Reflectance of a General Lumped Waveguide Termination
Calculate the reflectance of the terminated waveguide. That is, find the Laplace transform of the return wave divided by the Laplace transform of the input wave going into the waveguide. In general, the reflectance of an impedance step for force waves (voltage waves in the electrical case) is
This is easily derived from continuity constraints across the junction. Specifically, referring to Fig.F.1b, let
![$ F_R(s) =
F^{+}_R(s) + F^{-}_R(s)$](http://www.dsprelated.com/josimages_new/pasp/img4753.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ +$](http://www.dsprelated.com/josimages_new/pasp/img2035.png)
![$ V(s) =
V^{+}(s)+V^{-}(s)$](http://www.dsprelated.com/josimages_new/pasp/img4754.png)
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
![$ V_R(s) =
V^{+}_R(s)+V^{-}_R(s)$](http://www.dsprelated.com/josimages_new/pasp/img4755.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ +$](http://www.dsprelated.com/josimages_new/pasp/img2035.png)
![\begin{eqnarray*}
F(s) &=& F_R(s)\\
0 &=& V(s) + V_R(s)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4756.png)
By the definition of wave impedance in a waveguide, we have
![\begin{eqnarray*}
F^{+}(s) &=& \quad\! R_0 V^{+}(s)\\
F^{-}(s) &=& - R_0 V^{-}(...
... &=& \quad\! R(s) V^{+}_R(s)\\
F^{-}_R(s) &=& - R(s) V^{-}_R(s)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4757.png)
Thus,
![\begin{eqnarray*}
0 &=& V(s) + V_R(s)\\
&=& \left[V^{+}(s)+V^{-}(s)\right] + ...
...s)}\right]
&=& \frac{2}{R_0}F^{+}(s) + \frac{2}{R(s)}F^{+}_R(s)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4758.png)
Defining
and
, we have
Now that we've solved for the junction force
![$ F(s)$](http://www.dsprelated.com/josimages_new/pasp/img64.png)
![$ F(s)
= F^{+}(s)+F^{-}(s) = F^{+}_R(s)+F^{-}_R(s)$](http://www.dsprelated.com/josimages_new/pasp/img4762.png)
Finally, the force-wave reflectance of an impedance step from
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ F^{-}(s)/F^{+}(s)$](http://www.dsprelated.com/josimages_new/pasp/img4767.png)
![$ F^{+}_R(s)$](http://www.dsprelated.com/josimages_new/pasp/img4768.png)
![\begin{eqnarray*}
\hat{\rho}(s) &\isdef & \frac{F^{-}(s)}{F^{+}(s)} = \frac{F(s)...
...mma_0-\Gamma(s)}{\Gamma_0+\Gamma(s)}
= \frac{R(s)-R_0}{R(s)+R_0}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4769.png)
as claimed.
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Reflectances of Elementary Impedances
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Interior Scattering Junctions