Field-Extracted Capacitance and Inductance Matrices
Node role: Concept.
Reader task: Decide what a field-extracted capacitance or inductance file means, which coordinate basis it uses, and whether it is eligible to enter a node-flux energy model.
Reading map
Begin with electric-energy coordinates and the reference boundary. Compare principal-block replay with named-ground projection before reading the mutual-form netlist view. Magnetic energy then needs a different step: an oriented current/flux-linkage map. The final sufficiency table shows what static matrices cannot determine about an open response.
Canonical matrix contract
Reorganized explanation. Field energies determine capacitance and inductance coefficients in their declared excitation bases (Palace, n.d.-a). The circuit substitutions below are this-page algebra; the named-ground projection is a separate engineering choice.
For a finite conservative circuit, the field solver’s Maxwell capacitance matrix is the raw electric-energy evidence. Before using it in the node-flux Lagrangian, resolve the reference conductor and apply the project projection convention defined below. A magnetostatic inductance matrix needs one additional oriented branch or loop map because its native coordinates are source currents and flux linkages, not node fluxes:
\[ \boxed{ \mathbf C_{\mathrm{Maxwell}} \text{ owns electric energy}, \qquad (\mathbf L_{\mathrm{Magnetostatic}}, \mathbf B_{\mathrm{inductive}}) \text{ own magnetic energy}. } \]
The inverse and mutual-form files are derived views of those same quadratic forms. They are not additional components to stamp again.
This asymmetry is physical. Conductor voltage is exactly the time derivative of node flux, while a source current lives on an oriented branch or loop. Calling both outputs “Maxwell matrices” does not put them in the same basis.
This Knowledge Base writes semantic matrix labels in full: \(\mathbf C_{\mathrm{Maxwell}}\), \(\mathbf L_{\mathrm{Magnetostatic}}\), \(\mathbf C_{\mathrm{Mutual}}\), and, only after an actual mode transform, \(\mathbf C_{\mathrm{Mode}}\). A bare subscript or symbol \(M\) is not used to mean Maxwell, magnetostatic, mutual, or mode. The Palace filenames terminal-M.csv, terminal-Minv.csv, and terminal-Mm.csv remain literal filenames, not our mathematical notation.
Electrostatic matrix in the node-flux Lagrangian
This-page derivation. Substitute integrated node voltage into the Maxwell quadratic electric energy (Palace, n.d.-a). Preserve the same terminal order and reference; no mode transform has occurred.
Write the ordered non-reference conductor potentials and their node-flux relation as
\[ \vec V_T=(V_1,\ldots,V_N)^T, \qquad \vec V_T=\dot{\vec\Phi}_T. \]
Charge and electric energy obey
\[ \boxed{ \vec Q_T=\mathbf C_{\mathrm{Maxwell}}\vec V_T, \qquad T_E=\frac12\vec V_T^T\mathbf C_{\mathrm{Maxwell}}\vec V_T. } \]
Therefore, in the identical ordered terminal/node basis,
\[ \boxed{ T_E =\frac12\dot{\vec\Phi}_T^T \mathbf C_{\mathrm{Maxwell}} \dot{\vec\Phi}_T. } \]
In ordinary language, terminal-C.csv already contains the coefficient of the capacitive kinetic term. Its diagonal contains each conductor’s complete electric self-loading under the declared ground boundary; its off-diagonals contain electrostatic coupling and are normally non-positive.
Ground included versus ground already imposed
Reorganized explanation. Palace’s electrostatic terminals are excited relative to separately grounded boundaries (Palace, n.d.-b, Electrostatic problems). The principal-block operation below instead describes an export that explicitly includes its reference conductor.
Some solvers export a listed-conductor matrix \(\mathbf C_{\mathrm{Maxwell},E}\) that includes the named reference conductor. It is an all-conductor matrix with a common-potential null direction only when the listed set \(E\) is electrostatically closed. If conductor \(0\) is fixed to \(\Phi_0=0\), the selector \(\mathbf P_{\neg0}\) retains every non-reference listed conductor, so the node matrix is the principal block
\[ \boxed{ \mathbf C_{\mathrm{Maxwell},T} =\mathbf P_{\neg0}^T \mathbf C_{\mathrm{Maxwell},E} \mathbf P_{\neg0}. } \]
Palace instead defines Ground as a boundary and exports only the declared Terminal basis. In that case terminal-C.csv is already \(\mathbf C_{\mathrm{Maxwell},T}\); do not remove another row or column.
Named ground and implicit outer-reference residual
This-page derivation. Decompose the explicit-conductor row into named pairs and its signed row sum. The decomposition alone does not identify the physical origin of the residual.
