# Three-Phase Induction Motor Operation Explained

## This article describes the operating principles and equivalent circuits of a three-phase induction motor.

An induction motor is an AC machine in which alternating current is directly supplied to the stator armature windings and indirectly to the rotor windings by induction or transformer action from the stator. Hence, it is also referred to as a rotating transformer. Its stator windings are similar to those of synchronous machines. However, the induction motor's rotor can be one of two types:

- Wound rotors have three windings, similar to stator armature windings. The rotor winding terminals are connected to insulated slip rings installed on the rotor shaft. The carbon brushes mounted on these rings give the machine operator access to the rotor terminals. During steady-state operation, these terminals are shorted.
- In squirrel-cage rotors, conducting bars are fixed in slots in the rotor magnetic core, and the bars are short-circuited at both ends by conducting end rings. Since the rotor bars and short-circuit rings are shaped like a squirrel cage, the rotor is termed a squirrel-cage rotor.

Most induction motors have squirrel-cage rotors, however, the two types of rotors are similar from a modeling point of view.

The three balanced alternating voltages applied to the stator cause balanced stator currents to flow. These stator currents generate a rotating mmf that can be characterized as a rotating magnetic field. According to Faraday's law of electromagnetic induction, this rotating magnetic field induces voltages in the rotor windings. These induced voltages cause balanced currents to flow through the short-circuited rotor. These rotor currents generate a rotor mmf, which can also be characterized as a rotating magnetic field. As a result of the interaction between these two rotating magnetic fields, an electromagnetic torque T_{e} is produced, which is used to turn a mechanical load T_{m}. In a steady state condition, when the motor losses are ignored, T_{m} and T_{e} are equal.

The motor windings are inductors, and energy is stored in the windings through magnetic fields.

The speed of rotation of the rotor magnetic field, when viewed from a stationary position on the stator, is equal to the speed of rotation of the stator magnetic field, which is the synchronous speed n_{s}. However, the rotor speed n_{r} is different from the synchronous speed. If n_{r} were equal to n_{s}, there would be no variation of flux linkage in the rotor, or there would be no net flux cutting; hence, no voltage would be induced in the rotor. Therefore, rotor speed has to be less than synchronous speed. The difference in speed is represented by the slip s, which is defined as:

\[s=\frac{n_{s}-n_{r}}{n_{s}}\,\,\,(1)\]

Where

n_{r} = rotor speed (rpm)

n_{s} = \(\frac{120f}{p}\) = synchronous speed (rpm)

p = number of poles

The difference between the speed of the rotor magnetic field and the speed of the stator expressed in revolutions per minute is called slip rpm and is equal to (n_{s} – n_{r}). Therefore, the frequency of the rotor currents is determined by:

\[f_{r}={\big(}n_{s}-n_{r}{\big)}\frac{p}{120}=s{\Big(}\frac{pm_{s}}{120}{\Big)}\,\,\,(2)\]

Since \(\frac{pm_{s}}{120}\) is equal to the frequency of the stator currents, Equation 2 can also be written as:

f_{r} = sf (3)

For steady-state operation, the slip has a normal range of values between 1% and 5%.

### Development of Equivalent Circuit in a Three-Phase Induction Motor

An equivalent circuit is invaluable in the performance analysis of the three-phase induction motor. Therefore, the equivalent circuit, as well as approximate equivalent circuits, are developed in this section.

The general form of the equivalent circuit for a three-phase induction motor can be derived from the equivalent circuit of a three-phase transformer. The induction motor can be considered a three-phase transformer whose secondary, or the rotor, is short-circuited and revolves at the motor speed. Since the motor usually operates at balanced conditions, only a single-phase equivalent circuit is needed.

When balanced three-phase currents flow in both stator and rotor windings, the resultant synchronously rotating air-gap flux wave induces balanced three-phase voltages in both stator windings and rotor windings. The stator-induced voltage has a frequency equal to the frequency f of the applied voltage, while the rotor-induced voltage has a frequency f_{r} given by Equation 3.

Consider the stator first. The applied voltage per phase across the stator terminals is equal to the sum of the stator-induced voltage per phase, plus the voltage drop across the stator winding resistance, plus the voltage drop across the stator leakage reactance due to the leakage flux, which links only the stator winding. Mathematically, in phasor form, the relationship may be expressed as:

V_{1} = E_{1} + R_{1}I_{1} + jX_{1}I_{1} = E_{1} + I_{1}(R_{1} + jX_{1}) (4)

Where

V_{1} = stator terminal voltage per phase

E_{1} = stator-induced voltage per phase

I_{1} = stator current per phase

R_{1} = stator winding resistance

X_{1} = stator winding reactance

The magnetic core can be modeled as a parallel combination of a resistance R_{c} to account for hysteresis and eddy current losses and a reactance X_{m} to account for the magnetizing current required to produce the air-gap magnetic flux. The magnetizing current in an induction motor is much larger than that in a transformer because of the presence of the air gap in a motor.

