UNIT 2: POWER ELECTRONICS - EXAM-FOCUSED SHORT NOTES

I. POWER SEMICONDUCTOR DEVICES & CHARACTERISTICS

A. Power Diodes

• Types & Applications:

  • Fast Recovery Diode: Reverse recovery time $$\displaystyle t_{rr} < 5\mu s $$. Used in high-frequency switching circuits (e.g., choppers, inverters).

  • Schottky Diode: Low forward voltage drop ($\approx 0.2-0.4V$), very fast switching, low reverse voltage rating. Used in low-voltage, high-frequency SMPS.

  • Snubber Diode: Used across inductive loads to absorb voltage spikes from $di/dt$.

• Static V-I Characteristic:

  • Forward: Similar to signal diode but with higher current/voltage ratings.

  • Reverse: Blocking region until breakdown voltage $$\displaystyle V_{BR} $$.

B. Thyristor (SCR)

• Basic Structure & Two-Transistor Analogy:

  • Four-layer (PNPN), three-terminal (Anode A, Cathode K, Gate G) device.

  • Analogy: Equivalent to an NPN and PNP transistor coupled regeneratively. Gate current $$\displaystyle I_G $$ triggers the NPN, which latches the PNP, turning on the SCR.

  [!TIP] SCR is latching; once on, gate loses control. Turn-off requires external circuit (commutation).

• Static & Dynamic Characteristics:

  • Static: Forward blocking ($$\displaystyle V_{AK} < V_{BO} $$), forward conducting (low $$\displaystyle V_{AK} $$), reverse blocking.

  • Dynamic: Turn-on time $$\displaystyle t_{on} $$ (delay $$\displaystyle t_d $$, rise $$\displaystyle t_r $$), turn-off time $$\displaystyle t_{off} $$ (reverse recovery $$\displaystyle t_{rr} $$, decay $$\displaystyle t_{gr} $$). $dv/dt$ and $di/dt$ ratings specify maximum allowable rates to avoid false triggering or damage.

• Turn-On Methods:

  1. Gate Triggering: Normal method (positive $$\displaystyle I_G $$).

  2. $dv/dt$ Triggering: Excessive rate of voltage rise causes capacitive turn-on (undesirable).

  3. $di/dt$ Triggering: High initial $di/dt$ can damage the device.

  4. Thermal Triggering: High temperature reduces $$\displaystyle V_{BO} $$.

  5. Light Triggering (LASCR): Light photons generate carriers in the junction.

• Turn-Off (Commutation) Methods:

  • Natural Commutation: AC circuit; current goes to zero naturally.

  • Forced Commutation: External circuit forces current to zero.

    • External Pulse: Auxiliary SCR discharges capacitor into main SCR.

    • Load Commutation: Load is resonant (underdamped), current naturally commutates.

• Protection Circuits:

  • $dv/dt$ Protection: Snubber circuit (R-C across SCR).

  • $di/dt$ Protection: Series inductor.

  • Over-current: Fuses, fast-acting circuit breakers.

  • Over-voltage: Surge arrestors (Metal Oxide Varistors - MOVs).

• Series & Parallel Operation:

  • Need: To handle higher voltage (series) or current (parallel) than a single device.

  • Problems:

    • Series: Static/dynamic voltage imbalance due to unequal leakage currents ($$\displaystyle I_{CEO} $$) and junction capacitances ($$\displaystyle C_j $$).

    • Parallel: Static/dynamic current imbalance due to $$\displaystyle V_{GT} $$ (gate trigger voltage) and $$\displaystyle V_{TM} $$ (on-state voltage) mismatches.

  • Solutions:

    • Static Equalizing: Shunt resistor $R$ across each SCR. Derivation: For voltage sharing, $$\displaystyle R \ll r_d $$ (dynamic resistance). Condition: $$\displaystyle V_{S1} = V_{S2} \Rightarrow I_{S1}R_1 = I_{S2}R_2 $$. If $$\displaystyle I_{S1} \neq I_{S2} $$, choose $R$ such that voltage drop across $R$ compensates.

    • Dynamic Equalizing: Shunt capacitor $C$ across each SCR (with optional $R$). During $dv/dt$, capacitor voltage lags, helping balance transient voltages.

• Firing Circuits:

  • R-Firing: Simple, limited to $$\displaystyle \alpha > 90^\circ $$. $$\displaystyle I_G = \frac{V_m - V_{GT}}{R} $$ at $$\displaystyle \omega t = \alpha $$.

  • RC-Firing: Provides $\alpha$ from $$\displaystyle 0^\circ $$ to $$\displaystyle 180^\circ $$. $$\displaystyle V_C $$ across capacitor triggers when $$\displaystyle V_C > V_{GT} $$.

  • UJT Firing: UJT acts as a relaxation oscillator. Pulse from UJT's emitter triggers SCR. Provides sharp, isolated pulses.

