Electrical Machine-I (EX-402) - Unit 5 Short Notes

UNIT 5: TRANSFORMERS AND THREE-PHASE INDUCTION MOTORS

I. TRANSFORMERS

Constructional Features

• Core: Made of high-grade silicon steel laminations (0.35-0.5 mm thick) to reduce eddy currents. Two types:

  • Core-Type: Windings surround the core limbs. Easier to insulate, better cooling.

  • Shell-Type: Core surrounds the windings. More mechanically robust, shorter magnetic path.

• Windings: Made of high-conductivity copper/aluminum. Primary (high-voltage usually) and secondary (low-voltage usually). Insulated with paper, cloth, or enamel.

• Tank & Oil: Filled with insulating and cooling oil. Provides insulation and aids heat dissipation.

• Conservator & Breather: Accommodates oil expansion/contraction and prevents moisture ingress.

• Tap Changer: For voltage regulation (off-circuit or on-load).

• Three-Phase Transformers: Can be three single-phase units or a single three-phase unit. Core has three limbs or five-limb construction.

[!TIP] Exam Focus: Core vs. Shell type differences, reasons for laminations, and three-phase core construction (3-limb vs 5-limb for unbalanced loads) are frequently asked.

EMF Equation of Transformer

Derivation:

1. Flux per pole: $$\displaystyle \phi = B_m \times A $$ (Wb), where $$\displaystyle B_m $$ is max flux density, $A$ is core area.

2. Flux linkage change per turn: $d\phi/dt$ induces EMF.

3. Average EMF per turn: $$\displaystyle E_{avg} = 4.44 f \phi $$ (for sinusoidal flux).

4. Primary EMF: $$\displaystyle E_1 = 4.44 f N_1 \phi $$

5. Secondary EMF: $$\displaystyle E_2 = 4.44 f N_2 \phi $$

6. Voltage Transformation Ratio: $$\displaystyle a = \frac{E_2}{E_1} = \frac{N_2}{N_1} \approx \frac{V_2}{V_1} $$ (on no-load).

$$\boxed{E = 4.44 f N \phi}$$

$$\boxed{a = \frac{N_2}{N_1}}$$

[!TIP] Common Pitfall: Remember 4.44 comes from $\sqrt{2} \times \pi \approx 4.44$. For non-sinusoidal flux, this factor changes.

Excitation Phenomenon & No-Load Current

• When primary is connected to AC supply and secondary is open, primary draws no-load current $$\displaystyle I_0 $$.

• $$\displaystyle I_0 $$ has two components:

  1. Active Component ($$\displaystyle I_w $$ or $$\displaystyle I_c $$): Supplies core losses (hysteresis + eddy current). In phase with $$\displaystyle V_1 $$.

  2. Magnetizing Component ($$\displaystyle I_m $$): Creates mutual flux $$\displaystyle \phi_m $$. Lags $$\displaystyle V_1 $$ by 90°.

• Phasor: $$\displaystyle I_0 = I_w + I_m $$. $$\displaystyle I_0 $$ lags $$\displaystyle V_1 $$ by angle $$\displaystyle \phi_0 $$ (no-load power factor).

• Significance: $$\displaystyle I_m $$ is large (20-50% of rated current) due to high reluctance of air gap.

[!TIP] Exam Focus: Draw the no-load phasor diagram clearly showing $$\displaystyle I_w $$, $$\displaystyle I_m $$, and $$\displaystyle I_0 $$. $$\displaystyle I_w $$ is small, $$\displaystyle I_m $$ is large.

Core Losses (Iron Losses)

1. Hysteresis Loss ($$\displaystyle P_h $$): Due to reversal of magnetization in core material.

   • $$\displaystyle P_h \propto f B_m^{1.6} \propto f V^{1.6} $$ (since $$\displaystyle B_m \propto V/f $$).

   • Reduced by using high-grade silicon steel (narrow hysteresis loop).

2. Eddy Current Loss ($$\displaystyle P_e $$): Induced currents in core laminations.

   • $$\displaystyle P_e \propto f^2 B_m^2 \propto f V^2 $$.

   • Reduced by laminating core and using thin, insulated sheets.

Separation from Frequency Variation:

Given total core loss $$\displaystyle P_i = P_h + P_e $$.

At constant voltage $V$, $$\displaystyle B_m \propto V/f $$.

• $$\displaystyle P_h \propto f $$, $$\displaystyle P_e \propto f^2 $$.

If $$\displaystyle P_i $$ is measured at two frequencies $$\displaystyle f_1, f_2 $$ at same $V$: $$\displaystyle P_{i1} = k_h f_1 + k_e f_1^2 $$ $$\displaystyle P_{i2} = k_h f_2 + k_e f_2^2 $$

Solve for constants $$\displaystyle k_h, k_e $$ to separate $$\displaystyle P_h $$ and $$\displaystyle P_e $$.

[!TIP] Past Paper Pattern: Direct numerical problems on loss separation with changed frequency/voltage are common (e.g., Jun 2025 Q3).

Copper Losses & Load Dependence

• $$\displaystyle I^2R $$ loss in primary and secondary windings.

