Network Trace Views and Extraction Authority
Node role: Procedure.
Reader task: Start from one declared network artifact, choose an \(S/Y/Z\) view, and decide which extracted quantities are eligible for promotion.
Why it matters
An external FEM result is not just a plot. A network exchange format such as Touchstone carries frequency-indexed parameter data and reference metadata (IBIS, n.d.); the particular solver/export contract still determines which terminal or wave model was exported. It is a frequency-indexed network model with port order, reference impedance, units, and provenance. Equivalent-circuit modeling starts by choosing the view that makes the physics visible.
Physical basis
First identify the same physical experiment in three representations. Power-wave scattering depends on wave references (Kurokawa 1965), while terminal admittance and impedance describe voltage/current response. The comparisons below reorganize that distinction; the internal-susceptibility substitution is this page’s use of the coupled-mode model (Suh et al. 2004).
The model is linear and its terminal basis, loads, and reference planes are declared. A frequency-converting response needs mode indices in addition to port indices. Choose the view only after those assumptions are explicit.
This engineering page consumes, but does not redefine, the physical boundary: Physical Circuit Coordinates and Open-EOM Reduction defines the coordinate/open-operator construction, and Ports, Waves, Reference Planes, and Internal-Basis Invariance defines the port maps and wave convention. This page adds only the choice and promotion meaning of a declared terminal \(S/Y/Z\) view.
One physical network, three terminal representations
| View | Useful for | Main risk |
|---|---|---|
| S matrix | Measurement/FEM exchange, port matching, and SAX-style downstream use. | Reference-impedance and port-order mistakes. |
| Z matrix | Impedance peaks, series behavior, and differential/common-mode reductions. | Ill-conditioned conversion near singular responses. |
| Y matrix | Shunt branches, capacitance/conductance behavior, and admittance poles and zeros. | Unstable interpretation when the S-to-Y conversion assumptions are wrong. |
Default S/Y/Z Reporting Convention
This is a retained site/workflow reporting choice, not a definition imposed by microwave physics or the cited wave theory. It separates a wave-reference operation from removal of proven explicit solver shunts. The algebraic PTC operation itself is derived in its linked node.
SCQ Design uses one solver-independent reporting convention, aligned with the ANSYS HFSS distinction between wave response and terminal-matrix post-processing:
| Unqualified symbol | Meaning |
|---|---|
| \(\mathbf S\) | Solver wave response at the declared reference planes and reference impedances. No PTC is implied. |
| \(\mathbf Y\), \(\mathbf Z\) | Design-facing terminal matrices after evidence-authorized PTC. |
| \(\mathbf Y^{\mathrm{raw}}\), \(\mathbf Z^{\mathrm{raw}}\) | Matrices before PTC, including the declared solver/probe termination scaffold. |
This is an output convention, not permission to subtract a port impedance. Every solver adapter must first establish whether its returned terminal matrix contains an explicit shunt and must satisfy the PTC authorization contract before publishing unqualified \(\mathbf Y\) or \(\mathbf Z\). A deliberately uncompensated terminal matrix must carry the raw qualifier. A compensated \(\mathbf S\) is non-default and must be labeled explicitly with its conversion path and reference impedance.
Changing among valid \(S\), \(Y\), and \(Z\) representations does not change the physical network, its loads, or its reference planes. It changes the terminal variables used to expose that network. An internal coupled-mode susceptibility \(\boldsymbol\chi=\mathbf M^{-1}\) becomes an \(S\) matrix only after direct propagation and port projections are applied:
\[ \mathbf S =\mathbf S_{\mathrm{dir}} +\mathbf D_{\rm port}\boldsymbol\chi\mathbf K_{\rm port}. \]
Likewise, a reduced driving-point \(Y\) or transfer \(Z\) requires its own terminal coordinate and boundary derivation. None is automatically one entry of \(\boldsymbol\chi\) or \(\boldsymbol\chi^{-1}\).