Next, a model of the rotor is developed. Let E** _{2}** denote the rotor-induced voltage at a standstill, that is, s= 1.0. At a standstill, the induction motor may be viewed as a transformer with an air gap, and the stator per-phase induced voltage E

**is related to the rotor per-phase induced voltage E**

_{1}_{2}by the turns ratio (N

_{1}/N

_{2}) that is:

\[E_{1}={\Big(}\frac{N_{1}}{N_{2}}{\Big)}E_{2}\,\,\,(5)\]

Where

N_{1} = number of turns in the stator winding

N_{2} = number of turns in the rotor winding

The voltage induced in the rotor of the induction motor is directly proportional to the relative motion of the rotor and the synchronously rotating air-gap magnetic field. When the induction motor is rotating at a speed n or a slip s, the rotor-induced voltage E_{2s} equals the induced voltage at standstill E_{2} multiplied by the slip. In the short-circuited rotor circuit, the induced voltage E_{2s} appears as a voltage drop across the rotor resistance and leakage reactance.

The rotor resistance does not depend on the slip. However, the rotor leakage reactance does and is equal to X_{r} = 2πf_{r}L_{r} = s2πfL_{r}, where L_{r} is the leakage inductance of the rotor winding due to flux linking the rotor winding only. Thus, the rotor-induced voltage at slips may be expressed mathematically as follows:

E_{2s} = sE_{2} = I_{r}R_{r} + j(2πf_{r}L_{r})I_{r}

=IrRr+js2πfLrIr

= I_{r}R_{r} + jsX_{2}'I_{r} (6)

Where

E_{2s} = rotor-induced voltage at slip s

E_{2} = rotor induced voltage at standstill (s = 1.0)

I_{r} = rotor phase current

R_{r} = rotor resistance per phase

X2' = 2πfL_{r }= rotor leakage reactance per phase at a standstill

Dividing both sides of Equation 6 by the slip s and referring rotor quantities to the stator side, as in a transformer, yields

\[E_{2}={\Big(}\frac{R_{r}}{s}+jX^{'}_{2}{\Big)}I_{r}\]

\[{\Bigg(}\frac{N_{1}}{N_{2}}{\Bigg)}E_{2}={\Bigg(}\frac{N_{1}}{N_{2}}{\Bigg)}^{2}{\Bigg(}\frac{R_{r}}{s}+jX^{'}_{2}{\Bigg)}{\Bigg(}\frac{N_{2}}{N_{1}}{\Bigg)}I_{r}\]

\[E_{1}={\Bigg(}\frac{R_{2}}{s}+jX_{2}{\Bigg)}I_{2}\,\,\,(7)\]

Where

E_{1} = (N_{1}/N_{2}) E_{2 }= rotor-induced voltage referred to the stator

I_{2} = (N_{2}/N_{1}) I_{r} = rotor current referred to the stator

R_{2 }= (N_{1}/ N_{2}) R_{1} = rotor resistance referred to the stator

X_{2} = (N_{1}/ N_{2})^{2} X2' = rotor leakage reactance referred to the stator

The stator circuit represented by Equation 4 and the rotor circuit represented by Equation 7 are at the same frequency f of the applied voltage. Therefore, these stator and rotor circuits can be joined together and combined with the model of the magnetic core into the per-phase equivalent circuit of the induction motor, which is shown in Figure 1.

**Figure 1.** Per-phase equivalent circuit of a three-phase induction motor. Image used courtesy of Amna Ahmad

**Figure 1.**Per-phase equivalent circuit of a three-phase induction motor. Image used courtesy of Amna Ahmad

An approximate equivalent circuit for an induction motor is derived by moving the shunt elements, R_{c} and X_{m,} in parallel, representing the core to the motor terminals. This simplification introduces little error but greatly reduces the computational effort. This approximate equivalent circuit is shown in Figure 2. Also shown in the figure is the equivalent rotor resistance R_{2}/s, which has been decomposed into R_{2} and R_{2}[(1-s)/s]. The first resistance component, R_{2}, represents the rotor copper loss, and the second component represents the power developed by the motor.

**Figure 2.** Approximate equivalent circuit of an induction motor. Image used courtesy of Amna Ahmad

**Figure 2.**Approximate equivalent circuit of an induction motor. Image used courtesy of Amna Ahmad

### Induction Generators

Just like DC and synchronous machines, the induction machine may be used as a generator or motor. Because their performance cannot compare with synchronous machines, induction generators have yet to be very popular. In recent years, however, induction generators have found use in wind power plants. Because of its wide use and popularity, the induction motor is called the workhorse of the power industry, which works on the principle of electromagnetic induction.

*Featured image used courtesy of Adobe Stock*

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