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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 [437]. 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 © 2014-10-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA