UNIT 1: POWER SEMICONDUCTOR DEVICES & POWER CONVERTERS

I. POWER SEMICONDUCTOR DEVICES

A. Thyristor (SCR)

Definition: A four-layer (PNPN), three-junction, three-terminal (Anode, Cathode, Gate) semiconductor device acting as a bistable switch.

Basic Structure & Two-Transistor Analogy:

• Structure: Alternating P-N-P-N layers. Terminals: Anode (P1), Cathode (N2), Gate (P2).

• Analogy: Equivalent to an NPN transistor (Q2) and a PNP transistor (Q1) coupled. Gate current triggers Q2, which injects current into Q1's base, providing positive feedback and latching the device ON.

  DiagramCANVAS: SCR cross-section with layers J1, J2, J3 labeled and two-transistor equivalent circuit

Static V-I Characteristics:

• Forward Blocking (OFF): Anode (+ve) w.r.t Cathode. J1 & J3 forward biased, J2 reverse biased. Small forward leakage current. Device blocks voltage until breakover voltage (V_BO) or gate trigger.

• Forward Conduction (ON): Once triggered, anode-cathode voltage drops to ON-state voltage (V_T), typically 1-2V. Current = holding current (I_H) minimum to maintain conduction.

• Reverse Blocking: Cathode (+ve) w.r.t Anode. J1 & J3 reverse biased, J2 forward biased. Blocks reverse voltage up to reverse breakdown (V_RRM).

  Key Points: Latching current (I_L) > Holding current (I_H). Device turns ON at gate but turns OFF only when anode current < I_H.

Dynamic Switching Characteristics:

• Turn-on Time (t_on): Delay time (t_d) + Rise time (t_r). Gate trigger pulse must be longer than t_d.

• Turn-off Time (t_off): Reverse recovery time (t_rr) + Gate recovery time (t_gr). Critical for high-frequency operation.

  Exam Tip: t_off is typically 50-100µs for standard SCRs. GTOs have much shorter t_off.

Methods of Turning ON (Triggering):

1. Gate Triggering: Most common. Positive gate current pulse.

2. dv/dt Triggering: Excessive rate of voltage rise across J2 can cause false turn-on. Requires snubber circuit.

3. di/dt Triggering: Excessive anode current rise can damage device. Requires inductor in series.

4. Thermal Triggering: High temperature increases leakage current, may cause thermal runaway.

5. Light Triggering (LASCR): Radiation (IR/UV) on silicon creates carriers.

Methods of Commutation (Turning OFF):

• Natural (Line) Commutation: AC circuit. Anode current naturally goes to zero (e.g., in AC controllers, phase-controlled rectifiers).

• Forced Commutation: DC circuit. Additional circuitry forces anode current to zero.

  • Class A: Self-commutation (load resonant). Used in inverters.

  • Class B: External pulse commutation (auxiliary SCR).

  • Class C: Complementary commutation (auxiliary SCR in parallel).

  • Class D: Anode auxiliary commutation.

  • Class E: Impulse commutation.

  Common Pitfall: Confusing commutation classes. Remember Class B uses an auxiliary SCR to send a pulse to main SCR's cathode.

Ratings & Protection:

• dv/dt Rating: Max allowable rate of voltage rise without false triggering. Protected by RC snubber.

• di/dt Rating: Max allowable rate of current rise. Protected by series inductor.

• Over-current: Protected by fast-acting fuses (I²t rating).

• Over-voltage: Protected by varistors, spark gaps, RC snubbers.

• Thermal: Heat sink required.

Series & Parallel Operation:

• Series: For high voltage. Need static voltage sharing (shunt resistor R) and dynamic voltage sharing (shunt capacitor C) due to unequal junction capacitances/recovery times.

  • Derivation for R: \( V_{R1} = V_{R2} \Rightarrow R = \frac{V_{TM}}{I_{RM} - I_{TM}} \) (simplified).

  • Derivation for C: Ensure equal voltage distribution during transients.

• Parallel: For high current. Need static current sharing (small series resistor R_s) due to unequal V_T characteristics.

  Exam Focus: Derive R and C for series string. String efficiency = (Total rating) / (Sum of individual ratings).

