Definition: Circuits that convert AC voltage to DC voltage

Purpose: Power supply for DC loads, motor drives, battery charging

Types:

• Uncontrolled (Diode-based)

• Controlled (Thyristor-based)

• Bidirectional

Key Parameters: Ripple factor, efficiency, regulation, THD

Classification by Phases: Single-phase, Three-phase, Polyphase

Classification by Connection: Half-wave, Full-wave, Bridge

Circuit: Single diode with AC source and load

Output Voltage: \(V_o = \dfrac{V_m}{\pi}\) (average)

RMS Output: \(V_{rms} = \dfrac{V_m}{2}\)

Peak Inverse Voltage (PIV): \(V_m\)

Ripple Factor: \(r = 1.21\)

Efficiency: \(\eta = 40.6\%\)

Form Factor: \(FF = 1.57\)

Transformer Utilization Factor: \(TUF = 0.287\)

Disadvantages: High ripple, poor transformer utilization, DC saturation

Circuit: Two diodes with center-tapped transformer

Output Voltage: \(V_o = \dfrac{2V_m}{\pi}\) (average)

RMS Output: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\)

PIV: \(2V_m\)

Ripple Factor: \(r = 0.48\)

Efficiency: \(\eta = 81.2\%\)

Form Factor: \(FF = 1.11\)

TUF: \(0.693\)

Ripple Frequency: \(2f\) (twice input frequency)

Circuit: Four diodes arranged in bridge configuration

Output Voltage: \(V_o = \dfrac{2V_m}{\pi}\) (average)

RMS Output: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\)

PIV: \(V_m\) (lower than center-tap)

Ripple Factor: \(r = 0.48\)

Efficiency: \(\eta = 81.2\%\)

TUF: \(0.81\)

Advantages: No center-tapped transformer required

Disadvantage: Two diode drops in conduction path

Circuit: Three diodes connected to three-phase supply

Output Voltage: \(V_o = \dfrac{3\sqrt{3}V_m}{2\pi}\) (average)

RMS Output: \(V_{rms} = \sqrt{\dfrac{3}{2\pi}}\left(\dfrac{3\sqrt{3}V_m}{2}\right)\)

Ripple Factor: \(r = 0.17\)

Ripple Frequency: \(3f\)

Conduction Angle: \(120^{\circ}\) per diode

Efficiency: \(\eta = 96.8\%\)

Better performance than single-phase rectifiers

Circuit: Six diodes in bridge configuration

Output Voltage: \(V_o = \dfrac{3\sqrt{3}V_m}{\pi}\) (average)

RMS Output: \(V_{rms} = \sqrt{\dfrac{3}{2\pi}}\left(3\sqrt{3}V_m\right)\)

Ripple Factor: \(r = 0.04\)

Ripple Frequency: \(6f\)

Conduction Angle: \(120^{\circ}\) per diode

Efficiency: \(\eta = 99.5\%\)

Advantages: Low ripple, high efficiency, no transformer center-tap

Input Current Waveform: Discontinuous, rich in harmonics

Harmonic Order: \(n = 2, 3, 4, 5, 6, ...\) (all harmonics present)

Magnitude: \(I_n = \dfrac{4I_1}{n\pi}\) for \(n = 3, 5, 7, ...\)

Phase: All harmonics are in phase with fundamental

THD: \(THD_i = \sqrt{\dfrac{I_{rms}^2 - I_1^2}{I_1^2}} = 48.4\%\) (bridge)

Dominant Harmonics: 3rd, 5th, 7th order

Power Factor: \(PF = 0.9\) (for resistive load)

Displacement Factor: \(\cos\phi_1 = 1.0\) (resistive load)

Characteristic Harmonics: \(n = 6k \pm 1\) (where \(k = 1, 2, 3, ...\))

Harmonic Order: \(n = 5, 7, 11, 13, 17, 19, ...\) (no triplen harmonics)

Magnitude: \(I_n = \dfrac{I_1}{n}\) for characteristic harmonics

Phase Sequence:

• \(n = 6k + 1\): Positive sequence (5th, 11th, 17th)