Engineering use
Choose the domain that makes the modeling question easiest to inspect. Do not fit blindly in only one domain: a candidate should explain the response in the most revealing domain and remain plausible after conversion.
Preserve this metadata before modeling:
- frequency axis and unit;
- complex trace convention;
- port names and port order;
- reference impedance when S-parameters are involved;
- upstream solver/export provenance; and
- known symmetry, reciprocity, or grounding assumptions.
From a declared circuit to an eligible parameter
A response feature and a circuit coefficient answer different questions. The observable is obtained by solving and projecting the declared model; recovering an internal coefficient additionally requires that model’s structure. The detailed promotion vocabulary in this section preserves the earlier engineering workflow. It is not a general rational-fitting policy; generic Vector Fitting and Passivity distinguishes data approximation from physical interpretation without numeric eligibility rules.
Two authorities must remain separate:
- a declared coupling-on circuit and its coordinate map define its bare-coordinate Hamiltonian coefficients; and
- the complex \(S\)-, \(Y\)-, or \(Z\)-matrix response of that same topology and coupling state validates or identifies those coefficients and owns its observable open-response quantities.
Bare-coordinate frequencies and coherent couplings may come from declared circuit matrices or from a topology-constrained physical-response fit when some circuit values are unknown. Either route must close against the relevant complex \(S/Y/Z\) response before promotion. Open-system pole frequencies, linewidths, transfer zeros, and port quantities belong to the response layer and must be extracted from that declared response.
The topology labels are
\[ \mathcal T\equiv\text{the declared physical topology}, \qquad \mathcal T'\equiv\text{a modified reference topology}. \]
\(\mathcal T'\) may provide initial estimates, bounds, or continuation evidence, but it does not own parameters of \(\mathcal T\). Geometry, element values, capacitance/stiffness matrices, and labeled RLGC matrices are valid physical model inputs. A closed-circuit generalized eigenfrequency is nevertheless a conservative normal-mode result, not an open-response pole or linewidth.
A scalar Vector Fitting pole may own an eligible visible response-pole frequency and total linewidth. The fixture and pole/residue continuation—not the pole location alone—own mode identity. Vector fitting does not identify bare-coordinate diagonals, \(g\), \(J\), \(Q_i/Q_c\), channel decomposition, or design-objective membership from a scalar trace. It can represent zeros of that rational projection, but interpreting an intrinsic physical transfer zero requires the named observable and boundary below. The generic Vector Fitting and Passivity now explains rational approximation and same-data physical-model comparison; it is not a current authority for the earlier workflow’s promotion policy.
Within that earlier workflow, every promoted artifact names its circuit and topology, coordinate or mode layer, observable and port basis, retained loads and terminations, estimator, units, phasor convention, residuals, and provenance.
When a declared model is lossless, missing \(R\) and \(G\) must be recorded as unavailable and explicitly assumed zero, not presented as extracted zero matrices. Only when that model is the complete loss model may it establish \(\kappa_{\mathrm{int}}=0\). If one proven external path is the sole remaining decay channel, topology may establish equality between total and external linewidth. That equality is topology evidence, not a channel decomposition performed by Vector Fitting, and it does not identify left/right rates.
Canonical Differential-Admittance Pipeline
The useful result is a driving-point admittance in a declared differential coordinate, with unwanted external injections set to zero. Each step answers a different question: compensation chooses retained loads, transformation chooses variables, and reduction chooses boundary conditions. This is an engineering ordering built from the linked algebra, not a further source theorem.
A floating differential-port observable is constructed in this order:
raw solver port-basis network
-> Y in the raw port basis
-> evidence-authorized PTC
-> real power-conjugate coordinate transformation
-> zero-external-current Kron reduction
-> retained differential driving-point YPTC removes only a compiled shunt that is both proven to exist and explicitly declared to be probe scaffold. A port reference impedance is not automatically such a shunt. Physical feedline terminations and environment loads remain in the Circuit Model; their coordinates are eliminated by the declared network boundary, not compensated away.
