Admittance Coordinate Transforms

How voltage coordinates, power-conjugate currents, and admittance matrices change between port and physical-mode bases.

Node role: Method.

Reader task: Express one already-declared terminal network in new voltage coordinates while preserving the dual current and power pairing.

Why it matters

A floating or differential circuit coordinate is usually not identical to one solver port. The circuit topology must first define which voltage combination is physically meaningful; this page owns only the mathematical representation change that expresses the same terminal network in that declared coordinate. It does not choose the coordinate or change the circuit. The upstream basis construction belongs to Circuit Models to Bare Coordinates, Open EOM, and Normal Modes.

The normalization changes the numerical value of a modal admittance, so a result such as \(Y_{dm}\) is not auditable unless both its voltage and power-conjugate current definitions are stated.

The coordinate contract

The model is a reversible change of terminal variables, not elimination. The same physical power must be expressible in either set of variables: substitute \(\mathbf v=A^{-1}\mathbf v_m\) into \(\mathbf v^\dagger\mathbf i\), then read off the dual current. The congruence below is this page’s derivation from that pairing. Mixed-mode network analysis is related background (Bockelman and Eisenstadt 1995), not a source for the particular external-cut weighting chosen later.

Start from a port-basis relation

\[ \mathbf i = \mathbf Y\mathbf v. \]

For a real, nonsingular \(A\), the voltage-coordinate map is

\[ \mathbf v_m=A\mathbf v. \]

Preserving the voltage-current power pairing requires

\[ \mathbf i_m=A^{-T}\mathbf i. \]

Therefore

\[ \boxed{\mathbf Y_m=A^{-T}\mathbf Y A^{-1}}. \]

For a genuinely complex transformation under the usual complex-power pairing, replace \(A^{-T}\) with \(A^{-H}\). A real-transform implementation must reject a genuinely complex \(A\) rather than silently apply the wrong dual-current map.

Floating-pair coordinates from the full external cut

Why not always average the two voltages? In an asymmetric pair the exterior capacitance weights differ. The construction below chooses a common voltage that removes the common/differential cross term of the isolated pair block. It is a declared coordinate choice followed by this page’s matrix algebra, not a universal floating-circuit prescription. The full matrix can still couple either new coordinate to exterior conductors.

For a declared conductor pair \((1,2)\), let \(E\) contain every conductor outside the pair, including the fixed reference conductor when it is represented by a branch capacitance. Write \(C_{je}^{(b)}\geq0\) for the physical branch capacitance between pair conductor \(j\) and exterior conductor \(e\). The two full-external-cut sums are

\[ c_1^{\mathrm{cut}}=\sum_{e\in E}C_{1e}^{(b)}, \qquad c_2^{\mathrm{cut}}=\sum_{e\in E}C_{2e}^{(b)}, \qquad C_{\mathrm{cut}}=c_1^{\mathrm{cut}}+c_2^{\mathrm{cut}}>0. \]

The mutual branch \(C_{12}^{(b)}\) lies inside the selected pair and therefore does not enter either cut sum. It remains present in the complete capacitance matrix and in every transformed result.

The external-cut weighting is

\[ \alpha=\frac{c_1^{\mathrm{cut}}}{C_{\mathrm{cut}}}, \qquad \beta=\frac{c_2^{\mathrm{cut}}}{C_{\mathrm{cut}}}, \qquad \alpha+\beta=1. \]

The corresponding pair transform is

\[ A_p= \begin{bmatrix} \alpha & \beta \\ 1 & -1 \end{bmatrix}, \qquad A_p^{-1}= \begin{bmatrix} 1 & \beta \\ 1 & -\alpha \end{bmatrix}, \qquad A_p^{-T}= \begin{bmatrix} 1 & 1 \\ \beta & -\alpha \end{bmatrix}. \]

The transformed common and differential voltages are

\[ V_{cm}=\alpha V_1+\beta V_2, \qquad V_{dm}=V_1-V_2. \]

The power-conjugate currents are then

\[ I_{cm}=I_1+I_2, \qquad I_{dm}=\beta I_1-\alpha I_2. \]

For the symmetric choice \(\alpha=\beta=1/2\), this gives \(I_{dm}=(I_1-I_2)/2\). Choosing a different scaling for \(V_{dm}\) requires the reciprocal change in \(I_{dm}\) and changes the numerical \(Y_{dm,dm}\).

