Reluctance

reluctance (less commonly), reluctancy

Physics a measure of the resistance of a closed magnetic circuit to a magnetic flux, equal to the ratio of the magnetomotive force to the magnetic flux

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reluctance [ri′lək·təns]

(electromagnetism)

A measure of the opposition presented to magnetic flux in a magnetic circuit, analogous to resistance in an electric circuit; it is equal to magnetomotive force divided by magnetic flux. Also known as magnetic reluctance.

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Reluctance

A property of a magnetic circuit analogous to resistance in an electric circuit.

Every line of magnetic flux is a closed path. Whenever the flux is largely confined to a well-defined closed path, there is a magnetic circuit. That part of the flux that departs from the path is called flux leakage.

For any closed path of length l in a magnetic field H, the line integral of H cos α dl around the path is the magnetomotive force (mmf) of the path, as in

(1) 

Eq. (1), where α is the angle between H and the path. If the path encloses N conductors, each with current I, Eq. (2) holds.

(2) 

Consider the closely wound toroid shown in the illustration. For this arrangement of currents, the magnetic field is almost entirely within the toroidal coil, and there the flux density or magnetic induction B is given by

(3) 

Eq. (3), where l is the mean circumference of the toroid and μ is the permeability. The flux Φ within the toroid of cross-sectional area A is given by either form of

(4) 

Eqs. (4), which is similar in form to the equation for the electric circuit, although nothing actually flows in the magnetic circuit. The factor l/μA is called the reluctance of the magnetic circuit. The reluctance is not constant because the permeability μ varies with changing flux density.

A toroidal coil

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Reluctance 

a characteristic of a magnetic circuit; the reluctance R m is equal to the ratio of the magnetomotive force F acting on a magnetic circuit to the magnetic flux Φ developed in the circuit. For a uniform section of a magnetic circuit the reluctance can be calculated from the formula R_m = l/μμ₀S, where l and S are the length and cross section of the section of the magnetic circuit, respectively; μ is the relative magnetic permeability of the material in the circuit; and μ₀ is the magnetic constant.

In the case of a nonuniform magnetic circuit composed of successive uniform sections having different values of l, S, and μ, the reluctance is equal to the sum of the values of R_m for the uniform sections. Such a computation is approximate, since the formula does not take into account “magnetic leakage” (the diffusion of magnetic flux into the space surrounding the magnetic circuit), the nonuniformity of the magnetic field in the circuit, and the nonlinear dependence of the reluctance on the field. In a variable magnetic field the reluctance is a complex quantity, because in this case μ, is a function of the frequency of the electromagnetic oscillations. The unit of reluctance in the International System of Units is the ampere or ampere-turn per weber (A/Wb), and in the cgs system of units, the gilbert per maxwell (Gb/Mx); 1 A/Wb = 4 × 10^-9 Gb/Mx ≈ 1.2566 × 10^-8 Gb/Mx.