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When to use I²R, V²/R or VI while calculating Electric Power?

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I came across a question that was asked to me quite a couple of time: When to use I²R, V²/R or VI while calculating power? At first, the question looks simple and as if it needs no formal explanation but that's not the case. Use of any of them will give the correct result provided that you apply it correctly. Correct application is possible only when you understand the type of circuit you are dealing with. Another important point is to use the one which is going to take less time and help you calculate easily. All of the three equations tell the same thing but their usage is subject to the type of problem you're trying to solve. Let us understand when, how an what to use while calculating power in an electrical circuit.
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Effect of external rotor resistance on torque of an induction Motor.

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What causes change in rotor resistance externally in a three phase slip-ring induction motor to change its Torque?

Slip-Ring Induction Motor (Credit: http://yourelectricalguide.com)

While studying about induction motor types you must have come across this machine whereit has three conductors for rotor connected externally to a rheostat. But how exactly they help? Why do we need those rotor external resistance? Let's find out.

The one line answer to this question most of the time people suggest is, "To increase the Motor Torque". Well I willn't say it's incorrect but it's vague or incomplete.

 Torque in an induction motor depends on two things, air gap flux and rotor magnetic flux generated due to induced emf. When rotor cuts air gap flux, a current flows through rotor conductors. As a result, the rotor generates its own flux and hence under the effect of this stator and rotor flux, there is a resultant flux which us responsible for torque.
From the above statement we can also say that Max Torque during start in an induction motor is directly proportional to maximum air gap flux and rotor current (as rotor current is responsible for rotor flux).

Now, coming to the question. Think about a practical machine where 3 phase supply has just been given. The rotor is still at standstill whereas the stator air gap flux begins to rotate at synchronous speed. As a result the rotor conductors cut these magnetic field lines which are rotating at synchronous speed and hence an emf and current (as rotor conductors are shorted) of supply frequency is generated in the rotor. As the current is at supply frequency in the rotor during start, the corresponding inductance of the rotor also increases (inductance is directly proportional to frequency). Because of this high inductance in the rotor conductors, the rotor current begins to lag behind the stator air gap flux. Now as I said above maximum torque in an induction motor is directly proportional to maximum air gap flux and rotor current which means

Tm=kØmIr



Where Tm is the maximum starting torque, Øm is maximum air gap flux and Ir is the rotor current. K is some constant.
Now if the rotor current lags behind the air gap flux by an angle let say θ then it is the cosine component of the current which aligns with the air gap flux. So, the equation of maximum starting torque becomes

Tm=kØmIrcosθ




As cosθ value is always less than 1 (inductive circuit) the effective maximum starting torque reduces.
You can simply make it out that higher the lag angle θ , greater will Torque reduction. During start of an induction motor, the inductance is very high than the rotor resistance for the reason I meantioned above and hence the rotor current lags with the maximum air gap flux by a greater value of θ .
Therefore to decrease this lag angle or you can say to improve the rotor circuit power factor, external resistance is added to the rotor conductors. This increases the value of cosθ and hence increases the starting Torque bringing it closer to maximum torque. Thus higher the resistance, greater will be starting torque. The resistance is cut out in steps as the motor gains optimum speed as with increase in speed, the rotor conductor cut less amount of stator air gap flux compared to during start hence reducing the rotor reactance. Keeping these resistors intact is of no use then and infact result in ohmic loss as well as reduction in torque.

That's how we get high starting torque in a slip-ring or wound rotor type induction motor compared to squirrel cage rotor type induction motor. However, the advent of VVVF Drives and cost, complexity and maintainence problem of slip ring IM has made them a machine of rare application. In most of the places we use squirrel cage type IM.
Note: Maximum starting torque is not the maximum torque of a motor. Maximum torque of a machine is fixed and can't be changed. By maximum starting torque we mean the closness by which the starting Torque gets to maximum torque. That's why simply stating "increase the Torque" is a vague statement.

I hope you liked this post and it was quite helpful for you. There is a lot of equation and mathematics behind the explanation of everything but to really enjoy and understand things, you should look at the practical side of the explanation than mathematical. This blog provides you such practical posts on Electrical Engineering. If you want to get more such informative posts on here, then subscribe to this blog to get the feed directly in your E-mail. Also you can Subscribe me on YouTube where I post informative videos on Electrical Engineering topics. To subscribe me on YouTube, click on the Subscribe button below.

Thanks all. Have a great day!



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