Electrical Drives (EX-801) - Unit 1 Short Notes

UNIT 1: ELECTRICAL DRIVES – FUNDAMENTALS AND CONTROL TECHNIQUES

1. INTRODUCTION AND BASIC CONCEPTS

Definition: An electrical drive is a system that uses electrical energy to control the motion and torque of a motor for a specific load. It provides precise speed, position, and torque control.

Need: Replaces mechanical drives with advantages like:

• Flexible control (wide speed range, quick response)

• Energy efficiency (especially in variable load applications)

• Automatic control and remote operation

• Regenerative braking capability

• Better power-to-weight ratio

Block Diagram:

[Power Supply] → [Power Converter] → [Electric Motor] → [Load]

         ↑              ↑               ↑

    [Controller] ← [Sensors/Feedback]

• Power Supply: AC mains or DC source.

• Power Converter: Converts supply power to required voltage/current/frequency (e.g., rectifier, chopper, inverter).

• Electric Motor: Converts electrical to mechanical energy (DC, IM, SM, special motors).

• Load: The mechanical system being driven (fans, pumps, conveyors, etc.).

• Controller: Generates switching signals for converter based on reference and feedback.

• Sensors: Measure current, speed, position (encoders, tachogenerators, Hall sensors).

Classification:

Advantages over Mechanical Drives:

• Smooth speed control over wide range.

• High efficiency at part-load.

• Automatic fault protection.

• Remote control and automation compatibility.

• Energy recovery during braking.

Four-Quadrant Operation:

• Quadrant I: Forward motoring ($$\displaystyle \omega > 0 $$, $$\displaystyle T > 0 $$)

• Quadrant II: Forward braking ($$\displaystyle \omega > 0 $$, $$\displaystyle T < 0 $$)

• Quadrant III: Reverse motoring ($$\displaystyle \omega < 0 $$, $$\displaystyle T < 0 $$)

• Quadrant IV: Reverse braking ($$\displaystyle \omega < 0 $$, $$\displaystyle T > 0 $$)

• **Two-quadrant drives:**通常用于提升机、机床（需正反转但制动能量小）。

• Four-quadrant drives: 用于起重机、电梯、轧机（需快速制动和反转）。

Typical Load Requirements:

• Constant Torque Loads: Conveyors, extruders ($$\displaystyle T_L \propto \text{constant} $$).

• Constant Power Loads: Machine tools (milling, drilling) ($$\displaystyle T_L \propto 1/\omega $$).

• Fan/Pump Loads: $$\displaystyle T_L \propto \omega^2 $$, $$\displaystyle P \propto \omega^3 $$.

2. DC MOTOR DRIVES

2.1 DC Motor Characteristics

Types:

• Separately Excited: $$\displaystyle V_f $$ independent of $$\displaystyle V_a $$ → easy control, used in drives.

• Shunt: $$\displaystyle V_f = V_a $$ → constant speed characteristic.

• Series: $$\displaystyle I_f = I_a $$ → high starting torque, variable speed.

• Compound: Cumulative (high starting torque, better speed regulation) or differential (constant speed).

Torque-Speed Equation (Separately Excited):

$$ T = K_t \phi I_a \quad \text{and} \quad E_b = K_e \phi \omega_m = V_a - I_a R_a $$

$$ \Rightarrow \omega_m = \frac{V_a}{K_e \phi} - \frac{R_a}{K_e K_t \phi^2} T $$

• Linear torque-speed characteristic for constant $\phi$.

• Effect of Armature Reaction: Demagnetizing effect reduces flux $\phi$ at high torque → speed rises slightly.

Basic Speed Control:

• Armature Voltage Control ($$\displaystyle V_a $$): Speed $$\displaystyle \propto V_a $$. Used for speeds below base speed ($$\displaystyle \phi = \text{rated} $$). Requires variable voltage supply (converter/chopper).

• Field Control ($$\displaystyle I_f $$): Speed $\propto 1/\phi$. Used for speeds above base speed ($$\displaystyle V_a = \text{rated} $$). Reduces flux → reduced torque capability (constant power region).

[!TIP]

Armature control gives constant torque capability; field control gives constant power capability. Never use field weakening at low speeds (risk of instability).

2.2 Converter-Fed DC Drives

Single-Phase Semi-Controlled Converter (Half-Controlled)

• Circuit: Two SCRs (T1, T2) and two diodes (D1, D2) in bridge configuration.

• Operation: SCRs fired at angle $\alpha$ ($0 \le \alpha \le \pi$). Freewheeling diode (FD) across motor armature allows current continuity.

• Waveforms: Output voltage $$\displaystyle V_{dc} $$ has positive half-cycles only; current continuous due to FD.

• Average Output Voltage:

$$ V_{dc} = \frac{V_m}{\pi} (1 + \cos \alpha) \quad \text{(with FD, continuous conduction)} $$

• Speed: $$\displaystyle \omega_m \propto V_{dc} $$.

• Discontinuous Conduction: Occurs at high $\alpha$ or low load. $$\displaystyle V_{dc} $$ formula changes; need to calculate extinction angle $\beta$.

Single-Phase Fully Controlled Converter

• Circuit: Four SCRs (T1–T4).

• Operation: SCRs fired in pairs (T1,T2 at $\alpha$, T3,T4 at $\alpha+\pi$). Bidirectional power flow possible (regeneration with $$\displaystyle \alpha > 90^\circ $$).

• Average Output Voltage:

$$ V_{dc} = \frac{2V_m}{\pi} \cos \alpha \quad \text{(continuous conduction)} $$

• Power Factor: Degrades as $\alpha$ increases (displacement and distortion).

• Harmonics: Rich in odd harmonics (5th, 7th, ...).

