SCADA Systems & Applications (EX-802 (C)) - Unit 3 Short Notes

UNIT 3: SCADA SYSTEMS & APPLICATIONS - SHORT NOTES

I. STEPPER MOTORS

Definition: A stepper motor is a synchronously rotating electric motor that converts a train of input pulses into discrete angular displacements. The step angle is the minimum rotation per input pulse.

Types & Classification:

Construction & Working Principle (Hybrid Type):

• Schematic: Stator has multiple phases (e.g., 4-phase). Rotor is a cylindrical PM with fine teeth. Stator also has teeth.

• Principle: When a phase is energized, the stator's magnetic field attracts the nearest rotor teeth. Sequential energization causes step-by-step rotation. Step angle determined by:

$$ \theta_s = \frac{360^\circ}{N_r \cdot N_s} \quad \text{or} \quad \theta_s = \frac{360^\circ}{N_r \cdot m} $$

where $$\displaystyle N_r $$ = rotor teeth, $$\displaystyle N_s $$ = stator teeth per phase, $m$ = number of phases.

Driver Circuits & Control:

• Unipolar Drive: Each winding center-tapped. Current flows in one direction per half-winding. Simple driver (e.g., ULN2003).

• Bipolar Drive: Whole winding used. Current can flow in both directions. Requires H-bridge driver. Higher torque.

• Dual Voltage Driver (Two-Phase-On Drive for 4-phase motor):

  • Uses two supply voltages ($$\displaystyle V_H $$ for fast current rise, $$\displaystyle V_L $$ for holding).

  • Schematic: Two transistors per phase. $$\displaystyle V_H $$ applied initially for quick step, then switched to $$\displaystyle V_L $$ to maintain current with less power loss.

  • Current Build-up: Fast rise with $$\displaystyle V_H $$, slower decay/steady with $$\displaystyle V_L $$.

Advanced Control:

• Microstepping: Divides one full step into many microsteps by proportionally controlling phase currents (sinusoidal/cosine waveforms). Reduces vibration, noise, and increases resolution.

• Load Angle Control: Maintaining a constant load angle (rotor position relative to stator field) by adjusting pulse rate. Prevents missed steps under load.

Characteristics:

• Static: Torque vs. Rotor Position curve shows detent torque (in unenergized PM/Hybrid) and holding torque (energized).

• Dynamic:

  • Pull-in Torque: Max torque at which motor can start/stop instantly without losing synchronism.

  • Pull-out Torque: Max torque motor can maintain once at speed.

  • Torque-Speed Curve: Inverted-U shape. Torque decreases as speed increases due to back-EMF and inductance.

Torque Equation (Approximate for Hybrid):

$$ T = k_t \cdot I \cdot \sin(\phi) $$

where $$\displaystyle k_t $$ = torque constant, $I$ = phase current, $\phi$ = load angle (deviation from equilibrium).

Operational Modes:

• Full-step: One phase energized at a time (or two-phase-on for higher torque). Step angle = $$\displaystyle \theta_s $$.

• Half-step: Alternating single-phase and two-phase-on excitation. Step angle = $$\displaystyle \theta_s/2 $$.

• Microstepping: Continuous current control. Step angle << $$\displaystyle \theta_s $$.

Speed Control Methods:

1. Pulse Rate Control: Primary method. Speed $\propto$ pulse frequency.

2. Voltage/Current Control: Higher voltage/current increases available torque at higher speeds.

3. Winding Switching: Series/parallel switching of windings changes inductance and time constant, altering max speed.

Applications:

• CNC machines, 3D printers, robotics (precise open-loop positioning).

• Printers/Plotters (paper feed, carriage).

• PV Water Pumping (direct drive, efficient at variable speeds from PV).

[!TIP] Exam Focus: Stepping angle calculation for hybrid VR motor with castellated poles is a high-frequency problem. Remember: $$\displaystyle \theta_s = \frac{360^\circ}{N_r \cdot N_s} $$ where $$\displaystyle N_s $$ is teeth per phase on stator.

