Special Machine (EX-802 (B)) - Unit 2 Short Notes

UNIT 2: SPECIAL MACHINES – EXAM-FOCUS NOTES

1. Magnetic Materials and Fundamental Concepts

Soft vs. Hard Ferromagnetic Materials

Exam Tip: Questions often ask for applications. Link soft materials to AC applications (low core loss) and hard materials to PM machines (high remanence).

B-H Relationship and Hysteresis Loop

• The B-H curve shows the non-linear relationship between magnetic flux density (B) and magnetic field intensity (H).

• Hysteresis Loop: Traced by varying H cyclically. Key points:

  • Br (Remanence): Residual flux density at H=0.

  • Hc (Coercivity): Reverse field required to reduce B to zero.

  • Area of loop ∝ Energy loss per cycle (hysteresis loss).

  • μ = B/H (Slope at any point).

Leakage Flux, Fringing, and Stacking Factor

• Leakage Flux: Flux that does not follow the intended magnetic path in the core but takes a path through surrounding air. Reduces coupling between windings.

• Fringing: Bulging of magnetic flux lines at the air-gap edges. Effectively increases air-gap area, reducing flux density in the gap.

• Stacking Factor (K_st): Ratio of net cross-sectional area of core (after accounting for insulation/varnish between laminations) to gross cross-sectional area.

  \begin{equation}

  \boxed{K_{st} = \frac{A_{net}}{A_{gross}} < 1}

  \end{equation}

  Used in core design to calculate effective flux-carrying area.

Magnetic Reluctance (ℛ)

• Analogy to electrical resistance. Opposition to magnetic flux creation.

  \begin{equation}

  \boxed{\mathcal{R} = \frac{l}{\mu A}} \quad \text{(for uniform path)}

  \end{equation}

  where l = length of magnetic path, A = cross-sectional area, μ = permeability.

• Reluctance Torque Principle: Torque is produced in a machine when there is a tendency to minimize the reluctance of the magnetic circuit. This is the fundamental principle of Variable Reluctance (VR) and Switched Reluctance (SR) motors. Torque is proportional to the rate of change of inductance with rotor position ($dL/dθ$).

Energy Conversion via Electric Field

• Conversion using electrostatic forces (Coulomb's Law).

• Examples:

  1. Electrostatic Loudspeaker/Microphone: Vibrating diaphragm changes capacitance, inducing current (microphone) or voltage causes diaphragm movement (speaker).

  2. Electrometer: Measures charge/voltage using electrostatic attraction/repulsion.

  3. Capacitive Actuators/MEMS: Used in precise positioning systems.

2. Stepper Motors

Types and Classification

1. Permanent Magnet (PM) Stepper: Rotor made of permanent magnet. Simple, low cost, but lower resolution and torque.

2. Variable Reluctance (VR) Stepper: Toothed, non-magnetized rotor (soft iron). Aligns with stator field to minimize reluctance. High torque at low speed, no detent torque when unpowered.

3. Hybrid Stepper (HV/HY): Combines PM and VR principles. Castellated poles on stator and rotor. Highest resolution, torque, and speed performance. Most common in precision applications.

Hybrid Stepper Motor – Construction & Working (Neat Diagram)

• Stator: Has two or more phases (commonly 4 or 5). Each phase winding is on a "castellated" pole (multiple teeth).

• Rotor: Made of soft iron with teeth, and often has a permanent magnet axially magnetized (for hybrid type). The rotor teeth are misaligned with stator teeth by a fraction of the tooth pitch.

• Working: When a stator phase is excited, its teeth become north/south poles. The rotor, having opposite polarity (due to PM) or seeking minimum reluctance (due to teeth), aligns its teeth with the stator teeth. Sequential excitation of phases causes stepping rotation.

  • Stepping Angle (β): For a hybrid motor with N_r rotor teeth and m number of phases:

    \begin{equation}

    \boxed{\beta = \frac{360^\circ}{N_r \times m}} \quad \text{(Full-step)}

    \end{equation}

    For multi-stack or different pole counts, formula varies.

