Uncontrolled Rectifiers: Single & Three-Phase
Performance Parameters & Analysis

Introduction

Introduction to Rectifiers

  • Rectifiers convert alternating current (AC) to direct current (DC)

  • Two main categories:

    • Uncontrolled (uses diodes only)

    • Controlled (uses thyristors/SCRs)

  • Classification by phase:

    • Single-phase rectifiers (residential, low power)

    • Three-phase rectifiers (industrial, high power)

  • Uncontrolled rectifiers are simpler but have fixed output

  • Wide applications in power supplies, battery chargers, DC drives

Single-Phase Rectifiers

Types of Single-Phase Rectifiers

Single-Phase Rectifier Configurations

  • Half-Wave Rectifier

    • Uses 1 diode

    • Conducts for half cycle only

    • Poor efficiency and high ripple

  • Full-Wave Center-Tapped Rectifier

    • Uses 2 diodes with center-tapped transformer

    • Conducts for full cycle

    • Better efficiency than half-wave

  • Full-Wave Bridge Rectifier

    • Uses 4 diodes in bridge configuration

    • Best transformer utilization

    • Most commonly used configuration

Three-Phase Rectifiers

Types of Three-Phase Rectifiers

Three-Phase Rectifier Configurations

  • Three-Phase Half-Wave Rectifier (M3)

    • Uses 3 diodes connected to three-phase supply

    • Each diode conducts for 120°

    • Simple but poor transformer utilization

  • Three-Phase Full-Wave Bridge Rectifier (B6)

    • Uses 6 diodes in bridge configuration

    • Each diode conducts for 120°

    • Highest power applications

    • Best performance parameters

  • Three-Phase Full-Wave with Center-Tap

    • Uses 6 diodes with center-tapped transformer

    • Good performance but requires special transformer

Performance Parameters

Overview of Performance Parameters

Key Performance Parameters

The quality of rectification is judged by:

  • Rectification ratio (\(\sigma\)) - DC power efficiency

  • Form factor (FF) - Shape of output waveform

  • Ripple factor (RF) - AC content in output

  • Transformer utilization factor (TUF) - Transformer efficiency

  • Peak inverse voltage (PIV) - Diode voltage stress

  • Displacement factor (DF) - Phase relationship

  • Total harmonic distortion (THD) - Harmonic content

  • Power factor (PF) - Overall power efficiency

  • Conduction efficiency (\(\eta\)) - Voltage regulation

  • Voltage regulation (VR) - Load stability

  • Crest factor (CF) - Peak to RMS ratio

Rectification Ratio

Rectification Ratio (\(\sigma\))

Definition

Ratio of DC output power to AC input power

\[\sigma = \frac{P_{\text{dc}}}{P_{\text{ac}}} = \frac{V_{\text{dc}} \cdot I_{\text{dc}}}{V_{\text{rms}} \cdot I_{\text{rms}}}\]
Rectifier Type Rectification Ratio
Single-phase half-wave \(\frac{2}{\pi^2} \approx 0.203\)
Single-phase full-wave (CT) \(\frac{4}{\pi^2} \approx 0.405\)
Single-phase bridge \(\frac{8}{\pi^2} \approx 0.812\)
Three-phase half-wave (M3) \(\frac{3\sqrt{3}}{2\pi^2} \approx 0.264\)
Three-phase bridge (B6) \(\frac{27}{2\pi^2} \approx 1.366\)

Note

Higher rectification ratio indicates better power conversion efficiency

Form Factor

Form Factor (FF)

Definition

Ratio of RMS value to average value of output voltage/current

\[\text{FF} = \frac{V_{\text{rms}}}{V_{\text{dc}}} = \frac{I_{\text{rms}}}{I_{\text{dc}}}\]
Rectifier Type Form Factor
Single-phase half-wave \(\frac{\pi}{2} \approx 1.57\)
Single-phase full-wave (CT) \(\frac{\pi}{2\sqrt{2}} \approx 1.11\)
Single-phase bridge \(\frac{\pi}{2\sqrt{2}} \approx 1.11\)
Three-phase half-wave (M3) \(\frac{\pi}{3\sqrt{3}} \approx 1.017\)
Three-phase bridge (B6) \(\frac{\pi\sqrt{2}}{3\sqrt{3}} \approx 1.014\)

