# Torque RPM Characteristics of Single Phase Induction Motor within Maxwell's Equations

A single phase induction motor is constructed of a stator, which contains the stationary main winding, and a rotor, which is a conductive cage which rotates. The stationary main winding is a two pole coil. The rotor is not a permanent magnet, and it is not connected to the AC supply voltage: it is a brushless motor. However, due to the AC current in the main winding of the stator, an oscillating magnetic field is created, which induces an EMF in the rotor bars, driving a current through the rotor. By Lorentz Force law, a force and thus torque is exerted on the rotor.

I am now attempting to compute the torque in terms of angular speed of the motor. For low angular speeds this formula is at least qualitatively correct, it is expected that torque is linearly proportional to the angular speed of the rotor, but at the full load torque, the torque no longer varies linearly, but looks like the black line in the image (ignore the red and blue lines for now): The cause of the torque no longer being linearly proportional to the speed is probably due to losses, and in the literature the rotor is modelled as a secondary winding of a transformer circuit, with the mechanical load on the shaft as resistance, and additional rotor reactance to model flux losses.

My analysis seems to miss out the zero torque near synchronous speed criterion. What phenomenon is responsible for this which I have not included?

My question is: within the context of Maxwell's Equations (physically), how can I calculate these losses using the geometry of the motor?

However, due to the AC current in the main winding of the stator, an oscillating magnetic field is created, which induces an EMF in the rotor bars, driving a current through the rotor.

My analysis seems to miss out the zero torque near synchronous speed criterion. What phenomenon is responsible for this which I have not included?

Although a transformer equivalent circuit is used to analyze induction motors, the EMF induced in the rotor depends on viewing the stator as producing a rotating magnetic field. The rotor emf requires a difference between the rotor speed and the speed of the stator field. Unless the motor is acting as a generator, slip has a value between -1 and +1. It is calculated as s = (n1-n)/n1 where n is the operating speed and n1 is the synchronous speed.

If you look at the equivalent circuit of the motor, you can see that the resistor that models the conversion of electrical energy to mechanical energy has a value of infinity when s = 0. When s = 1, the value of that resistor is zero. At hat point, all of the electrical power received by the motor is lost as heat in the motor. The majority of the heat is dissipated in the physical resistance of the rotor, Rr, a lot of heat is dissipated in the stator resistance, Rs, and the normal amount of iron losses are dissipated.

The torque at zero speed is less than the peak torque mostly due to the voltage drop across Xs and Xr due to the high current drawn by the rotor.

Note that the equivalent circuit below is for one phase of a three-phase motor. A single phase motor is a bit more complicated and essentially requires two such circuits that give you the red and blue curves in your diagram. Diagram adapted from Malcom Barnes "Practical Variable Speed Drives and Power Electronics"

My question is: within the context of Maxwell's Equations (physically), how can I calculate these losses using the geometry of the motor?

That is not really what is needed.

• Thank you for the answer! In my case I am interested in the design of the motor and not the operation. Do you know of any books which detail the physics of single-phase induction motors in detail? May 9, 2021 at 16:22
• There is the one that I credited below the diagram. I mentioned some more in a comment to another question that may be hidden. I will find that and add another comment.
– user80875
May 9, 2021 at 16:41
• Some leading texts that I know of are: Fitzgerald & Kingsley's Electric Machinery by Stephen Umans; Electric Machinery Fundamentals by Stephen J. Chapman; Introduction to Ac Machine Design by Thomas A. Lipo.
– user80875
May 9, 2021 at 17:00