Induction motor equations
Aim : to find parameters that define 50hz or 400hz operation etc
Ref: Electric Machines Charles Hubert
Macmillan Pulishing Company
621.31042 H878 Levels Library

Synchronous Speed ns, where fs is frequency of supply, and P is no. of poles formed by
stator winding

Slip speed n where nr is speed of rotor:

Slip in Per Units:

Frequency of voltage induced in a rotor loop by a rotating magnetic field:

Blocked rotor tests freeze the rotor, hence slip is 1.0 and frequency of voltage generated in the rotor
is identical to the frequency of the applied stator voltage:

Voltage generated in rotor loop (formed by 2 rotor bars and the end-connections) as it is swept by the rotating stator flux, is given by:

Rotor impedance is in terms of slip and blocked rotor quantities:

Rotor current is in terms of rotor impedance and blocked rotor voltage times slip :

Ir magnitude is :

Ir phase angle is :

Air-gap power :

Electrical power transferred across the air-gap from stator to rotor is converted to mechanical power
the remainder being heat-power losses in the rotor conductors

All of the air-gap power is expended as heat losses in the equivalent resistance :

Heat power dissipated in the real rotor conductors for all 3 phases is (rotor core loss) :

Developed Torque :

in units of watts

in units of hp

Developed Torque in pound-foot units :

Torque Speed Characteristic

Power flow, Pscl refers to stator core loss :

Useful Shaft-power output :

Efficiency :

Maximum Torque occurs where breakdown of the torque slip curve occurs, ie at slope = 0
slip is given by differentiating TD wrt s.

Note torque at which TDmax occurs is directly proportional to rotor resistance R2. Thus where very high starting torque is required, the rotor circuit is designed with sufficient resistance so TDmax occurs at blocked rotor (s = 1).

Maximum Torque

Note: maximum torque (breakdown torque) that a given induction motor can develop is independent of rotor resistance. Changing the value of rotor resistance will change the slip at which TDmax occurs, but will not change the value of TDmax.

Approximate Running Torque equation , s<=0.03

TD is proportional to Voltage squared, slip
TD is inversely proportional to frequency, assuming R and X don't vary.

Effect on Locked Rotor Current
Current is directly proportional to voltage and inversely proportional to frequency.

Using motors of different frequency. For example, can i use a 50 Hz motor on a 60Hz supply system?

Operating 60hz motors on 50hz systems
This is possible provided the voltage is reduced to 50/60 and the output power correspondingly reduced. As long as the V/F ratio is kept constant, no problem.

Mix of 25hz and 60hz systems
Where mix occurs, if a 25hz motor is attached to 60hz, a dangerous overfrequency situation occurs. If 60hz motor is attachedto the 25hz system, the motor will burn out.

Conclusion,
a 400hz motor will have 50/400 the resistance and inductance to give the same torque. However, since less wire and steel is used, I would expect the motor to be <= 8 times smaller/lighter.