Single-Phase Half-Controlled Bridge Rectifiers

Analysis and Performance Characteristics

Single-Phase Half-Controlled Bridge Rectifier with R Load

Circuit Configuration

Key Features

  • Formed by replacing two thyristors of a full converter with diodes

  • Cost-effective solution with reduced control complexity

Configuration Types:

  • Symmetrical configurations:

    • Common cathode

    • Common anode

  • Asymmetrical configurations

Advantages:

  • Common cathode type most commonly used

  • Single triggering circuit for both thyristors

  • Simplified control scheme

Circuit diagrams: (a) common-cathode, (b) common anode, (c) asymmetrical
Circuit diagrams: (a) common-cathode, (b) common anode, (c) asymmetrical

Operation Principle

Positive Half Cycle

  • Thyristor \(T_1\) is forward biased

  • At \(\omega t = \alpha\): triggering pulse applied

  • \(T_1\) turns ON, current flows through \(T_1\) and \(D_2\)

  • At \(\omega t = \pi\): \(T_1\) reverse biased and turns OFF

Negative Half Cycle

  • Thyristor \(T_3\) is forward biased

  • At \(\omega t = \pi + \alpha\): triggering pulse applied

  • \(T_3\) turns ON, current flows through \(T_3\) and \(D_4\)

  • At \(\omega t = 2\pi\): \(T_3\) reverse biased and turns OFF

Key Point

Each thyristor conducts for \((\pi - \alpha)\) duration per cycle

Voltage and Current Waveforms

Voltage and current waveforms of single-phase half-controlled bridge-rectifier with R-load
Voltage and current waveforms of single-phase half-controlled bridge-rectifier with R-load

Waveform Characteristics

  • positive half-cycles

    Output voltage waveform consists of of input voltage

  • Load current waveform follows voltage waveform (resistive load)

  • Conduction periods:

    • \(T_1\): \(\alpha\) to \(\pi\) in positive half cycle

    • \(T_3\): \(\pi + \alpha\) to \(2\pi\) in negative half cycle

Performance Parameters

Average Output Voltage:

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

Average Load Current:

\[I_o = I_{dc} = \frac{V_o}{R}\]
\[= \frac{V_m}{\pi R}(1 + \cos\alpha)\]

RMS Output Voltage:

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

RMS Output Current:

\[I_{rms} = \frac{V_{rms}}{R}\]

Quality Factors

Form Factor:

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

Ripple Factor:

\[RF = \sqrt{FF^2 - 1}\]
\[= \sqrt{\frac{\pi\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)}{2(1 + \cos\alpha)^2} - 1}\]

Note

  • Form factor increases with firing angle \(\alpha\)

  • Higher ripple factor indicates more AC content in output

Single-Phase Half-Controlled Bridge Rectifier with RL Load

Circuit Configuration and Key Differences

Configuration Types

Similar to R load case: Common cathode, Common anode, Asymmetrical

Impact of Inductance

  • Load current continues to flow at the end of each half cycle

  • Freewheeling action occurs through diodes

  • Current cannot change instantaneously due to \(L\frac{di}{dt}\) constraint

Circuit configurations for RL load: (a) common-cathode, (b) common anode, (c) asymmetrical
Circuit configurations for RL load: (a) common-cathode, (b) common anode, (c) asymmetrical

Operation Principle - Detailed Analysis

Positive Half Cycle (\(\alpha < \omega t < \pi\))

  • Thyristor \(T_1\) and diode \(D_2\) conduct

  • Supply voltage directly connected across load

  • Current builds up according to RL circuit dynamics

Freewheeling Period (\(\pi < \omega t < \pi + \alpha\))

  • Input voltage becomes negative, but current cannot reverse instantly

  • Diode \(D_1\) comes into conduction (freewheeling)

  • Load current flows through \(T_1\) and \(D_1\)

  • Current decays exponentially with time constant \(\tau = L/R\)

Negative Half Cycle (\(\pi + \alpha < \omega t < 2\pi\))

  • Thyristor \(T_3\) and diode \(D_4\) conduct

  • Process repeats for negative half cycle

Continuous vs Discontinuous Mode

Continuous Mode

  • Load current never reaches zero

  • Rectangular input current waveform

  • Magnitude equals load current during conduction

Discontinuous Mode

  • Load current reaches zero during each half cycle

  • Output voltage equals zero when current is zero

  • Conduction period depends on \(\tau = L/R\)

Performance Parameters for RL Load

Average Output Voltage:

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

RMS Output Voltage:

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

Important Note

For continuous conduction mode, voltage expressions remain same as R load case, but current analysis becomes more complex due to inductive effects.

