Electric Drives · Lecture 4A

Single-Phase Half-Controlled Rectifiers with Resistive and Resistive-Inductive Loads (RE & RLE)

Chopper-Controlled DC Motor Drives

Prof. Mithun Mondal BITS Pilani, Hyderabad Campus
SECTION 01

Firing Angle Analysis for Transient Current

Firing Angle to Ensure No Transient Current

Key Formula

The firing angle to ensure no transient current is:

\[\boxed{\alpha = \tan^{-1}\left(\frac{\omega L}{R}\right)}\]

  • Where:

    • \(\omega = 2\pi f\) is the angular frequency (rad/s)

    • \(L\) is the inductance (H)

    • \(R\) is the resistance (\(\Omega\))

  • This angle corresponds to the load impedance angle

  • At this firing angle, the transient component becomes zero

Maximum Transient Current Condition

Maximum Transient Condition

For maximum transient current:

\[\begin{aligned} \sin(\alpha - \phi) &= 1 = \sin 90^\circ\\ \therefore \quad \alpha &= 90^\circ + \phi \end{aligned}\]

  • Where \(\phi = \tan^{-1}\left(\frac{\omega L}{R}\right)\) is the load impedance angle

  • This represents the worst-case scenario for transient current

  • Practical implication: Avoid firing at this angle for smooth operation

SECTION 02

Half-Controlled Converters with RL Load and Freewheeling Diode

Circuit Configuration

  • Components:

    • Single-phase AC source

    • Thyristor (\(T_1\))

    • RL load (R, L)

    • Freewheeling diode (\(D_f\))

  • Purpose of freewheeling diode:

    • Provides path for inductive current

    • Improves converter performance

    • Reduces voltage stress on thyristor

Single-phase half-controlled converter with RL load and freewheeling diode
Single-phase half-controlled converter with RL load and freewheeling diode
SECTION 03

Discontinuous Mode Operation

Discontinuous Mode: Operating Principle

Mode I: Thyristor Conduction (\(\alpha < \omega t < \pi\))

  • Thyristor \(T_1\) is forward biased during positive half-cycle

  • Triggering pulse applied at \(\omega t = \alpha\)

  • Current flows: Source \(\rightarrow\) \(T_1\) \(\rightarrow\) Load

Mode II: Freewheeling Diode Conduction (\(\pi < \omega t < \pi + \beta\))

  • Input polarity reverses at \(\omega t = \pi\)

  • Inductive current continues through \(D_f\)

  • Current flows: \(D_f\) \(\rightarrow\) Load (decaying)

  • Current becomes zero at \(\omega t = \pi + \beta\)

Waveforms of a single-phase half-controlled converter with RL load and freewheeling diode (Discontinuous Mode)
Waveforms of a single-phase half-controlled converter with RL load and freewheeling diode (Discontinuous Mode)

Discontinuous Mode: Performance Analysis

Average Output Voltage

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

Average Output Current

\[\boxed{I_o = I_{dc} = \frac{V_m}{2\pi R} (1 + \cos \alpha)}\]

  • Note: \(V_m = \sqrt{2}V\) is the peak input voltage

  • Output voltage is higher compared to without freewheeling diode

  • Current ripple is reduced due to freewheeling action

SECTION 04

Continuous Mode Operation

Continuous Mode: Mode I Analysis

Mode I: Thyristor \(T_1\) Conduction (\(\alpha < \omega t < \pi\))

Voltage equation:

\[\sqrt{2} V \sin \omega t = L \frac{di_o}{dt} + Ri_o\]

Output current solution:

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

  • Where: \(Z = \sqrt{R^2 + (\omega L)^2}\) and \(\phi = \tan^{-1}\left(\frac{\omega L}{R}\right)\)

  • \(I_o\) is the initial current at \(\omega t = \alpha\)

Continuous Mode: Mode II Analysis

Mode II: Freewheeling Diode \(D_f\) Conduction (\(\pi < \omega t < 2\pi + \alpha\))

Voltage equation:

\[0 = L \frac{di_o}{dt} + Ri_o\]

Output current solution:

\[\boxed{i_o(t) = I_\pi e^{-\frac{R}{L}(\omega t - \pi)}}\]

  • \(I_\pi\) is the current at \(\omega t = \pi\) (end of Mode I)

  • Current decays exponentially with time constant \(\tau = \frac{L}{R}\)

  • Mode II continues until the next firing pulse at \(\omega t = 2\pi + \alpha\)

Waveforms of a single-phase half-controlled converter with RL load and freewheeling diode (Continuous Mode)
Waveforms of a single-phase half-controlled converter with RL load and freewheeling diode (Continuous Mode)

Advantages of Freewheeling Diode

Performance Improvements

  1. Higher output voltage: Better voltage utilization

  2. Improved power factor: Reduced reactive power demand

  3. Better current waveform: Reduced current ripple

  4. Enhanced efficiency: Energy stored in inductance is transferred to load resistance

  5. Reduced voltage stress: Lower reverse voltage across thyristor

Key Benefit

The stored magnetic energy in inductance \(L\) is efficiently transferred to the resistive load \(R\) during freewheeling, improving overall converter efficiency.

