Skip to main content

Ferrite toroid at radio frequencies

When you need this

At radio frequencies the permeability of ferrite becomes complex, and the self-capacitance of the winding stops being a detail. So what is worked out here is not an «inductance» but the full impedance of the equivalent circuit.

coil on a toroidLsESRCs
Ls — equivalent inductance, ESR — loss resistance, Cs — self-capacitance of the winding
Number of turns N
Operating frequency f (MHz)
Ferrite material (Fair-Rite)
Toroid size
Toroid chamfer C (mm)
Wire diameter dw (mm)
Use the thickest wire that fits in one layer
Result
Equivalent inductance Ls
Loss resistance ESR
Structural Q
Impedance magnitude |Z|
Self-capacitance of the winding Cs
AL at this frequency
Wire length
Winding layers
Recommended wire for a single layer
Ferrite at the operating frequency
Real part of the permeability μ′
Imaginary part of the permeability μ″ (loss)
Initial permeability μi (datasheet)

📐 Formulas​

1. Permeability at the working frequency. The μ′(f) and μ″(f) tables for 19 Fair-Rite materials are interpolated with monotone cubic interpolation against the logarithm of frequency. Monotone rather than a «natural» spline: an ordinary spline on a grid like this overshoots unphysically — negative losses and jumps of an order of magnitude.

2. Inductance:

C1 = 2π / (he·ln(OD/ID))
AL [nH/N²] = μ₀·μ′·10⁶ / C1
L [µH] = AL·N²·10⁻³

3. Self-capacitance of the winding (the G3YNH model): the «turn-to-turn» part does not depend on N, while the «end-to-end» part falls as 1/N²:

lp = OD − ID + 2h
le = 2π·ln(OD/ID) / (2/ID − 2/OD)
Cs [pF] = 1.13·10⁻³·C1·lp²/(0.4π) + 1.13·10⁻¹·C1·(0.9·le)²/(0.4π) / N²

4. Impedance:

X_L = 2π·f·L
Z_coil = X_L·(μ″/μ′) + j·X_L        the ferrite losses → the real part
Z = 1 / ( 1/Z_coil + j·2π·f·Cs )

Ls  = Im(Z) / (2πf)
ESR = Re(Z) + R_DC
Q   = Im(Z) / ESR

Notation:​

  • N — number of turns
  • f — working frequency [MHz]
  • μ′ — real part of the permeability (the magnetic behaviour)
  • μ″ — imaginary part (the losses in the ferrite)
  • Ls — equivalent inductance [µH]
  • ESR — loss resistance [Ω]
  • Cs — self-capacitance of the winding [pF]
Limits of the calculator:
  • Wire losses are counted at DC only: skin and proximity effects are not modelled, so the real ESR will be higher and the Q lower
  • Each table has its own frequency range; outside it the calculator refuses to extrapolate
  • The self-capacitance model is empirical: the real value depends on how the turns are laid and how far apart the ends of the winding are
  • If the reactive part comes out negative, the coil is working above its self-resonance and an inductance figure means nothing there

🎯 Where it is used​

  • Interference-suppression chokes: a high Re(Z) is not a defect, it is the working mechanism.
  • Tuned-circuit coils on toroids, where the Q is still acceptable.
  • Checking that the coil is not working above its self-resonance — the commonest hidden mistake.
Practical notes:
  1. Note how far AL at the working frequency departs from the datasheet value: the datasheet figure is given for low frequencies. That is the whole reason this calculator exists.
  2. A low Q together with a high impedance magnitude is normal for suppression materials (31, 43, 73, 75). In a tuned circuit such a coil is useless; as a choke it is excellent.
  3. The calculator immediately suggests the thickest wire that still fits in a single layer: a single-layer winding has the lowest self-capacitance.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Ferrite toroid at radio frequencies». CyberDev.Space. https://cyberdev.space/en/radio/calculators/coils/rf_ferrite_toroid (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

Last updated on