Three-Phase Half-Wave Controlled Rectifiers (R and RL Loads)

Need for Three-Phase Rectifiers

Limitations of Single-Phase Rectifiers

High ripple content in output voltage
Significant voltage variation with firing angle
Higher harmonic content requiring extensive filtering
Limited power handling capability
Poor power quality and low transformer utilization

Solution

Three-phase controlled rectifiers offer superior performance for medium to high power applications

Advantages of Three-Phase Controlled Rectifiers

Three-phase rectifiers provide:

Electrical Benefits:

Significantly reduced ripple voltage
Lower harmonic amplitude
Smoother DC output
Improved power factor

System Benefits:

Better transformer utilization
Higher power capability (\(>10 \, \mathrm{kW}\))
Reduced filter requirements
Cost-effective for industry

Ideal for high-power industrial applications

Industrial Applications

Motor Drives & Traction:

DC motor speed control
Variable speed drives
Railway traction systems

Power Systems:

HVDC transmission
Battery charging stations

Industrial Processes:

Electroplating
Electrolysis
Welding equipment
Manufacturing drives

Power Range

Typically \(>10 \, \mathrm{kW}\), extending to MW levels

Classification of Three-Phase Rectifiers

Classification of three-phase controlled rectifiers

Classification Categories

By Configuration:

Half-wave (3-pulse, M-3)
Full-wave (6-pulse, M-6)

By Control Devices:

Fully controlled (all thyristors)
Semi-controlled (thyristors + diodes)

By Pulse Number:

3-pulse converter
6-pulse converter
12-pulse converter

Key Principle

Higher pulse number \(\Rightarrow\) Lower ripple and better performance

Circuit Configuration

Key Components:

Three thyristors: \(T_1\), \(T_2\), \(T_3\)
Three-phase transformer
- Delta-connected primary
- Star-connected secondary
Common neutral point N
Load between output and neutral

Designation: 3-phase, 3-pulse converter (M-3)

Three-phase half-wave controlled rectifier

Three-Phase Voltage System

Balanced three-phase voltages:

\[\begin{aligned} v_{RN}(t) &= V_m\sin(\omega t) \\ v_{YN}(t) &= V_m\sin\left(\omega t - \frac{2\pi}{3}\right) = V_m\sin(\omega t - 120^\circ) \\ v_{BN}(t) &= V_m\sin\left(\omega t - \frac{4\pi}{3}\right) = V_m\sin(\omega t - 240^\circ) \end{aligned}\]

Parameters:

\(V_m = \sqrt{2}V_{\text{ph}}\) – peak phase voltage
\(V_{\text{ph}}\) – RMS phase voltage
\(\omega = 2\pi f\) – angular frequency (rad/s)
Phase displacement: \(120^\circ\) or \(2\pi/3\) rad

Principle of Operation

Operating Principle:

At any instant, the thyristor connected to the most positive phase is forward biased
When triggered, a thyristor conducts and connects its phase to the load
Natural commutation occurs when the next thyristor is fired
Load voltage follows the envelope of the most positive phase voltage

Operating Parameters

Firing sequence: \(T_1 \to T_2 \to T_3 \to T_1\) (repeats)
Firing interval: \(120^\circ\) between consecutive gate pulses
Conduction angle: \(120^\circ\) per thyristor (when \(\alpha \leq 30^\circ\))

Natural Commutation

Natural Commutation:

Occurs when incoming phase voltage exceeds conducting phase voltage
Conducting thyristor turns off automatically
Line commutation – no external circuit needed

Commutation Points at \(\alpha = 0^\circ\):

\(T_1 \to T_2\): \(\omega t = 150^\circ\)
\(T_2 \to T_3\): \(\omega t = 270^\circ\)
\(T_3 \to T_1\): \(\omega t = 30^\circ\)

