Multiconductor RLGC Matrices: Basis, Modes, and Physical Meaning
Node role: Concept.
Reader task: determine what an RLGC or lossless LC matrix means before a procedure uses it as a modal or circuit input.
A multiconductor line is not defined by four unlabeled arrays. The numbers only become an electrical model after the conductor order, reference conductor, voltage and current directions, matrix representation, units, frequency, and provenance are fixed. Losing any one of those declarations can produce a plausible circuit with the wrong coupling.
This node is the review boundary between a field extractor such as Q2D and a distributed circuit model. It explains the reusable physics; producer schemas, consumer APIs, and notebook workflows remain in their owning repositories.
The conductor geometry, reference conductor, terminal fields, and stored or dissipated energy are physical. Terminal/modal basis matrices, eigenvectors, and matrix factorizations are mathematical representations of that same line; they acquire physical mode meaning only after voltage/current pairing, power normalization, and propagation direction are fixed.
The inherited selected-source handoff description records an OrPen-to-Workbench path carrying only \(L\) and \(C\), with a Workbench coupled-line type named MTLCoupledRLGCSpec whose described matrix model stores no \(R\) or \(G\) matrices. This is a scoped handoff description, not certification of today’s producer or consumer capabilities. Establish executable support from the owning repositories’ selected-version resources. Missing loss matrices must not silently mean measured zero loss.
The Human-selected v1 contract is an explicitly lossless/quasi-static LC matrix artifact. Its quantity-availability record must state that \(R/G\) are unavailable and assumed_zero_for_v1; those zeros are modeling assumptions, not extracted evidence. A true four-matrix RLGC artifact is later scope and requires a new producer/consumer contract.
Reading map
Start with paired terminal voltages and currents; propagate that basis through the wave eigenproblem. Then distinguish Maxwell branch lowering from series inductance, and use even/odd modes only for the symmetric special case. The handoff checklist records model meaning; it does not make a source count or solver label into proof of accuracy.
1 — Declare the terminal basis
Reorganized explanation. Multiconductor line equations act on paired terminal-voltage and current vectors (Reibiger 2003). General modal normalization requires preserving their physical power relation (Barth and Iyer 2017).
Choose \(N\) non-reference conductors and one declared reference conductor or reference-conductor group. In a terminal basis,
\[ \mathbf V = \begin{bmatrix}V_1&\cdots&V_N\end{bmatrix}^{\mathsf T}, \qquad V_i = V(\text{conductor }i)-V(\text{reference}), \]
and every \(I_i\) has a declared positive direction along the propagation axis \(+z\). With the canonical \(e^{-i\omega t}\) convention from Dynamics Conventions and the Flux-to-Mode Symbol Bridge, the multiconductor telegrapher equations are
\[ -\frac{\partial \mathbf V}{\partial z} = \underbrace{(\mathbf R-i\omega\mathbf L)}_{\mathbf Z'}\mathbf I, \qquad -\frac{\partial \mathbf I}{\partial z} = \underbrace{(\mathbf G-i\omega\mathbf C)}_{\mathbf Y'}\mathbf V. \]
All four matrices are per-unit-length quantities in the same ordered basis.
| Matrix | Meaning | SI unit |
|---|---|---|
| \(\mathbf R\) | series conductor-loss matrix | \(\Omega/\mathrm m\) |
| \(\mathbf L\) | series inductance matrix | \(\mathrm H/\mathrm m\) |
| \(\mathbf G\) | shunt dielectric/conductive-loss matrix | \(\mathrm S/\mathrm m\) |
| \(\mathbf C\) | shunt capacitance matrix | \(\mathrm F/\mathrm m\) |
The matrices may depend on frequency. An extraction frequency is therefore part of the value’s identity, not optional notebook decoration.
What the matrices predict
This-page derivation. Differentiate the uniform-line matrix equations. The voltage and current products have different ordering; an elementwise scalar square root does not solve either eigenproblem.
