ADR-0005: Carson correction for the earth-return path¶


Status	Accepted
Date	2026-05-06
Deciders	Project maintainers
Scope	`groundfield`

Context¶

ADR-0004 added the Neumann self- and mutual-inductance integrals between distributed-conductor segments and assembled them into a per-frequency branch-impedance block

\[ Z_b(\omega) \;=\; R \;+\; j\omega\, L_\text{Neumann}. \]

For the magnetic image the earth was treated as a perfect mirror ($\sigma_\text{earth}\to\infty$). That is the same simplification Sunde 1968, ch. 7.4 uses for buried filaments and is known to be accurate when the relevant geometric dimension is small compared to the electromagnetic skin depth in soil

\[ \delta(\omega) \;=\; \sqrt{\frac{2}{\omega\,\mu_0\,\sigma_\text{earth}}} \;\approx\; 503\,\sqrt{\rho_\text{earth} \big/ f}\quad[\text{m}]. \]

For the quasi-static frequency window ($f \le 1\,\mathrm{kHz}$) and typical soil resistivities ($\rho_\text{earth} \in [50, 5000]\,\Omega\,\mathrm{m}$) the skin depth $\delta$ ranges from roughly 350 m (50 Hz, 50 Ωm) to 35 km (50 Hz, 5000 Ωm). The relevant geometric dimensions of an the TN low-voltage distribution network — house-to-station distance < 200 m, PEN-cable length up to 1 km, separation between measurement-lead and current injection 10–100 m — are not always small compared to $\delta$. The perfect-mirror model therefore systematically underestimates the resistive part of the earth-return impedance and slightly overestimates the reactive part. The discrepancy is the very quantity that the research question

"diffusion field and Carson relevance for earth currents"

asks us to put a number on.

This ADR adds the Carson 1926 correction to close that gap. The result is a frequency-dependent, complex-valued addition to the branch-impedance block,

\[ Z_b(\omega) \;=\; \underbrace{R}_{\text{ohmic}} \;+\;\underbrace{j\omega\, L_\text{Neumann}^{\text{(perfect mirror)}}}_{\text{ADR-0004}} \;+\;\underbrace{\Delta Z_\text{Carson}(\omega, \sigma_\text{earth}, h_i, h_j, d_{ij})}_{\text{this ADR}}, \]

evaluated per frequency on the same per-branch endpoint geometry that ADR-0004 already exposes.

Decision¶

Physical model¶

The starting point is John R. Carson, "Wave propagation in overhead wires with ground return", Bell Syst. Tech. J. 5(4) (1926), pp. 539–554. Carson solves Maxwell's equations for a straight horizontal wire above a homogeneous semi-infinite conductive half-space and writes the ground-return self-impedance correction as

\[ Z'_\text{Carson} \;=\; 4\omega\,J(2h\sqrt{\alpha},\,0), \qquad \alpha \;=\; 4\pi\lambda\omega \;\;[\text{CGS-emu}], \]

with

\[ J(p, q) \;=\; \int_0^{\infty}\!\!\bigl(\sqrt{\mu^2 + j} \;-\; \mu\bigr)\, e^{-p\mu}\cos(q\mu)\;d\mu \quad\text{(Carson eq. 29).} \]

The mutual-impedance correction between two parallel wires at heights $h_1, h_2$ with horizontal separation $x$ is

\[ Z'_{12,\text{Carson}} \;=\; 4\omega\,J\bigl((h_1+h_2)\sqrt{\alpha},\,x\sqrt{\alpha}\bigr). \]

In SI units this becomes (Carson eq. 30/31, converted by $4\omega \to \omega\mu_0/\pi$ and $\alpha = 4\pi\lambda\omega \to \omega\mu_0\sigma_\text{earth}$):

\[ \boxed{\; \Delta Z_\text{Carson}(\omega) \;=\; \frac{\omega\mu_0}{\pi}\, \bigl[P(a,\theta) \;+\; jQ(a,\theta)\bigr], \;} \]

with the dimensionless Carson parameters

\[ a \;=\; D\sqrt{\omega\mu_0\sigma_\text{earth}} \;=\; \frac{D\sqrt{2}}{\delta(\omega)}, \qquad D \;=\; \begin{cases} 2h_i & \text{(self impedance, $\theta = 0$)} \\ \sqrt{(h_i+h_j)^2 + d_{ij}^2} & \text{(mutual impedance)} \end{cases}, \qquad \theta \;=\; \arctan\!\bigl(d_{ij}\big/(h_i+h_j)\bigr). \]

The functions $P(a,\theta)$, $Q(a,\theta)$ are the real and imaginary parts of the Carson integral $J$. Their evaluation is split into three regimes following Carson's own discussion in section III of the original paper.

