## Problem 1, part b

A developed diagram approach is used, "unrolling" the machine such that increasing $\phi$ (counterclockwise as drawn) is to the left.

Magnetomotive force (MMF) for a machine with concentrated windings may be found as $\mathcal{F} = \int_{\phi = 0}^{360^\circ} N(\phi) i \, d\phi$.

It is given that $N_{as} = \mp 1$ at an inferred position of $\phi_s = 0^\circ$ and $\phi_s = 180^\circ$. The current is $i_{as} = 1 \, \textrm{A}$.

Because $\oiint_S \vec{B} \cdot d\vec{S} = 0$, including on the interior surface of the stator and ends of the machine, the average value in the developed diagram of $B(\phi_s)$ must be 0. The average value of MMF must also be 0 because the airgap is constant, meaning that $\mathcal{F}(\phi_s)$ and $B(\phi_s)$ are proportional to each other in the airgap from $B(\phi_s) = \frac{\mu_0 \mathcal{F}(\phi_s)}{g}$. Thus, the offset of -1.0 A is removed.

## Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras. 