Differentiating once more gives
\[ \frac{\partial^2\mathbf V}{\partial z^2} = \mathbf Z'\mathbf Y'\mathbf V, \qquad \frac{\partial^2\mathbf I}{\partial z^2} = \mathbf Y'\mathbf Z'\mathbf I. \]
The modal propagation constants satisfy an eigenproblem such as
\[ \mathbf Z'\mathbf Y'\,\mathbf t_{V,m} = \gamma_m^2\mathbf t_{V,m}. \]
For a general asymmetric or lossy line, voltage and current transformations must be paired and normalized so terminal-domain and modal-domain power agree; Admittance Coordinate Transforms states the same power-conjugate rule for terminal admittance data. Modal characteristic impedance is therefore not obtained by taking an elementwise square root of \(\mathbf L/\mathbf C\).
Only after a mode has been defined and its scalar \(R_m,L_m,G_m,C_m\) are valid may one use
\[ \gamma_m=\sqrt{(R_m-i\omega L_m)(G_m-i\omega C_m)}, \qquad Z_{c,m}=\sqrt{\frac{R_m-i\omega L_m}{G_m-i\omega C_m}}. \]
Choose the passive forward-wave branch
\[ \gamma_m=\alpha_m-i\beta_m, \qquad e^{-\gamma_m z}=e^{-\alpha_m z}e^{+i\beta_m z}, \qquad \alpha_m\ge0,\quad\beta_m=-\operatorname{Im}\gamma_m\ge0. \]
An unlabeled matrix entry such as \(\sqrt{L_{12}/(-C_{12})}\) is a heuristic ratio, not the general definition of a multiconductor modal impedance.
2 — Interpret Maxwell matrices
Reorganized explanation. The Maxwell matrix is the charge–potential coefficient matrix (Reibiger 2003). This-page derivation: expanding the quadratic branch energy gives the row-sum and negative-off-diagonal rules below under the stated complete-reference convention.
For a Maxwell capacitance matrix,
\[ \mathbf Q=\mathbf C_{\mathrm{Maxwell}}\mathbf V. \]
In the usual passive reciprocal convention, diagonal entries are positive and off-diagonal entries are non-positive. The physical capacitors used by a lumped ladder are
\[ C_{ij}^{\mathrm{branch}}=-C_{\mathrm{Maxwell},ij}\quad(i\ne j), \qquad C_{i0}^{\mathrm{reference}}=\sum_j C_{\mathrm{Maxwell},ij}. \]
The row sum is the capacitance from conductor \(i\) to the declared reference. This interpretation is valid only when the matrix contains the complete declared set of non-reference conductors and its reduction history is known.
For two conductors,
\[ \mathbf C_{\mathrm{Maxwell}} = \begin{bmatrix} C_{g1}+C_m & -C_m\\ -C_m & C_{g2}+C_m \end{bmatrix}, \]
so
\[ C_m=-C_{\mathrm{Maxwell},12}, \qquad C_{g1}=C_{\mathrm{Maxwell},11}+C_{\mathrm{Maxwell},12}, \qquad C_{g2}=C_{\mathrm{Maxwell},22}+C_{\mathrm{Maxwell},21}. \]
Using \(C_{\mathrm{Maxwell},11}\) as a ground shunt and adding the cross capacitor \(-C_{\mathrm{Maxwell},12}\) double-counts mutual capacitance on the nodal diagonal.
When \(\mathbf G\) is supplied as the analogous nodal shunt matrix, the same off-diagonal branch and row-sum interpretation applies. Series matrices \(\mathbf R\) and \(\mathbf L\) do not use this Maxwell-capacitance row-sum rule. In particular, the sign of \(L_{ij}\) is meaningful only with declared current references and branch orientation.