The mathematical steps are defined once elsewhere. Use the power-conjugate voltage/current transform, then apply the zero-injection Kron boundary. This page fixes their physical order and promotion meaning; it does not define another transform or Schur complement. A qubit frequency extracted from this pipeline is a linearized differential resonance, never \(f_{01}\) unless a separate quantization contract establishes that mapping.
This paragraph states the intended retained-view meaning, not a certification of a current executable package. Version-bound APIs and execution evidence remain in the implementing repository.
At the Knowledge layer, an SCNSim result is a labeled retained view of one declared circuit. The runtime starts from a source-backed terminal network, applies evidence-authorized PTC, constructs the declared physical coordinates, carries the dual current and wave-reference normalization, and eliminates unretained coordinates under the declared boundary. Direct or HB analysis then reports the response in that retained view.
The result is not a second circuit, a universal floating-node weighting rule, or a native \(S\) submatrix relabeled as Kron reduction. Use the full-external-cut coordinate rule, the matrix-reference wave contract, and the zero-current versus matched-wave boundary for those meanings. SCNSim method names, request fields, receipts, and result accessors remain product documentation rather than Knowledge authority.
The real-axis root contract is
\[ \operatorname{Im}Y_{\mathrm{diff}}(2\pi f_{q,\mathrm{lin}})=0. \]
Root eligibility requires mode continuation, the convention-aware crossing slope, \(\operatorname{Re}Y_{\mathrm{diff}}\) at the root, the complete candidate-root list, neighboring poles or singular samples, and the selection anchor. A nearest-zero fallback is not eligible. In a multi-view topology-constrained fit, this root is an initial q-like branch marker. When a complete forward observation map is defined and the optional second view is included, the final parameter fit also consumes the local complex \(Y_{\mathrm{diff}}\) trace.
Probe And Physical Loading
The following probe-sweep prescription is retained as an earlier engineering workflow, not a universal sampling minimum. Its artificial observation hook and physical environment must remain distinguishable in every interpretation.
\(C_{\mathrm{probe}}\) is an artificial observation hook used to expose an otherwise dark resonator in feedline \(S_{21}\). Physical filter-to-feedline loading is a layout-owned component. For an interdigitated capacitor it is generally one geometry-linked block containing the mutual capacitance and both terminal-to-ground capacitances. A scalar \(C_{\mathrm{ext}}\) is allowed only as an explicitly declared special case of that block. Observation and physical loading may occupy a similar circuit location, but must never share a field name or semantic role. See Localized Electrostatic Components.
An isolated dark resonator is evaluated at no fewer than five distinct, strictly positive \(C_{\mathrm{probe}}\) values. Each trace receives its own eligible scalar Vector Fitting pole extraction or topology-specific complex- \(S_{21}\) fit, with pole ownership and fit-quality evidence, and the fitted frequency is extrapolated to \(C_{\mathrm{probe}}\to0^+\). A failed point, repeated value, or untracked mode rejects the extraction. In the lossless LC v1, a readout whose probe is its only decay path also satisfies \(\kappa_{r,\mathrm{tot,off\text{-}ref}}(C_{\mathrm{probe}}=0)=0\).
The earlier probe workflow additionally recorded scalar-pole resolution, grid-refinement, stability, continuation, nearby-pole ownership, and complex-residual evidence. These are retained engineering choices here, not a second public vector-fitting method or universal numerical rules.
Complex Transfer-Zero Contract
This page’s definition is a zero of a named complex observable. Inspecting only a signed imaginary crossing can locate a candidate but cannot show cancellation of the whole complex response. The promotion terminology below belongs to the retained workflow, not a new generic fit Gate.
