Electrical Machine-I (EX-402) - Unit 4 Short Notes

UNIT 4: TRANSFORMERS AND INDUCTION MOTORS

I. TRANSFORMERS

A. Construction and EMF Equation

• Core Types:

  • Core Type: Windings surround the core limbs. Used for high voltage, high kVA.

  • Shell Type: Core surrounds the windings. More mechanically robust, better short-circuit strength.

• Core Material: High-grade silicon steel laminations (0.35-0.5 mm thick) to reduce eddy current loss. Grain-oriented for power transformers.

• Windings: Copper or aluminum conductors with paper, varnish, or cotton insulation. Arrangements: concentric (common), helical, disc.

• EMF Equation (Single-Phase):

$$E = 4.44 f N \Phi_{\text{max}}$$

Where:

*   $E$ = RMS induced EMF (Volts)

*   $f$ = Frequency (Hz)

*   $N$ = Number of turns

*   $$\displaystyle \Phi_{\text{max}} $$ = Maximum flux in core (Wb)

*   **4.44** = Form factor ($$\displaystyle \sqrt{2} \times \frac{\pi}{2} $$) for sinusoidal flux.

• Transformation Ratio:

$$K = \frac{E_1}{E_2} \approx \frac{V_1}{V_2} \approx \frac{N_1}{N_2} \approx \frac{I_2}{I_1}$$

• Excitation Phenomenon: Primary current $$\displaystyle I_0 $$ has two components:

  1. Magnetizing Component ($$\displaystyle I_m $$): Creates core flux $\Phi$, lags $$\displaystyle V_1 $$ by 90°.

  2. Core Loss Component ($$\displaystyle I_c $$): Supplies hysteresis & eddy current losses, in phase with $$\displaystyle V_1 $$.

$$I_0 = \sqrt{I_m^2 + I_c^2}$$

At low voltage, core is unsaturated, $$\displaystyle I_m $$ is sinusoidal. As voltage increases, core saturates, $\Phi$ distorts, $$\displaystyle I_m $$ becomes non-sinusoidal with rich **3rd harmonic** content.

[!TIP] Exam Focus: Derivation of EMF equation from $$\displaystyle \frac{d\Phi}{dt} $$ is a common 5-7 mark question. Remember the form factor 4.44.

B. Performance Tests

Separation of Core Losses: Perform OC test at different frequencies while keeping $V/f$ constant (to keep $$\displaystyle \Phi_{\text{max}} $$ constant).

• Total core loss $$\displaystyle P_0 = P_h + P_e $$

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

• Plot $$\displaystyle P_0 $$ vs $f$. Intercept on $$\displaystyle P_0 $$ axis gives $$\displaystyle P_h $$, slope gives $$\displaystyle P_e/f $$.

C. Losses and Efficiency

Efficiency:

$$\eta = \frac{\text{Output Power}}{\text{Input Power}} = \frac{P_{\text{out}}}{P_{\text{out}} + P_{\text{losses}}}$$

Condition for Maximum Efficiency:

$$\frac{d\eta}{dP_{\text{out}}} = 0 \quad \Rightarrow \quad \text{Core Loss} = \text{Copper Loss}$$

Let $x$ = fraction of full load.

$$P_{cu} = x^2 P_{cu(\text{fl})}$$

At max efficiency: $P_0 = x^2 P_{cu(\text{fl})} \quad \Rightarrow \quad x = \sqrt{\frac{P_0}{P_{cu(\text{fl})}}}

$$ > [!TIP] **Common Pitfall**: Maximum efficiency occurs when **variable losses** (copper) equal **constant losses** (core). Do not confuse with maximum power transfer. **All-Day Efficiency** (for Distribution Transformers): $$

\eta_{\text{all-day}} = \frac{\text{Energy Output (24h)}}{\text{Energy Input (24h)}}

$$

Important for transformers with low load factor.

D. Voltage Regulation

Definition: Percentage change in secondary terminal voltage from no-load to full-load at constant primary voltage and same power factor. $$ \% \text{Reg} = \frac{V_{2(\text{NL})} - V_{2(\text{FL})}}{V_{2(\text{FL})}} \times 100\% $$

• Phasor Diagram Derivation (Using Exact Equivalent Circuit):

  $$ V_1 = V_2 + I_2(R_2' \cos\phi_2 \pm I_2 X_2' \sin\phi_2) + I_2 R_c \| jX_m \text{ (often neglected)} $$

  Approximate (ignoring $$\displaystyle I_0 $$ & excitation current):

  $$ \text{Reg} \approx \frac{I_2 R_2' \cos\phi_2 \pm I_2 X_2' \sin\phi_2}{V_2} $$

  Sign: + for lagging PF, - for leading PF.