Reduced matrices and physical health checks
A full electrostatic capacitance matrix that still includes every conductor, including the reference, has a common-voltage gauge direction and is normally singular positive semidefinite. The Workbench does not consume that full matrix. It consumes the reduced matrix after the declared reference has been removed; its present two-line contract requires
\[ \mathbf L=\mathbf L^{\mathsf T}\succ0, \qquad \mathbf C_{\mathrm{Maxwell}} =\mathbf C_{\mathrm{Maxwell}}^{\mathsf T}\succ0, \]
non-positive capacitance off-diagonals, and positive physical reference shunts. For a reciprocal dissipative model at a fixed frequency, at minimum the symmetric dissipative parts must satisfy
\[ \frac{\mathbf R+\mathbf R^{\mathsf T}}{2}\succeq0, \qquad \frac{\mathbf G+\mathbf G^{\mathsf T}}{2}\succeq0. \]
These checks do not by themselves prove broadband passivity or causality for frequency-dependent matrices.
This-page derivation. The following symmetry assumptions make the stated orthonormal transform diagonalize both lossless matrices. These formulas are not attributed to a general asymmetric-line solution.
For a reciprocal, lossless, geometrically symmetric two-line model, write
\[ \mathbf L= \begin{bmatrix}L_s&L_m\\L_m&L_s\end{bmatrix}, \qquad \mathbf C_{\mathrm{Maxwell}}= \begin{bmatrix}C_g+C_m&-C_m\\-C_m&C_g+C_m\end{bmatrix}. \]
The orthonormal transform
\[ \mathbf T=\frac{1}{\sqrt 2} \begin{bmatrix}1&1\\1&-1\end{bmatrix} \]
separates the even and odd modes:
\[ \mathbf V=\mathbf T\mathbf V_{eo}, \qquad \mathbf I=\mathbf T\mathbf I_{eo}, \]
where both terminal currents use the same \(+z\) direction, so the orthonormal transform preserves \(\mathbf V^{\dagger}\mathbf I\) in this real lossless case. Then
\[ \mathbf T^{\mathsf T}\mathbf L\mathbf T =\operatorname{diag}(L_e,L_o), \qquad \mathbf T^{\mathsf T}\mathbf C_{\mathrm{Maxwell}}\mathbf T =\operatorname{diag}(C_e,C_o), \]
with
\[ L_e=L_s+L_m, \quad L_o=L_s-L_m, \quad C_e=C_g, \quad C_o=C_g+2C_m. \]
The lossless modal values are then
\[ Z_e=\sqrt{\frac{L_e}{C_e}}, \quad Z_o=\sqrt{\frac{L_o}{C_o}}, \quad v_e=\frac{1}{\sqrt{L_eC_e}}, \quad v_o=\frac{1}{\sqrt{L_oC_o}}, \qquad n_e=\frac{c_0}{v_e}, \quad n_o=\frac{c_0}{v_o}. \]
Conversely, a declared symmetric even/odd specification gives
\[ L_e=\frac{Z_e}{v_e}, \quad L_o=\frac{Z_o}{v_o}, \quad C_e=\frac{1}{Z_ev_e}, \quad C_o=\frac{1}{Z_ov_o}, \]
followed by
\[ L_{11}=L_{22}=\frac{L_e+L_o}{2}, \quad L_{12}=L_{21}=\frac{L_e-L_o}{2}, \]
\[ C_{11}=C_{22}=\frac{C_e+C_o}{2}, \quad C_{12}=C_{21}=\frac{C_e-C_o}{2}. \]
These formulas are a useful specialization, not a general modal solver. Neither “odd mode is slower” nor any fixed sign for the mutual-inductance entry is a theorem without the relevant matrix values and reference directions. Positive definiteness requires
\[ L_s+L_m>0, \quad L_s-L_m>0, \quad C_g>0, \quad C_g+2C_m>0. \]
The corresponding dimensionless coefficient and passive bound follow the canonical self/mutual-inductance conversion.
This specialization is useful for checking a symmetric extractor result. The separate pi-ladder procedure does not require even/odd symmetry.
From matrices to a circuit ladder
This concept page stops once the matrix representation, basis, directions, and physical checks are clear. To build a finite circuit, use the Lossless MTL Matrix to Pi-Ladder procedure. It owns section values, orientation, endpoint splitting, and refinement convergence.
Artifact interpretation checklist
Engineering choice. These fields describe a reconstructible extraction handoff. They are not a new universal schema, numerical tolerance or scientific eligibility criterion; exact consumer requirements remain package-owned.
