Single-Phase Full-Wave Controlled Rectifiers with Centre-Tapped Transformers

Analysis and Performance Characteristics

Introduction

Introduction to Single-Phase Controlled Rectifiers

Key Concepts

  • Controlled rectifiers convert AC to DC using thyristors (SCRs)

  • The output voltage can be controlled by varying the firing angle ( \(\alpha\))

  • Thyristors turn OFF when current becomes zero (natural commutation)

  • Conduction angle = \(\beta - \alpha\)

  • Circuit turn OFF time depends on the conduction angle

Applications

DC motor drives, battery chargers, electroplating, welding equipment

Single-Phase Full-Wave Controlled Rectifiers Using Centre Tap Transformer

Circuit Configuration

Circuit Features

  • Similar to single-phase full-wave uncontrolled rectifier but with SCRs instead of diodes

  • Uses a center-tap transformer

  • SCR cathodes are commonly connected

  • Single gate drive circuit can turn ON both SCRs

  • SCRs must withstand twice the peak supply voltage (\(2\sqrt{2}V\))

Single-phase full-wave controlled rectifier with centre-tapped transformer
Single-phase full-wave controlled rectifier with centre-tapped transformer

Note

The center-tap configuration provides better transformer utilization compared to bridge configuration

Operation with R Load

Operation with R Load - Positive Half Cycle

Positive Half Cycle Operation

  • During positive half cycle:

    • Thyristor \(T_1\) is forward biased

    • Thyristor \(T_2\) is reverse biased

    • When gate pulse is applied at \(\omega t = \alpha\), \(T_1\) conducts

    • Current flows through load resistance R and \(T_1\)

  • \(T_1\) conducts from \(\omega t = \alpha\) to \(\omega t = \pi\)

Operation with R Load - Negative Half Cycle

Negative Half Cycle Operation

  • During negative half cycle:

    • Thyristor \(T_2\) is forward biased

    • Thyristor \(T_1\) is reverse biased

    • When trigger pulse is applied at \(\omega t = \pi + \alpha\), \(T_2\) conducts

    • Current flows through load resistance R and \(T_2\)

  • \(T_2\) conducts from \(\omega t = \pi + \alpha\) to \(\omega t = 2\pi\)

  • Current flows through load in same direction for both half cycles

Waveform of a single-phase full-wave controlled rectifier with R-load
Waveform of a single-phase full-wave controlled rectifier with R-load

Analysis for R Load

Voltage and Current Analysis

Average output voltage:

\[V_{av} = V_{dc} = \frac{2\sqrt{2}V}{\pi}\cos\alpha\]

  • When \(\alpha = 0^{\circ}\): \(V_{av} = \frac{2\sqrt{2}V}{\pi}\) (maximum)

  • When \(\alpha = 90^{\circ}\): \(V_{av} = 0\)

  • When \(\alpha = 180^{\circ}\): \(V_{av} = -\frac{2\sqrt{2}V}{\pi}\) (minimum)

Average load current:

\[I_{av} = I_{dc} = \frac{V_{dc}}{R} = \frac{2\sqrt{2}V}{\pi R}\cos\alpha\]

Further Analysis for R Load

RMS Values and Power

RMS output voltage:

\[V_{rms} = V\sqrt{\frac{1}{\pi}\left[(\pi - \alpha) + \frac{1}{2}\sin 2\alpha\right]}\]

RMS output current:

\[I_{rms} = \frac{V_{rms}}{R} = \frac{V}{R}\sqrt{\frac{1}{\pi}\left[(\pi - \alpha) + \frac{1}{2}\sin 2\alpha\right]}\]

Power and Performance Parameters

Power Analysis

DC output power:

\[P_{dc} = V_{dc}I_{dc} = \frac{8V^2}{\pi^2 R}\cos^2\alpha\]

AC output power:

\[P_{ac} = V_{rms}I_{rms} = \frac{V^2}{\pi R}\left[(\pi - \alpha) + \frac{1}{2}\sin 2\alpha\right]\]

Transformer utilisation factor (TUF):

\[TUF = \frac{P_{dc}}{VA_{rating}} = \frac{4\cos^2\alpha}{\pi\left[(\pi - \alpha) + \frac{1}{2}\sin 2\alpha\right]}\]

Operation with RL Load

Operation with RL Load

RL Load Characteristics

  • Similar to R load but with inductive load

  • Load current may be continuous or discontinuous depending on inductance value

  • Inductance opposes changes in current, causing current to lag voltage

Single-phase full-wave centre-tapped controlled-rectifier with RL load
Single-phase full-wave centre-tapped controlled-rectifier with RL load