• Full-load copper loss: $$\displaystyle P_{c,FL} = I_1^2 R_1 + I_2^2 R_2 $$.

• At any load $x$ (fraction of full load): $$\displaystyle P_c = x^2 P_{c,FL} $$.

• Dependence: Varies with square of load current, independent of power factor.

Efficiency of Transformer

• $$\displaystyle \eta = \frac{\text{Output Power}}{\text{Input Power}} = \frac{x S \cos\phi}{x S \cos\phi + P_i + x^2 P_{c,FL}} $$

• Condition for Maximum Efficiency:

  • $$\displaystyle \frac{d\eta}{dx} = 0 \Rightarrow $$ Variable losses = Constant losses.

  • $$\displaystyle x^2 P_{c,FL} = P_i $$

  • $$\displaystyle \boxed{x = \sqrt{\frac{P_i}{P_{c,FL}}}} $$

  • At max efficiency, $$\displaystyle \eta_{max} = \frac{x S \cos\phi}{x S \cos\phi + 2P_i} $$.

• Efficiency is maximum when core loss equals copper loss.

[!TIP] Key Point: Max efficiency condition is independent of power factor. It depends only on the ratio of constant to full-load variable losses.

Voltage Regulation

• Definition: Percentage change in secondary terminal voltage from no-load to full-load, at constant primary voltage and same power factor.

$$VR = \frac{E_2 - V_2}{V_2} \times 100\%$$

(For transformer, usually $$\displaystyle V_2 $$ is rated terminal voltage).

• Derivation using Approximate Equivalent Circuit (Referred to primary/secondary):

$$VR \approx \frac{I_2 (R_{02} \cos\phi_2 \pm X_{02} \sin\phi_2)}{V_2} \times 100\%$$

(+ for lagging PF, - for leading PF).

• Conditions:

  • Maximum Regulation: Occurs when $$\displaystyle \frac{d(VR)}{d\phi_2} = 0 \Rightarrow \tan\phi_2 = \frac{X_{02}}{R_{02}} $$. PF = $$\displaystyle \cos\phi_2 = \frac{R_{02}}{Z_{02}} $$ (lagging).

  • Zero Regulation: Occurs when $$\displaystyle R_{02} \cos\phi_2 + X_{02} \sin\phi_2 = 0 \Rightarrow \tan\phi_2 = -\frac{R_{02}}{X_{02}} $$. PF is leading.

[!TIP] Critical: Zero regulation is possible only for leading power factor loads. Max regulation always occurs at lagging PF.

Open-Circuit Test (OC Test)

• Purpose: Determine core loss ($$\displaystyle P_i $$) and excitation parameters ($$\displaystyle R_0, X_0 $$).

• Procedure: Apply rated voltage to primary, keep secondary open. Measure $$\displaystyle V_1, I_0, P_0 $$.

• Circuit: Primary connected to supply, secondary open.

• Calculations:

  • $$\displaystyle P_0 = \text{Core loss (hysteresis + eddy)} $$.

  • $$\displaystyle I_w = \frac{P_0}{V_1} $$, $$\displaystyle I_m = \sqrt{I_0^2 - I_w^2} $$.

  • $$\displaystyle R_0 = \frac{V_1}{I_w} $$, $$\displaystyle X_0 = \frac{V_1}{I_m} $$.

• Performed on: Usually LV side for safety and convenience.

Short-Circuit Test (SC Test)

• Purpose: Determine equivalent resistance ($$\displaystyle R_{eq} $$) and reactance ($$\displaystyle X_{eq} $$) for voltage drop calculation.

• Procedure: Apply reduced voltage to primary until full-load current flows in both windings. Secondary is shorted.

• Circuit: Primary connected to variable low-voltage supply, secondary terminals shorted.

• Measurements: $$\displaystyle V_{sc}, I_{sc}, P_{sc} $$ (all on primary side).

• Calculations:

  • $$\displaystyle P_{sc} = I_{sc}^2 R_{eq} $$ (copper loss at full-load current).

  • $$\displaystyle R_{eq} = \frac{P_{sc}}{I_{sc}^2} $$.

  • $$\displaystyle Z_{eq} = \frac{V_{sc}}{I_{sc}} $$.

  • $$\displaystyle X_{eq} = \sqrt{Z_{eq}^2 - R_{eq}^2} $$.

• Performed on: Usually HV side to circulate full-load current with lower test voltage.

Sumpner's Test (Back-to-Back Test)

• Purpose: To determine temperature rise and combined efficiency under actual loading conditions, without needing separate loading.

• Setup: Two identical transformers connected back-to-back. Primary of T1 connected to supply, secondary of T1 connected to primary of T2, secondary of T2 supplies a load. Both transformers share the load current.

• Measurements: Total input power to both primaries. Output power from T2 secondary.

• Calculations:

  • Total losses = Input - Output.

  • Core loss = $2 \times$ (OC test loss of one transformer).

  • Copper loss = Total losses - Core loss.

  • Efficiency of each transformer can be found.

[!TIP] Distinguish: OC test gives core loss & excitation params. SC test gives copper loss & impedance params. Sumpner's test gives temperature rise and combined efficiency under load.