Single-Stack vs. Multi-Stack Steppers

• Single-Stack: One stack of stator and rotor. Simple, limited steps/rev.

• Multi-Stack: Several identical stacks (stator & rotor) mounted on a common shaft, axially displaced. Each stack has its own winding. Increases number of steps per revolution without reducing tooth size. Common in high-resolution motors.

Static & Dynamic Characteristics

• Static:

  • Pull-in Torque: Maximum torque at which the motor can start/stop synchronously without losing steps (starting torque). Depends on drive method (full/half-step).

  • Pull-out Torque: Maximum torque the motor can withstand while running at a given speed without losing synchronism.

  • Detent Torque: Torque required to rotate the shaft when phases are de-energized (present in PM and Hybrid types due to PM).

• Dynamic:

  • Resonance: Motor can vibrate violently at certain stepping rates (natural frequencies). Avoid operating in this slew range.

  • Slew Range: Speed range where motor runs smoothly without resonance.

  • Maximum Starting/Slewing Speed: Limits of reliable operation.

  • Torque-Speed Curve: Torque decreases as speed increases. Pull-in curve is below Pull-out curve.

Torque Equation & Load Angle Control

• Torque Expression (Approximate):

  \begin{equation}

  \boxed{T = K_t \cdot I \cdot \sin(\delta)}

  \end{equation}

  where K_t = torque constant, I = phase current, δ = load angle (angular displacement between rotor's equilibrium position and actual position under load).

• Load Angle Control: To maintain synchronism, δ must be kept within a limit (typically < 90° for stability). Increasing current I increases torque capacity for a given δ. Control systems adjust current to maintain δ within safe limits under varying load.

Drive Systems

• Unipolar Drive: Center-tapped winding, current flows in only one direction through half-winding. Simple driver, lower torque per amp.

• Bipolar Drive: Full winding energized, current can flow in both directions. Higher torque per amp, more complex driver (H-bridge).

• Chopper Drive (Constant Current): Maintains constant phase current regardless of speed by rapidly switching voltage (PWM). Improves high-speed torque, reduces heating.

• Dual Voltage Drive (for 4-phase): Uses two voltage levels (V_H high, V_L low). At low speed, high voltage (V_H) for fast current rise. At high speed, switches to low voltage (V_L) to prevent excessive back-EMF and current overshoot. Current build-up is faster with V_H.

Microstepping

• Principle: Divide a full step into many smaller steps by proportionally controlling the current in two simultaneously energized phases.

• Implementation: Use a DAC or PWM to set precise current levels in each phase (e.g., 1.0A in phase A, 0.5A in phase B for a 1/4 step between full-step positions).

• Advantages:

  • Smoother motion, reduced vibration/noise.

  • Higher resolution without mechanical changes.

  • Reduced resonance problems.

  • Precise positioning.

Torque Pulsation Reduction

• Microstepping (primary method).

• Optimized current shaping (non-sinusoidal profiles to compensate for inductance effects).

• Increasing number of phases (e.g., 5-phase motors have less ripple than 4-phase).

• Mechanical: Precise machining, using detent torque advantageously.

Applications: Printers, plotters, CNC machines, robotics, camera lenses, 3D printers, valve control.

3. Switched Reluctance Motors (SRM)

Construction

• Double Saliency: Both stator and rotor have salient poles (teeth).

• Stator: Has concentrated windings on each pole. Number of stator poles (S) ≠ number of rotor poles (R) typically (e.g., 6/4, 8/6).

• Rotor: Solid, non-magnetized, non-laminated (often just a stack of iron laminations). No windings, no magnets, no commutator. Robust and low-cost.

• Pole Arcs: Stator and rotor pole arcs are less than pole pitch to ensure only one pair of poles aligns at a time.