For a complete dynamic coordinate order \((1,2,e_1,\ldots,e_m)\), leave every exterior dynamic coordinate unchanged with

\[ \mathcal A=A_p\oplus I_m. \]

The same dual map applies to every quadratic terminal matrix. In particular,

\[ \boxed{ \mathbf C_m=\mathcal A^{-T}\mathbf C\mathcal A^{-1}, \qquad \mathbf Y_m=\mathcal A^{-T}\mathbf Y\mathcal A^{-1}. } \]

For the isolated pair block

\[ C_{pp}= \begin{bmatrix} C_{12}^{(b)}+c_1^{\mathrm{cut}} & -C_{12}^{(b)} \\ -C_{12}^{(b)} & C_{12}^{(b)}+c_2^{\mathrm{cut}} \end{bmatrix}, \]

Multiplying the pair block by the two maps gives the exact identity

\[ \boxed{ A_p^{-T}C_{pp}A_p^{-1} =\operatorname{diag}\!\left( C_{\mathrm{cut}}, C_{12}^{(b)}+ \frac{c_1^{\mathrm{cut}}c_2^{\mathrm{cut}}}{C_{\mathrm{cut}}} \right). } \]

This diagonalizes the isolated pair block only. Couplings between the generated coordinates and exterior dynamic coordinates may remain in the complete transformed matrix.

This is a coordinate construction for one declared pair and one declared external cut. Equal weights follow only from equal cut sums. A different topology, retained conductor set, reference choice, or differential-voltage scaling generally changes the weights; \(1/2,1/2\) is not a universal floating-node rule. If \(C_{\mathrm{cut}}=0\), the external-cut rule does not select a common coordinate and an implementation must not invent one.

Terminations and operation order

Use the transformed matrix to compute currents in the new coordinates, not to infer a new physical circuit. In particular, a diagonal load in the old coordinates need not be diagonal in the new ones. This page’s algebra shows why load subtraction and coordinate transformation commute only when the load is transformed too.

Suppose a known shunt matrix \(D\) is to be removed in the original port basis. Subtraction and a correctly transformed coordinate change are algebraically compatible:

\[ A^{-T}(Y-D)A^{-1} =Y_m-A^{-T}DA^{-1}. \]

The error is not performing the coordinate transform first. The error is subtracting the original, usually diagonal \(D\) after the basis has changed without transforming \(D\) as well. In practice, compensation is often done in the port basis because that is where each physical or artificial shunt is identified.

  • Is \(A\) written down and nonsingular?
  • Are both voltage and power-conjugate current coordinates defined?
  • Is the common/differential normalization explicit?
  • Is the transform real, or is a complex-power convention declared?
  • Are load or compensation matrices transformed with the network matrix?
  • Are port order, units, and reference semantics preserved?

The 2026-07-10 Workbench evidence recorded below concerned a shared helper applying this operation to a frequency-indexed admittance artifact. Its focused test uses unequal \(\alpha,\beta\), complex port voltages and currents, and verifies both

\[ \mathbf i_m=\mathbf Y_m\mathbf v_m \]

and preservation of the complex-power pairing. A separate negative test gives the helper a genuinely complex \(A\) and requires a visible failure. The floating LC Notebook 02 then records the exact real \(A\), labels, weights, quantity domain, and source lineage used on a real solver result.

This historical evidence supports the reported real-transform mechanics at that snapshot; it is not current package-capability certification. It does not approve the notebook’s capacitance-derived weights as a project-wide floating-qubit normalization.

