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07 — Comparison of measurement distances on a synthetic grounding system

A fall-of-potential test is the classical field method to measure the grounding impedance of a substation: inject a known current \(I\) between the substation and an auxiliary current electrode (Hilfserder) placed at some distance \(D\), then measure the substation potential against a voltage probe (Spannungssonde) placed between the substation and the Hilfserder (or 90° from the current-feed axis). The reading is biased by the \(U_\text{probe}\) contribution; the bias vanishes only as \(D\) grows toward "remote earth".

This example builds a synthetic grounding world, runs the measurement at several Hilfserder distances and two probe geometries (0° inline vs 90° perpendicular), and shows the bias disappearing as the loop expands.

Build the synthetic world

import groundfield as gf
from groundfield.generators import OrtsnetzLayout

# Synthetic footprints lining a single street, plus a substation
# at the west end.
def make_house(cx, cy, dx=10.0, dy=10.0, osm_id=0):
    return gf.BuildingFootprint(
        polygon_xy_m=[
            (cx - dx / 2.0, cy - dy / 2.0),
            (cx + dx / 2.0, cy - dy / 2.0),
            (cx + dx / 2.0, cy + dy / 2.0),
            (cx - dx / 2.0, cy + dy / 2.0),
        ],
        building_use="residential",
        osm_id=osm_id,
    )

footprints = [make_house(40.0 + 25.0 * i,
                         +20.0 if i % 2 == 0 else -20.0,
                         osm_id=i)
              for i in range(8)]
layout = OrtsnetzLayout.from_footprints(
    footprints,
    substation_xy=(0.0, 0.0),
    obstacle_clearance_m=1.0,
)
layout.add_pen_cable(start="substation", end_xy=(250.0, 0.0),
                    min_segment_length_m=8.0)
layout.connect_buildings()

Configure the measurement loop

layout.add_auxiliary_electrode(distance_m=D, direction_deg=0.0) drops the Hilfserder (default geometry: a 3 × 0.5 m rod triangle, parallel-bonded) at radius \(D\) east of the substation. The matching layout.to_world(...) then wires a second current source with phase_deg=180 at the Hilfserder anchor so the engine sees a proper closed-loop pair (the v0.7 image solver ignores Source.return_to, so the workaround injects the return current explicitly).

Sweep over \(D\) with two probe positions

import math
import numpy as np

distance_sweep = [25.0, 50.0, 75.0, 100.0, 150.0, 200.0,
                  300.0, 500.0, 800.0]
PENETRATION = 0.30
SOURCE_MAGNITUDE_A = 1.0

sx, sy = layout.substation_xy
theta = math.radians(0.0)                    # Hilfserder direction = east
u_inline = (math.cos(theta), math.sin(theta))
u_perp = (-u_inline[1], u_inline[0])         # 90 deg CCW

z_inline, z_perp = [], []
fixed_mask = layout.foundation_mask(PENETRATION)
engine = gf.create_engine(backend="image",
                          segment_length=0.5,
                          frequencies=[50.0])

for D in distance_sweep:
    layout.add_auxiliary_electrode(distance_m=D, direction_deg=0.0)
    world = layout.to_world(
        foundation_mask=fixed_mask,
        source_magnitude_A=SOURCE_MAGNITUDE_A, seed=0,
    )
    result = engine.solve(world)

    half_D = 0.5 * D
    probe_inline = (sx + half_D * u_inline[0],
                    sy + half_D * u_inline[1])
    probe_perp = (sx + half_D * u_perp[0],
                  sy + half_D * u_perp[1])

    z_inline.append(layout.measured_grounding_impedance(
        result, source_magnitude_A=SOURCE_MAGNITUDE_A,
        probe_xy=probe_inline,
    ))
    z_perp.append(layout.measured_grounding_impedance(
        result, source_magnitude_A=SOURCE_MAGNITUDE_A,
        probe_xy=probe_perp,
    ))

for D, zi, zp in zip(distance_sweep, z_inline, z_perp):
    delta = abs(zp) - abs(zi)
    rel = 100.0 * delta / abs(zp) if abs(zp) else 0.0
    print(f"D = {D:5.0f} m  |Z_inline| = {abs(zi):7.3f}  "
          f"|Z_perp| = {abs(zp):7.3f}  delta = {delta:+.3f} ({rel:+.1f} %)")

What you should see

The two curves both converge to the same asymptote (the true substation grounding impedance) as \(D\) grows. For small \(D\) the inline probe sits in the substation's near field and reads too low, while the perpendicular probe stays closer to remote earth and reads closer to the asymptote. Above roughly \(D = 5 \cdot r_\text{sub}\) (where \(r_\text{sub}\) is the largest dimension of the substation grounding) the bias drops below a few percent for both placements.

Reproducibility and physics check

  • layout.foundation_mask(p) is deterministic in \(p\) and the footprint OSM id — the same penetration \(p = 0.30\) always picks the same houses, and increasing \(p\) only adds houses to the equipped subset.
  • layout.measured_grounding_impedance(result, probe_xy=...) reads \(\varphi_\text{sub} - \varphi_\text{probe}\) from a single solve — both probe geometries share the same engine call so the comparison is apples-to-apples.
  • layout.verify_current_balance(result) is the Kirchhoff plausibility check: positive side (substation + KVS + foundation electrodes) sums to \(+I_\text{src}\), auxiliary side sums to \(-I_\text{src}\).

Where to go next