AC induction motor constant power

An AC induction motor is supposed to be a constant power motor, which implies it draws more current on low voltage. Consider a motor running a constant torque load at a particular speed. Suppose now the voltage is reduced, which should cause it to settle down at a lower speed supplying the same torque as per the new torque speed characteristic. If we consider the electrical side, higher slip will cause more current to be drawn that too at higher pf, which should maintain the power justifying the above theory. But on the mechanical side the new output power Torque x speed is supposed to be lesser now as speed is less now. Is it this contradiction?

The following guidelines prove there is no contradiction since your question about Motor under running condition:

1. Torque / Slip characteristic for Induction Motor has three Zones.
a)- Starting Torque @ S=1, selection of this torque depends on the application. The starting should be greater the system torque at time of starting.
b)- Unstable Zone during which acceleration and torque development took place. This zone up to the Max. Torque can be developed.
In this regard, it may be necessary to mention that the seventh harmonics to be considered otherwise crawling / clogging may occur.
c)- Normal Operating Zone. NOZ about which your query raised. NOZ ranged as " 0 < S< 1" ie up to the Max. Torque. It is worth mentioning that Max. torque always remain the same regardless to its location of occurrence.

2. The torque is directly proportional to rotor resistance "r2" & varies with slip "S". hence increase of rotor resistance is the most practical method of changing the torque (ie wound rotor Slip ring Motors). Moreover, the Max torque achieved when rotor Resistance "r2" = The Stator impedance, At starting S=1.

3. Accordingly, the ration r2/x2 gives the location of the max. Torque w.r.t Slip (if the max. torque is required at starting (S=!) then r2/x2 should equal "1".

4. load being constant. Mechanical output = Electrical input - losses.

5. Tmax Propotional to Sq(v). decrease of 50% of the supply voltage generate a reduction of 20% in the max. running Torque (zone c) , increase in slip and also Full load current and temperature raise increase while the full load speed decrease. the status of the above parameters will be opposite if the voltage increases by 10%.

Based on the above, in all cases since the Motor is running within the operating range will be no issue unless the supply voltage falls behind the above limits (-50%, +110%). Accordingly, variable frequency drives provided by under/Over voltage protection relay to avoid damage to insulation due to Heat/temperature rise that will be generated due to excessive current intend to composite load.