📚 Introduction
AC-DC Converters (Rectifiers) convert alternating current (AC) to direct current (DC). They are classified as:
- Uncontrolled Rectifiers: Use diodes only
- Controlled Rectifiers: Use thyristors (SCRs) for controllable output
🔌 Uncontrolled Rectifiers
Single-Phase Half-Wave Rectifier
Configuration: Single diode, conducts only during positive half-cycle
Average (DC) Output Voltage: $$V_{dc} = \frac{V_m}{\pi} = 0.318 \, V_m$$ where \(V_m\) is the peak input voltage
RMS Output Voltage: $$V_{rms} = \frac{V_m}{2}$$
Form Factor (FF): $$FF = \frac{V_{rms}}{V_{dc}} = \frac{\pi}{2} = 1.57$$
Ripple Factor (RF): $$RF = \sqrt{FF^2 - 1} = 1.21$$
Peak Inverse Voltage (PIV): $$PIV = V_m$$
Efficiency: $$\eta = \frac{P_{dc}}{P_{ac}} = \frac{4}{\pi^2} = 40.6\%$$
Single-Phase Full-Wave Rectifier (Center-Tap)
Configuration: Two diodes with center-tapped transformer
Average (DC) Output Voltage: $$V_{dc} = \frac{2V_m}{\pi} = 0.636 \, V_m$$
RMS Output Voltage: $$V_{rms} = \frac{V_m}{\sqrt{2}}$$
Form Factor: $$FF = 1.11$$
Ripple Factor: $$RF = 0.48$$
Peak Inverse Voltage: $$PIV = 2V_m$$
Efficiency: $$\eta = 81.2\%$$
Single-Phase Full-Wave Bridge Rectifier
Configuration: Four diodes in bridge configuration
Average Output Voltage: $$V_{dc} = \frac{2V_m}{\pi} = 0.636 \, V_m$$
RMS Output Voltage: $$V_{rms} = \frac{V_m}{\sqrt{2}}$$
Ripple Factor: $$RF = 0.48$$
Peak Inverse Voltage: $$PIV = V_m$$ (Half of center-tap configuration)
Advantage: Lower PIV requirement compared to center-tap, no need for center-tapped transformer
Three-Phase Half-Wave Rectifier
Configuration: Three diodes connected to three-phase supply
Average Output Voltage: $$V_{dc} = \frac{3\sqrt{3}}{\pi} \, V_{m(ph)} = 1.654 \, V_{m(ph)}$$ where \(V_{m(ph)}\) is peak phase voltage
For Line Voltage \(V_L\): $$V_{dc} = \frac{3\sqrt{3}}{2\pi} \, V_L = 0.827 \, V_L$$
Ripple Factor: $$RF = 0.17$$
Peak Inverse Voltage: $$PIV = \sqrt{3} \, V_{m(ph)}$$
Three-Phase Full-Wave Bridge Rectifier
Configuration: Six diodes in bridge configuration
Average Output Voltage: $$V_{dc} = \frac{3\sqrt{3}}{\pi} \, V_{m(ph)} = 1.654 \, V_{m(ph)}$$
For Line Voltage: $$V_{dc} = \frac{3}{\pi} \, V_L = 0.955 \, V_L$$
Ripple Factor: $$RF = 0.04$$ (Very low ripple - nearly pure DC)
Peak Inverse Voltage: $$PIV = \sqrt{3} \, V_{m(ph)} = V_L$$
Most Common: This is the most widely used industrial rectifier due to low ripple and high efficiency
🎛️ Controlled Rectifiers (Phase-Controlled)
Firing Angle (α): The angle at which thyristor is triggered after zero crossing. Controls output voltage.
