Series Combination of One-Ports

Figure J.6 shows the series combination of two one-ports.

Figure J.6: Two one-port networks combined in series. The impedance of the series combination is $R(s) = F(s)/V(s) = R_1(s) + R_2(s)$.

Impedances add in series, so the aggregate impedance is $R(s) = R_1(s) + R_2(s)$, and the admittance is

$$\Gamma(s) = \frac{1}{\frac{1}{\Gamma_1(s)} + \frac{1}{\Gamma_2(s)}} = \frac{\Gamma_1(s) \Gamma_2(s) }{\Gamma_1(s) + \Gamma_2(s)}$$

The latter expression is the handy product-over-sum rule for combining admittances in series.

In a physical situation, if two elements are connected in such a way that they share a common velocity, then they are in series. An example is a mass connected to one end of a spring where the other end is attached to a rigid support and the force is applied to the mass, as shown in Fig. J.7.

Figure J.7: A mass and spring combined as one-ports in series.