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Physical Audio Signal Processing
    Digital Waveguide Theory
       Waveguide Transformers and Gyrators


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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... ...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... ...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).


Subsections
Previous: General Conditions for Losslessness
Next: Gyrators

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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