Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

4.3.9 Recap

Recap

Two rules: around every closed path $\sum H_{i} l_{i} = \sum N I$ (Ampère's law), and at every node $\sum Φ = 0$ (flux lines do not end).
Reluctance $Z = l / (μ S)$ $[H^{- 1}]$ , mmf $F_{m} = N I$ $[At]$ , and the loop rule $\sum Z Φ = \sum N I$ , Eq. (4.3.16). The dictionary of Table 4.3.1 turns every magnetic circuit into a resistor network: series $Z_{1} + Z_{2}$ (same flux), parallel $Z_{1} Z_{2} / (Z_{1} + Z_{2})$ (same nodes), mesh equations (4.3.13)–(4.3.14) for several loops.
The air gap: $Z_{g} / Z_{c} = μ_{r} l_{g} / l_{c}$ , so a gap of $l_{c} / μ_{r}$ already equals the whole iron path, and in most gapped circuits the gap sets the flux, $H_{g} \approx N I / l_{g}$ .
Nonlinear iron: forget $μ$ , read $H$ from the $B$ – $H$ curve at the $B = Φ/ S$ of each branch, write $H l$ for the iron and $B l_{g} / μ_{0}$ for the gaps, and add around the loop, Eq. (4.3.21). Saturation near $2 T$ is the ceiling, the hysteresis loop area is the loss per cycle, soft materials for cores and hard ones for magnets.
Devices: the transformer ( $N_{1} i_{1} - N_{2} i_{2} = Z_{c} Φ$ , ideal when $Z_{c} \to 0$ , $V_{rms} = 4.44 f N B_{m a x} S$ , magnetising inductance $N_{1}^{2} / Z_{c}$ ), the motor (two gaps in series with the yoke and the rotor), the write head (a nanometre-scale gapped ring, $H_{g} \approx N I / l_{g}$ ), the reluctance motor ( $T = \frac{1}{2} i^{2} d L / d ϑ$ , pulled to minimum reluctance) and the lifting magnet ( $F = B^{2} S /2 μ_{0}$ per pole face).
Fringing and leakage: fringing enlarges the gap area, $S_{eff} = (a + l_{g}) (b + l_{g})$ , and lowers $Z_{g}$ . Leakage is a reluctance in parallel, and the coil carries $k_{ℓ} \approx 1.1$ to $1.4$ times the useful flux.