[!TIP]

Semi-controlled converters cannot regenerate (unidirectional current). Fully controlled converters can regenerate by firing $$\displaystyle \alpha > 90^\circ $$, making $$\displaystyle V_{dc} $$ negative.

Three-Phase Converters (Overview):

• Half-wave: Rarely used.

• Full-wave (bridge): Six SCRs. Better performance (less ripple, higher $$\displaystyle V_{dc} $$).

• Average Voltage: $$\displaystyle V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos \alpha = 1.654 V_{LL} \cos \alpha $$.

2.3 Chopper-Controlled DC Drives

Principle: Switch DC supply ON/OFF rapidly. Time Ratio Control (TRC): Vary duty cycle $$\displaystyle D = T_{ON}/(T_{ON}+T_{OFF}) $$.

Types:

• Step-down (Buck): $$\displaystyle V_o = D V_{in} $$ (motoring).

• Step-up (Boost): $$\displaystyle V_o = V_{in}/(1-D) $$ (used in regeneration).

• Reversible (H-bridge): Four switches; can reverse voltage polarity.

Motoring Control (Buck Chopper):

• Circuit: Switch (SCR/MOSFET) in series with motor, diode across motor for freewheeling.

• Operation: Switch ON → motor gets $$\displaystyle V_{in} $$; Switch OFF → motor current freewheels through diode.

• Average Voltage: $$\displaystyle V_a = D V_{in} $$.

• Speed Control: $$\displaystyle \omega_m \propto D $$.

Regenerative Braking Control (Boost Chopper):

• Circuit: Switch (SCR) and diode in series with motor, inductor in parallel.

• Operation: Motor acts as generator. Switch ON → motor current builds in inductor; Switch OFF → inductor discharges through diode into supply → energy fed back.

• Average Voltage: $$\displaystyle V_a = V_{in}/(1-D) $$ → $$\displaystyle V_a > V_{in} $$, enabling regeneration.

• Braking Torque: $$\displaystyle T_b = K_t \phi I_a $$ (opposite to rotation).

Advantages over Converter Drives:

• Higher efficiency (less switching losses in DC-DC).

• Faster dynamic response.

• Simpler control for regeneration.

• Suitable for battery-powered vehicles (DC source).

2.4 Braking Methods for DC Motors

1. Plugging (Reverse Current Braking)

• Circuit: Reverse armature polarity while keeping field same. Add external resistance $$\displaystyle R_{ext} $$.

• Operation: $$\displaystyle V_a $$ reversed → $$\displaystyle E_b $$ and $$\displaystyle V_a $$ additive → high reverse current → high braking torque.

• Braking Torque: $$\displaystyle T_b = K_t \phi I_{a,b} $$ (negative).

• Current at start of plugging: $$\displaystyle I_{a,b0} = \frac{V_a + E_{b0}}{R_a + R_{ext}} $$ (very high!).

• Resistance for current limit: $$\displaystyle R_{ext} = \frac{V_a + E_{b0}}{I_{a,\max}} - R_a $$.

• Energy: All kinetic energy dissipated in resistors → inefficient.

2. Dynamic Braking

• Circuit: Disconnect $$\displaystyle V_a $$, connect braking resistor $$\displaystyle R_b $$ across armature. Field remains excited.

• Operation: Motor acts as generator, energy dissipated in $$\displaystyle R_b $$.

• Torque-Speed Curve: $$\displaystyle T = -\frac{K_t^2 \phi^2}{R_a+R_b} \omega_m $$ (straight line through origin).

• Torque zero at $$\displaystyle \omega_m=0 $$ because $$\displaystyle E_b=0 $$ → $$\displaystyle I_a=0 $$ → $$\displaystyle T=0 $$. Motor stops smoothly.

• Braking time: Depends on inertia and $$\displaystyle R_b $$.

3. Regenerative Braking

• Condition: $$\displaystyle E_b > V_a $$ (speed high enough for generated voltage to exceed supply).

• Operation: Power flows from motor to supply. Requires converter that allows reverse power flow (fully controlled converter or chopper with boost mode).

• Energy: Fed back to supply → efficient.

• Torque: Negative (braking).

[!TIP]

Comparison:

• Plugging: High torque, high energy loss, used for quick stop (e.g., cranes).

• Dynamic: Medium torque, energy wasted, simple, used for controlled deceleration.

• Regenerative: Energy recovery, requires suitable converter, used in traction/elevators.

2.5 Stability and Dynamic Analysis

Steady-State Stability

• Criterion: Drive stable if $$\displaystyle \frac{d(T_m - T_L)}{d\omega_m} < 0 $$ at equilibrium point.

• Interpretation: For small disturbance in speed, net torque should restore equilibrium.

• Dependence on Characteristics: Stability determined by relative slopes of $$\displaystyle T_m(\omega) $$ and $$\displaystyle T_L(\omega) $$, not individually.

  • If $$\displaystyle T_m $$ curve steeper than $$\displaystyle T_L $$ → stable.

  • If $$\displaystyle T_L $$ steeper → unstable.

  • If equal slopes → marginally stable.

Load Equalization:

• Purpose: Limit peak motor torque during cyclic loads (e.g., punch press) by storing/releasing energy in flywheel.

• Flywheel Design:

  • Motor torque $$\displaystyle T_m $$ limited to $$\displaystyle T_{m,\max} $$.

  • Load torque $$\displaystyle T_L $$ varies cyclally: $$\displaystyle T_{L1} $$ (high) for $$\displaystyle t_1 $$, $$\displaystyle T_{L2} $$ (low) for $$\displaystyle t_2 $$.

