S-Parameter Workbench — 2-Port Networks (S / Z / Y / ABCD)

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Edit the four S-parameters (magnitude + angle), and watch the four 2-port matrices [S], [Z], [Y], [ABCD] convert in lockstep. The Physical meaning panel ties S₂₁ to forward gain, S₁₂ to reverse isolation (crosstalk), S₁₁/S₂₂ to input/output reflection, and reports reciprocity (S₂₁ = S₁₂?), correlation (noise + envelope), and Rollett's stability. Smith chart shows S₁₁/S₂₂ with input/output stability circles.

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S-parameter inputs

Z₀ (Ω):

Input form:

S₁₁:

S₁₂:

S₂₁:

S₂₂:

S₂₁ = forward gain path (|S₂₁| > 1 = amplifier). S₁₂ = reverse isolation / crosstalk.

Presets

① The four 2-port matrices (live conversion)

Edit the S-parameters on the left; the [Z], [Y], [ABCD] matrices are computed from S using the standard conversion formulas. All four are equivalent representations of the same 2-port network.

[S] scattering

[Z] impedance (Ω)

[Y] admittance (S)

[ABCD] transmission

② Polar plots of the four S-parameters

Each S_ij is a complex number inside the unit disk. |S_ij| is the distance from the origin; ∠S_ij is the angle (CCW from +x). The dashed unit circle is the |S| = 1 boundary.

S₁₁ (input reflection) S₁₂ (reverse / crosstalk) S₂₁ (forward / gain) S₂₂ (output reflection)

③ Smith chart of S₁₁ & S₂₂ with stability circles

The Smith chart background is the standard reflection-coefficient plane. The two stability circles bound the region of source / load impedances for which the 2-port becomes unstable. If the origin (matched) lies outside the unstable region, the network is unconditionally stable.

S₁₁ (red dot) S₂₂ (blue dot) input stability circle (red) output stability circle (blue)

Physical meaning (S_ij)

S₂₁ forward gain |S₂₁|²

— dB

S₁₂ isolation

— dB

S₁₁ input match (return loss)

— dB

S₂₂ output match (return loss)

— dB

Reciprocity

|S₂₁ − S₁₂|

—

Verdict

—

For a passive / linear reciprocal 2-port, S₂₁ = S₁₂. Active devices (amplifiers) typically violate reciprocity.

Correlation

|ρ| noise correl. (eq. inputs)

—

|ρ_env| envelope correl.

—

ρ_env < 0.5 → low-correlation diversity antennas (good for MIMO). |ρ| = 1 → fully correlated (single-port behavior).

Stability (Rollett)

Δ = S₁₁S₂₂ − S₁₂S₂₁

—

|Δ|

—

K (Rollett)

—

Unconditional stability

—

Unconditional: K > 1 AND |Δ| < 1. Otherwise, the network can oscillate for some source / load impedances (the unstable region is inside the red/blue stability circles on the Smith chart above).