4.3.2 The magnetic circuit

A core with two windows

Figure 4.3.3 shows an iron core with two windows and two windings, drawn as the currents $I_1$ and $I_2$ that they carry around the windows. The dashed blue line is the mean flux path, the line running along the middle of every branch of the core. It has six corners, labelled A to F, and seven branches: the outer ones $DA$, $AF$, $FE$, $EB$, $BC$ and $CD$, and the central limb $AB$ that the two windows share. Each branch has its own mean length, cross-sectional area and permeability, which for the branch $DA$ we call $l_{DA}$, $S_{DA}$ and ${\mu }_{DA}$ (labelled in the figure).

The core guides the flux. Because ${\mu }_{\mathrm{iron}}\gg {\mu }_0$, almost all lines of $B$ stay inside the iron and run along it (we shall return to the "almost" in Section 4.3.8), so inside the branch $DA$ the field $B$ is parallel to the mean path and, if the branch is thin compared with its length, nearly uniform over its cross-section. Two approximations are therefore made:

• in every branch the flux is $\Phi =BS$, with $B$ the field on the mean path, and
• the line integral of $H$ along the branch is $Hl$, with $l$ the length of the mean path.

One branch at a time

Take the branch $DA$. Its flux is

Substituting the constitutive relation $B_{DA}={\mu }_{DA}{H}_{DA}$ of Section 4.2, we can express the field intensity in the branch through its flux:

The contribution of the branch to the circulation of $H$ in loop 1 is therefore

Similarly on the branch $AB$,

and on the branches $BC$ and $CD$

Adding the four pieces, exactly as in Eq. (4.3.2), and equating the sum to the current linked by loop 1, we obtain the circulation of $H$ around loop 1:

Every term has the same form: the flux of the branch multiplied by a factor $l/(\mu S)$ that depends only on the geometry and the material of the branch. That factor deserves a name.

The unit follows from $Z\Phi =Hl$: amperes divided by webers, and since $Wb/A=\mathrm{H}$ this is ${\mathrm{H}}^{-1}$. Long, thin branches of poor material have large reluctance. Short, fat branches of good iron have small reluctance. With reluctances the loop equation (4.3.6) becomes

and the same reasoning around loop 2 ($A\to F\to E\to B\to A$), which links the current $I_2$, gives

Here ${\Phi }_{BA}$ is the flux of the central branch counted in the direction $B\to A$, the direction of travel of loop 2, so that ${\Phi }_{BA}=-{\Phi }_{AB}$ while $Z_{BA}=Z_{AB}$.

The fluxes at the corners

Equations (4.3.8) and (4.3.9) contain seven unknown fluxes, one per branch. The node rule (4.3.3) reduces them to two. Along the outer branches of loop 1 there is no junction, so the same flux runs through $BC$, $CD$ and $DA$,

and in the same way along the outer branches of loop 2,

At the corner A the flux ${\Phi }_1$ arrives from D, the flux ${\Phi }_2$ leaves towards F, and the rest goes down the central limb towards B. "What flows in must flow out" at A gives

and the same balance at B is automatically satisfied. Substituting Eqs. (4.3.10) to (4.3.12) into Eqs. (4.3.8) and (4.3.9), and using ${\Phi }_{BA}=-({\Phi }_1-{\Phi }_2)$ in the second, we obtain two equations for the two loop fluxes.