Power Electronics -PE-EC505C- Module2 ( MAKAUT-Syllabus)

Today, Our Communication Elements Are: - Controlled Rectifiers – Single Phase Study, Analysis of Load Voltage and Input Current – Derivations of Form Factor and Ripple Factor, Effect of Source Impedance in Controlled Rectifiers, Input Current Fourier Series Analysis of Input Current to Derive Input Supply Power Factor, Displacement Factor and Harmonic Factor in Power Electronics.

Controlled Rectifiers – Single Phase Study: - 

Introduction to Controlled Rectifiers:

Controlled rectifiers are power electronic circuits that convert alternating current (AC) into direct current (DC) using semiconductor devices, such as thyristors (SCRs). Unlike uncontrolled rectifiers, which use diodes, controlled rectifiers enable us to control the output voltage and current by varying the firing angle of thyristors. They are widely used in motor control, power supplies, battery charging, and industrial drives.

Types of Converters:

• Semi-controlled rectifiers: A combination of diodes and thyristors provides limited control.
• Fully controlled rectifiers: Use only thyristors and offer complete control of output DC.
• They can be half-wave, semi-bridge, or full-bridge depending on the configuration.

Single-Phase Semi-Bridge Converter:

• A semi-converter uses a combination of SCRs and diodes.
• Generally, it has two SCRs and two diodes in a bridge form.
• Works as a half-controlled rectifier → only partial control over the output.

Working:

During the positive half cycle, the thyristor is triggered at angle α, allowing current to flow through the load. During the negative half cycle, the diode conducts, maintaining current through the RL load.

Load Conditions

R load (Resistive) – Output voltage follows the input sine wave when the SCR is triggered. Current and voltage are in phase, with no energy storage.

RL load (Inductive) - Current lags behind voltage. When the input goes negative, current continues through the freewheeling diode, ensuring continuous load current.

RLE load (DC motor) – The back EMF (E) opposes the current. The controlled rectifier adjusts the average voltage, thus controlling the motor speed.

Average DC output voltage -  Vdc = (Vm/π) (1 + cos α)

Single-Phase Full-Bridge Converter:

• A full-bridge converter has four SCRs arranged in bridge form.
• It is a fully controlled rectifier – complete control over output voltage and current

Working:

Two thyristors conduct at a time, depending on the firing angle α. Output is available during both half-cycles, providing full-wave rectification.

Load Conditions

R load – Output waveform follows AC input but only in positive half cycles (controlled by firing angle). Output is pulsating DC. proportional to cos α

RL load – Current lags, so freewheeling paths are needed. The load current can remain continuous even when the voltage is zero.

RLE load – Back EMF causes regenerative operation (power flows back to the AC source if the firing angle is large). Useful in motor braking. Can provide inversion if α > 90°.

Average DC output voltage - Vdc = (2Vm/π) cos α

Analysis of Different Loads:

R load – Current waveform same as voltage, no energy storage.

RL load – Inductor stores energy, causing delayed current; output smoother.

RLE load – Includes back EMF (E). If Vdc < E, inverter mode occurs.

Level Load – Represents constant current drawn by certain devices, approximates practical systems.

Performance Parameters

Form Factor (FF): Ratio of RMS to the average value of output voltage.
Ripple Factor (RF): Measures AC ripple in the output DC.
Input Current: Contains harmonics; affects power quality.
Power Factor: Product of displacement factor (cos φ) and distortion factor.
Source Impedance Effect: Causes a voltage drop, reduces output.

Comparison: Semi vs Full Bridge

• Semi-Bridge: Lower output, simpler design, one controlled device.
• Full-Bridge: Higher output, complete control, more devices, and cost.

Practical Applications

• DC motor speed control
• Battery charging
• Welding equipment
• Power supplies for electronic devices
• HVDC transmission systems

Conclusion

Controlled rectifiers are essential in modern power electronics. Single-phase semi- and full-bridge converters provide a variable DC output for different types of loads. Their performance depends on the load characteristics and firing angle. They form the backbone of many industrial and domestic applications requiring DC power.

Analysis of Load Voltage and Input Current –

 Derivations of Form Factor and Ripple

 Factor: -

Introduction

In controlled rectifiers, it is essential to analyze both the output voltage and the input current
to evaluate the performance of the converter. Depending on the load (R, RL, or RLE), the output
The waveform changes significantly. Performance parameters such as form factor and ripple factor help
in quantifying the smoothness and quality of the DC output.

