Electrical Machine-I (EX-402) Unit 2 Notes

UNIT 2: TRANSFORMERS & THREE-PHASE INDUCTION MOTORS

A. TRANSFORMERS

1. Fundamental Construction & Principle

Construction:
- Core: Made of high-grade silicon steel laminations (to reduce eddy current loss). Types: Core-type (windings surround core) & Shell-type (core surrounds windings).
- Windings: Primary (input) & Secondary (output). Materials: Copper or Aluminium. Insulated from core and each other.
- Tank: Filled with insulating oil (for cooling & insulation).
- Bushings: Provide insulated exit for terminals.
- Conservator & Breather: (See Section 6).
Working Principle: Based on mutual induction. AC supply in primary creates alternating flux in core, linking secondary and inducing EMF.
EMF Equation (Single-phase):

$$E_1 = 4.44 f N_1 \Phi_m \quad \text{and} \quad E_2 = 4.44 f N_2 \Phi_m$$

where $f$ = frequency, $N$ = turns, $$\displaystyle \Phi_m $$ = max flux.

Transformation Ratio: $$\displaystyle k = \frac{E_2}{E_1} = \frac{N_2}{N_1} = \frac{V_2}{V_1} $$ (on no-load).
Ideal vs. Practical:
- Ideal: No losses (core/copper), $$\displaystyle \mu = \infty $$ (no magnetizing current), $$\displaystyle R_1=R_2=X_1=X_2=0 $$.
- Practical: Has core loss, copper loss, leakage flux, winding resistance & leakage reactance.

2. Transformer Tests & Equivalent Circuit

Open Circuit (OC) Test:
- Purpose: Determine core loss ($$\displaystyle P_i $$) and magnetizing parameters ($$\displaystyle R_m $$, $$\displaystyle X_m $$).
- Procedure: LV side energized at rated voltage, HV side open. Measure $$\displaystyle V_1 $$, $$\displaystyle I_0 $$, $$\displaystyle P_0 $$.
- Parameters: $$\displaystyle P_0 = P_i $$ (core loss). $$\displaystyle I_w = \frac{P_0}{V_1} $$ (working component), $$\displaystyle I_\mu = \sqrt{I_0^2 - I_w^2} $$ (magnetizing component).

$$R_m = \frac{V_1}{I_w}, \quad X_m = \frac{V_1}{I_\mu}$$

*   **Phasor Diagram:** $$\displaystyle I_0 $$ lags $$\displaystyle V_1 $$ by $$\displaystyle \phi_0 $$ (large angle, low pf).

> [!TIP] OC test is performed on **LV side** for safety and convenience (low voltage, low current).

Short Circuit (SC) Test:
- Purpose: Determine full-load copper loss ($$\displaystyle P_c $$) and equivalent resistance & reactance ($$\displaystyle R_{eq} $$, $$\displaystyle X_{eq} $$).
- Procedure: HV side (or higher voltage side) energized at reduced voltage until rated current flows. LV side shorted. Measure $$\displaystyle V_{sc} $$, $$\displaystyle I_{sc} $$, $$\displaystyle P_{sc} $$.
- Parameters: $$\displaystyle P_{sc} = P_c $$ (copper loss at rated current). $$\displaystyle Z_{eq} = \frac{V_{sc}}{I_{sc}} $$, $$\displaystyle R_{eq} = \frac{P_{sc}}{I_{sc}^2} $$, $$\displaystyle X_{eq} = \sqrt{Z_{eq}^2 - R_{eq}^2} $$.
- Phasor Diagram: $$\displaystyle I_1 $$ (rated) in phase with $$\displaystyle V_2' $$ (referred voltage). $$\displaystyle V_{sc} $$ is small, pf is low.
[!TIP] SC test is on HV side because rated current is lower there, making the test setup easier.
Sumpner's Test (Back-to-Back Test):
- Purpose: Determine efficiency & regulation under loaded conditions simultaneously, without actual loading.
- Procedure: Two identical transformers connected back-to-back. One (T1) fed from supply, other (T2) supplies load impedance. Measure total input power.
- Calculation: Total input power = Core loss of both + Copper loss of both. Efficiency $$\displaystyle \eta = \frac{\text{Output of T2}}{\text{Input to T1}} $$. Regulation can be found from voltage drop.
Equivalent Circuit:
- Exact Circuit: Referred to either side. Includes $$\displaystyle R_1 $$, $$\displaystyle X_1 $$, $$\displaystyle R_2' $$, $$\displaystyle X_2' $$, $$\displaystyle R_m $$, $$\displaystyle X_m $$.
- Approximate Circuit: $$\displaystyle R_1 $$ & $$\displaystyle X_1 $$ combined with $$\displaystyle R_2' $$ & $$\displaystyle X_2' $$ into $$\displaystyle R_{01} $$, $$\displaystyle X_{01} $$. $$\displaystyle R_m \parallel X_m $$ moved to input side (valid for high efficiency).
DiagramCANVAS: Draw per-phase exact equivalent circuit of a single-phase transformer showing all components: V1, I1, R1, X1, magnetizing branch (Rm || Xm), and ideal transformer with secondary open or loaded, showing R2', X2' on secondary side referred to primary.

