S-parameter fundamentals¶
Scattering parameters (S-parameters) describe how a linear network responds to traveling waves. At each port, an incident wave a and a reflected (or transmitted) wave b are related by the S-matrix:
b = S · a
For a 2-port device the four entries are:
| Parameter | Stimulus | Response | Everyday name |
|---|---|---|---|
| S11 | Port 1 | Port 1 | Input return loss / reflection |
| S21 | Port 1 | Port 2 | Forward transmission (insertion) |
| S12 | Port 2 | Port 1 | Reverse transmission |
| S22 | Port 2 | Port 2 | Output return loss / reflection |
Index convention: Sij is the response at port i when the stimulus is at port j, with every other port terminated in the reference impedance. That is “to, from” — S21 is to 2 from 1.
What an S-parameter is not¶
- It is not a power meter reading. S-parameters are complex ratios of wave amplitudes (magnitude and phase, or real and imaginary).
- It is not “the cable.” A measurement includes fixtures, launches, adapters, and the instrument’s calibration. De-embedding and fixture removal are separate steps.
- A single S21 trace is not a dataset. In CloudSprite the dataset is the measurement event; S21 is one trace parsed from a file on that dataset.
Reference impedance¶
Unless a file header says otherwise, VNA / Touchstone S-parameters are 50 Ω single-ended. Mixed-mode (differential / common-mode) quantities are usually interpreted in 100 Ω differential / 25 Ω common-mode for a pair of 50 Ω lines. Convert to mixed-mode after you have confirmed port mapping; do not change Z0 first and then mix the formulas.
Frequency, not time¶
S-parameters are a frequency-domain description. Each frequency point is an independent complex sample. Time-domain views (TDR, gating) are transforms of that same data. Correlation QC and mixed-mode conversion both operate on the frequency-domain samples unless you explicitly transform first.
Passive, linear, time-invariant¶
The usual conversion identities (single-ended ↔ mixed-mode) assume the DUT is linear and time-invariant. A compressor, a mixer, or a part that changes while you sweep is outside that model — measure it as such, and do not treat mixed-mode algebra as a substitute for a true 4-port, nonlinear characterization.