Operating Modes with Resistive Load

Continuous Conduction Mode (CCM)
- Condition: \(0^\circ \leq \alpha \leq 30^\circ\)
- Each thyristor conducts for full \(120^\circ\)
- Output voltage always positive
- No gaps between conduction periods
Discontinuous Conduction Mode (DCM)
- Condition: \(\alpha > 30^\circ\)
- Conduction angle \(< 120^\circ\)
- Output voltage periodically zero
- Gaps appear between thyristor conductions

Continuous Conduction Mode Waveforms

Features:

Three output pulses per AC cycle
Ripple frequency: \(3f\)
Each thyristor: \(120^\circ\) conduction

Discontinuous Conduction Mode Waveforms

Characteristics:

Conduction angle: \((150^\circ - \alpha)\)
Non-conduction period: \((\alpha - 30^\circ)\)
Discontinuous pulses with zero periods

Average DC Output Voltage

Continuous Conduction Mode (\(\alpha \leq 30^\circ\)):

\[\begin{aligned} V_{\text{dc}} &= \frac{3}{2\pi}\int_{\pi/6+\alpha}^{5\pi/6+\alpha} V_m\sin\theta \, d\theta \\[0.3cm] &= \frac{3V_m}{2\pi}\left[-\cos\theta\right]_{\pi/6+\alpha}^{5\pi/6+\alpha} \\[0.3cm] &= \frac{3\sqrt{3}}{2\pi}V_m\cos\alpha \end{aligned}\]

Key Results

\(\boxed{V_{\text{dc}} = 0.827\,V_m\cos\alpha}\)
At \(\alpha = 0^\circ\): \(V_{\text{dc(max)}} = 0.827\,V_m = 1.17\,V_{\text{ph}}\)

Discontinuous Mode (\(\alpha > 30^\circ\)):

\[\boxed{V_{\text{dc}} = \frac{3\sqrt{3}}{2\pi}V_m\left[1 + \cos(30^\circ + \alpha)\right]}\]

RMS Output Voltage

Continuous Conduction Mode (\(\alpha \leq 30^\circ\)):

\[V_{\text{rms}} = V_m\sqrt{\frac{3}{2\pi}\left[\frac{\pi}{3} + \frac{\sqrt{3}}{4}\cos 2\alpha\right]}\]

Discontinuous Conduction Mode (\(\alpha > 30^\circ\)):

\[V_{\text{rms}} = V_m\sqrt{\frac{3}{2\pi}\left[\frac{5\pi}{6} - \alpha + \frac{1}{2}\sin(30^\circ + 2\alpha)\right]}\]

Important

Both \(V_{\text{dc}}\) and \(V_{\text{rms}}\) decrease as firing angle \(\alpha\) increases

Performance Parameters

Form Factor:

\[FF = \frac{V_{\text{rms}}}{V_{\text{dc}}}\]

Ripple Factor:

\[RF = \sqrt{FF^2 - 1} = \sqrt{\left(\frac{V_{\text{rms}}}{V_{\text{dc}}}\right)^2 - 1}\]

Average Load Current (resistive load):

\[I_{\text{dc}} = \frac{V_{\text{dc}}}{R}\]

DC Output Power:

\[P_{\text{dc}} = V_{\text{dc}} \cdot I_{\text{dc}} = \frac{V_{\text{dc}}^2}{R}\]

Rectification Efficiency:

\[\eta = \frac{P_{\text{dc}}}{P_{\text{ac}}} \times 100\%\]

Thyristor Current Ratings

For continuous conduction mode with resistive load:

Current Parameters

Average Thyristor Current:

\[I_{T(\text{avg})} = \frac{I_{\text{dc}}}{3}\]

RMS Thyristor Current:

\[I_{T(\text{rms})} = \frac{I_{\text{dc}}}{\sqrt{3}}\]

Peak Thyristor Current:

\[I_{T(\text{peak})} = \frac{V_m}{R}\]

Conduction Period: \(120^\circ\) or \(2\pi/3\) rad per thyristor

Three-Phase Half-Wave with RL Load

Key Differences:

Inductance maintains continuous current flow
Load current cannot change instantaneously
Higher firing angles possible (up to \(150^\circ\))
Enables inverter mode (\(\alpha > 90^\circ\))
\(120^\circ\) conduction over wide \(\alpha\) range

