Unit‑3 Alternators (Synchronous Generators)

3.1 Construction – Salient and Cylindrical Rotor

• Salient Pole Rotor:

  • Large diameter, short axial length.

  • Poles project out; field windings wound on pole shoes.

  • Suitable for low‑speed (100–400 rpm) hydro machines.

  • Many poles (4–60).

  • Diagram:

    

• Cylindrical Rotor

• Small diameter, long axial length.

• Smooth steel forging with slots for field windings.

• Suitable for high‑speed (1500–3000 rpm) turbo‑alternators.

• Few poles (2–4).

• Diagram: Smooth cylinder with slots along periphery.

3.2 Rotating Magnetic Field and Working

• Principle: Electromagnetic induction – EMF induced when flux cuts conductors.

• Rotating Magnetic Field: DC excitation on rotor → rotating flux → induces alternating EMF in stator.

• Synchronous Speed:

$$N_s=\frac{120f}{P}$$

• Slip Speed: $N_s-N_r$. For synchronous machines, slip = 0.

• Diagram: Stator with 3‑phase windings, rotor field flux rotating.

3.3 Equivalent Circuit Model

• Internal EMF $E$ in series with armature resistance $R_a$ and synchronous reactance $X_s$.

• Terminal voltage $V$ = $E-(R_a+j{X}_s)I_a$.

• Diagram: Voltage source E, series Ra and Xs, output V.

• Phasor Diagram:
  

  
  
  
  

3.4 EMF Equation

$$E=4.44f\varphi T{k}_p{k}_b$$

Where:

• $f$ = frequency,

• $\varphi$ = flux per pole,

• $T$ = turns per phase,

• $k_p$ = chording factor,

• $k_b$ = breadth factor.

3.5 Winding Factors

Chording Factor (kₚ)

• Definition: The chording factor (also called the pitch factor) accounts for the reduction in generated EMF when the coil sides are short‑pitched instead of being placed exactly 180° electrical apart.

• Formula:

$$k_p=\cos \left(\frac{\alpha }{2}\right)$$

where α = short‑pitch angle (electrical degrees).

• Purpose:

  • Reduces harmonic components in the generated EMF.

  • Improves waveform quality.

  • Slightly reduces the magnitude of the fundamental EMF but enhances sinusoidal output.

• Typical Value: Between 0.9 and 1.0 depending on coil pitch.

• Example: If a coil is short‑pitched by 30°,

$$k_p=\cos (15\mathrm{°})=0.9659$$

→ EMF reduced by 3.4%, harmonics minimized.

Breadth Factor (k_b)

• Definition: The breadth factor (also called the distribution factor) accounts for the distribution of stator winding across several slots per pole per phase instead of concentrating all conductors in one slot.

• Formula:

$$k_b=\frac{\sin (m\beta \mathrm{/}2)}{m\sin (\beta \mathrm{/}2)}$$

where

• m = number of slots per pole per phase,

• β = slot angle = $\frac{180\mathrm{°}}{\text{slots per pole}}$.

• Purpose:

  • Improves the sinusoidal nature of the generated EMF.

  • Reduces harmonic content.

  • Slightly decreases the total EMF magnitude but enhances waveform smoothness.

• Typical Value: Between 0.95 and 0.98.

• Example: For 6 slots per pole per phase,

$$\beta =30\mathrm{°},k_b=\frac{\sin (90\mathrm{°})}{6\sin (15\mathrm{°})}=0.96$$

→ EMF reduced by 4%, harmonics suppressed.

• 
  

Improves sinusoidal EMF.

3.6 Armature Reaction

Definition

Armature reaction is the effect of the magnetic field produced by the armature (stator) current on the main field flux of the alternator. When the alternator supplies load current, the armature current produces its own magnetic field that interacts with the rotor’s field flux, altering the resultant flux distribution.

1️⃣ Unity Power Factor (Cross‑Magnetizing)

• Condition: Load current is in phase with terminal voltage.

• Effect: The armature flux is perpendicular to the main field flux.

• Result: The flux is distorted but not strengthened or weakened — hence cross‑magnetizing.

• Circuit Diagram: Shows field winding (DC excitation) and armature winding carrying AC current.

• Phasor Diagram: Field flux (Φ_F) horizontal; armature flux (Φ_A) vertical — 90° apart.

2️⃣ Lagging Power Factor (Demagnetizing)

• Condition: Load current lags terminal voltage (inductive load).

• Effect: Armature flux opposes the main field flux.

• Result: Net flux decreases — demagnetizing effect.

• Circuit Diagram: Same as unity PF, but current phasor lags voltage.

• Phasor Diagram: Armature flux (Φ_A) directed opposite to field flux (Φ_F).

3️⃣ Leading Power Factor (Magnetizing)

• Condition: Load current leads terminal voltage (capacitive load).

• Effect: Armature flux aids the main field flux.

• Result: Net flux increases — magnetizing effect.

• Circuit Diagram: Same as unity PF, but current phasor leads voltage.

• Phasor Diagram: Armature flux (Φ_A) in same direction as field flux (Φ_F).

• Unity PF: Cross‑magnetizing.

• Lagging PF: Demagnetizing.

• Leading PF: Magnetizing.

• Diagram: Phasor showing flux of armature vs field.

3.7 Open Circuit & Short Circuit Characteristics

• OCC: Plot of generated EMF vs field current at no load. Shows saturation.

• SCC: Shorted armature, current vs field current. Linear.

• Diagram: OCC curve (nonlinear), SCC curve (straight line).