  • Speed variation $\Delta \omega$ determined by energy balance:

$$ \text{Energy surplus during light load} = \text{Energy deficit during heavy load} $$

$$ (T_{m,\avg} - T_{L2}) t_2 = (T_{L1} - T_{m,\avg}) t_1 $$

• Flywheel inertia $$\displaystyle J_f $$:

$$ J_f = \frac{(T_{L1} - T_{m,\avg}) t_1}{\Delta \omega} - J_m $$

where $\Delta \omega$ = allowable speed variation (from motor $$\displaystyle T_m-\omega $$ curve).

• Assumptions: Motor $$\displaystyle T_m-\omega $$ linear, losses negligible.

Equilibrium Points & Stability:

Given $$\displaystyle T_m = f(\omega_m) $$ and $$\displaystyle T_L = g(\omega_m) $$:

1. Solve $$\displaystyle f(\omega) = g(\omega) $$ → equilibrium speeds $$\displaystyle \omega_{eq} $$.

2. Compute $$\displaystyle \frac{d}{d\omega}(f-g) $$ at each $$\displaystyle \omega_{eq} $$.

3. If negative → stable; positive → unstable.

Referred Quantities:

• Refer load inertia to motor side: $$\displaystyle J_{eq} = J_m + J_L \left( \frac{N_m}{N_L} \right)^2 $$

• Refer load torque: $$\displaystyle T_{L,eq} = T_L \left( \frac{N_m}{N_L} \right) $$

2.6 Starting of DC Motors

Need for Starters:

• High inrush current: At start, $$\displaystyle E_b=0 $$ → $$\displaystyle I_a = V_a/R_a $$ (5–10× rated).

• Low starting torque: $$\displaystyle T \propto I_a^2 $$? Actually $$\displaystyle T \propto I_a \phi $$, but high $$\displaystyle I_a $$ may saturate core → $\phi$ constant → $$\displaystyle T \propto I_a $$. Still, high current damages commutator, stresses supply.

Types of Starters:

• Manual: Series resistors manually stepped.

• Automatic (Face-plate): Pre-set resistors, contractor sequence.

• Electronic: Thyristor-based soft starter (ramp up $$\displaystyle V_a $$).

Starting Torque & Current:

• With starting resistor $$\displaystyle R_{st} $$:

$$ I_{a,\max} = \frac{V_a}{R_a + R_{st}} \quad ; \quad T_{st} = K_t \phi I_{a,\max} $$

• Choose $$\displaystyle R_{st} $$ to limit $$\displaystyle I_{a,\max} $$ to 1.5–2× rated.

3. INDUCTION MOTOR DRIVES

3.1 Induction Motor Fundamentals

Construction:

• Squirrel Cage: Rotor bars short-circuited by end rings → rugged, low cost.

• Wound Rotor: Three-phase windings connected to slip-rings → allows external rotor resistance insertion.

Equivalent Circuit (Per Phase, Stator Referred):

      Rs      jXs     jXm

V1 ──►──[ ]───[ ]───[ ]───

            │     │

            │     └───► (Rr'/s) ──[ ]───

            │               │

            └───────────────┘

• $$\displaystyle R_s $$: Stator resistance

• $$\displaystyle X_s $$: Stator leakage reactance

• $$\displaystyle R_r' $$: Rotor resistance referred to stator

• $$\displaystyle X_r' $$: Rotor leakage reactance referred to stator

• $$\displaystyle X_m $$: Magnetizing reactance

Torque-Speed Characteristic:

$$ T = \frac{3}{\omega_s} \frac{V_1^2 (R_r'/s)}{(R_s + R_r'/s)^2 + (X_s + X_r')^2} $$

• Maximum Torque (Breakdown Torque):

$$ T_{max} = \frac{3}{\omega_s} \frac{V_1^2}{2(R_s + \sqrt{R_s^2 + (X_s+X_r')^2})} \approx \frac{3 V_1^2}{2\omega_s X_{eq}} \quad \text{if } R_s \text{ small} $$

• $$\displaystyle T_{max} \propto V_1^2 $$, independent of $$\displaystyle R_r' $$.

• Slip at $$\displaystyle T_{max} $$: $$\displaystyle s_{max} = \frac{R_r'}{X_{eq}} $$ (increases with $$\displaystyle R_r' $$).

• Effect of Rotor Resistance: Shifts $T-s$ curve rightwards → higher starting torque, reduced efficiency.

Influence of Voltage & Frequency:

• Voltage: $$\displaystyle T \propto V^2 $$ (both $$\displaystyle T_{max} $$ and starting torque).

• Frequency: $$\displaystyle \omega_s = 120f/P $$. Changing $f$ shifts torque curve horizontally. With constant $V/f$, flux constant → torque capability maintained.

3.2 Variable Frequency Control (V/f Control)

Principle: Maintain constant air-gap flux $$\displaystyle \phi_m $$ to avoid saturation (core loss) and maintain torque capability.

• Flux $$\displaystyle \phi_m \propto V_1/f $$.

• So, keep $$\displaystyle V_1/f = \text{constant} $$ below base speed.

Below Base Speed (Constant V/f):

• $$\displaystyle V_1/f = \text{const} $$ → $$\displaystyle \phi_m = \text{const} $$.

• Torque equation: $$\displaystyle T \propto \frac{V_1^2}{\omega_s} \cdot \frac{s/R_r'}{(R_s/R_r' + s)^2 + (X_{eq}/R_r')^2} $$

• Since $$\displaystyle V_1 \propto f $$, $$\displaystyle \omega_s \propto f $$ → $$\displaystyle T \propto f \cdot \frac{s/R_r'}{...} $$.

• Constant torque region: Can maintain rated current → rated torque up to base speed.

Above Base Speed (Constant Voltage):

• $$\displaystyle V_1 = \text{rated} $$ (cannot increase beyond insulation limit).

• $f$ increases → $$\displaystyle V_1/f $$ decreases → $$\displaystyle \phi_m $$ decreases (field weakening).