Auto-Transformer

• Construction: Single continuous winding with a tap. Provides voltage transformation with a single coil.

• Comparison with Two-Winding Transformer:

  | Feature | Two-Winding Transformer | Auto-Transformer | | :--- | :--- | :--- | | Windings | Two separate, electrically isolated. | One continuous winding with tap. | | Size/Weight | Larger, heavier for same rating. | Smaller, lighter, cheaper. | | Efficiency | Slightly lower (more copper). | Higher (less copper). | | Isolation | Provides electrical isolation. | No isolation between primary & secondary. | | Application | General purpose, isolation needed. | Voltage regulation (buck/boost), motor starters. |

• Advantages: Smaller size, lower cost, higher efficiency for similar rating.

• Disadvantage: No electrical isolation, higher short-circuit current.

• Copper Saving Derivation:

  Let $$\displaystyle a = \frac{V_2}{V_1} $$ (step-down, $$\displaystyle a<1 $$).

  Weight of two-winding transformer $$\displaystyle \propto (N_1 + N_2) $$.

  Weight of auto-transformer $$\displaystyle \propto (N_1 + (N_2 - N_1)) = N_2 $$.

  % Saving $$\displaystyle = \frac{(N_1+N_2) - N_2}{N_1+N_2} \times 100 = \frac{N_1}{N_1+N_2} \times 100 = \frac{1}{1+1/a} \times 100 $$ (for step-down).

  For step-up: % Saving $$\displaystyle = \frac{N_2}{N_1+N_2} \times 100 $$.

Three-Phase Transformer Connections

[!TIP] Scott Connection: Crucial to remember the 2:1 turns ratio between the main and teaser transformers for balanced 3-phase to 2-phase conversion. Derive using phasors.

Parallel Operation of Transformers

• Necessary Conditions:

  1. Same voltage ratio (turns ratio).

  2. Same phase sequence (for three-phase).

  3. Same polarity.

  4. Similar impedance voltage (%Z) for proper load sharing.

• Circulating Current: Arises if voltage ratios are unequal. Flows even at no-load, causes extra losses.

$$I_{cir} = \frac{V_{1a} - V_{1b}}{Z_{eq,a} + Z_{eq,b}}$$

• Load Sharing (Different Ratings & Impedances):

  • Transformers share load in proportion to their kVA ratings if their % impedances (on their own bases) are equal.

  • If %Z differ, load sharing is inversely proportional to %Z.

  • KVA loading of Transformer A: $$\displaystyle S_A = \frac{S_{total} \times (kVA_A / Z_A)}{\sum (kVA_i / Z_i)} $$

  • Where $$\displaystyle Z_A $$ is per-unit impedance of A.

[!TIP] Past Paper Favorite: Problems involving two transformers with different ratings and %Z supplying a common load (e.g., Jun 2023 Q6, Nov 2023 Q7-9).

Conservator and Breather

• Conservator: A small auxiliary tank mounted above the main tank. It provides space for oil expansion when the transformer heats up and contraction when it cools. It is connected to the main tank via a pipeline.

• Breather: Contains silica gel (blue when dry, pink when moist). It is connected to the conservator. Its function is to absorb moisture from the air that enters the conservator during oil contraction (cooling). Prevents moisture from entering the main tank and deteriorating oil insulation.

• Why breather on conservator? Because air enters/exits the conservator during thermal cycles, not the main tank (which is sealed except for the conservator link).

[!TIP] Direct Question: "Why breather is connected with conservator and not with main tank?" – Answer: Air only enters/exits through the conservator during oil volume changes.

Tap Changers

• Off-Circuit Tap Changer (OCTC): Used when transformer is isolated from supply. Manual or motor-driven. For coarse voltage adjustment.

• On-Load Tap Changer (OLTC): Used while transformer is on-load. Complex mechanism with selector switches and diverter switches to prevent arcing during tap change. Used for fine, continuous voltage regulation (e.g., in power transformers).

Cooling Methods of Transformers

1. ONAN: Oil Natural, Air Natural (Small distribution).

2. ONAF: Oil Natural, Air Forced (Forced air fans).

3. OFAF: Oil Forced, Air Forced (Pumps + fans).

4. OFWF: Oil Forced, Water Forced (For large power transformers, water coolers).

5. ODAF: Oil Directed, Air Forced (Oil directed by baffles).

6. Hydrogen Gas Cooled: For very large generators/transformers.

Pulse Transformers & High-Frequency Transformers

• Pulse Transformer: Used in digital circuits, SCR/T thyristor triggering. Characteristics: Low leakage inductance, low inter-winding capacitance, fast response, high isolation. Core material: Ferrite.

• High-Frequency Transformer: Used in switch-mode power supplies (SMPS). Operates at 20 kHz - 1 MHz. Core: Ferrite or powdered iron. Smaller size due to higher frequency. Key params: Low losses at high frequency, high saturation flux density.

Power Transformers vs Distribution Transformers

Inrush Currents

• Nature: Very high transient current (5-30 times full-load) drawn at energization.

• Causes: Sudden application of voltage to an unfluxed core. The core may have residual flux. To establish flux, the transformer draws a large magnetizing current with a DC offset.

