Calculation of electromagnetic parameters of toroidal core — for composite soft magnetic material

1. Composite Structure Assumptions

Assume the toroidal core consists of two materials ‌concentrically laminated‌ (e.g., ferrite in the outer layer and nanocrystalline in the inner layer, or vice versa), with the magnetic field distributed along the closed annular path, ignoring edge effects.

Define:

Inner layer material‌: Nanocrystalline soft magnetic material (thickness t1, permeability μ1, volume fraction V1)

‌Outer layer material‌: Ferrite soft magnetic material (thickness t2, permeability μ2, volume fraction V2)

Total magnetic path length‌:

consistent with a single-material ring.

‌2. Equivalent Permeability

The ‌effective permeability‌ (μeff) of the composite material depends on the lamination configuration:

*Series Model (same magnetic field direction)‌: If the materials are layered in series along the magnetic path (e.g., concentric inner/outer layers), the total magnetic reluctance is the sum of individual reluctances:

where 

*Parallel Model (perpendicular magnetic field direction)‌: If the materials are axially laminated in parallel (e.g., stacked vertically), the effective permeability is the volume-weighted average:

‌3. Inductance Calculation

Using the effective permeability μeff, the inductance formula is:

where

4. Loss Model

Total losses include ‌eddy current losses‌ and ‌hysteresis losses‌ from both materials:

The loss for each material is expressed as:

5. Saturation Characteristics

The ‌saturation flux density‌ (Bsat) of the composite core depends on the saturation thresholds of both materials:

where Bs1, Bs2 are the saturation flux densities of nanocrystalline and ferrite, respectively.

6. High-Frequency Corrections

At high frequencies, account for ‌skin effect‌ and ‌dielectric losses‌ with a correction factor:

Example Calculation‌

Assume the illustrated  toroidal core  parameters:

Step 1: Calculate effective permeability (series model)

Step 2: Calculate inductance