reluctances

Reluctance is the opposition to the passage of magnetic flux within a magnetic circuit. It is the magnetic analogue of electrical resistance, linking the magnetomotive force to the resulting flux. In a simple, uniform path of length l and cross-sectional area A within a material of permeability μ, the reluctance is Rm = l/(μ A), where μ = μ0 μr and μ0 is the vacuum permeability. The relationship between flux Φ (in webers) and magnetomotive force F (in ampere-turns) is F = Φ Rm. Since Φ = B A and B = μ H, with H = F/l for a uniform leg, the flux depends on the material properties, geometry, and the applied MMF.

The reciprocal of reluctance is permeance, Pm = 1/Rm = μ A / l. In a magnetic circuit with multiple paths, reluctances add in series (R_total = Σ Rm) and in parallel combine as 1/R_total = Σ (1/Rmi). This mirrors Ohm’s law in electrical circuits.

Practical considerations include the impact of air gaps, which have very low μ and therefore produce large reluctance even for small gaps. Reluctance can be nonlinear: the permeability μ varies with flux density, especially near saturation, causing the effective reluctance to change with operating point. Designers consider reluctance in the cores of transformers and inductors, magnetic circuit paths in electric machines, and reluctance-based sensors and motors.

Overall, reluctance provides a framework for analyzing magnetic circuits using geometry, material properties, and a form of circuit law analogous to resistance.