• Torque capability decreases: $T \propto 1/f$ (approx) → Constant power region.

• Speed range: Typically 2:1 to 4:1 above base speed.

Derivation: Max Torque vs Frequency

From torque equation with $$\displaystyle V_1/f = \text{const} $$:

$$ T_{max} = \frac{3}{\omega_s} \frac{V_1^2}{2X_{eq}} \approx \frac{3}{2\omega_s} \frac{(k f)^2}{X_{eq}} = \text{constant} \quad (\text{since } V_1 \propto f, \omega_s \propto f) $$

Thus $$\displaystyle T_{max} $$ constant below base speed.

[!TIP]

For fan/pump loads ($$\displaystyle T_L \propto \omega^2 $$), V/f control gives energy savings at reduced speed because $$\displaystyle P = T\omega \propto \omega^3 $$.

3.3 Stator Voltage Control

Application: Fan/pump loads ($$\displaystyle T_L \propto \omega^2 $$).

• Method: Reduce $$\displaystyle V_1 $$ using autotransformer or thyristor AC voltage controller.

• Effect on Torque-Speed: $$\displaystyle T \propto V_1^2 $$, so curve shifts downwards.

• Speed Reduction: For given $$\displaystyle T_L $$, speed reduces with voltage.

• Disadvantage: Slip increases → rotor copper loss increases → efficiency low at low speeds.

Numerical Example Approach:

Given motor parameters and load $$\displaystyle T_L = k \omega^2 $$:

1. At rated: $$\displaystyle \omega_{n} $$, $$\displaystyle V_{rated} $$ → find $$\displaystyle k = T_n / \omega_n^2 $$.

2. At reduced speed $\omega$: $$\displaystyle T_L = k \omega^2 $$.

3. From $$\displaystyle T = \frac{3}{\omega_s} \frac{V^2 (R_r'/s)}{(R_s+R_r'/s)^2 + X_{eq}^2} $$, solve for $s$ and $V$ simultaneously (usually iterative).

3.4 Rotor Resistance Control (Wound Rotor IM)

Principle: Insert external resistance $$\displaystyle R_{ext} $$ in rotor circuit via slip-rings.

• Effect: Increases slip for same torque → speed reduces.

• Torque-Speed: $$\displaystyle T_{max} $$ unchanged, but $$\displaystyle s_{max} $$ increases.

• Disadvantage: $$\displaystyle I_r^2 R_{ext} $$ loss in external resistor → efficiency drops.

Closed-Loop Speed Control:

• Block Diagram:

  
  Speed Ref ──►[Controller]──►[Rotor Resistance Circuit]──►Motor
  
                ↑                               ↓
  
                └──────────[Speed Sensor]───────┘
  
  

• Operation: PI controller adjusts $$\displaystyle R_{ext} $$ (via converter/contactors) to maintain speed.

Numerical:

Given initial rotor copper loss $$\displaystyle P_{cu2} $$ at slip $$\displaystyle s_1 $$:

$$ P_{cu2} = s \cdot P_{ag} \quad \text{(air-gap power)} $$

With $$\displaystyle R_{r,new} = R_r + R_{ext} $$:

• New slip for same torque: $$\displaystyle s_2 = s_1 \cdot \frac{R_{r,new}}{R_r} $$.

• New rotor copper loss: $$\displaystyle P_{cu2,new} = s_2 \cdot P_{ag} = s_1 \cdot \frac{R_{r,new}}{R_r} \cdot P_{ag} = P_{cu2} \cdot \frac{R_{r,new}}{R_r} $$.

3.5 Slip Power Recovery Drives

Static Kramer Drive

• Power Circuit:

  
  Rotor Slip Rings → Diode Bridge (uncontrolled) → DC Link (inductor L) → Chopper → Fixed DC Source (or Grid via inverter in some versions)
  
  

• Operation: Slip power rectified to DC, then fed back to supply via chopper (or to DC source). Rotor resistance effectively increased → speed control below synchronous.

• Speed Range: $$\displaystyle 0 < s < s_{max} $$ (typically up to 30% below sync speed).

• Torque-Speed: Similar to rotor resistance control but no external resistor loss → efficient.

• Closed-Loop: Speed sensor → controller → chopper duty cycle.

Static Scherbius Drive

• Power Circuit:

  
  Rotor → Thyristor Bridge (controlled rectifier) → DC Link → Inverter (thyristor) → Fixed Frequency AC Supply
  
  

• Operation: Slip power converted to fixed frequency AC and fed back. Allows super-synchronous operation ($$\displaystyle s < 0 $$) as well.

• Range: Both sub- and super-synchronous speeds.

• Comparison with Kramer: Scherbius allows regeneration and wider speed range (including above sync), but more complex (needs inverter).

Conventional Scherbius System (Historical): Used wound rotor with separate converter set connected to grid → bulky, expensive. Scherbius drive is static version.

3.6 Current Source Inverter (CSI) Fed Drives

Power Circuit:

• DC Link: Large inductor $$\displaystyle L_d $$ to maintain constant current $$\displaystyle I_d $$.

• Inverter: Thyristors (or GCTs/IGCTs) with forced commutation (or load commutation if motor has high $$\displaystyle X_d $$).

• Motor: IM with high leakage reactance (to limit di/dt).

Operation:

• $$\displaystyle I_d $$ constant → torque $$\displaystyle T \propto I_d \cdot \text{power factor} $$.

• Frequency varied by inverter switching.

• Advantage: Inherently current-limited → short-circuit proof.

• Disadvantage: Requires motor with high leakage reactance, poor power factor at high speeds, complex commutation.

Speed Control: Vary frequency while maintaining $$\displaystyle I_d $$ constant (adjust $$\displaystyle V_d $$ via converter).

