Why does a 90° phase shift maximize induction motor torque?

[linxcat]

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.

 

[anorlunda]

Wikipedia says

Larger single phase motors are split-phase motors and have a second stator winding fed with out-of-phase current; such currents may be created by feeding the winding through a capacitor or having it receive different values of inductance and resistance from the main winding. In capacitor-start designs, the second winding is disconnected once the motor is up to speed

That suggests that it is more complex. Not only do you have a phase shift, you have two windings with different phases interacting with each other.

 

[Frogghorn]

Linxcat,

This is a fascinating a esoteric question, indeed.

You basically answered your own question. With a 90-degree phase shift, the linear change in field angle maximizes torque throughout the cycle. This varying rate of change in field angle for lesser phase shifts is why motors will sometimes start with a missing run capacitor. If the power switch is closed during a part of the cycle with a large change in angle, the motor will have enough torque to start, but most of the time that doesn't happen. A motor with no phase shift would never start, but there is enough difference in inductance between start and run windings in a typical motor that starting is sometimes still possible without a capacitor. Even so, this type of motor should never be run without a run capacitor because excessive current will flow through the windings.

You are also correct about the vibration at low speeds. However, the vibration is reduced by an effect called proton depletion resonance. As the rotor reaches a high speed, magneto-centrifugal force causes protons to escape their nuclei and concentrate at the outer surface of the rotor. As this happens, an electro static force tries to pull protons back to where they belong, and they begin oscillating. This causes an oscillation in the magnetic field that almost completely counteracts the vibratory forces. The reason for this almost perfect counteraction is beyond the scope of this explanation, but the effect is really quite extraordinary.

I hope this explanation satisfies your curiosity, but feel free to ask additional questions!