Load Voltage Analysis: 

The average DC output voltage is obtained by integrating the instantaneous voltage over one cycle.
For a single-phase full bridge rectifier, the expression is:

Vdc = (2Vm/π) cos α

Where:
- Vm = maximum value of input voltage
- α = firing angle of thyristors

The RMS value of the load voltage is given as:
Vrms = √(1/π ∫(α to π) Vm² sin²θ dθ)

Input Current Analysis: 

The input current drawn from the supply is not sinusoidal because the thyristors conduct only
for a part of the input cycle. This current contains harmonics, which can be analyzed using
Fourier series expansion. These harmonics affect the input power factor, which is defined as:

Power Factor = Displacement Factor × Distortion Factor

Form Factor (FF):

Form factor is the ratio of the RMS value to the average value of the output voltage:
FF = Vrms / Vdc

For an ideal single-phase full-bridge rectifier:
Vrms = Vm/√2, Vdc = (2Vm/π)
So, FF = (Vm/√2) / (2Vm/π) = π / (2√2) ≈ 1.11

Ripple Factor (RF):

Ripple Factor measures the unwanted AC component in the output and is given as:
RF = √((Vrms/Vdc)² – 1)

For a single-phase full-bridge rectifier:
RF = √((π/(2√2))² – 1) ≈ 0.48

This means that 48% of the output contains ripple. Filters are used to reduce this ripple for
a smoother DC output.

Diagrams and Waveforms:

Below is a comparison of AC input and rectified output voltage waveforms:

Conclusion

From the analysis of load voltage and input current, we observe that the quality of DC output depends on the firing angle and the type of load. Form factor and ripple factor are important indices to evaluate the smoothness of the DC output. A lower ripple factor indicates a better quality DC output. To improve the performance, filters are used in practical rectifier circuits.

Effect of Source Impedance in Controlled

 Rectifiers: -

Introduction

In practical power systems, the AC supply feeding a controlled rectifier is not ideal.
It always contains some amount of source impedance, consisting of resistance (Rs) and inductance (Ls) of the transformer and transmission line.
This source impedance affects the performance of the rectifier by reducing the output voltage and changing the shape of the current and voltage waveforms.

What is the source impedance

Source impedance is the internal opposition offered by the power source to the flow of current.
In AC supply systems, this impedance can be represented as:
   
    Zs = Rs + jωLs
   
Where Rs is the resistance, Ls is the source inductance, and ω is the angular frequency of the AC source.
The presence of this impedance causes a voltage drop (Is * Zs), where Is is the supply current.

Effect on Controlled Rectifiers

When source impedance is present, the following effects can be observed in single-phase controlled rectifiers:

1. Reduction in Output Voltage: 
   The average DC output voltage decreases because part of the input voltage is dropped across Rs and Ls.

2. Delayed Commutation: 
   The inductive component Ls delays the transfer of current from one thyristor to another, leading to an overlap period.

3. Overlap Angle (μ): 
   The time during which both incoming and outgoing thyristors conduct simultaneously is called the overlap angle μ.
  This angle increases with higher source inductance.

4. Harmonics in Input Current: 
   Source impedance distorts the input current waveform, introducing harmonics and reducing power quality.

5. Reduction in Power Factor: 
   Due to phase shift and waveform distortion, the power factor of the system decreases.

Mathematical Representation

The average DC output voltage of a single-phase full bridge rectifier with source inductance Ls is given as:

    Vdc = (2Vm/π) cos(α + μ)
   
Where: 
- Vm = Peak input voltage 
- α = Firing angle 
- μ = Overlap angle (depends on Ls and load current)

Thus, the higher the source impedance, the greater the overlap angle and the lower the output DC voltage.

Practical Considerations

• Transformers and transmission lines always introduce some impedance. 
• In high-power converters, the effect of source inductance is significant. 
• Designers often minimise Rs and Ls using proper conductor sizing and transformer design. 
• Snubber circuits and commutation circuits are used to mitigate the adverse effects of source impedance.

Conclusion

The presence of source impedance in controlled rectifiers reduces output voltage, causes an overlap angle, and deteriorates the input power factor. Proper design and compensation techniques are required to reduce its impact in practical power electronic systems.

Input Current Fouxrier Series Analysis of Input Current to Derive Input Supply Power Factor: -

Introduction

In controlled rectifiers, the input current is not a pure sinusoidal waveform.
Instead, it is a non-sinusoidal, periodic waveform due to the switching nature of thyristors.
This current contains harmonics, which must be analyzed to evaluate the true power factor.
Fourier series analysis is the most effective method for representing and studying these waveforms.