Operating Ranges with RL Load

Range 1: \(0^\circ \leq \alpha \leq 30^\circ\)

Continuous conduction naturally maintained
Similar to resistive load behavior

Range 2: \(30^\circ < \alpha < 90^\circ\) (Rectifier Mode)

Modified rectifier operation
Output has negative portions, but \(V_{\text{dc}} > 0\)
Inductance maintains current continuity

Range 3: \(90^\circ < \alpha < 150^\circ\) (Inverter Mode)

Average voltage negative: \(V_{\text{dc}} < 0\)
Power flows from DC to AC supply
Requires DC voltage source (e.g., back EMF)

Waveforms for RL Load: \(\alpha = 60^\circ\)

Observations:

Each thyristor: \(120^\circ\) conduction
Output voltage contains negative portions
Current remains continuous
Average voltage:

\[V_{\text{dc}} = 0.414\,V_m~\text{ (positive)}\]

Rectifier Mode: \(\alpha < 90^\circ\)

Characteristics:

Average output voltage positive: \(V_{\text{dc}} > 0\)
Power flows from AC supply to DC load
Energy delivered to the load

Average Output Voltage

\[\boxed{V_{\text{dc}} = \frac{3\sqrt{3}}{2\pi}V_m\cos\alpha = 0.827\,V_m\cos\alpha}\]

Boundary Condition at \(\alpha = 90^\circ\):

\[V_{\text{dc}} = 0.827\,V_m\cos 90^\circ = 0\]

This represents the transition between rectifier and inverter modes.

Inverter Mode: \(90^\circ < \alpha < 150^\circ\)

Characteristics:

Average voltage negative: \(V_{\text{dc}} < 0\)
Power flows DC to AC (regeneration)
Requires DC voltage source
Used for regenerative braking

\[V_{\text{dc}} = 0.827\,V_m\cos\alpha < 0 ~ \text{for}~\alpha > 90^\circ\]

Voltage and Current Relations with RL Load

For highly inductive load (continuous current):

Average Output Voltage:

\[V_{\text{dc}} = 0.827\,V_m\cos\alpha\]

Average Load Current (with back EMF \(E\)):

\[I_{\text{dc}} = \frac{V_{\text{dc}} - E}{R}\]

For Pure Inductive Load (\(R \approx 0\))

Voltage equation: \(V_{\text{dc}} = E\)
Required firing angle: \(\alpha = \cos^{-1}\left(\dfrac{E}{0.827\,V_m}\right)\)
Conduction: \(120^\circ\) per thyristor when \(\alpha \leq 150^\circ\)

Delta-Zigzag Transformer Connection

Problem

DC component flows through transformer secondary:

Causes core saturation
DC magnetization
Reduced efficiency
Increased losses

Delta-Zigzag Transformer – Working Principle

Construction:

Each secondary phase has two half-windings
Half-windings wound on different transformer legs
Connected in zigzag (interconnected-star) configuration

DC Component Cancellation:

Each half-winding carries \(I_{\text{dc}}/3\) DC current
Currents flow in opposite directions
Magnetic fluxes cancel each other
Net DC magnetization becomes zero
Prevents core saturation

Benefits

Improved transformer performance, efficiency, and reliability

Free-Wheeling Diode Configuration

Purpose of FWD:

Prevents negative output voltage
Alternative path for inductive current
Improves average voltage
Reduces PIV on thyristors
Improves load current waveform

Operation with Free-Wheeling Diode

Operation (\(\alpha > 30^\circ\)):

FWD conducts when output tends negative
Thyristor conduction: \((150^\circ - \alpha)\)
FWD conduction: \((\alpha - 30^\circ)\) per cycle
Output clamped at zero
Current freewheels through FWD

Output Voltage with Free-Wheeling Diode

Average DC Voltage (with FWD)

\[\boxed{V_{\text{dc}} = \frac{3\sqrt{3}}{2\pi}V_m\left[1 + \cos(30^\circ + \alpha)\right]}\]