Single-Phase Half-Wave Controlled Rectifier
Configuration: Single SCR with resistive load
Average Output Voltage: $$V_{dc} = \frac{V_m}{\pi} (1 + \cos\alpha)$$
RMS Output Voltage: $$V_{rms} = \frac{V_m}{2} \sqrt{\frac{1}{\pi} (\pi - \alpha + \frac{\sin 2\alpha}{2})}$$
Firing Angle Range: \(0° \leq \alpha \leq 180°\)
At α = 0°, behaves like uncontrolled rectifier
At α = 0°, behaves like uncontrolled rectifier
Single-Phase Full-Wave Controlled Rectifier
Configuration: Four SCRs in bridge (or two SCRs in center-tap)
Average Output Voltage (Resistive Load): $$V_{dc} = \frac{2V_m}{\pi} \cos\alpha$$
RMS Output Voltage: $$V_{rms} = \frac{V_m}{\sqrt{2}} \sqrt{\frac{1}{\pi} (\pi - \alpha + \frac{\sin 2\alpha}{2})}$$
With RL Load (Continuous Current): $$V_{dc} = \frac{2V_m}{\pi} \cos\alpha$$ Same as resistive load when current is continuous
Firing Angle Range: \(0° \leq \alpha \leq 90°\) for rectification
\(90° < \alpha \leq 180°\) for inversion mode
\(90° < \alpha \leq 180°\) for inversion mode
Three-Phase Half-Wave Controlled Rectifier
Average Output Voltage: $$V_{dc} = \frac{3\sqrt{3}}{2\pi} \, V_m \cos\alpha$$
For Line Voltage: $$V_{dc} = \frac{3\sqrt{3}}{2\pi} \, V_L \cos\alpha$$
Firing Angle Range: \(0° \leq \alpha \leq 90°\)
Three-Phase Full-Wave Controlled Rectifier (6-Pulse)
Most Important Industrial Converter
Average Output Voltage: $$V_{dc} = \frac{3V_m}{\pi} \cos\alpha = \frac{3\sqrt{2}}{\pi} \, V_{ph} \cos\alpha$$
For Line Voltage: $$V_{dc} = \frac{3\sqrt{3}}{\pi} \, V_L \cos\alpha = 1.654 \, V_L \cos\alpha$$
Maximum DC Voltage (α = 0°): $$V_{dc(max)} = 1.654 \, V_L$$
Firing Angle Range:
Rectification: \(0° \leq \alpha \leq 90°\)
Inversion: \(90° < \alpha < 180°\)
Rectification: \(0° \leq \alpha \leq 90°\)
Inversion: \(90° < \alpha < 180°\)
Ripple Frequency: \(f_r = 6f\) where \(f\) is supply frequency
Single-Phase Semi-Converter
Configuration: Two SCRs + Two Diodes (or One SCR + One Diode for half-wave)
Average Output Voltage: $$V_{dc} = \frac{V_m}{\pi} (1 + \cos\alpha)$$
Feature: Provides one-quadrant operation, inherently has freewheeling action, output voltage always positive
📊 Performance Parameters
Form Factor (FF)
$$FF = \frac{V_{rms}}{V_{dc}}$$
Measures the shape of the waveform. For pure DC, FF = 1
Ripple Factor (RF)
$$RF = \sqrt{FF^2 - 1} = \frac{V_{ac}}{V_{dc}}$$
Indicates AC component in output. Lower is better.