Single-Phase Half-Controlled Bridge Rectifier with RLE Load

Circuit Configuration with Back EMF

Load Components

  • R: Resistance representing losses

  • L: Inductance providing current smoothing

  • E: Back EMF (e.g., from DC motor or battery)

Single-phase half-controlled bridge rectifier with RLE load
Single-phase half-controlled bridge rectifier with RLE load

Operation Principle

  • Load current flows through either \(T_1\) or \(T_3\) and one diode

  • Positive half cycle: \(T_1\) and \(D_2\) conduct (\(\alpha < \omega t < \pi\))

  • Negative half cycle: \(T_3\) and \(D_4\) conduct (\(\pi + \alpha < \omega t < 2\pi\))

  • Freewheeling through \(T_1\) & \(D_1\) or \(T_3\) & \(D_3\)

Continuous Load Current Mode

Operating Conditions

Load current \(i_o\) is always greater than zero

Voltage Equation During Conduction:

\[\sqrt{2}V\sin\omega t = L\frac{di_o}{dt} + Ri_o + E \quad \text{for } \alpha < \omega t < \pi\]

Current Expression:

\[i_o(t) = \frac{\sqrt{2}V}{Z}\left[\sin(\omega t - \phi) - \sin(\theta - \phi)e^{-\frac{R}{L}t}\right]\]
Where:
\[\begin{aligned} Z &= \sqrt{R^2 + (\omega L)^2} & \tan\phi &= \frac{\omega L}{R} \\ \sin\theta &= \frac{E}{\sqrt{2}V} & \tau &= \frac{L}{R} \end{aligned}\]

Continuous Mode Characteristics

Key Characteristics

  • Load current is continuous and always positive

  • Output voltage follows input voltage during thyristor conduction

  • Output voltage is zero during freewheeling periods

  • \(T_1\) conducts from \(\alpha\) until \(T_3\) is fired at \(\pi + \alpha\)

Discontinuous Load Current Mode

Operating Conditions

Load current \(i_o\) becomes zero for certain time periods

Operation Sequence

  • When \(T_1\) triggered at \(\omega t = \alpha\) and output voltage \(>\) E:

    • Current starts from zero and increases

    • Current increases until \(\omega t = \pi - \theta\) (where output voltage = E)

    • After \(\omega t = \pi - \theta\): output voltage \(<\) E, current decreases

    • At \(\omega t = \beta\): current becomes zero before \(T_3\) is triggered

  • During \(\beta < \omega t < \pi + \alpha\):

    • No devices conduct

    • Output voltage equals back EMF E

Discontinuous Mode Waveforms

Important Observations

  • Load current starts and ends at zero during each half cycle

  • Output voltage equals back EMF E when no devices conduct

  • Conduction angle \(\gamma = \beta - \alpha\) depends on:

    • Firing angle \(\alpha\)

    • Load parameters (R, L, E)

    • Input voltage magnitude

Performance Analysis with High Inductive Load

High Inductance Approximation

Assumptions

When inductance is very high: load current is continuous with negligible ripple

Input Current Expression:

\[i_s(t) = \begin{cases} I_o & \text{for } \alpha < \omega t < \pi \\ -I_o & \text{for } \pi + \alpha < \omega t < 2\pi \\ 0 & \text{otherwise} \end{cases}\]

Fourier Series Representation:

\[i_s(t) = \sum_{n=1,3,5...}^{\infty} \frac{4I_o}{n\pi} \cos\frac{n\alpha}{2} \sin\left(n\omega t - \frac{n\alpha}{2}\right)\]

RMS Fundamental Current:

\[I_{s1} = \frac{2\sqrt{2}I_o}{\pi} \cos\frac{\alpha}{2}\]

Power Quality Parameters

Displacement Factor:

\[DF = \cos\phi_1 = \cos\frac{\alpha}{2}\]

Distortion Factor:

\[CDF = \frac{I_{s1}}{I_s} = \frac{2\sqrt{2}}{\pi} \cos\frac{\alpha}{2}\]

Power Factor:

\[PF = CDF \times DF\]
\[= \frac{2\sqrt{2}}{\pi} \cos^2\frac{\alpha}{2}\]

Harmonic Factor:

\[HF = \sqrt{\left(\frac{I_s}{I_{s1}}\right)^2 - 1}\]