Three evaluation regimes¶

The functions $P, Q$ are evaluated by selecting one of three expansions according to the magnitude of $a$:

Regime 1 — small $a$ ($a \le 0.25$, leading-term form)¶

Carson eqs. 34–35:

\[ P_\text{small}(a,\theta) \;=\; \frac{\pi}{8} \;-\; \frac{a}{3\sqrt{2}}\cos\theta \;+\; \frac{a^2}{16}\cos(2\theta)\, \bigl(0.6728 + \ln(2/a)\bigr) \;+\; \frac{a^2}{16}\theta\,\sin(2\theta), \]

\[ Q_\text{small}(a,\theta) \;=\; -0.0386 \;+\; \tfrac{1}{2}\,\ln(2/a) \;+\; \frac{a}{3\sqrt{2}}\cos\theta. \]

Regime 2 — intermediate $a$ ($0.25 < a \le 5$, direct quadrature)¶

(Rewritten after the 2026-07-08 audit, WP-B2. The originally planned $\sigma_i$-series implementation was never built; the code evaluates Carson's integral $J(p, q)$ numerically.)

Two quadrature paths, selected by the oscillation parameter $q = a\sin\theta$:

$\theta \approx 0$ (self-correction geometry): 64-node Gauss–Legendre on $[0, 30/p]$ — the integrand decays purely exponentially and the historic evaluation is machine-accurate (verified against adaptive references).
$q > 0$ (mutual geometries): QUADPACK's Fourier integrator (scipy.integrate.quad with weight='cos') on $[0, \infty)$. This replaced the historic fixed-node Gauss–Legendre on $[0, 30/p]$, which failed for $\theta \to \pi/2$ (small $p$, unresolved $\cos(q\mu)$ oscillation over a huge interval): the audit measured errors up to $-36\,\%$ in $P$ and $+67\,\%$ in $Q$ at buried-pair mutual geometries ($a = 0.44$–$1.6$, $\theta \approx 89°$). The Fourier path also covers $p = 0$ ($\theta = \pi/2$) exactly, where the imaginary part converges only conditionally through the oscillation. Verified to $10^{-11}$ against an independent composite per-half-period Gauss rule with Euler acceleration (tests/test_audit_quadratures.py).

Regime 3 — large $a$ ($a > 5$, asymptotic expansion)¶

Carson eqs. 36–37:

\[ P_\text{large}(a,\theta) \;=\; \frac{1}{\sqrt{2}}\,\frac{\cos\theta}{a} \;-\; \frac{\cos(2\theta)}{a^2} \;+\; \frac{1}{\sqrt{2}}\,\frac{\cos(3\theta)}{a^3} \;+\; \frac{3}{\sqrt{2}}\,\frac{\cos(5\theta)}{a^5} \;-\; \dots, \]

\[ Q_\text{large}(a,\theta) \;=\; \frac{1}{\sqrt{2}}\,\frac{\cos\theta}{a} \;-\; \frac{\cos(3\theta)}{a^3} \;+\; \frac{3}{\sqrt{2}}\,\frac{\cos(5\theta)}{a^5} \;-\; \dots \]

The truncation matches Carson's published curves (Fig. 2/3 of the original paper) within plotting accuracy for $a \ge 5$; the regime-2/regime-3 boundary at $a = 5$ carries the asymptote's truncation error ($\lesssim 5\cdot10^{-4}$ absolute in $P, Q$ at $\theta = 80°$, smaller for steeper geometries). Two audit corrections (2026-07-08, WP-B2): the $a^{-3}$ term of $Q$ carries the factor $1/\sqrt{2}$ (previously missing — $\sim 6\cdot10^{-4}$ absolute at $a = 8$), and for $\theta > 1.4$ rad ($\approx 80°$) the expansion degrades and the dispatcher routes through the regime-2 quadrature instead.

