AoNArt of Nanoelectronics
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05 · Advanced

From V/I ports to normalized power waves

Synthesize equal and unequal real/complex reference impedances, derive Z↔S and Kurokawa transforms, quantify passive power and plan calibrated VNA renormalization.

250–310 minAdvanced
In this lesson
  • Recover voltage and inward current vectors from incident/outgoing waves.
  • Derive peak-phasor power and distinguish RMS wave normalization.
  • Convert Z to S with unequal real reference impedances.
  • Invert the S-to-Z transformation and check its condition.
  • Calculate S for a reciprocal network under unequal real references.
  • Define Kurokawa complex-reference power waves with correct conjugation.
  • Derive general complex-reference Z↔S including ordering and reference factors.
  • Distinguish conjugate power matching from pseudo-wave matching.
  • Audit passivity using I−S†S under correctly normalized waves.
  • Design a reproducible VNA renormalization and fixture de-embedding workflow.

This final RF-04 chapter links measured port voltages and inward currents to incident and outgoing power waves. It derives complete real/complex reference transformations, checks a reciprocal two-port with unequal references, and connects scattering matrices with the calibrated VNA and uncertainty.

0105.1 — Peak phasors and real-reference port waves

Start with complex peak terminal phasors V and I, currents both entering the positive port terminals. The physical average power is P_abs=(1/2)Re(V†I). Choosing positive real diagonal port reference R defines incident and outgoing wave amplitudes in √W, not dimensionless quantities.

Define a=R^(−1/2)(V+RI)/2 and b=R^(−1/2)(V−RI)/2. Adding and subtracting gives V=R^(1/2)(a+b) and I=R^(−1/2)(a−b). Hence P_abs=(a†a−b†b)/2. This factor one half is essential for peak rather than RMS phasors.

Figure 5.1 — Signed electrical references of a two-port

Both I₁ and I₂ are defined ENTERING the network at their positive terminals; each voltage arrow points from minus to plus.

I₁I₂−+V₁−+V₂N

I₁ enters the left port pointing right, while I₂ enters the right port pointing left. V₁ and V₂ are positive on the upper terminals. The conventional output current toward an attached load equals −I₂. These arrows define references, not actual solved current flow.

a=12R−1/2(V+RI)b=12R−1/2(V−RI)\begin{aligned}\mathbf a&=\frac{1}{2}\mathbf R^{-1/2}(\mathbf V+\mathbf R\mathbf I)\\[3pt]\mathbf b&=\frac{1}{2}\mathbf R^{-1/2}(\mathbf V-\mathbf R\mathbf I)\end{aligned}V=R1/2(a+b)I=R−1/2(a−b)\begin{aligned}\mathbf V&=\mathbf R^{1/2}(\mathbf a+\mathbf b)\\[3pt]\mathbf I&=\mathbf R^{-1/2}(\mathbf a-\mathbf b)\end{aligned}Pabs=a†a−b†b2=12Re⁡(V†I)P_{\mathrm{abs}}=\frac{\mathbf a^\dagger\mathbf a-\mathbf b^\dagger\mathbf b}{2}=\frac{1}{2}\operatorname{Re}(\mathbf V^\dagger\mathbf I)

0205.2 — Impedance to scattering and back

If V=ZI, introduce the symmetrically normalized impedance ζ=R^(−1/2) Z R^(−1/2). Eliminating V and I from their wave definitions gives S=(ζ−I)(ζ+I)⁻¹, provided ζ+I is invertible at the chosen frequency.

For the inverse map, ζ=(I+S)(I−S)⁻¹ and Z=R^(1/2)ζR^(1/2), requiring I−S invertible. The square-root reference factors must appear on both sides when the port resistances are unequal. Do not replace them with a scalar 50-ohm shortcut.