Power Analysis

Active Input Power:

\[P_i = V \times I_{s1} \times \cos\frac{\alpha}{2} = \frac{2\sqrt{2}VI_o}{\pi} \cos^2\frac{\alpha}{2}\]

Reactive Power Input:

\[Q_i = V \times I_{s1} \times \sin\frac{\alpha}{2} = \frac{\sqrt{2}VI_o}{\pi} \sin\alpha\]

Power Factor Implications

  • Power factor decreases as firing angle \(\alpha\) increases

  • At \(\alpha = 0^{\circ}\): Maximum power factor = \(\frac{2\sqrt{2}}{\pi} \approx 0.9\)

  • At \(\alpha = 90^{\circ}\): Power factor = \(\frac{\sqrt{2}}{\pi} \approx 0.45\)

Centre Tap Transformer with Leakage Inductance

Effect of Transformer Leakage Inductance

Circuit Components

  • Centre tap transformer with leakage inductance \(L_c\)

  • RL load

  • Two thyristors

Single-phase full-wave controlled rectifier using centre-tapped transformer with RL load and leakage inductance
Single-phase full-wave controlled rectifier using centre-tapped transformer with RL load and leakage inductance

Commutation Process

  • Current cannot transfer instantaneously between thyristors

  • Finite commutation time required

  • Commutation overlap interval (\(u\)) occurs

  • During overlap: both thyristors conduct simultaneously

Physical Interpretation

Leakage inductance opposes sudden current changes, causing gradual transfer of current from outgoing to incoming thyristor.

Commutation Analysis

Commutation Process

  • Initially: one thyristor conducts full load current

  • At \(\omega t = \alpha\): second thyristor is triggered

  • During overlap (\(\alpha\) to \(\alpha + u\)):

    • Both thyristors conduct

    • Current transfers gradually from first to second thyristor

    • Short circuit occurs in transformer secondary

  • At \(\omega t = \alpha + u\): commutation complete

Overlap Angle Relationship:

\[\cos(\alpha + u) = \cos\alpha - \frac{2\omega L_c I_a}{\sqrt{2}V}\]

Voltage Reduction Due to Commutation

Voltage Drop During Commutation:

\[\Delta V_{av} = \frac{\sqrt{2}V}{\pi} [\cos\alpha - \cos(\alpha+u)] = \frac{2\omega L_c I_a}{\pi}\]

Actual Average Output Voltage:

\[V_{dc} = \frac{2\sqrt{2}V}{\pi} \cos\alpha - \frac{2\omega L_c I_a}{\pi}\]

Key Insights

  • Average output voltage is reduced by commutation reactance

  • Voltage drop is proportional to load current

  • Higher leakage inductance \(\Rightarrow\) greater voltage drop

Bridge Rectifier with Source Inductance

Full-Wave Controlled Bridge with Source Inductance

Configuration

  • Four thyristors in bridge configuration

  • RL load with source inductance \(L_s\)

  • Assumptions: continuous conduction, ripple-free load current

Single-phase full-wave controlled bridge rectifier with source inductance and RL load
Single-phase full-wave controlled bridge rectifier with source inductance and RL load

Source Inductance Effects

  • Output voltage not constant during conduction

  • Input current waveform changes significantly

  • Commutation overlap occurs between thyristor pairs

Commutation Process in Bridge Rectifier

Commutation Sequence

  • Initially: \(T_3\) and \(T_4\) conduct

  • At \(\omega t = \alpha\): \(T_1\) and \(T_2\) are triggered

  • During overlap (\(\alpha\) to \(\alpha + u\)):

    • All four thyristors conduct

    • Load current freewheels through all devices

    • Output voltage becomes zero

    • Input current changes polarity gradually

  • At \(\omega t = \alpha + u\): \(T_3\) and \(T_4\) turn OFF

Mathematical Analysis of Commutation

Voltage Equation During Overlap:

\[v_s = L_s \frac{di}{dt} \quad \text{for} \quad \alpha < \omega t < \alpha+u\]
where \(v_s = \sqrt{2}V \sin(\omega t)\)

Current During Overlap:

\[i = \frac{\sqrt{2}V}{\omega L_s} (\cos\alpha - \cos(\omega t)) - I_a\]

Overlap Angle:

\[\cos(\alpha+u) = \cos\alpha - \frac{\omega L_s I_a}{\sqrt{2}V}\]

Corrected Average Output Voltage:

\[V_{dc} = \frac{2\sqrt{2}V}{\pi} \cos\alpha - \frac{2\omega L_s I_a}{\pi}\]

