# Phasor Diagram for Synchronous Generator

In the present article we are going to discuss one of the easiest methods of making the **phasor diagram for synchronous generator**. Now, let us write the various notations for each quantity at one place, this will help us to understand the phasor diagram more clearly. In this phasor diagram we are going to use:

E_{f} which denotes excitation voltage
V_{t} which denotes terminal voltage
I_{a} which denotes the armature current
θ which denotes the phase angle between V_{t} and I_{a}
ᴪ which denotes the angle between the E_{f} and I_{a}
δ which denotes the angle between the E_{f} and V_{t}
r_{a} which denotes the armature per phase resistance

In order to draw the phasor diagram we will use V_{t} as reference. Consider these two important points which are written below:

- We already know that if a machine is working as a synchronous generator then direction of I
_{a}will be in phase to that of the E_{f}. - Phasor E
_{f}is always ahead of V_{t}.

**phasor diagram of synchronous generator**. Given below is the phasor diagram of synchronous generator:

In this phasor diagram we have drawn the direction of the I_{a} is in phase with that of the E_{f} as per the point number 1 mentioned above. Now let us derive expression for the excitation emf in each case. We have three cases that are written below:

- Generating operation at lagging power factor.
- Generating operation at unity power factor.
- Generating operation at leading power factor.

Given below are the phasor diagrams for all the operations.

(a) Generating operation at lagging power factor:
We can derive the expression for the E_{f} by first taking the component of the V_{t} in the direction of I_{a}. Component of V_{t} in the direction of I_{a} is V_{t}cosΘ, hence the total voltage drop is along the I_{a}. Similarly we can calculate the voltage drop along the direction perpendicular to I_{a}. The total voltage drop perpendicular to I_{a} is . With the help of triangle BOD in the first phasor diagram we can write the expression for E_{f} as
(b) Generating operation at unity power factor:
Here also we can derive the expression for the E_{f} by first taking the component of the V_{t} in the direction of I_{a}. But in this case the value of theta is zero and hence we have ᴪ = δ.

With the help of triangle BOD in the second phasor diagram we can directly write the expression for E_{f} as
(c) Generating operation at leading power factor:
Component in the direction of I_{a} is V_{t}cosΘ. As the direction of Ia is same to that of the V_{t} thus the total voltage drop is . Similarly we can write expression for the voltage drop along the direction perpendicular to I_{a}. The total voltage drop comes out to be . With the help of triangle BOD in the first phasor diagram we can write the expression for E_{f} as