• Condition for Maximum Regulation:

  $$ \frac{d(\text{Reg})}{d\phi_2} = 0 \quad \Rightarrow \quad \tan\phi_2 = \frac{X_2'}{R_2'} \quad (\text{PF lagging}) $$

  Max Reg occurs at lagging PF where impedance angle $$\displaystyle \phi_z = \tan^{-1}(X_2'/R_2') $$.

• Condition for Zero Regulation:

  $$ \text{Reg} = 0 \quad \Rightarrow \quad \cos\phi_2 = \frac{R_2'}{Z_2'} $$

  Occurs at a leading power factor.

• Condition for Negative Regulation:

  Occurs when leading PF is higher than zero-regulation PF. $$\displaystyle V_{2(\text{NL})} < V_{2(\text{FL})} $$. Possible only if $$\displaystyle X_2' > R_2' $$.

[!TIP] Exam Key: Be able to derive regulation expression from phasor diagram. Zero/negative regulation is a leading PF phenomenon.

E. Auto-transformer

• Construction: Single winding with a tap. Common winding + series winding.

• Working: $$\displaystyle V_1 $$ applied to common winding, $$\displaystyle V_2 $$ taken across common+series winding.

• Comparison with Two-Winding Transformer:

  | Feature | Two-Winding | Auto-transformer | | :--- | :--- | :--- | | Copper Saving | Less | More (Saving = $$\displaystyle \frac{K-1}{K} \times 100\% $$ for step-down, K=V1/V2) | | Size/Weight | Larger | Smaller for same kVA | | Efficiency | Lower | Higher (less copper) | | Short-circuit Current | Lower | Higher (impedance low) | | Isolation | Yes | No | | Applications | General | Starter for IM, voltage regulators, interconnection. |

• Copper Saving Derivation:

  Let $$\displaystyle K = V_1/V_2 > 1 $$ (Step-down).

  Mass of copper in 2-winding $$\displaystyle \propto (V_1 + V_2) $$.

  Mass in auto $$\displaystyle \propto (V_1 - V_2) $$.

  Saving = $$\displaystyle \frac{(V_1+V_2) - (V_1-V_2)}{V_1+V_2} = \frac{2V_2}{V_1+V_2} = \frac{2/K}{1+1/K} = \frac{2}{K+1} $$.

  Fractional saving = $$\displaystyle \frac{K-1}{K+1} $$. % saving = $$\displaystyle \frac{K-1}{K+1} \times 100\% $$.

  For step-up (K<1), saving formula differs.

F. Three-phase Transformer Connections & Parallel Operation

1. Connections & Vector Groups

• Vector Group: Indicates phase displacement between corresponding line voltages (e.g., Dy1 means secondary line voltage lags primary by 30°).

• Parallel Operation Conditions:

  1. Same voltage ratio (turns ratio).

  2. Same impedance voltage (%Z) and impedance ratio (R/X).

  3. Same polarity and phase sequence.

  4. Same vector group (or phase shift must be zero).

• Circulating Currents: Arise if voltage ratios or impedances are unequal.

  • Due to unequal $$\displaystyle V_2 $$: $$\displaystyle I_c \propto \frac{\Delta V}{Z_{eq}} $$. Causes overheating, reduces capacity.

  • Due to unequal %Z: Load sharing inversely proportional to impedance. Transformer with lower %Z takes more load.

• Load Sharing Calculation (For two transformers in parallel, same vector group):

  Let $$\displaystyle S_{A(\text{rated})} $$, $$\displaystyle S_{B(\text{rated})} $$, $$\displaystyle Z_A $$, $$\displaystyle Z_B $$ (in p.u. on their own bases).

  Load shared: $$\displaystyle S_A : S_B = \frac{S_{A(\text{rated})}}{Z_A} : \frac{S_{B(\text{rated})}}{Z_B} $$.

• Operation with One Transformer Out (e.g., Δ-Δ bank):

  • Original: 3 transformers, each carries $$\displaystyle \frac{1}{3} $$ load.

