Waveguide Transformers

The ideal transformer, depicted in Fig. G.21 a, is a lossless two-port electric circuit element which scales up voltage by a constant $g$ [102]. 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 G.21: a) Two-port description of the ideal transformer with ``turns ratio'' $g$. b) Corresponding wave digital transformer.

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

$$\begin{eqnarray*} F_2(s) &=& g F_1(s) \\ V_2(s) &=& \frac{1}{g} V_1(s) \end{eqnarray*}$$

We now want to see what happens when we convert the transformer describing equations to the wave variable formulation. Let $R_1$ and $R_2$ denote the reference impedances chosen 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

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

eliminates the scattering terms and gives the simple relations

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

The corresponding wave flow diagram is shown in Fig. G.21 b.

Thus, a transformer with a voltage gain $g$ corresponds to simply changing the reference impedance from $R_1$ to $R_2$, where $g=\sqrt{R_2/R_1}$. Note that the transformer implements a change in reference impedance without scattering, unlike what happens when wave impedance is changed in a waveguide.