Current Modes

Discontinuous: Low inductance, current reaches zero
Continuous: High inductance, current never reaches zero

Operation with RL Load - Discontinuous Current

Discontinuous Current Mode

  • When inductance value is low:

    • Load current is discontinuous

    • Each thyristor conducts for less than \(180^{\circ}\)

    • \(T_1\) conducts from \(\omega t = \alpha\) to \(\omega t = \beta\)

    • \(T_2\) conducts from \(\omega t = \pi + \alpha\) to \(\omega t = \pi + \beta\)

Average output voltage:

\[V_{av} = V_{dc} = \frac{\sqrt{2}V}{\pi}(\cos\alpha - \cos\beta)\]

where \(\beta\) is the extinction angle determined by the circuit parameters.

Analysis for RL Load - Discontinuous Current

Current and Voltage Analysis

Average load current:

\[I_{av} = I_{dc} = \frac{V_{dc}}{R} = \frac{\sqrt{2}V}{\pi R}(\cos\alpha - \cos\beta)\]

RMS output voltage:

\[V_{rms} = V\sqrt{\frac{1}{\pi}\left[(\beta - \alpha) + \frac{1}{2}(\sin 2\alpha - \sin 2\beta)\right]}\]

RMS output current:

\[I_{rms} = \frac{V_{rms}}{R}\]
RL Load - voltage and current waveforms in discontinuous load current
RL Load - voltage and current waveforms in discontinuous load current

Operation with RL Load - Continuous Current

Continuous Current Mode

  • When inductance value is very large:

    • Load current is continuous

    • Each thyristor conducts for \(180^{\circ}\) duration

    • \(T_1\) conducts from \(\omega t = \alpha\) to \(\omega t = \pi + \alpha\)

    • \(T_2\) conducts from \(\omega t = \pi + \alpha\) to \(\omega t = 2\pi + \alpha\)

Average output voltage:

\[V_{av} = V_{dc} = \frac{2\sqrt{2}V}{\pi}\cos\alpha\]

Analysis for RL Load - Continuous Current

Voltage Control Range

  • When \(\alpha = 0^{\circ}\): \(V_{av} = \frac{2\sqrt{2}V}{\pi}\) (rectifier mode)

  • When \(\alpha = 90^{\circ}\): \(V_{av} = 0\)

  • When \(\alpha = 180^{\circ}\): \(V_{av} = -\frac{2\sqrt{2}V}{\pi}\) (inverter mode)

DC output voltage range:

\[-\frac{2\sqrt{2}V}{\pi} \leq V_{av} \leq \frac{2\sqrt{2}V}{\pi}\]

RMS output voltage:

\[V_{rms} = V\]

Current Expression for Continuous Mode

Output Current Analysis

Output current equation:

\[i_o(t) = \frac{\sqrt{2}V}{Z}\sin(\omega t - \phi) + A e^{-\frac{R}{L}(\omega t - \alpha)/\omega}\]

Where:

  • \(Z = \sqrt{R^2 + (\omega L)^2}\) is the load impedance

  • \(\phi = \tan^{-1}(\omega L/R)\) is the load phase angle

  • \(A\) is determined by boundary conditions

The constant \(A\) is found using the condition that current at the end of conduction equals the current at the beginning of the next conduction period.

RL Load - voltage and current waveforms in continuous load current
RL Load - voltage and current waveforms in continuous load current

Operation with RL Load and Free Wheeling Diode

Free Wheeling Diode Configuration

Circuit with Free Wheeling Diode

  • Free wheeling diode connected across RL load

  • Also known as half-controlled converter or semi-converter

  • Provides path for inductive current when SCRs are OFF

  • Prevents negative voltage across load

Single-phase full-wave controlled rectifier with RL load and FWD
Single-phase full-wave controlled rectifier with RL load and FWD

Advantages

  • Improves power factor

  • Reduces ripple in output current

  • Prevents reverse voltage across load

  • Improves converter efficiency

Operation with Free Wheeling Diode

Positive Half Cycle

  • \(T_1\) conducts when triggered at \(\omega t = \alpha\)

  • Current flows through \(T_1\) and load

  • At \(\omega t = \pi\), supply voltage becomes negative

  • \(T_1\) turns OFF, free wheeling diode \(D_F\) conducts

  • Inductive current continues through \(D_F\)