Principle of Operation (Reluctance Torque)

• Fundamental Principle: Rotor moves to a position where the magnetic circuit reluctance (ℛ) is minimized.

• When a stator pole is energized, it creates a magnetic field. The rotor pole aligns with this energized stator pole to minimize the air-gap reluctance.

• Phase Energizing Sequence: Phases are energized in sequence as rotor poles approach and then move away from alignment. This creates a rotating "magnetic field" that pulls the rotor along.

Torque Analysis

• Instantaneous Torque Expression (for one phase, unsaturated):

  \begin{equation}

  \boxed{T = \frac{1}{2} i^2 \frac{dL(\theta)}{d\theta}}

  \end{equation}

  where i = phase current, L(θ) = phase inductance as a function of rotor position θ.

  • Torque is produced only when dL/dθ ≠ 0 (i.e., when inductance is changing with position).

  • Positive torque when dL/dθ > 0 (inductance increasing as rotor moves into alignment).

  • Negative torque ( braking) when dL/dθ < 0.

• Total Torque: Sum of torques from all active phases.

• Energy & Co-energy: Useful for deriving torque when saturation is considered. Co-energy W' is more convenient for systems with stored magnetic energy.

  \begin{equation}

  T = \frac{\partial W'}{\partial \theta} \bigg|_i

  \end{equation}

• Angle-Torque Characteristics: Torque vs. rotor position for a given current. Shows positive torque region (around aligned position) and negative torque region (around unaligned). Average torque is the integral over one stroke.

Numerical Problem Approach (Given L(θ), i):

1. Identify the position range where the phase is energized.

2. Use T = ½ i² (dL/dθ). Calculate dL/dθ from the given L(θ) function.

3. Instantaneous Torque: Plug i and dL/dθ at the specific θ.

4. Maximum Energy Conversion (per stroke): Energy converted = area under T(θ) curve over the positive torque region (stroke angle θ_s). Alternatively, compute co-energy difference between aligned and unaligned positions at max current I_max:

   \begin{equation}

   W_{max} = \int_{\theta_{unaligned}}^{\theta_{aligned}} T_{max}(\theta) d\theta = \frac{1}{2} I_{max}^2 (L_{aligned} - L_{unaligned})

   \end{equation}

   (if saturation neglected and L constant with i).

5. Average Torque: T_avg = (Energy per stroke) / (Stroke angle in radians).

Position Sensing

• Essential for control (commutation timing).

• Shaft Position Sensors:

  • Hall Effect Sensors: Most common. Placed on stator to detect rotor pole passage.

  • Optical Encoders: High resolution.

  • Resolvers: Robust, for harsh environments.

• Sensorless Control: Indirect methods:

  • Inductance Detection: Measuring phase inductance variation.

  • Back-EMF Detection: During "freewheeling" periods.

  • Current Waveform Analysis.

Performance & Characteristics

• Torque-Speed: Wide constant-torque region at low speed. Torque drops at high speed due to insufficient time for current build-up (inductance effect).

• Acoustic Noise & Vibration: Major drawback. Caused by:

  • Torque ripple (inherent due to salient poles).

  • Radial force pulsations.

  • Mitigation: Advanced control (current profiling), optimized pole geometry, skewing.

Advantages & Disadvantages

• Advantages:

  • Extremely robust (solid rotor, no magnets, no brushes/commutator).

  • Low cost (simple construction).

  • High efficiency over wide speed range.

  • Safe failure (if a phase fails, motor continues with reduced torque).

  • High starting torque.

• Disadvantages:

  • High torque ripple & acoustic noise.

  • Complex control (requires precise rotor position and current control).

  • Higher current ripple (needs large DC bus capacitor).

  • Not suitable for very high-speed (due to inductance).

Applications: Industrial drives (pumps, fans), traction (EVs/HEVs – e.g., Cobra), appliances (washing machines), aerospace (fuel pumps).