Note: Identical to discontinuous mode with resistive load

Current Ratings with FWD:

Average Thyristor Current:

\[I_{T(\text{avg})} = I_{\text{dc}}\left[\frac{5\pi/6 - \alpha}{2\pi}\right]\]

Average FWD Current:

\[I_{\text{FWD(avg)}} = I_{\text{dc}}\left[\frac{\alpha - \pi/6}{2\pi}\right]\]

Six-Pulse Half-Wave Converter Circuit

Configuration:

Six-phase star secondary
Six thyristors (one per phase)
Phase displacement: \(60^\circ\)
Designated M-6 converter

Advantages of Six-Pulse Converter

Compared to 3-pulse:

Advantages:

Six pulses per cycle
Lower ripple (\(f_r = 6f\))
Reduced filter requirements
Better TUF
Lower harmonic distortion
Easier commutation

Disadvantages:

Requires six-phase transformer
More complex construction
Higher number of thyristors
Increased cost

Six-Phase Voltage System

Six-phase voltages with \(60^\circ\) phase displacement

Phase Voltages (double-star configuration):

\[\begin{aligned} v_1 &= V_m\sin(\omega t) \\ v_2 &= V_m\sin(\omega t - 60^\circ) \\ v_3 &= V_m\sin(\omega t - 120^\circ) \end{aligned}\]

\[\begin{aligned} v_4 &= -v_1 \\ v_5 &= -v_2 \\ v_6 &= -v_3 \end{aligned}\]

Principle of Operation

Operating Characteristics:

Each thyristor conducts for \(60^\circ\) (\(\pi/3\) rad)
Firing sequence: \(T_1 \to T_2 \to T_3 \to T_4 \to T_5 \to T_6 \to T_1\)
Firing interval: \(60^\circ\) between consecutive gate pulses
Output follows most positive phase voltage
Natural commutation every \(60^\circ\)

Output voltage waveforms at \(\alpha = 30^\circ\) and \(\alpha = 60^\circ\)

Output Voltage Equations

Average DC Output Voltage:

\[\begin{aligned} V_{\text{dc}} &= \frac{6}{2\pi}\int_{\pi/6+\alpha}^{\pi/3+\alpha} V_m\sin\theta \, d\theta \\[0.3cm] &= \frac{3}{\pi}V_m\cos\alpha \end{aligned}\]

Key Result

\(\boxed{V_{\text{dc}} = 0.955\,V_m\cos\alpha}\) At \(\alpha = 0^\circ\): \(V_{\text{dc(max)}} = 0.955\,V_m = 1.35\,V_{\text{ph}}\)

RMS Output Voltage: \(V_{\text{rms}} = V_m\sqrt{\frac{1}{2\pi}\left[\frac{\pi}{6} + \frac{1}{2}\sin 60^\circ + \frac{1}{2}\sin 2\alpha\right]}\)

Ripple Frequency: \(f_{\text{ripple}} = 6f\) (300 Hz for 50 Hz supply)

Operating Modes of Six-Pulse Converter

With Resistive Load:

Continuous Conduction Mode

\(0^\circ \leq \alpha \leq 60^\circ\)

Each thyristor conducts for full \(60^\circ\)
No gaps in output voltage waveform

Discontinuous Conduction Mode

\(60^\circ < \alpha \leq 120^\circ\)

Conduction angle less than \(60^\circ\)
Gaps appear in output voltage

With RL Load: Continuous conduction maintained up to \(\alpha = 90^\circ\)

Thyristor Current Ratings

For continuous conduction with six-pulse converter:

Current Parameters

Average Thyristor Current: \(I_{T(\text{avg})} = \frac{I_{\text{dc}}}{6}\)

RMS Thyristor Current: \(I_{T(\text{rms})} = \frac{I_{\text{dc}}}{\sqrt{6}} = 0.408\,I_{\text{dc}}\)

Peak Inverse Voltage: \(\text{PIV} = 2V_m\)