B. Gate Turn-Off Thyristor (GTO)

Structure: Similar to SCR but with highly doped P+ gate layer for efficient hole extraction. Operation: Turned ON by positive gate pulse (like SCR). Turned OFF by high-current negative gate pulse (5-6x I_GT). V-I Characteristics: Similar to SCR but with specified turn-off current (I_GT(off)). Comparison with SCR:

C. Power MOSFET

Structure (n-channel enhancement): Source (N+), Drain (N-), Gate (metal/oxide), Body (P). Vertical structure. Transfer Characteristics (I_D vs V_GS): \( I_D = K \left( (V_{GS} - V_{TH})^2 \right) \) for \( V_{GS} > V_{TH} \) in saturation. Output Characteristics (I_D vs V_DS): Shows ohmic, saturation, and cut-off regions. Switching Characteristics: Very fast (ns). No minority carrier storage. Input is capacitive (gate charge). Advantages: High input impedance, fast switching, simple drive, no commutation circuitry. Applications: High-frequency (kHz-MHz), low-voltage (<500V), low-power converters.

D. Insulated Gate Bipolar Transistor (IGBT)

Structure: Combines MOSFET gate with BJT output. P+ substrate, N- drift, P body, N+ source, and P+ collector. Operation: MOSFET input controls BJT base. Voltage-controlled. Transfer Characteristics: Similar to MOSFET but with higher current density. Output Characteristics: Shows BJT-like saturation. Switching Characteristics: Faster than BJT (µs), slower than MOSFET. Tail current during turn-off. Comparison:

E. Other Devices

• Power Diode:

  • Standard (PN): Slow reverse recovery (µs). Used in 50Hz rectifiers.

  • Fast Recovery: Gold/platinum doping, P+ layer. t_rr ~ 0.1-5 µs. Used in high-frequency converters.

  • Schottky: Metal-semiconductor junction. No minority carriers. Very fast, low V_F (0.3-0.5V), low reverse voltage (<200V). Used in low-voltage SMPS.

• DIAC: Two-terminal, bidirectional trigger diode. Conducts when |V| > Breakover voltage (V_BO). Used to trigger TRIACs.

  • V-I Char: Symmetrical, no gate. Used in dimmer circuits.

• TRIAC: Three-terminal (MT1, MT2, Gate), bidirectional thyristor. Conducts in both polarities when triggered. Used in AC phase control (light dimmers, motor speed control).

  • Modes: MT2 (+ve), Gate (+ve) → Q1 & Q3 on. MT2 (-ve), Gate (-ve) → Q2 & Q4 on.

II. RECTIFIERS / AC-DC CONVERTERS (Line Frequency Phase-Controlled)

A. Single-Phase Converters

1. Half-Wave Rectifier (R, RL, RLE Load)

• R Load: Output V_o = (V_m/π)(1+cosα) for α ≤ π. Current in phase.

• RL Load (Continuous Conduction, α ≤ β): β = extinction angle. \( V_o = \frac{V_m}{2\pi}(1 + \cos\alpha) \). Current delayed.

• RLE Load: Forced commutation needed if E large. β > π.

  Waveforms: Show source v_s, gate pulse, load v_o, i_o. For RL, i_o continuous if α small.

2. Fully Controlled Bridge Converter (4 SCRs)

• Operation (R Load): T1,T2 conduct (0-π), T3,T4 conduct (π-2π). V_o = V_m|sinωt| for α=0.

  • Average Output Voltage: \( V_{dc} = \frac{2V_m}{\pi} \cos\alpha \) for α ≤ π.

  • Rectification Mode: 0 ≤ α ≤ 90°, V_dc positive, power P_ac→P_dc.

  • Inversion Mode: 90° < α ≤ 180°, V_dc negative, power P_dc→P_ac (requires DC source E).

• RL Load (Continuous Conduction, α ≤ β ≤ π+α): \( V_{dc} = \frac{2V_m}{\pi} \cos\alpha \). β determined from \( \cos\beta = \cos\alpha - \frac{2\omega L I_{avg}}{V_m} \).

• Effect of Source Inductance (L_s): Causes overlap angle (μ). Two pairs conduct simultaneously during μ.

  • Voltage Drop: \( V_{dc} = \frac{2V_m}{\pi} \cos(\alpha + \frac{\mu}{2}) \).

  • Overlap Angle: \( \mu = \frac{2\omega L_s I_{avg}}{V_m - V_{dc}} \) (approx). μ increases with I_dc.

  Key Formula with L_s: \( V_{dc} = \frac{2V_m}{\pi} \cos\alpha - \frac{2\omega L_s I_{avg}}{\pi} \).

3. Half-Controlled Bridge Converter (2 SCRs + 2 Diodes)

• Operation: T1,D2 (0-π), T3,D4 (π-2π). Freewheeling diode (D across load) conducts when SCRs off.