ζ=R−1/2ZR−1/2\boldsymbol{\zeta}=\mathbf R^{-1/2}\mathbf Z\mathbf R^{-1/2}S=(ζ−I)(ζ+I)−1\mathbf S=(\boldsymbol{\zeta}-\mathbb I)(\boldsymbol{\zeta}+\mathbb I)^{-1}Z=R1/2(I+S)(I−S)−1R1/2\mathbf Z=\mathbf R^{1/2}(\mathbb I+\mathbf S)(\mathbb I-\mathbf S)^{-1}\mathbf R^{1/2}

0305.3 — Example: reciprocal Z, unequal port references

Consider a reciprocal physical impedance matrix Z=[[75,15],[15,100]] Ω and positive real reference resistances R1=50 Ω and R2=75 Ω. Its normalized diagonal terms are 75/50=1.5 and 100/75=4/3, and its cross-terms are 15/√3750.

Exact inversion produces S=[[83,15√6],[15√6,58]]/433. Numerically, S11≈0.191686, S12=S21≈0.0848553 and S22≈0.133949. The transmissions remain equal because of reciprocal physical Z and symmetric power normalization, but the reflection coefficients differ.

Z=(751515100) Ω ,R=(500075) Ω\mathbf Z=\begin{pmatrix}75&15\\15&100\end{pmatrix}\,\Omega\,,\qquad\mathbf R=\begin{pmatrix}50&0\\0&75\end{pmatrix}\,\Omegaζ=(3215375015375043)\boldsymbol{\zeta}=\begin{pmatrix}\dfrac{3}{2}&\dfrac{15}{\sqrt{3750}}\\[6pt]\dfrac{15}{\sqrt{3750}}&\dfrac{4}{3}\end{pmatrix}S=1433(8315615658)\boxed{\mathbf S=\frac{1}{433}\begin{pmatrix}83&15\sqrt6\\15\sqrt6&58\end{pmatrix}}

0405.4 — Complex-reference Kurokawa power waves

With a complex port reference Z_ref=R+jX having a positive real part R, the real-reference pseudo-voltage wave definition does not by itself preserve the simple power difference. Kurokawa defines a_p using Z_ref and b_p using its complex conjugate Z_ref*.

Expanding |a_p|²−|b_p|² cancels the pure voltage and current terms and leaves Re(VI*). Thus net average power is again half their difference for AoN peak phasors. Omitting the conjugate on the outgoing wave defines a different pseudo-wave coordinate.

Zref=R+jX,R=Re⁡(Zref)>0Z_{\mathrm{ref}}=R+jX,\qquad R=\operatorname{Re}(Z_{\mathrm{ref}})>0ap=V+ZrefI2Ra_p=\frac{V+Z_{\mathrm{ref}}I}{2\sqrt R}bp=V−Zref∗I2Rb_p=\frac{V-Z_{\mathrm{ref}}^{*}I}{2\sqrt R}

0505.5 — General complex-reference matrix transformations

For a diagonal complex port reference Z_ref=R+jX, where R>0 and X are real diagonal matrices, insert V=ZI into the incident power-wave definition. The physical port current is then I_port=2(Z+Z_ref)⁻¹R^(1/2)a_p.

Substitute that current into the outgoing wave to obtain S_p=R^(−1/2)(Z−Z_ref*)(Z+Z_ref)⁻¹R^(1/2). Conversely, invert the wave definitions, preserve matrix order and derive Z=R^(1/2)(I+S_p)(I−S_p)⁻¹R^(1/2)−jX. Both conversions require their displayed inverses.

Sp=R−1/2(Z−Zref∗)(Z+Zref)−1R1/2\mathbf S_p=\mathbf R^{-1/2}(\mathbf Z-\mathbf Z_{\mathrm{ref}}^{*})(\mathbf Z+\mathbf Z_{\mathrm{ref}})^{-1}\mathbf R^{1/2}Z=R1/2(I+Sp)(I−Sp)−1R1/2−jX\mathbf Z=\mathbf R^{1/2}(\mathbb I+\mathbf S_p)(\mathbb I-\mathbf S_p)^{-1}\mathbf R^{1/2}-j\mathbf XZref=R+jX\mathbf Z_{\mathrm{ref}}=\mathbf R+j\mathbf X

0605.6 — Conjugate match and pseudo-wave mismatch

For a 50+j10 Ω complex reference and a physical 50−j10 Ω load, the Kurokawa power reflection coefficient is Γ_p=(Z_L−Z_ref*)/(Z_L+Z_ref)=0. The load is conjugate matched to the selected complex power-wave reference.