Conduction Period: \(60^\circ\) or \(\pi/3\) rad per thyristor

Performance Comparison: 3-Pulse vs 6-Pulse

Comparison of 3-pulse and 6-pulse converters
Parameter	3-Pulse (M-3)	6-Pulse (M-6)
Number of thyristors	3	6
Conduction angle	\(120^\circ\)	\(60^\circ\)
Output pulses per cycle	3	6
Ripple frequency	\(3f\)	\(6f\)
\(V_{\text{dc}}\) at \(\alpha=0^\circ\)	\(0.827\,V_m\)	\(0.955\,V_m\)
\(I_{T(\text{avg})}\)	\(I_{\text{dc}}/3\)	\(I_{\text{dc}}/6\)
\(I_{T(\text{rms})}\)	\(I_{\text{dc}}/\sqrt{3}\)	\(I_{\text{dc}}/\sqrt{6}\)
Ripple factor	Higher	Lower
Filter size	Larger	Smaller

6-pulse provides superior performance with lower ripple

Generalized m-Phase Half-Wave Converter

Concept:

Extension of 3-pulse and 6-pulse converters
\(m\)-phase star-connected secondary winding
\(m\) thyristors (one per phase)
Phase displacement: \(360^\circ/m\)

General Characteristics

Each thyristor conducts: \(360^\circ/m\) or \(2\pi/m\) rad
Output pulses per cycle: \(m\)
Higher \(m\) \(\Rightarrow\) Smoother output and lower ripple
Ripple frequency: \(m \times f\)

Average Output Voltage – General Formula

For \(m\)-phase half-wave converter

\(\boxed{V_{\text{dc}} = \frac{m}{2\pi}V_m\sin\left(\frac{\pi}{m}\right)\cos\alpha}\)

Verification for specific cases:

\(m = 3\): \(V_{\text{dc}} = \dfrac{3}{2\pi}V_m\sin 60^\circ \cos\alpha = 0.827\,V_m\cos\alpha \quad \checkmark\)
\(m = 6\): \(V_{\text{dc}} = \dfrac{6}{2\pi}V_m\sin 30^\circ \cos\alpha = 0.955\,V_m\cos\alpha \quad \checkmark\)
\(m = 12\): \(V_{\text{dc}} = \dfrac{12}{2\pi}V_m\sin 15^\circ \cos\alpha = 0.988\,V_m\cos\alpha \quad \checkmark\)

RMS Output Voltage – General Formula

General expression for \(m\)-phase converter:

\(V_{\text{rms}} = V_m\sqrt{\frac{1}{2\pi}\left[\frac{\pi}{m} + \frac{1}{2}\sin\left(\frac{2\pi}{m}\right)\cos 2\alpha\right]}\)

Key Observations:

As \(m\) increases, \(V_{\text{rms}}\) approaches \(V_m\)
Ripple content decreases with increasing \(m\)
Form factor approaches unity for large \(m\)
Better power quality with higher pulse numbers

Current Ratings – General Formulas

For \(m\)-phase half-wave converter:

Current Parameters

Average Thyristor Current: \(I_{T(\text{avg})} = \frac{I_{\text{dc}}}{m}\)

RMS Thyristor Current: \(I_{T(\text{rms})} = \frac{I_{\text{dc}}}{\sqrt{m}}\)

Peak Thyristor Current: \(I_{T(\text{peak})} = \frac{V_m}{R}\)

Conduction Angle per Thyristor: \(\theta_c = \frac{2\pi}{m} \text{ rad} = \frac{360^\circ}{m}\)

Effect of Pulse Number on Performance

Performance metrics vs. pulse number
Pulse No. (\(m\))	3	6	12	24
\(V_{\text{dc}}/V_m\) at \(\alpha=0^\circ\)	0.827	0.955	0.988	0.997
Ripple frequency	\(3f\)	\(6f\)	\(12f\)	\(24f\)
Conduction angle	\(120^\circ\)	\(60^\circ\)	\(30^\circ\)	\(15^\circ\)
Filter size	Large	Medium	Small	Very Small
Relative ripple	High	Medium	Low	Very Low
Complexity	Low	Medium	High	Very High

