05 — Analysing inductive coupling

Below 1 kHz the static / quasi-static formulation of groundfield correctly captures the resistive part of every grounding installation, but the longitudinal-branch impedance of a cable picks up a significant \(j\omega L\) contribution. groundfield supports two inductance corrections (ADR-0004 / ADR-0005 / ADR-0006):

inductance_model	Kernel	Suitable for
None (default)	DC, purely resistive	Quick parameter sweeps
"neumann"	Neumann self / mutual integral	Cable + above-ground / shallow lines

The earth-return contribution (Carson 1926 and the rigorous Sommerfeld correction) lives on the Engine instead of on the conductor; it is controlled via the earth_inductive_model attribute.

Setup

Two parallel buried cables, 5 m apart, 30 m long, both at 0.6 m depth. Cable A carries a 1 A test current (return through remote earth via a 1.5 m driven rod at one end); cable B is a passive observer whose induced voltage drop we want to read out across a frequency sweep.

import groundfield as gf

def build_two_parallel_cables(omega_frequencies):
    world = gf.create_world(soil=gf.HomogeneousSoil(resistivity=100.0))

    # Cable A: driven at x = 0, returns at x = 30.
    gf.create_electrode(world, "rod", name="A_start",
                        position=(0.0, 0.0, 0.5), length=1.5,
                        wire_radius=0.01)
    gf.create_electrode(world, "rod", name="A_end",
                        position=(30.0, 0.0, 0.5), length=1.5,
                        wire_radius=0.01)
    gf.create_conductor(
        world, name="cable_A",
        start="A_start", end="A_end",
        conductor_type="pen",
        wire_radius=0.005,
        cross_section="from_radius",
        coupling_to_soil="isolated",
        discretize_segment_length=1.0,
        inductance_model="neumann",      # activate Neumann kernel
    )

    # Cable B: parallel observer.
    gf.create_electrode(world, "rod", name="B_start",
                        position=(0.0, 5.0, 0.5), length=1.5,
                        wire_radius=0.01)
    gf.create_electrode(world, "rod", name="B_end",
                        position=(30.0, 5.0, 0.5), length=1.5,
                        wire_radius=0.01)
    gf.create_conductor(
        world, name="cable_B",
        start="B_start", end="B_end",
        conductor_type="pen",
        wire_radius=0.005,
        cross_section="from_radius",
        coupling_to_soil="isolated",
        discretize_segment_length=1.0,
        inductance_model="neumann",
    )

    # Source on cable A only.
    gf.create_source(world, attached_to="A_start", magnitude=1.0,
                     return_to="A_end")
    return world

frequencies = [50.0, 150.0, 250.0, 500.0, 1000.0]
world = build_two_parallel_cables(frequencies)
engine = gf.create_engine(
    backend="image",
    segment_length=0.5,
    frequencies=frequencies,
    earth_inductive_model="carson_series",   # Carson earth-return
)
result = engine.solve(world)

# Read the induced potential drop on cable B at every frequency.
phi_b_start = result.electrode_potentials["B_start"]
phi_b_end = result.electrode_potentials["B_end"]
for f, ps, pe in zip(frequencies, phi_b_start, phi_b_end):
    induced = pe - ps
    print(f"f = {f:6.0f} Hz   |U_B| = {abs(induced):8.4f} V   "
          f"phase = {induced and __import__('cmath').phase(induced):+6.2f} rad")

Earth-return models

The Carson 1926 series accounts for the current return through the soil and dominates the inductive coupling at typical LV / MV frequencies. The rigorous Sommerfeld kernel (earth_inductive_model="sommerfeld") is the absolute reference for layered soil; for homogeneous soil and frequencies well below 1 kHz the cheaper Carson series and the perfect-mirror baseline agree closely.

What to look at

• The induced cable-B end-to-end potential rises linearly with \(\omega\) — the magnitude doubles when frequency doubles.
• The phase is close to \(+\pi/2\) relative to the source current, consistent with \(j\omega M\,I_A\) inductive coupling through the mutual inductance \(M\).
• Switching earth_inductive_model from "perfect_mirror" to "carson_series" reduces the induced voltage by a factor that depends on \(\delta = 1/\sqrt{\pi f \mu_0 \sigma}\) (the Carson penetration depth). For 100 Ω·m at 50 Hz that's \(\delta \approx 1.4\;\mathrm{km}\) — much larger than the cable separation, so the correction is moderate.

Where to go next

• Read the engine theory pages for the full inductance kernels: docs/engines/mom_sommerfeld.md for the rigorous reference, docs/engines/image.md and docs/engines/image_2layer.md for the closed-form layered backends. The ADRs ADR-0004 (inductive coupling), ADR-0005 (Carson) and ADR-0006 (Sommerfeld) document the physics in depth.
• Visualise the magnetic-coupling pattern via the postprocess helpers — see 09 — All plots in action.