3.7 AC Dynamic Braking

Two-Lead Connection:

• Circuit: Disconnect two supply leads from motor, connect them to a resistor $$\displaystyle R_b $$ (across two phases). Supply disconnected.

• Operation: Motor acts as induction generator. Rotor cuts stator residual flux → generates AC voltage across $$\displaystyle R_b $$ → energy dissipated.

• Torque-Speed: Braking torque $$\displaystyle T_b \propto \omega_m R_b/(R_b^2 + (sX_{eq})^2) $$? Actually, with two leads, equivalent circuit changes.

• Characteristic: Braking torque zero at $$\displaystyle \omega_m=0 $$ and at $$\displaystyle \omega_m = \omega_s $$? Actually, maximum braking torque occurs at $$\displaystyle s \approx R_b/X_{eq} $$ (positive slip for generator action?).

• Note: Requires residual magnetism in rotor (cage motor has it) or initial DC injection.

3.8 Starting of Induction Motors

Direct-On-Line (DOL):

• Current: $$\displaystyle I_{start} = 5–7 \times I_{rated} $$.

• Torque: $$\displaystyle T_{start} = 1.5–2.5 \times T_{rated} $$.

• Used for: Small motors (< 5 kW) where supply capacity is high.

Reduced Voltage Starting:

1. Star-Delta:

   • Start in star: $$\displaystyle V_{phase} = V_{line}/\sqrt{3} $$ → $$\displaystyle I_{start} $$ and $$\displaystyle T_{start} $$ reduce to 1/3.

   • After time, switch to delta.

2. Autotransformer:

   • Tap at 50%, 65%, 80% → $$\displaystyle I_{start} \propto V^2 $$, $$\displaystyle T_{start} \propto V^2 $$.

3. Stator Resistor/Air-Cooled Reactor:

   • Series impedance limits current.

   • Calculations: For voltage $$\displaystyle V_{applied} $$:

$$ I_{start} = \frac{V_{applied}}{Z_{eq}} \quad ; \quad T_{start} \propto V_{applied}^2 $$

4. SYNCHRONOUS MOTOR DRIVES

4.1 Synchronous Motor Fundamentals

Construction:

• Salient Pole: Large diameter, few poles → hydro generators.

• Cylindrical (Round Rotor): Small diameter, many poles → turbo generators.

Principle: DC field on rotor locks with rotating stator field → synchronous speed $$\displaystyle \omega_s = 120f/P $$.

• Torque Production: $$\displaystyle T = \frac{3}{\omega_s} \frac{V E_f}{X_s} \sin \delta $$ (for negligible $$\displaystyle R_a $$), where $\delta$ = load angle.

• V-Curves: Plot of armature current $$\displaystyle I_a $$ vs field current $$\displaystyle I_f $$ at constant load. Shows power factor variation (over-excited → leading, under-excited → lagging).

4.2 VSI-Fed Synchronous Drives

Block Diagram:

Speed Ref ──►[Controller]──►[PWM Inverter (VSI)]──►Synchronous Motor

                ↑                              ↓

                └──────────[Speed/Position Sensor]┘

• Operation: Inverter provides variable voltage/frequency. $V/f$ controlled to maintain flux.

• Speed-Torque: Similar to IM V/f control but with synchronous operation (no slip).

• Regenerative Braking: Motor operates as generator → power flows back through inverter to DC bus → requires regenerative converter (or braking resistor).

4.3 Self-Controlled Synchronous Drives

Load-Commutated Inverter (LCI)

• Construction: Thyristor bridge (6-pulse or 12-pulse) directly connected to synchronous motor.

• Commutation: Uses motor back EMF (synchronous motor acts as commutating source). Requires leading power factor → motor over-excited.

• Switching Scheme: 120° conduction (like thyristor CSI).

• Advantage: No forced commutation needed → simple for high power.

• Disadvantage: Requires leading power factor, poor power factor at light loads, low speed operation difficult (back EMF small → commutation fails).

Comparison LCI vs VSI:

4.4 Variable Frequency Control

Below Base Speed (Constant V/f):

• $$\displaystyle V/f = \text{const} $$ → constant flux $\phi$.

• Torque $$\displaystyle T \propto I_a $$ (since $$\displaystyle T \propto \phi I_a \sin \delta $$, $\phi$ const).

• Can operate at rated current → constant torque region.

Above Base Speed (Constant Voltage):

• $$\displaystyle V = \text{rated} $$ → $\phi \propto 1/f$ (field weakening).

• Torque $$\displaystyle T \propto \phi I_a \propto (1/f) I_a $$.

• To limit $$\displaystyle I_a $$ to rated, $T \propto 1/f$ → constant power region.

Range of Constant Power Operation:

• Limited by:

  1. Current limit: $$\displaystyle I_a \le I_{rated} $$.

  2. Voltage limit: $$\displaystyle V \le V_{rated} $$.

  3. Stability: $\delta$ must be less than pull-out angle (typically 70°–80°).

• Typically 2:1 to 3:1 above base speed.

4.5 Synchronous Reluctance Motor (SynRM)

Construction: Salient pole rotor without field winding (or with DC excitation for d-axis). $$\displaystyle X_d \ne X_q $$ (difference due to saliency).

• Torque Production: $$\displaystyle T = \frac{3}{\omega_s} \frac{V^2}{2} \left( \frac{1}{X_q} - \frac{1}{X_d} \right) \sin 2\delta $$

  • Reluctance torque due to $$\displaystyle X_d > X_q $$ (d-axis aligned with field, higher reluctance).

• Control: Adjust load angle $\delta$ via $$\displaystyle I_d $$ and $$\displaystyle I_q $$ (vector control).

• Power Factor: Can be improved by proper $\delta$.

