Parallel connection of two-port networks

A two-port network is a network with two pairs of terminals (an input port 1 and an output port 2), where the current entering one terminal of a port equals the current leaving the other. It is described by the four variables $V_1, I_1, V_2, I_2$ and a set of parameters (z, y, h, ABCD) that relate them.

   I1 -->                    <-- I2
  o-------+-------------+-------o
  +       |             |       +
  V1      |  Two-port   |       V2
  -       |  network    |       -
  o-------+-------------+-------o

Parallel connection of two two-ports

In a parallel connection, the input ports of networks $N_a$ and $N_b$ are connected in parallel, and the output ports are also connected in parallel.

  I1 -->     +--------+      <-- I2
 o-----+-----|   Na   |-----+-----o
       |     +--------+     |
  V1   |                    |   V2
       |     +--------+     |
       +-----|   Nb   |-----+
             +--------+
 (both input ports joined in parallel,
  both output ports joined in parallel)

Derivation. Each network is best described by its short-circuit admittance (y) parameters:

$$\begin{bmatrix} I_{1a} \\ I_{2a} \end{bmatrix} = \begin{bmatrix} y_{11a} & y_{12a} \\ y_{21a} & y_{22a} \end{bmatrix} \begin{bmatrix} V_{1a} \\ V_{2a} \end{bmatrix}, \qquad \begin{bmatrix} I_{1b} \\ I_{2b} \end{bmatrix} = [Y_b] \begin{bmatrix} V_{1b} \\ V_{2b} \end{bmatrix}$$

Because the ports are in parallel, the voltages are common:

$$V_1 = V_{1a} = V_{1b}, \qquad V_2 = V_{2a} = V_{2b}$$

and the port currents add (KCL):

$$I_1 = I_{1a} + I_{1b}, \qquad I_2 = I_{2a} + I_{2b}$$

Substituting:

$$\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \left([Y_a] + [Y_b]\right)\begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$$

So the overall y-parameters are the sum of the individual ones:

$$y_{11} = y_{11a}+y_{11b}, \quad y_{12} = y_{12a}+y_{12b}, \quad y_{21} = y_{21a}+y_{21b}, \quad y_{22} = y_{22a}+y_{22b}$$

Validity condition. This addition rule holds only if the interconnection does not change the port currents of either network (the current entering one terminal of each port still leaves by the other). This is always true for three-terminal (common-ground) networks; otherwise it is checked by the Brune test, or an ideal 1:1 isolating transformer is used.

Use in filter design. Parallel connection is used to realize transmission zeros: for example, the twin-T notch network is two T-sections in parallel whose $y_{21}$ terms cancel at the notch frequency.