• Effect of Freewheeling Diode:

  • Load current continuous for α > 90°.

  • Improved input power factor (displacement angle = α, but current fundamental in phase with voltage? Actually PF improves because no negative voltage).

  • \( V_{dc} = \frac{V_m}{\pi}(1 + \cos\alpha) \).

  Comparison: V_dc(Half-controlled) < V_dc(Full-controlled) for same α.

B. Three-Phase Converters

1. Fully Controlled Bridge Converter (6 SCRs)

• Operation (Resistive Load, Continuous Conduction): Each SCR conducts for 120°. Natural commutation from supply.

• Waveforms: Line-to-line voltage (v_ab) appears at load. V_o = v_ab of conducting pair.

• Average Output Voltage:

  • For α ≤ 60°: \( V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos\alpha = \frac{3\sqrt{2}}{\pi} V_{LN} \cos\alpha \approx 2.34 V_{LN} \cos\alpha \).

  • For 60° < α ≤ 90°: \( V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \left(1 + \cos\left(\alpha + \frac{\pi}{3}\right)\right) \).

  • Rectification: 0° ≤ α ≤ 90°. Inversion: 90° < α ≤ 180° (max 120° typically).

• Effect of Source Inductance (L_s): Overlap angle μ. Each SCR conducts for 120°+μ.

  • Voltage Drop: \( V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos(\alpha + \frac{\mu}{2}) \).

  • Overlap Angle: \( \mu = \frac{\omega L_s I_{dc}}{V_{LL}} \) (approx for 3-phase).

  Numerical Problem: Given V_dc, I_dc, V_LL, α, find L_s or R. Use \( V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos\alpha - \frac{3\omega L_s I_{dc}}{\pi} \).

2. Half-Controlled Bridge (3 SCRs + 3 Diodes): Similar operation, freewheeling occurs. V_dc expression different.

C. AC Voltage Controllers (Phase Control)

1. Single-Phase

• Half-Wave (1 SCR): V_o_rms = \( V_s \sqrt{\frac{1}{2\pi}(2\pi - \alpha + \sin2\alpha)} \). PF = \( \frac{V_o}{V_s} \cos\alpha \) (approx).

• Full-Wave (Anti-parallel SCRs or TRIAC):

  • R Load: V_o_rms = \( V_s \sqrt{\frac{1}{\pi}(\pi - \alpha + \frac{\sin2\alpha}{2})} \).

  • RL Load: Current delayed. Conduction angle γ = π - α (for α < φ). Dead angle (δ) may appear if α > φ.

    • Waveforms: Show v_s, gate pulses, v_o, i_o. i_o starts at ωt = α+φ.

    • Power: \( P = \frac{V_m^2}{2\pi R} (\cos\alpha - \cos\beta) \), where β = α+γ.

    • Displacement PF: cosφ (load). Distortion PF: <1 due to non-sinusoidal current.

  • Two-Stage Sequence Control for RL Load: To improve PF at high α.

    • Concept: Use two pairs of SCRs (T1,T4 & T2,T3) with different firing angles α1 and α2 (α1 < α2).

    • Waveforms: First stage (α1) conducts for full half-cycle. Second stage (α2) conducts only near peak, reducing negative current.

    Why? Reduces phase shift between fundamental current and voltage, improving PF.

2. Three-Phase: Basic concept using back-to-back SCRs per phase or 6-SCR bridge. V_o_rms expression complex.

D. Cycloconverters (AC-AC Converters)

Principle: Direct AC-AC conversion without DC link. Uses natural commutation. Output frequency f_o < input frequency f. Grouping: Positive group (conducts +ve half-cycle), Negative group (conducts -ve half-cycle).

1. Single-Phase to Single-Phase

• Mid-Point Type:

  • Two transformers (center-tapped). Four SCRs per group.

  • Operation: For f_out = f/2, each SCR conducts for 180° of output cycle. For f_out = f, each SCR conducts for 90°.

  • Waveforms (f_out = f/2): Output is full-wave rectified sine. Frequency halved.

• Bridge Type:

  • One transformer, 8 SCRs (4 per group). More flexible.

  • Same principle, SCRs conduct in pairs.

  Key: Output voltage is segment of input sine waves. f_out = f/(number of segments per cycle).

2. Three-Phase to Single-Phase:

• Mid-Point: 3 transformers, 12 SCRs. Output from one secondary.

