Magnetic circuit, similarities with electric circuit, Ohm's law

A magnetic circuit is the closed path followed by magnetic flux. It is usually made of high-permeability iron (sometimes with air gaps) and is excited by a coil carrying current, e.g. the core of a transformer, relay or electrical machine.

     +-----------------+
     |    phi -->      |
   ==|== N turns       |  iron core,
   ==|== I             |  mean length l,
     |                 |  area A
     +-----------------+

Similarities between magnetic and electric circuits

Points of difference: flux does not truly flow, no energy is spent to maintain a constant flux (but $I^2R$ loss occurs with current), and $\mu$ is not constant because iron saturates, while R is nearly constant.

Ohm's law for a magnetic circuit

Take a uniform ring of mean length $l$ (m), area $A$ (m²), relative permeability $\mu_r$, wound with $N$ turns carrying $I$.

By Ampere's circuital law, $\oint H\,dl = NI$, so

$$H = \frac{NI}{l}$$

Then

$$\begin{aligned} B &= \mu_0\mu_r H = \frac{\mu_0\mu_r NI}{l} \\ \phi &= BA = \frac{\mu_0\mu_r A\, NI}{l} \\ \phi &= \frac{NI}{\dfrac{l}{\mu_0\mu_r A}} = \frac{F}{S} \end{aligned}$$

So flux = mmf ÷ reluctance, which is Ohm's law for the magnetic circuit ($I = E/R$). For a composite circuit (iron + air gap) the reluctances add: $\phi = NI/(S_i + S_g)$.