Trade-off

Higher pulse number provides better performance but increases system complexity and cost

Selection Criteria for Pulse Number

3-pulse converter:

Simple, low-cost applications
Low to medium power (\(<\) 10 kW)

6-pulse converter:

Most common for industry
Medium power (10–100 kW)
Good balance: performance vs. cost

12-pulse converter:

High power applications
HVDC transmission
Strict harmonic requirements

24-pulse converter:

Special applications
Very low harmonics required

Comprehensive Comparison

Complete comparison of multi-phase converters
Parameter	3-pulse	6-pulse	\(m\)-pulse
Number of thyristors	3	6	\(m\)
Output pulses/cycle	3	6	\(m\)
Ripple frequency	\(3f\)	\(6f\)	\(mf\)
Conduction angle	\(120^\circ\)	\(60^\circ\)	\(360^\circ/m\)
\(V_{\text{dc}}\) at \(\alpha=0^\circ\)	\(0.827\,V_m\)	\(0.955\,V_m\)	\(\frac{m}{2\pi}V_m\sin\frac{\pi}{m}\)
\(I_{T(\text{avg})}\)	\(I_{\text{dc}}/3\)	\(I_{\text{dc}}/6\)	\(I_{\text{dc}}/m\)
\(I_{T(\text{rms})}\)	\(I_{\text{dc}}/\sqrt{3}\)	\(I_{\text{dc}}/\sqrt{6}\)	\(I_{\text{dc}}/\sqrt{m}\)
Filter requirement	Large	Medium	Small
Transformer	Simple	Center-tap	Complex
Cost	Low	Medium	High

Key Performance Metrics Comparison

Performance at \(\alpha = 0^\circ\)
Parameter	3-pulse	6-pulse
Avg. output voltage	\(1.17\,V_{\text{ph}}\)	\(1.35\,V_{\text{ph}}\)
Form factor	\(\approx 1.016\)	\(\approx 1.001\)
Ripple factor	\(\approx 0.18\) (18%)	\(\approx 0.04\) (4%)
Ripple frequency	150 Hz @ 50 Hz	300 Hz @ 50 Hz
Conduction/thyristor	\(120^\circ\)	\(60^\circ\)

6-pulse: 4% ripple vs. 3-pulse: 18% ripple

Effect of Firing Angle on Output Voltage

Voltage-Angle Characteristic: \(V_{\text{dc}} = K\,V_m\cos\alpha\) where \(K = 0.827\) (3-pulse), \(K = 0.955\) (6-pulse)

Key Observations:

All converters follow \(\cos\alpha\) characteristic
Higher pulse \(\Rightarrow\) higher output voltage
Maximum voltage at \(\alpha = 0^\circ\)
Zero voltage at \(\alpha = 90^\circ\) (boundary)
Negative voltage (inverter) for \(\alpha > 90^\circ\) with RL load
Practical range: \(0^\circ \leq \alpha \leq 150^\circ\)

Advantages and Disadvantages

Advantages:

Simple topology
Natural commutation
Lower component count
Lower voltage stress
Bidirectional power flow
Cost-effective

Disadvantages:

DC component in transformer
Higher ripple vs. full-wave
Larger filter requirements
Unidirectional current
Limited to lower power

Industrial Applications

Three-Phase (3-pulse):

Battery charging systems
Electroplating processes
Simple DC power supplies
Low power DC motor drives
( \(<\) 10 kW)

Six-Phase (6-pulse):

Industrial DC motor drives
(10–100 kW)
Controlled rectification
Better power quality needs
Medium power DC supplies

Note

For high power applications (\(>\) 100 kW), full-wave converters are generally preferred

Harmonics and Power Quality

Output Voltage Harmonics:

Dominant frequency: \(mf\) (where \(m\) is pulse number)
Harmonic orders: \(m, 2m, 3m, 4m, \ldots\)
Amplitude decreases with harmonic order