• Problems: Maloperation of protection relays (differential, overcurrent), mechanical stress on windings, voltage dip in supply system.

• Mitigation: Controlled switching, pre-insertion resistors, inrush restrainers in relays.

Impact of Harmonics on Transformer Performance

1. Increased Core Loss: Harmonics increase $f$ and $$\displaystyle B_m $$ (if voltage is non-sinusoidal but RMS same). $$\displaystyle P_e \propto f^2 $$, so eddy current loss rises significantly.

2. Copper Loss Increase: Harmonic currents cause additional $$\displaystyle I^2R $$ loss.

3. Neutral Current: In Y-connected secondary with balanced 3-phase harmonic load (e.g., rectifiers), triplen harmonics (3rd, 9th) are in-phase and add in neutral, causing overheating.

4. Resonance: With system capacitance, harmonic frequencies can cause parallel/series resonance, leading to overvoltages.

5. Reduced Efficiency & Life: Extra heating accelerates insulation aging.

Effect of Supply Voltage & Frequency

• Voltage Variation:

  • Flux $\phi \propto V/f$. If $V$ increases, $\phi$ increases → Core saturation → $$\displaystyle I_0 $$ increases sharply → $$\displaystyle P_h, P_e $$ increase.

  • If $V$ decreases, flux decreases, but for same load, current increases to maintain output → Copper loss increases.

• Frequency Variation:

  • At constant voltage, $f \downarrow \Rightarrow \phi \uparrow$ → Saturation → $$\displaystyle I_0 \uparrow $$, $$\displaystyle P_h \uparrow $$, $$\displaystyle P_e \uparrow $$.

  • At constant flux ($V/f$ constant), $f \downarrow \Rightarrow$ for same $V$, $V$ must be reduced. Copper loss may change depending on load.

Consequences of Winding Short-Circuit

• Partial Short-Circuit in Primary:

  • Reduces effective turns → Increased flux density → Core saturation.

  • $$\displaystyle I_0 $$ increases drastically.

  • Local overheating at shorted spot.

  • May cause Buchholz relay operation (if oil-filled).

• Partial Short-Circuit in Secondary:

  • Reduces effective turns → Lower induced EMF → High current drawn from primary to maintain load → Overloading, overheating.

  • Voltage regulation becomes poor.

Calculation of Number of Turns and Core Area

Given: Per turn EMF $$\displaystyle E_t $$ (V/turn), flux density $$\displaystyle B_m $$ (T), frequency $f$ (Hz), primary voltage $$\displaystyle V_1 $$ (V).

1. $$\displaystyle N_1 = \frac{V_1}{E_t} $$.

2. $$\displaystyle E_t = 4.44 f \phi \Rightarrow \phi = \frac{E_t}{4.44 f} $$.

3. Core area $$\displaystyle A = \frac{\phi}{B_m} $$ (ensure units: $\phi$ in Wb, $$\displaystyle B_m $$ in T, $A$ in m²).

4. $$\displaystyle N_2 = a \times N_1 $$ (where $$\displaystyle a = V_2/V_1 $$).

[!TIP] Standard Value: Per turn EMF is typically 1-2 V/turn for distribution transformers, higher for power transformers to reduce turns.

II. THREE-PHASE INDUCTION MOTORS

Construction & Working Principle

• Stator: Three-phase AC supply creates a rotating magnetic field (RMF). Laminated core with 3-phase distributed winding.

• Rotor: Two types:

  • Squirrel Cage: Aluminum/copper bars short-circuited by end rings. Simple, rugged, cheap.

  • Slip Ring (Wound Rotor): Three-phase winding (star-connected) with slip rings. External resistances can be added for starting/control.

• Air Gap: Small (0.5-2 mm) uniform gap. Crucial for performance.

• Working: RMF sweeps across rotor conductors → induces EMF → current → torque → rotor rotates. Slip ($s$) is essential for torque production.

Squirrel Cage vs Slip Ring Induction Motor

Production of Rotating Magnetic Field (RMF)

• Three-phase balanced sinusoidal currents in stator windings, spatially displaced by 120° electrical, produce a uniform rotating magnetic field.

• Synchronous Speed: $$\displaystyle N_s = \frac{120 f}{P} $$ (rpm).

• Direction: Reverses if any two supply terminals are interchanged.

• Magnitude: Constant = $$\displaystyle \frac{3}{2} \times \frac{4}{\pi} \times \frac{N I_m}{\sqrt{2}} \times k_w \times $$ (flux per pole) – but for balanced supply, it's constant.

Slip and Rotor Frequency

• Slip: $$\displaystyle s = \frac{N_s - N_r}{N_s} $$ (fractional slip).

• Rotor Frequency: $$\displaystyle f_r = s f $$.

  • At start ($$\displaystyle N_r=0 $$): $$\displaystyle s=1 $$, $$\displaystyle f_r = f $$.

  • At synchronous speed ($$\displaystyle N_r=N_s $$): $$\displaystyle s=0 $$, $$\displaystyle f_r = 0 $$.

• Rotor EMF: $$\displaystyle E_2 = s E_{20} $$ (where $$\displaystyle E_{20} $$ is standstill rotor EMF).