II. SWITCHED RELUCTANCE MOTORS (SRM)

Construction & Design:

• Salient Poles: Both stator and rotor have salient poles. No windings or PMs on rotor.

• Pole Arcs: Stator pole arc ($$\displaystyle \beta_s $$) < Rotor pole arc ($$\displaystyle \beta_r $$) to ensure overlapping inductance profile during rotation. Typical: $$\displaystyle \beta_s \approx 30^\circ $$, $$\displaystyle \beta_r \approx 32^\circ $$ for 6/4 SRM.

• Cross-section:

  DiagramSEARCH: "SRM 6 stator 4 rotor poles cross section"

Principle of Operation:

• Variable Reluctance Principle: Rotor aligns to position of minimum reluctance (maximum inductance) when stator phase is energized.

• Sequence: Phase A energized → rotor aligns → Phase A off, Phase B on → rotor moves to next position. Continuous rotation requires sequential excitation based on rotor position.

Torque Production:

• Derived from co-energy ($$\displaystyle W'_{fld} $$):

$$ T = \frac{\partial W'_{fld}}{\partial \theta} \bigg|_{i=const} = \frac{1}{2} i^2 \frac{dL(\theta)}{d\theta} $$

where $L(\theta)$ = phase inductance profile, $i$ = phase current, $\theta$ = rotor position.

• Instantaneous Torque: Positive when $$\displaystyle dL/d\theta > 0 $$ (inductance increasing, rotor pulled in). Negative when $$\displaystyle dL/d\theta < 0 $$ (inductance decreasing, rotor pushed out).

• Angle-Torque Characteristics: For a given current, torque is zero at aligned ($$\displaystyle L_{max} $$) and unaligned ($$\displaystyle L_{min} $$) positions, peaks in between.

Shaft Position Sensing:

• Necessity: Commutation must be synchronized with rotor position.

• Methods:

  • Resolvers / Encoders: Accurate, robust, used in high-performance drives.

  • Hall Effect Sensors: Simple, low-cost.

  • Sensorless: Estimates position from phase inductance or voltage/current signatures (e.g., detecting zero-crossing of induced voltage).

Performance & Characteristics:

• Torque-Speed: High starting torque. Torque ripple is significant due to doubly salient structure and single-phase excitation.

• Advantages:

  • Simple, rugged, low-cost rotor (no PMs, no windings).

  • High efficiency, wide speed range.

  • Fault-tolerant (phase failures don't cause locking).

• Disadvantages:

  • High torque ripple & acoustic noise.

  • Requires precise position sensing.

  • Non-sinusoidal torque.

Applications:

• Traction drives (EVs, locomotives).

• Industrial drives (pumps, fans, compressors).

• Aerospace (due to reliability).

[!TIP] Exam Focus: Torque calculation problems are very common. Given $L(\theta)$ profile, current $i$, and rotor position $\theta$, use $$\displaystyle T = \frac{1}{2} i^2 \frac{dL}{d\theta} $$. For maximum energy per stroke: $$\displaystyle E_{max} = \frac{1}{2} i^2 (L_{max} - L_{min}) $$. Average torque $$\displaystyle T_{avg} = \frac{E_{max}}{\text{stroke angle}} $$.

III. BRUSHLESS DC MOTORS (BLDC)

Construction & Topology:

• Stator: Similar to AC motor. Three-phase windings (star/delta). Winding Pattern:

  • Series: Higher voltage, lower current.

  • Parallel: Lower voltage, higher current.

• Rotor: Surface-mounted or interior permanent magnets (NdFeB, SmCo). Creates trapezoidal back-EMF.

• Comparison with Brushed DC: No commutator/brushes → no maintenance, no sparking, higher speed, better reliability.

Principle of Operation:

• Electronic Commutation: Six-step (120° conduction) commutation. Inverter switches phases based on rotor position (from Hall sensors or back-EMF).

• Rotating Magnetic Field: Sequential energization of stator phases creates a rotating field that pulls the PM rotor.

Torque Production:

• Lorentz Force: $$\displaystyle F = i (\vec{l} \times \vec{B}) $$.

