\[\begin{array}{|l|l|} \hline \textbf{Symbol} & \textbf{Description} \\ \hline V_t & \text{Terminal voltage (line or phase)} \\ E & \text{Induced EMF per phase} \\ f & \text{Frequency (Hz)} \\ n_s & \text{Synchronous speed (RPM)} \\ p & \text{Number of poles} \\ I_a & \text{Armature current per phase} \\ R_a & \text{Armature resistance per phase} \\ X_s & \text{Synchronous reactance per phase} \\ Z_s & \text{Synchronous impedance} \\ \phi & \text{Power factor angle} \\ \delta & \text{Load angle (power angle)} \\ \phi_m & \text{Flux per pole (Weber)} \\ k_w & \text{Winding factor} \\ K_p & \text{Pitch factor} \\ K_d & \text{Distribution factor} \\ k_h & \text{Harmonic factor for harmonic order \( h \)} \\ P & \text{Output power (kW)} \\ T & \text{Torque (Nm)} \\ \eta & \text{Efficiency (%)} \\ B_m & \text{Peak flux density (T)} \\ l & \text{Active length of armature conductor (m)} \\ D & \text{Diameter of the armature (m)} \\ z & \text{Number of slots} \\ \hline \end{array}\]

\[\begin{aligned} &\text{Synchronous Speed:} \quad n_s = \frac{120 f}{p} \quad \text{(RPM)} \\ &\text{Frequency in Terms of Speed:} \quad f = \frac{p n_s}{120} \quad \text{(Hz)} \end{aligned}\]

\[\begin{aligned} &\text{Induced EMF per Phase (Fundamental):} \quad E = 4.44 f \phi_m T_p k_w \\ &\text{Winding Factor:} \quad k_w = K_p \cdot K_d \\ &\text{Pitch Factor (\( K_p \)):} \quad K_p = \cos \left( \frac{\alpha}{2} \right) \\ &\text{Distribution Factor (\( K_d \)):} \quad K_d = \frac{\sin \left( \frac{m \beta}{2} \right)}{m \sin \left( \frac{\beta}{2} \right)} \\ &\text{No-Load Induced EMF (\( E_0 \)):} \quad E_0 = \sqrt{3} E \end{aligned}\]

\[\begin{aligned} &\text{Synchronous Impedance:} \quad Z_s = \sqrt{R_a^2 + X_s^2} \\ &\text{Voltage Regulation (at No Load):} \quad \text{Voltage Regulation} = \frac{|E| - |V_t|}{|V_t|} \times 100\% \\ &\text{Armature Current:} \quad I_a = \frac{E - V_t}{Z_s} \end{aligned}\]

\[\begin{aligned} &\text{Harmonic EMF (for \( h^{th} \) order harmonic):} \quad E_h = E \cdot k_h \\ &\text{Harmonic Factor:} \quad k_h = K_p \cdot K_d \cdot \frac{1}{h}, \quad h = 1, 3, 5, \dots \\ &\text{Harmonic Frequency:} \quad f_h = h \cdot f \\ &\text{Chording Angle for Eliminating \( h \)-order Harmonic:} \quad \alpha = \frac{180^\circ}{h} \end{aligned}\]

\[\begin{aligned} &\text{Power Output of Alternator:} \quad P = \sqrt{3} V_t I_a \cos \phi \\ &\text{Power Developed by the Alternator:} \quad P_d = \frac{3 E V_t}{X_s} \sin \delta \\ &\text{Electromagnetic Torque:} \quad T = \frac{P}{\omega_m} = \frac{3 E V_t}{\omega_m X_s} \sin \delta \end{aligned}\]

\[\begin{aligned} &\text{Mechanical Power Output:} \quad P_m = T \cdot \omega_m \\ &\text{Input Power:} \quad P_{in} = \sqrt{3} V_t I_a \cos \phi \\ &\text{Power Factor Adjustment:} \quad \text{Overexcited: Leading, Underexcited: Lagging} \end{aligned}\]

\[\begin{aligned} &\text{Losses in Alternators:} \quad P_{loss} = P_{cu} + P_{core} + P_{mech} \\ &P_{cu} = 3 I_a^2 R_a, \quad P_{core} = k_c f^2 B_m^2, \quad P_{mech} = k_m n^3 \\ &\text{Efficiency of Alternator:} \quad \eta = \frac{P_{out}}{P_{in}} \times 100\% \end{aligned}\]

\[\begin{aligned} &\text{Synchronizing Power Coefficient:} \quad P_{sync} = \frac{dP}{d\delta} = \frac{3 V_t^2}{\omega_s X_s} \cos \delta \\ &\text{Reactance Voltage:} \quad V_x = I_a X_s \\ &\text{Flux per Pole:} \quad \phi_m = \frac{B_m \cdot \pi D l}{p} \end{aligned}\]