• Bridge (Commonly used): 3-phase input, single-phase output. 18 SCRs (6 per group). Each group has 3 SCRs in parallel per phase.

• Operation: At any instant, one SCR from each phase (positive group) conducts, connecting that phase to output. Sequence rotates to synthesize sine wave.

• Waveforms: Output voltage is quasi-sine with steps. f_out < f.

III. INVERTERS (DC-AC CONVERTERS)

A. Voltage Source Inverters (VSI)

Basic Principle: DC input voltage (stiff voltage source). Switches (SCRs, MOSFETs, IGBTs) commutate to produce AC output.

1. Single-Phase Bridge Inverter (4 switches)

• 180° Conduction Mode:

  • Each switch conducts 180°. T1,T2 ON (0-π), T3,T4 ON (π-2π).

  • R Load: V_o = ±V_s. Square wave. i_o in phase.

  • L Load (or R-L): i_o triangular (for pure L) or lagging. i_o continuous.

  • Performance: V_o_rms = V_s. Fundamental component \( V_{o1} = \frac{4V_s}{\pi} \). THD = 48.34% for square wave.

  Waveforms: v_o (square), i_o (sinusoidal for R, triangular for L).

2. Three-Phase Bridge Inverter (6 switches)

• 180° Conduction Mode:

  • Each switch conducts 180°. Sequence: T1(T6) → T2(T1) → T3(T2) → T4(T3) → T5(T4) → T6(T5).

  • Star Load: Line-to-neutral voltages (v_AN, v_BN, v_CN) are 120° apart, each is 120° wide flat-top.

    • \( V_{LN,rms} = \frac{V_s}{\sqrt{3}} \) (for square wave).

    • Fundamental: \( V_{LN1} = \frac{2\sqrt{3}V_s}{\pi} \).

  • Delta Load: Line currents 120° apart, 120° wide. \( I_{L,rms} = \frac{V_s}{R} \) (for resistive delta).

• 120° Conduction Mode:

  • Each switch conducts 120°. Two switches ON at any time.

  • Waveforms: V_AN, V_BN, V_CN are 120° wide, 60° gap. More steps, lower harmonic content than 180° mode.

  Numerical: Given V_s, R (star), find I_phase_rms, P_total. Use \( I_{ph,rms} = \frac{V_{ph,rms}}{R} \), \( P = 3 I_{ph,rms}^2 R \).

3. Pulse Width Modulated (PWM) Inverters

• Principle: High-frequency switching (carrier) vs. sinusoidal reference. Modulates pulse width to control output voltage magnitude and reduce harmonics.

• Single-Phase PWM:

  • Unipolar: One leg referenced to midpoint. Output voltage switches between +V_s/2 and -V_s/2. Lower harmonic distortion.

  • Bipolar: Both legs switch together. Output switches between +V_s and -V_s.

• Sinusoidal PWM: Triangular carrier vs. sinusoidal reference. Fundamental amplitude controlled by modulation index (m_a = V_ref/V_car).

  Advantage: Fundamental voltage control, significant harmonic reduction (carrier frequency harmonics).

B. Current Source Inverters (CSI)

Basic Principle: DC input current (stiff current source, large inductor L_dc). Switches must commutate naturally (load commutation) or forced. Comparison with VSI:

McMurray-Bedford Inverter:

• Circuit: DC inductor L_dc, 4 SCRs (T1-T4), commutating capacitor C, commutating inductors L1, L2.

• Operation (180° mode):

  1. T1,T2 ON: Load current i_o from L_dc via T1,T2. C charges to V_s.

  2. T3 triggered: C discharges through L1,T1,T3 → T1 turns OFF (forced).

  3. T4 triggered: C discharges through L2,T2,T4 → T2 turns OFF.

  4. T3,T4 ON: i_o flows via T3,T4. C recharges opposite polarity.

• Key: Capacitor C provides turn-off path. Load must be inductive for natural commutation of T3,T4.

C. Harmonics & Reduction

Need: Harmonics cause heating, torque ripple, EMI, filter size. Techniques:

1. PWM: Shifts harmonics to high frequency (carrier freq), easy filtering.

2. Multi-Pulse Inverters: Use phase-shifting transformers (12-pulse, 18-pulse) to cancel lower harmonics.

3. Passive Filters: LC tuned to specific harmonic frequencies (e.g., 5th, 7th).

4. Active Filters: Inject opposite harmonic currents.

D. Resonant Inverters

Series Resonant Inverter:

• Circuit: DC source V_s, switch (SCR), series RLC load, diode across load.