3. Performance Characteristics

Losses:
- Core (Iron) Losses: Constant (independent of load). $$\displaystyle P_i = P_h + P_e $$.
  - Hysteresis Loss: $$\displaystyle P_h \propto f B_m^{1.6} $$ (Steinmetz equation).
  - Eddy Current Loss: $$\displaystyle P_e \propto f^2 B_m^2 t^2 $$ (t = lamination thickness).
- Copper Loss: $$\displaystyle P_c \propto I^2 $$. Varies with load fraction $x$ as $$\displaystyle x^2 P_{c,FL} $$.
- Stray Loss: Due to leakage flux causing eddy currents in tank, etc. Part of load-dependent loss.
- Effect of V/f Variation: To keep $$\displaystyle \Phi_m $$ constant (avoid saturation), $V/f$ must be constant. If $f$ increases, $V$ must increase proportionally to keep losses same. If $V$ increases at constant $f$, $$\displaystyle \Phi_m $$ increases, hysteresis loss increases ($$\displaystyle B_m^{1.6} $$), eddy current increases ($$\displaystyle B_m^2 $$).
Efficiency:

$$\eta = \frac{\text{Output}}{\text{Input}} = \frac{x S \cos\phi}{x S \cos\phi + P_i + x^2 P_c}$$

where $x$ = load fraction, $S$ = rated kVA.

*   **Condition for Maximum Efficiency:** $$\displaystyle \frac{d\eta}{dx}=0 $$ gives $$\displaystyle x = \sqrt{\frac{P_i}{P_c}} $$. At max efficiency, **variable copper loss = constant core loss**.

*   **All-day Efficiency:** $$\displaystyle \eta_{ad} = \frac{\text{Energy output (24h)}}{\text{Energy input (24h)}} $$. Used for distribution transformers with varying load.

Voltage Regulation:
- Definition: Change in secondary terminal voltage from no-load to full-load, expressed as % of rated voltage, at constant primary voltage & frequency.

$$\text{Percentage Reg.} = \frac{V_{2,NL} - V_{2,FL}}{V_{2,FL}} \times 100\%$$

*   **Equation (Approx):** $$\displaystyle \text{Reg.} \approx \frac{I_1 R_{01} \cos\phi_2 \pm I_1 X_{01} \sin\phi_2}{V_2} $$ (+ for lagging, - for leading pf).

*   **Condition for Zero Regulation:** $$\displaystyle \phi_2 = -\tan^{-1}\left(\frac{X_{01}}{R_{01}}\right) $$. (Leading pf).

*   **Condition for Maximum Regulation:** $$\displaystyle \phi_2 = \tan^{-1}\left(\frac{X_{01}}{R_{01}}\right) $$. (Lagging pf).

*   **Effect of Power Factor:** Regulation is **maximum at lagging pf**, **zero at a leading pf**, and **intermediate at unity pf**.