C. TRIAC

• Structure: Two SCRs connected in inverse parallel with common gate.

• Modes of Operation (with equivalent circuits):

  • Mode I ($$\displaystyle I^+ $$ quadrant): $$\displaystyle V_2 > 0, I_1 > 0 $$. Gate $$\displaystyle G^+ $$. Equivalent: $$\displaystyle T_1 $$ (PNP) and $$\displaystyle T_2 $$ (NPN) regenerative pair.

  • Mode III ($$\displaystyle III^- $$ quadrant): $$\displaystyle V_2 < 0, I_1 < 0 $$. Gate $$\displaystyle G^- $$. Equivalent: $$\displaystyle T_3 $$ (NPN) and $$\displaystyle T_4 $$ (PNP).

• Applications: AC power control (light dimmers, motor speed control).

D. Power MOSFET (n-channel Enhancement)

• Structure: Vertical structure with source, drain, gate, body. Channel formed by positive $$\displaystyle V_{GS} $$.

• Transfer Characteristic: $$\displaystyle I_D = k[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}] $$ (triode), $$\displaystyle I_D = \frac{k}{2}(V_{GS} - V_{th})^2 $$ (saturation). $$\displaystyle V_{th} $$ ~ 2-4V.

• Output Characteristic: $$\displaystyle I_D $$ vs $$\displaystyle V_{DS} $$ for various $$\displaystyle V_{GS} $$. Linear region (switch ON), saturation (constant current).

• Switching: Voltage-controlled, majority carrier device → very fast switching ($$\displaystyle t_{on}, t_{off} < 100ns $$), high input impedance.

E. IGBT

• Structure: MOSFET gate controlling a BJT (PNP). Combines MOSFET input with BJT output.

• Static V-I: Similar to BJT but with MOSFET gate. Latch-up possible at high $$\displaystyle I_C $$.

• Transfer Char: $$\displaystyle I_C $$ vs $$\displaystyle V_{GE} $$ (gate-emitter voltage). Threshold $$\displaystyle V_{GE(th)} $$.

• Switching: Faster than BJT, slower than MOSFET. $$\displaystyle t_{on} $$ dominated by MOSFET, $$\displaystyle t_{off} $$ by BJT storage time.

F. GTO (Gate Turn-Off Thyristor)

• Working: Like SCR but with highly doped $$\displaystyle p^+ $$ gate. Large negative $$\displaystyle I_G $$ ($$\displaystyle -I_{GO} $$) extracts carriers from $p$-base, turning off.

• V-I Characteristics: Similar to SCR but with specified turn-off gate current. Requires complex gate drive.

G. DIAC

• Structure: Two-terminal, bidirectional, $$\displaystyle p^+ - n - p^+ - n^+ $$ (or $$\displaystyle n^+ - p - n^+ - p^+ $$).

• V-I Characteristic: Symmetrical, breaks over at $$\displaystyle V_{BO} $$ (typically 30V) in both polarities. Negative resistance region.

• Operation & Application: Used to trigger TRIACs (e.g., in light dimmers).

II. PHASE CONTROLLED RECTIFIERS (AC-DC CONVERTERS)

A. Single-Phase Converters

• Half-Wave Rectifier (R & RL Load):

  • R Load: $\omega t \in [\alpha, \pi]$ conduction. $$\displaystyle v_o = V_m \sin \omega t $$, $$\displaystyle i_o = \frac{v_o}{R} $$.

  • RL Load (Continuous): Conduction from $\alpha$ to $$\displaystyle \beta > \pi $$. $$\displaystyle i_o = \frac{V_m}{Z} \sin(\omega t - \theta) - \frac{V_m}{Z} \sin(\alpha - \theta) e^{-(R/L)(\omega t - \alpha)} $$, where $$\displaystyle Z = \sqrt{R^2 + (\omega L)^2} $$, $$\displaystyle \theta = \tan^{-1}(\omega L/R) $$.

  • Average Output Voltage: $$\displaystyle V_{dc} = \frac{V_m}{2\pi}(1 + \cos \alpha) $$ (R load). For RL, $$\displaystyle V_{dc} = \frac{V_m}{2\pi}(\cos \alpha - \cos \beta) $$.

• Full-Wave Converters:

  • Half-Controlled Bridge (2 SCRs, 2 Diodes): Unidirectional output current. Freewheeling Diode (FWD) conducts during negative half-cycle when SCRs are off, providing continuous $$\displaystyle i_o $$ and improving PF.

  • Fully Controlled Bridge (4 SCRs):

    • R Load: $$\displaystyle V_{dc} = \frac{2V_m}{\pi} \cos \alpha $$. $\alpha \in [0, \pi]$.

    • RL Load (Continuous): $$\displaystyle V_{dc} = \frac{2V_m}{\pi} \cos \alpha $$. $$\displaystyle \beta = \pi + \alpha $$ for highly inductive (constant $$\displaystyle i_o $$).

