Two-port impedance model and series connection

Impedance model of a two-port

In the impedance (z-parameter) model the port voltages are expressed in terms of the port currents:

$$\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$$ $$z_{11} = \left.\frac{V_1}{I_1}\right|_{I_2=0},\quad z_{12} = \left.\frac{V_1}{I_2}\right|_{I_1=0},\quad z_{21} = \left.\frac{V_2}{I_1}\right|_{I_2=0},\quad z_{22} = \left.\frac{V_2}{I_2}\right|_{I_1=0}$$

They are called open-circuit impedance parameters because each is measured with the other port open. $z_{11}, z_{22}$ are driving-point impedances; $z_{12}, z_{21}$ are transfer impedances. For a reciprocal network $z_{12}=z_{21}$ and the network can be represented by a T-equivalent:

     z11-z12        z22-z12
o----[Za]----+----[Zb]----o
             |
           [z12]
             |
o------------+------------o

Series combination 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.