A two-phase permanent-magnet stepper motor has 50 rotor...

A two-phase permanent-magnet stepper motor has 50 rotor teeth. When the rotor is driven by an external mechanical source at $\omega=100 \mathrm{rad} / \mathrm{s}$, the measured open circuit phase voltage is 25 V , peak-to-peak. Calculate $\lambda$. If $i_a=1 \mathrm{~A}$ and $i_b=0$, express the developed torque. Assume the winding resistance is $0.1 \Omega$.

Key Concepts

Back?EMF and Flux Linkage

In a rotating permanent?magnet machine the induced (back) electromotive force (EMF) is produced due to the time rate of change of magnetic flux linkage. When a phase winding is open?circuited and the rotor rotates at a constant speed, the amplitude of the sinusoidal induced voltage is directly related to the product of the rotational speed and the coil’s flux linkage. This concept allows one to determine the flux linkage constant (?) from a measurement of the open?circuit voltage and the known angular speed using the relation e = ?·?.

Rotor Geometry and Step Angle

Stepper motors are characterized by discrete rotor positions formed by features such as teeth. The number of rotor teeth determines the step angle (angular displacement per step), which is the mechanical angle corresponding to one full electrical cycle or step. This step angle is crucial not only to the motor’s resolution but also enters into derivations of the torque?producing characteristics since the change in flux linkage with respect to angle (d?/d?) is often approximated over the span of one step.

Electromagnetic Torque Production

The developed torque in permanent?magnet machines is based on the interaction between the current in the stator windings and the permanent magnet field of the rotor. In general, the electromagnetic torque can be computed by taking the derivative of the magnetic energy (or co?energy) with respect to the rotor position. When a winding is excited so that its current is known, the torque is proportional to the rate at which the flux linkage changes with rotor angle (d?/d?). In problems where the flux linkage variation is assumed to have a sinusoidal or linear form over a step, the step angle (derived from the rotor’s geometry) is used in the calculation.

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