    • RLE Load: $$\displaystyle V_{dc} = \frac{2V_m}{\pi} \cos \alpha - E $$ (with back-EMF $E$). Inversion possible if $$\displaystyle E > \frac{2V_m}{\pi} \cos \alpha $$.

  • Waveforms: Load voltage $$\displaystyle v_o $$ (pulsating), load current $$\displaystyle i_o $$ (continuous/discontinuous), source current $$\displaystyle i_s $$ (discontinuous for R load, continuous for RL).

  • Effect of Source Inductance (Overlap):

    • Overlap Angle $\mu$: Period during which two SCRs conduct simultaneously.

    • Average Output Voltage: $$\displaystyle V_{dc} = \frac{2V_m}{\pi} \cos(\alpha + \frac{\mu}{2}) $$ (approx. for small $\mu$). More precisely: $$\displaystyle V_{dc} = \frac{1}{\pi} [V_m(\cos \alpha + \cos(\alpha + \mu)) + \omega L_s I_{dc}] $$.

    • RMS Output Voltage: Decreases with $\mu$.

• Role of Freewheeling Diode (FWD):

  • Provides path for $$\displaystyle i_o $$ when SCRs are reverse-biased.

  • Improves PF: Reduces reactive power drawn from source by making $$\displaystyle i_s $$ unidirectional and more in phase with $$\displaystyle v_s $$.

  • Improves Load Waveform: Makes $$\displaystyle i_o $$ continuous, reducing ripple.

B. Three-Phase Converters (Fully Controlled Bridge)

• Operation with Continuous & Constant Load Current ($$\displaystyle \alpha = 45^\circ $$ example):

  • Each SCR conducts for $$\displaystyle 120^\circ $$.

  • Line-to-Neutral Voltages: $$\displaystyle v_{an}, v_{bn}, v_{cn} $$.

  • Output Voltage $$\displaystyle v_o $$: Six-pulse waveform. $$\displaystyle v_o = v_{ab} $$ when T1,T2 on; $$\displaystyle v_{bc} $$ when T3,T2 on; etc.

  • Average Output Voltage: $$\displaystyle V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos \alpha = \frac{3\sqrt{2}}{\pi} V_L \cos \alpha $$, where $$\displaystyle V_{LL} $$ = line voltage RMS, $$\displaystyle V_L $$ = phase voltage RMS.

• Analysis with Source Inductance:

  • Overlap Angle $\mu$: Occurs when commutation from one pair to next.

  • Average Output Voltage: $$\displaystyle V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos(\alpha + \frac{\mu}{2}) $$ (approx.).

  • Given $$\displaystyle V_{dc}, I_s, \alpha, V_s $$: Use $$\displaystyle V_{dc} = \frac{3\sqrt{6}}{\pi} V_{LL} \cos(\alpha + \frac{\mu}{2}) $$ and $$\displaystyle \mu = \frac{\omega L_s I_{dc}}{V_m} $$ (where $$\displaystyle V_m = \sqrt{2} V_{LL} $$) to solve for $$\displaystyle L_s $$ and $R$.

• Waveforms: For balanced resistive/inductive loads, $$\displaystyle v_o $$ is six-step, $$\displaystyle i_o $$ is more ripple-free.

III. AC VOLTAGE CONTROLLERS (AC-AC CONVERTERS)

A. Principles

• Phase Control: Thyristors triggered at delay angle $\alpha$ each half-cycle. Output $$\displaystyle V_{rms} $$ controlled by $\alpha$.

• On-Off (Integral Cycle) Control: Whole cycles are switched on/off. Used for high-power heating.

B. Single-Phase AC Voltage Controllers

• With Resistive Load:

  • Half-Wave: Single SCR. $$\displaystyle v_o = v_s $$ for $\omega t \in [\alpha, \pi]$, 0 otherwise.

  • Full-Wave (Anti-Parallel): Two SCRs in parallel opposite. $$\displaystyle v_o = |v_s| $$ for $|\omega t| \in [\alpha, \pi]$.

  • RMS Output Voltage: $$\displaystyle V_{o(rms)} = V_s \sqrt{\frac{1}{\pi} (\pi - \alpha + \frac{1}{2}\sin 2\alpha)} $$.

• With Inductive (RL) Load (Full-Wave):

  • Need for FWD: Without FWD, $$\displaystyle i_o $$ becomes discontinuous and lags $$\displaystyle v_s $$, causing negative $$\displaystyle v_o $$ when SCRs off. FWD provides path, making $$\displaystyle v_o \geq 0 $$.

  • Waveforms: $$\displaystyle v_o $$ follows $$\displaystyle v_s $$ when SCRs on, zero when FWD conducts. $$\displaystyle i_o $$ continuous, lags $$\displaystyle v_o $$.

  • Firing Angle Limit: $$\displaystyle \alpha > \phi = \tan^{-1}(\omega L/R) $$ for continuous $$\displaystyle i_o $$.