Numerical Problem (Given $$\displaystyle X_d, X_q $$, Load Torque):

1. Torque equation: $$\displaystyle T = \frac{3}{\omega_s} \frac{V^2}{2} \left( \frac{1}{X_q} - \frac{1}{X_d} \right) \sin 2\delta $$.

2. Solve for $\delta$.

3. Current phasor: $$\displaystyle I = \frac{V}{jX_d \cos \delta + jX_q \sin \delta} $$? Actually, from phasor diagram:

$$ I_d = \frac{V}{X_d} \sin \delta \quad ; \quad I_q = \frac{V}{X_q} \cos \delta - \frac{E_f}{X_d} \text{ (if excited)} $$

For unexcited SynRM, $$\displaystyle E_f=0 $$: $$\displaystyle I_q = \frac{V}{X_q} \cos \delta $$.

4. Line current $$\displaystyle I_L = \sqrt{I_d^2 + I_q^2} $$.

5. Power factor: $$\displaystyle \cos \phi = I_q / I_L $$.

4.6 Starting of Synchronous Motors

Methods:

1. Using Damper Windings (Amortisseur):

   • Damper bars act like squirrel cage → motor starts as IM.

   • At near sync speed ($\approx 95\%$), apply DC field → pull-in into synchronism.

   • Pull-in Torque: Torque at which motor locks into synchronism.

2. Using Pony Motor: Small auxiliary motor brings rotor near sync speed.

3. Variable Frequency Starting: Start with low $f$ → smooth acceleration to sync speed.

4. Direct On-Line (with damper): DOL starting, damper provides starting torque.

Why field applied near sync speed?

• At low speed, back EMF $$\displaystyle E_f $$ small → stator current $$\displaystyle I_a $$ large when field applied (high $$\displaystyle V - E_f $$).

• Near sync speed, $$\displaystyle E_f \approx V $$ → small inrush current, smooth pull-in.

5. SPECIAL MOTORS AND DRIVES

5.1 Switched Reluctance Motor (SRM)

Construction:

• Stator: Salient poles with concentrated windings.

• Rotor: Salient poles, no windings or magnets (laminated iron).

• Number of poles: Typically 6/4, 8/6, etc. (stator/rotor poles).

Principle of Operation:

• Torque Production: Due to variation of inductance with rotor position.

  • Inductance $L(\theta)$ maximum when poles aligned (d-axis), minimum when unaligned (q-axis).

  • Energize phase when rotor moves from unaligned to aligned → rising inductance → positive torque.

  • Sequence: Energize phases in sequence to produce continuous rotation.

• Rotor Position Sensing: Hall sensors, resolvers, or sensorless control (using inductance measurement).

Power Converter: Typically asymmetric bridge (4 switches per phase) for independent phase control.

Advantages over IM/SM:

• Simple, rugged, low cost (no magnets, no windings on rotor).

• High torque/inertia ratio → good for high acceleration.

• Wide speed range.

• Fault tolerant (phase failure still operates).

• High efficiency at high speeds.

Torque-Speed Characteristics:

• Low speed: Constant torque region (current limited by voltage).

• High speed: Constant power (flux weakening by advancing turn-on angle).

5.2 Permanent Magnet Synchronous Motor (PMSM)

Construction:

• Rotor: Permanent magnets (NdFeB, SmCo) surface-mounted or interior (IPM).

• Stator: Three-phase windings (similar to SM).

Working Principle:

• Back EMF $$\displaystyle E \propto \omega \phi_{pm} $$.

• Torque $$\displaystyle T = \frac{3}{\omega_s} \frac{V E}{\omega_s L_d} \sin \delta + \frac{3}{\omega_s} \frac{3}{2} (L_d - L_q) I_d I_q $$ (for IPM, reluctance torque).

• Comparison with BLDC: PMSM sinusoidal back EMF, sinusoidal current; BLDC trapezoidal back EMF, rectangular current.

5.3 Brushless DC Motor (BLDC)

Construction:

• Stator: Three-phase windings.

• Rotor: Permanent magnets.

• Electronic Commutator: Position sensors (Hall) + power converter (3-phase bridge).

Operation (Six-Step Commutation):

• 60° conduction per phase.

• Hall sensors determine rotor position → switch phases in sequence.

• Torque ripple higher than PMSM.

Switching Scheme:

• 6-step (120°) or sinusoidal (FOC).

• Drive Circuit: DC bus → 3-phase inverter (MOSFET/IGBT) → motor.

5.4 Stepper Motor

Variable Reluctance Stepper Motor

• Construction: Stator with multiple phases (e.g., 2-phase), rotor with teeth (no windings).

• Operation: Energize phases sequentially → rotor aligns to minimum reluctance position.

• Step Angle: $$\displaystyle \theta_s = \frac{360^\circ}{N_r \cdot N_s} $$? Actually, for VR stepper:

$$ \theta_s = \frac{(N_r - N_s) \times 360^\circ}{N_r N_s} $$

where $$\displaystyle N_r $$ = rotor teeth, $$\displaystyle N_s $$ = stator teeth per phase.

• Micro-stepping: Divide full step into smaller steps by proportioning phase currents sinusoidally.

  • Example: Half-stepping → alternate between one-phase-on and two-phase-on.

  • Currents: $$\displaystyle I_a = I_m \sin \theta $$, $$\displaystyle I_b = I_m \cos \theta $$ for smooth rotation.

Load Angle Control:

• Torque $$\displaystyle T \propto \sin(\theta_{load}) $$, where $$\displaystyle \theta_{load} $$ = displacement from equilibrium position.

• Maximum torque at $$\displaystyle \theta_{load} = 90^\circ $$ (pull-out torque).

6. BRAKING METHODS (COMPREHENSIVE)

DC Motor Braking Calculations (Plugging):

• Initial braking current: $$\displaystyle I_{b0} = \frac{V_a + E_{b0}}{R_a + R_{ext}} $$

• Braking torque: $$\displaystyle T_b = K_t \phi I_b $$ (negative).

