Port Reference Impedance Semantics

How intrinsic network matrices, explicit port shunts, wave references, and reference planes produce different S, Y, and Z observables.

Node role: Concept.

Reader task: distinguish an intrinsic network, an explicit load, a wave reference, and a reference plane before interpreting or converting \(S/Y/Z\).

Why this distinction decides whether a trace is usable

A \(50\,\Omega\) number can name a wave reference, a resistor in a compiled netlist, or a physical load. Those objects can have the same numerical value without having the same meaning. Confusing them can produce a smooth, well-shaped, completely wrong trace.

The non-negotiable rule

Changing a wave reference renormalizes \(S\); it does not change the underlying network \(Z\) or \(Y\). Adding a shunt changes the network. Subtracting a shunt is valid only when both its compiled presence and its removal intent are proven.

Four objects, four meanings

Intrinsic network

\(V=ZI\) and \(I=YV\) describe the selected DUT terminals, current orientation, and reference plane. At fixed definitions, \(Z\) and \(Y\) do not depend on a wave reference.

Explicit termination or load

A resistor or other branch in the circuit changes its nodal matrix. For a parallel load, \(Y_{\mathrm{loaded}}=Y_{\mathrm{DUT}}+Y_L\).

Wave reference

\(Z_{0i}\) defines how port voltage and current become incident and outgoing waves. Renormalizing \(Z_0\) changes \(S\), not the DUT.

Reference plane

The plane states where port voltage, current, and waves are defined. Moving it is an embedding or de-embedding operation, not a change of normalization.

Real-positive wave-reference contract

The real-reference power-wave definitions follow (Kurokawa 1965). The conversion equations below are this page’s algebra: insert \(V=ZI\) or \(I=YV\) into those definitions and solve for the outgoing wave. All terminal currents enter the DUT; this convention is as important as the matrix order.

The v1 conversion contract is limited to linear ports with real positive references:

\[ R_0=\operatorname{diag}(Z_{01},\ldots,Z_{0N}), \qquad D=R_0^{1/2}, \]

LaTeX source

R_0=\operatorname{diag}(Z_{01},\ldots,Z_{0N}),
\qquad
D=R_0^{1/2},

Every port current enters the DUT. Here \(V\) and \(I\) are RMS-equivalent phasors, so the wave-power difference equals average power. Using unscaled peak phasors instead introduces the corresponding factor of one half in average power. The power waves are

\[ a=\frac{1}{2}\left(D^{-1}V+DI\right), \qquad b=\frac{1}{2}\left(D^{-1}V-DI\right), \qquad b=Sa. \]

LaTeX source

a=\frac{1}{2}\left(D^{-1}V+DI\right),
\qquad
b=\frac{1}{2}\left(D^{-1}V-DI\right),
\qquad
b=Sa.

Here \(a,b\) are the standard RF power-wave symbols, not internal oscillator operators. Elsewhere in this Knowledge Base the unambiguous aliases are

\[ \boxed{ \vec a_{\rm pw}\equiv\vec s_{\rm in}, \qquad \vec b_{\rm pw}\equiv\vec s_{\rm out}, \qquad \vec b_{\rm pw}=\mathbf S\vec a_{\rm pw}. } \]

LaTeX source

\boxed{
\vec a_{\rm pw}\equiv\vec s_{\rm in},
\qquad
\vec b_{\rm pw}\equiv\vec s_{\rm out},
\qquad
\vec b_{\rm pw}=\mathbf S\vec a_{\rm pw}.
}

Normalize the network impedance as

\[ \widetilde Z=D^{-1}ZD^{-1}. \]

LaTeX source

\widetilde Z=D^{-1}ZD^{-1}.

Where the required inverses exist,

\[ S=(\widetilde Z+I)^{-1}(\widetilde Z-I), \qquad Z=D(I-S)^{-1}(I+S)D. \]

LaTeX source

S=(\widetilde Z+I)^{-1}(\widetilde Z-I),
\qquad
Z=D(I-S)^{-1}(I+S)D.