Rectification Efficiency
$$\eta = \frac{P_{dc}}{P_{ac}} = \frac{V_{dc}^2 / R}{V_{rms}^2 / R}$$
Ratio of DC output power to AC input power
Transformer Utilization Factor
$$TUF = \frac{P_{dc}}{VA \, \text{rating}}$$
Measure of transformer utilization
Power Factor
For Controlled Rectifiers: $$PF = \frac{P_{input}}{V_{rms} \times I_{rms}}$$
Displacement Power Factor: $$DPF = \cos\phi_1$$ where \(\phi_1\) is fundamental displacement angle
Distortion Factor: $$DF = \frac{I_{s1}}{I_s}$$ where \(I_{s1}\) is fundamental component and \(I_s\) is RMS supply current
Total Power Factor: $$PF = DPF \times DF$$
⚙️ Effect of Load on Performance
Resistive Load (R)
• Current waveform follows voltage waveform
• Current goes to zero when voltage is zero
• Discontinuous current possible
• Simple analysis
• Current goes to zero when voltage is zero
• Discontinuous current possible
• Simple analysis
Inductive Load (RL)
• Current lags voltage
• Current may remain continuous even when voltage is zero
• Extinction angle (\(\beta\)) >\(\pi\) for half-wave
• Freewheeling diode can be used to improve performance
• Current may remain continuous even when voltage is zero
• Extinction angle (\(\beta\)) >\(\pi\) for half-wave
• Freewheeling diode can be used to improve performance
Extinction Angle (\(\beta\)):
Angle at which current falls to zero
For RL load: \(\beta > \pi + \alpha\)
Angle at which current falls to zero
For RL load: \(\beta > \pi + \alpha\)
Highly Inductive Load (L → ∞)
• Current is essentially constant (ripple-free)
• No discontinuous conduction
• Simplified analysis possible
• Represents worst-case for supply harmonics
• No discontinuous conduction
• Simplified analysis possible
• Represents worst-case for supply harmonics
Freewheeling Diode
Connected across the load, provides path for inductive current during non-conducting periods
Benefits:
• Improves average output voltage
• Reduces ripple
• Improves input power factor
• Prevents negative voltage across load
• Improves average output voltage
• Reduces ripple
• Improves input power factor
• Prevents negative voltage across load
With Freewheeling Diode (Single-Phase): $$V_{dc} = \frac{V_m}{\pi} (1 + \cos\alpha)$$
📈 Quick Comparison Table
| Rectifier Type | Vdc/Vm | Ripple Factor | Efficiency | PIV |
|---|---|---|---|---|
| 1-φ Half-Wave | 0.318 | 1.21 | 40.6% | Vm |
| 1-φ Full-Wave (CT) | 0.636 | 0.48 | 81.2% | 2Vm |
| 1-φ Bridge | 0.636 | 0.48 | 81.2% | Vm |
| 3-φ Half-Wave | 1.654 Vph | 0.17 | ~97% | √3 Vph |
| 3-φ Bridge | 0.955 VL | 0.04 | ~99% | VL |
🌊 Harmonics
Input Current Harmonics
Rectifiers draw non-sinusoidal current, introducing harmonics into supply
Harmonic Order:
Single-phase: \(n = 2k \pm 1\) (odd harmonics: 3rd, 5th, 7th, ...)
Three-phase: \(n = 6k \pm 1\) (5th, 7th, 11th, 13th, ...)
Single-phase: \(n = 2k \pm 1\) (odd harmonics: 3rd, 5th, 7th, ...)
Three-phase: \(n = 6k \pm 1\) (5th, 7th, 11th, 13th, ...)
Harmonic Magnitude (for 6-pulse): $$I_n = \frac{I_1}{n}$$ where \(I_1\) is fundamental component
Total Harmonic Distortion (THD): $$THD = \frac{\sqrt{\sum_{n=2}^{\infty} I_n^2}}{I_1}$$
Output Voltage Harmonics
Dominant Ripple Frequency:
Single-phase full-wave: \(f_r = 2f\)
Three-phase half-wave: \(f_r = 3f\)
Three-phase full-wave (6-pulse): \(f_r = 6f\)
12-pulse: \(f_r = 12f\)
Single-phase full-wave: \(f_r = 2f\)
Three-phase half-wave: \(f_r = 3f\)
Three-phase full-wave (6-pulse): \(f_r = 6f\)
12-pulse: \(f_r = 12f\)
Higher pulse numbers result in lower harmonic content and smoother DC output
🔄 Commutation and Overlap
Source Inductance Effect
Practical AC sources have inductance (transformer leakage, line inductance). This causes commutation overlap.
