Magnetic Field¶

Accurate magnetic field distribution calculations are essential for: - Winding loss predictions (proximity effect depends on local field) - Leakage inductance calculations - Thermal analysis (loss distribution) - Saturation checking (local flux density)

MKF implements several magnetic field models, each with different trade-offs between accuracy and computational cost.

Available Models¶

Binns-Lawrenson¶

The Binns-Lawrenson method uses Fourier series expansion to solve the magnetic field in the winding window:

\[H(x,y) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} H_{nm} \sin(n\pi x/a) \sin(m\pi y/b)\]

Advantages: - Handles arbitrary conductor positions - Good accuracy for typical transformer geometries - Computationally efficient

Default model in MKF for most calculations.

Reference: Binns, Lawrenson. 'Analysis and Computation of Electric and Magnetic Field Problems.' Pergamon Press

Dowell¶

Dowell's 1D field model assumes uniform current sheets:

\[H(y) = \frac{N \cdot I}{w} \cdot \frac{y}{h}\]

Where \(w\) is winding width and \(h\) is window height.

Limitations: 1D approximation, best for full-width layers. Advantage: Very fast computation.

Reference: Dowell, P.L. 'Effects of eddy currents in transformer windings.' Proc. IEE, 1966

Wang¶

Wang's model provides improved 2D solutions for planar magnetic structures, accounting for end effects and non-ideal layer positioning.

Reference: Wang et al. 'Analysis of Planar E-Core Transformer.' IEEE Trans. Magnetics, 2009

Albach¶

Albach's 2D analytical solution for round conductors provides accurate field distributions even with partial layers and non-uniform spacing.

Reference: Albach et al. 'Calculating core losses in transformers.' IEEE Trans. Magnetics, 2011

Fringing Field Models¶

Gap fringing affects the field distribution near air gaps, causing: - Increased losses in conductors near the gap - Modified flux density distribution - Potential localized saturation

Roshen Fringing Model¶

Roshen's analytical model for fringing field around air gaps:

\[H_{fringe}(r) = H_g \cdot \frac{2}{\pi} \arctan\left(\frac{l_g}{2r}\right)\]

Where \(r\) is the distance from gap edge.

Reference: Roshen, W. "Fringing Field Formulas and Winding Loss Due to Fringing Field." IEEE Trans. Magnetics, 2007. IEEE

Albach Fringing Model¶

Albach's 2D field solution includes fringing effects in the overall field calculation, providing smooth transitions between gap and non-gap regions.

Design Tip: Keep conductors at least 2-3 gap lengths away from the air gap to minimize fringing-induced losses.

Model Comparison¶

Model	Error	Reference

Model Selection Guide¶

Application	Recommended Model	Notes
Standard transformers	Binns-Lawrenson	Best general accuracy
Quick estimates	Dowell	Fast 1D approximation
Planar magnetics	Wang or Albach	Handle wide, flat windings
Near air gaps	Include fringing models	Critical for loss accuracy

Grid Resolution Settings: - magnetic_field_number_points_x: X-axis grid points (default: 20) - magnetic_field_number_points_y: Y-axis grid points (default: 20)

Increase for more accurate leakage inductance; decrease for faster computation.