4. Parallel Operation of Transformers

Need: Increase capacity, improve reliability (backup).
Conditions for Successful Parallel Operation:
1. Same Voltage Ratio & Turns Ratio: Else circulating currents.
2. Same Phase Sequence & Polarity: Else short circuit.
3. Same Phase Shift: (For 3-phase, same vector group like Dyn11, Yyn0).
4. Impedance Voltages (%Z) should be equal: For proportional load sharing. If unequal, lower %Z transformer gets overloaded.
Load Sharing with Different Impedances:
- Load shared inversely proportional to impedance.

$$\frac{S_1}{S_2} = \frac{Z_2}{Z_1} \quad \text{(in pu)}$$

*   **kVA Loading:** $$\displaystyle S_1 = S_{total} \times \frac{Z_2}{Z_1 + Z_2} $$.

*   **Power Factor:** Same for both transformers (as voltage is common).

Circulating Current at No-load: Caused by unequal voltage ratios or different %Z. $$\displaystyle I_{cir} = \frac{V_1 - V_2}{Z_1 + Z_2} $$ (per phase).
Effect of One Transformer Removed (Δ-Δ Bank): Remaining transformers take more than their share of load (overloaded). Total capacity reduces.

5. Special Transformers & Connections

Auto-transformer:
- Construction: Single continuous winding with a tap. Common winding & series winding.
- Working: Voltage transformation by conduction (common winding) and induction (series winding).
- Comparison with Two-winding:
  
  | Feature | Two-winding | Auto-transformer | | :--- | :--- | :--- | | Copper Saving | Less | More (saving $$\displaystyle \propto (1 - \frac{1}{k}) $$ ) | | Efficiency | Lower | Higher (less copper, less loss) | | Voltage Regulation | Poorer | Better | | Short-circuit Impedance | Higher | Lower (higher fault current) | | Isolation | Yes (galvanic) | No (common winding) |
- Applications: Booster transformers, furnace transformers, motor starters, interconnection transformers.
- Saving of Copper: Weight of auto-transformer $$\displaystyle W_a = W_t (1 - \frac{1}{k}) $$, where $$\displaystyle W_t $$ is weight of equivalent two-winding transformer, $k$ = voltage ratio.
[!TIP] Auto-transformers are not used for large voltage ratios (e.g., 230V/11kV) due to high fault current risk and lack of isolation.
Scott Connection:
- Purpose: 3-phase to 2-phase (or vice versa) conversion.
- Circuit: Two single-phase transformers. Main transformer (center-tapped on primary) connected line-to-line. Teaser transformer connected from midpoint of main primary to third line.
- Working & Neutral Division: Primary currents are balanced 3-phase. Teaser primary voltage = $$\displaystyle \frac{V_{LL}}{\sqrt{3}} $$. To balance teaser secondary, its turns ratio must be $$\displaystyle \frac{N_t}{N_m} = \frac{1}{\sqrt{3}} $$. Neutral point divides teaser primary in 2:1 ratio (proven by phasor diagram).
- Balanced Load: Both secondaries supply equal 2-phase loads. Primary currents are balanced 3-phase.
- Unbalanced Load: Primary currents become unbalanced, but still 3-phase (no zero sequence).
DiagramCANVAS: Draw Scott connection. Show 3-phase supply (R,Y,B). Main transformer: primary across R-Y, center-tap on R-Y. Teaser: primary from center-tap to B. Secondaries: two independent single-phase outputs, 90° apart.
Three-winding Transformer: Primary, Secondary, Tertiary. Used for interconnecting systems at different voltage levels (e.g., power plant auxiliaries).
Pulse & High-Frequency Transformers:
- Construction: Very low leakage inductance & inter-winding capacitance. Ferrite core (high resistivity, low loss at high f).
- Applications: Pulse transformers (digital circuits, gate drives), HF inverters, SMPS, isolation.