Input Current in Controlled Rectifiers

When a controlled rectifier draws current from an AC source,
the current waveform depends on the firing angle (α) and the load type. The waveform is discontinuous,
non-sinusoidal, and rich in harmonics. Mathematically, it can be expressed as a Fourier series containing
a fundamental component (same frequency as the source) and higher-order harmonics.

Fourier Series Representation

A non-sinusoidal current waveform i(t) with period T can be written as:

i(t) = I0 + Σ [ (an cos(nωt)) + (bn sin(nωt)) ] , n = 1, 2, 3 ...

Where:

• I0 = DC component (usually zero for AC side)
• an, bn = Fourier coefficients
• n = harmonic order

The fundamental component (n=1) contributes to real power transfer, while harmonics (n=3,5,7…)
increase losses and distortions.

Power Factor Concept

Power factor (PF) is defined as the ratio of real power (P)
to apparent power (S). In controlled rectifiers, it is influenced by:
Displacement Factor (cos φ1): Angle between fundamental current and voltage.
Distortion Factor: Due to harmonic components.
Thus, the overall power factor is:

PF = Displacement Factor × Distortion Factor

PF = cos(φ1) × (I1 / Irms)

Where:

• I1 = RMS value of fundamental current
• Irms = RMS value of total current (including harmonics)

Derivation of Supply Power Factor

If input current is expressed as:

i(t) = I1sin(ωt - φ1) + Σ In sin(nωt - φn)

Then,

• Real power: P = V1 I1 cos φ1
• Apparent power: S = Vrms × Irms
• Power Factor: PF = P / S

Hence,
PF = (V1 I1 cos φ1) / (Vrms Irms)

This shows that PF is lowered not only by phase displacement but also by harmonic distortion.

Applications & Importance

• Evaluating PF helps in designing rectifiers with improved efficiency.
• Harmonic analysis allows estimation of filter requirements.
• Poor PF increases losses and voltage drops in supply systems, requiring correction devices such as capacitors or filters.

Conclusion

Fourier series analysis of input current is a powerful tool to evaluate
the impact of harmonics and displacement on input supply power factor.
For controlled rectifiers, both displacement angle and harmonic distortion
must be considered to obtain the true PF, which is essential for efficient power system design.

Displacement Factor and Harmonic Factor in

 Power Electronics: -

Introduction:

In power electronic circuits like controlled rectifiers, the input current drawn from the AC supply is not a pure sine wave.

• It may be phase-shifted because of the firing angle delay of thyristors.
• It may also contain harmonics due to switching and non-linear current flow.

This results in a low power factor, which reduces the efficiency of power transfer. To analyse this, we divide the power factor into two important parts:

• Displacement Factor (DF) → relates to the phase shift between current and voltage.
• Harmonic Factor (HF) → relates to distortion caused by harmonics.

 

Displacement Factor (DF):

Definition: The displacement factor is the cosine of the angle between the fundamental component of the input current and the supply voltage. 

DF=cos(ϕ1​)

Where:

• ϕ1​ = phase angle between supply voltage and fundamental current.

Key Points:

• In an uncontrolled rectifier (diode), DF ≈ 1 (since current is almost in phase with voltage).
• In a controlled rectifier (thyristor), DF decreases as the firing angle (α) increases, because the current waveform shifts.

Harmonic Factor (HF):

Definition: The harmonic factor measures the distortion in the current waveform due to harmonics.

$$HF = \frac{\sqrt{I_{rms}^2 - I_1^2}}{I_1}$$

Where:

• $I_{rms}$ = RMS value of total current.

• $I_1$ = RMS value of fundamental current.

Key Points:

• A low HF means fewer harmonics and better power quality.
• A high HF indicates strong harmonic distortion, leading to losses, heating, and poor efficiency.

Impact of High HF:

• Causes additional copper and iron losses in machines.
• Increases heating in transformers and transmission lines.
• Introduces electromagnetic interference (EMI).

Overall Power Factor:

he overall Power Factor (PF) of a converter is given by:

PF=DF×Distortion Factor

• If DF is high but HF is large, PF still becomes poor.
• Both DF and HF must be optimized for efficient rectifier operation.

Practical Applications:

• In controlled rectifiers, DF and HF determine how effectively AC power is converted to DC.
• Filters are used to minimize HF.
• Firing angle control is optimized to balance DF and efficiency.
•  In industries, reducing harmonics prevents overheating in motors, transformers, and cables.

Conclusion:

• The Displacement Factor (DF) shows the effect of the phase shift between current and voltage.
• Harmonic Factor (HF) reflects distortion due to harmonics.
• Together, they determine the overall power factor of the system.
• Maintaining high DF and low HF ensures efficient, reliable, and high-quality power conversion.

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