Definition and extraction of S-parameters

In RF-04.02 we defined normalized peak-phasor wave amplitudes a and b at a real positive reference resistance. At a specified frequency, a linear, source-free two-port relates outgoing wave vector b to incident wave vector a through four dimensionless complex coefficients. This assumes one controlled propagating port mode at each reference plane.

The first column describes the outgoing response to an incident excitation at port 1 alone; the second is the response to port 2 alone. Linearity allows the responses to add when both ports are driven coherently. S is a model of port waves, not a new microscopic electromagnetic law.

To determine S11 and S21, excite port 1 with nonzero a1 and arrange the port-2 external environment so that no wave is incident on the DUT from that side: a2=0. The reflected b1 and transmitted b2 may both be nonzero. With both currents referenced entering, the outward transmitted wave is a b2 wave, not an inward I2 current.

Repeat with a1=0 and a2 nonzero to determine S12 and S22. A matched external termination at the chosen real R0 removes re-reflections from the environment; it neither opens port 2 nor shorts it. VNA source and receiver error correction is a separate measurement matter.

For equal real positive R0 at both ports define a=(V+R0 I)/(2√R0), b=(V−R0 I)/(2√R0), with each I entering the DUT. Hence V=√R0(a+b) and I=(a−b)/√R0. Inserting V=ZI gives (Z+R0 I)b=(Z−R0 I)a, where I within a matrix expression denotes the identity.

When Z+R0I is invertible, S=(Z−R0I)(Z+R0I)^−1; when Y exists S=(I−R0Y)(I+R0Y)^−1. For unequal real references chapter 02 requires the symmetrically normalized ζ, not this equal-R0 shorthand. For complex reference impedances, power-wave definitions must change.

RF-03 uses the chain convention [V1,I1]ᵀ=[A B;C D][V2,−I2]ᵀ. Substitute the normalized waves and eliminate the unknown outgoing wave amplitudes. Under the same real positive R0 at both ports, the denominator is Δ=A+B/R0+C R0+D.

The four numerators follow from that same algebra. If AD−BC=1, reciprocity gives S12=S21. This result depends on keeping −I2 in the ABCD output state; silently switching to [V2,I2] makes the conversion formulas wrong. A unit determinant does not imply absence of loss.

Consider a passive 25 Ω resistance between two 50 Ω measurement ports, with common return conductor and no additional shunt branches. The ABCD matrix is [1 25 Ω;0 1]; therefore Δ=2+25/50=2.5 and S11=S22=0.2, S21=S12=0.8.

With port 1 excited and a2=0, the incident wave is divided between reflection and transmission: 4% of incident power reflects and 64% is transmitted. The missing 32% is dissipated in the series resistance. A circuit can be reciprocal, symmetric and still dissipative.

Take shunt input conductance G1=0.010 S, output conductance G2=0.005 S, and bridge conductance Gb=0.020 S. Kirchhoff yields Y=[[0.030,−0.020],[−0.020,0.025]] S, reciprocal but not port-interchange symmetric.

At R0=50 Ω, invert I+R0Y and multiply I−R0Y to find S11=−1/37, S21=S12=16/37 and S22=3/37. The transfer coefficients agree but the reflection coefficients differ. This is a worked counterexample to the claim that reciprocity requires identical input/output matching.

Move the measurement planes away from the DUT through additional known lossless uniform transmission-line sections. The incident and exiting waves each collect a propagation phase. Under the +jωt convention and outward distances d1,d2, port-1 reflection acquires exp(−j2β1d1); forward transmission acquires exp[−j(β1d1+β2d2)].

The magnitudes remain constant for ideal lossless added sections, but measured phases change. In real work, cables and fixture losses alter magnitudes too. Consequently comparing two S2P files without common port planes and frequency-dependent phase references can be physically misleading.

For an excitation at port 1 only, incoming wave power is |a1|²/2 in our peak-wave convention. The two measured outgoing powers are |S11|²|a1|²/2 and |S21|²|a1|²/2. Passivity imposes |S11|²+|S21|²≤1; equality requires no dissipation or power exported to unmeasured channels.

Reciprocity gives S12=S21 for the real properly power-normalized references used here, but no requirement that S11 equal S22. A VNA may report small passivity violations due to noise or calibration uncertainty. The next chapter gives the full S†S criterion, modal losses and the distinction between a matched source and a matched DUT.

1. Derive b1 and b2 from the 2×2 scattering map for arbitrary a1,a2.
2. Explain precisely what a2=0 means physically in a forward S-parameter measurement.
3. Calculate S11 and S21 if a1=2√W (peak wave amplitude), a2=0, b1=0.4√W and b2=1.6√W.
4. Starting from Z and the real reference-wave definitions, derive S=(Z−R0I)(Z+R0I)^−1.
5. State the correct ABCD-to-S21 formula with both port currents entering, and its denominator.
6. Calculate all S parameters for a 25 Ω series resistor with R0=50 Ω.
7. Verify the dissipated-power fraction of the series-resistor network independently using circuit V/I.
8. Convert the reciprocal asymmetric π-network admittance matrix into S and compare S11,S22.
9. If S11=0.4∠30° and the port-1 plane moves outward by d=λg/8, find new S11.
10. Explain why a port-2 external match means a2=0 but cannot imply S21=0 or b2=0.

Next: use S†S, eigenvalues and modal power to distinguish reciprocal, passive, lossless and nonreciprocal RF networks.