📋 Nomenclature
| Symbol | Description |
|---|---|
| \(V_t\) | Terminal voltage (line or phase) |
| \(E\), \(E_f\) | Induced EMF per phase / Field EMF |
| \(f\) | Frequency (Hz) |
| \(n_s\) | Synchronous speed (RPM) |
| \(p\) | Number of poles |
| \(I_a\) | Armature current per phase |
| \(I_f\) | Field current |
| \(R_a\) | Armature resistance per phase |
| \(X_s\) | Synchronous reactance per phase |
| \(X_d\), \(X_q\) | Direct & Quadrature axis reactances |
| \(Z_s\) | Synchronous impedance |
| \(\phi\) | Power factor angle |
| \(\delta\) | Load angle (power angle) |
| \(\psi\) | Internal angle |
| \(\phi_m\) | Flux per pole (Weber) |
| \(k_w\) | Winding factor |
| \(T_p\) | Turns per phase |
| \(\omega_s\), \(\omega_m\) | Synchronous & mechanical angular velocity |
1️⃣ Fundamental Relationships
Synchronous Speed:
\[n_s = \frac{120 f}{p} \quad \text{(RPM)}\]
\[\omega_s = \frac{4\pi f}{p} \quad \text{(rad/s)}\]
Frequency:
\[f = \frac{p n_s}{120} = \frac{p \omega_m}{2\pi} \quad \text{(Hz)}\]
Mechanical Speed:
\[\omega_m = \frac{2\pi n_s}{60} \quad \text{(rad/s)}\]
2️⃣ EMF Equation of Alternator
Induced EMF per Phase:
\[E = 4.44 f \phi_m T_p k_w\]
Winding Factor:
\[k_w = K_p \cdot K_d\]
Pitch Factor:
\[K_p = \cos\left(\frac{\alpha}{2}\right)\]
α = chording angle
Distribution Factor:
\[K_d = \frac{\sin\left(\frac{m\beta}{2}\right)}{m\sin\left(\frac{\beta}{2}\right)}\]
m = slots per pole per phase
Line-to-Line Voltage:
\[V_{LL} = \sqrt{3} E_{\text{phase}}\]
3️⃣ Voltage Equation & Phasor Relations
Voltage Equation for Alternator:
\[\vec{E} = \vec{V_t} + \vec{I_a}(R_a + jX_s)\]
Voltage Equation for Motor:
\[\vec{V_t} = \vec{E} + \vec{I_a}(R_a + jX_s)\]
For Cylindrical Rotor:
\[Z_s = R_a + jX_s\]
\[|Z_s| = \sqrt{R_a^2 + X_s^2}\]
For Salient Pole (Two-Reaction Theory):
\[\vec{E} = \vec{V_t} + \vec{I_a}R_a + j\vec{I_d}X_d + j\vec{I_q}X_q\]
Where: \(I_d = I_a\sin(\psi)\), \(I_q = I_a\cos(\psi)\)
4️⃣ Armature Reaction & Voltage Regulation
Voltage Regulation:
\[\text{VR} = \frac{|E_{0}| - |V_t|}{|V_t|} \times 100\%\]
E₀ = No-load EMF, V_t = Full-load terminal voltage
Methods to Find Voltage Regulation:
Synchronous Impedance Method:
\[E_0 = \sqrt{(V_t\cos\phi + I_aR_a)^2 + (V_t\sin\phi + I_aX_s)^2}\]
MMF Method:
\[E_0 = \sqrt{F_f^2 + F_{ar}^2 + 2F_fF_{ar}\cos\theta}\]
Zero Power Factor Method (Potier):
\[X_L = \text{Potier reactance (leakage reactance)}\]
More accurate than synchronous impedance method
5️⃣ Power and Torque
For Cylindrical Rotor Machines:
Power Output (3-phase):
\[P = 3V_t I_a \cos\phi = \sqrt{3}V_L I_L \cos\phi\]
Power Developed:
\[P_d = \frac{3EV_t}{X_s}\sin\delta\]
Maximum Power:
\[P_{max} = \frac{3EV_t}{X_s} \quad \text{(at } \delta = 90°\text{)}\]
Electromagnetic Torque:
\[T_e = \frac{P_d}{\omega_s} = \frac{3EV_t}{\omega_s X_s}\sin\delta\]
For Salient Pole Machines:
Power Developed:
