Calculation of impedance of a ferrite toroidal inductor – from first principles

A toroidal inductor is a resonator, though it can be approximated as a simple inductor at frequencies well below its self resonant frequency (SRF). Lets take a simple example, a ferrite toroid of rectangular cross section.

From the basic definition $$\mu=B/H$$ we can derive the relationship that the flux density in the core with current I flowing through N turns is given by $$B=\frac{\mu_0 \mu_r N I}{2 \pi r}$$.

The incremental flux at any incremental radius is proportional to the flux path length $$2 \pi r$$, so the total flux due to B(r) is $$\Phi_B=\int_{a}^{b}Bc \, dr=\frac{\mu_0 \mu_r N I}{2 \pi r}c \, dr=\frac{\mu_0 \mu_r N I}{2 \pi} c \, ln \frac b a$$.

Note that the core geometry is captured in the term $$c \, ln \frac b a$$.

From that we can calculate inductance $$L \equiv \frac{N \Phi_B}I=\frac{\mu_0 \mu_r N^2}{2 \pi} c \, ln \frac b a$$ where $$\mu_0=4\pi 10^{-7}$$, the permeability of a vacuum.

Ferrite datasheets commonly give $$\mu_r=\mu^{\prime}-\jmath \mu^{\prime\prime}$$, a complex value (note it is usually a frequency dependent parameter). The imaginary term represents the core loss.

We can calculate the impedance at frequency f by substituting the values.

$$Z=\jmath 2 \pi f L=\jmath 2 \pi f \frac{\mu_0 (\mu^{\prime}-\jmath \mu^{\prime\prime}) N^2}{2 \pi}c \, ln \frac b a \\$$

The model can be improved for frequencies approaching SRF by addition of a small equivalent shunt capacitance $$C_s$$.

$$Z=\frac1{\frac1{\jmath 2 \pi f \frac{\mu_0 (\mu^{\prime}-\jmath \mu^{\prime\prime}) N^2}{2 \pi} c \,ln \frac b a}+ \jmath 2 \pi f C_s}$$

Calculator

The calculator Calculate ferrite cored inductor – rectangular cross section does exactly this calculation. Note that a real FT240-43 has chamfered corners, so these calculations based on sharp corners will very slightly overestimate L, but the error is trivial in terms of the tolerance of µr.

µr comes from the datasheet, but you may find Ferrite permeability interpolations convenient.

The value Cs is best obtained by observation of SRF of a particular winding, it is sensitive to winding layout.

The calculated value $$\sum{\frac{A}{l}}=\frac{c \, ln \frac b a}{2 \pi}$$ and captures the core geometry in a more general form. It or its inverse often appear in datasheets and can be used to calculate Z (Calculate ferrite cored inductor – ΣA/l or Σl/A).