The equivalent admittance form uses \(\widetilde Y=DYD\):

\[ S=(I+\widetilde Y)^{-1}(I-\widetilde Y), \qquad Y=D^{-1}(I+S)^{-1}(I-S)D^{-1}. \]

LaTeX source

S=(I+\widetilde Y)^{-1}(I-\widetilde Y),
\qquad
Y=D^{-1}(I+S)^{-1}(I-S)D^{-1}.

For one port these reduce to

\[ S_{11}=\Gamma =\frac{Z_L-Z_0}{Z_L+Z_0} =\frac{1-Z_0Y_L}{1+Z_0Y_L}. \]

LaTeX source

S_{11}=\Gamma
=\frac{Z_L-Z_0}{Z_L+Z_0}
=\frac{1-Z_0Y_L}{1+Z_0Y_L}.

Setting all other incident waves to zero is the matched-boundary definition of an \(S\)-parameter column. It does not prove that equal shunt resistors are inside the DUT matrix.

Real-SPD matrix-reference power waves

This is this page’s matrix-coordinate extension of the real-positive wave pairing, not a formula attributed wholesale to Kurokawa. The SPD assumption ensures a real principal square root and a positive reference metric. Carrying that metric alongside the voltage/current transform makes the wave map orthogonal; merely mixing voltage entries does not preserve power.

The diagonal contract above is the special case of a real symmetric positive- definite (SPD) reference matrix \(R_0\). Let

\[ D=R_0^{1/2} \]

LaTeX source

D=R_0^{1/2}

be its unique real-SPD principal square root. Matrix-reference power waves use the same definitions,

\[ a=\frac12\left(D^{-1}V+DI\right), \qquad b=\frac12\left(D^{-1}V-DI\right), \qquad b=Sa. \]

LaTeX source

a=\frac12\left(D^{-1}V+DI\right),
\qquad
b=\frac12\left(D^{-1}V-DI\right),
\qquad
b=Sa.

The square root is the principal matrix square root, not an elementwise square root and not an arbitrary Cholesky factor. With every current entering the DUT, \(a^\dagger a-b^\dagger b=\operatorname{Re}(V^\dagger I)\).

Now apply a real nonsingular voltage-coordinate transform \(V_m=AV\) with its power-conjugate current map \(I_m=A^{-T}I\). The reference matrix and its principal root in the new coordinates are

\[ R_{0m}=AR_0A^T, \qquad D_m=R_{0m}^{1/2}. \]

LaTeX source

R_{0m}=AR_0A^T,
\qquad
D_m=R_{0m}^{1/2}.

Define the wave-coordinate map

\[ \boxed{Q=D_m^{-1}AD}. \]

LaTeX source

\boxed{Q=D_m^{-1}AD}.

\(Q\) is real orthogonal, and therefore

\[ a_m=Qa, \qquad b_m=Qb, \qquad \boxed{S_m=QSQ^T}. \]

LaTeX source

a_m=Qa,
\qquad
b_m=Qb,
\qquad
\boxed{S_m=QSQ^T}.

Here \(Q\) is a wave-coordinate map, not electrical charge. Transforming terminal voltages without transforming the dual currents and reference matrix does not preserve power. The physical construction and normalization of \(A\) are owned by Admittance Coordinate Transforms.

Complex references are a separate contract

For complex \(Z_0\), power, pseudo, and traveling waves are not interchangeable. Kurokawa power waves (Kurokawa 1965) require \(\operatorname{Re}Z_{0i}>0\) at every port and use

\[ a_i=\frac{V_i+Z_{0i}I_i}{2\sqrt{\operatorname{Re}Z_{0i}}}, \qquad b_i=\frac{V_i-Z_{0i}^{*}I_i}{2\sqrt{\operatorname{Re}Z_{0i}}}. \]

LaTeX source

a_i=\frac{V_i+Z_{0i}I_i}{2\sqrt{\operatorname{Re}Z_{0i}}},
\qquad
b_i=\frac{V_i-Z_{0i}^{*}I_i}{2\sqrt{\operatorname{Re}Z_{0i}}}.

Do not apply the real-\(Z_0\) equations unless the artifact declares a real positive reference. A complex-reference artifact must state its wave definition and conversion implementation. Complex references alone do not destroy reciprocity symmetry: for a reciprocal terminal matrix \(Z=Z^{\mathsf T}\) and consistently oriented Kurokawa waves with diagonal references satisfying \(\operatorname{Re}Z_{0i}>0\), the scattering matrix remains symmetric (Kurokawa 1965, sec. IV, Eqs. (21)–(23)). Other wave definitions need their own reciprocity relation.

