---
title: "Z and g parameters, neither reciprocal nor symmetrical"
subject: "Electrical Circuit Theory"
chapter: 7
key: "semester_3/electrical_circuit_theory:z-and-g-parameters-neither-reciprocal"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/z-and-g-parameters-neither-reciprocal
site: BCT Engineering
---

# Z and g parameters, neither reciprocal nor symmetrical

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

- Asked in: 2068 Shrawan, 8 marks
- Video: [Condition for Reciprocity in Two−Port Networks (Part 1)](https://www.youtube.com/watch?v=zk8Fzg6ahAA) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/z-and-g-parameters-neither-reciprocal
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7

## Question

For the network shown below, find the Z and g parameters and show that the network is neither reciprocal nor symmetrical.
[Figure: port 1 (V1, I1) with a 1 Ω shunt resistor, then a 1 Ω series resistor, then a dependent voltage source 3V1 (+ at top) as a middle shunt branch, then a 1 Ω series resistor and a 2 Ω shunt resistor at port 2 (V2, I2)]

## Short answer

The 1 Ω shunt and dependent source 3V1 give V1 = −I1, so z11 = −1 Ω, z12 = 0. At port 2, V2 = (2/3)I2 − 2I1 giving z21 = −2 Ω, z22 = 2/3 Ω. g: g11 = −1 S, g12 = 0, g21 = 2, g22 = 2/3 Ω. Not reciprocal and not symmetrical.

## Answer

Z-parameters: V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2. g-parameters (hybrid): I1 = g11 V1 + g12 I2, V2 = g21 V1 + g22 I2.

**Circuit (from the figure).** 1 Ω shunt at port 1; 1 Ω series resistor; a dependent voltage source 3V1 (+ at top) as the middle shunt branch, so the middle node is VM = 3V1; 1 Ω series resistor; 2 Ω shunt at port 2.

#### Port 1

```text
I1 = V1/1 + (V1 − VM)/1 = V1 + V1 − 3V1 = −V1      →   V1 = −I1
```

#### Port 2

The current from M into the port 2 node is (VM − V2)/1; KCL at the port 2 node gives I2 + (3V1 − V2) = V2/2:

```text
1.5 V2 = I2 + 3V1
V2 = (2/3) I2 + 2 V1 = (2/3) I2 − 2 I1
```

#### Z-parameters

```text
z11 = −1 Ω     z12 = 0
z21 = −2 Ω     z22 = 2/3 Ω
```

#### g-parameters

Use I1 = −V1 and V2 = 2V1 + (2/3) I2:

```text
g11 = I1/V1 |I2=0 = −1 S      g12 = I1/I2 |V1=0 = 0
g21 = V2/V1 |I2=0 = 2         g22 = V2/I2 |V1=0 = 2/3 Ω
```

(Also g11 = 1/z11, g12 = −z12/z11, g21 = z21/z11, g22 = Δz/z11, which give the same values.)

#### Reciprocity and symmetry

- Reciprocal needs z12 = z21: here z12 = 0 ≠ z21 = −2 (equivalently g12 = −g21 fails: 0 ≠ −2). **Not reciprocal.**
- Symmetrical needs z11 = z22 (together with reciprocity): −1 ≠ 2/3. **Not symmetrical.**

**Answer: Z = [−1 0; −2 2/3] Ω, g = [−1 0; 2 2/3]; the network is neither reciprocal nor symmetrical.**

## Check yourself

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

1. What does KCL at port 1 give?
   A. V1 = -I1
   B. V1 = I1
   C. V1 = 3I1
   D. V1 = -3I1
   Answer: A. I1 = V1 + (V1 - 3V1) = -V1, so V1 = -I1. (Key point: Port 1 equation)

2. Which Z matrix is found?
   A. [-1 0; -2 2/3] ohm
   B. [1 0; 2 2/3] ohm
   C. [-1 -2; 0 2/3] ohm
   D. [-1 0; 2 -2/3] ohm
   Answer: A. V1 = -I1 and V2 = -2I1 + (2/3)I2. (Key point: Z-parameters)

3. Which relations convert Z to g?
   A. g11 = 1/z11, g12 = -z12/z11, g21 = z21/z11, g22 = delta z/z11
   B. g11 = z11, g12 = z12, g21 = z21, g22 = z22
   C. g11 = z22/delta z
   D. g11 = z11/z21
   Answer: A. These standard relations reproduce the directly found g values. (Key point: Z to g conversion)

4. Why is the network neither reciprocal nor symmetrical?
   A. z12 = 0 differs from z21 = -2, and z11 = -1 differs from z22 = 2/3
   B. z11 = z22
   C. z12 = z21
   D. Delta z = 0
   Answer: A. Both reciprocity (z12 = z21) and symmetry (z11 = z22) fail. (Key point: Reciprocity and symmetry)

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