ABCD Matrix Cascade
The ABCD (chain) matrix makes cascaded two-ports trivial: the matrix of a chain is simply the product of the individual matrices, in order. This tool builds the ABCD matrix for each element you list — a series impedance, a shunt admittance, an ideal transmission line, or an ideal transformer — multiplies them, and converts the result to S-parameters at your reference impedance. It reports the reciprocity check (AD − BC = 1) as an internal consistency test.
Equations & Element syntax ▸
series Z: \(\begin{bmatrix}1&Z\\0&1\end{bmatrix}\) shunt Y: \(\begin{bmatrix}1&0\\Y&1\end{bmatrix}\) line: \(\begin{bmatrix}\cos\theta&jZ_0\sin\theta\\ j\sin\theta/Z_0&\cos\theta\end{bmatrix}\) xfmr: \(\begin{bmatrix}n&0\\0&1/n\end{bmatrix}\)
\(S_{21}=\dfrac{2}{A+B/Z_0+CZ_0+D},\ S_{11}=\dfrac{A+B/Z_0-CZ_0-D}{A+B/Z_0+CZ_0+D}\)
\(S_{21}=\dfrac{2}{A+B/Z_0+CZ_0+D},\ S_{11}=\dfrac{A+B/Z_0-CZ_0-D}{A+B/Z_0+CZ_0+D}\)
Zs R X | Series impedance Z = R + jX (Ω). |
Yp G B | Shunt admittance Y = G + jB (S). |
TL Zc deg | Ideal line: characteristic impedance Zc (Ω), electrical length (degrees). |
XFMR n | Ideal transformer, turns ratio n:1. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §4.4 (Tables 4.1, 4.2).
Inputs
Ω
For S-parElements (one per line, port 1 → port 2)
Total ABCD matrix
S-parameters
Reflection / transmission
S11—
S21—
|S21| (insertion)—
Consistency
S22—
|S11| (return loss)—
det = AD − BC—
Cascade