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Cascaded two-ports: ABCD matrices multiply

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.

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.

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

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:

 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

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

Using V2a = V1b and −I2a = I1b:

[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:

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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