A Q2D/FEM matrix artifact is eligible for circuit consumption only when a reviewer can answer every row below from the artifact itself.
| Evidence | Acceptance question |
|---|---|
| Identity | Which schema version, case, parameter point, and source files produced this value? |
| Basis | What is the ordered non-reference conductor list, and what conductor or group is the reference? |
| Directions | How are \(+z\), positive terminal current, and terminal voltage defined? |
| Representation | Is each matrix Maxwell/nodal, terminal-domain series, physical branch, Spice, or a dimensionless coupling-coefficient matrix? |
| Quantity availability | Is each of \(R,L,G,C\) extracted, assumed zero, or unavailable? Absence is not zero. |
| Units and frequency | Are source units, SI per-meter values, distributed-length basis, and extraction frequency explicit? |
| Common provenance | Do matrices share the same geometry, materials, solver setup, solution, reduction, and frequency? |
| Shape and labels | Do ordered row and column labels agree, with no missing, duplicate, truncated, or non-finite entry? |
| Physical checks | Are declared reciprocity, symmetry, passivity/positivity, Maxwell signs, and reference row sums satisfied? |
| Consumer capability | Does the receiving implementation support this conductor count, representation, loss model, and frequency dependence? |
A dimensionless solver Couple matrix is not an RLGC matrix. A reduced Spice matrix is not automatically a Maxwell matrix. Representation names must remain attached through every conversion.
Expected evidence
Before a matrix-backed design claim is accepted, show:
- the labeled source artifact and provenance;
- the exact terminal-to-model mapping;
- a reconstruction check for Maxwell \(C\) (and \(G\) when present);
- comparison against a direct solver trace, analytic limit, or another independently justified reference when available.
If the artifact is lowered into a finite ladder, section refinement and endpoint accounting are owned by the lowering procedure.
Failure modes
- Sorting terminal names after extraction and calling the result “solver order.”
- Treating a missing \(R\) or \(G\) matrix as measured zero loss.
- Feeding terminal-domain matrices into scalar \(Z=\sqrt{L/C}\) formulas.
- Using Maxwell diagonal \(C_{ii}\) directly as a ground shunt while also adding \(-C_{ij}\) cross capacitors.
- Assuming an \(L_{12}\) sign without carrying current direction and spatial orientation.
- Mixing matrices from different frequencies, parameter points, reductions, or solver solutions.
References
- Multiconductor Transmission-Line Characterization: Representations, Approximations, and Accuracy supports the conductor/modal distinction, per-unit-length matrix model, and need for explicit normalization and approximation boundaries.
- Reibiger, On the matrices of capacitance and inductance coefficients for lossless homogeneous multiconductor transmission lines gives a field-theoretic treatment of lossless \(L\) and \(C\) matrices.
- Barth and Iyer, On the Unique Determination of Modal Multiconductor Transmission-Line Properties explains why modal transformation normalization matters.
- Ansys documents the distinction between Maxwell matrices and the Maxwell-to-Spice relationship.
| Field | Value |
|---|---|
| Status | Source-backed |
| Review need | Verify the selected lossless LC v1 quantity-availability state, then fix the supported conductor count, representation set, current orientation, single-frequency versus frequency-indexed scope, and whether the noncanonical public CoupledWindowSpec path is removed. True RLGC remains a later contract. Readability and visual acceptance remain separate Human gates. |
| Used by | CPW Quarter-Wave Resonator. |
| Producer evidence | The inherited handoff description identifies the OrPen Q2D case exporter as an \(L/C\) producer. The moving source link is an entrypoint; use the owner’s selected-version resources to establish actual quantity availability. |
| Consumer implementation | The inherited handoff description identifies Workbench transmission-line components, MTLCoupledRLGCSpec and couple_transmission_window!. The moving source link does not bind a consumer version. |
| Consumer boundary | The documented scoped coupled-window model is reciprocal \(2\times2\) lossless \(L/C\); this description provides no implementation evidence for \(R/G\) matrices or general \(N\)-conductor lowering. Current capability belongs to the selected-version consumer resources. |