Reluctance and permeance expressions

Reluctance ($S$ or $\mathcal{R}$) is the opposition that a magnetic circuit offers to the setting up of flux. It is the ratio of mmf to flux, and its unit is AT/Wb (or H$^{-1}$). Permeance ($P$) is the reciprocal of reluctance. It measures the ease with which flux is set up, and its unit is Wb/AT (henry).

Derivation for a uniform circuit

Take a core of mean length $l$, cross-section $A$ and permeability $\mu = \mu_0\mu_r$, carrying a coil of $N$ turns and current $I$.

$$\begin{aligned} \text{mmf } F &= NI \\ H &= \frac{NI}{l},\qquad B = \mu H = \frac{\mu NI}{l} \\ \Phi &= BA = \frac{\mu A\,NI}{l} = \frac{NI}{\dfrac{l}{\mu A}} \end{aligned}$$

Comparing with $\Phi = \dfrac{\text{mmf}}{\text{reluctance}}$:

$$S = \frac{l}{\mu A} = \frac{l}{\mu_0\mu_r A}, \qquad P = \frac{1}{S} = \frac{\mu A}{l}$$

For other types of circuit

• Air gap of length $l_g$: $S_g = \dfrac{l_g}{\mu_0 A_g}$.
• Series circuit (a core of several sections, or a core with a gap): the same flux flows through each part, so

$$S = S_1 + S_2 + \dots,\qquad NI = \Phi(S_1 + S_2 + \dots)$$

• Parallel circuit (for example the legs of a three-limb core): the mmf across the branches is common, so

$$P = P_1 + P_2 + \dots \Rightarrow \frac{1}{S} = \frac{1}{S_1} + \frac{1}{S_2} + \dots$$

• Non-uniform section: in general, add up the reluctances of small elements: $S = \displaystyle\int \frac{dl}{\mu A(l)}$.

Points to note: $S$ is small for iron (high $\mu_r$) and large for air, so even a short air gap contributes a large share of the total reluctance. Unlike resistance, reluctance does not dissipate energy; the energy used to set up the flux is stored in the magnetic field.