  • One failed: Remaining two in V-V (Open-Δ) connection.

  • Capacity reduces to $$\displaystyle \frac{2}{3} $$ of original bank rating.

  • Each transformer now carries $$\displaystyle \frac{1}{2} $$ of reduced load = $$\displaystyle \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} $$ of original full load. So each operates at 100% of its individual rating if originally loaded to 66.7%.

2. Scott Connection (3-φ to 2-φ)

• Purpose: Convert balanced 3-phase supply to balanced 2-phase (90° apart).

• Circuit:

  • Main Transformer: Center-tapped primary on one line (say, B). Primary: $$\displaystyle N_1 $$ on each half. Secondary: $$\displaystyle N_2 $$ (1:1 ratio).

  • Teaser Transformer: Primary between tap point (T) and third line (C). Turns ratio: $$\displaystyle N_{1t} : N_2 = \frac{\sqrt{3}}{2} : 1 \approx 0.866 : 1 $$. (Or primary divided in 2:1 ratio to get equal secondary voltages).

• Phasor Diagram (Balanced Load):

  • $$\displaystyle V_{BB'} = V_{ph} $$ (main primary).

  • $$\displaystyle V_{TC} = V_{line} $$ (teaser primary).

  • Secondary voltages $$\displaystyle V_{2m} $$ and $$\displaystyle V_{2t} $$ are equal and 90° apart.

• Line Currents (Balanced Loads $$\displaystyle Z_m $$, $$\displaystyle Z_t $$):

  $$ I_A = I_m \angle 30° - I_t \angle 90° $$

  $$ I_B = I_m \angle -90° - I_t \angle 30° $$

  $$ I_C = I_m \angle 150° - I_t \angle -30° $$

  Magnitudes equal, 120° apart.

[!TIP] Scott connection is a frequent 7-mark question. Be able to draw the circuit, explain the 2:1 division on teaser primary, and derive line currents for balanced load.

G. Components and Protection

• Conservator: Cylindrical tank above main tank. Accommodates oil expansion due to temperature. Maintains constant oil pressure.

• Breather: Contains silica gel (blue when dry, pink when wet). Connected to conservator, not main tank because:

  1. Conservator is in direct contact with atmosphere via breather.

  2. Main tank is sealed; oil expansion/contraction happens in conservator.

  3. Prevents moisture entry into main tank oil during breathing.

• Tap Changer:

  • Off-Circuit (OCTC): De-energized operation. For minor voltage adjustments.

  • On-Load Tap Changer (OLTC): Operates under load. Complex, with selector & diverter switches. For large voltage regulation.

• Buchholz Relay: Gas-actuated relay. Protects against internal faults (incipient faults, arcing). Detects gas accumulation (slow fault) or sudden oil surge (severe fault). Gives alarm/trip.

• Cooling Methods (ANSI/IEC codes):

  • ONAN: Oil Natural, Air Natural (radiators).

  • ONAF: Oil Natural, Air Forced (fans).

  • OFAF: Oil Forced, Air Forced (pumps + fans).

  • OFWF: Oil Forced, Water Forced (for very large transformers).

H. Special Transformers

• Pulse Transformer: For digital/pulse circuits. Requirements: fast response (low leakage inductance, low interwinding capacitance), good isolation, linearity.

• High-Frequency Transformer: For SMPS. Uses ferrite cores (low loss at high f). Operates at 20 kHz - 1 MHz. Smaller size due to $$\displaystyle B_{max} $$ constant ($$\displaystyle E \propto f N \Phi_{max} $$).

I. Operational Aspects

• Excitation Phenomenon: (See A above). Saturation causes magnetizing inrush.

• Inrush Currents:

  • Nature: Unidirectional, high magnitude (10-20× $$\displaystyle I_n $$), contains DC offset and harmonics (2nd dominant).

  • Cause: Residual flux + applied voltage at zero crossing → core saturation.

  • Problems: Maloperation of protective relays (differential, overcurrent), mechanical stress, fuse blowing.

• Supply Variations:

  • Voltage ↑: $\Phi \uparrow$, $$\displaystyle B_m \uparrow $$ → Core loss ↑↑ (hysteresis ∝ $$\displaystyle B_m^{1.6} $$, eddy ∝ $$\displaystyle B_m^2 $$), magnetizing current ↑ (saturation).