An explicit-conductor export may list a conductor \(G\) named GND while the solver also retains an implicit outer or reference-boundary potential. Let \(E=\{G,1,\ldots,N\}\) be the explicit conductor set. For each retained conductor \(i\), define
\[ \boxed{ C_{iG}^{\mathrm{named}}=-C_{iG}^{E}, \qquad C_{ij}^{\mathrm{pair}}=-C_{ij}^{E}, \qquad \delta_i^{\mathrm{outer}} =\sum_{j\in E}C_{ij}^{E}. } \]
Then
\[ C_{ii}^{E} =C_{iG}^{\mathrm{named}} +\sum_{\substack{j\ne i\\j\ne G}}C_{ij}^{\mathrm{pair}} +\delta_i^{\mathrm{outer}}. \]
The signed row sum \(\delta_i^{\mathrm{outer}}\) is the coefficient left after decomposing the row into branches between explicitly named conductors. A nonzero value may contain implicit outer/reference-boundary loading, unlisted grounded geometry, and numerical extraction residue. The matrix alone does not distinguish those causes, so do not silently clip the value or describe it as a separately identified physical capacitor without boundary evidence.
Fixing \(V_G=0\) and retaining the principal block preserves both the named-GND and implicit-reference loading:
\[ \boxed{ \mathbf C_{\mathrm{node}}^{\mathrm{direct\text{-}retained}} =\mathbf P_{\neg G}^{T}\mathbf C_{\mathrm{Maxwell},E} \mathbf P_{\neg G}. } \]
This is the energy-exact replay of the complete solver boundary, but it mixes the explicitly named ground with whatever physical or numerical loading is contained in \(\boldsymbol\delta_T^{\mathrm{outer}}\).
Engineering choice. SCQ_Design instead uses the named-conductor-only projection as its unqualified project convention:
\[ \boxed{ \mathbf C_{\mathrm{named}} =\mathbf P_{\neg G}^{T}\mathbf C_{\mathrm{Maxwell},E} \mathbf P_{\neg G} -\operatorname{diag}(\boldsymbol\delta_T^{\mathrm{outer}}). } \]
That projection preserves all pairwise off-diagonals and uses \(C_{iG}^{\mathrm{named}}\) as the ground-shunt branch. It intentionally omits the outer-reference coefficient from the circuit diagonal. This is a project-level modeling convention, not an identity of the raw Maxwell matrix. Its artifact must retain the raw matrix, conductor order, named ground, signed row sums, policy authority, and response-impact review.
Therefore:
- an unqualified projected node capacitance matrix means \(\mathbf C_{\mathrm{node}}\equiv\mathbf C_{\mathrm{named}}\);
- direct principal-block replay must be written \(\mathbf C_{\mathrm{node}}^{\mathrm{direct\text{-}retained}}\); and
- a solver export with ground already imposed, such as Palace
terminal-C.csv, must declare that direct-retained processing state when it is used without a separately identified named-ground projection.
The two matrices are not interchangeable:
\[ \boxed{ \mathbf C_{\mathrm{node}}^{\mathrm{direct\text{-}retained}} -\mathbf C_{\mathrm{node}} =\operatorname{diag}(\boldsymbol\delta_T^{\mathrm{outer}}). } \]
Because inversion follows projection, this difference can change \(\mathbf C^{-1}\), downstream oscillator normalization, anchored-bare open-EOM roots and couplings, and every hybridized response.
Embed a terminal block into a larger circuit
This-page derivation. Substitute the terminal embedding map into electric energy. Stamping this complete block twice would double its stored energy.
The embedding matrix maps the extracted terminal ordering into the larger physical-node vector:
\[ \vec V_T=\mathbf E_T^T\dot{\vec\Phi}_{\rm node}. \]
The complete matrix-valued stamp is
\[ \boxed{ \Delta\mathbf C_{\rm node} =\mathbf E_T\mathbf C_{\mathrm{Maxwell},T}\mathbf E_T^T. } \]
This is the precise meaning of a multi-conductor electrostatic block. It is not one capacitor with many pins, and it must not be stamped together with a second set of scalar capacitors derived from the same matrix.
When the block represents only a local layout region, its terminal-to-ground and terminal-to-terminal entries travel together. The surrounding distributed model must exclude the same stored energy. The region ledger and geometry-to-matrix procedure are owned by localized electrostatic component modeling.
Mutual-form capacitance is a netlist view
Reorganized explanation / this-page derivation. Palace exposes Maxwell, inverse and mutual-form files (Palace, n.d.-b, Electrostatic problems). Expand the scalar branch energies below to recover the Maxwell matrix; the table of branch values is not the quadratic coefficient matrix itself.