• Two-Stage Sequence Control (for RL Load):

  • Purpose: Improve PF at low output power.

  • Operation: Two sets of thyristors (T1,T2 and T3,T4). First stage (T1,T2) triggered at $$\displaystyle \alpha_1 $$, second stage (T3,T4) triggered at $$\displaystyle \alpha_2 > \alpha_1 $$.

  • Waveforms: $$\displaystyle v_o $$ consists of two pulses per half-cycle. By controlling $$\displaystyle \alpha_1 $$ and $$\displaystyle \alpha_2 $$, output power and PF are optimized.

C. Performance Calculations

• Firing Angle $\alpha$ for Given Power $P$ & Resistance $R$ (R Load):

$$P = \frac{V_{o(rms)}^2}{R} = \frac{V_s^2}{R} \left[ \frac{1}{\pi} (\pi - \alpha + \frac{1}{2}\sin 2\alpha) \right]$$

Solve for $\alpha$.

• Input Power Factor: $$\displaystyle PF = \frac{P}{V_s I_{s(rms)}} $$. For R load, $$\displaystyle I_{s(rms)} = I_{o(rms)} $$.

• Example (Half-Wave RL): Given $$\displaystyle R, L, V_s, \alpha $$, find $$\displaystyle V_{o(rms)} $$, PF, $$\displaystyle I_{avg} $$.

  1. Find $$\displaystyle \phi = \tan^{-1}(\omega L/R) $$.

  2. Check if $$\displaystyle \alpha > \phi $$ for discontinuous $$\displaystyle i_o $$.

  3. Use integration over conduction interval to find $$\displaystyle V_{o(rms)} $$ and $$\displaystyle I_{s(rms)} $$.

IV. DC CHOPPERS (DC-DC CONVERTERS)

A. Classification & Basic Topologies

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

• Circuit & Operation: Switch S (MOSFET/SCR) in series with load ($R,L,E$). Diode D across load for freewheeling.

  • Ton (S ON): $$\displaystyle v_o = V_s $$, $$\displaystyle i_o $$ increases linearly (if $L$ large).

  • Toff (S OFF): $$\displaystyle v_o = 0 $$, $$\displaystyle i_o $$ freewheels through D, decreases linearly.

• Waveforms: $$\displaystyle v_o $$ = rectangular (0 or $$\displaystyle V_s $$), $$\displaystyle i_o $$ = triangular (continuous) or pulsed (discontinuous), $$\displaystyle i_s $$ = rectangular (only during Ton).

• Average Output Voltage: $$\displaystyle V_{dc} = \alpha V_s $$, where $$\displaystyle \alpha = T_{on}/T $$ (duty cycle).

• Problem Solving (Given $$\displaystyle V_s, T_{on}, T_{off}, R, L, E $$):

  1. Check Continuous/Discontinuous Conduction:

     • Calculate $$\displaystyle I_{min} $$ (at end of Toff). For continuous, $$\displaystyle I_{min} \geq 0 $$.

     • During Toff: $$\displaystyle i_o(t) = I_{max} - \frac{(0 - E)}{L}t $$? Actually, during Toff, $$\displaystyle v_L = -E $$? Wait, careful:

       • Ton: $$\displaystyle v_L = V_s - E - i_o R $$, $$\displaystyle di_o/dt = (V_s - E - i_o R)/L $$.

       • Toff: $$\displaystyle v_L = -E - i_o R $$, $$\displaystyle di_o/dt = (-E - i_o R)/L $$.

     • For steady-state, $$\displaystyle I_{min} $$ at end of Toff, $$\displaystyle I_{max} $$ at end of Ton.

     • Condition for CCM: $$\displaystyle I_{min} > 0 $$. Solve using average $$\displaystyle I_{dc} = V_{dc}/R $$ (if $$\displaystyle E=0 $$) or from energy balance.

  2. Average Output Current: $$\displaystyle I_{avg} = \frac{V_{dc} - E}{R} $$ (if CCM). For DCM, more complex.

  3. Imax & Imin:

     • CCM: $$\displaystyle I_{max} = I_{avg} + \frac{\Delta i}{2} $$, $$\displaystyle I_{min} = I_{avg} - \frac{\Delta i}{2} $$.

     • $$\displaystyle \Delta i = \frac{(V_s - E) T_{on} - E T_{off}}{L} $$? Actually, from volt-sec balance:

       • During Ton: $$\displaystyle \Delta i_1 = \frac{(V_s - E - I_{avg}R) T_{on}}{L} \approx \frac{(V_s - E) T_{on}}{L} $$ (if $$\displaystyle I_{avg}R $$ small).

       • During Toff: $$\displaystyle \Delta i_2 = \frac{(-E - I_{avg}R) T_{off}}{L} \approx \frac{-E T_{off}}{L} $$.

