Gyrators
Another way to define the ideal waveguide transformer is to ask for a
two-port element that joins two waveguide sections of differing wave
impedance in such a way that signal power is preserved and no
scattering occurs. From Ohm's Law for traveling waves
(Eq.(6.6)), and from the definition of power waves
(§C.7.5), we see that to bridge an impedance
discontinuity between
and
with no power change and no scattering requires the
relations
![$\displaystyle \frac{[f^{{+}}_i]^2}{R_i} = \frac{[f^{{+}}_{i-1}]^2}{R_{i-1}}, \qquad\qquad
\frac{[f^{{-}}_i]^2}{R_i} = \frac{[f^{{-}}_{i-1}]^2}{R_{i-1}}.
$](http://www.dsprelated.com/josimages_new/pasp/img4109.png)
where
Choosing the negative square root for
![$ g_i^{-1}$](http://www.dsprelated.com/josimages_new/pasp/img3695.png)
The dualizer is readily derived from Ohm's Law for traveling waves:
![\begin{eqnarray*}
f^{{+}}\eqsp Rv^{+}, \qquad
f^{{-}}\eqsp -Rv^{-}\\ [5pt]
\Lon...
...i\eqsp Rv^{+}_{i-1}, \qquad
v^{-}_{i-1} \eqsp -R^{-1} f^{{-}}_i
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4112.png)
In this case, velocity waves in section are converted to force
waves in section
, and vice versa (all at wave impedance
). The
wave impedance can be changed as well by cascading a transformer with
the dualizer, which changes
to
(where we assume
). Finally, the velocity waves in section
can be scaled to equal their corresponding force waves by
introducing a transformer
on the left, which then
coincides Eq.
(C.126) (but with a minus sign in the second equation).
Next Section:
Additive Synthesis
Previous Section:
General Conditions for Losslessness