  • Frequency ↓: $$\displaystyle n_s \downarrow $$, $\Phi \uparrow$ (if V constant) → Same as voltage ↑.

• Impact of Harmonics:

  • Additional Core Loss (due to higher frequency harmonics).

  • Additional Copper Loss (due to skin effect, harmonic currents).

  • Neutral Current in Δ-connected 3-phase transformer (circulating harmonic currents).

  • Torque Pulsations if feeding induction motor.

• Partial Winding Short-Circuit:

  • Localized high current in shorted turns.

  • Rapid heating, possible fire/explosion.

  • Causes voltage unbalance, negative sequence currents if severe.

II. THREE-PHASE INDUCTION MOTORS

A. Construction and Working Principle

• Stator: Laminated core, 3-phase distributed winding (double-layer common). Produces Rotating Magnetic Field (RMF).

• Rotor:

  • Squirrel Cage: Conductive bars (Al/Cu) short-circuited by end rings. Simple, rugged, cheap.

  • Slip Ring (Wound): 3-phase winding, star-connected, connected to external resistors via slip rings & brushes. High starting torque, variable speed.

• RMF Production: 3 balanced sinusoidal currents 120° apart → Resultant flux rotates at Synchronous Speed:

  $$ n_s (\text{rpm}) = \frac{120 f}{P} $$

  Where $P$ = number of poles.

• Slip:

  $$ s = \frac{n_s - n}{n_s} $$

  • $$\displaystyle s=1 $$ at standstill, $$\displaystyle s=0 $$ at synchronous speed.

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

B. Equivalent Circuit

• Exact Equivalent Circuit (Per Phase, Stator Referred):

  
  V1 --[R1]--[jX1]--+--[Rc]--[jXm]--+
  
                     |               |
  
                     +--[R2'/s]--[jX2']--
  
  

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

  • $$\displaystyle R_c $$, $$\displaystyle X_m $$: Magnetizing branch (core loss & magnetizing reactance).

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

  • $$\displaystyle R_2'/s $$: Represents rotor power conversion ($$\displaystyle \frac{1-s}{s}R_2' $$ is mechanical load equivalent).

• Approximate Equivalent Circuit (Thevenin's):

  • Move $$\displaystyle R_c \| jX_m $$ to input. Compute Thevenin voltage $$\displaystyle V_1' $$ and impedance $$\displaystyle Z_{th} = R_{th} + jX_{th} $$.

  • $$\displaystyle Z_{th} = (jX_m) \parallel (R_1 + jX_1) $$.

  • Simplified circuit: $$\displaystyle V_1' $$ in series with $$\displaystyle Z_{th} $$ and $$\displaystyle R_2'/s + jX_2' $$.

• Parameter Determination:

  • No-Load Test (OC on stator): $$\displaystyle I_0 $$, $$\displaystyle P_0 $$, $$\displaystyle V_1 $$. → $$\displaystyle R_c = V_1^2 / P_0 $$, $$\displaystyle X_m = V_1 / \sqrt{I_0^2 - (P_0/V_1)^2} $$.

  • Blocked Rotor Test (SC on stator, rotor locked): $$\displaystyle V_{sc} $$, $$\displaystyle I_{sc} $$, $$\displaystyle P_{sc} $$. → $$\displaystyle Z_{eq} = V_{sc}/I_{sc} $$, $$\displaystyle R_{eq} = P_{sc}/I_{sc}^2 $$, $$\displaystyle X_{eq} = \sqrt{Z_{eq}^2 - R_{eq}^2} $$.

  • $$\displaystyle R_{eq} = R_1 + R_2' $$, $$\displaystyle X_{eq} = X_1 + X_2' $$. Need $$\displaystyle R_1 $$ (DC test) to separate $$\displaystyle R_2' $$.

C. Torque-Slip Characteristics

• Torque Equation (Developed Torque):

  $$ T = \frac{3 I_2'^2 R_2'}{s \omega_s} = \frac{3}{\omega_s} \cdot \frac{(E_2'/s)^2 R_2'}{(R_1 + R_2'/s)^2 + (X_1 + X_2')^2} $$

  Where $$\displaystyle \omega_s = 2\pi n_s/60 $$ (rad/s).