Linear-system integration¶

The Carson correction enters the existing per-branch impedance block as one extra complex contribution per branch pair $(i, j)$:

\[ Z_b^{(i,j)}(\omega) \;=\; \delta_{ij}\,R^{(i)} \;+\; j\omega\, L_\text{Neumann}^{(i,j)} \;+\; \Delta Z_\text{Carson}^{(i,j)}(\omega). \]

The matrix $\Delta Z_\text{Carson}$ is dense and symmetric (the Carson kernel is reciprocal). Assembly cost is the same as the Neumann matrix — one closed-form evaluation per $M(M-1)/2$ pairs — but it must be rebuilt at every frequency, because the kernel itself depends on $\omega$ through $a$. The Neumann part stays frequency-independent and is built once, exactly as in ADR-0004.

The branch block consumed by _solve_cluster_currents is therefore formed at run time as

Z_b(omega) = R_diag + 1j * omega * L_neumann + dZ_carson(omega)

with R_diag, L_neumann precomputed and dZ_carson recomputed per frequency. The complex linear system

\[ \begin{pmatrix} Z & -C & 0 \\ C^\top & 0 & B^\top \\ 0 & B & -Z_b(\omega) \end{pmatrix} \begin{pmatrix} I_e \\ \varphi_n \\ I_b \end{pmatrix} \;=\; \begin{pmatrix} 0 \\ I_\text{in} \\ 0 \end{pmatrix} \]

is solved per frequency, exactly as in ADR-0004.

Earth-conductivity source¶

The earth conductivity $\sigma_\text{earth}$ is taken from the soil model:

Soil model	Source for $\sigma_\text{earth}$
`HomogeneousSoil`	$\sigma = 1/\rho$ — exact
`TwoLayerSoil`	$\sigma = 1/\rho_1$ (upper layer) with a runtime warning; see "Layered earth" below
`MultiLayerSoil`	$\sigma = 1/\rho_1$ with a warning, identical strategy

For the homogeneous case the Carson series is exact (within the quasi-static / low-frequency assumption that Carson himself states: $\omega \to 0$ and displacement currents in the dielectric neglected). For layered soils the pure Carson series is approximate; we keep it as a default for usability and document the escape hatch (mom_sommerfeld) explicitly.

Engine-side switch¶

A new field is added to Engine:

class Engine(BaseModel):
    ...
    earth_inductive_model: Literal["perfect_mirror", "carson_series"] = "perfect_mirror"

Conductor.inductance_model = "neumann" continues to mean use the Neumann self/mutual integrals — no change. Whether the earth behaves like a perfect mirror or like a finite-σ Carson half-space is now a property of the engine rather than of each conductor, because every distributed conductor in the same world sits above the same earth. Default is "perfect_mirror" for backward compatibility (every existing test and notebook keeps its exact result).

Conductor.inductance_model = None is unchanged — the system stays real and the DC fast path is preserved bit-exact, regardless of earth_inductive_model.

What about the electric Green's function?¶

The Carson correction in this ADR addresses the magnetic image only — i.e. the mutual-impedance contribution to the longitudinal branch block. The electric image (charge image used by the existing image / image_2layer / mom kernels for the leakage current to soil) is not affected. That part already uses the appropriate static Green's function for the soil model (homogeneous, two-layer, n-layer). The two pieces are physically decoupled in the quasi-static regime: the conductive current in the wires sees the earth via the magnetic Green's function (Carson territory), the leakage current sees it via the electric Green's function (already implemented). For full electromagnetic coupling one would have to upgrade mom_sommerfeld to the Pollaczek kernel; that is left as a follow-up — see ADR-0006 (deferred).

Alternatives considered¶

Deri/Semlyen complex depth¶

The Deri & Semlyen 1981 approximation replaces Carson's integral with the closed-form expression

\[ \Delta Z_\text{Deri-Semlyen}(\omega) \;=\; \frac{j\omega\mu_0}{2\pi}\, \ln\!\Bigl(\frac{D' + p}{D}\Bigr), \qquad p \;=\; 1\Big/\sqrt{j\omega\mu_0\sigma_\text{earth}}, \]

where the image is moved by a complex depth $p$ below the soil surface. It is appealingly cheap (no series), but its accuracy degrades for short wires and for $a \in [0.5, 5]$. We keep it as an internal sanity check in coupling/carson.py (deri_semlyen_correction) and use it in tests as a second opinion against the series implementation; we do not use it as the production code path.

