1. Transformers

Principle: Static device that transfers AC electrical power between circuits via electromagnetic induction (Faraday’s Law) without changing frequency.

• Core: Laminated silicon steel (0.3–0.5 mm) to minimize eddy current losses.
• Types: Step-up/Step-down, Core-type / Shell-type, Single/Three-phase.

🔑 Key Formulas

• Faraday’s Law: $$e = -N \frac{d\phi}{dt}$$
• Turns Ratio (Ideal): $$\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1} = K$$
• EMF Equation: $$E_1 = 4.44 f N_1 \Phi_m$$and $$E_2 = 4.44 f N_2 \Phi_m$$ (where $f$ = frequency, $N$ = turns, $\Phi_m$ = max flux)
• Voltage Regulation: $$\%VR = \frac{V_{nl} - V_{fl}}{V_{fl}} \times 100\%$$
• Efficiency: $$\eta = \frac{P_{out}}{P_{in}} \times 100\% = \frac{P_{out}}{P_{out} + P_{cu} + P_{core}} \times 100\%$$
• Max Efficiency Condition: Copper Loss ( $$I^2R$$) = Core (Iron) Loss

🧪 Testing

• Open Circuit (O.C.) Test: Done on HV side. Finds core losses, no-load current ( $$I_0$$), and shunt parameters ( $$G_0, B_0$$).
• Short Circuit (S.C.) Test: Done on LV side. Finds full-load copper losses and series impedance ( $$R_{01}, X_{01}$$).

🔌 Special Transformers

• Auto-transformer: Single winding acts as both primary and secondary. Economical for small voltage adjustments.
• 3-Phase Connections: Y-Y, Y- $$\Delta$$, $$\Delta$$-Y, $$\Delta$$- $$\Delta$$. ( $$\Delta$$- $$\Delta$$can operate as “Open Delta” or V-V connection if one transformer fails).

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2. Induction Motors (IM)

Principle: Stator’s 3-phase supply creates a Rotating Magnetic Field (RMF), which induces voltage and current in the short-circuited rotor, producing torque.

• Construction:
  • Stator: Stationary, laminated core with 3-phase slots.
  • Rotor: Squirrel-cage (robust, low maintenance) or Wound-rotor (slip rings, external resistance possible).

🔑 Key Formulas

• Synchronous Speed: $$n_{sync} = \frac{120 f_e}{P}$$(in rpm)
• Slip ($s$): $$s = \frac{n_{sync} - n_m}{n_{sync}}$$ (where $n_m$ = mechanical rotor speed; $s=1$ at standstill, $s=0$ at sync speed)
• Rotor Frequency: $$f_r = s \cdot f_e$$
• Induced Torque: $$\tau_{ind} = k (B_R \times B_S)$$
• Starting Torque Relation: $$T_{start} \propto V_{start}^2 \propto I_{start}^2$$

🚦 Starting Methods (to limit high inrush current)

1. D.O.L. (Direct-On-Line): Full voltage applied. Simple, cheap, but high starting current. Used for motors $$< 5$$kW.
2. Star-Delta: Starts in Star (voltage per phase = $$1/\sqrt{3}$$of rated), switches to Delta at ~80% speed. Reduces starting current and torque to 1/3 of DOL values.
3. Auto-transformer Starter: Uses taps to reduce starting voltage. Suitable for large motors ( $$> 20$$kW).
4. Rotor Resistance Starter: (Slip-ring only) Adds external resistance to increase starting torque and reduce starting current.
5. Soft Starter: Uses back-to-back SCRs to smoothly ramp up voltage/current.

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3. Synchronous Machines (Generators & Motors)

🔄 Synchronous Generators (Alternators)

Principle: Converts mechanical energy to 3-phase AC electrical energy. Field winding is on the rotor (DC excited via 2 slip rings), armature winding is on the stator (easier to insulate for high voltage, no large slip rings needed).

Rotor Types:

Feature	Salient Pole	Non-Salient (Cylindrical)
Speed	Low/Medium (120–400 rpm)	High (1500–3000 rpm)
Prime Mover	Hydro/Water turbine	Steam turbine (Turboalternator)
Geometry	Large diameter, short length	Small diameter, long length
Air Gap	Non-uniform ($X_d \neq X_q$)	Uniform ($X_d = X_q = X_s$)
Poles	$> 4$ poles	$\le 4$ poles

🔑 Key Formulas

• Frequency: $$f = \frac{P \cdot N}{120}$$(Hz)
• Induced EMF per phase: $$E_{ph} = 4.44 \Phi f T K_w$$ (where $K_w = K_p K_d$ is the winding factor)
• Line Voltage (Star-connected): $$E_L = \sqrt{3} E_{ph}$$

⚙️ Automatic Voltage Regulator (AVR)

• Function: Maintains constant generator terminal voltage under varying loads by automatically adjusting the DC excitation current to the rotor field winding.

🔄 Synchronous Motors

Principle: Stator RMF locks with the DC-excited rotor poles.

• Critical Trait: Not self-starting. Net average starting torque is zero because the torque reverses every half-cycle ( $$T \propto \sin\delta$$), and rotor inertia prevents it from following the rapidly reversing field.

Starting Methods:

1. Damper Winding (Amortisseur): Copper bars embedded in rotor poles (acts like a squirrel-cage IM). Motor starts as an IM, and once near $$n_{sync}$$, DC is applied to “pull into synchronism”.
2. Auxiliary Motor: A small AC/DC motor brings the synchronous motor to near synchronous speed before DC excitation is applied.
3. Slip-Ring Induction Start: High external resistance is used to start as a wound-rotor IM, then switched to DC supply for synchronous operation.

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💡 Quick Cross-Machine Comparisons

• Transformer vs. IM: Both work on mutual induction. In a transformer, the secondary is stationary; in an IM, the secondary (rotor) rotates, causing the rotor frequency to be $$f_r = s \cdot f_e$$.
• Max Efficiency: Achieved in transformers and motors when Variable Losses (Copper) = Constant Losses (Core/Iron).
• Slip: Unique to induction machines. $$s=0$$means rotor is at synchronous speed (no induced rotor current, no torque).