       • Net $$\displaystyle \Delta i = 0 $$ in steady state: $$\displaystyle \Delta i_1 + \Delta i_2 = 0 \Rightarrow \frac{(V_s - E)T_{on}}{L} = \frac{E T_{off}}{L} \Rightarrow V_s T_{on} = E T $$? That's for $E \neq 0$? Actually, volt-sec across $L$ must be zero: $$\displaystyle (V_s - E)T_{on} + (-E)T_{off} = 0 \Rightarrow V_s T_{on} = E T $$. But that gives $$\displaystyle V_{dc} = \alpha V_s = E $$, which is only when $$\displaystyle I_{avg}=0 $$? Wait, I'm confusing.

     • Correct Approach: For CCM, $$\displaystyle I_{avg} = \frac{V_{dc} - E}{R} = \frac{\alpha V_s - E}{R} $$.

     • $$\displaystyle \Delta i = \frac{(V_s - E) T_{on}}{L} - \frac{E T_{off}}{L} $$? No, during Toff, $$\displaystyle v_L = -E $$ only if diode drop negligible? Actually, during Toff, $$\displaystyle v_o = 0 $$, so $$\displaystyle v_L = -E - i_o R $$? But for $\Delta i$ calculation, we often neglect $iR$ drop for ripple calc.

     • Simplified Ripple: $$\displaystyle \Delta i = \frac{(V_s - E) T_{on}}{L} $$ (if $E$ constant) and during Toff, $$\displaystyle di/dt = -E/L $$, so $$\displaystyle \Delta i_{Toff} = -\frac{E T_{off}}{L} $$. Net zero: $$\displaystyle \frac{(V_s - E)T_{on}}{L} = \frac{E T_{off}}{L} \Rightarrow V_s T_{on} = E T \Rightarrow \alpha = E/V_s $$. That's only at boundary where $$\displaystyle I_{min}=0 $$.

     • General CCM: $$\displaystyle I_{max} = I_{avg} + \frac{(V_s - E - I_{avg}R)T_{on}}{2L} $$? Actually, $$\displaystyle i_o $$ is linear if $R$ small. Exact: $$\displaystyle i_{max} = I_{avg} + \frac{\Delta i}{2} $$, $$\displaystyle i_{min} = I_{avg} - \frac{\Delta i}{2} $$, with $$\displaystyle \Delta i = \frac{(V_s - E - I_{avg}R)T_{on}}{L} = \frac{(E + I_{avg}R)T_{off}}{L} $$.

     • From given numbers: Calculate $$\displaystyle I_{avg} $$ first assuming CCM, then check $$\displaystyle I_{min} > 0 $$.

C. Step-Up (Boost) Chopper (Type-B)

• Circuit & Operation: Switch S across source. Inductor $L$ in series with source, diode D to load.

  • Ton (S ON): $$\displaystyle v_L = V_s $$, $$\displaystyle i_L $$ increases, energy stored in $L$. Load supplied by capacitor $C$.

  • Toff (S OFF): $$\displaystyle v_L = V_s - v_o $$, $$\displaystyle i_L $$ decreases, transfers energy to load via D.

• Waveforms: $$\displaystyle v_o $$ = DC with ripple, $$\displaystyle i_s $$ (through S) = rectangular, $$\displaystyle i_L $$ = triangular.

• Output Voltage Derivation:

  • Volt-sec balance on $L$: $$\displaystyle V_s T_{on} + (V_s - V_o) T_{off} = 0 $$.

  • $$\displaystyle \Rightarrow V_s T_{on} = (V_o - V_s) T_{off} $$.

  • $$\displaystyle \Rightarrow V_o = V_s \left(1 + \frac{T_{on}}{T_{off}}\right) = \frac{V_s}{1 - \alpha} $$.

D. Buck-Boost Chopper

• Principle: Output voltage polarity opposite to input. $$\displaystyle V_o = \frac{\alpha}{1-\alpha} V_s $$. Can be > or < $$\displaystyle V_s $$ depending on $\alpha$.

E. Special Chopper Circuits

• Morgan Chopper: Uses two SCRs and a commutating capacitor. Forced commutation by transferring capacitor charge. Used for motor control.

• Current Limit Control: Switch ON until $$\displaystyle i_o $$ reaches $$\displaystyle I_{max} $$, then OFF until $$\displaystyle i_o $$ falls to $$\displaystyle I_{min} $$. Controls average $$\displaystyle i_o $$ and limits ripple.

F. Type-C (Reversible) Chopper

• Circuit: Two switches (S1,S2) and two diodes (D1,D2) in bridge-like form across load.

• Operation:

  • Motoring (Positive $$\displaystyle V_o $$): S1 ON → $$\displaystyle v_o = V_s $$, $$\displaystyle i_o $$ positive. S1 OFF, D2 conducts for freewheeling.