Direct numerical Sommerfeld quadrature¶

Pollaczek 1926 generalises Carson to the layered half-space; the resulting integral has to be evaluated numerically. This is already implemented for the electric Green's function in solver/mom_sommerfeld.py. The magnetic Pollaczek kernel would slot into the same backend and would be the rigorous answer for the layered case. Cost: every frequency requires a 2-D Sommerfeld quadrature per segment pair — orders of magnitude slower than the closed-form Carson series. We defer this to a follow-up ADR (ADR-0006) and use mom_sommerfeld (electric only) plus Carson series (magnetic) as the typical working configuration.

Layered earth via effective resistivity¶

A pragmatic middle ground: when the soil is layered we replace $\rho_\text{earth}$ in the Carson series by an effective resistivity $\rho_\text{eff}(\omega)$ that interpolates between $\rho_1$ (when $\delta(\omega) \ll h_1$, the first-layer regime) and $\rho_2$ (when $\delta(\omega) \gg h_1$, the deep regime). We keep this as a future option and currently emit a runtime warning instead. The reason: the depth-blending is itself an approximation whose validity range is the same as the Pollaczek result we are trying to avoid. Better to be explicit about the approximation than to hide it inside an interpolation rule.

Validation¶

Every test below is implemented in tests/test_carson_coupling.py unless noted otherwise.

Unit tests against Carson 1926¶

Wave-antenna example (Carson p. 552) $h = 30\,\mathrm{ft}$, $f = 5 \cdot 10^4\,\mathrm{Hz}$, $\lambda = 10^{-12}$ emu → $a = 4.0$ → $J = 0.126 + j0.168$. Tolerance: $\le 0.5\,\%$ relative.
Wave-antenna example, second case Same geometry, $\lambda = 10^{-14}$ emu → $a = 0.4$ → $J = 0.323 + j0.871$. Tolerance: $\le 0.5\,\%$.
Railway example (Carson p. 553) $f = 25\,\mathrm{Hz}$, $h = 30\,\mathrm{ft}$, $x = 120\,\mathrm{ft}$, $\lambda = 10^{-12}$ emu → $J = 0.369 + j1.135$, $\theta \approx 63°30'$. Tolerance: $\le 0.5\,\%$.
Tabulated curves Fig. 2/3 Spot checks at $a \in \{0.5, 1, 2, 4\}$ and $\theta \in \{0, \pi/4, \pi/2\}$. Tolerance: $\le 1\,\%$ (Carson plotted on a small grid).

Regime-boundary continuity¶

At $a = 0.25$ the small-$a$ form (regime 1) and the full series (regime 2) agree to $\le 10^{-9}$ in $P, Q$.
At $a = 5$ the full series (regime 2) and the asymptotic expansion (regime 3) agree to $\le 10^{-6}$ in $P, Q$.

Limit checks¶

$\sigma_\text{earth} \to \infty$ (perfect conductor) → $\Delta Z_\text{Carson} \to 0$. The full assembled branch block reproduces ADR-0004 bit-exact (regression test).
$\omega \to 0$ → $\Delta Z_\text{Carson} \to 0$. The DC solution reproduces ADR-0003 bit-exact.
Frequency monotonicity — for fixed $\sigma_\text{earth}$ and a 1 km PEN over earth the real part of the diagonal $Z_b(\omega)$ grows monotonically with $\omega$ in $[10\,\mathrm{Hz}, 1\,\mathrm{kHz}]$. Sanity check: ohmic earth-return contribution increases with frequency.

Engineering benchmarks¶

1 km PEN self impedance vs. Oeding + Carson lookup. Build a horizontal PEN conductor at $h = 0.6\,\mathrm{m}$ above a homogeneous earth ($\rho = 100\,\Omega\,\mathrm{m}$), 1 km long, radius 4 mm, run all distributed-capable backends at $f \in \{50, 150, 500, 1000\}\,\mathrm{Hz}$. Verify $Z_\text{self,per km}$ within 5 % of the Oeding/Tleis closed-form Carson R/X tabulation. Implementation: scripts/benchmarks/pen_1km_carson.py plus a regression test.
Loop coupling open-circuit voltage with vs. without Carson. Two parallel galvanic conductors, 1 km long, separation 50 m. The injected current is 1 A on conductor 1, conductor 2 is open. The open-circuit voltage at $f = 50\,\mathrm{Hz}$ must increase by $\ge 5 \%$ (Carson on) compared to the perfect-mirror result. Test: test_carson_coupling.py:: test_loop_coupling_carson_increases_open_voltage.
Cross-engine consistency at 50 Hz with Carson active. All distributed-capable backends (image, image_2layer, mom, cim, bem) must agree on the cluster impedance to within 5 %. fem continues to log a warning and fall back to the resistive solution.