• At $$\displaystyle \omega=0 $$, $$\displaystyle E_b=0 $$ → $$\displaystyle I_b = V_a/(R_a+R_{ext}) $$ (still high).

7. STABILITY AND LOAD CHARACTERISTICS

Steady-State Stability Criterion:

For drive with motor torque $$\displaystyle T_m(\omega) $$ and load torque $$\displaystyle T_L(\omega) $$:

• Equilibrium: $$\displaystyle T_m(\omega) = T_L(\omega) $$.

• Stable if $$\displaystyle \frac{d(T_m - T_L)}{d\omega} < 0 $$ at equilibrium.

• Physical Meaning: If speed increases slightly, net torque should become negative (decelerating) to restore equilibrium.

Effect of Characteristics:

• Stable: $$\displaystyle T_m $$ curve steeper than $$\displaystyle T_L $$.

• Unstable: $$\displaystyle T_L $$ steeper than $$\displaystyle T_m $$.

• Marginally stable: Equal slopes.

Load Equalization (Flywheel Design):

Given:

• Motor torque limit $$\displaystyle T_{m,\max} $$.

• Load torque $$\displaystyle T_{L1} $$ for time $$\displaystyle t_1 $$, $$\displaystyle T_{L2} $$ for $$\displaystyle t_2 $$ ($$\displaystyle T_{L1} > T_{L2} $$).

• Motor inertia $$\displaystyle J_m $$, no-load speed $$\displaystyle \omega_0 $$, slip at $$\displaystyle T_{L1} $$? Actually, from linear $$\displaystyle T_m-\omega $$:

$$ T_m = T_{m0} - \frac{T_{m0} - T_{L1}}{\omega_0 - \omega_1} (\omega - \omega_1) \quad \text{?} $$

Standard approach:

1. Average motor torque: $$\displaystyle T_{m,\avg} = \frac{T_{L1} t_1 + T_{L2} t_2}{t_1 + t_2} $$.

2. Energy balance:

$$ \text{Energy stored in flywheel during light load} = \frac{1}{2} (J_m + J_f) (\omega_{max}^2 - \omega_{min}^2) \approx (J_m+J_f) \omega_0 \Delta \omega $$

$$ \text{Energy deficit during heavy load} = (T_{L1} - T_{m,\avg}) t_1 $$

Equate: $$\displaystyle (J_m+J_f) \omega_0 \Delta \omega = (T_{L1} - T_{m,\avg}) t_1 $$

3. From motor characteristic, $\Delta \omega$ related to torque deviation:

$$ \Delta T = \frac{R_a}{K_e K_t \phi^2} \Delta \omega \quad \text{or from linear approx:} \quad \Delta \omega = \frac{\Delta T}{K} $$

where $$\displaystyle K = \text{slope of } T_m-\omega $$.

4. Given: slip $s$ at torque $T$ → from $$\displaystyle T = K_t \phi I_a $$, and $$\displaystyle I_a = (V_a - K_e \phi \omega)/R_a $$, derive slope.

Example Problem (May 2024):

Given $$\displaystyle T_m = (1+2a_m) $$? That seems odd. Possibly $$\displaystyle T_m = (1+2a)\omega_m $$? Actually, past paper: $$\displaystyle T_m = (1+2a_m) $$ and $$\displaystyle T_l = 3\sqrt{\omega_m} $$. Likely typo? Should be $$\displaystyle T_m = f(\omega_m) $$. Assume $$\displaystyle T_m = a_0 + a_1 \omega_m $$? Let's interpret: Maybe $$\displaystyle T_m = (1 + 2\alpha) \omega_m $$? Not clear. Standard method: equate $$\displaystyle T_m = T_L $$, solve for $\omega$, then check derivative.

8. CLOSED-LOOP AND DIGITAL CONTROL

Closed-Loop Schemes:

• Static Kramer Drive: Speed sensor → slip power controller (chopper duty cycle) → maintain speed.

• Scherbius Drive: Speed sensor → inverter frequency/phase control.

• VSI-Synchronous Drive: Speed/position sensor → current controllers (FOC) → PWM.

Digital Control of Drives:

• Block Diagram:

  
  Ref ──►[Digital Controller (DSP/FPGA)]──►[PWM Generator]──►[Power Converter]──►Motor
  
                ↑                                      ↓
  
                └──────────[ADC: Current, Speed, Position]┘
  
  

• Advantages:

  • Flexible control algorithms (FOC, DTC).

  • Precise tuning, adaptive control.

  • Diagnostics, communication (fieldbus).

  • Cost-effective for complex control.

• Implementation: Sample times, quantization, anti-aliasing filters.

Constant Torque & Constant Power Operation:

• Constant Torque: Below base speed (DC: armature control; IM: V/f constant; SM: V/f constant). Maintain rated current.

• Constant Power: Above base speed (DC: field control; IM: constant V; SM: constant V). Torque inversely proportional to speed.

• Range: Determined by maximum voltage and current limits, and stability limits (for SM, $$\displaystyle \delta_{max} $$).

9. INDUSTRIAL APPLICATIONS AND SYSTEM CONSIDERATIONS

Cement Industries:

• Drives: Crushers, mills, fans, conveyors.

• Requirements: High torque, dusty environment, constant speed (for mills) or variable speed (for fans).

• Schemes: IM drives with V/f control for fans; DC drives or slip-power recovery for mills (high torque, low speed).

Steel Industries:

• Drives: Rolling mills (reversing, non-reversing), cranes, conveyors.

• Requirements: High power (MW), high torque, rapid reversal, regenerative braking.