Interpretation

Lower form factor indicates output closer to pure DC

Ripple Factor

Ripple Factor (RF)

Definition

Measure of the ripple content in the output (AC component relative to DC)

\[\text{RF} = \sqrt{\text{FF}^2 - 1} = \frac{V_{\text{ac}}}{V_{\text{dc}}}\]
Rectifier Type Ripple Factor Ripple Frequency
Single-phase half-wave \(1.21\) \(f\)
Single-phase full-wave (CT) \(0.482\) \(2f\)
Single-phase bridge \(0.482\) \(2f\)
Three-phase half-wave (M3) \(0.172\) \(3f\)
Three-phase bridge (B6) \(0.042\) \(6f\)

Goal

Lower ripple factor indicates better DC output quality. Three-phase rectifiers have much lower ripple.

Conduction Efficiency

Conduction Efficiency (\(\eta\))

Definition

Ratio of DC output power to AC input power (related to Form Factor)

\[\eta = \frac{P_{\text{dc}}}{P_{\text{ac}}} = \frac{1}{\text{FF}^2} \times 100\%\]
Rectifier Type Conduction Efficiency
Single-phase half-wave \(40.6\%\)
Single-phase full-wave (CT) \(81.2\%\)
Single-phase bridge \(81.2\%\)
Three-phase half-wave (M3) \(96.8\%\)
Three-phase bridge (B6) \(97.3\%\)

Observation

Three-phase rectifiers have significantly higher conduction efficiency

Transformer Utilization Factor

Transformer Utilization Factor (TUF)

Definition

Ratio of DC power delivered to load to AC rating of transformer secondary

\[\text{TUF} = \frac{P_{\text{dc}}}{V_{s(\text{rms})} \cdot I_{s(\text{rms})}}\]
Rectifier Type TUF
Single-phase half-wave \(0.203\)
Single-phase full-wave (CT) \(0.405\)
Single-phase bridge \(0.812\)
Three-phase half-wave (M3) \(0.264\)
Three-phase bridge (B6) \(0.955\)

Significance

Three-phase bridge rectifier has the highest TUF, indicating best transformer utilization

Peak Inverse Voltage

Peak Inverse Voltage (PIV)

Definition

Maximum reverse voltage a diode must withstand when it is reverse-biased

Rectifier Type PIV
Single-phase half-wave \(V_m\)
Single-phase full-wave (CT) \(2V_m\)
Single-phase bridge \(V_m\)
Three-phase half-wave (M3) \(\sqrt{3}V_m\)
Three-phase bridge (B6) \(\sqrt{3}V_m\)

Design Consideration

PIV rating determines diode selection. For three-phase systems: PIV = \(\sqrt{3} \times\) Line voltage peak

Safety Factor

Practical diode rating = PIV \(\times\) Safety Factor (typically 2-3)

Crest Factor

Crest Factor (CF)

Definition

Ratio of peak value to RMS value of input current

\[\text{CF} = \frac{I_{\text{peak}}}{I_{\text{rms}}}\]
Rectifier Type Crest Factor
Single-phase half-wave \(\sqrt{2} \approx 1.414\)
Single-phase full-wave (CT) \(\sqrt{2} \approx 1.414\)
Single-phase bridge \(\sqrt{2} \approx 1.414\)
Three-phase half-wave (M3) \(\sqrt{2} \approx 1.414\)
Three-phase bridge (B6) \(\sqrt{2} \approx 1.414\)

Impact

High crest factor indicates high peak currents, requiring robust components and causing stress on supply system

Total Harmonic Distortion

Total Harmonic Distortion (THD)

Definition

Measure of harmonic content in the input current relative to the fundamental

\[\text{THD}_I = \dfrac{\sqrt{\displaystyle \sum_{n=2}^{\infty} I_{n(\text{rms})}^2}}{I_{1(\text{rms})}} \times 100\%\]
Rectifier Type THD (%)
Single-phase half-wave \(121.0\)
Single-phase full-wave (CT) \(48.4\)
Single-phase bridge \(48.4\)
Three-phase half-wave (M3) \(31.1\)
Three-phase bridge (B6) \(31.1\)