6. Transformer Components & Auxiliaries

Conservator & Breather:
- Conservator: Sealed drum above main tank. Allows oil expansion/contraction with temperature. Prevents oil contact with atmosphere.
- Breather: Contains silica gel (blue when dry, pink when wet). Connected between conservator and atmosphere. Absorbs moisture from air entering conservator during oil contraction.
[!TIP] Breather is on conservator, not main tank, to protect main oil from moisture; conservator oil is in contact with atmosphere via breather.
Tap Changer:
- Off-circuit (OLTC): Changes taps when transformer is de-energized. For minor voltage adjustments.
- On-load (OLTC): Changes taps while energized & loaded. Uses diverter switches & resistors to prevent arcing. For major voltage regulation (e.g., in substations).
Cooling Methods (ANSI/IEC Codes):
- ONAN: Oil Natural, Air Natural (radiators).
- ONAF: Oil Natural, Air Forced (fans on radiators).
- OFAF: Oil Forced, Air Forced (pump + fans).
- OFWF: Oil Forced, Water Forced (water-cooled heat exchangers).
Buchholz Relay: Gas-actuated relay in conservator pipe. Operates for internal faults (arcing generates gas). Gives alarm for minor faults, trip for major faults.
Inrush Current: High transient current (5-30x rated) during energization. Caused by remanent flux & saturation. Problems: maloperation of protection, mechanical stress, voltage dip.

7. Classification & Applications

Power Transformer: >200 MVA, high voltage (>33kV), used in transmission substations. Designed for maximum efficiency at full-load.
Distribution Transformer: <200 MVA, low voltage (<33kV), used at distribution end. Designed for maximum efficiency at about 50-70% load (all-day efficiency). Often oil-immersed with conservator.
Core-type vs. Shell-type:

| Feature | Core-type | Shell-type | | :--- | :--- | :--- | | Core Construction | Two vertical limbs, horizontal yokes | One central limb, two outer limbs, double shell | | Winding | Surrounds core limbs | Core surrounds windings | | Leakage Flux | Less (windings on limbs) | More (windings inside shell) | | Repair/Inspection | Easier | Difficult | | Application | Power transformers (large) | Distribution transformers (small), rectifier transformers |

B. THREE-PHASE INDUCTION MOTOR (IM)

1. Construction & Principle of Operation

Stator: Laminated core, 3-phase distributed winding (double-layer, full-pitch or chorded). Produces rotating magnetic field (RMF).
Rotor Types:

| Feature | Squirrel Cage | Slip Ring (Wound) | | :--- | :--- | :--- | | Construction | Al/Cu bars shorted by end rings | 3-phase star-connected winding, slip rings | | Resistance | Very low, fixed | High (external resistance addable) | | Starting Torque | Low | High (with rotor resistance) | | Starting Current | High | Low (with resistance) | | Speed Control | Not possible | Possible (rotor resistance) | | Cost/Maintenance | Cheap, robust | Costly, brushes/slip rings maintenance | | Applications | Fans, pumps, compressors | Cranes, mills, high starting torque loads |
Production of RMF: Balanced 3-phase currents in spatially displaced windings (120° apart) produce a constant magnitude RMF rotating at synchronous speed $$\displaystyle N_s = \frac{120f}{P} $$.
Slip: $$\displaystyle s = \frac{N_s - N_r}{N_s} $$. Rotor speed $$\displaystyle N_r = N_s(1-s) $$. Slip at start $$\displaystyle s=1 $$, at sync $$\displaystyle s=0 $$.

2. Equivalent Circuit & Performance Analysis

Per-phase Equivalent Circuit (Rotor side referred to stator):

DiagramCANVAS: Draw per-phase equivalent circuit. Supply V1, stator R1, X1, magnetizing branch (Rm || Xm), then ideal transformer gap, then referred rotor R2', X2' (in series) with slip-dependent resistor R2'(1-s)/s. Show all components clearly.
- Rotor Resistance Split: $$\displaystyle R_2' $$ (actual rotor copper loss) and $$\displaystyle \frac{R_2'(1-s)}{s} $$ (developed mechanical power).
Power Flow Diagram (in terms of slip $s$):

$$P_1 = 3 V_1 I_1 \cos\phi_1 \quad \text{(Input)}$$

$$P_2 = P_1 - (\text{Stator Cu loss} + \text{Stator core loss}) \quad \text{(Air-gap power / Rotor input)}$$

$$P_2 = \frac{R_2'}{s} \cdot I_2'^2 \quad \text{(Rotor input)}$$

$$P_{cu2} = s P_2 = R_2' I_2'^2 \quad \text{(Rotor copper loss)}$$

$$P_{dev} = (1-s) P_2 = \frac{R_2'(1-s)}{s} I_2'^2 \quad \text{(Mechanical power developed)}$$

$$P_{out} = P_{dev} - (\text{Friction & windage} + \text{Stray load loss})$$

> [!TIP] Remember: **Rotor input = Rotor Cu loss / s**.