Renormalization is not loading

The useful result is a separation of operations, not one preferred impedance. Wave renormalization changes coordinates (Kurokawa 1965); adding a circuit branch changes the object being described. This page’s comparison table organizes those meanings before the implementation-specific example.

Operation	What changes	What must remain explicit
Renormalize \(S\)	Wave coordinates and the numerical \(S\) matrix.	Original and new \(Z_0\), wave definition, and conversion.
Add a physical load	The network \(Y/Z\) and its response.	Load topology, value, and retained ownership.
Apply PTC	A proven explicit shunt is removed from \(Y\).	Removal intent, compiled row, value, and raw-versus-PTC evidence.
Move the reference plane	The embedded network seen at the port.	Plane locations and de-embedding model.
Convert \(S\leftrightarrow Y/Z\)	Representation of one declared network.	Parameter type, \(Z_0\), port order, wave definition, and numerical conditioning.

Port-Termination Compensation owns the conditional shunt-removal operation. Schur Complement and Kron Reduction owns port elimination under a declared boundary condition.

JosephsonCircuits photon-flux-to-power normalization

This solver-specific conversion preserves the earlier documented byte convention. Its starting normalization belongs to the version-bound solver source, not to Kurokawa’s RF power-wave definition. The diagonal-scaling algebra below is derived here and must not certify a later solver version without its own source evidence.

JosephsonCircuits stores solver-native HB amplitudes with a \(1/\sqrt{|\omega|}\) scaling. That byte convention must not be identified directly with a physical photon-flux amplitude. For the ordered input and output mode sets at nonzero signed physical frequencies, define

\[ W_{\rm in}=\operatorname{diag} \left(\sqrt{|\omega_{\rm in,\nu}|}\right), \qquad W_{\rm out}=\operatorname{diag} \left(\sqrt{|\omega_{\rm out,\mu}|}\right). \]

LaTeX source

W_{\rm in}=\operatorname{diag}
\left(\sqrt{|\omega_{\rm in,\nu}|}\right),
\qquad
W_{\rm out}=\operatorname{diag}
\left(\sqrt{|\omega_{\rm out,\mu}|}\right).

The exact solver-byte to classical power-wave matrix relation is

\[ \boxed{ S_{\rm pw}=W_{\rm out}S_{\rm JC}W_{\rm in}^{-1}. } \]

LaTeX source

\boxed{
S_{\rm pw}=W_{\rm out}S_{\rm JC}W_{\rm in}^{-1}.
}

For a square block with identical ordered input/output mode indices, this specializes to

\[ \boxed{ S_{\rm pw}=WS_{\rm JC}W^{-1}. } \]

LaTeX source

\boxed{
S_{\rm pw}=WS_{\rm JC}W^{-1}.
}

If a physical photon-flux amplitude is needed, define it separately as \(c_\gamma=c_{\rm JC}/\sqrt{\hbar}\). The factor \(\sqrt{\hbar}\) does not enter the byte-level matrix conversion above. Any selected \(\omega=0\) makes \(W^{-1}\) singular, so this conversion is undefined for that selection.

The same-index factor cancels for an entry whose input and output frequencies are equal. Cross-mode or frequency-converting \(S/Z\) must preserve:

input and output mode tuples;
signed physical frequencies;
solver version and mode ordering; and
the wave/frequency normalization used by the solver.

The absolute value in \(W\) supplies the positive scale; it does not erase the signed physical frequency carried by the mode label. Do not apply the same- mode reconstruction identity blindly to a signal-idler or other cross-frequency block. Even a diagonal mode block needs the full multi-mode inverse or an explicit Schur-complement boundary when off-diagonal mode coupling is nonzero. Use Harmonic Balance: Periodic Steady State and Mode Semantics for that boundary.

Imported and HFSS artifacts

Generalized waveguide-circuit theory distinguishes wave and terminal representations (Marks and Williams 1992). The checklist below is this page’s engineering import interpretation, not a claim that every export uses the same wave type.