Standards

IEEE 519 limits: THD \(< 5\%\) for systems \(> 1000V\), THD\(< 8\%\) for systems \(< 1000V\)

Power Factor

Power Factor (PF)

Definition

Ratio of real power to apparent power, considering both distortion and displacement

\[\text{PF} = \text{DF} \times \text{CF} = \frac{P}{S} = \frac{I_1}{I_{\text{rms}}}\]
Rectifier Type Power Factor
Single-phase half-wave \(0.45\)
Single-phase full-wave (CT) \(0.90\)
Single-phase bridge \(0.90\)
Three-phase half-wave (M3) \(0.827\)
Three-phase bridge (B6) \(0.955\)

Practical Impact

Low power factor results in higher utility charges and reduced system efficiency

Voltage Regulation

Voltage Regulation (VR)

Definition

Measure of how well the output voltage remains constant with load changes

\[\text{VR} = \frac{V_{\text{no-load}} - V_{\text{full-load}}}{V_{\text{full-load}}} \times 100\%\]

  • Factors affecting VR:

    • Source resistance and reactance

    • Diode forward voltage drop (typically 0.7V for Si)

    • Transformer resistance and leakage reactance

    • Load current magnitude

  • Three-phase systems: Generally better voltage regulation due to:

    • Lower ripple content

    • More continuous conduction

    • Better transformer utilization

Goal

Lower voltage regulation percentage indicates better load stability

Comprehensive Comparison

Complete Performance Comparison for Single-Phase Rectifiers

Single-Phase Rectifiers Comparison
Parameter Half-wave Full-wave CT Bridge
Rectification ratio 0.203 0.405 0.812
Form factor 1.57 1.11 1.11
Ripple factor 1.21 0.482 0.482
Conduction efficiency (%) 40.6 81.2 81.2
TUF 0.203 0.405 0.812
PIV \(V_m\) \(2V_m\) \(V_m\)
THD (%) 121.0 48.4 48.4
Power Factor 0.45 0.90 0.90
Crest Factor 1.414 1.414 1.414
No. of Diodes 1 2 4
Ripple Frequency \(f\) \(2f\) \(2f\)

Complete Performance Comparison for Three-Phase Rectifiers

Three-Phase Rectifiers Comparison
Parameter 3-Phase Half-wave (M3) 3-Phase Bridge (B6)
Rectification ratio 0.264 1.366
Form factor 1.017 1.014
Ripple factor 0.172 0.042
Conduction efficiency (%) 96.8 97.3
TUF 0.264 0.955
PIV \(\sqrt{3}V_m\) \(\sqrt{3}V_m\)
THD (%) 31.1 31.1
Power Factor 0.827 0.955
Crest Factor 1.414 1.414
No. of Diodes 3 6
Ripple Frequency \(3f\) \(6f\)
Conduction Angle 120° 120°

Harmonic Analysis

Single-Phase Harmonic Content

Single-Phase Harmonic Analysis

Half-wave output:

\[v_o(t) = \frac{V_m}{\pi} + \frac{V_m}{2}\sin(\omega t) - \frac{2V_m}{3\pi}\cos(2\omega t) - \cdots\]

Dominant harmonics:

  • DC: \(\frac{V_m}{\pi}\)

  • Fundamental: 50%

  • 2nd: 21.2%

  • 4th: 4.2%

Full-wave output:

\[v_o(t) = \frac{2V_m}{\pi} - \frac{4V_m}{3\pi}\cos(2\omega t) - \frac{4V_m}{15\pi}\cos(4\omega t) - \cdots\]

Dominant harmonics:

  • DC: \(\frac{2V_m}{\pi}\)

  • No fundamental

  • 2nd: 42.4%

  • 4th: 8.5%

  • 6th: 3.6%

Three-Phase Harmonic Content

Three-Phase Harmonic Analysis

Three-Phase Bridge (B6) Output:

\[v_o(t) = \frac{3\sqrt{3}V_m}{\pi} - \frac{12V_m}{\pi}\sum_{n=1}^{\infty}\frac{\cos(6n\omega t)}{36n^2-1}\]