Overlap Angle (μ or u): Period during which both incoming and outgoing devices conduct simultaneously
Average Output Voltage with Overlap (3-phase bridge): \(V_{dc} = \frac{3\sqrt{3}}{\pi} V_L \cos\alpha - \frac{3}{\pi} \omega L_s I_{dc}\) where \(L_s\) is source inductance per phase
Overlap Angle: \(\cos(\alpha + \mu) = \cos\alpha - \frac{2\omega L_s I_{dc}}{V_m}\)
Effects of Overlap:
• Reduces average output voltage
• Notches in AC line voltage
• Increases THD
• Limits maximum available DC voltage
• Reduces average output voltage
• Notches in AC line voltage
• Increases THD
• Limits maximum available DC voltage
Voltage Regulation
Voltage Drop due to Commutation: \(\Delta V = \frac{3}{\pi} \omega L_s I_{dc}\)
Equivalent DC Resistance: \(R_c = \frac{3}{\pi} \omega L_s\)
🔧 Filters and Smoothing
Output Filters
Capacitor Filter
\(V_{ripple(pp)} \approx \frac{I_{dc}}{fC}\)
• Used for light loads
• High peak currents
• Improves voltage regulation
Inductor Filter
\(L_{critical} = \frac{R}{2\pi f}\)
• For continuous current
• Smooths current waveform
• Used in controlled rectifiers
LC Filter Ripple Reduction: \(\frac{V_{ripple(out)}}{V_{ripple(in)}} = \frac{1}{1 - \omega^2 LC}\)
π-Filter: Combination of C-L-C provides excellent smoothing for high-current applications
🏭 Applications
Uncontrolled Rectifiers
Low Power Supplies Battery Chargers Consumer Electronics Voltage ReferencesFixed DC output, simple, low cost
Controlled Rectifiers
DC Motor Drives HVDC Transmission Electroplating Variable DC Supplies Battery ChargingVariable DC output, precise control
Specific Application Areas
Three-Phase Bridge: Industrial DC drives, HVDC systems, high-power applications (> 15 kW)
Single-Phase Bridge: Household appliances, low-power drives (< 5 kW), small battery chargers
Semi-Converter: One-quadrant DC drives where regeneration not required
Full Converter: Two-quadrant and four-quadrant drives with regenerative capability
Single-Phase Bridge: Household appliances, low-power drives (< 5 kW), small battery chargers
Semi-Converter: One-quadrant DC drives where regeneration not required
Full Converter: Two-quadrant and four-quadrant drives with regenerative capability
⚡ Operating Modes
Rectification Mode
Firing Angle: \(0° \leq \alpha < 90°\)
Power flows from AC to DC side
Average output voltage is positive
Power flows from AC to DC side
Average output voltage is positive
\(V_{dc} > 0, \quad P_{dc} > 0\)
Inversion Mode
Firing Angle: \(90° < \alpha \leq 180°\)
Power flows from DC to AC side
Average output voltage is negative
Requires DC source (back EMF) on load side
Power flows from DC to AC side
Average output voltage is negative
Requires DC source (back EMF) on load side
\(V_{dc} < 0, \quad P_{dc} < 0\)
Extinction Angle (\(\gamma\)): \(\gamma = 180° - \alpha\) Must be sufficient for thyristor turn-off (typically \(\gamma\) > 15°-20°)
Dual Converter Operation
Two converters connected back-to-back (anti-parallel) for four-quadrant operation
Modes:
• Positive voltage, positive current (Forward motoring)
• Positive voltage, negative current (Forward braking)
• Negative voltage, negative current (Reverse motoring)
• Negative voltage, positive current (Reverse braking)
• Positive voltage, positive current (Forward motoring)
• Positive voltage, negative current (Forward braking)
• Negative voltage, negative current (Reverse motoring)
• Negative voltage, positive current (Reverse braking)
Circulating Current Reactor: \(L_{reactor} = \frac{V_m}{\pi f I_{circ(max)}}\)
📐 Design Considerations
Device Ratings
Average Current Rating: \(I_{avg} = \frac{I_{dc}}{n}\) where \(n\) is number of devices conducting simultaneously
RMS Current Rating: \(I_{rms} = \frac{I_{dc}}{\sqrt{n}}\)
Peak Current: Must account for inrush and fault conditions
