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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 $ g$ [110,35]. In other words, the voltage at port 2 is always $ g$ times the voltage at port 1. Since power is voltage times current, the current at port 2 must be $ 1/g$ times the current at port 1 in order for the transformer to be lossless. The scaling constant $ g$ 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 $ g$ .

Figure C.37: a) Two-port description of the ideal transformer with ``turns ratio'' $ g$ . b) Corresponding wave digital transformer.
\includegraphics[width=\twidth]{eps/lTransformer}

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

\begin{eqnarray*}
F_2(s) &=& g F_1(s) \\ [5pt]
V_2(s) &=& \frac{1}{g} V_1(s)
\end{eqnarray*}

where $ F$ and $ V$ denote force and velocity, respectively. We now convert these transformer describing equations to the wave variable formulation. Let $ R_1$ and $ R_2$ denote the wave impedances on the port 1 and port 2 sides, respectively, and define velocity as positive into the transformer. Then

\begin{eqnarray*}
f^{{+}}_1(t) &=& \frac{f_1(t) + R_1 v_1(t)}{2} \\
f^{{-}}_1(t) &=& \frac{f_1(t) - R_1 v_1(t)}{2}
\eqsp \frac{\frac{1}{g}f_2(t) + R_1 g v_2(t)}{2} \\
&=& \frac{1}{g} \frac{f_2(t) + R_1 g^2 v_2(t)}{2}.
\end{eqnarray*}

Similarly,

\begin{eqnarray*}
f^{{+}}_2(t) &=& \frac{f_2(t) + R_2 v_2(t)}{2} \\
f^{{-}}_2(t) &=& \frac{f_2(t) - R_2 v_2(t)}{2}
\eqsp \frac{g f_1(t) + R_2 \frac{1}{g}v_1(t)}{2} \\
&=& g \frac{f_1(t) + R_2 \frac{1}{g^2} v_1(t)}{2}.
\end{eqnarray*}

We see that choosing

$\displaystyle g^2 = \frac{R_2}{R_1}
$

eliminates the scattering terms and gives the simple relations

\begin{eqnarray*}
f^{{-}}_2(t) &=& g f^{{+}}_1(t)\\ [5pt]
f^{{-}}_1(t) &=& \frac{1}{g}f^{{+}}_2(t).
\end{eqnarray*}

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

Thus, a transformer with a voltage gain $ g$ corresponds to simply changing the wave impedance from $ R_1$ to $ R_2$ , where $ g=\sqrt{R_2/R_1}$ . Note that the transformer implements a change in wave impedance without scattering as occurs in physical impedance steps (§C.8).



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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