A declared intrinsic transfer notch is the real-axis complex zero
\[ Z_{21}^{\mathrm{PTC}}(2\pi f_n)=0. \]
The response must be defined and finite at that frequency. A zero numerator with a nonsingular denominator defines a regular transfer zero; coincident numerator/denominator zeros require cancellation or an explicit analytic-limit calculation, not a numerator crossing alone. A rational fit’s zero is first a feature of that fitted projection, not automatic identification of this intrinsic network zero.
The signed \(\operatorname{Im}Z_{21}^{\mathrm{PTC}}\) crossing locates candidate roots. \(|\operatorname{Im}Z_{21}^{\mathrm{PTC}}|\) is only a plotting or sampled-residual view. Promotion also gates \(|\operatorname{Re}Z_{21}^{\mathrm{PTC}}|\), \(|\operatorname{Im}Z_{21}^{\mathrm{PTC}}|\), and \(|Z_{21}^{\mathrm{PTC}}|\) at the refined root, with continuation and complete candidate-root evidence. This intrinsic transfer zero is not claimed to be a feedline-\(S_{21}\) notch.
Review checklist
- Is the frequency axis present, with its unit?
- Is the complex trace convention explicit?
- Are port names and port order preserved?
- Is the reference impedance recorded for S-parameters?
- Can the trace be traced back to its solver or export?
- Are symmetry, reciprocity, and grounding assumptions stated where known?
- Does the candidate remain plausible after conversion to another useful view?
- Does every promoted quantity name its Circuit Model, mode layer, observable, estimator, convention, residual, and provenance?
- Are
C_probeand the complete physical loading component kept distinct? - Does one layout geometry update every capacitance entry owned by that local component?
- For parameters identified by a controlled sweep, does one unchanged topology report the pointwise circuit-to-Hamiltonian map, all complex-response residuals, continuation, stability, identifiability, and deltas from every modified-topology initializer?
- When a topology-constrained fit and VF use the same response, are pole branches paired by continuation, frequency and total-linewidth deltas reported, and every dark-pole exception justified by a projected residue?
- Does every promoted response pole publish its step, linewidth, samples-per-linewidth, refinement shifts, and declared convergence tolerances?
Limits and failure modes
- S-matrix interpretation can fail when reference impedance or port order is wrong.
- Z-matrix conversion can become ill-conditioned near singular responses.
- Y-matrix interpretation is unstable when the S-to-Y assumptions are wrong.
- A fit that looks useful in one domain is not enough by itself.
- A generalized eigenfrequency owns a conservative normal-mode result, but it cannot replace the estimator when the claimed quantity is an open pole or linewidth. A scalar Vector Fitting pole is eligible only for response-pole frequency and total linewidth under the isolation, resolution, stability, continuation, and residual gates above.
Connections
- From Node and Line Flux to Input–Output Scattering constructs exact open-circuit \(S/Y/Z\) from a physical node/line boundary before any modal or Markov reduction.
- Multimode Input–Output Scattering derives how internal susceptibility, direct propagation, and port coupling become a scattering matrix.
- Port Reference Impedance Semantics and Port-Termination Compensation define which raw or compensated network the trace represents.
- Admittance Coordinate Transforms turns a port-basis \(Y\) matrix into declared physical-mode coordinates.
- Schur Complement and Kron Reduction constructs the effective network seen at retained ports or modes.
- Poles, Zeros, and Residues turns trace features into evidence for an equivalent-circuit family.
- Vector Fitting and Passivity explains the rational response model and the limits of scalar versus full-network data.
- The Circuit Workbench External FEM Result Contract records the upstream network-result boundary.
References
- Touchstone 2.1 specification
- scikit-rf Networks tutorial
- SAX documentation
- External FEM Result Contract
| Field | Value |
|---|---|
| Status | Seed |
| Implementation links | Candidate audit entry points only; this Seed does not verify their behavior: S-parameter fitting module, Y11 fitting module. |