4. Brushless DC Motors (BLDC)

Construction

• Stator: Three-phase concentrated or distributed windings (like AC induction motor). Can be series (higher voltage, lower current) or parallel (lower voltage, higher current) connected.

• Rotor: Surface-mounted or interior-mounted permanent magnets (NdFeB, Ferrite). Creates constant magnetic field.

• Position Sensors: Hall Effect sensors (3 sensors, 120° electrical apart) mounted on stator to sense rotor magnet position.

Working Principle (Electronic Commutation)

• No brushes/commutator. Commutation is done electronically by a 3-phase inverter based on rotor position signals from Hall sensors.

• Six-Step (Trapezoidal) Commutation: At any time, two phases are energized (one positive, one negative), one phase is floating. Hall sensor states (6 combinations) determine which two phases to energize. Rotor rotates in 60° electrical steps.

• Back-EMF: Trapezoidal waveform (flat top for 120°). Used for sensorless control (zero-crossing detection).

Torque Production

• Torque Constant (K_t): Relates torque to phase current. For BLDC, K_t (Nm/A) = K_e (V/(rad/s)) in SI units.

  \begin{equation}

  \boxed{T = K_t \cdot I_{peak}}

  \end{equation}

  where I_peak is the current in the energized phase (for 2-phase-on, 120° commutation).

• Permeance Coefficient (P): Used in design to relate air-gap flux density to magnet properties.

  \begin{equation}

  \boxed{P = \frac{B_r / \mu_0}{H_c + \frac{g}{\mu_0 \mu_r A_m}}}

  \end{equation}

  where B_r = remanence, H_c = coercivity, g = air-gap length, A_m = magnet area, μ_r = relative permeability of magnet. P helps determine operating point on demagnetization curve.

Commutation & Armature Reaction

• Commutation Process: As rotor moves, Hall sensors change state at 60° e intervals. Inverter switches current to next phase pair. Ideal commutation occurs when rotor's magnetic axis is exactly between the two energized stator poles (60° e from alignment).

• Armature Reaction: The stator MMF distorts the rotor's main field (from PMs). In BLDC (trapezoidal back-EMF), it tends to flatten the trapezoid and can cause flux weakening at high currents. Less severe than in brushed DC motors due to distributed winding.

Control Methods

• Speed Control:

  1. Voltage Control: Varying DC bus voltage (inefficient).

  2. PWM Voltage Control: Most common. Keep DC bus constant, vary PWM duty cycle to control average voltage/current.

  3. Current Control: Inner current loop for torque control, outer speed loop.

• Sensorless Control (Back-EMF Detection):

  • Zero-Crossing Detection: Monitor back-EMF in the floating phase. When it crosses zero, it's time to commutate (60° e later). Only works above a minimum speed (where back-EMF is measurable).

  • Back-EMF Integration: For lower speeds.

Position Sensing

• Hall Sensors: 3 sensors provide 6 discrete positions (60° e each). Simple, reliable, works at zero speed. Block Diagram: Hall sensors → Logic circuit (Decoder) → Gate driver → Inverter → Motor.

• Sensorless: No hardware sensors. Uses back-EMF. Lower cost, but cannot start/stop precisely at zero speed without special starting algorithm.

Characteristics

• Torque-Speed Curve:

  • Constant Torque Region: Up to base speed. Torque ∝ Current. Controlled by current limit.

  • Constant Power Region: Above base speed. Back-EMF approaches DC bus voltage, limiting current. Torque ∝ 1/Speed (field weakening by advancing commutation angle).

• Efficiency: Very high (85-95%) due to no brush losses and permanent magnet excitation.

Permanent Magnet Materials

Advantages over Conventional Brushed DC Motors

• No mechanical commutator/brushes → No sparking, lower maintenance, longer life.

• Higher efficiency (no brush contact loss).

• Higher power density.

• Better speed range.

• Lower EMI.

• Can operate in explosive/hazardous environments.

