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## Main Question or Discussion Point

Capacitor-start induction motors use a capacitor in series with the start winding to create torque to start the motor. After the motor gets nearly up to full speed, a switch takes the capacitor out of the circuit. I understand the torque is caused by the phase shift created by having the capacitor in series with the start winding. I also understand that the phase shift causes the magnetic field generated by the stator to rotate and why that causes torque.

What I'm not sure I understand is why a 90° phase shift results in maximum torque. Even a small phase shift will cause the magnetic field to rotate at the frequency of the input current. So, why does a larger phase shift (up to 90°) result in more torque?

I've been thinking about this question in terms of a simple 2-pole induction motor with the start and run windings physically oriented 90° from each other. If the current in the run winding is given by sin(Θ), then the phase-shifted current in the start winding is given by sin(Θ + Φ). Therefore, the magnitude of the magnetic field is proportional to sqrt((sin(Θ))^2 + (sin(Θ + Φ))^2).

I decided to draw some plots (linked at the end of this paragraph) to help see what is going on. The x-axis is the electrical angle of the current in the run winding. The black line is the relative magnitude of the stator's magnetic field. The green line is the physical angle of the stator's magnetic field. I plotted it over half a cycle to avoid the arctangent's quadrant ambiguity. I made plots for the following phase angles: 0°, 5°, 30°, and 90°.

In the 90° phase shift case, the magnetic field angle changes linearly. The derivative of this angle is related to the change in magnetic flux, which is related (by Lenz's law) to how much current is induced in the rotor. Furthermore, in the 90° case, this slope is large throughout the plot compared to the other phase shifts. Is this why the torque is maximized with a 90° phase shift?

The other interesting thing I noticed is that the magnetic field doesn't rotate at a constant angular velocity (slope of the green line), except in the 90° phase shift case. My understanding is that PSC induction motors operate continuously with a phase shift much less than 90°. Wouldn't that, together with the varying magnetic field magnitude, cause a varying torque on the rotor? Wouldn't that varying torque cause the motor to vibrate?

Of course, my analysis ignores the presence of the rotor and assumes an empty stator. Perhaps the magnetic field behaves very differently in a real motor.

I appreciate any insight.

What I'm not sure I understand is why a 90° phase shift results in maximum torque. Even a small phase shift will cause the magnetic field to rotate at the frequency of the input current. So, why does a larger phase shift (up to 90°) result in more torque?

I've been thinking about this question in terms of a simple 2-pole induction motor with the start and run windings physically oriented 90° from each other. If the current in the run winding is given by sin(Θ), then the phase-shifted current in the start winding is given by sin(Θ + Φ). Therefore, the magnitude of the magnetic field is proportional to sqrt((sin(Θ))^2 + (sin(Θ + Φ))^2).

I decided to draw some plots (linked at the end of this paragraph) to help see what is going on. The x-axis is the electrical angle of the current in the run winding. The black line is the relative magnitude of the stator's magnetic field. The green line is the physical angle of the stator's magnetic field. I plotted it over half a cycle to avoid the arctangent's quadrant ambiguity. I made plots for the following phase angles: 0°, 5°, 30°, and 90°.

In the 90° phase shift case, the magnetic field angle changes linearly. The derivative of this angle is related to the change in magnetic flux, which is related (by Lenz's law) to how much current is induced in the rotor. Furthermore, in the 90° case, this slope is large throughout the plot compared to the other phase shifts. Is this why the torque is maximized with a 90° phase shift?

The other interesting thing I noticed is that the magnetic field doesn't rotate at a constant angular velocity (slope of the green line), except in the 90° phase shift case. My understanding is that PSC induction motors operate continuously with a phase shift much less than 90°. Wouldn't that, together with the varying magnetic field magnitude, cause a varying torque on the rotor? Wouldn't that varying torque cause the motor to vibrate?

Of course, my analysis ignores the presence of the rotor and assumes an empty stator. Perhaps the magnetic field behaves very differently in a real motor.

I appreciate any insight.