---
title: "Z and T' parameters with dependent sources"
subject: "Electrical Circuit Theory"
chapter: 7
key: "semester_3/electrical_circuit_theory:z-and-t-parameters-dependent-sources"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/z-and-t-parameters-dependent-sources
site: BCT Engineering
---

# Z and T' parameters with dependent sources

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

- Asked in: 2075 Asoj, 8 marks
- Video: [Find Z and T parameters](https://www.youtube.com/watch?v=HjGAFEpsHeU) — Engg-Course-Made-Easy
- 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-t-parameters-dependent-sources
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7

## Question

Find the Z-parameter and hence T'-parameter for the network shown in figure below also check if network is symmetrical.
[Figure: port 1 (V1, I1) followed by a 10 Ω series resistor, then a dependent voltage source 3I2 in series, to a middle node X. A 2 Ω resistor with voltage V3 across it (+ at top) is connected from X to the bottom wire, and a 2 Ω series resistor connects X to port 2 (V2, I2). A dependent current source 2V3 (arrow pointing toward port 1) is connected from the node after the 10 Ω resistor to the port 2 terminal, across the dependent voltage source and the 2 Ω series resistor.]

## Short answer

Z-parameters: Z = [12 5; −6 −4] Ω. Inverse transmission parameters: T' = [−4/5 −18/5; 1/5 12/5]. The network is not symmetrical (z11 ≠ z22) and not reciprocal.

## Answer

Z-parameters: V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2. T'-parameters (inverse transmission): V2 = A' V1 − B' I1, I2 = C' V1 − D' I1.

**Reading the figure (assumptions).** From port 1: a 10 Ω series resistor to node N, then the current-controlled voltage source 3I2 in series (+ on the port 1 side), reaching node X. A 2 Ω shunt resistor from X to the bottom wire carries V3 = VX (+ at the top). A 2 Ω series resistor goes from X to port 2. The current source 2V3, with its arrow pointing towards port 1, connects the port 2 terminal to node N (just before the 3I2 source).

#### Equations (all currents in A, resistances in Ω)

```text
(1) V1 = VN + 10 I1                       (10 Ω drop)
(2) VX = VN − 3 I2                        (3I2 source, + towards port 1)
(3) KCL at N: I1 + 2VX = Isrc             (2V3 flows into N; Isrc goes through the 3I2 source)
(4) KCL at X: Isrc = VX/2 + (VX − V2)/2 = VX − V2/2
(5) KCL at port 2 node: I2 + (VX − V2)/2 = 2VX
```

#### Solving

From (3) and (4): I1 + 2VX = VX − V2/2, so VX = −I1 − V2/2.
Equation (5) gives I2 = 1.5 VX + V2/2. Substituting VX = −I1 − V2/2: I2 = −1.5 I1 − 0.75 V2 + 0.5 V2 = −1.5 I1 − 0.25 V2, so V2 = −6 I1 − 4 I2.
Then VX = −I1 − V2/2 = 2 I1 + 2 I2. From (1) and (2): V1 = VX + 3 I2 + 10 I1 = 12 I1 + 5 I2.

```text
z11 = 12 Ω    z12 = 5 Ω
z21 = −6 Ω    z22 = −4 Ω
Δz  = z11 z22 − z12 z21 = −48 + 30 = −18
```

#### T'-parameters

Eliminate I2 from the Z equations: I2 = (V1 − z11 I1)/z12.

```text
A' = z22/z12 = −4/5          B' = Δz/z12 = −18/5
C' = 1/z12   = 1/5           D' = z11/z12 = 12/5
```

#### Symmetry check

A network is symmetrical if z11 = z22 (and it is reciprocal). Here z11 = 12 Ω ≠ z22 = −4 Ω, and also z12 = 5 ≠ z21 = −6 (the dependent sources make it non-reciprocal).

**Answer: Z = [12 5; −6 −4] Ω; T' = [A' B'; C' D'] = [−4/5 −18/5; 1/5 12/5]; the network is not symmetrical (and not reciprocal).**

## Check yourself

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

1. How are the T'-parameters (inverse transmission) defined in the answer?
   A. V1 = A'V2 - B'I2, I1 = C'V2 - D'I2
   B. V2 = A'V1 - B'I1, I2 = C'V1 - D'I1
   C. I1 = A'V1 + B'I2, V2 = C'I1 + D'V2
   D. V1 = A'I1 + B'I2, V2 = C'I1 + D'I2
   Answer: B. T' expresses the output-port quantities V2 and I2 in terms of the input-port quantities V1 and -I1. (Key point: T' parameter definition)

2. What Z matrix does the answer obtain for the network?
   A. [12 5; -6 -4] ohm
   B. [12 5; 6 4] ohm
   C. [10 3; 2 2] ohm
   D. [-4 -6; 5 12] ohm
   Answer: A. Solving the node equations gives V1 = 12 I1 + 5 I2 and V2 = -6 I1 - 4 I2. (Key point: Z-parameter result)

3. Given z12 = 5, z22 = -4 and delta z = -18, what is B'?
   A. 18/5
   B. -18/5
   C. -4/5
   D. 12/5
   Answer: B. B' = delta z / z12 = -18/5. (Key point: Z to T' conversion)

4. Why is the network declared neither symmetrical nor reciprocal?
   A. z11 = z22 but z12 differs from z21
   B. z11 differs from z22, and z12 = 5 differs from z21 = -6 because of dependent sources
   C. Because the determinant of Z is zero
   D. Because it has a shunt resistor
   Answer: B. Symmetry needs z11 = z22 and reciprocity needs z12 = z21; both fail here, the latter due to the dependent sources. (Key point: Symmetry and reciprocity test)

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