Reference impedance and wave renormalization

The forward traveling-wave voltage/current ratio Z_c is determined by the line or mode. The input impedance Z_in includes reflection from the termination and depends on the observation plane. The port reference R_ref is selected when defining scattering waves; it does not alter the physical DUT.

If a 75 Ω termination is attached to a 50 Ω line and the instrument uses 50 Ω references, the characteristic impedance and instrument reference coincide only by the chosen setup. A 75 Ω renormalization changes the reported reflection without changing the termination.

For real R_ref>0 define incident a and outgoing b using V and inward current I. Add and subtract the definitions to obtain V=√R_ref(a+b) and I=(a−b)/√R_ref. Their difference of squared magnitudes equals Re(VI*).

For AoN peak phasors, average absorbed power includes a factor 1/2. In other literature RMS-normalized wave amplitudes absorb this factor into definitions; the physics does not change. Never use a peak-wave amplitude while interpreting it as the square root of RMS power.

For two ports with unequal real references, define a diagonal positive matrix R of reference resistances. The normalized impedance is ζ=R^(−1/2)ZR^(−1/2). The scattering matrix follows as S=(ζ−I)(ζ+I)⁻¹ under regular inversion.

The symmetric normalization maintains reciprocity symmetry: if Z is symmetric, then ζ and S are symmetric. The simpler unscaled (Z−R)(Z+R)⁻¹ formula is not generally the same matrix when the diagonal references differ; do not silently interchange these definitions.

With a complex reference Z_ref=R+jX, R>0, power waves need distinct coefficients for incident and outgoing directions. Kurokawa defines a_p=(V+Z_ref I)/(2√R) and b_p=(V−Z_ref* I)/(2√R), with conjugation on the outgoing reference.

Expand |a_p|²−|b_p|² to recover Re(VI*). Thus positive net power is represented exactly under the same peak-phasor 1/2 factor. Omitting the star changes the wave definition and no longer preserves this simple magnitude-difference power identity for a complex reference.

For a load Z_L, Kurokawa's power reflection coefficient is Γ_p=(Z_L−Z_ref*)/(Z_L+Z_ref). It vanishes when the load is the complex conjugate of the reference, a conjugate power match.

Pseudo-voltage waves instead give Γ_q=(Z_L−Z_ref)/(Z_L+Z_ref). For Z_ref=50+j10 Ω and Z_L=50−j10 Ω, Γ_p=0 but Γ_q=−j0.2. One physical load therefore has different numbers under different wave definitions; specify the convention before judging matching.

Consider a fixed 100 Ω resistive load. With R_ref=50 Ω, Γ=(100−50)/(100+50)=1/3, reflected-power fraction 1/9, and VSWR=2. With R_ref=75 Ω, Γ=1/7, reflected fraction 1/49, and VSWR=4/3.

No resistor value changed. The reference normalization changed and thus so did the incident and reflected amplitude coordinates. Such renormalization is a representation conversion rather than a mechanical adjustment of the DUT or VNA calibration plane.

When a measured S is defined relative to positive real references, recover normalized impedance ζ=(I+S)(I−S)⁻¹ if I−S is invertible, then Z=R^(1/2)ζR^(1/2).

Near singularity the inversion magnifies measurement noise; check condition number, port order, reference and frequency grid before treating the converted Z as exact. A physical network need not fail because one coordinate representation is ill-conditioned.

A VNA calibration establishes its specified error-corrected planes. A reference-impedance display change alters the wave normalization, while de-embedding requires an independently known fixture model to relocate or remove physical launch/adapter responses.

For a repeatable experiment measure a calibrated known resistance, export Touchstone with correct frequency units and reference impedances, and compare VNA S11 with the predicted complex Γ. Report repeatability and the instrument's actual usable frequency range rather than assuming ideal standards.

1. Distinguish Z_c, Z_in and R_ref in a terminated line.
2. Derive V,I from a,b for real R_ref.
3. Prove P_abs=(|a|²−|b|²)/2 for peak phasors.
4. Write the correct normalized Z-to-S formula for unequal real port references.
5. Derive the complex Kurokawa power waves and explain the conjugate.
6. Calculate Γ_p and Γ_q for Z_ref=50+j10 Ω, Z_L=50−j10 Ω.
7. For Z_L=100 Ω compare reflection, VSWR and power under 50/75 Ω references.
8. Derive the inverse S-to-Z transformation and its singularity condition.
9. Explain why passive reciprocal Z gives symmetric power-normalized S under positive real references.
10. Describe a repeatable VNA test to distinguish reference renormalization from de-embedding.

Next: rigorous extraction of S-parameters from matched-port experiments.