Maximum Torque and Output Power
Using the Thevenin torque equation:
it is possible to plot the torque-speed curve of an induction machine. However, it is useful to know the magnitude of the maximum or "pullout" torque without plotting the full torque-speed curve. From the equation above, the peak torque could be found be differentiating with respect to slip to find the slip which gives maximum torque. However, a mathematically simpler and intuitively clearer answer can be found be considering the power flow in the Thevenin equivalent circuit
Analysing the full equivalent circuit it was observed that
Therefore, since synchonous speed is constant, maximum torque occurs at the same slip as maximum airgap power. Considering the Thevenin circuit, and applying maximum power transfer theory, maximum airgap power and maximum torque will occur when
Re-arranging it is possible to obtain the slip for maxiumum torque, or pullout torque. Note that this should not be called the "maxiumum slip".
Substituting the pullout slip into the Thevenin torque equation:
Discussion
From the two equations above it can be seen that
- The slip at which maximum torque occurs is proportional to rotor resistance
- The magnitude of the maxiumum torque is independent of rotor resistance
If all other parameters remain constant, increasing the rotor resistance will:
- Reduce the speed at which maximum toruqe occurs
- Increase the starting torque (until spo > 1, then it will reduce the start torque)
- Increase slip for a given torque
- Reduce the speed for a given torque
- Increase the rotor losses at a given torque
The last point above can be shown by considering that the torque equation
is actually rotor copper loss divided by slip speed in radians per second. If slip increases, losses must increase to maintaint the torque.
The diagram below plots torque speed curves for a motor with the following parameters, R2 is varied.
R1=0.5 Ω, X1=0.75 Ω, X2=0.5 Ω, Xm=100 Ω, f=60Hz, p=6, VLL=230V, Y-connection
