V
Is the velocity of the rotor relative to the magnetic field.
I
is the current induced in the rotor and
F
is the force on the rotor.
s is the slip. If the rotor is rotating at synchronous speed, s is zero.
If the rotor is blocked and not rotating, s = 1.
P,
is dissipated in the variable resistance.
_{2} |

February 19, 2018

Efficiency and power factor are derived as a function of motor speed as expressed by the slip. This is done using the standard equivalent circuit, shown in Figure 4. Efficiency and power factor increase as slip decreases and the motor speed approaches the synchronous speed.

A representation of a single phase induction motor is shown in Figure 1.
The line voltage * V* causes a current to flow in the stator winding. This
current creates a magnetic field,

Figure 2 shows a representation of the sinusoidally varying magnetic field, *B _{o}cos(wt).*
The magnetic field is in the vertical direction. It consists of two components, one rotating clockwise and the other
rotating counterclockwise. Motor operation requires a rotating magnetic field. Rotor currents interacting with the rotating
magnetic field generate the motor torque.

The physical mechanism that generates rotor torque is depicted in Figure 3. The clockwise rotating component of the magnetic field, *B*
is shown.
The rotor also rotates clockwise but at a slower rate.
The relative motion of the rotor coil and the magnetic field causes a force on charges in the coil that results in coil current. The force on charges that causes coil current is,

where

(1)

The rotor current flowing in the magnetic field experiences a second force given by,

where

(2)

**Starting**

When the magnetic field is not rotating, there is no torque generated. The magnetic field shown in Figure 3 is stationary.
True, it has two rotating
components, but their sum is stationary. An additional coil, the starting coil, is necessary. A starting stator
winding is employed to produce a rotating magnetic field. The starter winding is perpendicular to the main
stator winding. If the current in the starter winding were 90^{o} out of phase with the main
winding and if it produced a magnetic field of the same magnitude, the magnetic field in the, say x, direction would be
* B cos(w t)* and the magnetic field in the y direction
would be * B sin(w t) *. The result would be a rotating magnetic field.
Typically the starter winding has a more resistive impedance than the main winding. The current
in the starter winding is out of phase with the current in the main winding. This results in a rotating stator field
that causes the motor to start.
Once the rotor is rotating, it will interact with one of the rotating components of the magnetic field,
shown in Figure 2,
to produce torque. Typically the starter winding is switched out once the motor comes up to speed.

**Slip**

The rotating components of the magnetic field, shown in Figure 2, rotate at synchronous speed.
Synchronous speed for the double pole motor shown in Figure 1 is 60 Hz or 3600 rpm. The rotor
rotates at a speed less than the synchronous speed. The rotor sees a changing magnetic field
that results in voltages and currents that produce torque. If the rotor is rotating at
*w* radians per second
in the forward direction and the synchronous speed is
* w _{s}*,
then the rotor is
traveling at a speed

(3)

**Motor Model**

The interaction of the stator and the rotor windings is similar to that of the primary and secondary windings of
a transformer. The rotor windings act like the secondary.
The speed of the rotor relative to the rotating magnetic field is
* w _{s} - w *.
If the rotor
were rotating at synchronous speed,

The motor is like a transformer where impedances attached to the secondary can be referred to the primary.
Rotor resistance *r _{2}*
and reactance

Since the induced voltage in the rotor is proportional to the frequency seen by the rotor and the
voltage induced in the stator is proportional to
* w _{s}*,
the rotor voltage referred to the stator is divided by the slip,

The impedance of the rotor is

where the reactance,

(4)

Since the rotor voltage is also less by a factor of *s*,

Solving for the rotor impedance referred back to the stator.

(5)

If the rotor is rotating at synchronous speed,

(6)

An equivalent circuit for the motor is shown in Figure 4.
* x _{1}* is the stator leakage reactance.

**Efficiency**

Power is torque multiplied by angular frequency. The magnetic field rotates at the synchronous angular velocity and exerts a torque. The power produced by this torque is proportional to the synchronous angular velocity.

where

(7)

The rotor experiences this torque, but it rotates at the operating speed of,
*w*.
Therefore the mechanical power produced by the rotor is,

where

(8)

The inherent rotor efficiency is the mechanical power,
* P _{2}* (Equation(8))
divided by the power delivered to the rotor,

where equation 3 has been used. Equation 9 does not include ohmic and core losses.

(9)

Ohmic and core losses can be included using the
circuit in Figure 5. In Figure 5 there is just one
loop. The same current flows in the elements representing the stator
and in the elements representing the rotor.
One loop has been achieved by Thevenizing
the stator circuit
looking toward the source from the terminal ab. Also, the rotor resistance
* r _{2} /s*,
is divided into two resistances. Mechanical power to the load is
dissipated in the variable resistance. Power divides between the two resistors
as predicted by Equation 9.

Since
X_{m}
is large,
the Thevenin resistance
* r _{a}*
is nearly equal to the stator resistance,

The power input to the motor from the power line is,

(10)

The efficiency is the power to the load,

(11)

where

(12)

Figure 6 Efficiency as a function of slip is plotted using Equation 12. |
Figure 7 Since motor speed is a function of slip, w / w ,
efficiency as a function of motor speed is shown by just flipping
Figure 6's horizontal axis.
_{s} = 1 - s |

**Power Factor**

The power factor is the ratio of the power absorbed from the power line to the volt-amperes absorbed. The power factor is the ratio of the resistance to the magnitude of the impedance that the motor presents to the power line. The power factor is,

From the circuit in Figure 6,

(13)

(14)

Figure 8 Power factor as a function of slip is plotted using Equation 14. |
Figure 9 Power factor as a function of motor speed is shown. |

**Summary**

Efficiency and power factor as a function of slip are given by Equations 12 and 14.
The difference between the synchronous speed and the motor speed causes power loss.
This results in an inherent efficiency of *1 - s *. As slip increases efficiency decreases.
Power factor also decreases as slip increases. The effect of motor parameters can
be determined using Equations 12 and 13. Motor parameter values for the plots were chosen to represent
typical cases.