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 $\Gamma (s) = \Gamma _1(s) + \Gamma _2(s)$ .

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

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

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

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