The ideal transformer, depicted in Fig. G.21 a, is a
lossless two-port electric circuit element which scales up voltage by
a constant [102]. 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
.
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In the case of mechanical circuits, the two-port transformer relations appear as
We now want to see what happens when we convert the transformer describing
equations to the wave variable formulation. Let and
denote
the reference impedances chosen on the port 1 and port 2 sides,
respectively, and define velocity as positive into the transformer. Then
Similarly,
We see that choosing
The corresponding wave flow diagram is shown in Fig. G.21 b.
Thus, a transformer with a voltage gain corresponds to simply
changing the reference impedance from
to
, where
. Note that the transformer implements a change
in reference impedance without scattering, unlike what happens
when wave impedance is changed in a waveguide.