This module focuses on solving numerical problems related to synchronous generators, a crucial topic for the GATE Electrical Engineering exam, particularly within the Power Systems and Machines syllabus. We will cover key concepts and common problem types to build your problem-solving skills.

Before diving into problems, it's essential to be familiar with the key parameters of a synchronous generator. These include: synchronous reactance ($X_s$), armature resistance ($R_a$), open-circuit voltage ($E_f$), terminal voltage ($V_t$), armature current ($I_a$), field current ($I_f$), and power factor (PF).

The generated EMF ($E_f$) is the voltage produced internally by the machine. It's often calculated using the terminal voltage ($V_t$), armature current ($I_a$), armature resistance ($R_a$), and synchronous reactance ($X_s$). The phasor equation is key here: $E_f = V_t + I_a(R_a + jX_s)$.

Problems often involve calculating input power, output power, and efficiency. Output power ($P_{out}$) is typically $3V_t I_a \cos(\phi)$, where $\phi$ is the power factor angle. Input power ($P_{in}$) is output power plus losses. Efficiency ($\eta$) is $(P_{out} / P_{in}) imes 100\%$.

Voltage regulation quantifies how well a generator maintains its terminal voltage under varying load conditions. It's calculated as: Voltage Regulation = $\frac{|E_f| - |V_t|}{|V_t|} \times 100\%$. This requires calculating the no-load voltage ($E_f$) for a given load condition.

The load angle ($\delta$) is the angle between the rotor's magnetic field axis and the rotating magnetic field of the stator. The power output of a synchronous generator is given by $P = \frac{3 |E_f| |V_t|}{X_s} \sin(\delta)$ (assuming $R_a$ is negligible). Understanding how $\delta$ changes with load is important.

Typical GATE problems involve:

1. Calculating $E_f$ given $V_t$, $I_a$, $R_a$, and $X_s$ at a specific power factor.
2. Calculating voltage regulation.
3. Determining power output and efficiency.
4. Finding the load angle ($\delta$) for a given power output.
5. Problems involving synchronous motors operating as synchronous condensers (though this module focuses on generators).

Strategy: Always draw a phasor diagram. Clearly identify the given quantities and what needs to be calculated. Use the appropriate formulas and ensure consistent units.

Consider a problem asking to find the generated EMF of a synchronous generator. You'll be given the terminal voltage, armature current, power factor, armature resistance, and synchronous reactance. First, represent $V_t$ as a phasor (e.g., $V_t \angle 0^\circ$). Then, determine the phasor for $I_a$ based on the power factor (e.g., $I_a \angle -\phi$ for lagging PF). Calculate the armature impedance drop $I_a(R_a + jX_s)$ as a phasor. Finally, add this phasor drop to the terminal voltage phasor $V_t$ to obtain the generated EMF phasor $E_f$. The magnitude of $E_f$ is your answer.