---
title: "Z parameters, reciprocity and symmetricity with 3V1"
subject: "Electrical Circuit Theory"
chapter: 7
key: "semester_3/electrical_circuit_theory:z-parameters-reciprocity-symmetry-3v1-1f"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/z-parameters-reciprocity-symmetry-3v1-1f
site: BCT Engineering
---

# Z parameters, reciprocity and symmetricity with 3V1

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

- Asked in: 2068 Chaitra, 6 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/z-parameters-reciprocity-symmetry-3v1-1f
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7

## Question

Find the Z-parameters in the network shown below and also check for its reciprocity and symmetricity.
[Figure: port 1 (V1, I1); a 1 H inductor as a shunt at port 1; a 1 H inductor in the series arm; a dependent voltage source 3V1 (+ at top) as a middle shunt branch; a 1 F capacitor in the series arm; a 1 H inductor as a shunt at port 2 (V2)]

## Short answer

The 1 H shunt and dependent source 3V1 give V1 = −s I1, so z11 = −s, z12 = 0. KCL at port 2 with 1 F capacitor yields z21 = −3s³/(s²+1), z22 = s/(s²+1). Not reciprocal (z12 ≠ z21) and not symmetrical (z11 ≠ z22).

## Answer

Z-parameters: V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2.

**Circuit (from the figure).** A 1 H shunt inductor at port 1; a 1 H inductor in the series arm; a dependent voltage source 3V1 (+ at top) as the middle shunt branch; a 1 F capacitor in the series arm; a 1 H shunt inductor at port 2. The middle node M is therefore held at VM = 3V1.

#### Port 1

The current through the series 1 H into M is (V1 − VM)/s, and the shunt inductor takes V1/s:

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

So z11 = −s and z12 = 0 (V1 does not depend on I2).

#### Port 2

The 1 F capacitor (impedance 1/s) carries current (VM − V2)·s towards port 2. KCL at the port 2 node (I2 flows in) gives I2 + s(3V1 − V2) = V2/s:

```text
V2 (s + 1/s) = I2 + 3s V1 = I2 − 3s² I1
V2 = [ s/(s² + 1) ] I2 − [ 3s³/(s² + 1) ] I1
```

```text
z11 = −s                 z12 = 0
z21 = −3s³/(s² + 1)      z22 = s/(s² + 1)
```

#### Reciprocity and symmetry

- Reciprocal if z12 = z21: here 0 ≠ −3s³/(s² + 1), so the network is **not reciprocal** (the dependent source).
- Symmetrical if z11 = z22 (and reciprocal): here −s ≠ s/(s² + 1), so it is **not symmetrical**.

**Answer: Z = [−s 0; −3s³/(s²+1) s/(s²+1)] Ω; neither reciprocal nor symmetrical.**

## Check yourself

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

1. What is the voltage at the middle node M?
   A. VM = V1
   B. VM = 3V1
   C. VM = V2
   D. VM = 3V2
   Answer: B. The dependent source 3V1 is the middle shunt branch, holding VM = 3V1. (Key point: Middle node voltage)

2. What do z11 and z12 equal?
   A. z11 = -s, z12 = 0
   B. z11 = s, z12 = 0
   C. z11 = -s, z12 = s
   D. z11 = 1/s, z12 = -s
   Answer: A. I1 = V1/s + (V1 - 3V1)/s = -V1/s so V1 = -s I1, independent of I2. (Key point: z11 and z12)

3. Why is the network not reciprocal?
   A. z12 = 0 but z21 = -3s^3/(s^2+1), so z12 differs from z21
   B. z11 = z22
   C. It has a capacitor
   D. z22 = 0
   Answer: A. The dependent source breaks the equality z12 = z21. (Key point: Reciprocity verdict)

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