Torque-Slip Characteristics:
- Torque Equation:

$$T = \frac{k s R_2'}{R_2'^2 + (s X_2')^2}$$

where $$\displaystyle k = \frac{3}{\omega_s} \frac{V_1^2}{(R_1 + \frac{R_2'}{s})^2 + (X_1 + X_2')^2} $$ approximately.

*   **Simplified (neglecting R1, X1):**

$$T \propto \frac{s R_2'}{R_2'^2 + (s X_2')^2}$$

*   **Slip at Maximum Torque:** $$\displaystyle s_m = \frac{R_2'}{X_2'} $$ (for simplified). **Maximum torque $$\displaystyle T_m \propto \frac{1}{X_2'} $$** (independent of R2').

*   **Effect of Rotor Resistance:** Increases $$\displaystyle s_m $$ (shifts curve right), $$\displaystyle T_m $$ unchanged.

*   **Effect of Leakage Reactance:** Decreases $$\displaystyle T_m $$, shifts $$\displaystyle s_m $$ left.

*   **Shape:** Stable region (0 to $$\displaystyle s_m $$), unstable region ($$\displaystyle s_m $$ to 1). Normal operation in stable region.

Losses & Efficiency:
- Stator: Copper loss ($$\displaystyle 3I_1^2 R_1 $$), Core loss ($$\displaystyle 3 I_m^2 R_m $$ or from OC test).
- Rotor: Copper loss ($$\displaystyle s P_2 $$).
- Mechanical: Friction & windage (almost constant).
- Stray Load Loss: Due to harmonics, space harmonics, saturation (1-2% of output).
- Efficiency: $$\displaystyle \eta = \frac{P_{out}}{P_1} = \frac{P_{dev} - \text{mech losses}}{P_1} $$.

3. Starting Methods

Squirrel Cage IM:
1. Direct-on-Line (DOL): Full voltage applied. High starting current (5-8x), low starting torque. Simple, cheap.
2. Star-Delta (Y-Δ) Starter: Stator winding star-connected at start (reduces voltage/current by √3), then delta. Starting torque = 1/3 of DOL. Used for normal delta-connected motors.
3. Autotransformer Starter: Reduced voltage via autotransformer taps (e.g., 50%, 65%, 80%). Starting torque ∝ V². Flexible.
4. Soft Starters: Solid-state (thyristors) control voltage ramp. Reduces mechanical stress.
5. Variable Frequency Drives (VFD): Most efficient. V/f control for smooth start & speed control.
Slip Ring IM:
- Rotor Resistance Starting: External resistances added to rotor circuit via slip rings. Increases starting torque, reduces starting current. Resistances cut out gradually as motor accelerates.
Improvement via Rotor Design:
- Double Cage Rotor:
  - Construction: Outer cage (high resistance, low inductance - narrow bars), inner cage (low resistance, high inductance - deep bars).
  - Working: At start (high slip, high freq), outer cage dominates (high R → high starting torque). At run (low slip, low freq), inner cage dominates (low R → good efficiency).
  - Characteristics: Higher starting torque, lower starting current, poorer speed regulation than single cage.
- Deep Bar Rotor: Skin effect increases effective resistance at start (high freq). Similar effect to double cage but continuous.

4. Speed Control Methods

From Stator Side:
- Varying Supply Voltage: Torque ∝ V². Limited range, high slip, inefficient.
- Varying Supply Frequency (V/f Control): Maintain constant V/f to avoid saturation. Used in VFDs for wide speed range.
- Pole Changing Method: Change number of poles by reconnecting stator winding (e.g., 2/4 pole). Step change in speed.
- Cascade Operation: Two motors on same shaft. Supply one from mains, other from slip rings. Gives two speeds.
From Rotor Side (Slip Ring IM only):
- Inserting Rotor Resistance: Increases slip for given torque. Simple, inefficient (losses in resistor).
- Cascade Operation (Concatenation): Two IMs coupled. Rotor of main motor supplies stator of auxiliary. Gives two speeds.
- Scherbius Drive (Power Recovery): Rotor slip power fed back to supply via rectifier-inverter. Used for large pumps/fans.

