S-Parameter Analysis
S-Parameter: Theoretical Foundations
What are S-Parameters?
Professor, what quantity do S-parameters represent?
They represent the input/output characteristics of high-frequency circuits using the ratio of reflected and transmitted waves. For a 2-port case:
$a_i$: incident wave, $b_i$: reflected wave. $S_{11}$: reflection coefficient, $S_{21}$: transmission coefficient.
So a smaller $|S_{11}|$ indicates better impedance matching, right?
Correct. $S_{11} = -20$ dB means reflected power is 1%. $S_{21} = -3$ dB means transmitted power is halved (3 dB loss). S-parameters are functions of frequency and are measured with a VNA (Vector Network Analyzer).
Summary
- $S_{11}$: Reflection coefficient — Indicator of impedance matching
- $S_{21}$: Transmission coefficient — Indicator of insertion loss
- Function of frequency — Measured with VNA, calculated with FEM
The Birth of S-Parameters——How the Scattering Matrix Changed "Inter-Port Relationships"
The concept of S-parameters (scattering parameters) was formalized by K. Kuroki and D. M. Pozar, among others. They provided a unified description of the relationships between incident, reflected, and transmitted waves in microwave circuits, which were difficult to handle with Z-parameters or Y-parameters. In particular, the "reflection coefficient Γ (=S₁₁)" and "transmission coefficient S₂₁," which are measurable on transmission lines, directly connected experiment and theory, becoming the common language of high-frequency design alongside the spread of network analyzers. S-parameter calculation in CAE is realized through the process of eigenmode expansion → port mode normalization.
Computational Methods for S-Parameter
S-Parameter Extraction in FEM
How do you extract S-parameters from FEM?
1. Set mode patterns (e.g., TE10) on ports
2. Excite an incident wave from one port
3. Calculate reflected/transmitted waves at each port
4. Calculate via $S_{ij} = b_i/a_j$
HFSS's adaptive mesh uses $\Delta S$ (change in S-parameters) for convergence judgment.
What about multi-port cases?
For $N$ ports, it's an $N \times N$ S-matrix. Solve sequentially by exciting from each port ($N$ times, Direct Solver). Or solve all ports simultaneously and extract $S$ via matrix operations (Fast Frequency Sweep).
Summary
- Port Mode Setting — Definition of incident wave
- $\Delta S$ Convergence Criterion — Indicator for adaptive mesh
- Fast Frequency Sweep — Fast acquisition of S-parameters across the entire band
De-embedding——The Technology to "Remove Port Influence from Measured Values"
When actually measuring S-parameters with a VNA, the influence of connectors, cables, and fixtures gets mixed in. "De-embedding" removes this by measuring known reference structures (Open/Short/Thru) and removing the port model via inverse matrix operations. A similar de-embedding concept is used in CAE, extracting actual device characteristics from port waveforms within the analysis domain. TRL (Thru-Reflect-Line) calibration is the standard method for VNAs, and the port settings in CST/HFSS automate this procedure.