• Torque Expression:

$$ T = \frac{3}{\omega_m} \left( e_a i_a + e_b i_b + e_c i_c \right) \approx k_t \cdot I $$

where $$\displaystyle e_{a,b,c} $$ = phase back-EMF (trapezoidal), $$\displaystyle i_{a,b,c} $$ = phase currents, $$\displaystyle \omega_m $$ = mechanical speed, $$\displaystyle k_t $$ = torque constant.

In steady-state, $T \propto I$ (for constant $$\displaystyle k_t $$).

Control Strategies:

• Speed Control: Vary DC bus voltage (PWM) or phase current amplitude.

• Commutation Logic:

  • Sensor-based: Hall sensors provide absolute position.

  • Sensorless: Detect zero-crossing of back-EMF in unenergized phase.

Position Sensing:

• Hall Effect Sensors: Three sensors spaced 120° electrical apart. Provide six commutation states per revolution.

• Sensorless: Back-EMF integration or filtering. Works only above certain speed (cannot start from zero).

Performance Analysis:

• Torque-Speed: Constant torque region up to base speed, constant power beyond (field weakening possible).

• Armature Reaction: Demagnetizing effect, can distort air-gap flux. Less severe than in brushed DC due to distributed windings.

• Advantages over Brushed DC: High efficiency (>90%), low maintenance, high power-to-weight ratio, wide speed range.

• Disadvantages: Higher cost, complex controller, torque ripple.

Comparison with PMSM:

[!TIP] Exam Focus: "Differentiate BLDC vs PMSM" is a classic question. Focus on back-EMF shape and control strategy (trapezoidal vs sinusoidal).

IV. PERMANENT MAGNET SYNCHRONOUS MOTORS (PMSM)

Construction & Materials:

• Stator: Three-phase AC winding, laminated core.

• Rotor:

  • Surface-mounted (SPM): Magnets on surface. Low reluctance difference ($$\displaystyle L_d \approx L_q $$).

  • Interior (IPM) / Inset: Magnets embedded. Salient ($$\displaystyle L_d > L_q $$), provides reluctance torque.

• PM Materials: NdFeB (highest energy product), SmCo (high temp), Ferrite (low cost).

Principle & Operation:

• Synchronous: Rotor locks to rotating stator magnetic field. Speed $$\displaystyle n_s = \frac{120f}{P} $$.

• Back-EMF Generation: Rotor PMs cut stator windings, inducing sinusoidal EMF.

EMF Equation & Analysis:

• Fundamental EMF per phase:

$$ E_f = 4.44 \cdot f \cdot N \cdot \phi \cdot k_w $$

where $f$ = frequency, $N$ = turns/phase, $\phi$ = flux per pole, $$\displaystyle k_w $$ = winding factor.

• Phasor Diagram (No-load): $$\displaystyle \vec{V} = \vec{E_f} + jI_a(X_s) $$ (synchronous reactance $$\displaystyle X_s $$).

• Circle Diagram (Torque-Current):

$$ T = \frac{3}{\omega_s} \left[ \frac{E_f V_t}{X_s} \sin\delta + \frac{V_t^2}{2X_s} \left( \frac{1}{L_q} - \frac{1}{L_d} \right) \sin 2\delta \right] $$

First term = mutual torque, second = reluctance torque (only for salient rotors, IPM).

Torque-Speed Characteristics:

• Constant Torque Region: Below base speed, $V/f$ constant, maximum torque limited by current.

• Constant Power Region: Above base speed, field weakening (reduce $$\displaystyle E_f $$) to maintain voltage limit.

Control Methods:

• V/f Control: Scalar control, simple, for constant load.

• Vector Control (FOC): Decouples torque & flux. Requires rotor position (sensor/sensorless). Enables fast dynamic response.

• Torque Pulsation Reduction: Skewing, fractional-slot winding, optimal current profiling in FOC.

Position Sensing & Control:

• Sensorless: High-frequency injection (for IPM at standstill), back-EMF (above base speed).

• Role: Position feedback is essential for FOC to align d-q axes with rotor flux.

