V/f speed control of 3 phase Induction motors

[makkena]

When the voltage / frequency is maintained constant in 3 phase Induction motor i.e. flux is maintained constant by changing the voltage and frequency in same proportion, What is the torque proportionality equation.
Whether

Torque is proportional to Slip X Frequency

or

Torque is proportional to only Slip ?

Based on the above equation I need to solve a question given in GATE EE paper 2006.

The speed of a 4-pole induction motor is controlled by varying the supply frequency while maintaining the ratio of supply voltage to supply frequency (V/f) constant. At rated frequency of 50 Hz and rated voltage of 400 V its speed is 1440 rpm. Find the speed at 30 Hz, if the load torque is constant.
(A) 882rpm
(B) 864 rpm
(C) 840 rpm
(D) 828 rpm

GK publishers Gate previous papers Book is given the answer as 864 rpm. This answer will come if we use T proportional to Slip.

But I saw in some textbooks Torque proportional to slip X (square of Voltage) / f.
From this eauation when V/f is constant, T should be proportional to slip X Voltage i.e. T proportional to slip X frequency.

I am confusing on this which Torque equation to follow when V/f is maintained constant.
Please help me in this.

Thanks

 

[uart]

If the flux is maintained at the same level then for the same torque you require the same slip frequency. So the answer is "C".

 

[makkena]

thanks uart.