The positive pair and reference capacitances are
\[ \boxed{ C_{ij}^{\mathrm{Mutual}}=-C_{\mathrm{Maxwell},ij}, \qquad C_{i0}^{\mathrm{Mutual}}=\sum_j C_{\mathrm{Maxwell},ij}. } \]
Palace stores \(C_{i0}^{\mathrm{Mutual}}\) on the diagonal of terminal-Cm.csv and \(C_{ij}^{\mathrm{Mutual}}\) on its off-diagonals. The inverse conversion is
\[ \boxed{ C_{\mathrm{Maxwell},ij}=-C_{ij}^{\mathrm{Mutual}}\quad(i\ne j), \qquad C_{\mathrm{Maxwell},ii} =C_{i0}^{\mathrm{Mutual}}+\sum_{j\ne i}C_{ij}^{\mathrm{Mutual}}. } \]
The equivalent scalar-branch energy is
\[ T_E =\frac12\sum_i C_{i0}^{\mathrm{Mutual}}\dot\Phi_i^2 +\frac12\sum_{i<j}C_{ij}^{\mathrm{Mutual}} (\dot\Phi_i-\dot\Phi_j)^2. \]
The array in terminal-Cm.csv is therefore a table of scalar branch values; it is not the matrix to substitute as \(\dot{\vec\Phi}^T\mathbf C\dot{\vec\Phi}/2\).
Magnetostatic matrix and the missing branch map
Reorganized explanation. Magnetic coefficients require stated source and return-path boundaries (Palace, n.d.-a). The equations below assume one finite reciprocal current/flux-linkage matrix. This-page derivation: substitute its oriented branch map into magnetic energy to obtain node-flux stiffness; the current matrix alone cannot specify that map.
Palace returns ordered source currents and their flux linkages. Their magnetostatic relation and stored energy are
\[ \boxed{ \vec\lambda_{\mathrm{inductive}} =\mathbf L_{\mathrm{Magnetostatic}}\vec I_{\mathrm{inductive}}, \qquad U_{\mathrm{Magnetostatic}} =\frac12\vec I_{\mathrm{inductive}}^T \mathbf L_{\mathrm{Magnetostatic}}\vec I_{\mathrm{inductive}} =\frac12\vec\lambda_{\mathrm{inductive}}^T \mathbf L_{\mathrm{Magnetostatic}}^{-1} \vec\lambda_{\mathrm{inductive}}. } \]
To use it with node flux, declare an oriented branch/loop incidence matrix \(\mathbf B_{\mathrm{inductive}}\) whose columns use exactly the same ordering and current directions as the magnetostatic sources:
\[ \boxed{ \vec\lambda_{\mathrm{inductive}} =\mathbf B_{\mathrm{inductive}}^T\vec\Phi_{\rm node}. } \]
Then
\[ \boxed{ U_{\mathrm{Magnetostatic}} =\frac12\vec\Phi_{\rm node}^T \underbrace{ \mathbf B_{\mathrm{inductive}} \mathbf L_{\mathrm{Magnetostatic}}^{-1} \mathbf B_{\mathrm{inductive}}^T }_{\displaystyle\mathbf K_{\Phi,\mathrm{Magnetostatic}}} \vec\Phi_{\rm node}. } \]
Thus terminal-M.csv is sufficient only together with the return paths, source ordering, current orientations, and \(\mathbf B_{\mathrm{inductive}}\). Reversing one source orientation changes the signs of its corresponding row and column; the branch map must change with it.
terminal-Mm.csv applies the same algebraic “mutual form” used for capacitance:
\[ L_{ij}^{\mathrm{MutualForm}} =-L_{\mathrm{Magnetostatic},ij}\quad(i\ne j), \qquad L_{ii}^{\mathrm{MutualForm}} =\sum_j L_{\mathrm{Magnetostatic},ij}. \]
It is a difference-current coefficient view. It must not be confused with the ordinary full flux-linkage matrix \(\mathbf L_{\mathrm{Magnetostatic}}\), whose off-diagonal entries are the signed mutual inductances used in \(\vec\lambda_{\mathrm{inductive}} =\mathbf L_{\mathrm{Magnetostatic}}\vec I_{\mathrm{inductive}}\). The conversion from those signed entries and the diagonal self inductances to dimensionless \(k_{ij}\) is owned by Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance.
