Synchronous Machines GATE EE Exam Quick Notes

Fundamentals

Synchronous Machine Basics

Definition: AC machines operating at synchronous speed

Synchronous Speed:

\[N_s = \frac{120f}{P} \text{ rpm}\]

Types:

  • Cylindrical Rotor: \(X_d = X_q\), high speed (steam/gas turbines)

  • Salient Pole: \(X_d > X_q\), low speed (hydro generators)

Key Parameters:

  • \(X_d\) = Direct axis reactance

  • \(X_q\) = Quadrature axis reactance

  • \(E_f\) = Field induced EMF

  • \(V_t\) = Terminal voltage

Generator Analysis

Synchronous Generator - Fundamental Equations

Voltage Equation:

\[V_t = E_f - I_a(R_a + jX_s)\]

For calculations, often \(R_a << X_s\):

\[V_t \approx E_f - jI_aX_s\]

EMF Equation:

\[E_f = 4.44 f \phi N_{ph} K_w\]

Power Relations

Cylindrical Rotor:

\[P = \frac{E_f V_t}{X_s} \sin\delta\]

Salient Pole:

\[P = \frac{E_f V_t}{X_d} \sin\delta + \frac{V_t^2}{2}\left(\frac{1}{X_q} - \frac{1}{X_d}\right)\sin 2\delta\]

Reactive Power:

\[Q = \frac{E_f V_t \cos\delta - V_t^2}{X_s}\]

Important:

  • \(\delta\) = Load angle (power angle)

  • Maximum power at \(\delta = 90°\)

  • Stability limit: \(\delta < 90°\)

Voltage Regulation

Voltage Regulation Methods

Definition:

\[\text{Voltage Regulation} = \frac{E_0 - V_t}{V_t} \times 100\%\]

Methods (GATE Important):

1. Synchronous Impedance Method:

  • \(Z_s = \sqrt{R_a^2 + X_s^2}\)

  • \(E_0 = V_t + I_a Z_s\)

  • Simple but overestimates regulation

2. MMF Method:

  • Considers saturation

  • Uses OCC and SCC curves

  • More accurate than impedance method

3. Zero Power Factor (ZPF) Method:

  • Potier reactance method

  • Most accurate for practical machines

Regulation Calculations

For Lagging Power Factor:

\[E_0 = \sqrt{(V_t\cos\phi + I_aR_a)^2 + (V_t\sin\phi + I_aX_s)^2}\]

For Leading Power Factor:

\[E_0 = \sqrt{(V_t\cos\phi + I_aR_a)^2 + (V_t\sin\phi - I_aX_s)^2}\]

Approximate Formula (when \(R_a << X_s\)):

\[\begin{aligned} \text{Lagging: } E_0 &\approx V_t + I_aX_s\cos\phi \\ \text{Leading: } E_0 &\approx V_t - I_aX_s\cos\phi \end{aligned}\]

Note: Leading power factor gives negative regulation

Parallel Operation

Synchronizing Conditions

Four Conditions for Paralleling:

  1. Equal voltage magnitudes: \(|V_1| = |V_2|\)

  2. Equal frequencies: \(f_1 = f_2\)

  3. Same phase sequence: ABC = ABC

  4. Zero phase difference: \(\phi_1 - \phi_2 = 0°\)

Synchronizing Methods:

  • Dark lamp method

  • Bright lamp method

  • Synchroscope method

Circulating Current:

\[I_{circ} = \frac{E_1 - E_2}{Z_1 + Z_2}\]

Load Sharing

Real Power Sharing:

  • Controlled by prime mover (governor)

  • Frequency droop: \(f = f_0 - k_p \cdot P\)

  • Change in fuel/steam input changes real power

Reactive Power Sharing:

  • Controlled by field excitation

  • Voltage droop: \(V = V_0 - k_q \cdot Q\)

  • Change in excitation changes reactive power

GATE Key Points:

