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 .
In the case of mechanical circuits, the two-port transformer relations appear as
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
We see that choosing
eliminates the scattering terms and gives the simple relations
The corresponding wave flow diagram is shown in Fig. C.39 b.
Thus, a transformer with a voltage gain corresponds to simply changing the wave impedance from to , where . Note that the transformer implements a change in wave impedance without scattering as occurs in physical impedance steps (§C.8).