Conversion (S to Z)
From Scattering Waves to Open-Circuit Impedance
S-parameters describe a network in terms of incident and reflected traveling waves normalized to a fixed reference impedance, which is exactly what a vector network analyzer reports because matched-load terminations are easy to realize at microwave frequencies. Z-parameters, by contrast, describe the same network in terms of port voltages and currents measured under open-circuit conditions. The two sets contain identical information about a linear network, so a deterministic algebraic transformation links them. That transformation is what engineers call the S-to-Z conversion, and it is one of a family of network-parameter conversions used to move freely between S, Z, Y, ABCD, and hybrid representations.
The conversion matters because each parameter set is convenient for a different task. S-parameters are ideal for measurement and for cascading matched components, but they hide the series and shunt structure of a circuit. The impedance matrix exposes that structure directly: a series element shows up as an addition to a Z entry, which makes Z the natural form for building lumped-element equivalent circuits, for series-connecting two networks, and for extracting physical component values during de-embedding. Software such as scikit-rf, ADS, and AWR performs this conversion internally every time you request a Z-parameter view of measured data.
Numerical care is essential. The formula inverts the matrix (I − S), so the conversion is only valid when that matrix is non-singular. Near a singularity the Z entries grow without bound and become extremely sensitive to measurement noise, which is why a network that is close to a series-open condition is usually kept in S form. The reference impedance used during measurement must also be carried through unchanged; applying the wrong z0 produces a Z matrix that does not correspond to the physical hardware.
The Governing Equations
Z = √z0 × (I + S)(I − S)−1 × √z0
Single real reference Z0 (all ports equal):
Z = Z0 × (I + S)(I − S)−1
One-port special case:
Z11 = Z0 × (1 + S11) / (1 − S11)
Where S = normalized scattering matrix, I = identity matrix, z0 = diagonal matrix of port reference impedances, √z0 = diagonal matrix of their square roots. Example: a one-port with S11 = 0.2∠0° at Z0 = 50 Ω gives Z11 = 50 × 1.2 / 0.8 = 75 Ω. The conversion is undefined when det(I − S) ≈ 0.
Choosing the Right Parameter Set
| Parameter set | Measured under | Best for | Convert from S via | Singular when |
|---|---|---|---|---|
| Z (impedance) | Open-circuit ports | Series connection, equivalent circuits | Z = √z0(I+S)(I−S)−1√z0 | det(I − S) = 0 |
| Y (admittance) | Short-circuit ports | Shunt (parallel) connection | Y = √y0(I−S)(I+S)−1√y0 | det(I + S) = 0 |
| ABCD (chain) | Mixed V and I | Cascading two-ports | Two-step via Z or Y | 2-port only |
| h (hybrid) | Mixed open/short | Transistor low-frequency models | Two-step via Z | S22 = 1 case |
| S (scattering) | Matched loads | Measurement, cascading matched parts | (native form) | Always defined |
Frequently Asked Questions
How do I write the S-to-Z conversion for an n-port network with mixed reference impedances?
Z = √z0(I + S)(I − S)−1√z0, where I is the identity matrix and z0 is the diagonal matrix of port reference impedances. With one common real reference such as 50 Ω this reduces to Z = Z0(I + S)(I − S)−1. The √z0 factor restores the ohmic scaling that normalization removed.
Which networks make the S-to-Z conversion blow up or return garbage Z values?
The formula inverts (I − S). When det(I − S) = 0 the Z matrix is undefined, which occurs for an ideal series open or any network with an eigenvalue of S equal to +1. Measured data near that point yields huge, ill-conditioned Z entries that are very sensitive to noise, so the data is usually kept in S form or handled with ABCD parameters instead.
Does the reference impedance have to be 50 ohms for S-to-Z conversion?
No. Any positive real reference works, and ports may even use different values via the diagonal z0 matrix. You must apply the same z0 used at measurement time; converting 75 Ω data with a 50 Ω reference gives a Z matrix that does not match the hardware. VNAs typically export at 50 Ω, but the math itself is impedance-agnostic as long as z0 is consistent.