Conversion (S to Y)
From Travelling Waves to Shunt Admittances
Scattering parameters describe a network in terms of reflected and incident power waves at each port, which is the natural measurement basis for a vector network analyzer at microwave and millimeter-wave frequencies. Admittance parameters, by contrast, relate the terminal currents to the terminal voltages with every other port short-circuited. Converting between the two does not change the physics of the device; it simply re-expresses the same measured behavior in a basis that is easier to work with for certain tasks. The admittance form is especially convenient because Y-matrices of two networks placed in parallel (sharing the same node pair) add element by element, a property the scattering matrix does not have.
The conversion is purely algebraic and frequency-by-frequency. At each measurement point the complex S-matrix is combined with the identity matrix and scaled by the reference admittance Y0. Because the formula contains a matrix inverse, it is performed on the full N-by-N matrix rather than entry by entry, and the reference impedance Z0 recorded in the data file must be carried through so the answer comes out in siemens. The same machinery converts S to impedance (Z) parameters by swapping the roles of the plus and minus terms.
The Governing Transform
Y = Y0 × (I − S) × (I + S)−1
2-port closed form (denominator Δ):
Δ = (1 + S11)(1 + S22) − S12S21
Y11 = Y0 × [(1 − S11)(1 + S22) + S12S21] / Δ
Y12 = Y0 × [−2 S12] / Δ
Y21 = Y0 × [−2 S21] / Δ
Y22 = Y0 × [(1 + S11)(1 − S22) + S12S21] / Δ
Where Y0 = 1/Z0 (≈ 0.02 S for Z0 = 50 Ω), I is the identity matrix, and all S terms are complex at each frequency. The transform is singular when Δ → 0, i.e. an eigenvalue of S equals −1 (a short-circuit condition).
Network Parameter Conversion Reference
| From / To | Core operation | Reference scale | Singular when | Typical use |
|---|---|---|---|---|
| S → Y | Y0(I−S)(I+S)−1 | Y0 = 1/Z0 | eig(S) = −1 | Shunt de-embedding, FET gm extraction |
| S → Z | Z0(I+S)(I−S)−1 | Z0 (50 Ω) | eig(S) = +1 | Series de-embedding |
| Y → S | (Y0I−Y)(Y0I+Y)−1 | Y0 = 1/Z0 | det(Y0I+Y) = 0 | Re-normalize to VNA ports |
| S → ABCD | Wave-to-chain mapping | Z0 per port | S21 = 0 | Cascading 2-ports |
Frequently Asked Questions
What is the formula to convert a 2-port S-matrix to a Y-matrix?
The general form is Y = Y0(I − S)(I + S)−1, with I the identity matrix and Y0 = 1/Z0 (0.02 S at 50 Ω). Because of the matrix inverse, (I + S) is inverted first and then pre-multiplied by (I − S). It is valid only when all ports share the same real reference impedance; complex or mismatched references need the generalized power-wave form.
When does the S to Y conversion become singular or numerically unstable?
It blows up when (I + S) is singular, which occurs when an eigenvalue of S equals −1, the matrix signature of a short circuit at a port or mode. A series-resonant null pushes the determinant Δ toward zero and inflates the condition number, so Y entries lose precision. Guard against it by testing det(I + S) before inverting, or convert through the Z-matrix when a short is expected.
Do I need the reference impedance to convert S-parameters to Y-parameters?
Yes. S-parameters are dimensionless ratios normalized to Z0, so the admittance scale is undefined without it. The Y0 = 1/Z0 factor restores units of siemens. A Touchstone .s2p file stores Z0 in its option line (for example # GHz S RI R 50); ignoring it scales every Y value by the wrong ratio. VNAs default to 50 Ω, but 75 Ω and on-wafer setups must carry the correct value through the math.