Limits and failure modes

  • Transforming only voltages and reusing the old currents breaks the network relation and power pairing.
  • A singular \(A\) does not define a reversible coordinate change.
  • Applying \(A^{-T}\) to an arbitrary complex transform is not the usual complex-power-preserving operation.
  • A direct transform of solver \(S\)-parameters needs wave and reference- impedance normalization; the admittance formula above does not silently provide that contract.
  • An automatically chosen \(\alpha,\beta\) is a model assumption and needs its own physical or artifact evidence.
  • Missing, ambiguous, non-finite, or negative compiled branch data; a non-positive cut sum; or nonreciprocity outside the declared model scope must remain a visible failure rather than producing fallback weights.

Connections

This retained engineering record specifies how the earlier workflow made its coordinate choice auditable. It is not a physical law or a new universal eligibility rule; child-package documentation owns current executable fields.

Required field Meaning
Pair declaration Ordered conductor/port pair, fixed-reference treatment, voltage definitions, and current orientations.
Branch provenance Source identity and units for the reciprocal compiled capacitance branches, with every included exterior branch and excluded direct-mutual branch named.
Cut and maps \(c_1^{\mathrm{cut}}\), \(c_2^{\mathrm{cut}}\), \(C_{\mathrm{cut}}\), \(\alpha\), \(\beta\), \(A_p\), \(A_p^{-T}\), and the complete-coordinate map \(\mathcal A\).
Input quantities Original ordered \(\mathbf C\) and/or \(\mathbf Y\), units, frequency axis where applicable, and source artifact identity.
Output quantities Transformed ordered \(\mathbf C_m\) and/or \(\mathbf Y_m\), output labels, and the physical meaning and normalization of every coordinate.
Retained loading Every load or compensation matrix present before the transform and the basis in which it was applied.
Validation Branch reciprocity evidence; transform reconstruction, network-relation, and power-pairing residuals; and explicit rejection evidence for an unsupported complex \(A\). Residuals are reported measurements, not a new threshold or Gate.
Failure record Visible failure identity for incomplete, ambiguous, non-finite, or negative branch data; \(C_{\mathrm{cut}}\leq0\); out-of-scope nonreciprocity; or a singular/non-finite map.
Lineage Input quantity and source kind, transform operation, output quantity and source kind, and implementation revision or working-tree state.

References

  • D. E. Bockelman and W. R. Eisenstadt, “Combined Differential and Common-Mode Scattering Parameters: Theory and Simulation,” IEEE Transactions on Microwave Theory and Techniques 43(7), 1530–1539 (1995), doi:10.1109/22.392911.
Field Value
Status Artifact-backed
Semantic role Define the voltage/current dual-coordinate contract for a real basis change of admittance data.
Used by Schur Complement and Kron Reduction, Qubit Charging-Energy Targeting and Local-System Reduction, and Network Trace Views.
Workbench evidence The shared helper declares an admittance quantity domain, rejects unsupported complex transforms, and preserves quantity and source lineage. Its focused test covers asymmetric weights, the dual-current relation, power pairing, and the complex-transform failure boundary; Notebook 02 publishes the exact live-workflow transform contract.
Validation record On 2026-07-10, julia --startup-file=no scripts/test/test_pluto_port_matrix_post_processing.jl passed 52/52 focused checks, and a top-to-bottom include of Notebook 02 passed all 14 published sanity gates. These were working-tree results at that date, not immutable released artifacts or a current rerun.
Implementation links Real-valued coordinate implementation, focused helper tests, and floating LC executable workflow.
Known implementation gap PortMatrixStack preserves the quantity and source kind but does not yet persist the exact \(A\), coordinate definitions, or source-artifact identity; Notebook 02 therefore publishes those fields beside the result. A Human reviewer must also decide whether a generic real \(A\) is acceptably conditioned for its intended use.
Human review gate Can a reviewer reconstruct both voltage and current coordinates from the published \(A\), verify the power pairing, explain why complex \(A\) is rejected, and separately decide whether the local differential normalization should become project-wide?

Bibliography

Bockelman, D. E., and W. R. Eisenstadt. 1995. “Combined Differential and Common-Mode Scattering Parameters: Theory and Simulation.” IEEE Transactions on Microwave Theory and Techniques 43 (7): 1530–39. https://doi.org/10.1109/22.392911.