The declared reciprocal surface-current model uses the following file views (Palace, n.d.-b, Electrostatic and magnetostatic problems). This table is a representation map, not a certification of every Palace version or magnetostatic excitation option:
| Palace output | Basis and equation | Proper role |
|---|---|---|
terminal-C.csv |
Terminal voltages/charges, \(\vec Q=\mathbf C_{\mathrm{Maxwell}}\vec V\) | Canonical electric-energy input after the reference and ordering are verified. |
terminal-Cinv.csv |
Charge-to-voltage map, \(\mathbf C_{\mathrm{Maxwell}}^{-1}\) | Derived elastance/Hamiltonian charge coefficient; use a linear solve numerically. |
terminal-Cm.csv |
Reference and pair capacitance table | Optional scalar-capacitor netlist view. |
terminal-M.csv |
Source currents/flux linkages, \(\vec\lambda_{\mathrm{inductive}}=\mathbf L_{\mathrm{Magnetostatic}}\vec I_{\mathrm{inductive}}\) | Canonical magnetostatic energy input before mapping to node flux. |
terminal-Minv.csv |
Flux-linkage energy coefficient, \(\mathbf L_{\mathrm{Magnetostatic}}^{-1}\) | Derived view used inside \(\mathbf B_{\mathrm{inductive}}\mathbf L_{\mathrm{Magnetostatic}}^{-1}\mathbf B_{\mathrm{inductive}}^T\); use a solve numerically. |
terminal-Mm.csv |
Difference-current coefficient table | Optional derived representation, not another inductance block. |
Electrostatic runs excite one terminal at unit voltage while grounding the other terminals. The magnetostatic description above assumes reciprocal surface-current sources with inactive ports open. Other inactive-port boundaries and flux-loop or mixed excitations are not covered by this reciprocal surface-current model; their producer-version output semantics must be established separately. Preserve solver version, excitation type, inactive-source treatment and terminal/source Index order rather than relying on CSV row position alone.
Is this sufficient for a Hamiltonian?
| Intended result | Sufficient input? |
|---|---|
| Finite electrostatic kinetic term | Yes: verified \(\mathbf C_{\mathrm{Maxwell}}\) in the retained non-reference conductor basis. |
| Finite linear magnetic potential | Yes: verified \(\mathbf L_{\mathrm{Magnetostatic}}\), its oriented branch/loop map \(\mathbf B_{\mathrm{inductive}}\), and any additional inductive branches not represented by the solve. |
| Complete conservative linear Hamiltonian | Yes only after both electric and magnetic terms, coordinate constraints/reductions, and every Josephson or kinetic inductance contribution are included. |
| Open linewidth, radiation, propagation, or broadband \(S/Y/Z\) | No. Static matrices do not replace the environment or driven/distributed response model. |
The practical answer is therefore: make Maxwell capacitance the electric SoT; make Palace’s absolute-current inductance matrix plus its branch map the magnetic SoT; treat inverse and mutual-form matrices as derived evidence.
Engineering use. The items below make the model reconstructible. They supply no numerical rank, symmetry or positivity tolerance and do not replace the owning model’s accepted contract.
Before either matrix enters a Lagrangian:
- record the exact conductor or source ordering and physical directions;
- record the reference/ground and return-path boundary;
- verify symmetry, units, rank/nullspace, and positive semidefinite energy;
- distinguish an all-conductor matrix from an already grounded terminal matrix;
- prove that no scalar branch duplicates energy already present in the matrix block; and
- state which kinetic, Josephson, geometric, or distributed inductive effects are absent and must be added separately.
Connections
- Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance owns \(k_{ij}\leftrightarrow M_{ij}\) after the magnetostatic basis and orientations are valid.
- Circuit Models to Bare Coordinates, Open EOM, and Normal Modes owns the complete modeling procedure.
- Multiconductor RLGC Matrices owns the distinct per-unit-length transmission-line use of Maxwell and mutual capacitance.
References
- Open-source Highly Parallel Electromagnetic Simulations, the local supporting research node for Palace-based EM workflows.
- Palace, Problem Types, electrostatic and magnetostatic output contracts.
- Palace, Electrostatic Simulations, the live reference for boundary-conditioned electric-energy extraction. The moving stable reference is not certification of a pinned producer version and does not authorize the named-ground projection used on this page.
- Palace, Capacitance Matrix for Two Spheres.
- Palace, Inductance Matrix for a Pair of Concentric Rings.
| Field | Value |
|---|---|
| Status | Source-backed |
| Review state | Equations and Palace file semantics verified against the official documentation and local Palace source; not yet Human-reviewed. |
| Open review question | Can a reader explain why terminal-C.csv enters the node-flux Lagrangian directly, while terminal-M.csv still requires \(\mathbf B_{\mathrm{inductive}}\)? |
| Used by | Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance and Circuit Models to Bare Coordinates, Open EOM, and Normal Modes. |