Waveguide Transformers and Gyrators

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

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The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

Wave Digital Filters (WDF) digitize analog circuits element-by-element using the bilinear transform and virtual traveling waves. — Click for https://ccrma.stanford.edu/~jos/pasp/Wave_Digital_Filters.html

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Waveguide Transformers and Gyrators

The ideal transformer, depicted in Fig. C.39 a, is a lossless two-port electric circuit element which scales up voltage by a constant

[110,35]. 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

**Figure C.39:** a) Two-port description of the ideal transformer with ``turns ratio'' . 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

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

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

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

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

Thus, a transformer with a voltage gain

corresponds to simply changing the wave impedance from

, 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).

Waveguide Transformers and Gyrators

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University