Voltage harmonics:

  • DC: \(\frac{3\sqrt{3}V_m}{\pi} = 1.654V_m\)

  • 6th: 4.0%

  • 12th: 1.0%

  • 18th: 0.44%

  • Only \((6n)^{th}\) harmonics present

Current harmonics:

  • 5th: 20.0%

  • 7th: 14.3%

  • 11th: 9.1%

  • 13th: 7.7%

  • Characteristic: \((6n \pm 1)^{th}\)

Advantage

Three-phase rectifiers have much lower harmonic content, especially at lower frequencies

Practical Design Considerations

Single-Phase vs Three-Phase Selection

Single-Phase Applications:

  • Power < 5 kW typically

  • Residential applications

  • Small motor drives

  • Battery chargers

  • Electronic equipment

  • Where 3-phase supply unavailable

Advantages:

  • Simple control

  • Lower cost

  • Readily available supply

Three-Phase Applications:

  • Power > 5 kW typically

  • Industrial applications

  • Large motor drives

  • DC transmission systems

  • High-power supplies

  • Where low ripple required

Advantages:

  • Higher efficiency

  • Lower ripple

  • Better power factor

  • Smaller filter requirements

Component Selection Guidelines

  • Diode Selection:

    • Average current: \(I_{F(avg)} \geq 1.5 \times I_{\text{load}}\)

    • Peak current: \(I_{FSM} \geq 10 \times I_{\text{load}}\) (for capacitive loads)

    • Reverse voltage: \(V_{RRM} \geq 2-3 \times \text{PIV}\)

    • Recovery time: Critical for high-frequency switching

  • Transformer Rating:

    • VA rating based on TUF and safety margin

    • Voltage regulation under full load

    • Temperature rise and insulation class

    • Short-circuit withstand capability

  • Filter Design:

    • Capacitor: \(C = \frac{I_{\text{load}}}{4fV_{\text{ripple}}}\) (approx.)

    • Inductor: Reduces current ripple, improves PF

    • LC combination: Best performance but higher cost

Filter Circuits

Filter Circuit Requirements

Single-Phase Filters:

  • Higher ripple content (48.4%)

  • Larger filter components needed

  • 2nd harmonic dominant (100/120 Hz)

  • Capacitor filter most common

Design equations:

\[C = \frac{I_{\text{load}}}{4f \cdot V_{\text{ripple}}}\]
\[L = \frac{V_{\text{ripple}}}{4\pi f \cdot I_{\text{ripple}}}\]

Three-Phase Filters:

  • Much lower ripple (4.2%)

  • Smaller filter components

  • 6th harmonic dominant (300/360 Hz)

  • Often no filter needed for some applications

Advantages:

  • 10-30x smaller capacitors

  • Lower cost and size

  • Better dynamic response

  • Less energy storage

Economic Impact

Three-phase systems require significantly smaller and cheaper filter components

Advanced Topics

Commutation and Overlap

  • Commutation Process:

    • Transfer of current from one diode to another

    • Affected by source inductance

    • Creates commutation notches in voltage

  • Overlap Angle (\(\mu\)):

    • Period during which two diodes conduct simultaneously

    • Reduces output voltage: \(V_{dc} = V_{dc0}\cos(\mu/2)\)

    • More significant in three-phase systems

  • Effects of Source Inductance:

    • Increases overlap angle

    • Reduces average output voltage

    • Improves current continuity

    • Creates voltage notches

Practical Consideration

Source inductance is unavoidable and must be considered in real designs

Conclusion

Key Takeaways

  • Single-Phase Rectifiers:

    • Bridge rectifier best choice for most applications

    • Efficiency up to 81.2%, but high ripple (48.4%)

    • Suitable for low-power applications (< 5 kW)

  • Three-Phase Rectifiers:

    • Superior performance in all parameters

    • Bridge (B6) offers 97.3% efficiency, 4.2% ripple

    • Essential for high-power industrial applications

  • Selection Criteria:

    • Power level (major deciding factor)

    • Supply availability (1-phase vs 3-phase)