There is no single safe “HFSS \(Y/Z\)” rule without the export contract. Before interpreting or converting an imported artifact, record:

solution type and modal or terminal basis;
generalized, pseudo-wave, power-wave, or renormalized \(S\) definition;
per-port reference or characteristic impedances;
reference-plane and de-embedding settings;
whether \(Y/Z\) is direct solver output or derived from \(S\); and
parameter type and reference metadata in the exchange file.

Until those fields are known, keep the artifact at Seed and do not compare it numerically with Workbench returned \(Z\) or derived \(Y\).

Limits of the conversions

The real-positive and real-SPD models do not silently cover arbitrary complex references. A singular conversion matrix can make a particular \(Y\) or \(Z\) representation unavailable even when another response view remains meaningful. Cross-frequency solver blocks require their own frequency normalization, and selected zero-frequency modes lie outside the displayed inverse-\(W\) map. None of these conversions decides which physical loads should be removed.

Decision procedure

flowchart TB
    A["Identify direct output family"] --> B["Inspect compiled or exported port elements"]
    B --> C["Record plane, order, Z0, and wave definition"]
    C --> D["Choose intrinsic, raw, loaded, or wave observable"]
    D --> E["Apply one declared conversion or PTC operation"]
    E --> F["Reconstruct a second view and record residual"]

Stop if any box cannot be answered from the artifact. A plausible resonance or notch is not substitute evidence.

Review checklist

Are intrinsic network, explicit shunts, wave reference, and reference plane named separately?
Is each \(S\), \(Y\), or \(Z\) labeled direct or derived?
Are port order, current direction, units, solver version, and manifest tree preserved?
Is \(Z_0\) recorded per port and, when needed, per frequency?
Is the wave definition real-positive power wave, complex power wave, pseudo-wave, or traveling wave?
Does PTC name the exact rows removed and the loads retained?
Are input/output HB modes and signed frequencies present for cross-mode data?
Does a reconstruction residual prove the chosen conversion?

Historical JosephsonCircuits and Workbench evidence

This section preserves the earlier recorded implementation evidence, not a universal microwave rule or current capability certification. The observation concerned these two JosephsonCircuits environments:

Execution surface	JosephsonCircuits version	Manifest tree
Julia Core project	`0.5.1`	`952c00edef77b1fc5cc3e85da4bb5081eb588b03`
Pluto `v1.12` environment	`0.5.2`	`e7834579316d9e50119b713bf44e9d90cf7db73c`

The recorded inspection found the same relevant boundary in both versions:

Every P branch must have exactly one resistor on the same branch; that resistor defines the solver port impedance.
The resistor enters the solved nodal conductance matrix.
returnZ forms port voltage divided by source current for the complete solved circuit.
returnS is constructed separately from Kurokawa power waves.
The solver does not return a direct \(Y\) family here. Workbench zero_mode_y_matrix derives \(Y_{\mathrm{raw}}\) by inverting returned \(Z\).

The Workbench compiler emits a port row and a colocated resistor for every ExternalPort. For real-positive, pump-off or verified mode-decoupled same-mode ports, the port-resistance objects are

\[ r_{\mathrm{port}}=(R_1,\ldots,R_N), \qquad R_{\mathrm{port}}=\operatorname{diag}(r_{\mathrm{port}}), \qquad D_{\mathrm{port}}=R_{\mathrm{port}}^{1/2}. \]

LaTeX source

r_{\mathrm{port}}=(R_1,\ldots,R_N),
\qquad
R_{\mathrm{port}}=\operatorname{diag}(r_{\mathrm{port}}),
\qquad
D_{\mathrm{port}}=R_{\mathrm{port}}^{1/2}.