  • Regeneration (Negative $$\displaystyle V_o $$): S2 ON → $$\displaystyle v_o = -V_s $$, $$\displaystyle i_o $$ negative (flows from load to source via D1).

• Waveforms: $$\displaystyle v_o $$ = $$\displaystyle \pm V_s $$ depending on which switch is ON.

V. INVERTERS (DC-AC CONVERTERS)

A. Classification

• VSI vs CSI: VSI has constant DC voltage source, CSI has constant DC current source (large inductor).

• Conduction Modes: 180° (each device conducts 180°), 120° (each device conducts 120°).

• Control: Square-wave (fixed $$\displaystyle \alpha=180^\circ $$), PWM (vary pulse width).

B. Single-Phase Inverters

• Bridge Inverter (Full-Bridge, 180° Conduction):

  • Switches: T1,T4 ON → $$\displaystyle v_o = V_s $$; T2,T3 ON → $$\displaystyle v_o = -V_s $$.

  • Resistive Load: $$\displaystyle i_o $$ in phase with $$\displaystyle v_o $$, sinusoidal.

  • Inductive Load (Continuous $$\displaystyle i_o $$): $$\displaystyle i_o $$ quasi-square wave (approximates sine), $$\displaystyle v_o $$ square wave. $$\displaystyle i_o $$ lags $$\displaystyle v_o $$ by $\phi$.

• PWM Inverter:

  • Circuit: Full-bridge with PWM control on all switches.

  • Principle: Compare sine reference ($$\displaystyle f_o $$) with high-frequency triangle carrier. Unipolar: one leg switches at carrier, other at reference. Bipolar: both legs switch together.

  • Advantages: Harmonic spectrum shifted to high frequencies (around $$\displaystyle f_c $$), easier filtering, fundamental amplitude control via modulation index $$\displaystyle m_a $$.

C. Three-Phase Inverters

• Bridge Configuration (180° Conduction):

  • Switching Sequence: T1(T4) → T2(T5) → T3(T6) → T1(T4)... Each conducts 180°, always one from top and one from bottom leg.

  • Line-to-Line Voltages: $$\displaystyle v_{ab} = v_{an} - v_{bn} $$ = square wave with $$\displaystyle 120^\circ $$ flat top, amplitude $$\displaystyle \frac{3}{2}V_s $$? Actually, for star load, $$\displaystyle v_{an} $$ = square wave, $$\displaystyle v_{ab} $$ = stepped waveform (6-level).

  • Waveforms for Star-Resistive Load: $$\displaystyle v_{an}, v_{bn}, v_{cn} $$ are $$\displaystyle 120^\circ $$ displaced square waves. $$\displaystyle i_{an} = v_{an}/R_{ph} $$.

  • Calculations:

    • RMS Phase Current: $$\displaystyle I_{ph(rms)} = \frac{V_s}{R_{ph}} \cdot \frac{1}{\sqrt{3}} $$? Actually, for 180° mode, $$\displaystyle V_{ph(rms)} = \frac{V_s}{\sqrt{2}} $$? Wait:

      • $$\displaystyle v_{an} $$ is square wave of amplitude $$\displaystyle V_s $$, so $$\displaystyle V_{an(rms)} = V_s $$.

      • But for star load, $$\displaystyle v_{ab} = \sqrt{3} V_{ph} \angle 30^\circ $$, and $$\displaystyle v_{ab} $$ has RMS = $$\displaystyle \sqrt{\frac{2}{3}} V_s $$? Let's derive:

      • In 180° mode, each phase voltage is a square wave of amplitude $$\displaystyle V_s $$, but only 120° of the cycle is $$\displaystyle +V_s $$, 120° is $$\displaystyle -V_s $$, 120° is 0? No, in 180° conduction, each device conducts 180°, so each phase terminal is connected to $$\displaystyle +V_s $$ for 120°, to $$\displaystyle -V_s $$ for 120°, and floating (0) for 120°? Actually, with 180° conduction, at any instant two devices conduct (one top, one bottom), so three phase voltages are:

        • T1,T6: $$\displaystyle v_a = V_s $$, $$\displaystyle v_b = 0 $$, $$\displaystyle v_c = -V_s $$? Wait, need switching table.

      • Standard Result: For 180° mode, phase voltage $$\displaystyle V_{ph(rms)} = \frac{V_s}{\sqrt{2}} $$? No, for square wave of amplitude $$\displaystyle V_s $$ and duty 50%, RMS = $$\displaystyle V_s $$. But here, each phase voltage is not a 50% square wave; it's a 120° pulse of $$\displaystyle +V_s $$, 120° pulse of $$\displaystyle -V_s $$, 120° zero. So RMS = $$\displaystyle V_s \sqrt{\frac{120}{360}} = V_s / \sqrt{3} $$.

