $ g_i^{-1}$ 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 [438]. 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:

\begin{eqnarray*}
f^{{+}}\eqsp Rv^{+}, \qquad
f^{{-}}\eqsp -Rv^{-}\\ [5pt]
\Longleftrightarrow\quad
f^{{+}}_i\eqsp Rv^{+}_{i-1}, \qquad
v^{-}_{i-1} \eqsp -R^{-1} f^{{-}}_i
\end{eqnarray*}

In this case, velocity waves in section $ i-1$ are converted to force waves in section $ i$ , and vice versa (all at wave impedance $ R$ ). The wave impedance can be changed as well by cascading a transformer with the dualizer, which changes $ R$ to $ R\sqrt{R_i/R}=\sqrt{RR_i}$ (where we assume $ R=R_{i-1}$ ). Finally, the velocity waves in section $ i-1$ can be scaled to equal their corresponding force waves by introducing a transformer $ g=\sqrt{1/R}$ on the left, which then coincides Eq.(C.123) (but with a minus sign in the second equation).


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2019-03-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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