The pseudo-wave reflection coefficient uses Z_ref without conjugation and gives Γ_q=(Z_L−Z_ref)/(Z_L+Z_ref)=−j0.2. These descriptions correspond to the same physical load and differ only in the chosen wave coordinates; measurements must declare which definition they use.

Zref=(50+j 10) Ω,ZL=(50−j 10) ΩZ_{\mathrm{ref}}=(50+j\,10)\,\Omega,\qquad Z_L=(50-j\,10)\,\OmegaΓp=ZL−Zref∗ZL+Zref=0\Gamma_p=\frac{Z_L-Z_{\mathrm{ref}}^*}{Z_L+Z_{\mathrm{ref}}}=0Γq=ZL−ZrefZL+Zref=−j 0.2\Gamma_q=\frac{Z_L-Z_{\mathrm{ref}}}{Z_L+Z_{\mathrm{ref}}}=-j\,0.2

0705.7 — Reciprocity and passive real power

With physically consistent power-wave normalization, b=Sa implies P_abs=(1/2)a†(I−S†S)a. A passive network requires I−S†S positive semidefinite for every incident vector, while an ideal lossless complete network has S†S=I.

Under real positive reference impedances and correctly paired modes, reciprocal physical networks give symmetric S. For unequal complex power-wave references, naive matrix symmetry may not hold without the proper reciprocity transformation. A reduced two-port can exclude radiated or higher-order channels.

Pabs=12a†(I−S†S)aP_{\mathrm{abs}}=\frac{1}{2}\mathbf a^\dagger(\mathbb I-\mathbf S^\dagger\mathbf S)\mathbf aI−S†S⪰0(passivity)\mathbb I-\mathbf S^\dagger\mathbf S\succeq\mathbf 0\qquad(\text{passivity})S†S=I(complete, lossless network)\mathbf S^\dagger\mathbf S=\mathbb I\qquad(\text{complete, lossless network})

0805.8 — A reproducible VNA renormalization workflow

An S-parameter data file must be interpreted with its port order, frequency units, wave/reference impedances, calibration planes and sample bias or temperature. If I−S is well conditioned, reconstruct Z at each sampled frequency and recompute scattering parameters using the desired new references.

This mathematical renormalization does not remove cables, adapters or probe-launch parasitics. De-embedding requires an independently characterized fixture model. A 25 Ω series resistance should nominally yield S11=0.2 and S21=0.8 under ideal 50 Ω ports; physical measurements must include parasitic and uncertainty checks.

S(f;Rref)  ⟶  Z(f)  ⟶  S(f;Rnew)\mathbf S(f;\mathbf R_{\mathrm{ref}})\;\longrightarrow\;\mathbf Z(f)\;\longrightarrow\;\mathbf S(f;\mathbf R_{\mathrm{new}})det⁡(I−S)≠0and check matrix conditioning\det(\mathbb I-\mathbf S)\ne 0\qquad\text{and check matrix conditioning}renormalization⏟new wave reference  ≠  de-embedding⏟remove modeled fixture\underbrace{\text{renormalization}}_{\text{new wave reference}}\;\ne\;\underbrace{\text{de-embedding}}_{\text{remove modeled fixture}}

Practice — impedance and power-wave transformations

  1. Derive the port-wave inversion and average power identity for unequal real reference resistances.
  2. Derive the forward and inverse impedance/scattering matrix transformations using diagonal real references.
  3. Calculate ζ and S for Z=[[75,15],[15,100]] Ω under R=diag(50,75) Ω.
  4. Verify reciprocal scattering for the numerical example despite unequal S11 and S22.
  5. Prove Kurokawa's complex-reference wave power identity, including the conjugation.
  6. Derive the full complex-reference multiport S_p formula with its two reference-normalization factors.
  7. Derive the inverse S_p-to-Z relation and explain why the −jX term is required.
  8. Compare Kurokawa and pseudo-wave reflections for Z_ref=50+j10 Ω and Z_L=50−j10 Ω.
  9. Explain the Hermitian passivity condition and why individual column power tests are insufficient.
  10. Design a calibrated VNA/Touchstone renormalization workflow that does not confuse fixture de-embedding.
Worked solutions

