Wrong answer key for this Induction Motor question?

[cnh1995]

So this question was asked in GATE-2019 (EE). (GATE is the national Engineering post-graduate entrance exam in India and is conducted every year by one of the 7 top IITs (Indian Institute of Technology) and IISc Bangalore. This year, it was conducted by IIT Madras.)

The answer I found in the official answer key released by IITM is c).
Most of the "coaching institutes" for this exam had also provided c) as the correct answer in their post-exam question paper analysis.
Their reasoning was as follows:
Since flux ∝ V/f and we are reducing rms value of voltage, flux is also reduced.
Also, magnetizing inductance L_m is proprtional to the flux. Hence, magnetizing inductance will also reduce. Therefore, c) is the correct option.

But how can magnetizing inductance change here as the magnetic coupling and self inductances of the coils remain unchanged throughout? If you reduce V 'x' times, the magnetizing current also reduces 'x' times and their ratio (X_magnetizing) remains the same. Assuming absence of saturation and other nonidealities, isn't this simple Ohm's law in action? How can the answer be c)?

My reasoning tells me it should be b) instead. I'll post my reasoning later, but first, I need to know whether c) is right or wrong.

Thanks in advance!

 

[eq1]

What model does the question assume for the three-phase motor? The answer probably follows directly from that.

In the Steinmetz (IEEE) model of an induction motor none of those are a function of V.
https://en.wikipedia.org/wiki/Induction_motor#Steinmetz_equivalent_circuit
I am interested to see your reasoning. I am not sure what line of reasoning makes the answer B or C and not B and C. :)

 

[anorlunda]

magnetizing inductance Lm is proprtional to the flux.

Not true. In fact, a non-energized coil of wire has the same inductance.

 

[cnh1995]

Not true. In fact, a non-energized coil of wire has the same inductance.

Yes, even I don't understand from where they got it.

In the Steinmetz (IEEE) model of an induction motor none of those are a function of V.

Yes, the question assumes the IEEE model where the rotor reactance is made free of slip 's'. So I agree with your conclusion that none of them would change with voltage.

But practically, since we are not reducing the supply frequency, stator impedance and magnetizing reactance should be fixed (as they operate on the supply frequency).
Assuming constant load torque, T∝sV² and if we change the voltage x times, the slip s should change 1/x² times to maintain the necessary load torque. As slip is changing, rotor leakage reactance should also change by the same proportion. Hence, my answer was B).

 

[jim hardy]

The answer key seems a mess to me. Somebody didn't pay attention to his basics.If we allow for the non-linearity of B-H curve, ie include effects of saturation, then inductance and reactance both will increase when you lower the voltage.
Remember - inductance is flux linkages per amp, NΦ/I ,
and at lower flux it takes less current to magnetize the iron because you are farther away from the knee where saturation becomes an issue.
so since I is in the denominator not the numerator i'd say their reasoning is inverted.

But C is still the right answer.

Since flux ∝ V/f and we are reducing rms value of voltage, flux is also reduced.
Also, magnetizing inductance Lm is an inverse function of flux.
Hence, magnetizing inductance and reactance will also reduce increase . Therefore, c) is the correct option.

anybody see a mistake in that ?

old jim

 

[eq1]

If we allow for the non-linearity of B-H curve, ie include effects of saturation, then inductance and reactance both will increase when you lower the voltage.

This assumes constant output power from the motor though right?

But C is still the right answer.

If we include saturation effects in the inductor model won't rotor leakage reactance also be affected? So the answer should be B and C? I'm actually rusty on this topic so I'm not confident that it would be but since the circuit is linear it's hard for me to see how they're not both "affected", which is all the question requires.

 

[jim hardy]

This assumes constant output power from the motor though right?

i don't think so, at least i didnt assume that. It should be true for any stationary iron core inductor.

If we include saturation effects in the inductor model won't rotor leakage reactance also be affected?

I don't disagree with your physics , because
as flux density approaches saturation the fraction of it that gets squeezed out into the air changes as well.
But i don't wholeheartedly agree with B because in the context of that test question i think the author didnt go that deep. It's a smaller effect that also depends on slip.

My two cents, and overpriced at that.

old jim

 

[cabraham]

Answer is C. The magnetizing reactance is L*omega. But L = N*phi/I. Phi decreases with decreasing supply voltage, as does magnetizing current I. But as Jim said, when supply voltage decreases, L increases, moving away from saturation.
Only C can be correct. As far as leakage reactance goes, the path for leakage flux is air. Not affected by how close or far from saturation.
It can only be C. No other choice can be correct.

Claude Abraham

 

[jim hardy]

Thank you, @cabraham !