Safety Factor: Typically use 1.5-2.0 times calculated ratings
Transformer Rating
VA Rating (Single-Phase): \(VA = V_{rms} \times I_{rms}\)
VA Rating (Three-Phase): \(VA = \sqrt{3} \, V_L \times I_L\)
Consider derating for harmonic heating (typically 10-20%)
Protection Requirements
Overcurrent Protection: Fast-acting fuses or circuit breakers
Overvoltage Protection: Snubber circuits (R-C), MOVs
di/dt Protection: Series inductance for thyristors
dv/dt Protection: R-C snubber across thyristors
Gate Protection: Zener diodes, gate resistors
Overvoltage Protection: Snubber circuits (R-C), MOVs
di/dt Protection: Series inductance for thyristors
dv/dt Protection: R-C snubber across thyristors
Gate Protection: Zener diodes, gate resistors
Snubber Design: \(C = \frac{t_{rr} \times I_T}{dv/dt_{max}}$ $R = \sqrt{\frac{L}{C}}\)
📋 Quick Formula Reference
| Parameter | Single-Phase | Three-Phase (6-pulse) |
|---|---|---|
| Uncontrolled Vdc | \(0.636 \, V_m\) | \(1.654 \, V_L\) or \(0.955 \, V_L\) |
| Controlled Vdc | \(\frac{2V_m}{\pi} \cos\alpha\) | \(\frac{3\sqrt{3}}{\pi} V_L \cos\alpha\) |
| Ripple Frequency | \(2f\) | \(6f\) |
| Ripple Factor | 0.48 | 0.04 |
| Efficiency | 81.2% | ~99% |
🎯 Key Points to Remember
1. Voltage Control: In controlled rectifiers, output voltage decreases with increasing firing angle (α)
2. Inversion: Possible only when α > 90° and DC side has voltage source (back EMF)
3. Pulse Number: Higher pulse numbers → Lower ripple, smoother output
4. Power Factor: Controlled rectifiers have poor PF, especially at large firing angles
5. Harmonics: Rectifiers inject harmonics into AC supply; require filters
6. Continuous Current: Requires sufficient inductance: \(L > L_{critical}\)
7. PIV Selection: Devices must withstand PIV with safety margin
8. Three-Phase Advantage: Better utilization, lower ripple, higher efficiency than single-phase
⚠️ Common Mistakes to Avoid
❌ Confusing peak voltage (Vm) with RMS voltage (Vrms)
✅ Remember: \(V_m = \sqrt{2} \, V_{rms}\)
✅ Remember: \(V_m = \sqrt{2} \, V_{rms}\)
❌ Using line voltage instead of phase voltage for three-phase formulas
✅ Check if formula requires VL or Vph
✅ Check if formula requires VL or Vph
❌ Forgetting that controlled rectifier formulas assume continuous current
✅ Verify continuous current condition or use appropriate formula
✅ Verify continuous current condition or use appropriate formula
❌ Ignoring commutation overlap in practical calculations
✅ Include source inductance effect for accurate voltage prediction
✅ Include source inductance effect for accurate voltage prediction
❌ Assuming inversion is possible without back EMF
✅ Inversion requires DC voltage source on load side
✅ Inversion requires DC voltage source on load side
🧮 Numerical Problem-Solving Tips
Step-by-Step Approach
Step 1: Identify rectifier type and configuration
Step 2: Determine if controlled or uncontrolled
Step 3: Check load type (R, RL, or highly inductive)
Step 4: Calculate peak voltage from given RMS value
Step 5: Apply appropriate formula for Vdc
Step 6: Calculate other parameters (Idc, power, etc.)
Step 7: Verify units and reasonableness of answer
Step 2: Determine if controlled or uncontrolled
Step 3: Check load type (R, RL, or highly inductive)
Step 4: Calculate peak voltage from given RMS value
Step 5: Apply appropriate formula for Vdc
Step 6: Calculate other parameters (Idc, power, etc.)
Step 7: Verify units and reasonableness of answer
Unit Conversions
Voltage: \(V_{m} = \sqrt{2} \, V_{rms} = \sqrt{2} \times 230 = 325.3 \, V\)
Angle: \(\alpha_{rad} = \alpha_{deg} \times \frac{\pi}{180}\)
Frequency to Angular Frequency: \(\omega = 2\pi f\)
📚 Quick Reference Guide
AC-DC Converters: Controlled & Uncontrolled Rectifiers
Essential formulas and concepts for power electronics