    • Ripple requirements and filter cost

    • Power factor and harmonic regulations

    • Initial cost vs operating efficiency

Performance Summary

Best Performing Rectifiers by Category
Parameter Single-Phase Three-Phase
Highest Efficiency Bridge (81.2%) Bridge B6 (97.3%)
Lowest Ripple Bridge (48.4%) Bridge B6 (4.2%)
Best TUF Bridge (0.812) Bridge B6 (0.955)
Best Power Factor Bridge/CT (0.90) Bridge B6 (0.955)
Lowest THD Bridge/CT (48.4%) Both (31.1%)
Lowest PIV Stress Half-wave/Bridge Both equal
Most Economic Half-wave Half-wave M3
Most Practical Bridge Bridge B6

Recommendation

For most applications: Single-phase bridge for P < 5kW, Three-phase bridge for P > 5kW

Modern Applications and Trends

Contemporary Applications

Renewable Energy:

  • Solar PV inverters (DC-AC)

  • Wind turbine rectifiers

  • Battery charging systems

  • Grid-tie applications

Electric Vehicles:

  • On-board chargers

  • DC fast charging stations

  • Motor drive systems

  • Regenerative braking

Industrial Applications:

  • Variable frequency drives

  • DC motor drives

  • Electroplating/electrolysis

  • Welding power supplies

Power Quality:

  • Active power filters

  • Harmonic mitigation

  • Power factor correction

  • Grid support systems

Problem-Solving Examples

Design Example: Single-Phase Bridge Rectifier

Given: 230V, 50Hz supply, 2kW resistive load

Required: Calculate all performance parameters

Solution:

\[\begin{aligned} V_m &= 230\sqrt{2} = 325.3\text{ V} \\ V_{dc} &= \frac{2V_m}{\pi} = \frac{2 \times 325.3}{\pi} = 207.1\text{ V} \\ I_{dc} &= \frac{P}{V_{dc}} = \frac{2000}{207.1} = 9.66\text{ A} \\ I_{rms} &= I_{dc} \times \text{FF} = 9.66 \times 1.11 = 10.72\text{ A} \\ \text{RF} &= 0.482 \text{ (from table)} \\ \text{PIV} &= V_m = 325.3\text{ V} \\ \eta &= 81.2\% \text{ (from table)} \\ \text{TUF} &= 0.812 \text{ (from table)} \end{aligned}\]

Design Example: Three-Phase Bridge Rectifier

Given: 400V (line-to-line), 50Hz supply, 10kW load

Required: Calculate key parameters and compare with single-phase

Solution:

\[\begin{aligned} V_{m(line)} &= 400\sqrt{2} = 565.7\text{ V} \\ V_{dc} &= \frac{3\sqrt{3}V_{m(line)}}{\pi} = \frac{3\sqrt{3} \times 565.7}{\pi} = 540.5\text{ V} \\ I_{dc} &= \frac{10000}{540.5} = 18.5\text{ A} \\ I_{rms} &= I_{dc} \times 1.014 = 18.76\text{ A} \\ \text{RF} &= 0.042 \text{ (11.5x better than single-phase)} \\ \text{PIV} &= \sqrt{3}V_{m(phase)} = \sqrt{3} \times 326.6 = 565.7\text{ V} \\ \eta &= 97.3\% \text{ (vs 81.2\% for single-phase)} \\ \text{Filter size} &\approx \frac{1}{10} \text{ of single-phase requirement} \end{aligned}\]

Laboratory Exercises

Suggested Laboratory Experiments

  1. Single-Phase Rectifier Comparison:

    • Build half-wave, center-tap, and bridge rectifiers

    • Measure all performance parameters

    • Compare theoretical vs experimental results

    • Observe waveforms with oscilloscope

  2. Three-Phase Rectifier Analysis:

    • Construct 3-phase bridge rectifier

    • Measure ripple factor and efficiency

    • Compare with single-phase bridge

    • Harmonic analysis using spectrum analyzer

  3. Filter Circuit Investigation:

    • Effect of different filter types (C, L, LC, RC)

    • Ripple reduction vs filter component values

    • Voltage regulation with different loads

    • Current waveform distortion analysis