Layered-earth handling¶

Two-layer warning. Building an Engine with earth_inductive_model = "carson_series" against a TwoLayerSoil world emits a UserWarning containing the string "Carson series uses upper-layer rho_1".
Carson(approx layered) vs. mom_sommerfeld. For a TwoLayerSoil with $\rho_1 = 100\,\Omega\,\mathrm{m}$, $\rho_2 = 1000\,\Omega\,\mathrm{m}$, $h_1 = 1\,\mathrm{m}$, the Carson approximation differs from the Pollaczek/Sommerfeld reference by less than the documented bound (≤ 20 % at 50 Hz, ≤ 10 % at 1 kHz). The bound itself is a documented test artefact, not a guarantee — validated by the project's internal Carson harness.

Cross-references¶

DC reproducibility — inductance_model = None keeps the system real for any setting of earth_inductive_model (regression).
Bit-exact perfect-mirror regression — the full ADR-0004 test suite continues to pass with earth_inductive_model = "perfect_mirror" (default).

Consequences¶

Positive¶

Quantifies the relevance of the Carson correction for earth currents: "at which frequency / soil resistivity does the perfect-mirror approximation break down" can now be answered numerically — validated by the project's internal Carson harness.
The closed-form Carson series adds essentially zero cost ($\mathcal{O}(M^2)$ scalar evaluations per frequency, vectorised via numpy). Frequency sweeps with $N_f \le 20$ on relevant geometries ($M \le 5\,000$) remain inside the workstation budget.
The implementation is physically transparent: the same series Carson published in 1926, with the same regime split. Tests against Carson's own worked examples are the gold standard.
The earth conductivity now enters the inductance assembly explicitly, which makes the penetration depth $\delta$ a first-class diagnostic. The new FieldResult.metadata["penetration_depth"] exposes $\delta(\omega)$ per frequency for every engine.

Negative¶

For layered soil the series is an approximation. We document the limit and provide a quantified comparison against mom_sommerfeld in the validation notebook. A user who needs rigor on layered soil must switch backends.
The branch block is rebuilt at every frequency. With Carson on and $N_f$ frequencies, $N$ segments, the assembly cost is $\mathcal{O}(N_f \cdot M^2)$ closed-form evaluations on top of the existing $\mathcal{O}(N_f \cdot N^3)$ LU. For typical ($M \le 5\,000$, $N_f \le 20$) the assembly is still $\le 10\,\mathrm{s}$ in the worst case; the LU dominates.

Neutral¶

Default behaviour (earth_inductive_model = "perfect_mirror") is unchanged. Existing notebooks, tests and benchmarks produce identical results to ADR-0004.
The Pollaczek kernel for the magnetic Green's function on a layered soil is deferred to ADR-0006 (open). That ADR will add solver/mom_sommerfeld.py magnetic-kernel support and will reuse the same engine.earth_inductive_model switch with a third value "sommerfeld".

References¶

Carson, J. R. (1926). Wave propagation in overhead wires with ground return. Bell Syst. Tech. J. 5(4), 539–554. Primary source — equations 23, 25–37 are the production code path of this ADR.
Pollaczek, F. (1926). Über das Feld einer unendlich langen wechselstromdurchflossenen Einfachleitung. Elektrische Nachrichtentechnik 3(9), 339–360. Layered-earth extension (deferred).
Deri, A.; Tevan, G.; Semlyen, A.; Castanheira, A. (1981). The complex ground return plane: a simplified model for homogeneous and multi-layer earth return. IEEE Trans. PAS 100(8), 3686–3693. Used as a sanity check.
Oeding, D. & Oswald, B. R. (2016). Elektrische Kraftwerke und Netze, 8. Aufl., Springer. §9.4 "Erdrückleiter".
Tleis, N. D. (2008). Power Systems Modelling and Fault Analysis: Theory and Practice, Newnes. Ch. 3 — modern Carson series in SI units.
Sunde, E. D. (1968). Earth Conduction Effects in Transmission Systems, Dover. Ch. 7 — buried-wire equivalent.