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Inductance of a Gapped E-Core Inductor Compared with Measurements

TRAFOLO computes the inductance of a gapped E-core inductor within 5 to 7 % of the measurement, at gaps of 1.0 to 2.0 mm. The classic gap formula, which leaves out the fringing flux, is 31 to 43 % too low.

  • Result: 2.21, 1.65 and 1.35 mH against 2.07, 1.58 and 1.26 mH measured, at gaps of 1.0, 1.5 and 2.0 mm.
  • Source: J. Mühlethaler, J. W. Kolar, A. Ecklebe, "A Novel Approach for 3D Air Gap Reluctance Calculations", 8th International Conference on Power Electronics – ECCE Asia 2011, pp. 446–452, doi:10.1109/ICPE.2011.5944575.
The gapped E-core inductor in TRAFOLO as a half model, flux density per element in the cores and the winding
The inductor in TRAFOLO, half model at 1 mm gaps: B per element in the cores and the winding.

Case

The inductor as modelled at lg = 1 mm, to scale: symmetry plane z = 0 (left) and top view (right).
Part Material in the model Size [mm]
Core, two halves N27 mur2000: μr = 2000, σ = 0, no core loss TDK E 55/28/21: 55 × 27.8 × 21 each, centre leg 17.2 × 21
Air gap, in all three legs Air (any non-magnetic spacer) lg = 1.0, 1.5, 2.0
Winding Copper, Stranded & Litz, 80 turns On the TDK B66252 coil former, centred on the gaps
  • Excitation: 0.1 A rms at 1 and 10 kHz. The model is linear and has no eddy currents, so the inductance depends on neither.
  • Measured: one inductance per gap (paper, Table I).
  • Air box: automatic, 3 × the part.

Results

Inductance against air gap from 1 to 2 mm, measured, TRAFOLO and the classic gap formula
Inductance against the gap in each leg: measured, TRAFOLO and the classic gap formula.
Gap lg [mm] Measured [mH] TRAFOLO [mH] Difference [%] Classic gap formula [mH]
1.0 2.07 2.212 +6.9 1.42
1.5 1.58 1.652 +4.6 0.96
2.0 1.26 1.353 +7.4 0.72

TRAFOLO follows the measured fall of the inductance with the gap. The classic formula counts only the gap cross-sections and misses the fringing flux, so it is 31 to 43 % low.

Flux density per element on the symmetry plane around the gaps of the centre leg and the outer leg at a 1 mm gap
B per element on the symmetry plane at lg = 1 mm: centre leg (left), window with the winding, outer leg (right).

Near the gaps part of the flux leaves the core faces and bypasses the gap through the window and around the outer leg. This fringing widens the effective gap area: the measured inductance is 46 to 75 % above the classic formula, more at larger gaps.

TRAFOLO model

Built from STEP files of the two core halves and the winding (attached), one project per gap, in TRAFOLO 1.66.1beta.

MUH_lg1p0.stepMUH_lg1p5.stepMUH_lg2p0.step

Tab Setting
Setup DOMAIN TYPE Symmetric: a half model, cut at the mid-depth plane. ELECTROMAGNETICS Inductance.
Geometry CAD import of the two core halves, lg apart, and the winding, scale mm. The gaps are air.
Materials Cores: N27 mur2000 (μr = 2000, the initial permeability of N27). Winding: Copper.
Coil Stranded & Litz, Turns per solid 80.
ElMag Winding on the primary, current excitation.
Waveform Inductance Sweep, SWEEP BY Frequency: 1 and 10 kHz, 0.1 A rms.
Mesh Max Element Edge Size: cores 1 mm, winding 2 mm, air 10 mm; about 130 000 elements. The default sizes are too coarse for millimetre gaps.
Solve Linear Solver MUMPS on 16 cores: about 15 s per gap.

Checks

  • Mesh: 4.5 times the elements, at the smallest gap, change the inductance by 0.06 %.
  • Air box: doubling the automatic air box, at the largest gap, changes it by 0.5 %.
  • Core permeability: the 1 to 2 mm gaps hold 95 % or more of the reluctance, so ±25 % on μr moves the inductance by about 1 %.
All runs
Run Gap [mm] Mesh: cores / winding / air [mm] Air box, x and y [mm] Elements Mesh / solve time [s] L [mH] vs measured [%] Note
R0 1.0 1 / 2 / 10 ±84.9 126 102 28 / 13 2.212 +6.9 Reference run
R0 1.5 1 / 2 / 10 ±85.65 130 278 28 / 14 1.652 +4.6 Reference run
R0 2.0 1 / 2 / 10 ±86.4 131 017 27 / 15 1.353 +7.4 Reference run
R1 1.0 0.6 / 1.2 / 6 ±84.9 562 854 126 / 105 2.213 +6.9 Mesh check
R2 2.0 1 / 2 / 10 ±172.8 210 515 47 / 22 1.360 +8.0 Air box doubled

Each run solves 1 and 10 kHz on 16 cores; the half model reaches in z from 0 to the x value of the air box.

Limits

  • Core dimensions: nominal values of the current TDK datasheet; within its tolerance the centre-leg area can be up to 2.9 % smaller.
  • Permeability: μr = 2000 assumed; the measurement temperature is not stated.
  • Winding: wire and layers are not stated, so the winding is a uniform 80-turn block on the coil former.
  • Saturation: not compared; the paper's saturation current needs an N27 B-H curve at 25 °C.