5. Testing & Graphical Analysis

No-load Test: Similar to transformer OC test. Run motor at no-load, rated voltage. Measures friction & windage, core loss, no-load current. $$\displaystyle I_0 $$ is small, low pf.
Blocked Rotor Test: Similar to SC test. Rotor locked, reduced voltage applied to stator to draw rated current. Measures short-circuit impedance ($$\displaystyle R_{01}, X_{01} $$) and blocked-rotor copper loss.
Circle Diagram:
- Construction: From OC & SC test data. Plot locus of input current phasor for varying voltage at constant frequency.
- Steps: 1) Draw axes (real/imag). 2) Plot OC point (no-load current, power factor). 3) Draw line from origin through SC point (scaled current, power factor). 4) Draw perpendicular from OC point to SC line, find center of circle. 5) Draw circle.
- Readings from Diagram:
  - Full-load current & power factor (at rated voltage on horizontal axis).
  - Maximum power (highest point on circle).
  - Starting torque (at zero voltage axis).
  - Efficiency & power factor for any load.
  - Slip (from mechanical power scale).
DiagramCANVAS: Draw typical circle diagram for IM. Label axes: I (real axis), I sinφ (imag axis). Show OC point (I0, cosφ0), SC point (Isc, cosφsc), line from origin to SC point, perpendicular from OC to this line meeting at O'. Circle with center on horizontal line through O'. Mark points for FL, max power, starting current.

6. Phenomena & Abnormal Operations

Crawling: Motor runs at ~1/7 of synchronous speed (due to 7th harmonic RMF). Caused by space harmonics (especially 7th, which has 7 pole pairs). More prominent in squirrel cage.
Cogging (Magnetic Locking): Motor fails to start. Rotor tends to lock in position with stator. Caused by alignment of stator and rotor slot harmonics (e.g., same number of slots). Prevented by skewing rotor bars or using fractional slot winding.
Effect of Supply Variations:
- Voltage Change: Torque ∝ V². Large voltage drop causes high slip, high current, overheating.
- Frequency Change: $$\displaystyle N_s \propto f $$. V/f constant needed for safe operation. Overfluxing if V constant, f↓.
Harmonics: Cause torque pulsations, additional losses (heating), noise, cogging/crawling.
Braking Methods:
- Plugging (Reverse Current): Reverse supply sequence (or phase). High braking torque, high energy loss.
- Dynamic Braking: Disconnect supply, connect stator to resistor. Kinetic energy dissipated in stator & rotor copper.
- Regenerative Braking: Motor speed > synchronous speed (e.g., downhill). Acts as generator, feeds power back to supply. Requires VFD or special arrangement.

7. Single-Phase Induction Motors

Why Not Self-starting? Double Revolving Field Theory: Single-phase supply produces pulsating (not rotating) field. Can be resolved into two equal opposite rotating fields. Starting torque = 0 (fields cancel). Needs auxiliary means to create starting torque.
Classification:
- Split-phase: Resistance start (auxiliary winding high R), Capacitor start (capacitor in auxiliary).
- Capacitor start-capacitor run: Two capacitors (start & run). Best pf & torque.
- Shaded-pole: Shading ring on pole. Low starting torque, cheap. Used in fans, small appliances.
- Reluctance, Hysteresis: Special types.
- AC Series Motor (Universal): Works on AC/DC. Series field & armature.
Equivalent Circuit (ignoring core loss): Similar to 3-phase but with only one winding. Double revolving field theory gives two equivalent circuits in parallel.
Construction & Working (Capacitor-start): Main winding + auxiliary winding (with series capacitor). Capacitor makes auxiliary current lead voltage, creating phase difference → starting torque. Centrifugal switch disconnects auxiliary at ~75% speed.
AC Series Motor (Universal Motor):
- Construction: Series-connected field & armature. Compensating winding (interpole) to reduce reactance voltage drop & sparking.
- Working: Same as DC series motor. High starting torque, high speed (no-load dangerous). Applications: Portable tools, vacuum cleaners, appliances.
- Characteristics: Speed decreases with load (similar to DC series).