  Simplified (neglecting $$\displaystyle R_1 $$):

  $$ T \propto \frac{s R_2'}{(R_2'^2/s^2) + X^2} \quad \text{or} \quad T \propto \frac{R_2'/s}{X^2 + (R_2'/s)^2} $$

• Maximum Torque (Breakdown Torque):

  Condition: $$\displaystyle \frac{dT}{ds} = 0 \quad \Rightarrow \quad R_2'/s_m = \sqrt{R_1^2 + X^2} $$

  $$ T_{max} = \frac{3}{2\omega_s} \cdot \frac{E_2'^2}{R_1 + \sqrt{R_1^2 + X^2}} \quad \text{or} \quad T_{max} \propto \frac{1}{2X} \text{ (if } R_1 \text{ neglected)} $$

  Key Point: $$\displaystyle T_{max} $$ independent of $$\displaystyle R_2' $$ if $$\displaystyle R_1 $$ is neglected.

• Slip at Max Torque:

  $$ s_m = \frac{R_2'}{\sqrt{R_1^2 + X^2}} \quad (\text{if } R_1 \neq 0) \quad \text{or} \quad s_m = \frac{R_2'}{X} \text{ (if } R_1 \text{ neglected)}$$

• Torque-Speed Curve:

  • Stable Region: $$\displaystyle 0 < s < s_m $$ (dT/dn < 0). Motor operates here.

  • Unstable Region: $$\displaystyle s_m < s < 1 $$ (dT/dn > 0). Generator or braking region.

  • Effect of Rotor Resistance ($$\displaystyle R_2' $$ ↑): $$\displaystyle T_{max} $$ unchanged, $$\displaystyle s_m $$ ↑, starting torque $$\displaystyle T_{st} $$ ↑.

[!TIP] Crucial: $$\displaystyle T_{max} $$ does not depend on $$\displaystyle R_2' $$ (if $$\displaystyle R_1=0 $$). Starting torque increases with $$\displaystyle R_2' $$. Maximum torque slip $$\displaystyle s_m \propto R_2' $$.

D. Starting Methods

Double Cage Principle:

• At start ($$\displaystyle s=1 $$, $$\displaystyle f_r=f $$): Skin effect → outer cage R ↑, $$\displaystyle X_{2o}↑ $$ → high $$\displaystyle T_{st} $$.

• At run ($s≈0$, $$\displaystyle f_r $$ low): Skin effect negligible → outer cage R ↓, inner cage dominates → low R, high efficiency.

E. Speed Control

From Stator Side:

1. Voltage Control: $$\displaystyle T \propto V^2 $$. Narrow speed range, poor efficiency at low speed.

2. Frequency Control (V/f Control): Maintain constant $V/f$ to keep $\Phi$ constant. Wide speed range. Used in VFDs.

3. Pole Changing: Change stator winding connections to alter $P$. Step change in $$\displaystyle n_s $$. For cage motors only (multiple windings or consequent pole).

From Rotor Side (Slip-ring only):

1. Rotor Resistance Control: Insert $$\displaystyle R_{ext} $$ in rotor circuit. $n \downarrow$ as $$\displaystyle R_{ext}↑ $$. Poor efficiency (losses in $$\displaystyle R_{ext} $$).

2. Slip Power Recovery (SPR):

   • Static Kramer Drive (Super-synchronous): Rectify rotor slip frequency AC → DC → feed back to supply via inverter. (Recovers slip power).

   • Scherbius Drive (Sub-synchronous): Cascade connection. Rotor slip power fed to auxiliary motor (W) mechanically coupled to main motor. $$\displaystyle n = n_s \pm n_W $$.

F. Performance Analysis

1. Circle Diagram

• Construction from No-Load (Point A) and Blocked Rotor (Point B) tests.

  • Plot $$\displaystyle I_1 $$ on X-axis, $$\displaystyle P_{in} $$ on Y-axis.

  • Draw line AB.

  • Draw perpendicular from point of maximum power (C) on AB. Locate center O.

  • Draw circle with center O.

• Readings from Diagram (at rated voltage $$\displaystyle V_1 $$):

  • Full-load current & PF: Vertical line at $$\displaystyle I_{fl} $$.

  • Maximum power (stall torque): $$\displaystyle P_{max} $$.

  • Starting torque: $$\displaystyle P_{st} $$ at $$\displaystyle I_{st} $$.

  • Efficiency: Chord from O to load point.