Power Electronic Controllers:

• Inverter: Typically 2-level IGBT inverter. Multilevel (NPC, T-type) for higher voltage.

• PWM: Sinusoidal PWM (SPWM) or Space Vector PWM (SVPWM) for sinusoidal back-EMF.

Applications:

• High-performance drives: Robotics, aerospace, machine tools, EVs.

• PV Water Pumping: High efficiency at partial loads, good low-speed torque.

[!TIP] Exam Focus: EMF equation derivation and circle diagram (torque equation) are critical. Remember reluctance torque term $\propto \sin 2\delta$ exists only for salient pole PMSMs (IPM).

V. POWER QUALITY FUNDAMENTALS & DISTURBANCES

Definition: Power Quality is the concept of maintaining voltage, current, and frequency within specified limits to ensure reliable operation of customer equipment.

Importance: Increased due to:

• Sensitive electronic loads (computers, PLCs).

• Renewable integration (inverter-based, fluctuating).

• Economic impact (downtime, equipment damage).

Major Power Quality Issues:

1. Voltage Variations:

• Fluctuation: Repeated voltage variations (amplitude modulation). Cause: Cyclic loads (arc furnaces, welding). Effect: Light flicker.

• Sag (Dip): Short-duration (0.5-30 cycles) voltage reduction to 10-90% of nominal. Cause: Faults, motor starting.

• Swell: Short-duration (0.5-30 cycles) voltage increase to 110-180% of nominal. Cause: Faults (esp. single-line-to-ground), large load switching off.

• Surge vs. Swell: Surge is a transient (µs-ms), fast rise, caused by lightning, capacitor switching. Swell is a sustained (cycles) rms increase.

• Spike/Transient: Very short-duration (<1ms), high-amplitude overvoltage. Cause: Lightning, switching of inductive/capacitive circuits.

2. Waveform Distortion:

• Harmonics: Sinusoidal components at integer multiples of fundamental frequency ($$\displaystyle h=2,3,4... $$).

  • Sources: Non-linear loads: Rectifiers (AC-DC), UPS, arc furnaces, fluorescent lamps, saturated transformers.

  • Industrial: Rectifiers, variable-speed drives.

  • Commercial: IT equipment, lighting.

• Harmonic Indices:

  • THD (Total Harmonic Distortion): $$\displaystyle THD_V = \frac{\sqrt{\sum_{h=2}^{\infty} V_h^2}}{V_1} \times 100\% $$ (voltage). Similar for current.

  • TDD (Total Demand Distortion): Current THD relative to demand current (15-30 min avg).

  • Crest Factor: $$\displaystyle \frac{I_{peak}}{I_{rms}} $$. High crest factor indicates peaky non-linear currents.

  • Interharmonics: Frequencies not integer multiples of fundamental (e.g., 150 Hz from 6-pulse rectifier). Cause: Cycloconverters, arcing loads.

3. Transients:

• Causes: Capacitor switching, fault clearing, lightning, load rejection.

• Transient Recovery Voltage (TRV): Voltage across circuit breaker poles after current interruption. Factors: system voltage, grounding, fault type, capacitor banks.

4. Interruptions & Outages:

• Momentary: < 5 sec (e.g., recloser operation).

• Temporary: 5 sec - 5 min (e.g., manual reset).

• Sustained: > 5 min.

Root Causes Analysis:

• Primary: Non-linear loads drawing non-sinusoidal currents → voltage distortion.

• System weaknesses: Weak short-circuit capacity, poor grounding.

• Renewable integration: Power electronic interfaces introduce harmonics, fluctuations.

[!TIP] Exam Focus: Be very clear on distinctions: Sag vs Swell (magnitude direction), Surge vs Swell (transient vs sustained), Harmonics vs Interharmonics (integer vs non-integer multiples).

VI. POWER QUALITY MITIGATION TECHNIQUES

Passive Compensation:

• Shunt Passive Filters (LC): Tuned to specific harmonic frequencies (e.g., 5th, 7th). Low cost, but can cause resonance, fixed compensation.

• Power Factor Correction Capacitors: Install at:

  • Load side: Individual correction.