8. Special Types & Applications

Linear Induction Motor (LIM):
- Construction: Stator (primary) "unrolled" to flat shape. Rotor (secondary) is flat conductive plate (aluminium/copper) or reaction rail.
- Principle: RMF travels linearly along primary → induces currents in secondary → thrust by Lorentz force.
- Applications: Maglev trains, traction (linear motors), conveyor systems, pumps (no moving parts), door openers.
AC Servo Motor:
- Constructional Features: Low inertia rotor (diameter small, length large), high torque/inertia ratio, often squirrel cage or drag-cup. Stator has two windings (main & control) displaced by 90°.
- Working: Control winding voltage magnitude/phase controls torque & direction. Used in position/speed control systems (robotics, CNC machines, radar).
Stepper Motor (Brief): Digital motor. Rotor moves in discrete steps (e.g., 1.8°/step). Used in open-loop position control (printers, plotters).

C. CROSS-CUTTING & APPLICATION TOPICS

Excitation Phenomenon in Transformer

Magnetization Current ($$\displaystyle I_m $$): Current required to establish flux in core. Lags voltage by ~90° (but has small active component due to core loss).
No-load Current ($$\displaystyle I_0 $$): Small (2-5% rated), highly non-sinusoidal due to core saturation. Contains harmonics (mainly 3rd).
Components: $$\displaystyle I_w $$ (in-phase, core loss component), $$\displaystyle I_\mu $$ (quadrature, magnetizing component).

Impact of Harmonics on Induction Motor

Torque Pulsations: Harmonic RMFs produce torque pulsations at harmonic slip frequencies → vibration, noise.
Additional Losses: Harmonic currents increase stator & rotor copper losses (especially rotor, where frequency is high).
Heating: Increased losses → temperature rise.
Reduced Torque: Fundamental torque may reduce due to distorted flux.
Cogging/Crawling: Harmonic orders (e.g., 5th, 7th) can cause locking or crawling at sub-synchronous speeds.

Negative Sequence Currents

Cause: Unbalanced supply (e.g., single-phase load) or stator winding fault.
Effect: Produces reverse-rotating RMF at synchronous speed. Relative speed with rotor = $$\displaystyle 2N_s $$ → induced EMF & currents at double frequency.
Heating: Double frequency currents cause severe additional heating in rotor (high frequency, skin effect). Can lead to rapid overheating even with small negative sequence current.
Performance: Reduced torque, increased vibration, possible protective relay operation.

Determination of Equivalent Circuit Parameters

For Transformer: From OC test ($$\displaystyle R_m, X_m $$) and SC test ($$\displaystyle R_{eq}, X_{eq} $$). Split $$\displaystyle R_{eq} $$ into $$\displaystyle R_1, R_2' $$ if needed (usually from design or assume equal).
For 3-phase IM: From no-load test (stator core loss + friction, $$\displaystyle R_m, X_m $$ approx) and blocked rotor test ($$\displaystyle R_{01}, X_{01} $$). Split $$\displaystyle R_{01}, X_{01} $$ into stator & rotor using design data or assume ratios.

Starting Performance Improvement

Squirrel Cage: Double cage, deep bar rotor, soft starters, VFD.
Slip Ring: External rotor resistance.
General: Reduce voltage (DOL not recommended for large motors), use star-delta/autotransformer.

> [!IMPORTANT] Exam Focus:

🔴 High Frequency: Derive torque-slip, conditions for zero/max regulation, circle diagram construction, parallel operation load sharing, Scott connection proof, double cage working, single-phase IM starting (double field theory).

🟡 Medium Frequency: Sumpner's test, auto-transformer saving, crawling/cogging, VFD, linear IM, servo motor, impact of harmonics/negative sequence.

🟢 Theory/Definitions: Construction differences (core/shell, SC/SCIM), excitation phenomenon, breather function, braking methods.

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Electrical Machine-I (EX-402) - Unit 2 Short Notes