\[P = \frac{V_tE_f}{X_d}\sin\delta + \frac{V_t^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin2\delta\]
First term = Excitation power, Second term = Reluctance power
Torque:
\[T = \frac{V_tE_f}{\omega_s X_d}\sin\delta + \frac{V_t^2}{2\omega_s}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin2\delta\]
6️⃣ Synchronous Motor Characteristics
Input Power:
\[P_{in} = \sqrt{3}V_L I_L \cos\phi = 3V_t I_a \cos\phi\]
Mechanical Power Output:
\[P_m = P_{in} - \text{Losses} = T\omega_m\]
Power Factor Control:
• Overexcited (E > V): Leading power factor
• Underexcited (E < V): Lagging power factor
• Normal excitation (E = V): Unity power factor
• Overexcited (E > V): Leading power factor
• Underexcited (E < V): Lagging power factor
• Normal excitation (E = V): Unity power factor
Pull-out Torque:
\[T_{max} = \frac{3EV_t}{\omega_s X_s}\]
V-Curves: Relationship between armature current (I_a) and field current (I_f) at constant load shows minimum current at unity power factor.
7️⃣ Losses and Efficiency
Total Losses:
\[P_{loss} = P_{cu,a} + P_{cu,f} + P_{core} + P_{mech} + P_{stray}\]
Copper Losses:
\[P_{cu,a} = 3I_a^2 R_a\]
\[P_{cu,f} = I_f^2 R_f\]
Core Loss:
\[P_{core} = P_{hysteresis} + P_{eddy}\]
\[\propto f^2 B_m^2\]
Efficiency:
\[\eta = \frac{P_{out}}{P_{in}} \times 100\% = \frac{P_{out}}{P_{out} + P_{loss}} \times 100\%\]
Condition for Maximum Efficiency:
\[\text{Variable losses} = \text{Constant losses}\]
8️⃣ Harmonics in Alternators
Harmonic EMF (h-th order):
\[E_h = 4.44 f_h \phi_h T_p k_{w,h}\]
Harmonic Frequency:
\[f_h = h \cdot f \quad (h = 1, 3, 5, 7, \ldots)\]
Winding Factor for h-th Harmonic:
\[k_{w,h} = K_{p,h} \cdot K_{d,h}\]
\[K_{p,h} = \cos\left(\frac{h\alpha}{2}\right)\]
Harmonic Elimination:
• For h-th harmonic elimination: \(\alpha = \frac{180°}{h}\)
• Triplen harmonics (3rd, 9th, etc.) cancel in 3-phase systems
• Use chorded windings and distributed windings
• For h-th harmonic elimination: \(\alpha = \frac{180°}{h}\)
• Triplen harmonics (3rd, 9th, etc.) cancel in 3-phase systems
• Use chorded windings and distributed windings
9️⃣ Synchronization & Parallel Operation
Conditions for Synchronization:
1. Equal terminal voltages: \(V_{gen} = V_{bus}\)
2. Equal frequencies: \(f_{gen} = f_{bus}\)
3. Same phase sequence
4. Voltages in phase (phase difference = 0)
1. Equal terminal voltages: \(V_{gen} = V_{bus}\)
2. Equal frequencies: \(f_{gen} = f_{bus}\)
3. Same phase sequence
4. Voltages in phase (phase difference = 0)
Synchronizing Power:
\[P_{sync} = \frac{dP}{d\delta} = \frac{3EV_t}{X_s}\cos\delta\]
Synchronizing Torque:
\[T_{sync} = \frac{P_{sync}}{\omega_s} = \frac{3EV_t}{\omega_s X_s}\cos\delta\]
Load Sharing:
\[P_1:P_2 = \frac{1}{X_{s1}}:\frac{1}{X_{s2}}\]
🔟 Synchronous Motor Starting
Why Synchronous Motors are Not Self-Starting:
At standstill, rotor cannot develop unidirectional torque due to inertia and alternating torque.
