Core Properties and Models

This page helps you choose how TRAFOLO models a magnetic core: the permeability model of the core material (constant, complex or a B-H curve), laminated stacks, the core-loss models, air gaps and permanent magnets. For each model it gives the runs that use it, what TRAFOLO computes and where the model stops being valid. You set the material models on the Materials tab, and the gaps and the magnet direction on the Core tab.

Choosing a permeability model

Each core material uses one permeability model, chosen under PERMEABILITY & LOSS MODELS on the General tab of the material form: Relative Permeability (1), Complex Permeability (2) or B-H Curve [T, A/m] (3). Core Loss Model (4) sets the core-loss formula of Transient runs: GSE or iGSE. In a Harmonic or Inductance project it offers only SE (Steinmetz + Eddy).

Core Material Type sets the kind of core: Ferrite (default), Powder and Composite are modelled the same way; Laminated adds the laminate model and Magnet adds a magnetization (both described further down this page).

Constant relative permeability

With Relative Permeability (1), B = μ₀·μ_r·H in the whole core, in every run type. The model is linear: it does not saturate and has no loss of its own; the core loss comes from the Core Loss tab.

The value to enter depends on the operating point. Datasheets quote several permeabilities, defined on the B-H loop below:

In a gapped core, the gap sets the inductance and the core permeability matters little. The effective permeability is μ_e = μ_r / (1 + μ_r·l_g/l_e), with l_g the total gap length and l_e the magnetic path length. For l_e = 50 mm and l_g = 0.5 mm, μ_r = 2000 gives μ_e = 95.2 and μ_r = 3000 gives 96.8: 50 % more μ_r, 1.6 % more inductance.

Complex permeability

Complex Permeability (2) describes a linear core with losses: μ = μ′ − j·μ″. The real part μ′ stores magnetic energy; the imaginary part μ″ holds the losses as measured (hysteresis and eddy currents in the grains). Ferrite datasheets give both against frequency.

Enter one row per frequency on the Complex μ tab: Frequency in Hz, Real μ′ and Imaginary μ″ as a positive number. For each run frequency, TRAFOLO interpolates linearly between the rows and passes the reluctivity ν = 1/(μ₀·μ) = (μ′ + j·μ″) / (μ₀·(μ′² + μ″²)) to the solver.

The core loss is part of the field solution: the loss density is p = π·f·μ₀·μ″·Ĥ² in W/m³, with Ĥ the peak field strength. The lossy core also shields the field, and the loss reaches the circuit: Rac Circuit (R AC source) and Q Factor on the Results tab include it. TRAFOLO does not use the Core Loss tab of a material with complex permeability.

B-H curve

What the curve describes

With B-H Curve [T, A/m] (3), the permeability follows the flux density, so the model captures saturation. TRAFOLO uses a single-valued (anhysteretic) curve: B averaged over the upper and lower branches of the hysteresis loop, without coercivity and remanence. The table holds its H ≥ 0 half.

The curve stores no magnetic history, so the hysteresis loss does not follow from the field solution: TRAFOLO computes it after the solve with the core-loss model (see Core loss models). Hysteresis models such as the Jiles-Atherton model need far more complex methods in 3D.

A B-H curve makes the problem nonlinear: the solver iterates until permeability and field agree, so runs take longer. If a run does not converge, set Solver Preset to Slow or lower the Nonlinear Relaxation Factor on the Solve tab.

Importing, fixing and extending the curve

On the B-H tab, Import Table reads a text file with two columns, B in T and H in A/m (file format on Materials). Both columns must rise from row to row. Manufacturers often give no curve or only a few points, and measured points can contain outliers that stop the nonlinear solver from converging. B-H Curve Optimization cleans and extends the curve.

Fixed Data removes:

• points where the curve stops rising (dB/dH ≤ 0);
• spikes: points whose segment is more than 10 times steeper than its two neighbouring segments together;
• the last point, when the last segment is steeper than the one before it.

Fixed + Extrapolated Data continues the fixed curve: the polarization J = B − μ₀·H approaches J_s exponentially, the differential relative permeability falls to 1, and the table ends at H = 10⁶ A/m with the slope of vacuum. Without the extension, a field beyond the last point of the table continues the curve at its last slope and can give too high a flux density.

The B-H curve in each run type

A harmonic run cannot show the non-sinusoidal flux of a saturating core. In the example (sinusoidal current, core driven into saturation), the transient run with the original curve (blue) clips the flux density at 0.31 T. The harmonic run with the same curve (green) gives a sine with about the same peak, so less flux at the fundamental and too low an inductance. A constant μ (orange) ignores saturation and reaches 1.04 T in both runs.

Effective B-H curves for harmonic runs

Some FEM tools replace the curve in harmonic runs with an effective curve that keeps the field energy of a transient solution, for example the Simple Energy curve B_SE(H) = (2/H)·∫₀^H B(H′) dH′ (G. Paoli et al., Complex representation in nonlinear time harmonic eddy current problems, IEEE Transactions on Magnetics, 1998; overview in Model Magnetic Materials in the Frequency Domain with an App). TRAFOLO does not compute such curves. You can import one as an ordinary B-H table, with two limits:

• it raises B above the saturation flux density (0.65 T against 0.34 T in the picture), so Steinmetz core losses computed from it come out too high;
• it holds only in harmonic runs: in transient runs it overshoots (SE transient, yellow, in the chart above).