Then the direct and derived quantities are

\[ Y_{\mathrm{raw}}=Z_{\mathrm{returned}}^{-1} =Y_{\mathrm{DUT}}+R_{\mathrm{port}}^{-1}, \]

LaTeX source

Y_{\mathrm{raw}}=Z_{\mathrm{returned}}^{-1}
=Y_{\mathrm{DUT}}+R_{\mathrm{port}}^{-1},

\[ Y_{\mathrm{DUT}} =Z_{\mathrm{returned}}^{-1}-R_{\mathrm{port}}^{-1}, \]

LaTeX source

Y_{\mathrm{DUT}}
=Z_{\mathrm{returned}}^{-1}-R_{\mathrm{port}}^{-1},

and

\[ S_{\mathrm{returned}} =2D_{\mathrm{port}}^{-1}Z_{\mathrm{returned}}D_{\mathrm{port}}^{-1}-I. \]

LaTeX source

S_{\mathrm{returned}}
=2D_{\mathrm{port}}^{-1}Z_{\mathrm{returned}}D_{\mathrm{port}}^{-1}-I.

An equivalent reconstruction is

\[ S_{\mathrm{returned}} =YtoS\!\left(Y_{\mathrm{DUT}}; \texttt{portimpedances}=r_{\mathrm{port}}\right). \]

LaTeX source

S_{\mathrm{returned}}
=YtoS\!\left(Y_{\mathrm{DUT}};
\texttt{portimpedances}=r_{\mathrm{port}}\right).

The removal step still needs the design declaration that the colocated resistors are solver scaffold for the selected DUT observable. A separate physical environment load remains in \(Y_{\mathrm{DUT}}\).

Two conversions that are wrong here

ZtoS(Z_returned; portimpedances=r_port) treats an already shunted raw \(Z\) as intrinsic \(Z\) and counts the port termination twice.
StoZ(S_returned) silently assumes the library default \(50\,\Omega\) at every port. It is wrong when the artifact has unequal or non-\(50\,\Omega\) references.

Unequal-port reconstruction probe

A pump-off, one-point two-port probe used \(R_1=25\,\Omega\), \(R_2=75\,\Omega\), a \(100\,\Omega\) link, \(200/300\,\Omega\) internal arms, and a \(400\,\Omega\) internal ground path. Both supported JosephsonCircuits versions returned the same matrices:

\[ Z_{\mathrm{returned}}= \begin{bmatrix} 21.0526 & 9.47368\\ 9.47368 & 43.2632 \end{bmatrix}\Omega, \]

LaTeX source

Z_{\mathrm{returned}}=
\begin{bmatrix}
21.0526 & 9.47368\\
9.47368 & 43.2632
\end{bmatrix}\Omega,

\[ S_{\mathrm{returned}}= \begin{bmatrix} 0.684211 & 0.437571\\ 0.437571 & 0.153684 \end{bmatrix}. \]

LaTeX source

S_{\mathrm{returned}}=
\begin{bmatrix}
0.684211 & 0.437571\\
0.437571 & 0.153684
\end{bmatrix}.

Check	Maximum discrepancy
Returned \(S\) versus raw-\(Z\) identity	\(3.89\times10^{-16}\)
Returned \(S\) versus PTC \(Y\rightarrow S\)	\(5.55\times10^{-16}\)
PTC \(Z\) versus \(S\rightarrow Z\) with \([25,75]\,\Omega\)	\(1.14\times10^{-13}\,\Omega\)
Wrong: standard \(Z_{\mathrm{returned}}\rightarrow S\)	\(0.788\)
Wrong: \(S\rightarrow Z\) with implicit \(50\,\Omega\)	\(533\,\Omega\)

The recorded wrong-path discrepancies expose a semantic distinction, not a negligible normalization choice. They are historical probe observations, not a new numerical acceptance threshold or a rerun of today’s package.

Historical Workbench product boundary

The unequal-port probe supports the adapter equations in its recorded setup. The following product limitations describe that snapshot only. It does not make every existing Workbench PTC trace artifact-backed:

compiled port_map currently preserves only logical port ID to solver index; the port node, R_port_<index> row, and resistance value remain distributed across netlist, node_map, and component_values rather than one structured port-evidence record;
the shared notebook helper now derives resistance only from exact compiled P/R_port branch evidence and requires an explicit removal intent, but that authorization record is still in memory rather than a first-class compiled port-evidence model, and the matrix stack does not itself carry the compiled artifact identity;
existing CSV trace artifacts do not persist the complete compiled authorization and raw-to-PTC lineage; and
the Product Runner currently publishes solver-native \(S\) but has no persisted PTC producer. New Product submissions with PTC enabled are rejected rather than being reported as available.

