4.3.10 Problems and solutions

A magnetic circuit is solved with two rules that are nothing but the two fundamental equations of magnetostatics written branch by branch: the loop rule ${\sum }_{\mathrm{loop}}{H}_i{l}_i=\sum NI$ from ${\oint }_{C}H\cdot dl=I$, and the node rule $\sum \Phi =0$ from ${\oint }_{S}B\cdot dS=0$. With $\Phi =BS$ and $B=\mu H$ each branch contributes $H_i{l}_i=Z_i{\Phi }_i$ with the reluctance $Z=l/(\mu S)$. In iron, however, $\mu$ is not a constant, so in Problems 4.3.1–4.3.3 the value of $H$ for a given $B$ is read from the $B$–$H$ curve of the material (the smooth curves of Figures 4.3.19, 4.3.21 and 4.3.23), and only the air gaps are treated with $H_g=B_g/{\mu }_0$. As in the lecture, the three problems take ${\mu }_0=1/800000H/m=1.25\times 10^{-6}H/m$, which keeps the gap terms round numbers. The home exercises and the further exercises use ${\mu }_0=4\pi \times 10^{-7}H/m$. In the circuit diagrams iron reluctances are orange and air-gap reluctances cyan.