Applications: Computer hard drives, CD/DVD drives, fans, pumps, Electric Vehicles (traction, power steering), aerospace actuators, industrial automation.

5. Permanent Magnet Synchronous Motors (PMSM)

Construction

• Rotor: PMs (Surface-mounted - SPM, or Interior-mounted - IPM). IPM provides reluctance torque component (saliency).

• Stator: Three-phase AC windings (distributed for sinusoidal back-EMF). Laminated core.

Working Principle

• Synchronous Operation: Rotor (PM field) locks in synchronism with the rotating magnetic field produced by the stator AC currents.

• Sinusoidal Back-EMF: Due to distributed windings and sinusoidal MMF distribution. E = E_m sin(ωt).

• Speed: N_s = 120f / P (synchronous speed), where f = supply frequency, P = number of poles.

EMF Equation

• Per Phase:

  \begin{equation}

  \boxed{E_{ph} = 4.44 \cdot f \cdot N \cdot \phi \cdot K_w}

  \end{equation}

  where:

  • f = frequency (Hz)

  • N = number of turns per phase

  • φ = flux per pole (Wb) from PM

  • K_w = winding factor (accounts for distribution & pitch)

• Line Voltage (Star): E_L = √3 E_ph

Torque-Speed Characteristics & Phasor Diagram

• Phasor Diagram (Steady State):

  • V = terminal voltage (reference)

  • E = internal induced EMF (leads V if over-excited, lags if under-excited).

  • I_a = armature current.

  • δ = torque angle (angle between E and V).

  • Power Angle: δ is key. For constant V and E, I_a cosφ (real power component) increases with δ.

• Circle Diagram (Power Angle Characteristic):

  • Plots P (or T) vs. δ.

  • Stable operating region: -90° < δ < 90°. Maximum torque at δ = 90° (for constant E and V).

  • Shows effect of loading: As load torque increases, δ increases to develop more torque. Pull-out occurs at δ = δ_max (usually < 90° due to saturation).

Speed Control Methods

1. V/f Control (Scalar Control): Maintain constant V/f ratio to keep flux constant. Simple, open-loop. Used for pumps/fans.

2. Vector Control (Field-Oriented Control - FOC): Decouples torque and flux producing components of stator current (like separately excited DC motor). High dynamic performance. Requires rotor position sensor (or sensorless estimator).

3. Direct Torque Control (DTC): Directly controls torque and flux by selecting optimal voltage vectors. Fast torque response, no PWM modulator, but higher torque ripple.

4. Sensorless Control: Estimates rotor position from:

   • Back-EMF (for speeds > ~10% base speed).

   • High-Frequency Signal Injection (for zero/low speed, especially for IPM).

Power Controllers

• Voltage Source Inverter (VSI): Standard. DC bus capacitor provides stiff voltage.

• PWM Techniques:

  • Sinusoidal PWM (SPWM): For smooth sinusoidal currents.

  • Space Vector PWM (SVPWM): Higher DC bus utilization (≈15% more), lower harmonic distortion.

• Current Regulated PWM: Inner current loop ensures phase currents follow reference precisely (essential for FOC).

Torque Pulsation Reduction

• Skewing: Stator or rotor slots/teeth skewed to distribute cogging torque.

• Fractional Slot Winding: Reduces space harmonics.

• Optimal Magnet Shaping/Placement.

• Current Shaping (in FOC): Injecting harmonic currents to compensate for non-sinusoidal back-EMF or inductance saliency.

Applications: Electric Vehicles (traction drives - e.g., Tesla Model 3 uses IPMSM), PV water pumping systems (high efficiency at part-load), high-performance servo drives, industrial robotics, compressors.

6. Brushed Permanent Magnet DC Motors (PMDC)

Construction

• Stator: Permanent magnets (radially mounted) creating constant stationary field.

• Rotor (Armature): Laminated core with windings, commutator, and brushes (carbon).

Working Principle

• Lorentz Force: Current-carrying conductor in magnetic field experiences force: F = B I L.