At standstill, rotor cannot develop unidirectional torque due to inertia and alternating torque.
Starting Methods:
1. Damper/Amortisseur Windings: Provides induction motor action
2. Reduced Voltage Starting: Use auto-transformer
3. Variable Frequency Starting: Gradually increase frequency
4. Auxiliary Motor: Bring to near synchronous speed
1. Damper/Amortisseur Windings: Provides induction motor action
2. Reduced Voltage Starting: Use auto-transformer
3. Variable Frequency Starting: Gradually increase frequency
4. Auxiliary Motor: Bring to near synchronous speed
1️⃣1️⃣ Testing Methods
Open Circuit Test (OCC):
Determines magnetization curve, E vs I_f relationship
Determines magnetization curve, E vs I_f relationship
Short Circuit Test (SCC):
\[X_s = \frac{V_{oc}}{I_{sc}} \quad \text{(at same } I_f\text{)}\]
\[X_s = \frac{V_{oc}}{I_{sc}} \quad \text{(at same } I_f\text{)}\]
Zero Power Factor Test (ZPF):
Used to find Potier reactance and leakage reactance
Used to find Potier reactance and leakage reactance
Slip Test (for Salient Pole):
Determines \(X_d\) and \(X_q\) separately
Determines \(X_d\) and \(X_q\) separately
1️⃣2️⃣ Additional Important Relations
Flux per Pole:
\[\phi_m = \frac{B_m \cdot \pi D l}{p}\]
EMF in Terms of Flux Density:
\[E = 2.22 B_{av} A_p f T_p k_w\]
Where \(A_p\) = pole area
Armature Ampere-Conductors:
\[AT_a = \frac{3 \sqrt{2} T_p I_a k_w}{\pi p}\]
Output Equation:
\[Q = C_o D^2 L n_s \times 10^{-3} \text{ (kVA)}\]
Where \(C_o\) = output coefficient
Specific Magnetic Loading:
\[B_{av} = \frac{\phi_m}{\tau \times l}\]
Where \(\tau = \frac{\pi D}{p}\) = pole pitch
Specific Electric Loading:
\[ac = \frac{\text{Total armature conductors}}{\pi D}\]
💡 Quick Revision Tips
Key Differences:
Cylindrical Rotor vs Salient Pole:
• Cylindrical: X_d = X_q, uniform air gap, high speed (2-4 poles)
• Salient Pole: X_d > X_q, non-uniform air gap, low speed (many poles)
Cylindrical Rotor vs Salient Pole:
• Cylindrical: X_d = X_q, uniform air gap, high speed (2-4 poles)
• Salient Pole: X_d > X_q, non-uniform air gap, low speed (many poles)
Remember:
• Load angle δ: Alternator (0° to 90°), Motor (0° to -90°)
• At UPF: Armature reaction is cross-magnetizing
• At zero PF lagging: Armature reaction is demagnetizing
• At zero PF leading: Armature reaction is magnetizing
• Load angle δ: Alternator (0° to 90°), Motor (0° to -90°)
• At UPF: Armature reaction is cross-magnetizing
• At zero PF lagging: Armature reaction is demagnetizing
• At zero PF leading: Armature reaction is magnetizing
Common Mistakes to Avoid:
• Confusing line and phase quantities in 3-phase
• Wrong sign convention for motor vs generator
• Forgetting to account for power factor in power calculations
• Not considering saliency in salient pole machines
• Confusing line and phase quantities in 3-phase
• Wrong sign convention for motor vs generator
• Forgetting to account for power factor in power calculations
• Not considering saliency in salient pole machines