Slot per Pole per Phase (SPP)

• Definition: $$\displaystyle SPP = \frac{\text{Total Number of Stator Slots}}{\text{Number of Poles} \times \text{Number of Phases}} $$.

• Significance:

  • Should be integer or fractional (e.g., 5/2, 7/3) for distributed winding.

  • Fractional SPP helps in reducing harmonics (chording), improves MMF waveform, and distributes winding evenly.

  • Must avoid integral SPP which can cause cogging (locking).

Equivalent Circuit of Three-Phase Induction Motor

Exact Equivalent Circuit (Referred to Stator):

• $$\displaystyle R_1, X_1 $$: Stator resistance & leakage reactance.

• $$\displaystyle R_m, X_m $$: Core loss resistance & magnetizing reactance (often combined or $$\displaystyle R_m $$ neglected for simplicity).

• $$\displaystyle R_2', X_2' $$: Rotor resistance & leakage reactance referred to stator.

• Rotor branch: $$\displaystyle R_2'/s $$ represents combined effect of rotor resistance and mechanical load.

Simplified Equivalent Circuit:

Often $$\displaystyle R_m $$ is moved to input or neglected for torque calculations. Split $$\displaystyle X_1 $$ and $$\displaystyle X_2' $$ into $X$ for simplicity.

[!TIP] Key: Understand how to reduce the exact circuit to the simplified form. The term $$\displaystyle R_2'(1-s)/s $$ represents the mechanical load equivalent resistance.

No-Load Test

• Procedure: Run motor at rated voltage, no mechanical load. Measure $$\displaystyle V_0, I_0, P_0 $$ (line values for 3-phase).

• Parameters Determined:

  • $$\displaystyle P_0 \approx P_{core} + P_{stray} $$ (friction, windage negligible at no-load? Actually $$\displaystyle P_0 $$ includes all no-load losses: core + friction + windage + small stator copper loss).

  • $$\displaystyle I_0 $$ gives excitation current.

  • $$\displaystyle R_m = \frac{V_0^2}{P_{core}} $$ (if core loss separated).

  • $$\displaystyle X_m = \frac{V_0}{\sqrt{I_0^2 - (V_0^2/P_0)^2}} $$ (approx).

• Purpose: Find $$\displaystyle R_m, X_m $$ and no-load losses.

Blocked Rotor Test

• Procedure: Lock rotor, apply reduced voltage to stator until full-load current flows. Measure $$\displaystyle V_{sc}, I_{sc}, P_{sc} $$.

• Parameters Determined:

  • $$\displaystyle P_{sc} = 3 I_{sc}^2 (R_1 + R_2') $$ (total copper loss at short-circuit).

  • $$\displaystyle Z_{sc} = \frac{V_{sc}}{I_{sc}} $$.

  • $$\displaystyle R_{eq} = R_1 + R_2' = \frac{P_{sc}}{3 I_{sc}^2} $$.

  • $$\displaystyle X_{eq} = X_1 + X_2' = \sqrt{Z_{sc}^2 - R_{eq}^2} $$.

• Purpose: Find total equivalent resistance and reactance (leakage impedances).

Circle Diagram

• Construction from Test Data:

  1. Draw horizontal axis (real power) and vertical axis (reactive power).

  2. Plot No-Load Point: $$\displaystyle P = P_0 $$, $$\displaystyle Q = \sqrt{3} V_0 I_0 \sin\phi_0 $$.

  3. Plot Short-Circuit Point: $$\displaystyle P = P_{sc} $$, $$\displaystyle Q = \sqrt{3} V_{sc} I_{sc} \sin\phi_{sc} $$.

  4. Draw line from origin through SC point. This is the input impedance line.

  5. Draw line from No-Load point parallel to input impedance line. Their intersection is the center of the circle.

  6. Draw circle with this center passing through SC point.

• Performance Determination:

  • Full-Load Current: Locate point on circle at distance = full-load current from origin along input impedance line.

  • Power Factor: Cosine of angle between OP (to point) and real axis.

  • Maximum Power: Point where tangent from origin touches circle (perpendicular to diameter).

  • Starting Torque: Torque at $$\displaystyle s=1 $$ (point on circle at $$\displaystyle I = I_{sc} $$).

  • Slip: $$\displaystyle s = \frac{\text{Stator input power at point} - \text{Constant losses}}{\text{Air-gap power}} $$.

[!TIP] Past Paper Favorite: Drawing circle diagram and determining full-load current, PF, maximum power, starting torque (e.g., Jun 2024 Q8).

Torque-Slip Equation & Maximum Torque

• Torque Development: $$\displaystyle T = \frac{3}{\omega_s} \frac{R_2'/s}{(R_1 + R_2'/s)^2 + (X_1 + X_2')^2} \times (E_1)^2 $$ (for exact circuit).

• Simplified (Neglecting $$\displaystyle R_1 $$): $$\displaystyle T \propto \frac{s R_2'}{(R_2'^2 + (s X_{eq})^2)} $$.

• Condition for Maximum Torque:

  • Differentiate $T$ w.r.t $s$ and set $$\displaystyle dT/ds=0 $$.