• Torque is produced by interaction of armature current with stator PM field.

• Commutation: Commutator segments and brushes mechanically reverse the direction of current in the armature coils as they pass under the poles, ensuring continuous unidirectional torque.

Torque and Speed Equations

• Torque:

  \begin{equation}

  \boxed{T = K_t \cdot I_a}

  \end{equation}

  where K_t = torque constant (Nm/A), I_a = armature current.

• Speed (N in rpm):

  \begin{equation}

  \boxed{N = \frac{V - I_a R_a}{K_e}}

  \end{equation}

  where V = supply voltage, R_a = armature resistance, K_e = back-EMF constant (V/(rad/s) or V/rpm).

  • K_e and K_t are numerically equal in SI units.

• Back-EMF: E_b = K_e ω (ω in rad/s). Opposes applied voltage.

Speed Control Methods

1. Armature Voltage Control (V): Most efficient. Varying V directly controls speed N. Used with DC-DC converters (choppers).

2. Field Flux Control (Weakening): Adding series resistance in the field winding (if separately excited or shunt). Reduces field flux φ, increasing speed N (since N ∝ V/φ). Used for above base speed operation. Not applicable for PMDC (field is fixed from PMs). This method is for wound-field DC motors.

Applications: Automotive (wipers, windows, seats), toys, small appliances, battery-powered tools, low-power drives.

7. AC Servomotors

Construction

• Stator: Two-phase (or three-phase with special control).

  • Reference (Excitation) Winding: Connected to constant voltage, constant frequency (AC) supply.

  • Control Winding: Connected to variable voltage, variable frequency (or phase-shifted) supply from servo amplifier. Phase-displaced (usually 90° electrical) from reference winding.

• Rotor: Squirrel cage (aluminum/copper bars) or solid iron (for high inertia). Low inertia preferred.

Working Principle

• Two AC fluxes (90° apart) produce a rotating magnetic field.

• The magnitude and direction of the net rotating field is controlled by the amplitude and phase of the voltage applied to the control winding relative to the reference winding.

• Rotor follows this field, producing torque proportional to the sine of the torque angle (angle between rotor axis and net field axis).

Torque-Speed Characteristics

• Linear Region (Small Torque Angle): For small loads, torque T ∝ V_control (voltage to control winding). Speed is nearly constant (like a synchronous motor). High stiffness.

• Nonlinear Saturation Region: At high loads (large torque angle), torque saturates and becomes non-linear with V_control. Speed drops.

• Pull-out Torque: Maximum torque before losing synchronism.

• Diagram: Shows torque vs. control voltage (linear then saturating) and torque vs. speed (vertical drop at pull-out).

8. Comparative Analysis and System Applications

BLDC vs. PMSM

Motors for Electric Vehicles (EVs)

Selection Criteria for EVs: Efficiency (range), Power Density (size/weight), Cost (rare earth dependency), Torque Ripple (drivability), Control Complexity, Fault Tolerance.

Motors for PV Water Pumping Systems

• Preferred: BLDC or PMSM.

• Why?

  1. Very High Efficiency over wide speed range → maximizes water output for given solar power.

  2. Good part-load performance (solar irradiance varies).

  3. Wide constant torque region → good for centrifugal pumps (torque ∝ N²).

  4. Can be directly coupled to pump (no gearbox needed if motor speed matches pump).

• System: PV Array → MPPT (Maximum Power Point Tracker) → DC-DC Converter/Inverter → PMSM/BLDC Motor → Pump.

• Control: MPPT controls motor speed to operate PV at MPP. Motor control is typically sensorless V/f or simple FOC.

General Applications Summary Table

Final Exam Tip: For comparative questions (BLDC vs PMSM, Motors for EVs/PV), structure your answer as a table first, then explain key points. Always link characteristics (back-EMF, control) to applications. For numericals (SRM torque, stepper angle), write the formula clearly, substitute values, and box the final answer.