3.8 Voltage Regulation – Synchronous Impedance Method

⚙️ Definition

Voltage Regulation of an alternator is the change in terminal voltage when the load is removed, keeping the speed and field excitation constant. It indicates how much the terminal voltage drops (or rises) under load compared to the no‑load condition.

$$\text{Voltage Regulation (\%)}=\frac{E-V}{V}\times 100$$

Where:

• $E$ = No‑load (open‑circuit) voltage per phase

• $V$ = Full‑load (rated) terminal voltage per phase

🧠 Concept

• When the alternator supplies load, the terminal voltage changes due to:

  • Armature resistance drop (IₐRₐ)

  • Reactance drop (IₐXₛ)

  • Armature reaction (demagnetizing or magnetizing effect depending on power factor)

• The combined effect causes the terminal voltage to differ from the generated EMF.

🧮 Synchronous Impedance Method (EMF Method)

This is the most common method used to determine voltage regulation.

Steps:

1. Perform Open Circuit Test (OCC): Record generated EMF vs field current at no load.

2. Perform Short Circuit Test (SCC): Record short‑circuit current vs field current.

3. Calculate Synchronous Impedance:

$$Z_s=\frac{E_{oc}}{I_{sc}}$$

4. Find Synchronous Reactance:

$$X_s=\sqrt{Z_s^2-R_a^2}$$

5. Compute Generated EMF (E):

$$E=\sqrt{(V\cos \varphi +I_a{R}_a{)}^2+(V\sin \varphi +I_a{X}_s{)}^2}$$

6. Calculate Voltage Regulation:

$$\mathrm{\%}VR=\frac{E-V}{V}\times 100$$

📊 Numerical Example

Given: $V=230V$, $I_a=20A$, $R_a=0.5\mathrm{\Omega }$, $X_s=2\mathrm{\Omega }$, power factor = 0.8 lagging.

$$E=\sqrt{(230\times 0.8+20\times 0.5{)}^2+(230\times 0.6+20\times 2{)}^2}$$

$$E=\sqrt{(184+10{)}^2+(138+40{)}^2}=\sqrt{(194{)}^2+(178{)}^2}=262.6V$$

$$\mathrm{\%}VR=\frac{262.6-230}{230}\times 100=14.2\mathrm{\%}$$

• Formula:

$$\mathrm{\%}VR=\frac{E-V}{V}\times 100$$

• Steps:

  1. Obtain OCC and SCC.

  2. Calculate synchronous reactance $X_s=\frac{E_{oc}}{I_{sc}}$.

  3. Compute E for given load.

  4. Find regulation.

• Numerical Example:

• OCC: 100 V at 2 A field current.

• SCC: 50 A at 2 A field current.

• $X_s=100\mathrm{/}50=2\mathrm{\Omega }$.

• For load current 20 A, Ra = 0.5 Ω, V = 230 V.

• Compute E = V + (Ra + jXs)Ia → magnitude.

• Regulation = (E – V)/V × 100.

3.9 Operating Characteristics

• Power Angle Equation:

$$P=\frac{EV}{X_s}\sin \delta$$

• Effect of Excitation:

  • Over‑excited → leading PF, supplies reactive power.

  • Under‑excited → lagging PF, absorbs reactive power.

• Diagram: Power‑angle curve.

3.10 Synchronization – Conditions

• Equal voltage magnitude.

• Equal frequency.

• Same phase sequence.

3.11 Synchronization Methods

• Two Bright & One Dark Lamp: Indicates synchronism and phase sequence.

• Synchroscope: Pointer shows phase difference and speed.

3.12 Parallel Operation of Alternators

• Advantages: Reliability, load sharing, maintenance flexibility.

• Conditions: Same voltage, frequency, phase sequence.

• Load Sharing: Active power controlled by prime mover input; reactive power controlled by excitation.

Applications

• Power generation in hydro, thermal, nuclear plants.

• Large synchronous generators in grid systems.

• Used for reactive power control in power systems.

Unity Power Factor Load

• Definition: When the load current is in phase with the terminal voltage.

• Circuit Diagram: Internal EMF $E$ in series with $R_a$ and $j{X}_s$, supplying terminal voltage $V$.

• Phasor Diagram:

  • $V$ and $I_a$ are in phase.

  • Armature reaction is cross‑magnetizing.

  • $E$ lies slightly ahead of $V$ due to $R_a{I}_a$ and $j{X}_s{I}_a$.

Lagging Power Factor Load

• Definition: When the load current lags the terminal voltage (inductive load).

• Circuit Diagram: Same equivalent circuit, but current phasor lags voltage.

• Phasor Diagram:

  • $I_a$ lags $V$ by angle $\varphi$.

  • Armature reaction is demagnetizing (reduces field flux).

  • $E$ is much ahead of $V$ to compensate for lagging current.

Leading Power Factor Load

• Definition: When the load current leads the terminal voltage (capacitive load).

• Circuit Diagram: Same equivalent circuit, but current phasor leads voltage.

• Phasor Diagram:

  • $I_a$ leads $V$ by angle $\varphi$.

  • Armature reaction is magnetizing (adds to field flux).

  • $E$ lies closer to $V$, sometimes even behind depending on excitation.

📌 Key Notes for Students:

• Always draw the equivalent circuit (E, Ra, jXs, V, Ia).

• In phasor diagrams:

  • Unity PF → $I_a$ in phase with $V$.

  • Lagging PF → $I_a$ behind $V$.

  • Leading PF → $I_a$ ahead of $V$.

• Label angles: load angle $\delta$ (between E and V) and power factor angle $\varphi$ (between V and Ia).