Negative Half Cycle

  • \(T_2\) conducts when triggered at \(\omega t = \pi + \alpha\)

  • Current flows through \(T_2\) and load

  • At \(\omega t = 2\pi\), \(T_2\) turns OFF

  • Free wheeling diode conducts again

Average Values with Free Wheeling Diode

Voltage and Current Analysis

Average output voltage:

\[V_o = V_{dc} = \frac{\sqrt{2} V_s}{\pi} (1 + \cos \alpha)\]

Average load current:

\[I_{dc} = \frac{V_o}{R} = \frac{\sqrt{2} V_s}{\pi R} (1 + \cos \alpha)\]

  • Output voltage is always positive

  • Better regulation compared to without free wheeling diode

  • Reduced current ripple

RMS Output Voltage Analysis

RMS Output Voltage Calculation

Mathematical Analysis

RMS output voltage:

\[V_o = \sqrt{\frac{1}{\pi} \int_{\alpha}^{\pi+\alpha} V_s^2 \sin^2 \omega t \, d(\omega t)}\]

After integration and simplification:

\[V_o = V_s \sqrt{\frac{1}{\pi} \left[ \pi - \alpha + \frac{\sin 2\alpha}{2} \right]}\]

Where:

  • \(V_s\) is the RMS input voltage per secondary winding

  • \(\alpha\) is the firing angle in radians

Circuit Analysis with Different Load Types

Voltage Equation for RL Circuit

Circuit Differential Equation

Voltage equation:

\[\sqrt{2} V_s \sin \omega t = L \frac{di_o}{dt} + R i_o\]

Valid for \(\alpha < \omega t < \pi + \alpha\) (conduction period)

Where:

  • \(V_s\) is the RMS input voltage

  • \(i_o\) is the instantaneous output current

  • \(L\) is the load inductance

  • \(R\) is the load resistance

Output Current Solution

Current Expression

Complete solution:

\[i_o(t) = \frac{\sqrt{2} V_s}{Z} \sin(\omega t - \phi) + A e^{-\frac{R}{L}(\omega t - \alpha)/\omega}\]

Where:

  • \(Z = \sqrt{R^2 + (\omega L)^2}\) is the load impedance

  • \(\phi = \tan^{-1}(\omega L/R)\) is the load phase angle

  • \(A\) is determined by boundary conditions

For continuous conduction: \(i_o(\alpha) = i_o(\pi + \alpha) = I_o\)

Operation with RLE Load

RLE Load Configuration

RLE Load Characteristics

  • Circuit operates similar to half-wave controlled rectifier with RLE load

  • Key difference: load current flows in both half cycles

  • Load current depends on:

    • Firing angle of thyristors (\(\alpha\))

    • Inductive load parameters (\(L\))

    • Ratio of battery voltage to transformer secondary voltage (\(E/V_s\))

Applications

Battery charging, DC motor drives with back EMF

Minimum Firing Angle for RLE Load

Critical Firing Angle

Minimum firing angle condition:

\[\sqrt{2} V_s \sin \alpha_{min} = E\]

Solving for minimum firing angle:

\[\alpha_{min} = \sin^{-1}\left(\frac{E}{\sqrt{2} V_s}\right)\]

Where:

  • \(E\) is the battery/back EMF voltage

  • \(V_s\) is the RMS secondary voltage per winding

  • For \(\alpha < \alpha_{min}\): No conduction occurs

Summary and Conclusions

Summary

Key Points

  • Single-phase full-wave controlled rectifiers use center-tap transformers with SCRs

  • Output voltage controlled by varying firing angle \(\alpha\) (\(0^{\circ}\) to \(180^{\circ}\))

  • Operation depends on load type:

    • R load: Simple operation, current follows voltage

    • RL load: Current may be continuous or discontinuous

    • RL with FWD: Improved performance, always positive output

    • RLE load: Battery charging applications

  • Free wheeling diode improves power factor and reduces ripple

  • Converter can operate in different modes based on firing angle

Performance Comparison

Load Type Comparison

Load Type Output Voltage Current Power Factor
R Load \(\frac{2\sqrt{2}V}{\pi}\cos\alpha\) Discontinuous Poor
RL Load \(\frac{2\sqrt{2}V}{\pi}\cos\alpha\) Continuous/Disc. Poor
RL + FWD \(\frac{\sqrt{2}V}{\pi}(1+\cos\alpha)\) Improved Better
RLE Load Variable Depends on E Variable