  • $$\displaystyle s_{m} = \frac{R_2'}{X_{eq}} $$ (for simplified case, neglecting $$\displaystyle R_1 $$).

  • $$\displaystyle T_{max} \propto \frac{E_1^2}{2 X_{eq}} $$ (if $$\displaystyle R_1 $$ neglected).

• Key Insight: $$\displaystyle T_{max} $$ is independent of $$\displaystyle R_2' $$ but $$\displaystyle s_m $$ is directly proportional to $$\displaystyle R_2' $$. Adding rotor resistance shifts $$\displaystyle s_m $$ to higher values but $$\displaystyle T_{max} $$ remains same (for slip ring motor).

[!TIP] Remember: $$\displaystyle T_{max} \propto 1/X_{eq} $$. To increase $$\displaystyle T_{max} $$, reduce leakage reactance (deep bar, double cage).

Torque-Speed Characteristics & Effect of Rotor Parameters

• Shape: Stable operating region (slip 0 to $$\displaystyle s_m $$). Unstable region ($$\displaystyle s > s_m $$).

• Effect of Rotor Resistance $$\displaystyle R_2' $$:

  • $$\displaystyle T_{max} $$ unchanged.

  • $$\displaystyle s_m \propto R_2' $$. Starting torque increases with $$\displaystyle R_2' $$ up to a point.

  • Slip ring: External resistance increases starting torque, reduces starting current.

  • Squirrel cage: $$\displaystyle R_2' $$ fixed → low starting torque.

• Effect of Leakage Reactance $$\displaystyle X_{eq} $$:

  • $$\displaystyle T_{max} \propto 1/X_{eq} $$.

  • Higher $$\displaystyle X_{eq} $$ reduces $$\displaystyle T_{max} $$ and shifts $$\displaystyle s_m $$ to lower values.

  • Deep bar/double cage rotors increase effective $$\displaystyle R_2' $$ at start (due to skin effect) → higher starting torque.

Crawling and Cogging

• Cogging (Locking):

  • Cause: Harmonic RMF (especially 5th, 7th) interacts with rotor. If SPP is integer, stator and rotor slots may align → no torque → motor refuses to start.

  • Prevention: Use fractional SPP (skewing, chording).

• Crawling:

  • Cause: Harmonic torques (especially 5th harmonic → negative sequence → reverse rotation at $$\displaystyle N_s/5 $$). Motor runs at about 1/5th of synchronous speed.

  • Prevention: Fractional SPP, proper design.

Double Cage and Deep Bar Rotors

• Purpose: Improve starting torque and reduce starting current for squirrel cage motors.

• Double Cage Rotor:

  • Two sets of bars: Outer cage (high resistance, low leakage reactance) and Inner cage (low resistance, high leakage reactance).

  • At start ($$\displaystyle s=1 $$): Skin effect forces current into outer cage (high $R$, low $L$) → high starting torque.

  • At full load ($s \approx 0$): Skin effect negligible, current splits → inner cage dominates (low $R$) → good efficiency.

• Deep Bar Rotor:

  • Single deep bar. Skin effect increases effective resistance at start → similar effect to double cage but less pronounced.

[!TIP] Explain: "How starting performance is improved by double cage rotor?" → Emphasize skin effect causing frequency-dependent impedance.

Starting Methods of Squirrel Cage Induction Motor

Speed Control Methods

Stator Side:

1. Voltage Control: $$\displaystyle T \propto V^2 $$. Used for small motors (fans). Poor regulation.

2. Frequency Control (V/f): Maintain constant $V/f$ to avoid saturation. Wide speed range. Requires converter (inverter).

3. Pole Changing: Change number of poles by reconnecting winding. Step change in speed. $$\displaystyle N_s \propto 1/P $$. Used for multi-speed motors.

Rotor Side (Only for Slip Ring):

4. Rotor Resistance Insertion: $$\displaystyle T_{max} $$ unchanged, $$\displaystyle s_m $$ increases → speed reduces. Simple but inefficient (losses in resistor).

5. Slip Power Recovery (SPR): Recovers slip power from rotor circuit.

   • Scherbius Drive: Rotor AC fed to rectifier → inverter → back to supply. Used for large motors (pumps).

   • Kramer Drive: Rotor AC rectified → DC motor coupled to main motor shaft. Mechanical recovery.

[!TIP] Past Question: "Explain slip power recovery schemes" – Describe Scherbius (electrical) and Kramer (mechanical).

Losses in Induction Motor & Efficiency

• Stator Copper Loss: $$\displaystyle 3 I_1^2 R_1 $$.

• Rotor Copper Loss: $s \times \text{Rotor Input}$.

• Core Loss (Stator): $$\displaystyle P_{core} $$ (from no-load test).

• Friction & Windage: Mechanical losses.

• Stray Load Losses: Miscellaneous (harmonic, etc.).

• Efficiency: $$\displaystyle \eta = \frac{\text{Output}}{\text{Input}} = \frac{\text{Rotor Input} - \text{Rotor Cu Loss} - \text{Stray Losses}}{\text{Rotor Input} + \text{Stator Losses}} $$.

• Loss Separation: From no-load test (core + friction + windage) and full-load test (total losses), subtract no-load core loss to get friction/windage/stray at full load.

