---
title: "Series connection of two-port networks"
subject: "Filter Design"
chapter: 4
key: "semester_5/filter_design:two-port-series-connection"
source: https://www.bctengineer.com/q/semester_5/filter_design/two-port-series-connection
site: BCT Engineering
---

# Series connection of two-port networks

Filter Design · Chapter 4: Properties and Synthesis of Passive Networks

- Asked in: 2081 Bhadra, 1+4 marks
- Video: [Series Connection of 2 port Network](https://www.youtube.com/watch?v=B4pD-9RD4i0) — Engineering Decoded
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/two-port-series-connection
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-4

## Question

What do you mean by 2-port network? Explain the series connection of two 2 port networks with figure and derivation.

## Short answer

A two-port network has two pairs of terminals (input and output) with port currents entering one terminal and leaving the other. In a series connection, input ports are in series and output ports in series, so currents are common and voltages add, yielding overall z-parameters as the sum of individual z-parameters: [Z] = [Za] + [Zb], provided Brune's test is satisfied.

## Answer

#### 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.

```text
 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**.

```text
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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A two-port network is one in which:
   A. Current entering one terminal of a port leaves by the other terminal of the same port
   B. Any current can enter at any terminal
   C. The network has exactly two elements
   D. Both ports must share a common terminal
   Answer: A. This port condition defines a two-port and lets the four variables V1, I1, V2, I2 be related by parameters. (Key point: Two-port definition)

2. In a series connection of two two-ports, which are common and which add?
   A. Currents are common and voltages add
   B. Voltages are common and currents add
   C. Both voltages and currents add
   D. Both are common
   Answer: A. I1 = I1a = I1b and V1 = V1a + V1b, and similarly at port 2. (Key point: Quantities in series connection)

3. Adding the z-equations of Na and Nb gives:
   A. [Z] = [Za][Zb]
   B. [Z] = [Za] + [Zb]
   C. [Y] = [Ya] + [Yb]
   D. [Z] = [Za]^-1 + [Zb]^-1
   Answer: B. Each z-parameter of the combination is the sum of the corresponding individual ones. (Key point: Result of derivation)

## Related in Filter Design

- [Parallel connection of two-port networks](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atwo-port-parallel-connection) — Filter Design, chapter 4
- [Two-port impedance model and series connection](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atwo-port-impedance-model-series) — Filter Design, chapter 4
- [Zero shifting for two-port synthesis](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Azero-shifting-two-port-synthesis) — Filter Design, chapter 4, asked 3×

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- [Two-port network model and parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atwo-port-network-model) — RF and Microwave Engineering, chapter 3
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