ADR-0004: Inductive coupling between conductor segments¶


Status	Accepted
Date	2026-05-04
Deciders	Project maintainers
Scope	`groundfield`

Context¶

ADR-0003 introduced the distributed-conductor model. Each conductor is split into \(n\) longitudinal sub-segments, and the resulting nodal-analysis system carries one branch per segment with series impedance

\[ Z_\text{long}^{(k)} \;=\; R^{(k)} \;+\; j\omega\, L^{(k)}. \]

ADR-0003 deferred \(L^{(k)}\) to a follow-up step. With \(L^{(k)} = 0\) the system is real and frequency-independent, and the engine.frequencies list returns the same DC solution for every entry. This is fine for DC-near grounding-impedance studies but not for analysing inductive coupling between the measurement lead and the current injection — the very effect that forced the distributed-conductor refactor in the first place.

This ADR settles how the inductive part is added on top of the distributed model.

Decision¶

Physical model¶

Self- and mutual-inductance of conductor segments are computed from Neumann's double-line integral (Grover 1946, Sunde 1968 ch. 7, Paul 2010 ch. 5):

\[ M_{ij} \;=\; \frac{\mu_0}{4\pi} \oint_{C_i} \oint_{C_j} \frac{d\vec{\ell}_i \cdot d\vec{\ell}_j}{r_{ij}}. \]

For two straight segments of equal length \(\ell\) that share the same axis direction at perpendicular distance \(d\), the closed form is

\[ M_\parallel(\ell, d) \;=\; \frac{\mu_0\,\ell}{2\pi} \Bigl[\ln\!\Bigl(\frac{\ell + \sqrt{\ell^2 + d^2}}{d}\Bigr) - \frac{\sqrt{\ell^2 + d^2} - d}{\ell}\Bigr]. \]

For arbitrary 3-D orientations and unequal lengths the implementation falls back to a two-point Gauss–Legendre quadrature of the double integral on each segment pair — sufficient because \(1/r_{ij}\) varies slowly over a single sub-segment, and the quadrature is cheap (constant cost per pair, total cost \(\mathcal{O}(M^2)\) for \(M\) segments). For the segment with itself we use the thin-wire approximation

\[ L_\text{self} \;\approx\; \frac{\mu_0\,\ell}{2\pi} \Bigl[\ln\!\Bigl(\frac{2\ell}{a}\Bigr) - 1\Bigr], \]

with \(a\) the wire radius. This formula is bounded between \(\ell/a > 10\) (thin-wire regime), which holds comfortably for the typical cable runs (\(\ell\) ≈ 1–5 m, \(a\) ≈ 4 mm).

Earth return ("ideal earth")¶

In this ADR the earth is treated as a perfect mirror for the magnetic field (\(\sigma_\text{earth} \to \infty\)): the image of a segment at depth \(z_s\) contributes its own Neumann integral against all other segments' real and image positions, exactly as in the electric image-charge sum used by the existing solvers. This is the same approximation already used by Sunde 1968 ch. 7.4 for buried power lines below 1 kHz; the finite earth conductivity correction (Carson) is the subject of a follow-up ADR.

Schema¶

Conductor gains one new field:

inductance_model: Literal[None, "neumann"]   # default None

None (default) — backwards-compatible: \(L^{(k)} = 0\), the longitudinal branch is purely resistive, and the system is real per frequency (existing behaviour).
"neumann" — the segments contribute self- and mutual-inductance to the branch impedance via the formulas above. The longitudinal block of the augmented system becomes complex:

\[ Z_b(\omega) \;=\; R + j\omega\, L \in \mathbb{C}^{M \times M}, \]

with \(L\) a dense, symmetric, positive-definite matrix over the distributed-conductor branches. Mutual coupling exists between every pair of segments — not only within the same conductor — so that a measurement lead routed parallel to the current-injection lead automatically picks up its mutual inductance.

Linear system¶

The block structure of the augmented system is unchanged from ADR-0003:

\[ \begin{bmatrix} Z_g & -C_s & 0 \\ C_s^{\top} & 0 & B^{\top} \\ 0 & B & -Z_b(\omega) \end{bmatrix} \begin{bmatrix} \mathbf{I}_\text{leak} \\ \boldsymbol{\varphi}_n \\ \mathbf{I}_\text{long} \end{bmatrix} \;=\; \begin{bmatrix} \mathbf{0} \\ \mathbf{I}_\text{in} \\ \mathbf{0} \end{bmatrix}. \]

The earth Green's function block \(Z_g\) stays real and frequency-independent (quasi-static, \(f < 1\,\mathrm{kHz}\)) — the finite-frequency correction would have to come from a magnetic Green's function on the layered soil, which is again Carson territory (ADR-0005). The \(Z_b(\omega)\) block carries all the frequency-dependent physics introduced here.