• Operation at Resonance (ω_0 = 1/√LC): Load current i_o sinusoidal, in phase with v_o. Switch turns ON when i_o=0 (ZCS - Zero Current Switching).

• Advantage: Reduced switching losses, high efficiency at resonance.

• Numerical: At resonance, \( I_{o,rms} = \frac{V_s}{R} \) (if ideal, lossless). For given V_s, R, L, C, compute I_o at f_res.

IV. CHOPPERS (DC-DC CONVERTERS)

A. Classification & Basic Topologies

1. Step-Down (Buck) Chopper (Type-A)

• Circuit: Switch (SCR/MOSFET) in series with load (R, L, E). Diode across load (freewheeling).

• Operation:

  • ON (T on): Source V_s applied to load. i increases.

  • OFF (T off): Energy in L maintains current via diode. i decreases.

• Waveforms (Continuous Conduction): i_o ripple. V_o = average of switched voltage.

• Average Output Voltage: \( V_{dc} = \alpha V_s \) (α = T_on/T, duty cycle).

• Continuity Condition: \( I_{min} > 0 \). For RLE load: \( I_{min} = I_{avg} - \frac{\Delta i}{2} \), \( \Delta i = \frac{(V_s - E)\alpha T}{L} \) (if T_off current decay linear).

  Numerical (Past Paper): Given V_s, T, T_on, R, L, E. Check continuity: Compute I_min. If I_min > 0, continuous.

  Calculate: \( I_{avg} = \frac{V_{dc} - E}{R} \), \( I_{max} = I_{avg} + \frac{\Delta i}{2} \), \( I_{min} = I_{avg} - \frac{\Delta i}{2} \).

2. Step-Up (Boost) Chopper

• Circuit: Switch in series with inductor L. Diode in series with load. Load across diode.

• Operation:

  • ON: V_s across L. i_L increases, storing energy. Load fed from capacitor C.

  • OFF: L current continues via diode, adding to V_s. V_o = V_s + L(di/dt).

• Average Output Voltage: \( V_{dc} = \frac{V_s}{1 - \alpha} \). (Derived from volt-sec balance on L: \( V_s T_{on} = (V_o - V_s) T_{off} \)).

• Note: V_o > V_s. Current discontinuous if L small.

3. Buck-Boost (Step-Up/Step-Down) Chopper

• Circuit: Switch, diode, L, C. Output polarity reversed.

• Operation: Similar to boost but output taken across diode/capacitor.

• Average Output Voltage: \( V_{dc} = -\frac{\alpha}{1-\alpha} V_s \). Magnitude: \( |V_{dc}| = \frac{\alpha}{1-\alpha} V_s \).

  Can step up or down. Inverting output.

B. Other Chopper Types

1. Type-C (Reversible) Chopper:

• Circuit: Two switches (T1, T2) in parallel with opposite diodes. Load can be motor (bidirectional).

• Operation:

  • Forward (Motoring): T1 ON → i positive, V_o = V_s.

  • Reverse (Regenerative): T2 ON → i negative, V_o = -V_s (energy back to source).

  • Both OFF: Freewheeling via D1 or D2.

• Two-Quadrant: I_o (+ve/-ve), V_o (+ve only). For motoring and regenerative braking.

2. Morgan Chopper (Current Source Chopper):

• Circuit: Uses two capacitors (C1, C2) and two SCRs (T1, T2). Commutation via capacitor discharge.

• Operation:

  • T1 ON: Load current from V_s via T1. C1 charges to V_s.

  • T2 triggered: C1 discharges through T2, T1 → T1 turns OFF.

  • T2 ON: Load current from V_s via T2. C2 charges.

  • T1 triggered: C2 discharges → T2 turns OFF.

• Key: Capacitors provide turn-off. Used for high-power DC drives.

C. Control Strategies

1. Current Limit Control (CLC):

• Constant Frequency: Switch ON until i reaches I_max, then OFF until i drops to I_min. T_on and T_off vary. Frequency constant.

• Variable Frequency: Fixed T_on, variable T_off (or vice versa). Frequency varies → filter design difficult. 2. Time Ratio Control (TRC): Fixed T, variable T_on (or T_off). α varied. Frequency constant.

