Toroid Inductor Formulas and Calculator

Toroidal inductors are often used in pulsed power and power conditioning applications since the magnetic fields are largely confined within the volume of the form. All of the formulas on this page are shown assuming an air core toroidal inductor. If one uses a magnetic core as a form to wind the toroid on, the inductance of the toroid can be found by calculating the value from the appropriate formula shown below for the air core inductor and then multiplying that value by the relative permeability of the magnetic core material.

Toroids may be wound on a circular form as shown in the figure below:

Diagram of a Circular Cross Section Toroid Inductor

Diagram of a Circular Cross Section Toroid Inductor

The inductance for such a toroid can be calculated from the equation below:

Equation for Circular Cross Section Toroid Inductor

Equation for Circular Cross Section Toroid Inductor

where N is the number of turns, R is the mean radius of the form as shown in the figure (in cm), and a is the radius of the windings on the form as shown in the figure (in cm).

Another formula for the inductance of a circular cross section toroid is shown below:

Alternate Formula for Circular Cross Section Toroid Inductor

Alternate Formula for Circular Cross Section Toroid Inductor

where N is the number of turns, D is the mean diameter of the form as shown in the figure (in inches), and d is the diameter of the windings as shown in the figure (in inches).

They may also be wound on a rectangular form as shown in the figure below:

Diagram of a Square Cross Section Toroid Inductor

Diagram of a Square Cross Section Toroid Inductor

The inductance for a rectangular cross section toroid can be found from the following equation (Terman, Frederick E., Radio Engineers Handbook, McGraw-Hill, New York, 1943, p58.):

Equation for a Square Cross Section Toroid Inductor

Equation for a Square Cross Section Toroid Inductor

where N is the number of turns, h is the height of the winding (in inches), d1 is the inner diameter (in inches), and d2 is the outer diameter (in inches).

A second formula for a rectangular form toroid is shown below:

Alternate Equation for a Square Cross Section Toroid Inductor

Alternate Equation for a Square Cross Section Toroid Inductor

where N is the number of turns, h is the height of the winding (in cm), r1 is the inner radius (in cm), and r2 is the outer radius (in cm).

The calculators below can be used to determine the proper parameters for either a circular or square cross section Toroid inductor. Credit for the initial Javascript code used in the calculator is given to Ray Allen who has a number of similar useful calculators on his website, Pulsed Power Portal.

Calculate Circular Cross Section Toroid Inductor Parameters

Parameter Value Units
Inputs: Coil Mean Radius (R)
Winding Radius (a)
N (# of Turns)
Outputs: L (Inductance)
Coil Mean Diameter (D)
Coil Inner Diameter
Coil Outer Diameter

Calculate Square Cross Section Toroid Inductor Parameters

Parameter Value Units
Inputs: Coil Winding Height (h)
Coil Inner Radius (r1)
Coil Outer Radius (r2)
N (# of Turns)
Outputs: L (Inductance)
Coil Mean Diameter (D)
Coil Inner Diameter
Coil Outer Diameter

Output Format:

Select Format: Scientific Engineering Fixed


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