Single-Phase Induction Motors

Classification:

1. Split Phase: Main winding + auxiliary winding (high resistance). Poor starting torque.

2. Capacitor-Start: Auxiliary winding with capacitor → high starting torque. Capacitor disconnected by centrifugal switch.

3. Capacitor-Run: Capacitor remains in circuit → improved PF and torque.

4. Shaded Pole: Shading ring on pole. Very low starting torque, cheap. Used in small fans.

5. Reluctance (not common).

Double Revolving Field Theory:

• Single-phase supply produces pulsating (not rotating) field.

• Can be resolved into two equal, opposite rotating fields ($$\displaystyle F_f $$ and $$\displaystyle F_b $$) at synchronous speed.

• At standstill, torques cancel → net torque zero → not self-starting.

• Auxiliary winding creates asymmetry → unequal fields → net starting torque.

Equivalent Circuit (Ignoring Core Loss):

Similar to 3-phase but with two windings (main $m$ and auxiliary $a$) in quadrature.

• Forward & Reverse Fields: Represented by two equivalent circuits in parallel.

• Net Torque: $$\displaystyle T = T_f - T_b $$.

Capacitor-Start Motor:

• Auxiliary winding has capacitor in series → high phase shift → large starting torque.

• Centrifugal switch disconnects capacitor at ~75% speed.

• Applications: Refrigerators, compressors.

Shaded Pole Motor:

• Construction: Salient poles, part of each pole is short-circuited by copper ring (shading coil).

• Working: Flux in shaded part lags → rotating effect → weak starting torque.

• Applications: Small fans, toys, instruments.

AC Series Motor (Universal Motor)

• Construction: Series DC motor armature & field windings. Commutator & brushes.

• Working: AC supply → alternating flux → alternating torque. Commutator acts as mechanical rectifier → unidirectional torque.

• Compensating Winding: Embedded in pole faces, connected in series with armature. Purpose: Neutralize armature reaction (cross-magnetizing), improve commutation, reduce sparking.

• Applications: Portable tools, vacuum cleaners, domestic appliances (high speed, high torque).

• Characteristics: Very high speed at no-load (dangerous), poor PF, low efficiency.

Linear Induction Motor (LIM)

• Principle: Unrolled version of rotary IM. Produces linear force instead of torque.

• Construction:

  • Short Primary: 3-phase winding on iron core (like stator).

  • Secondary (Reaction Plate): Aluminum sheet over iron (or just aluminum) – like squirrel cage.

• Working: Traveling magnetic field induces currents in plate → repulsion/attraction → linear motion.

• Applications: Maglev trains, conveyors, pumps, actuators.

Servo Motors

• Definition: Motors with fast response to control signals (voltage, pulse width).

• Types:

  1. AC Servo Motors: Squirrel cage induction motor with tachogenerator feedback. Simple, rugged.

  2. DC Servo Motors: Permanent magnet or separately excited. Better speed control.

  3. Brushless DC (BLDC) Servo: High performance, no brushes.

• Applications: Robotics, CNC machines, radar, automation.

Induction Generator

• Principle: Rotor driven above synchronous speed ($$\displaystyle s < 0 $$). Slip negative → power flows from rotor to stator → generates electricity.

• Self-Excited: Requires capacitor bank at terminals to provide reactive power (for isolated operation). Residual magnetism builds up voltage.

• Externally Excited: Connected to live grid (or synchronous condenser) → draws reactive power from grid.

• Applications: Wind power generation, regenerative braking.

Effect of Supply Voltage & Frequency Variation

• Voltage Variation:

  • Torque: $$\displaystyle T \propto V^2 $$. Large effect.

  • Current: For same torque, $I \propto V$.

  • Slip: Increases slightly if voltage drops (to maintain torque).

  • Efficiency: Decreases due to increased current and losses.

• Frequency Variation:

  • Synchronous Speed: $$\displaystyle N_s \propto 1/f $$.

  • Flux: $\phi \propto V/f$. If $f$ changes, to avoid saturation, $V$ must change proportionally.

  • If $V$ constant and $f \downarrow$: $\phi \uparrow$ → saturation → $$\displaystyle I_0 \uparrow $$, $$\displaystyle P_{core} \uparrow $$.

  • Torque-speed curve shifts horizontally (new $$\displaystyle N_s $$).

Braking Methods

1. Regenerative Braking: Motor runs as generator ($$\displaystyle s < 0 $$). Energy fed back to supply. Requires speed > $$\displaystyle N_s $$ and grid connection.

2. Plugging (Reverse Current Braking): Reverse supply sequence → reverse RMF → braking torque. High current → resistors needed. Stops quickly.

3. Dynamic Braking: Disconnect AC, connect DC to stator → stationary field → rotor generates, energy dissipated in rotor resistance (or external resistor). Requires slip ring motor or external resistor.

Impact of Harmonics on Performance

1. Torque Pulsations: Harmonic RMFs produce torques at harmonic synchronous speeds → torque ripple, noise, vibration.

2. Increased Losses: Harmonic currents increase copper loss; harmonic flux increases core loss.

3. Reduced Torque: Fundamental torque component may reduce due to distortion.

4. Heating: Additional losses cause overheating.

5. Starting Issues: Cogging/crawling exacerbated.

Unbalanced Supply & Negative Sequence Currents

• Unbalanced Voltage: Creates negative sequence rotating field (opposite direction to positive sequence).