  • Feeder side: Group correction.

  • Substation: Bulk correction.

  • Benefit: Reduces reactive power flow, lowers losses, improves voltage profile.

  • Risk: Can amplify harmonics if not detuned.

Flexible AC Transmission Systems (FACTS):

• Static Var Compensator (SVC):

  • Principle: Shunt-connected, TCR (Thyristor Controlled Reactor) + TSC (Thyristor Switched Capacitor) or TSR.

  • Provides variable reactive power (inductive/capacitive) by controlling TCR firing angle and TSC switching.

  • Advantages: Fast response (~1-2 cycles), wide range.

  • Limitations: Discrete steps (TSC), generates harmonics (TCR), limited overload capability.

• STATCOM (Static Synchronous Compensator):

  • Principle: VSC (Voltage Source Converter) based shunt device. Acts as a controllable voltage source behind a reactor.

  • Outputs current proportional to voltage deviation. Reactive power $$\displaystyle Q \propto V_{STATCOM} \cdot V_{sys} \cdot \sin\delta $$.

  • Comparison with SVC:

    | Feature | SVC | STATCOM | | :--- | :--- | :--- | | Response Time | 1-2 cycles | < 1 cycle (sub-cycle) | | Capability | Limited by capacitor/reactor ratings | Higher (limited by converter rating) | | Harmonics | Generates (needs filters) | Minimal (PWM) | | Low Voltage | $$\displaystyle Q \propto V^2 $$ (drops sharply) | $Q \propto V$ (better support) |

  • Difference from Active Power Filter: STATCOM's primary function is dynamic reactive power/voltage support. APF's primary function is harmonic current injection. STATCOM can be configured as APF with appropriate control.

Active Power Filters (APF):

• Shunt-Active Filter Principle:

  1. Detect load current harmonics (using FFT or instantaneous theory).

  2. Generate reference harmonic current to be injected.

  3. PWM Inverter injects this current at the PCC, canceling load harmonics.

• Advantages: Dynamic, selective, compensates interharmonics, can also compensate reactive power.

• Disadvantages: High cost, complex control, limited power rating (typically < 5 MVA).

Unified Power Quality Conditioner (UPQC):

• Configuration: Series APF + Shunt APF connected back-to-back via a DC capacitor.

• Principle:

  • Shunt APF: Injects current to compensate load harmonics/reactive power.

  • Series APF: Injects voltage to compensate supply voltage sag/swell/harmonics.

• Classification:

  • Shunt-Series (UPQC): Shunt at load side, series in supply line (most common).

  • Series-Shunt: Reverse order.

• Advantages: Comprehensive mitigation (voltage & current quality).

• Comparison with DSTATCOM:

  • DSTATCOM: Shunt-only device. Compensates current harmonics and reactive power (voltage support via reactive current).

  • UPQC: Handles both voltage and current disturbances. Series APF directly corrects voltage sags/swells.

Voltage Sag & Swell Mitigation:

• Estimation: Monitoring (rms, instantaneous), statistical analysis (SARFI index).

• Mitigation Devices:

  • DVR (Dynamic Voltage Restorer): Series-connected VSC, injects voltage to boost sag.

  • UPS (Uninterruptible Power Supply): Full isolation, battery backup.

  • Sag Protector: Solid-state switch that transfers load to backup source during sag.

• Protection Scheme Needed: Sensitive equipment (computers, PLCs) can malfunction or be damaged by sags/swells.

Transient & Surge Protection:

• Surge Protective Devices (SPDs): Metal Oxide Varistors (MOVs), Gas Discharge Tubes. Clamping voltage is the maximum voltage allowed across protected equipment.

• Prevention: Proper grounding, zero-voltage switching for capacitor banks (switching at voltage zero-crossing minimizes transient).

• Impulse Withstand: Insulation coordination - BIL (Basic Impulse Level) rating of equipment > expected surge levels.

Harmonic Mitigation:

• Best Methods:

  1. Active Filters (Shunt/Series/UPQC): Most effective, dynamic.

  2. Hybrid Filters: Passive + active (e.g., APF with passive detuned filter).

  3. Detuned Filters: Passive filters tuned away from system resonance.

• Passive Filter Design: Tuning frequency, damping resistance, reactive power rating.

