S-parameter Definitions

Scattering parameters describe a microwave network by relating incident and reflected waves at each port. In Flex RF, those wave amplitudes are computed from the port voltage $V$, port current $I$, and reference impedance $Z_\text{ref}$.

When $Z_\text{ref}$ is real, such as a standard 50-ohm reference, all supported S-parameter definitions produce the same S-matrix. However, when $Z_\text{ref}$ is complex, the definitions produce different S-matrices. The latter situation occurs commonly in lossy transmission lines when the line characteristic impedance is used as the reference.

S-Parameter Definitions

Use the s_param_def field in TerminalComponentModeler to set the convention.

Definition	s_param_def	Reflected-wave impedance	Main behavior
Pseudo-wave	pseudo	Z_ref	Default definition. For complex Z_ref, reciprocal devices are not guaranteed to have S12 = S21.
Power-wave	power	conj(Z_ref)	Relates wave amplitudes directly to delivered power and keeps passive reflection coefficients bounded.
Symmetric pseudo-wave	symmetric_pseudo	Z_ref	Keeps reciprocal S-matrices symmetric while using a complex scaling factor.

Wave Amplitudes

The incident and reflected wave amplitudes can be written as:

$$\begin{aligned} a &= F \, (V + Z_\text{ref} \, I) \\ b &= F \, (V - Z'_\text{ref} \, I) \end{aligned}$$

$F$ is a scaling factor. $Z'_\text{ref}$ is either $Z_\text{ref}$ or its complex conjugate, depending on the selected definition. The S-matrix relates these quantities as $b = S \, a$.

Power Relationship

For power waves, the net power delivered by a port is:

$$P = \frac{1}{2} \left( |a|^2 - |b|^2 \right)$$

For pseudo-waves and symmetric pseudo-waves, an additional cross term appears when $Z_\text{ref}$ is complex. This means the net power is not simply the difference between incident and reflected power. The cross term disappears when $Z_\text{ref}$ is real, so all definitions reduce to the familiar power relationship.

Passivity, Reciprocity, and Zero Reflection

Definition	Reciprocal S12 = S21	Condition for S11 = 0	Passive |S11| <= 1
Pseudo-wave	No	Z_in = Z_ref	No
Power-wave	Yes	Z_in = conj(Z_ref)	Yes
Symmetric pseudo-wave	Yes	Z_in = Z_ref	No

For pseudo-waves and symmetric pseudo-waves, $S_{11}$ follows the familiar impedance-match form and vanishes when the input impedance equals the reference impedance. For power waves, the conjugate in the reflected-wave definition means $S_{11} = 0$ requires a conjugate match when $Z_\text{ref}$ is complex.