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Transformer Scattering Formulas

General Two-Port:

\epsfig{file=eps/ltwoport.eps,width=0.8\textwidth }
The general 2-port.

Power conservation:

\begin{eqnarray*}
p_1 u_1
& = & - p_2 u_2 \\
\Leftrightarrow (p_1^+ + p_1^-)\left(\frac{p_1^+ - p_1^-}{R_1}\right)
& = & -(p_2^+ + p_2^-)\left(\frac{p_2^+ -p_2^-}{R_2}\right)
\end{eqnarray*}

Non-reflecting:

\begin{eqnarray*}
p_1^- & = & g_1 p_2^+ \\
p_2^- & = & g_2 p_1^+
\end{eqnarray*}

for some constants $ g_1$ , $ g_2$

Solution:

\begin{eqnarray*}
p_1^- & = & {\sqrt \frac{R_1}{R_2}}p_2^+ \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{1}{g} p_2^+ \\
p_2^- & = & {\sqrt \frac{R_2}{R_1}}p_1^+ \mathrel{\stackrel{\mathrm{\Delta}}{=}}g p_1^+
\end{eqnarray*}

where $ g \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ transformer ``turns ratio''



Wave-flow diagram:

\epsfig{file=eps/Transformer.eps,width=3in}
The ideal 2-port transformer


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Download Scattering.pdf
Download Scattering_2up.pdf
Download Scattering_4up.pdf

``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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