  • Slip at max power: $$\displaystyle s_m = P_{max} / (P_{in} \text{ at C}) $$.

2. Power Flow & Losses

• Air-gap Power (Rotor Input): $$\displaystyle P_{ag} = P_{in} - \text{Stator losses} $$

• Rotor Copper Loss: $$\displaystyle P_{rcu} = s \cdot P_{ag} $$

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

• Output Power: $$\displaystyle P_{out} = P_m - \text{Friction \& Windage losses} $$

• Relationship: $$\displaystyle P_{ag} = P_m + P_{rcu} = \frac{P_m}{1-s} $$

3. Losses

• Stator: Core loss (from OC), Copper loss ($$\displaystyle I_1^2 R_1 $$).

• Rotor: Copper loss ($$\displaystyle I_2'^2 R_2' $$).

• Friction & Windage: From OC test at rated V (since $$\displaystyle P_{rcu}≈0 $$).

• Stray Load Losses: Included in friction or core loss.

4. Phenomena

• Crawling: Tendency to run at ~$$\displaystyle n_s/2 $$ due to 5th & 7th harmonics producing torque dips.

• Cogging (Magnetic Locking): Motor fails to start at low slip. Due to harmonically related slots (stator/rotor slot ratio integer). Prevention: Skewing, proper slot combination (non-integer ratio), fractional pitch.

5. Braking Methods

• Regenerative: $$\displaystyle n > n_s $$ ($$\displaystyle s<0 $$). Power flows from rotor to stator. Used in downhill loads.

• Plugging (Reverse Voltage): Reverse any two stator phases. Produces opposite torque. High energy dissipation in rotor.

• Dynamic Braking: Disconnect AC, connect stator to resistors. Motor acts as generator, energy dissipated in resistors.

6. Effects of Supply Variations

• Voltage Variation: $$\displaystyle T \propto V^2 $$, $I \propto V$. Efficiency may drop at low V due to high $$\displaystyle I^2R $$ loss.

• Frequency Variation: $$\displaystyle n_s \propto f $$. If $V$ constant, $\Phi \propto 1/f$ → $$\displaystyle B_m \propto 1/f $$ → core loss ↓, but $$\displaystyle T \propto (V/f)^2 $$ if $V/f$ not constant.

• Harmonics: Additional losses, heating, torque pulsations, neutral current in Δ.

• Unbalanced Supply & Negative Sequence Currents:

  • Produce reverse rotating field.

  • Additional $$\displaystyle I^2R $$ loss, heating.

  • Reduced net torque, vibration, noise.

G. Tests

• No-Load Test: $$\displaystyle V_1 $$ rated, $$\displaystyle n≈n_s $$. → $$\displaystyle P_0 $$ (core loss + friction), $$\displaystyle I_0 $$, $$\displaystyle R_c $$, $$\displaystyle X_m $$.

• Blocked Rotor Test: $$\displaystyle V_{sc} $$ low, $$\displaystyle n=0 $$. → $$\displaystyle P_{sc} $$ (copper loss), $$\displaystyle Z_{eq} $$, $$\displaystyle R_{eq} $$, $$\displaystyle X_{eq} $$.

• DC Resistance Test: Measure $$\displaystyle R_1 $$, $$\displaystyle R_2 $$ (with rotor locked) using voltmeter-ammeter.

III. SPECIAL SINGLE-PHASE AND SPECIAL MOTORS

A. Single-Phase Induction Motors

• Not Self-Starting: Single-phase flux is pulsating (not rotating). Resolved into two equal and opposite rotating fields. Net torque at standstill = 0.

• Double Revolving Field Theory:

  • $$\displaystyle F(\theta) = F_m \cos\theta = \frac{F_m}{2} (\cos\omega t + \cos(-\omega t)) $$

  • Two fields: Forward (F) rotating at $$\displaystyle +\omega_s $$, Backward (B) at $$\displaystyle -\omega_s $$.

  • At standstill: $$\displaystyle s_f = s_b = 1 $$. Torques $$\displaystyle T_f $$ & $$\displaystyle T_b $$ equal & opposite → net T=0.

  • At speed $n$: $$\displaystyle s_f = (n_s - n)/n_s $$, $$\displaystyle s_b = (n_s + n)/n_s $$. $$\displaystyle T_b < T_f $$ → net forward torque.

