Transformer Scattering Formulas

General Two-Port:

The general 2-port.

Power conservation:

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

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:

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

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

Wave-flow diagram:

The ideal 2-port transformer