• \(n = 6k - 1\): Negative sequence (7th, 13th, 19th)

THD: \(THD_i = 31.1\%\) (six-pulse bridge)

Power Factor: \(PF = 0.955\) (for resistive load)

No Even Harmonics: Due to half-wave symmetry

Circuit: Single SCR with firing angle control

Output Voltage: \(V_o = \dfrac{V_m}{2\pi}(1 + \cos\alpha)\)

RMS Output: \(V_{rms} = \dfrac{V_m}{2}\sqrt{\dfrac{1}{2\pi}(\pi - \alpha + \dfrac{\sin 2\alpha}{2})}\)

Firing Angle: \(\alpha\) (\(0^{\circ}\) to \(180^{\circ}\))

Conduction Angle: \((180^{\circ} - \alpha)\)

Control: Output voltage controllable by firing angle

Disadvantage: DC component in transformer

Circuit: Four SCRs or two SCRs + two diodes

Output Voltage: \(V_o = \dfrac{2V_m}{\pi}\cos\alpha\)

RMS Output: \(V_{rms} = V_m\sqrt{\dfrac{1}{2\pi}(\pi - \alpha + \dfrac{\sin 2\alpha}{2})}\)

Firing Angle Range: \(0^{\circ}\) to \(180^{\circ}\)

Continuous Conduction: When \(\alpha < 60^{\circ}\)

Discontinuous Conduction: When \(\alpha > 60^{\circ}\)

Extinction Angle: \(\beta = 180^{\circ} - \alpha\) (for resistive load)

Applications: DC motor drives, battery charging

Circuit: Three SCRs with \(120^{\circ}\) phase difference

Output Voltage: \(V_o = \dfrac{3\sqrt{3}V_m}{2\pi}\cos\alpha\)

Firing Angle Range: \(0^{\circ}\) to \(180^{\circ}\)

Natural Commutation: Voltage commutated

Continuous Conduction: When \(\alpha < 60^{\circ}\)

Discontinuous Conduction: When \(\alpha > 60^{\circ}\)

Conduction Angle: \(120^{\circ}\) per SCR

Better ripple performance than single-phase

Circuit: Six SCRs in bridge configuration

Output Voltage: \(V_o = \dfrac{3\sqrt{3}V_m}{\pi}\cos\alpha\)

Firing Angle Range: \(0^{\circ}\) to \(180^{\circ}\)

Continuous Conduction: When \(\alpha < 60^{\circ}\)

Overlap Angle: Due to source inductance (\(\mu\))

Modified Equation: \(V_o = \dfrac{3\sqrt{3}V_m}{\pi}\cos\alpha - \dfrac{3\omega L_s I_o}{\pi}\)

Overlap Relation: \(\cos(\alpha + \mu) = \cos\alpha - \dfrac{2\omega L_s I_o}{V_m}\)

Input Current: Depends on firing angle \(\alpha\) and load type

Harmonic Magnitude: \(I_n = \dfrac{4I_1}{n\pi}\cos(n\alpha)\) for odd harmonics

Phase Shift: Each harmonic shifted by \(n\alpha\)

THD Variation: \(THD_i\) increases with firing angle

Power Factor: \(PF = \dfrac{P_{input}}{V_1 I_1} = \dfrac{\cos\phi_1}{\sqrt{1 + THD_i^2}}\)

Displacement Factor: \(\cos\phi_1\) depends on load impedance angle

Distortion Factor: \(DF = \dfrac{1}{\sqrt{1 + THD_i^2}}\)

Characteristic Harmonics: \(n = 6k \pm 1\) (same as uncontrolled)

Harmonic Magnitude: \(I_n = \dfrac{I_1}{n}\cos(n\alpha)\)

Phase Shift: \(n\alpha\) for each harmonic order

Dominant Harmonics: 5th and 7th order (lowest frequency)

THD at Different Firing Angles:

• \(\alpha = 0^{\circ}\): \(THD_i = 31.1\%\)

• \(\alpha = 30^{\circ}\): \(THD_i = 28.9\%\)

