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The number of poles does not affect directly the limiting values of either the air gap flux density Bg or the
continuous rotor linear current density Kr. Thus, the torque capability of the motor as given in equation 2.4 is not
changed with a change of pole number, and at the same speed, the rated mechanical power is relatively unchanged.
However, for the same speed, the required frequency of the supply will be increased in proportion to the number of
poles.
An increase in the number of poles above two reduces the yoke thickness that is required outside the stator slots to
accommodate the return paths of the radial tooth flux. This, in turn, allows for a larger rotor radius for a given
overall frame size. From equation 2.4, it may be noted that, for given values of flux density, linear current density,
and rotor shape (or ratio l/r), the rated torque is proportional to the cube of the rotor radius. Thus, the power rating
obtainable with a given frame size may be increased by increasing the pole number. Also, the required thickness of
the rotor core inside the rotor slots is decreased. With a large pole number, a semihollow spoked rotor may be used
reducing the rotor mass and inertia.
A disadvantage of increasing the number of poles is that, for a given shape l/r, the magnetizing component of the
stator current increases in proportion to the square of the number of poles. The power factor of the motor is
therefore reduced, the loss in the stator windings is increased, and the required rating of the supply system is
increased [8, 9]. Also, with increased numbers of poles, the coupling between the rotor and stator windings is
somewhat decreased thus increasing the leakage inductance. Conversely, two-pole motors tend to have mechanical
asymmetries that interact with the electromagnetic forces to produce rotor unbalance, shaft and bearing fluxes, and
other parasitic effects. To avoid these effects, higher precision is required in machine fabrication than for motors
with four or more poles. The optimum induction motor for a variable speed drive in the low- and medium-power
range usually has either two or four poles. For higher-power, low-speed applications, higher pole numbers may be
used together with shape ratios l/r considerably less than 1.
2.4.4. Torque Expressions
Suppose the motor voltage Vs supplied by the inverter is controlled so that the voltage Es induced in the stator
winding is kept proportional to the supply frequency ωs; that is, the stator flux linkage s is maintained constant
at all values of speed. Using the equivalent circuit of Figure 2-6, the torque is equal to the power into the effective
load resistance RRω0/ωR for all three phases divided by the actual mechanical angular velocity um. Alternatively,
the torque can be evaluated as the total power entering the rotor circuit resistance RRωs/ωR for all three phases
divided by the synchronous mechanical velocity (p/2)ωs. The value of the leakage inductance is typically in the
range 0.15-0.25 per unit. With small values of the rotor frequency ur, the rotor circuit resistance will be in excess
of 1.0 per unit. Then, the effect of the leakage inductance LL can be ignored and the torque can be approximated by
T  3Es2
R p
3 p 2 R

s
 S RR 2 s 2
RR
(2.6)
N.m
Note the linear relation between torque and slip frequency. As the motor is loaded more heavily, the rotor
frequency ωR increases, decreasing the effective total resistance of the rotor circuit as seen by the stator,
increasing the effect of the leakage inductance and shifting the rotor current distribution wave away from the flux
wave. For constant flux linkage s, maximum power is transferred to the rotor whenωR = RR/LL and the maximum
torque is given by
3 p 2s
Tˆ 
4 LL
N.m
(2.7)
This is just half of what it would have been at that rotor frequency if there were no leakage.
For regeneration, the slip frequency is negative. The same maximum reversed torque as in equation 2.7 is
produced with a value of slip frequency ωR = RR/LL. In most motor designs, this maximum torque T will be in the
range two to three times the continuous base torque Tb.
Most standard induction motors are made for operation on the standard utility constant voltage and constant
frequency supply. To provide them with adequate starting torque, their rotors are frequently designed with deep
bars or double cages of bars, making their effective resistance increase as the rotor frequency increases. This