• Schemes:

  • Rolling Mills: DC drives (armature control) or slip-power recovery IM drives for wide speed range and regeneration.

  • Cranes: Four-quadrant DC or AC drives with regenerative braking.

Electric Traction:

• Requirements: High starting torque, wide speed range, regenerative braking, ruggedness.

• DC vs AC: Historically DC (simple control). Now AC (IM with V/f or vector control) for lower maintenance.

• Control: Series DC motor with rheostatic or chopper control; IM with V/f and vector control.

Group vs Individual Drives:

• Group Drive: One motor drives multiple machines via line shaft. Advantages: lower cost, simpler. Disadvantages: inefficient, no individual control, all stop if one fails.

• Individual Drive: Each machine has its own motor. Advantages: flexible, efficient, independent control. Modern trend.

CNC Machines:

• Axes: Spindle (constant power) and feed axes (servo drives).

• Drives: AC servo motors (PMSM) with digital controllers, high resolution encoders.

• Integration: Motion controller coordinates axes, trajectory planning.

Energy Recovery Systems:

• Principle: Capture kinetic energy during braking and feed back to supply or store.

• Implementation: Regenerative converters (fully controlled rectifier, active front end), batteries, supercapacitors.

• Applications: Elevators, cranes, electric vehicles.

10. STARTING AND MISCELLANEOUS TOPICS

Starters for DC Motors:

• Necessity: Limit starting current, provide sufficient starting torque.

• Types:

  • Manual: Face-plate starter with series resistors.

  • Automatic: Time-delay relays, current relays to step out resistors.

  • Solid-state: Thyristor-based soft starter (ramp voltage).

Conventional Methods (Historical):

• Starting: Rheostats in armature/field.

• Braking: Mechanical brakes, plugging.

• Speed Control: Ward-Leonard (motor-generator set), metadyne, amplidyne.

Two-Phase and Three-Phase Converters:

• Two-Phase: Rarely used, for special applications.

• Three-Phase: Common for high power DC drives (6-pulse, 12-pulse to reduce harmonics).

Phase-Controlled vs Chopper-Controlled Drives:

Energy Recovery Systems (Detailed):

• Regenerative Braking: Motor as generator → power back to AC supply via inverter with regenerative capability (DC bus voltage rise controlled by PWM).

• Storage: Supercapacitors for short bursts, batteries for longer.

• Control: During braking, torque command negative, power flow monitored, DC bus voltage regulated.

Slip-Controlled IM Drives:

• Synonym for slip power recovery drives (Kramer, Scherbius).

• Control slip frequency/power to adjust speed.

EXAM TIPS:

1. Load Equalization: Always draw torque-time diagram, mark $$\displaystyle T_{m,\max} $$, $\Delta \omega$, use energy balance.

2. Converter Output Voltage: Remember formulas for semi/full converters with/without freewheeling.

3. Braking: Distinguish plugging (reverse voltage), dynamic (resistor across armature), regenerative ($$\displaystyle E_b > V_a $$).

4. V/f Control: Below base: constant $V/f$ → constant torque; above: constant $V$ → constant power.

5. Stability: Always compare slopes of $$\displaystyle T_m $$ and $$\displaystyle T_L $$ at equilibrium.

6. Special Motors: SRM (no magnets/rotor windings), PMSM (sinusoidal), BLDC (trapezoidal, Hall sensors).

KEY FORMULAS BOX:

• DC Motor Speed: $$\displaystyle \omega_m = \frac{V_a}{K_e \phi} - \frac{R_a}{K_e K_t \phi^2} T $$

• Semi-Converter $$\displaystyle V_{dc} $$ (with FD): $$\displaystyle V_{dc} = \frac{V_m}{\pi}(1+\cos\alpha) $$

• Full Converter $$\displaystyle V_{dc} $$: $$\displaystyle V_{dc} = \frac{2V_m}{\pi}\cos\alpha $$

• Chopper $$\displaystyle V_{avg} $$: $$\displaystyle V_{avg} = D V_{in} $$ (buck), $$\displaystyle V_{avg} = V_{in}/(1-D) $$ (boost)

• IM Torque: $$\displaystyle T = \frac{3}{\omega_s} \frac{V^2 (R_r'/s)}{(R_s+R_r'/s)^2 + X_{eq}^2} $$

• IM $$\displaystyle T_{max} $$: $$\displaystyle T_{max} \approx \frac{3 V^2}{2 \omega_s X_{eq}} $$

• SynRM Torque: $$\displaystyle T = \frac{3}{\omega_s} \frac{V^2}{2} \left( \frac{1}{X_q} - \frac{1}{X_d} \right) \sin 2\delta $$

• Flywheel Inertia: $$\displaystyle J_f = \frac{(T_{L1} - T_{m,\avg}) t_1}{\Delta \omega} - J_m $$

DIAGRAM REFERENCES:

• Block Diagram of Electrical Drives:

  DiagramSEARCH: "electrical drive block diagram power converter controller"

• Single-Phase Semi-Controlled Converter:

  DiagramCANVAS: "Two SCRs (T1,T2) and two diodes (D1,D2) in bridge, freewheeling diode across motor, AC source"

• DC Motor Torque-Speed Characteristics:

  DiagramSEARCH: "dc motor torque speed curve armature control field control"

• IM V/f Control Characteristics:

  DiagramSEARCH: "induction motor v/f control torque speed curves constant torque constant power"

• Static Kramer Drive Power Circuit:

  DiagramCANVAS: "Rotor slip rings → diode bridge → DC link inductor → chopper → DC source"

• SRM Construction:

  DiagramSEARCH: "switched reluctance motor stator rotor salient poles"

• Stepper Motor Micro-stepping Currents:

  DiagramSEARCH: "stepper motor microstepping sine cosine current waveforms"

END OF UNIT 1 SHORT NOTES