• Effects:

  • Negative sequence current flows → generates reverse torque → reduces net torque, increases heating.

  • Additional copper loss ($$\displaystyle I_2^2 R $$) in stator and rotor.

  • Vibration, noise.

  • Stator Winding Short Circuit Location: Negative sequence current magnitude depends on location of fault. By measuring negative sequence current from different terminals, fault location can be identified.

Rotor Input, Copper Loss, Output Relation (in terms of slip $s$)

• Rotor Input (Air-gap Power): $$\displaystyle P_{in,rotor} = P_{ag} $$.

• Rotor Copper Loss: $$\displaystyle P_{cu,rotor} = s \cdot P_{ag} $$.

• Mechanical Developed Power (Gross): $$\displaystyle P_{dev} = (1-s) P_{ag} $$.

• Relationship:

$$P_{ag} = P_{dev} + P_{cu,rotor}$$

$$\frac{P_{cu,rotor}}{P_{dev}} = \frac{s}{1-s}$$

$$\boxed{P_{dev} = P_{ag} (1-s)}$$

$$\boxed{P_{cu,rotor} = s P_{ag}}$$

[!TIP] Mnemonic: "Rotor input is divided into output (useful) and rotor loss (waste) in the ratio $(1-s):s$."

Separation of Losses from Tests

• No-Load Test: $$\displaystyle P_0 = P_{core} + P_{f&w} $$ (friction & windage). At no-load, rotor copper loss negligible.

• Full-Load Test: $$\displaystyle P_{in,FL} = P_{core} + P_{f&w} + P_{stator,cu} + P_{rotor,cu} + P_{stray} $$.

• Procedure:

  1. From no-load test, get $$\displaystyle P_{core} + P_{f&w} $$.

  2. From full-load test, subtract $$\displaystyle P_{core} + P_{f&w} $$ and $$\displaystyle P_{stator,cu} $$ (calculate from $$\displaystyle I_{FL}^2 R_1 $$) to get $$\displaystyle P_{rotor,cu} + P_{stray} $$.

  3. Often assume $$\displaystyle P_{stray} $$ is small or estimate from design.

Calculation of Slip, Rotor EMF, Current, PF from Motor Data

Given: $P$ (output), $V$, $I$, $\cos\phi$, $f$, $P$ (poles), $$\displaystyle R_2 $$, $$\displaystyle X_2 $$ (per phase), transformation ratio $$\displaystyle a = E_2/E_1 $$.

1. Synchronous Speed: $$\displaystyle N_s = 120f/P $$.

2. Input Power: $$\displaystyle P_{in} = \sqrt{3} V I \cos\phi $$.

3. Stator Losses: $$\displaystyle P_{stator,cu} = 3 I_1^2 R_1 $$ (if $$\displaystyle R_1 $$ known).

4. Air-gap Power: $$\displaystyle P_{ag} = P_{in} - P_{stator,cu} - P_{core} $$ (if core loss known).

5. Slip: $$\displaystyle s = \frac{P_{ag} - P_{dev}}{P_{ag}} $$, where $$\displaystyle P_{dev} = P_{out} + P_{f&w} $$ (or $$\displaystyle P_{dev} \approx P_{out} $$ if losses small).

6. Rotor EMF per phase: $$\displaystyle E_2 = s E_{20} = s \cdot a \cdot E_1 $$ (or from transformation ratio).

7. Rotor Current per phase: $$\displaystyle I_2 = \frac{E_2}{\sqrt{R_2^2 + (s X_2)^2}} $$.

8. Rotor Power Factor: $$\displaystyle \cos\phi_2 = \frac{R_2}{\sqrt{R_2^2 + (s X_2)^2}} $$.

Starting Current for Slip Ring Motor with Rotor Resistance

• At start ($$\displaystyle s=1 $$), rotor circuit: $$\displaystyle R_2 + R_{ext} $$ in series with $$\displaystyle X_2 $$.

• Rotor Current per phase: $$\displaystyle I_{2,start} = \frac{E_{20}}{\sqrt{(R_2 + R_{ext})^2 + X_2^2}} $$.

• Reflected to stator: $$\displaystyle I_{2,start}' = a I_{2,start} $$.

• Total Starting Current: $$\displaystyle I_{start} = I_1 + I_{2,start}' $$ (vector sum). Often approximated as $$\displaystyle I_{start} \approx \frac{V_1}{\sqrt{(R_1 + a^2(R_2+R_{ext}))^2 + (X_1 + a^2 X_2)^2}} $$.

[!TIP] Past Paper: "How much resistance must be inserted for maximum torque at start?" → Set $$\displaystyle s_m = 1 $$ (since start) → $$\displaystyle R_{2,ext} + R_2 = X_2 $$ (from $$\displaystyle s_m = R_2'/X_{eq} $$). Solve for $$\displaystyle R_{ext} $$. (e.g., Jun 2025 Q10).