• \(\alpha = 60^{\circ}\): \(THD_i = 66.8\%\)

Minimum THD: Around \(\alpha = 30^{\circ}\)

Input Power Factor: \(PF = \dfrac{P_{input}}{V_1 I_1}\)

Displacement Factor: \(\cos\phi_1 = \dfrac{I_{1,active}}{I_1}\)

Distortion Factor: \(DF = \dfrac{I_1}{I_{rms}} = \dfrac{1}{\sqrt{1 + THD_i^2}}\)

Relationship: \(PF = \cos\phi_1 \times DF\)

For Resistive Load: \(\cos\phi_1 = 1\), so \(PF = DF\)

For Inductive Load: Both displacement and distortion affect PF

Typical Values:

• Single-phase bridge: \(PF = 0.9\) (uncontrolled)

• Three-phase bridge: \(PF = 0.955\) (uncontrolled)

Single-Phase Controlled Bridge:

• \(PF = \dfrac{2\sqrt{2}}{\pi}\sqrt{\dfrac{\pi - \alpha + \sin(2\alpha)/2}{\pi}}\cos\phi_1\)

• Maximum at \(\alpha = 0^{\circ}\)

• Decreases with increasing \(\alpha\)

Three-Phase Controlled Bridge:

• \(PF = \dfrac{3\sqrt{3}}{2\pi}\sqrt{\dfrac{\pi - \alpha + \sin(2\alpha)/2}{\pi}}\cos\phi_1\)

• Better power factor than single-phase

• Minimum around \(\alpha = 30^{\circ}\) for THD, not PF

Power Factor Correction: Use passive or active filters

Principle: Current commutation using load/circuit inductance

Commutation Methods:

• Load commutation

• Complementary commutation

• External pulse commutation

Applications: High power applications, motor drives

Advantages: Inherent short-circuit protection

Disadvantages: Complex commutation circuits

Class A: Self-commutated with L-C circuit

Class B: Self-commutated with C-R circuit

Class C: Complementary commutation

Class D: Auxiliary commutation

Class E: External pulse commutation

Turn-off Time: \(t_q\) = Circuit turn-off time

Design Criteria: \(t_q >\) SCR turn-off time

Commutation Capacitor: \(C = \dfrac{I_o t_q}{V_c}\)

Rectifier Mode: AC to DC power conversion (\(P > 0\))

Inverter Mode: DC to AC power conversion (\(P < 0\))

Four-Quadrant Operation:

• Quadrant I: Forward motoring (\(+V_o, +I_o\))

• Quadrant II: Forward braking (\(+V_o, -I_o\))

• Quadrant III: Reverse motoring (\(-V_o, -I_o\))

• Quadrant IV: Reverse braking (\(-V_o, +I_o\))

Inversion Condition: \(\alpha > 90^{\circ}\) for inverter operation

Commutation Margin: \(\gamma = 180^{\circ} - (\alpha + \mu) > 15^{\circ}\)

Structure: Two controlled rectifiers in anti-parallel

Operation Modes:

• Non-circulating current mode (one converter ON)

• Circulating current mode (both converters ON)

Control Constraint: \(\alpha_1 + \alpha_2 = 180^{\circ}\)

Circulating Current Reactor: \(L_{cr} = \dfrac{2V_m}{\pi I_{cr,max}}\)

Advantages: Four-quadrant operation, regenerative braking

Disadvantages: Complex control, circulating current losses

Principle: High-frequency switching for bidirectional power flow

Advantages:

• Unity power factor operation

• Reduced harmonics (THD \(< 5\%\))

• Sinusoidal input current

• Bidirectional power flow

• Constant DC bus voltage

Control Methods:

• Voltage-oriented control (VOC)

• Direct power control (DPC)

• Model predictive control (MPC)

Applications: Active front-end rectifiers, grid-tie systems

Inductor Filter:

• Ripple factor: \(r = \dfrac{1}{12\sqrt{3}}\left(\dfrac{R}{\omega L}\right)\) (for 3-phase)

• Critical inductance: \(L_c = \dfrac{R}{2\pi f}\)

