How do Z, Y, ABCD and S parameters differ?
Two-port networks can be described by several equivalent parameter sets. Z and Y relate port voltages and currents and suit series and shunt combinations; ABCD (chain) parameters relate input to output and cascade by matrix multiplication; S parameters relate waves and are measured by VNAs; T parameters are the wave-domain cascade form. All convert into one another when they exist.[1]
What does each parameter set represent?
| Set | Relation | Natural use |
|---|---|---|
| Z | V = Z·I | Series-connected networks add |
| Y | I = Y·V | Parallel-connected networks add; transistor small-signal models |
| H | Hybrid | BJT low-frequency models |
| ABCD | [V1; I1] = [A B; C D][V2; I2] | Cascades multiply |
| S | b = S·a | Measurement, matching, stability |
| T | Waves at port 1 vs port 2 | Cascade in wave domain, de-embedding |
What are ABCD matrices of basic elements?
- Series impedance Z: [1 Z; 0 1].
- Shunt admittance Y: [1 0; Y 1].
- Lossless line (Z0, βℓ): [cos βℓ, jZ0 sin βℓ; j sin βℓ/Z0, cos βℓ].
- Ideal transformer N:1: [N 0; 0 1/N].
A reciprocal network has AD − BC = 1.[1]
How do you convert ABCD to S?
For equal real reference impedance Z0, with Δ = A + B/Z0 + CZ0 + D: S11 = (A + B/Z0 − CZ0 − D)/Δ, S21 = 2/Δ, S12 = 2(AD − BC)/Δ, S22 = (−A + B/Z0 − CZ0 + D)/Δ.[1]
Frequently asked questions
Which parameters do simulators use internally?
Linear simulators typically use nodal admittance (Y) matrices and convert results to S.
Why can't some parameters exist?
An ideal series element has no Z matrix; an ideal shunt element has no Y matrix.
What are T-parameters used for?
Cascading and de-embedding fixtures in the wave domain.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Microwave Engineering, 4th ed.”. D. M. Pozar, Wiley. 2012.