Series connection of two-port networks

Two-port network

A two-port is a network with two pairs of terminals (an input port 1-1′ and an output port 2-2′), where the current entering one terminal of a port leaves by the other terminal of the same port.

 I1 ->                 <- I2
 o------+---------+------o
 +      |         |      +
 V1     |   N     |      V2
 -      |         |      -
 o------+---------+------o
 1'                      2'

It is described by four parameters relating $V_1, I_1, V_2, I_2$, e.g. the z-parameters $V_1 = z_{11}I_1 + z_{12}I_2$, $V_2 = z_{21}I_1 + z_{22}I_2$.

Series connection of two two-ports

In a series connection the input ports of networks $N_a$ and $N_b$ are connected in series and the output ports are connected in series, so the same current flows through both networks at each port and the voltages add.

I1 ->                       <- I2
o----+---------------+----o
 V1a |      Na       | V2a
o----+---------------+----o
 |                        |
o----+---------------+----o
 V1b |      Nb       | V2b
o----+---------------+----o

Derivation. For each network:

$$\begin{aligned} V_{1a} &= z_{11a}I_{1a} + z_{12a}I_{2a}, & V_{2a} &= z_{21a}I_{1a} + z_{22a}I_{2a} \\ V_{1b} &= z_{11b}I_{1b} + z_{12b}I_{2b}, & V_{2b} &= z_{21b}I_{1b} + z_{22b}I_{2b} \end{aligned}$$

Series connection gives $I_1 = I_{1a} = I_{1b}$, $I_2 = I_{2a} = I_{2b}$, $V_1 = V_{1a}+V_{1b}$, $V_2 = V_{2a}+V_{2b}$. Adding:

$$\begin{aligned} V_1 &= (z_{11a}+z_{11b})I_1 + (z_{12a}+z_{12b})I_2 \\ V_2 &= (z_{21a}+z_{21b})I_1 + (z_{22a}+z_{22b})I_2 \end{aligned}$$ $$[Z] = [Z_a] + [Z_b]$$

So the z-parameters of a series combination are the sums of the individual z-parameters. This holds only if each port still carries equal and opposite currents in its two terminals after connection (Brune's test); it is always satisfied when both networks have a common ground line (e.g. three-terminal networks) or an ideal 1:1 transformer is used for isolation.