Capacitor Filter:

• Ripple factor: \(r = \dfrac{1}{4\sqrt{3}fRC}\) (for full-wave)

• Peak-to-peak ripple: \(V_{pp} = \dfrac{I_o}{fC}\)

L-C Filter:

• Resonant frequency: \(f_r = \dfrac{1}{2\pi\sqrt{LC}}\)

• Design: \(f_r < f_{ripple}\)

Ripple Factor: \(r = \dfrac{V_{ac,rms}}{V_{dc}} = \sqrt{\left(\dfrac{V_{rms}}{V_{dc}}\right)^2 - 1}\)

Form Factor: \(FF = \dfrac{V_{rms}}{V_{avg}}\)

Peak Factor: \(PF = \dfrac{V_{peak}}{V_{rms}}\)

Transformer Utilization Factor: \(TUF = \dfrac{P_{dc}}{VA_{transformer}}\)

Efficiency: \(\eta = \dfrac{P_{load}}{P_{input}}\)

Input Power Factor: \(PF = \dfrac{P_{input}}{V_1 I_1}\)

Voltage Regulation: \(\%VR = \dfrac{V_{nl} - V_{fl}}{V_{fl}} \times 100\)

Total Harmonic Distortion: \(THD = \sqrt{\dfrac{I_{rms}^2 - I_1^2}{I_1^2}}\)

Displacement Factor: \(\cos\phi_1 = \dfrac{I_{1,active}}{I_1}\)

Distortion Factor: \(DF = \dfrac{I_1}{I_{rms}} = \dfrac{1}{\sqrt{1 + THD^2}}\)

Characteristic Harmonics:

• Single-phase: All harmonics present

• Three-phase: \(n = 6k \pm 1\) (5th, 7th, 11th, 13th...)

IEEE Standards: THD \(< 5\%\) for grid-connected systems

Single-phase bridge: \(V_o = \dfrac{2V_m}{\pi}\cos\alpha\)

Three-phase bridge: \(V_o = \dfrac{3\sqrt{3}V_m}{\pi}\cos\alpha\)

Overlap angle: \(\cos(\alpha + \mu) = \cos\alpha - \dfrac{2\omega L_s I_o}{V_m}\)

Conduction angle: \(\gamma = 180^{\circ} - \alpha\) (for resistive load)

PIV calculations: Maximum reverse voltage across device

Power factor: \(PF = \cos\phi_1 \times DF\)

Harmonic magnitude: \(I_n = \dfrac{I_1}{n}\) (for three-phase)

Calculate average output voltage for given firing angle

Determine ripple factor and efficiency

Find PIV ratings of switching devices

Analyze continuous vs discontinuous conduction

Calculate overlap angle with source inductance

Determine harmonic content and THD

Four-quadrant operation analysis

Power factor and distortion factor calculations

Bidirectional power flow analysis

Commutation time calculations

Filter design for harmonic reduction

Harmonics: Three-phase has \(n = 6k \pm 1\), single-phase has all

Power Factor: \(PF = \cos\phi_1 \times \dfrac{1}{\sqrt{1 + THD^2}}\)

Bidirectional: \(\alpha > 90^{\circ}\) for inverter mode

Commutation: Voltage (natural) vs Current (forced)

Overlap: Always consider source inductance in practical circuits

THD: Minimum around \(\alpha = 30^{\circ}\) for three-phase

Dual Converter: \(\alpha_1 + \alpha_2 = 180^{\circ}\)

PWM Rectifiers: Unity PF, low THD, bidirectional

Always check units in your calculations

Harmonic analysis: Use RMS values for power calculations

Power factor: Separate displacement and distortion components

Three-phase: Remember \(V_L = \sqrt{3}V_{ph}\), \(I_L = I_{ph}\)

Bidirectional: Check quadrant of operation

Commutation: Consider overlap angle for practical circuits

THD calculations: Use \(THD = \sqrt{\sum_{n=2}^{\infty}(I_n/I_1)^2}\)

Filter design: Match resonant frequency with ripple frequency