V. SWITCHED-MODE POWER SUPPLY (SMPS) & SPECIAL TOPICS

A. Switched-Mode Power Supply (SMPS)

Principle: Switch operates at high frequency (20kHz-1MHz). Transformer size ↓. Regulation via feedback. Block Diagram: Input rectifier/filter → High-frequency switch → High-frequency transformer → Output rectifier/filter → Feedback loop (optocoupler) → PWM controller. Comparison with Linear Supply:

Fly-back SMPS (Single-ended):

• Circuit: Switch (MOSFET), transformer with primary & secondary, diode, output capacitor.

• Operation:

  • ON: Energy stored in core (primary). Secondary diode reverse-biased.

  • OFF: Core flux collapses, energy transferred to secondary. Diode conducts.

• Equivalent Circuits: During ON: primary circuit with L_p. During OFF: secondary circuit with reflected load.

• Waveforms: Primary current (ramp), secondary voltage (pulse), output capacitor ripple.

• Advantage: Simple, provides isolation, multiple outputs possible.

B. Firing Circuits for Thyristors

Need: Isolated, precise gate pulses. Pulse width > t_d. 1. R-Firing Circuit: Simple, poor stability (V_GT varies with temperature). 2. RC-Firing Circuit: Better stability. Adjusts firing angle with RC phase shift. 3. UJT Firing Circuit:

*   **Circuit:** UJT with capacitor charging through R. When V_UC = V_P (peak), UJT fires → pulse to SCR gate.

*   **Waveforms:** V_UC (ramp), V_E (pulse).

*   **Advantage:** Sharp pulse, good for SCRs.

4. LASCR: Light-triggered SCR. Gate replaced by light window. Used in high-voltage (HVDC) for electrical isolation. Isolation: Pulse transformers (for pulse transmission), Opto-couplers (for DC signals).

C. Cross-Cutting Topics

1. Analysis of RLE Load in Converters:

• Continuous Conduction: β > α. \( V_{dc} = \frac{2V_m}{\pi} \cos\alpha \) (single-phase full). β from \( \cos\beta = \cos\alpha - \frac{2\omega L I_{avg}}{V_m} \).

• Discontinuous Conduction: β < α+π. Load voltage zero for (α+π-β). V_dc expression includes zero interval.

  Past Paper: "Draw waveforms for single-phase full bridge with RLE load." Show v_o, i_o, with extinction angle β.

2. Power Factor Improvement:

• Use of Freewheeling Diode: In half-controlled bridge, eliminates negative voltage/current regions, improving PF.

• Phase Control: Displacement PF = cosα (for R load). For RL load, PF = cos(α+φ) or worse due to distortion.

• Other Methods: Use of PWM rectifiers, synchronous condensers.

BOXED KEY FORMULAS

Single-Phase Full Converter (R/L):

\[ \boxed{V_{dc} = \frac{2V_m}{\pi} \cos\alpha} \]

Three-Phase Full Converter (α ≤ 60°):

\[ \boxed{V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos\alpha \approx 2.34 V_{LN} \cos\alpha} \]

Single-Phase Full-Wave AC Controller (R Load):

\[ \boxed{V_{o,rms} = V_s \sqrt{\frac{1}{\pi}(\pi - \alpha + \frac{\sin2\alpha}{2})}} \]

Buck Chopper:

\[ \boxed{V_{dc} = \alpha V_s} \]

Boost Chopper:

\[ \boxed{V_{dc} = \frac{V_s}{1-\alpha}} \]

Buck-Boost Chopper:

\[ \boxed{V_{dc} = -\frac{\alpha}{1-\alpha} V_s} \]

Effect of Source Inductance (Single-Phase):

\[ \boxed{V_{dc} = \frac{2V_m}{\pi} \cos\alpha - \frac{2\omega L_s I_{avg}}{\pi}} \]

Exam Tips & Common Pitfalls:

1. Rectification vs. Inversion: α < 90° → Rectifier (V_dc > 0). α > 90° → Inverter (V_dc < 0, requires E > V_dc).

2. Continuity in Chopper: Always check I_min = I_avg - Δi/2. If I_min > 0 → continuous. If I_min < 0 → discontinuous (Δi = I_max).

3. Three-Phase VSI: Remember 180° mode gives square wave line voltages. 120° mode gives stepped wave with lower harmonics.

4. AC Voltage Controller with RL: Current always continuous for α < φ? No, if α large, may have dead angle. Conduction angle γ = π - α only if α < φ and continuous.

5. Overlap Angle μ: In converters, μ increases with load current. Inverters (CSI) need load inductance for natural commutation.

6. GTO vs. IGBT: GTO turn-off requires large negative gate current. IGBT turn-off is like MOSFET (remove gate drive).