• Harmonic Analyzer: Measures waveforms, performs FFT to identify harmonic magnitudes/phases, calculates THD, TDD.

[!TIP] Exam Focus: "SVC vs STATCOM" and "APF vs STATCOM" vs "UPQC vs DSTATCOM" are extremely frequent comparison questions. Know the topology (TCR/TSC vs VSC), primary function, response time, and harmonic generation.

VII. MAGNETIC & MATERIAL FUNDAMENTALS

Magnetic Circuit Analysis:

• B-H Relationship: Shows hysteresis (energy loss, remanence $$\displaystyle B_r $$, coercivity $$\displaystyle H_c $$) and saturation (non-linear, $\mu$ decreases).

• Permeance Coefficient ($$\displaystyle P_c $$): For PM, $$\displaystyle P_c = \frac{\mu_0 A_g}{l_g} $$ (air-gap permeance). Used in PM motor design to find operating point on B-H curve.

• Leakage Flux: Flux that does not follow intended path (e.g., across end-rings). Reduces effective flux linkage, causes losses.

• Fringing: Flux bulging at air-gap edges. Effect: Increases effective air-gap area, reduces average air-gap flux density $$\displaystyle B_g $$.

• Stacking Factor: Ratio of effective magnetic area to total physical area of laminated core (accounts for insulation between laminations). Typically 0.95-0.98.

Material Properties:

Reluctance & Torque:

• Magnetic Reluctance ($\mathcal{R}$): $$\displaystyle \mathcal{R} = \frac{l}{\mu_0 \mu_r A} $$. Analogous to electrical resistance.

• Reluctance Torque: In machines with saliency ($$\displaystyle L_d \neq L_q $$), torque produced by tendency to align with minimum reluctance. In SRM: $$\displaystyle T \propto \frac{dL}{d\theta} $$. In PMSM: second term in torque equation (reluctance torque $\propto \sin 2\delta$).

VIII. APPLICATIONS IN MODERN SYSTEMS

Electric Vehicles (EVs):

• Suitable Motor Types & Comparison:

  | Motor | Efficiency | Power Density | Cost | Control | EV Suitability | | :--- | :--- | :--- | :--- | :--- | :--- | | BLDC | High | High | Medium | Simple (6-step) | Good for cost-sensitive | | PMSM (IPM) | Very High | Very High | High | Complex (FOC) | Best for high-performance | | Induction (ACIM) | High | Medium | Low | Medium (FOC) | Robust, Tesla uses |

• Role in Powertrain: Converts battery DC power to mechanical drive. Requires inverter, often integrated with gearbox.

Photovoltaic (PV) Water Pumping:

• System Config: PV Array → DC-DC/AC-DC Converter → Inverter → Motor-Pump.

• Motor Types Used: BLDC or PMSM (high efficiency over wide speed range, good part-load efficiency). Induction motors also used with VFD.

• Advantage of PM Motors: High efficiency at partial loads (common in solar pumping), no field winding losses.

IX. COMPARATIVE STUDIES & KEY DIFFERENTIATIONS

Stepper Motor: Permanent vs. Hybrid

BLDC vs PMSM

SVC vs STATCOM

Active Power Filter vs STATCOM

UPQC vs DSTATCOM

Voltage Sag vs Swell vs Surge

Soft vs Hard Ferromagnetic Materials

Solid vs Laminated Rotor (Induction Motor)

X. DESIGN & CALCULATION-FOCUSED TOPICS

1. Stepper Motor Stepping Angle (Hybrid VR with Castellated Poles):

• For a hybrid VR motor with $$\displaystyle N_r $$ rotor teeth and $$\displaystyle N_s $$ stator teeth per phase:

$$ \theta_s = \frac{360^\circ}{N_r \cdot N_s} $$

• Example (May 2023): 8 main poles castellated to 5 teeth each, rotor 50 teeth.

  $$\displaystyle N_s = 5 $$ (teeth per phase), $$\displaystyle N_r = 50 $$.

$$ \theta_s = \frac{360}{50 \times 5} = 1.44^\circ $$

2. SRM Instantaneous Torque Calculation:

• Given: $L(\theta)$ profile, phase current $i$, rotor position $\theta$.

