---
title: "Equivalent Z-parameter of three series two-ports"
subject: "Electrical Circuit Theory"
chapter: 7
key: "semester_3/electrical_circuit_theory:equivalent-z-parameter-three-series-two"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/equivalent-z-parameter-three-series-two
site: BCT Engineering
---

# Equivalent Z-parameter of three series two-ports

Electrical Circuit Theory · Chapter 7: Two-port Parameter of Networks

- Asked in: 2068 Baisakh, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/equivalent-z-parameter-three-series-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7

## Question

With necessary circuit diagram, obtain the equivalent Z-parameter if three two port networks are connected in series.

## Short answer

In series connection, the same currents I1 and I2 flow through each network, so port voltages add. Equivalent Z-parameters are the sum: [Z] = [Za] + [Zb] + [Zc]. The port condition must be preserved; isolation transformers may be needed.

## Answer

When two-port networks are connected in series, their input ports are in series and their output ports are in series: the port currents are common and the port voltages add. The Z-parameters add.

```text
      I1       +------+        I2
  o-->---------|  Na  |-----<--------o
   +           +------+           +
  V1           +------+          V2
               |  Nb  |
               +------+
               +------+
               |  Nc  |
               +------+
  (I1 flows through all three at port 1, I2 through all three at port 2)
  o------------------------------------o
```

#### Derivation

For each network k = a, b, c: V1k = z11k I1 + z12k I2, V2k = z21k I1 + z22k I2, because the same currents I1 and I2 flow in every network.

The terminal voltages are the sums:

```text
V1 = V1a + V1b + V1c = (z11a + z11b + z11c) I1 + (z12a + z12b + z12c) I2
V2 = V2a + V2b + V2c = (z21a + z21b + z21c) I1 + (z22a + z22b + z22c) I2
```

Comparing with V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2:

```text
[Z] = [Za] + [Zb] + [Zc]
```

that is, z11 = z11a + z11b + z11c, z12 = z12a + z12b + z12c, z21 = z21a + z21b + z21c, z22 = z22a + z22b + z22c.

**Condition.** The individual Z-parameters stay unchanged only if the series connection keeps the port condition at every network (same current into one terminal of a port as out of the other); otherwise ideal isolating transformers are used.

## Check yourself

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

1. In a series connection of two-ports, which quantities are common?
   A. Port voltages
   B. Port currents I1 and I2
   C. Both voltages and currents
   D. Neither
   Answer: B. The same I1 and I2 flow through every network, while the port voltages add. (Key point: Series connection variables)

2. What is the equivalent Z for networks a, b and c in series?
   A. [Za] + [Zb] + [Zc]
   B. [Za][Zb][Zc]
   C. [Ya] + [Yb] + [Yc]
   D. ([Za]^-1 + [Zb]^-1 + [Zc]^-1)^-1
   Answer: A. Summing the port voltages gives each z-parameter as the sum of the three. (Key point: Equivalent Z matrix)

3. What is the condition for the Z-parameters to stay unchanged?
   A. The networks must be identical
   B. The port condition must hold at every network, otherwise isolating ideal transformers are used
   C. z12 must equal z21
   D. The networks must be symmetric
   Answer: B. The same current must enter one terminal of a port and leave the other. (Key point: Validity condition)

## Close variants set in other exams

- [Equivalent Y-parameter of parallel two-ports](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aequivalent-y-parameter-parallel-two-ports) — Electrical Circuit Theory, chapter 7, asked 3×

## Related in Electrical Circuit Theory

- [Equivalent T from Y parameters of series two-ports](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aequivalent-t-parameter-two-y-networks) — Electrical Circuit Theory, chapter 7
- [G21 and α21 of two-port terminated in 2 Ω](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aterminated-two-port-g21-alpha21) — Electrical Circuit Theory, chapter 5
- [Cascaded two-ports: ABCD matrices multiply](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Acascade-transmission-matrix-product) — Electrical Circuit Theory, chapter 7
- [G21(S) and Z21(S) of two-port with 0.5 F, 1 H](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ag21-z21-two-port-half-f) — Electrical Circuit Theory, chapter 5

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- [Series connection of two-port networks](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atwo-port-series-connection) — Filter Design, chapter 4
- [Equivalent ABCD of two networks in series](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aseries-abcd-equivalent) — Power System Analysis I, chapter 5
- [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