Definition
The ABCD (or transmission, or chain) matrix relates the voltages and currents at the two ports of a network. Its key advantage: cascaded networks multiply, so the overall ABCD of a series chain is simply [ABCD]₁ × [ABCD]₂ × … × [ABCD]ₙ.
[V₁] [A B] [V₂ ] [I₁] = [C D] [−I₂] A = V₁/V₂|I₂=0 (open-circuit voltage ratio) B = V₁/I₂|V₂=0 (short-circuit transfer impedance, Ω) C = I₁/V₂|I₂=0 (open-circuit transfer admittance, S) D = I₁/I₂|V₂=0 (short-circuit current ratio)
ABCD Matrices for Common Elements
| Element | A | B | C | D |
|---|---|---|---|---|
| Series impedance Z | 1 | Z | 0 | 1 |
| Shunt admittance Y | 1 | 0 | Y | 1 |
| T-line (Z₀, θ) | cos θ | jZ₀ sin θ | j sin θ/Z₀ | cos θ |
| Ideal transformer (n:1) | n | 0 | 0 | 1/n |
| Series L | 1 | jωL | 0 | 1 |
| Shunt C | 1 | 0 | jωC | 1 |
Converting ABCD to S-Parameters
S₁₁ = (A + B/Z₀ − C·Z₀ − D) / (A + B/Z₀ + C·Z₀ + D) S₂₁ = 2 / (A + B/Z₀ + C·Z₀ + D) S₁₂ = 2(AD−BC) / (A + B/Z₀ + C·Z₀ + D) S₂₂ = (−A + B/Z₀ − C·Z₀ + D) / (A + B/Z₀ + C·Z₀ + D)
Cascade Example: Matching Network + DUT
[ABCD_total] = [ABCD_input_match] × [ABCD_DUT] × [ABCD_output_match] Convert final [ABCD_total] → [S] to get overall S-parameters. This is exactly what RF View's Circuit Simulator does internally for series-connected elements in the simulation chain.
RF View Circuit Simulator: RF View uses ABCD cascade multiplication internally to combine R/L/C/T-line elements and S2P DUT blocks. Build your circuit, simulate instantly, and view combined S11/S21 — free on Android.