Equivalent Circuit Model

Simplified Model

  • DC voltage source: \(\frac{2\sqrt{2}V}{\pi} \cos\alpha\)

  • Commutation resistance: \(R_c = \frac{2\omega L_s}{\pi}\)

  • Series combination represents rectifier behavior

Physical Meaning

Commutation resistance represents:

  • Voltage drop across source inductance

  • Power loss during commutation

Equivalent circuit with commutation resistance
Equivalent circuit with commutation resistance

Waveforms with Source Inductance

Waveform Analysis

  • Input voltage: sinusoidal AC supply

  • Output voltage: reduced due to commutation overlap

  • Load current: continuous and smooth

  • Thyristor currents: show overlap periods

  • Input current: trapezoidal shape due to commutation

Voltage and current waveforms showing commutation overlap effects
Voltage and current waveforms showing commutation overlap effects

Performance Parameters of Controlled Rectifiers

Fundamental Performance Metrics

Rectification Efficiency: \(\eta = \frac{P_{dc}}{P_{ac}} \times 100\%\) where:

\[\begin{aligned} P_{dc} &= V_{dc} \times I_{dc} \\ P_{ac} &= V_{rms} \times I_{rms} \end{aligned}\]

Form Factor: \(FF = \frac{V_{rms}}{V_{dc}}\) Voltage Ripple Factor: \(RF = \sqrt{FF^2 - 1}\)

Quality Indicators

  • Lower ripple factor indicates better DC quality

  • Higher efficiency indicates better power conversion

  • For ripple-free load current: \(CRF = 0\)

Power Factor Analysis

RMS Fundamental Current: \(I_1 = \frac{2\sqrt{2}I_a}{\pi\sqrt{1 + \cos\alpha}}\)

Displacement Factor: \(DF = \cos\phi_1 = \cos\left(\frac{\alpha}{2}\right)\)

Distortion Factor: \(CDF = \frac{2\sqrt{2}}{\pi\sqrt{1 + \cos\alpha}}\)

Overall Power Factor: \(PF = CDF \times DF\)

Power Components:

\[\begin{aligned} P &= VI\cos\phi \\ Q &= VI\sin\phi \end{aligned}\]

Comparative Analysis

Performance Comparison

Parameter R Load RL Load RLE Load
Average Voltage \(\frac{\sqrt{2}V}{\pi}(1+\cos\alpha)\) Same Same
Current Mode Discontinuous Continuous/Disc. Continuous/Disc.
Freewheeling No Yes Yes
Back EMF Effect No No Yes
Control Range \(0^{\circ} \leq \alpha \leq 180^{\circ}\) \(0^{\circ} \leq \alpha \leq 180^{\circ}\) Limited by E

Key Differences

  • R Load: Simplest analysis, current follows voltage

  • RL Load: Freewheeling improves current continuity

  • RLE Load: Back EMF limits control range and affects mode of operation

Design Considerations

Advantages

  • Reduced cost (fewer thyristors)

  • Simpler control circuitry

  • Natural freewheeling through diodes

  • Good performance for many applications

Limitations

  • Lower power factor compared to full-controlled

  • Higher harmonic content

  • Limited control range with back EMF loads

  • Voltage drop due to commutation inductance

Applications

  • DC motor drives with moderate performance requirements

  • Battery charging systems

  • DC power supplies with cost constraints

Summary and Conclusions

Key Takeaways

Circuit Configurations

  • Half-controlled bridge rectifiers use two thyristors + two diodes

  • Common cathode configuration most widely used

  • Different load types require different analytical approaches

Operating Characteristics

  • R Load: Current discontinuous, follows voltage waveform

  • RL Load: Freewheeling action, continuous/discontinuous modes possible

  • RLE Load: Back EMF affects conduction, mode depends on load parameters

Practical Considerations

  • Leakage/source inductance causes commutation overlap

  • Results in voltage reduction and altered current waveforms

  • Can be modeled using equivalent circuit with commutation resistance

Performance Summary

Critical Performance Metrics

  • Average output voltage: \(V_{dc} = \frac{\sqrt{2}V}{\pi}(1 + \cos\alpha)\)

  • Power factor: Decreases with increasing firing angle

  • Harmonic distortion: Significant due to non-sinusoidal input current

  • Commutation effects: Reduce output voltage by \(\frac{2\omega L I_a}{\pi}\)

Design Guidelines

  • Choose firing angle based on voltage regulation requirements

  • Consider power factor implications for AC supply

  • Account for commutation effects in practical designs

  • Use appropriate filtering for output ripple reduction