---
title: "Cascaded two-ports: ABCD matrices multiply"
subject: "Electrical Circuit Theory"
chapter: 7
key: "semester_3/electrical_circuit_theory:cascade-transmission-matrix-product"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/cascade-transmission-matrix-product
site: BCT Engineering
---

# Cascaded two-ports: ABCD matrices multiply

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

- Asked in: 2074 Asoj, 4 marks
- Video: [ABCD Parameters #1. How to Derive the Total Cascade Matrix by Multiplying Individual ABCD Matrices](https://www.youtube.com/watch?v=rWZCpdr4Zt8) — Technologies Discussion
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/cascade-transmission-matrix-product
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7

## Question

Show that the overall transmission parameter matrix for cascaded two 2-port networks is simply the matrix product of transmission parameters for each individual 2-port network in cascade.

## Short answer

For cascaded networks a and b, the output of a is the input of b, so [V1; I1] = Ta [V2a; −I2a] = Ta [V1b; I1b] = Ta Tb [V2; −I2]. Hence the overall transmission matrix is the product T = Ta Tb.

## Answer

For a two-port, the transmission (ABCD) equations are:

```text
V1 = A V2 − B I2
I1 = C V2 − D I2
```

or, in matrix form, [V1; I1] = [A B; C D] [V2; −I2]. The minus sign on I2 means "current leaving the output port".

#### Cascade connection

Network a has parameters (Aa, Ba, Ca, Da) and network b has (Ab, Bb, Cb, Db). The output port of a is connected to the input port of b:

```text
 I1a      +------+  I2a    I1b      +------+   I2b
 o-->-----|  Na  |----<--o-->-------|  Nb  |---<--o
 V1a      +------+  V2a    V1b      +------+   V2b
```

Connection conditions: V2a = V1b and −I2a = I1b (the current leaving a is the current entering b). The overall quantities are V1 = V1a, I1 = I1a, V2 = V2b, I2 = I2b.

#### Derivation

```text
[V1a ; I1a] = Ta [V2a ; −I2a]
[V1b ; I1b] = Tb [V2b ; −I2b]
```

Using V2a = V1b and −I2a = I1b:

```text
[V1 ; I1] = [V1a ; I1a] = Ta [V2a ; −I2a] = Ta [V1b ; I1b] = Ta Tb [V2b ; −I2b]
          = Ta Tb [V2 ; −I2]
```

So the overall transmission matrix is T = Ta · Tb. In full:

```text
A = Aa Ab + Ba Cb      B = Aa Bb + Ba Db
C = Ca Ab + Da Cb      D = Ca Bb + Da Db
```

The order matters (matrix multiplication is not commutative): for n networks in cascade, T = T1 T2 ... Tn in the order from input to output. This is why transmission parameters are the natural choice for ladder networks.

## Check yourself

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

1. Which conditions describe network a cascaded into network b?
   A. V2a = V1b and -I2a = I1b
   B. V1a = V1b and I1a = I1b
   C. V2a = V2b and I2a = I2b
   D. V2a = -V1b and I2a = I1b
   Answer: A. The output voltage of a is the input voltage of b, and the current leaving a is the current entering b. (Key point: Cascade connection conditions)

2. What is the overall transmission matrix of the cascade?
   A. T = Ta + Tb
   B. T = Ta x Tb (in order from input to output)
   C. T = Tb x Ta
   D. T = Ta x Tb^-1
   Answer: B. Substituting the connection conditions gives [V1; I1] = Ta Tb [V2; -I2]. (Key point: Overall matrix)

3. Why must the order Ta Tb be kept?
   A. Matrix multiplication is not commutative
   B. Transmission parameters are always symmetrical
   C. Because currents add in cascade
   D. Because T matrices are diagonal
   Answer: A. Ta Tb generally differs from Tb Ta, so the matrices must be multiplied from input to output. (Key point: Order matters)

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