Until persisted lineage exists, only an in-memory result that passes the shared compiled-evidence helper may be promoted as local implementation evidence. Notebook 00 is the worked real-solver example.

Connections and references

From Node and Line Flux to Input–Output Scattering derives the real-positive traveling-wave and photon-flux boundary used by the fundamental circuit model; this page remains the authority for solver wave definitions, renormalization, and complex reference impedances.
Network Trace Views selects the useful representation after its semantics are known.
Ideal Parallel LC Resonator provides the simplest raw/PTC/\(S_{11}\) worked example.
Power Waves and the Scattering Matrix records the primary wave-definition source.
K. Kurokawa, “Power Waves and the Scattering Matrix,” doi:10.1109/TMTT.1965.1125964.
R. B. Marks and D. F. Williams, “A General Waveguide Circuit Theory,” NIST Journal of Research 97 (1992).
Touchstone 2.1 specification.
JosephsonCircuits v0.5.1 source.

Artifact Contract

This retained engineering record states how the earlier workflow documented its wave and solver choices. It introduces no universal reconstruction tolerance or new scientific Gate; executable schemas remain package-owned.

Required field	Meaning
Solver identity	Name, version, manifest tree or immutable revision.
Direct output	Exact solver-returned family before inversion or conversion.
Port definition	Order, terminal pair, current orientation, and reference plane.
Wave contract	Wave definition and per-port/per-frequency \(Z_0\).
Matrix-reference transform	Ordered coordinate map \(A\) and complete \(R_0\), \(R_{0m}\), \(D\), \(D_m\), and \(Q\) matrices with units and coordinate labels.
Matrix validation	Real-symmetry/SPD evidence; principal-root reconstruction, \(Q\) orthogonality, and voltage-current power-pairing residuals. Residuals are reported measurements, not a new threshold or Gate.
Explicit elements	Port shunt/load rows, values, and physical-versus-scaffold ownership.
Compensation authorization	Exact rows selected for removal, retained loads, design intent, and proof that the helper input matches the compiled artifact.
JosephsonCircuits mode contract	Ordered input/output mode tuples, signed physical frequencies, \(W_{\mathrm{in}}\), \(W_{\mathrm{out}}\), solver-native and power-wave matrix identities, and explicit exclusion of every selected zero-frequency mode.
Derived quantity	Formula, library/version, conditioning result, and target semantic layer.
Reconstruction	Independently reconstructed view and maximum residual.
Failure record	Visible failure identity for non-finite or non-SPD references, failed principal-root reconstruction or orthogonality, incompatible ordering, or a selected \(\omega=0\) in the JosephsonCircuits conversion.

Review Links

Field	Value
Status	Artifact-backed
Physics evidence	Starting wave definitions are supported by (Kurokawa 1965; Marks and Williams 1992); source context remains in Power Waves and the Scattering Matrix, Marks and Williams, and Touchstone 2.1.
Solver evidence	JosephsonCircuits `0.5.1` tree `952c00ed...` and `0.5.2` tree `e7834579...`; tagged power-wave construction, port-resistor discovery, and local unequal-port probes.
Used by	Ideal Parallel LC Resonator.
Implementation links	Workbench compiler, HB extraction helpers, PTC matrix helpers, and Product Runner publisher.
Historical product gap	At the recorded snapshot, the shared helper assembles evidence from current compiled fields, but a first-class compiled-port evidence record and persisted, evidence-authorized PTC producer do not yet exist end to end.
Human review gate	Can a reviewer explain why `inv(Z_returned)` is raw \(Y\), why `StoY(S_returned; portimpedances=r_port)` is DUT \(Y\), and why `ZtoS(Z_returned; portimpedances=r_port)` double-counts the same port termination?

Bibliography

Kurokawa, K. 1965. “Power Waves and the Scattering Matrix.” IEEE Transactions on Microwave Theory and Techniques 13 (2): 194–202. https://doi.org/10.1109/TMTT.1965.1125964.

Marks, Roger B., and Dylan F. Williams. 1992. “A General Waveguide Circuit Theory.” Journal of Research of the National Institute of Standards and Technology 97 (5): 533–62. https://doi.org/10.6028/jres.097.024.