For 3-pulse: Harmonics at \(3f, 6f, 9f, 12f, \ldots\)
For 6-pulse: Harmonics at \(6f, 12f, 18f, 24f, \ldots\)

Input Current Harmonics:

Non-sinusoidal current from AC supply
Contains DC component (star secondary)
Triplen harmonics present (3rd, 9th, 15th)
Power factor decreases with increasing \(\alpha\)
Significant Total Harmonic Distortion (THD)

Transformer Considerations

1. DC Magnetization:

DC current flows through secondary windings
Causes core saturation and increased magnetizing current
Reduced efficiency and increased losses
Solution: Delta-zigzag transformer connection

2. Transformer Utilization Factor (TUF):

Lower than full-wave converters
Each secondary winding partially utilized

3. Rating Requirements:

Must handle DC component and harmonics
Voltage rating based on peak phase voltage
Current rating based on load and harmonic content

Key Takeaways

Half-wave converters use one thyristor per phase
Firing angle \(\alpha\) controls average output: \(V_{\text{dc}} \propto \cos\alpha\)
Higher pulse number \(\Rightarrow\) smoother output, lower ripple
Inductance maintains current continuity in RL loads
Delta-zigzag transformer eliminates DC magnetization
Free-wheeling diode prevents negative output voltage
Ripple frequency = (pulse number) \(\times\) supply frequency
Trade-off: performance vs. complexity/cost
Natural commutation simplifies control

Introduction

Need for Three-Phase Rectifiers

Limitations of Single-Phase Rectifiers

Solution

Advantages of Three-Phase Controlled Rectifiers

Industrial Applications

Power Range

Classification of Three-Phase Rectifiers

Classification Categories

Key Principle

Three-Phase Half-Wave Controlled Rectifier

Circuit Configuration

Three-Phase Voltage System

Principle of Operation

Operating Parameters

Natural Commutation

Operation with Resistive Load

Operating Modes with Resistive Load

Continuous Conduction Mode Waveforms

Discontinuous Conduction Mode Waveforms

Average DC Output Voltage

Key Results

RMS Output Voltage

Important

Performance Parameters

Thyristor Current Ratings

Current Parameters

Operation with RL Load

Three-Phase Half-Wave with RL Load

Operating Ranges with RL Load

Waveforms for RL Load: \(\alpha = 60^\circ\)

Rectifier Mode: \(\alpha < 90^\circ\)

Average Output Voltage

Inverter Mode: \(90^\circ < \alpha < 150^\circ\)

Voltage and Current Relations with RL Load

For Pure Inductive Load (\(R \approx 0\))

Circuit Improvements

Delta-Zigzag Transformer Connection

Problem

Delta-Zigzag Transformer – Working Principle

Benefits

Free-Wheeling Diode Configuration

Operation with Free-Wheeling Diode

Output Voltage with Free-Wheeling Diode

Average DC Voltage (with FWD)

Six-Pulse Half-Wave Converter

Six-Pulse Half-Wave Converter Circuit

Advantages of Six-Pulse Converter

Six-Phase Voltage System

Principle of Operation

Output Voltage Equations

Key Result

Operating Modes of Six-Pulse Converter

Continuous Conduction Mode

Discontinuous Conduction Mode

Thyristor Current Ratings

Current Parameters

Performance Comparison: 3-Pulse vs 6-Pulse

Generalized m-Phase Converter

Generalized m-Phase Half-Wave Converter

General Characteristics

Average Output Voltage – General Formula

For \(m\)-phase half-wave converter

RMS Output Voltage – General Formula

Current Ratings – General Formulas

Current Parameters

Effect of Pulse Number on Performance

Trade-off

Selection Criteria for Pulse Number

Summary and Comparison

Comprehensive Comparison

Key Performance Metrics Comparison

Effect of Firing Angle on Output Voltage

Advantages and Disadvantages

Industrial Applications

Note

Harmonics and Power Quality

Transformer Considerations

Key Takeaways