• Formula:

$$ T = \frac{1}{2} i^2 \frac{dL}{d\theta} $$

• Example (May 2024): 6/4 SRM, $$\displaystyle \beta_s=30^\circ $$, $$\displaystyle \beta_r=32^\circ $$, $$\displaystyle L_{aligned}=10.7 $$ mH, $$\displaystyle L_{unaligned}=1.5 $$ mH, $$\displaystyle i=6 $$A, $$\displaystyle \theta = 30^\circ $$ before aligned.

  • Need $dL/d\theta$ at that position. Assuming linear inductance profile between unaligned ($$\displaystyle \theta_u $$) and aligned ($$\displaystyle \theta_a $$):

$$ \frac{dL}{d\theta} \approx \frac{L_{aligned} - L_{unaligned}}{\theta_a - \theta_u} $$

For 6/4, electrical stroke = $$\displaystyle 90^\circ $$ (mechanical). $$\displaystyle \theta_a - \theta_u \approx 30^\circ $$ (due to pole arcs). So:

$$ \frac{dL}{d\theta} \approx \frac{10.7 - 1.5}{30^\circ} \text{ mH/deg} = \frac{9.2}{30} \approx 0.3067 \text{ mH/deg} = 0.3067 \times 10^{-3} \text{ H/rad} \times \frac{180}{\pi} \approx 0.0175 \text{ H/rad} $$

$$ T = \frac{1}{2} \times (6)^2 \times 0.0175 \approx 0.315 \text{ Nm} $$

• Maximum Energy per Stroke: $$\displaystyle E_{max} = \frac{1}{2} i_{max}^2 (L_{max} - L_{min}) = \frac{1}{2} \times 7^2 \times (10.7 - 1.5) \times 10^{-3} = 0.2549 \text{ J} $$.

• Average Torque: Stroke angle (mechanical) for 6/4 = $$\displaystyle 90^\circ = \pi/2 $$ rad.

$$ T_{avg} = \frac{E_{max}}{\text{stroke}} = \frac{0.2549}{\pi/2} \approx 0.162 \text{ Nm} $$

3. PMSM EMF Equation Derivation:

• Flux per pole: $$\displaystyle \phi = B_g \cdot A \cdot k_f $$ (where $$\displaystyle k_f $$ = form factor).

• Average EMF per turn: $$\displaystyle e_{avg} = 4 f \phi $$ (for 2-pole, sinusoidal).

• For $N$ turns/phase, $$\displaystyle k_w $$ winding factor:

$$ E_f = 4.44 \cdot f \cdot N \cdot \phi \cdot k_w $$

4. PMBLDC Permeance Coefficient Derivation:

• Permeance coefficient $$\displaystyle P_c = \frac{\mu_0 A_g}{l_g} $$ (air-gap permeance).

• For PM: $$\displaystyle H_{pm} = \frac{B_{pm}}{\mu_0} - H_c $$ (demagnetization curve).

• Operating point on load line: $$\displaystyle B_g = \frac{\mu_0 N i}{l_g} + B_r \frac{l_m}{l_m + l_g \frac{\mu_r}{\mu_0}} $$ (simplified).

• $$\displaystyle P_c $$ relates $$\displaystyle B_g $$ and $$\displaystyle H_g $$ in air-gap: $$\displaystyle B_g = \mu_0 H_g + B_r \frac{l_m}{l_g} $$.

• In PM motor design, $$\displaystyle P_c $$ determines how much PM flux is "leaked" vs. crossing air-gap.

[!TIP] Exam Focus: SRM torque calculation and PMSM EMF derivation are almost guaranteed questions. Practice deriving $$\displaystyle T = \frac{1}{2}i^2 \frac{dL}{d\theta} $$ from co-energy $$\displaystyle W' = \int_0^i \lambda(\theta,i) di $$.