1. Define a=R^(−1/2)(V+RI)/2 and b=R^(−1/2)(V−RI)/2. Adding gives V=R^(1/2)(a+b), while subtracting gives I=R^(−1/2)(a−b). Expanding both squared norms yields a†a−b†b=Re(V†I), so peak-phase average absorbed power is half their difference.

2. Substitute V=R^(1/2)(a+b) and I=R^(−1/2)(a−b) into V=ZI and introduce ζ=R^(−1/2)ZR^(−1/2). Rearrangement gives (ζ+I)b=(ζ−I)a and S=(ζ−I)(ζ+I)⁻¹. Invert when det(I−S)≠0 to recover Z=R^(1/2)(I+S)(I−S)⁻¹R^(1/2).

3. The normalized diagonals are 75/50=1.5 and 100/75=4/3, while the transfer is 15/√(50×75). Evaluating (ζ−I)(ζ+I)⁻¹ yields the exact S=[[83,15√6],[15√6,58]]/433, or approximately [[0.1916859,0.0848553],[0.0848553,0.1339492]].

4. Because the physical impedance satisfies Z12=Z21=15 Ω and normalization by real positive R is symmetric, S12=S21=15√6/433. The reflections are S11=83/433 and S22=58/433, which differ because the port driving-point impedances and their reference values are not exchange symmetric.

5. Let Z_ref=R+jX with R>0. Then a_p=(V+Z_ref I)/(2√R), b_p=(V−Z_ref* I)/(2√R). Subtract squared magnitudes: the |V|² and |Z_ref I|² terms cancel, leaving |a_p|²−|b_p|²=Re(VI*). Average real power is one half of this because phasors are peak amplitudes.

6. From a_p=R^(−1/2)(Z+Z_ref)I_port/2, solve I_port=2(Z+Z_ref)⁻¹R^(1/2)a_p. Substitute in b_p=R^(−1/2)(Z−Z_ref*)I_port/2 to get S_p=R^(−1/2)(Z−Z_ref*)(Z+Z_ref)⁻¹R^(1/2). Both factors of R are essential when references vary by port.

7. Use I_port=R^(−1/2)(a_p−b_p) and V=R^(1/2)(a_p+b_p)−jX I_port. With b_p=S_p a_p, obtain a_p=(I−S_p)⁻¹R^(1/2)I_port and hence Z=R^(1/2)(I+S_p)(I−S_p)⁻¹R^(1/2)−jX. The order remains valid even when R and S do not commute.

8. With Z_ref=50+j10 Ω, its conjugate is Z_ref*=50−j10 Ω, exactly the load, so Γ_p=(Z_L−Z_ref*)/(Z_L+Z_ref)=0. Pseudo voltage waves instead give Γ_q=(Z_L−Z_ref)/(Z_L+Z_ref)=−j20/100=−j0.2. This is a wave-definition difference, not a physical load change.

9. Since b=Sa, P_abs=(a†[I−S†S]a)/2. For all complex incident a this remains nonnegative iff I−S†S is Hermitian positive semidefinite. Checking only each isolated input column misses cross terms; S=[[0.8,0.8],[0,0]] violates passivity under coherent a=(1,1).

10. Calibrate a known VNA plane and reference impedance, record port numbering and frequency units, and export S(f). For each frequency check invertibility/conditioning of I−S, reconstruct Z, then convert to new reference R' and verify a known resistor. Renormalization is mathematical; removing fixture parasitics requires separate calibrated de-embedding.

RF-05 will introduce practical VNA measurements, calibration methods, uncertainty budgets and de-embedding.

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