      • Actually, correct: In 180° conduction, each phase terminal is connected to either $$\displaystyle +V_s $$ or $$\displaystyle -V_s $$ for 120° each, and isolated for 120°. So waveform: $$\displaystyle +V_s $$ for $$\displaystyle 120^\circ $$, $$\displaystyle -V_s $$ for $$\displaystyle 120^\circ $$, 0 for $$\displaystyle 120^\circ $$. RMS = $$\displaystyle \sqrt{\frac{1}{2\pi} \left( \int_0^{2\pi/3} V_s^2 d\omega t + \int_{2\pi/3}^{4\pi/3} (-V_s)^2 d\omega t \right)} = V_s \sqrt{\frac{1}{2\pi} \cdot \frac{4\pi}{3}} = V_s \sqrt{\frac{2}{3}} $$.

      • So $$\displaystyle V_{ph(rms)} = \frac{\sqrt{6}}{3} V_s = \frac{V_s}{\sqrt{1.5}} $$.

    • Total Power: $$\displaystyle P = 3 \frac{V_{ph(rms)}^2}{R_{ph}} = 3 \cdot \frac{2}{3} \frac{V_s^2}{R_{ph}} = 2 \frac{V_s^2}{R_{ph}} $$? That seems high. Let's check standard formula: For three-phase inverter with DC source $$\displaystyle V_s $$, 180° mode, star load $$\displaystyle R_{ph} $$, output line-to-line RMS voltage $$\displaystyle V_{LL(rms)} = \frac{\sqrt{6}}{3} V_s $$? Actually, common formula: $$\displaystyle V_{LL(rms)} = \frac{\sqrt{6}}{3} V_s \approx 0.816 V_s $$. Then phase voltage $$\displaystyle V_{ph} = V_{LL}/\sqrt{3} = \frac{\sqrt{2}}{3} V_s \approx 0.471 V_s $$. So $$\displaystyle V_{ph(rms)} = \frac{V_s}{\sqrt{6}} $$? Wait, $$\displaystyle \frac{\sqrt{2}}{3} = \frac{1}{\sqrt{4.5}} $$. I'm messing up.

    • Standard Result: For 180° mode, fundamental component of phase voltage $$\displaystyle V_{ph1} = \frac{4V_s}{\pi\sqrt{2}} \sin(\omega t) $$? That's for square wave. Actually, the Fourier series of the phase voltage (120° pulse) has fundamental amplitude $$\displaystyle V_{ph1} = \frac{4V_s}{3\pi} \cdot \frac{\sqrt{3}}{2} $$? Let's not overcomplicate. For exam, remember:

      • Line-to-Line RMS Voltage: $$\displaystyle V_{LL(rms)} = \frac{\sqrt{6}}{3} V_s \approx 0.816 V_s $$.

      • Phase RMS Voltage (star): $$\displaystyle V_{ph(rms)} = \frac{V_{LL(rms)}}{\sqrt{3}} = \frac{V_s}{\sqrt{6}} \approx 0.408 V_s $$.

      • Total Power: $$\displaystyle P = 3 \frac{V_{ph(rms)}^2}{R_{ph}} = \frac{V_s^2}{2 R_{ph}} $$? Actually, $$\displaystyle 3 \cdot (V_s^2 / 6) / R_{ph} = V_s^2/(2 R_{ph}) $$. Yes.

• 120° Conduction Mode:

  • Only two devices conduct at any time (one top, one bottom).

  • Each device conducts $$\displaystyle 120^\circ $$.

  • Waveforms: Phase voltages are not independent; each is a flat-topped waveform of width $$\displaystyle 120^\circ $$.

  • Performance: Lower RMS current per device (since conduction period less), but output voltage has more harmonics.

D. Special Inverters

• Current Source Inverter (CSI):

  • Working: Constant DC current $$\displaystyle I_s $$ from large inductor. Uses thyristors (or GTOs) with forced commutation.

  • Circuit: Inductor $$\displaystyle L_d $$ in series with DC source, 6 SCRs in bridge.

  • Commutation: Requires external circuitry (e.g., capacitor-inductor network) to turn off SCRs.

  • Output: Current waveshape determined by load (usually inductive), voltage waveshape determined by switching.

• Series Resonant Inverter:

  • Operation at Resonance: $L$ and $C$ in series with load $R$. At $$\displaystyle \omega_0 = 1/\sqrt{LC} $$, load current sinusoidal, switches turn on/off at zero current (ZCS).

  • Output Current: $$\displaystyle I_o = \frac{V_s}{R} $$ at resonance (for series RLC).

• McMurray-Bedford Inverter:

  • Principle: Uses auxiliary commutating circuit (capacitor and inductor) to turn off main thyristors. Commutation is load-independent.

E. Harmonics & Reduction

• Sources: Non-sinusoidal switching (square waves) produce harmonics at $$\displaystyle nf_o $$ (odd harmonics dominant).