Laminated cores

Electrical-steel cores are stacks of thin, insulated sheets. TRAFOLO models the stack as one homogeneous body: set Core Material Type (8) to Laminated and enter the sheet data.

The sheet eddy currents enter through the low-frequency laminate model, in every run type: the field strength gets the term σ·d²/12·dB/dt, which reduces the flux inside the stack as the frequency rises. In the example below (0.48 mm sheets, σ = 2.7 MS/m), the highest flux density in the core falls from 0.170 T at 0 Hz to 0.100 T at 1 kHz (−41 %) and 0.012 T at 10 kHz (−93 %).

With a Stacking Factor k below 1, TRAFOLO homogenizes the stack:

• B-H curve: B = k·B_steel(H) + (1 − k)·μ₀·H; constant and complex permeability: μ = k·μ_r + (1 − k);
• core loss: K_h × k⁻ⁿ and K_e × k⁻², because the datasheet loss holds at the flux density in the steel, B/k;
• density k·ρ for the loss totals and the thermal mass, and laminate conductivity σ/k.

Background on the laminate model: journal article (Electromagnetics, 2016).

Core loss models

TRAFOLO computes the core loss in one of three ways, depending on the permeability model and the run type:

Steinmetz + Eddy (Harmonic and Inductance runs)

P = K_h·f^m·Bⁿ + K_e·(f·B)²

f is the frequency in Hz, B the peak flux density in T, and P the loss density in the Units of the Core Loss tab: W/kg, mW/cm3 or W/m3 (TRAFOLO converts with the Density of the material). K_h, m (frequency exponent) and n (flux-density exponent) describe the hysteresis loss, K_e·(f·B)² the eddy loss, which counts only when K_e is ticked on the Core Loss tab.

Example with the coefficients of Ferroxcube 3F3 100C in the Core Loss tab picture below (W/m3), at 100 kHz and 0.1 T: 229.93 × (10⁵)^1.06 × 0.1^3.02 = 43.8 kW/m³ hysteresis loss, plus 326.76 × 10⁻⁶ × (10⁴)² = 32.7 kW/m³ eddy loss: 76.5 kW/m³ in total.

GSE and iGSE (Transient runs)

Transient runs evaluate the loss at every time step from the flux density waveform B(t) of each element, with the same K_h, m and n. For a sinusoidal flux both reduce to the Steinmetz law above. Choose the formula with Core Loss Model (4) on the General tab:

• GSE (default): P = (1/T)·∫₀^T k_i·|dB/dt|^m·|B(t)|^n−m dt, with k_i = K_h / ((2π)^m−1·∫₀^2π |cos θ|^m·|sin θ|^n−m dθ)
• iGSE: P = (1/T)·∫₀^T k_i·|dB/dt|^m·(ΔB)^n−m dt, with k_i = K_h / ((2π)^m−1·2^n−m·∫₀^2π |cos θ|^m dθ)

T is the period and ΔB the peak-to-peak flux density over it. The iGSE takes the flux swing ΔB instead of the instantaneous |B|, which suits non-sinusoidal flux such as the triangular and trapezoidal waveforms of PWM converters. Neither model captures a DC bias or the relaxation loss after a switching edge.

Core Loss tab

The coefficients come from a measured loss table, P against B at several frequencies, usually from the manufacturer (file format on Materials).

The core temperature for the thermal factor is the Predicted Steady-State Temperature on the Heat tab in the first pass. With Temperature Iterations 2 or 3 on the Setup tab, it is the temperature field of the previous pass. Some library materials, for example Ferroxcube 3C95 100C and 3C97 100C, have no thermal coefficients: their loss is the 100 °C loss at any temperature.

Air gaps: physical and virtual

A gap a few tens of µm thick between two core halves needs very small elements, which makes the mesh large and the run slow. Gap Type on the Core tab chooses how TRAFOLO models the gaps of a core group:

The reluctance of a virtual gap is part of the field solution, so Circuit Inductance includes it. Settings, the gap table of physical gaps and the cut for imported cores: Core.

Permanent magnets

TRAFOLO models a permanent magnet as a linear magnet: a core body with a constant magnetization and its recoil permeability. Demagnetization and other nonlinear effects are not modelled.

1. On the Setup tab, with Harmonic selected, tick Magnets (18). Inductance runs do not offer it.

   

2. On the Materials tab, add a core material with Core Material Type Magnet (19).

3. Set Relative Permeability to the recoil permeability μ_rec of the magnet, for example 1.05 for sintered NdFeB.

4. Set Coercivity (Magnetization) [A/m] (20) to the coercivity H_cB of the magnet. For a linear magnet, H_cB = B_r / (μ₀·μ_rec): B_r = 1.2 T and μ_rec = 1.05 give 909 kA/m. The value must be above 0.

   

5. Make the magnet a body of a core group and give it the magnet material on the Assembly tab. TRAFOLO magnetizes only Magnet bodies of core groups.

6. On the Core tab, open the group with Configure and set Direction of Magnetization (21) as a vector x; y; z: 0; 1; 0 (default), the +y direction. TRAFOLO scales it to unit length.

TRAFOLO applies the magnetization M = H_cB along that direction to every Magnet body of the group.

Further reading

• Fixing B-H Curve in FEM Simulations (LinkedIn)
• Materials: every setting of the material form and the file formats
• Material library: the materials that come with TRAFOLO
• Core: gap types and core group settings