  • Real power → Governor → Frequency control

  • Reactive power → Excitation → Voltage control

  • Droop characteristics ensure stable operation

Synchronous Motors

Synchronous Motor Characteristics

Motor Equation:

\[V_t = E_f + I_a(R_a + jX_s)\]

Key Features:

  • Constant speed operation (no slip)

  • Power factor control capability

  • High efficiency

  • Self-starting not possible

Power Relations:

\[\begin{aligned} P &= \frac{E_f V_t}{X_s} \sin\delta \\ Q &= \frac{E_f V_t \cos\delta - V_t^2}{X_s} \end{aligned}\]

Note: Motor convention: \(\delta\) negative for motor operation

V-Curves and Power Factor Control

V-Curves: \(I_a\) vs \(I_f\) at constant load

  • Minimum current at unity power factor

  • Over-excited → Leading power factor

  • Under-excited → Lagging power factor

Power Factor Control:

  • Unity p.f.: \(E_f = V_t\) (minimum current)

  • Leading p.f.: \(E_f > V_t\) (over-excited)

  • Lagging p.f.: \(E_f < V_t\) (under-excited)

Applications:

  • Power factor improvement

  • Constant speed drives

  • Synchronous condensers

Starting Methods

Starting of Synchronous Motors

Problem: Cannot start by direct connection to AC supply

Starting Methods:

1. Damper Winding Method (Most Common):

  • Rotor acts as induction motor initially

  • Field winding short-circuited during starting

  • DC excitation applied at near-synchronous speed

  • Motor pulls into synchronism

2. Variable Frequency Starting:

  • Start with low frequency, gradually increase

  • Maintains synchronous operation throughout

  • Used with electronic drives

3. Separate Prime Mover:

  • DC motor brings to synchronous speed

  • Then AC supply connected

Important Formulas

Key Formulas for GATE

\[\begin{aligned} N_s &= \frac{120f}{P} \text{ rpm} \\ E_f &= 4.44 f \phi N_{ph} K_w \text{ V} \\ P &= \frac{E_f V_t}{X_s} \sin\delta \text{ W} \\ Q &= \frac{E_f V_t \cos\delta - V_t^2}{X_s} \text{ VAr} \\ \text{Regulation} &= \frac{E_0 - V_t}{V_t} \times 100\% \\ Z_s &= \sqrt{R_a^2 + X_s^2} \\ I_{circ} &= \frac{E_1 - E_2}{Z_1 + Z_2} \end{aligned}\]

Salient Pole Relations:

  • \(X_d > X_q\) (always)

  • Two-axis theory applies

GATE Problem Types

Common GATE Problem Areas

1. Synchronous Speed & EMF Calculations

  • Given: frequency, poles \(\to\) Find: \(N_s\)

  • EMF calculation from machine parameters

2. Voltage Regulation

  • Given: load conditions → Find: regulation

  • Different methods comparison

3. Power Calculations

  • Maximum power transfer

  • Load angle calculations

  • Efficiency calculations

4. Parallel Operation

  • Synchronizing conditions

  • Load sharing problems

  • Circulating current

5. Motor Characteristics

  • V-curves interpretation

  • Power factor control

  • Starting methods

Summary

Summary - Key Points for GATE

Generator Mode: \(V_t = E_f - I_a Z_s\)

  • Power = \(\frac{E_f V_t}{X_s} \sin\delta\)

  • Regulation methods: Impedance, MMF, ZPF

  • Parallel operation: 4 synchronizing conditions

Motor Mode: \(V_t = E_f + I_a Z_s\)

  • Power factor control by excitation

  • V-curves for performance analysis

  • Starting: Damper winding method

Common for Both:

  • Cylindrical rotor: \(X_d = X_q\)

  • Salient pole: \(X_d > X_q\)

  • Synchronous speed: \(N_s = \frac{120f}{P}\)

Focus Areas: Phasor diagrams, power calculations, regulation methods, parallel operation