• Reduction Techniques:

  1. PWM: Shifts harmonics to high frequencies (around carrier $$\displaystyle f_c $$).

  2. Multi-pulse (e.g., 12-pulse): Uses phase-shifting transformers to cancel lower harmonics.

  3. Selective Harmonic Elimination (SHE): Chooses switching angles to eliminate specific harmonics.

  4. Passive Filters: LC tuned to harmonic frequencies.

VI. CYCLOCONVERTERS (AC-AC CONVERTERS)

A. Principle of Operation

• Direct AC-AC conversion without intermediate DC link.

• Output frequency $$\displaystyle f_o < f_i $$ (step-down) typically. Step-up possible but rare.

• Uses phase-controlled rectifiers in reverse (as inverters) during negative half-cycle.

B. Single-Phase to Single-Phase Cycloconverters

• Mid-Point Configuration:

  • Circuit: Two single-phase full-wave converters (positive and negative) sharing a common load with center-tapped transformer.

  • Operation (Step-Down, $$\displaystyle f_o = f_i/2 $$):

    • Positive Converter: Active when $$\displaystyle v_o > 0 $$, triggered during positive half-cycles of input.

    • Negative Converter: Active when $$\displaystyle v_o < 0 $$, triggered during negative half-cycles of input.

  • Waveforms: For resistive load, output is rectified sine waves from each converter, forming a lower frequency sine-like wave.

• Bridge Configuration:

  • Circuit: Two bridge converters (positive and negative) in parallel across load.

  • Operation: Similar to mid-point but without center tap. Each bridge operates as a full-wave converter.

• Step-Up Cycloconverter: Possible but requires complex control and has poor performance; rarely used.

C. Three-Phase to Single-Phase Cycloconverter

• Circuit: Three-phase supply to three single-phase converters (one per phase), outputs connected in parallel to single-phase load.

• Working: Each phase converter is controlled independently to synthesize a single-phase output. More complex control (firing pulses from a master oscillator).

• Applications: High-power, low-speed AC motor drives (e.g., cement mills, ship propulsion).

• Limitations: Low output frequency (typically < 1/3 input frequency), poor PF, high harmonic content.

VII. SWITCHED-MODE POWER SUPPLIES (SMPS) & SPECIAL TOPICS

A. Switched-Mode Power Supply (SMPS)

• Principle vs. Linear Supply: Uses switching regulator (transistor operates in cutoff/saturation) → high efficiency (70-90%), smaller size, lighter weight. Linear supply uses transistor in active region → low efficiency, large heat sink.

• Basic Topologies: Buck, Boost, Buck-Boost (as choppers).

• Fly-back SMPS:

  • Circuit: Transformer with primary switch (MOSFET), secondary diode to output filter ($L,C$). Energy stored in core during Ton, released during Toff.

  • Equivalent Circuits:

    • DCM (Discontinuous Conduction Mode): $$\displaystyle i_m $$ (magnetizing current) reaches zero before next cycle.

    • CCM (Continuous Conduction Mode): $$\displaystyle i_m $$ never zero.

  • Waveforms: $$\displaystyle v_{DS} $$ has spike from leakage inductance, $$\displaystyle i_m $$ triangular, $$\displaystyle v_o $$ DC with ripple.

  • Operation: Isolation and multiple outputs possible.

B. Resonant Converters (Brief)

• Use $L$ and $C$ to create resonance.

• Soft Switching: Devices turn on/off at zero voltage (ZVS) or zero current (ZCS) → reduce switching losses.

• Types: Series resonant inverter (load in series with $L,C$), parallel resonant.

C. Quality & Performance Metrics

• Efficiency: $$\displaystyle \eta = \frac{P_{out}}{P_{in}} \times 100\% $$.

• Power Factor (PF): $$\displaystyle \frac{P_{in}}{V_s I_{s(rms)}} $$. Displacement PF (phase shift) + Distortion PF (harmonics).

• Total Harmonic Distortion (THD): $$\displaystyle THD = \frac{\sqrt{I_2^2 + I_3^2 + ...}}{I_1} \times 100\% $$. Lower THD → better power quality.

> [!TIP] EXAM STRATEGY

• For Numerical Problems: Always draw waveforms first. Identify conduction intervals. Use volt-sec balance for inductors, charge balance for capacitors.

• For Theory: Link device characteristics to application (e.g., MOSFET → high-frequency chopper, IGBT → medium-power inverter).

• Common Pitfalls:

  • Confusing 180° vs 120° conduction in inverters.

  • Forgetting freewheeling diode effect in RL loads of rectifiers/controllers.

  • Mixing up average vs RMS formulas for phase-controlled converters.

  • Overlooking continuous/discontinuous conduction condition in choppers.

• Past Paper Focus: Be prepared for derivations of $$\displaystyle V_{dc} $$ for single-phase full converter with RLE, three-phase converter with overlap, AC controller RMS voltage, chopper $$\displaystyle V_{dc} $$ and ripple. Waveforms for single-phase full bridge inverter with RL load and three-phase inverter 120° mode are frequently asked.