# Induction Motor Phasor Diagram

As we know that principle of induction motor is very similar to the transformer with some differences. An induction motor at standstill condition is similar to a transformer at no load condition. Therefore, the method of drawing the induction motor phasor diagram is also same as that of a transformer phasor diagram. In this post we will discuss the phasor diagram of induction motor at standstill condition and at full load slip.

**Induction Motor Phasor Diagram at Standstill Condition:**

Before going into the phasor diagram, there are some important points to be taken care:

- Per phase value of induced emf E
_{1}in the stator winding is given as below

E_{1} = √2πf_{1}k_{w1}N_{1}Ø

where f1 = supply frequency

N1 = Number of series turns per phase

Ø = resultant air gap flux per pole

k_{w1} = Stator winding factor

- Per phae value of induced emf E2 in rotor winding is given as

E_{2} = √2πf_{2}k_{w2}N_{2}Ø

where f_{2} = frequency of induced emf in rotor = sf_{1}

N_{2} = Number of series turns per phase

Ø = resultant air gap flux per pole

k_{w2} = Rotor winding factor

- Total air gap mmf F
_{r}of induction motor is the sum of stator mmf (F_{1}) and rotor mmf (F_{2}). - Magnetizing current I
_{m}taken by stator winding from the supply always remains in phase with the resultant flux Ø. - The induced emf always lags behind the resultant flux Ø by 90°.

Now we are at a stage to draw the induction motor phasor diagram. Let us take the resultant air gap flux Ø as the reference. This flux Ø will be in phase with the resultant mmf F_{r}. Also, the induced emf E_{1} and E_{2} in stator and rotor winding will lag behind the Ø by 90°. This is shown in the below phasor diagram of induction motor.

Since the rotor mmf counteracts the stator produced mmf as per lenz’s law, therefore stator takes extra current from supply to counterbalance the effect of rotor current. Therefore under normal condition,

Stator mmf = Rotor mmf

N_{1}’I_{2}^{’} = N_{2}’I_{2}

where N_{1}’ and N_{2}’ are the effective stator and rotor turns per phase.

This component of stator current is called the load component. In addition to load component, stator also takes magnetizing current I_{m} to build magnetic flux in the air gap. Thus the total stator current I_{1} = I_{2}’ + I_{m}. This is shown in the above phasor diagram. I_{2}’ is shown opposite to the rotor current I_{2} for the reason discussed above.

At standstill condition, E_{2} = I_{2} (r_{2} + jx_{2}). The core loss component of stator current I_{c} is in phase phase with V_{1}’ or –E_{1}. At standstill condition, the friction and windage loss is zero, therefore the stator no load current is given as

I_{0} = I_{m} + I_{c}

Since stator applied voltage V_{1} must balance the stator back emf V_{1}’ or -E1, stator impedance drop I_{1}(r_{1} + jx_{1}), therefore we can write

V_{1} = V_{1}’ + I_{1}(r_{1} + jx_{1}) ……(1)

Similar equation exists for rotor circuit and can be written as

E_{2} = I_{2} (r_{2} + jx_{2}) ……..(2)

The above equations are applied for drawing induction motor phasor diagram as shown in above figure. It can be easily seen from the above phasor diagram that, the power factor (* cosɵ*) of induction motor at starting is very poor as

*is large.*

**ɵ****Induction Motor Phasor Diagram at Full Load Slip:**

At full load, the slip s of induction is low. The stator voltage equation (1) do not changes when the motor is loaded. But the rotor voltage equation changes with slip. The rotor induced voltage at any slip s becomes sE_{2} and the rotor circuit reactance becomes sx_{2}. Therefore,

sE_{2} = I_{2} (r_{2} + jsx_{2})

The above rotor equation when implemented, the induction motor phasor diagram will becomes different from the phasor at standstill condition. The induction motor phasor at full load slip s is shown below.

Since at full load condition, some friction and windage loss will exists. This means that stator will have to draw some extra current from the supply main to provide this additional loss. Therefore the total no load current I_{0} taken by stator is the sum current I_{fc} and I_{m}. It can be seen from the above phasor diagram that power factor of induction motor improves. generally at full load the power factor ranges in between 0.8 to 0.9.