Frequency loop¶

When at least one conductor has inductance_model == "neumann", the solver:

Discretises the geometry once and assembles \(Z_g\), the resistance vector \(R\), and the inductance matrix \(L\) — all frequency-independent.
Loops over engine.frequencies. For each \(f_k\) the \(\omega = 2\pi f_k\) is plugged into \(Z_b(\omega)\) and the full complex linear system is solved with numpy.linalg.solve on the complex augmented matrix.
Returns one complex potential / current entry per frequency in FieldResult.electrode_potentials[name][k] etc. — the FieldResult shape is unchanged; entries that used to be real per frequency are now genuinely complex.

When all conductors have inductance_model is None (the default), the solver keeps the historic real-only fast path: one linalg.solve over the real augmented matrix, the result broadcast across engine.frequencies. Quasi-static DC studies stay bit-exact and at the same cost.

Self-inductance and shielded conductors¶

The thin-wire self-inductance assumes a bare conductor. For a cable with a conductor inside an outer sheath the relevant self-inductance is geometrically smaller; this distinction matters for shielded MV cables but not for the bare-copper / PEN typical study. We keep the model simple: inductance_model == "neumann" always uses the bare-conductor self formula; refinements come with the cable-shield work later.

Validation¶

Single isolated segment — the segment's self-L matches the closed-form Grover formula above to within 0.5 % (we sweep \(\ell/a\) from 10 to 1000).
Two parallel coaxial segments — the Neumann quadrature reproduces \(M_\parallel(\ell, d)\) to within 1 % for separations \(d / \ell \in [0.1, 10]\).
Two perpendicular segments crossing in the middle — the Neumann integral evaluates to zero by symmetry (test tolerance \(10^{-9}\)).
DC reproducibility — at \(\omega = 0\) the inductive system collapses bit-exact to the resistive system from ADR-0003 (regression).
Cross-engine at \(f = 50\,\mathrm{Hz}\) — image, mom, cim, bem agree on the cluster impedance of a galvanic distributed conductor with the inductive model active to within 5 %.
Loop coupling — two parallel galvanic conductors (one current injection, one open-circuit measurement lead) show the expected finite open-circuit voltage at 50 Hz that scales linearly with the source current and with \(\omega\).

Consequences¶

Positive¶

Covers inductive coupling between the measurement lead and the current injection. The model is now physically complete for \(f < 1\,\mathrm{kHz}\) except for the Carson earth-return correction.
The Neumann integral is geometric only — it works on top of any of the existing electric Green's-function backends without duplicating their kernel logic.
Frequency-resolved FieldResults without API change: the same electrode_potentials[name][k] shape now actually depends on \(f_k\).

Negative¶

The system becomes complex when the inductive model is enabled, doubling the per-frequency LU factorisation cost. A frequency sweep with \(N_f\) frequencies costs \(N_f\) complex solves (vs. one real solve in the resistive-only case). For a typical sweep (\(N_f \le 20\), \(N \le 5\,000\) segments) this is still well inside the 32-GB / 12-core workstation budget.
The mutual-inductance matrix is dense and symmetric; the same ACA/iterative roadmap that applies to the electric Z-matrix applies here. Both will be tackled together.

Neutral¶

Default behaviour (inductance_model is None) is unchanged. Existing notebooks and tests continue to produce DC results at identical cost.
The earth is still treated as a perfect mirror for the magnetic field — Carson finite-conductivity corrections are the subject of ADR-0005.

References¶

Grover, F. W. (1946). Inductance Calculations: Working Formulas and Tables. Dover (reprint 2004).
Sunde, E. D. (1968). Earth Conduction Effects in Transmission Systems, Dover, ch. 7.
Paul, C. R. (2010). Inductance: Loop and Partial. Wiley.
Carson, J. R. (1926). Wave propagation in overhead wires with ground return. Bell Syst. Tech. J. 5(4) — referenced for the follow-up ADR-0005.