Magnetic circuits

Magnetic circuit is a model to analyze magnetic flux flow, similar to an electric circuit

• Flux ($\Phi$): magnetic field lines passing through a cross sectional area $A_c$

• MMF (Magnetomotive force): $\text{MMF} = Ni$, where $N$ is the number of coil turns and $i$ is the current

  $$ \Phi = \frac{\text{MMF}}{\mathcal{R}} $$

  • $\text{MMF} = Hg = Hl$ by Ampere’s law, where $H = \frac{B_{core}}{\mu _0}$

• Reluctance ($\mathcal{R}$): opposition to flux, analogous to resistance in electric circuits

  $$ \mathcal{R} = \frac{l}{\mu A_c} $$

Reluctance and Magnetic Flux

To solve a magnetic circuit:

1. Compute $\mathcal{R}$ for each path (i.e., core, air gap)
2. Build the equivalent magnetic circuit
   1. Represent MMF as a voltage source
   2. Reluctances as resistors in series or parallel
3. Solve for flux
4. Use flux to calculate quantities like flux density $B$ or inductance $L$

Permeability

1. Air gaps or free space

   • Use $\mu _0$, because air’s relative permeability is $\mu _r \approx 1$

2. Magnetic cores

   • Use $\mu _c = \mu _r \mu _0$, where $\mu _r >> 1$

3. Materials with infinite permeability (idealized cases)

   • Assume $\mu _c → \infty$, resulting in zero reluctance

     $$ \mathcal{R}_{core} = \frac{l}{\mu _cA_c}= 0 $$

4. Mixed paths (core and air gap)

   • Calculate reluctance separately for the air fap $\mu _0$ and the core $\mu _c$, then combine in series

     $$ \mathcal{R}{total} = \mathcal{R}{core} + \mathcal{R}_{gap} $$

MMF vs EMF

MMF

EMF