• Classification:

  • Split-Phase: Main + Auxiliary winding (high R/L ratio). Good starting torque.

  • Capacitor-Start: Capacitor in series with auxiliary. High starting torque. Capacitor disconnected after start.

  • Capacitor-Run (Permanent Split Capacitor): Capacitor remains. Improved PF & torque.

  • Shaded-Pole: Shading ring on pole. Low starting torque, simple, cheap.

B. AC Series Motor (Universal Motor)

• Construction: Similar to DC series motor. Compensated or uncompensated.

• Working: $$\displaystyle T \propto \phi I_a $$. Both $\phi$ and $$\displaystyle I_a $$ reverse with AC → torque always in same direction.

• Compensating Winding: Embedded in pole faces. Neutralizes cross-magnetizing effect of armature reaction, improves commutation.

• Characteristics: High speed (up to 20,000 rpm), high starting torque, poor PF at light load.

• Applications: Portable tools, vacuum cleaners, mixers.

C. Linear Induction Motor (LIM)

• Construction: Primary (stator) = 3-phase winding. Secondary (reaction plate) = conductive plate (Al) with/without back iron.

  • Short Primary: Primary shorter than secondary (common).

  • Short Secondary: Secondary shorter (used in maglev).

• Principle: Traveling magnetic field from primary induces currents in secondary → Lorentz force → thrust.

• Types: Flat, cylindrical (tubular), disk.

• Applications: Maglev trains, conveyors, actuators, pumps (no moving parts).

D. Servo Motors

• Types: AC servo (2-phase or 3-phase with resolver/encoder), DC servo.

• Constructional Features: Low rotor inertia (long, thin rotor), high torque/inertia ratio, quick response.

• Working: Controlled by error signal (reference - feedback) in closed loop. Fast acceleration/deceleration.

• Applications: Robotics, CNC machines, radar, antenna positioning, automatic control systems.

E. Induction Generator

• Principle: When driven above synchronous speed ($$\displaystyle n > n_s $$, $$\displaystyle s<0 $$), rotor slip negative → power flows from rotor to stator → generation.

• Self-Excited: Requires capacitor bank across terminals for excitation (isolated system). Used in wind turbines.

• Externally Excited: Connected to grid or external source (provides reactive power).

• Applications: Renewable energy (wind, mini-hydro), emergency power, regenerative braking.

F. Other Topics (Frequently Asked Short Notes)

DC Shunt Motor

• Construction: Field winding in shunt with armature. High resistance, many turns.

• Working: $$\displaystyle I_f $$ constant (if V constant) → $\phi$ constant. $$\displaystyle T \propto I_a $$.

• Characteristics: Constant speed ($$\displaystyle n \propto (V - I_a R_a)/\phi $$). Speed drops slightly with load.

• Applications: Lathes, conveyors, fans, pumps (where constant speed needed).

Slip Power Recovery (SPR) Schemes

• Static Kramer Drive (Super-synchronous): Rotor slip frequency AC → rectifier → DC → inverter → feedback to supply. Recovers slip power.

• Scherbius Drive (Sub-synchronous): Rotor slip power → rectifier → DC → inverter → to auxiliary 3-phase motor (mechanically coupled to main motor). Main motor speed controlled by auxiliary motor speed.

Negative Sequence Currents

• Cause: Unbalanced supply (or unbalanced load/impedance).

• Effect: Produces reverse rotating field at synchronous speed.

  • Additional $$\displaystyle I^2R $$ loss → heating.

  • Reduced net torque.

  • Vibration, noise.

  • Can cause double frequency torque pulsations.

• Mitigation: Use phase sequence detectors, negative sequence relays, or static balancers.

[!TIP] Final Exam Strategy:

1. Transformers: Master EMF equation, tests (OC/SC), regulation derivation, auto-transformer saving, Scott connection, parallel operation conditions.

2. Induction Motors: Torque-slip derivation, circle diagram, double cage, starting methods, speed control (especially SPR), power flow (P_ag, P_rcu, P_m relationship).

3. Special Motors: Double revolving field theory (single-phase), compensating winding (AC series), LIM principle.

4. Diagrams: Practice phasor diagrams for transformer regulation (lag/lead PF), Scott connection, induction motor equivalent circuit, torque-slip curve.

5. Numericals: Efficiency at various loads, regulation for different PFs, parallel operation load sharing, torque calculation, slip & speed calculations.