Phasor Diagram for Synchronous Motor – A phasor diagram is used to Efficiently represent the Steady-state Interrelationship between Quantities that vary Sinusoidally in time on a visual level. We imagine that each phasor makes one complete Anticlockwise Rotation per cycle of the supply, Rotating at a constant speed.

Phasor Diagram for Synchronous Motor

Before we draw phasor diagram, let us write the various Notations for each Quantity at one place. Here we will use:
E_f to represent the excitation voltage
V_t to represent the terminal voltage
I_a to represent the armature current
Θ to represent the angle between terminal voltage and armature current
ᴪ to represent the angle between the excitation voltage and armature current
δ to represent the angle between the excitation voltage and terminal voltage
r_a to represent the armature per phase resistance.

In order to create the phasor diagram for the synchronous motor, we will use Vt as the reference phasor. Two crucial things that are listed below should be understood in order to draw the phasor diagram:
(1) We are aware that if a machine is designed to function as an asynchronous motor, the armature current will flow in a phase opposite to the excitation emf.
(2) The phasor terminal voltage is always behind the phasor excitation emf.
Drawing a phasor diagram for a synchronous motor only requires the aforementioned two points. The synchronous motor’s phasor diagram is provided below.

The direction of the armature current in the phasor one is in phase opposition to that of the excitation emf.
In the phasor of the synchronous motor, it is customary to omit the negative sign of the armature current; consequently, in the phasor two, we have done the same. For the synchronous motor, we will now draw a complete phasor diagram and also derive an expression for the excitation emf in each case. The following three cases are listed:

(a) Motoring operation at lagging power factor.
(b) Motoring operation at unity power factor.
(c) Motoring operation at leading power factor.
Given below are the phasor diagrams for all the operations.

(a) Motoring operation at lagging power factor: In order to derive the expression for the excitation emf for the lagging operation we first take the component of the terminal voltage in the direction of armature current I_a. Component in the direction of armature current is V_tcosΘ.
As the direction of armature is opposite to that of the terminal voltage therefore voltage drop will be –I_ar_a hence the total voltage drop is (V_tcosΘ – I_ar_a) along the armature current. Similarly we can calculate the voltage drop along the direction perpendicular to armature current. The total voltage drop comes out to be (V_tsinθ – I_aX_s). From the triangle BOD in the first phasor diagram we can write the expression for excitation emf as

(b) Motoring operation at unity power factor: In order to derive the expression for the excitation emf for the unity power factor operation we again first take the component of the terminal voltage in the direction of armature current I_a. But here the value of theta is zero and hence we have ᴪ = δ. From the triangle BOD in the second phasor diagram we can directly write the expression for excitation emf as

(c) Motoring operation at leading power factor: In order to derive the expression for the excitation emf for the leading power factor operation we again first take the component of the terminal voltage in the direction of armature current I_a. Component in the direction of armature current is V_tcosΘ. As the direction of armature is opposite to that of the terminal voltage therefore voltage drop will be (–I_ar_a) hence the total voltage drop is (V_tcosΘ – I_ar_a) along the armature current. Similarly we can calculate the voltage drop along the direction perpendicular to armature current. The total voltage drop comes out to be (V_tsinθ + I_aX_s). From the triangle BOD in the first phasor diagram we can write the expression for excitation emf as

Advantages of Drawing Phasor Diagrams for Synchronous Motor

(1) Phasors are very helpful for understanding how physically the synchronous motors work.
(2) Phasor diagrams make it simple to derive mathematical expressions for a variety of quantities.