Parallel Combination of One-Ports

Figure Fig.7.10 shows the parallel combination of two one-ports.

Figure 7.10: Two one-port networks combined in parallel. The admittance of the parallel combination is $ \Gamma (s) = \Gamma _1(s) + \Gamma _2(s)$.
\includegraphics[scale=0.9]{eps/lparallel}

Admittances add in parallel, so the combined admittance is $ \Gamma (s) = \Gamma _1(s) + \Gamma _2(s)$, and the impedance is

$\displaystyle R(s) = \frac{1}{\frac{1}{R_1(s)} + \frac{1}{R_2(s)}}
= \frac{R_1(s) R_2(s) }{R_1(s) + R_2(s)}
$

which is the familiar product-over-sum rule for combining impedances in parallel. This operation is often denoted by

$\displaystyle R= R_1 \vert\vert R_2
$


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