SECTION 05

Half-Wave Controlled Converters with RE Load

RE Load Configuration

  • Applications:

    • Battery charging systems

    • DC motor drives

    • Electroplating processes

  • Load components:

    • Resistance (R): Internal resistance

    • EMF source (E): Battery voltage

  • Operation principle:

    • Thyristor conducts when \(V_s > E\)

    • Current flows only during charging

Single-phase half-wave controlled rectifier with RE load
Single-phase half-wave controlled rectifier with RE load

RE Load: Operating Conditions

Conduction Conditions

Thyristor conducts when:

  • Input voltage \(V_s > E\) (forward bias condition)

  • Gate pulse is applied at \(\omega t = \alpha\)

Natural conduction angle:

\[\boxed{\beta_1 = \sin^{-1}\left(\frac{E}{\sqrt{2} V}\right)}\]

  • When \(V_s < E\): Thyristor is reverse biased and turns OFF

  • Extinction angle \(\beta_2 = \pi - \beta_1\) (due to symmetry)

  • Condition for conduction: \(\alpha \leq \beta_1\) for natural conduction

RE Load: Current and Power Analysis

Charging Current

During conduction period \(\beta_1 < \omega t < \beta_2\):

\[\boxed{i_o = \frac{\sqrt{2} V \sin \omega t - E}{R}}\]

Average Charging Current

\[\boxed{I_{dc} = \frac{1}{2\pi R} \int_{\beta_1}^{\beta_2} (\sqrt{2} V \sin \omega t - E) \, d(\omega t)}\]

  • Power delivered to battery: \(P_b = E I_{dc}\)

  • Power loss in resistance: \(P_R = I_{rms}^2 R\)

  • Rectifier efficiency: \(\eta = \frac{P_b}{P_b + P_R}\)

RE Load: Controlled Operation (\(\alpha > \beta_1\))

Modified Operating Range

When firing angle \(\alpha > \beta_1\):

  • Conduction period: \(\alpha < \omega t < \beta_2\)

  • Reduced charging time

  • Lower average charging current

Performance equations

Average charging current:

\[I_{dc} = \frac{1}{2\pi R} \int_{\alpha}^{\beta_2} (\sqrt{2} V \sin \omega t - E) \, d(\omega t)\]

RMS current:

\[I_{rms} = \sqrt{\frac{1}{2\pi R^2} \int_{\alpha}^{\beta_2} (\sqrt{2} V \sin \omega t - E)^2 \, d(\omega t)}\]
SECTION 06

Half-Wave Controlled Converters with RLE Load

RLE Load: Introduction and Applications

Load Components

  • R: Armature resistance / Internal resistance

  • L: Armature inductance / Circuit inductance

  • E: Back EMF / Counter EMF

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

Primary Applications

  • DC Motor Control: Armature circuit modeling

  • Battery Charging: With series inductance

  • Electrochemical Processes: Industrial applications

Key Characteristic

RLE load makes the most comprehensive load model for practical applications.

RLE Load: Operating Constraints

Minimum Firing Angle

\[\boxed{\delta_0 = \sin^{-1}\left(\frac{E}{\sqrt{2}V}\right)}\]

  • Constraint: \(\alpha \geq \delta_0\) for thyristor conduction

  • When \(\alpha < \delta_0\): Input voltage \(< E\), thyristor remains OFF

Operating Range

  • Minimum firing angle: \(\alpha_{min} = \delta_0\)

  • Maximum firing angle: \(\alpha_{max} = \pi - \delta_0\)

  • Practical range: \(\delta_0 \leq \alpha \leq \pi - \delta_0\)

RLE Load: Circuit equation and Solution

Voltage equation

When thyristor conducts (\(\alpha < \omega t < \beta\)):

\[\boxed{\sqrt{2}V \sin \omega t = L\frac{di}{dt} + Ri + E}\]

Complete Current Solution

\[\boxed{i(t) = \frac{\sqrt{2}V}{Z}\left[\sin(\omega t - \phi) - \sin(\alpha - \phi)e^{-\frac{R}{L}(t-\frac{\alpha}{\omega})}\right] - \frac{E}{R}\left[1 - e^{-\frac{R}{L}(t-\frac{\alpha}{\omega})}\right]}\]

  • Where: \(Z = \sqrt{R^2 + (\omega L)^2}\) and \(\tan\phi = \frac{\omega L}{R}\)

  • Current has both steady-state and transient components

RLE Load: Performance Parameters

Average Charging Current

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

Average Output Voltage

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

  • \(\beta\) is the extinction angle (when current becomes zero)

  • Both voltage and current depend on the conduction angle \((\beta - \alpha)\)

  • Control strategy: Vary \(\alpha\) to control average values

SECTION 07

Summary and Key Takeaways

Summary: Key Concepts

Firing Angle Considerations

  • No transient: \(\alpha = \tan^{-1}\left(\frac{\omega L}{R}\right)\)

  • Maximum transient: \(\alpha = 90° + \phi\)

Load Type Characteristics

  • RL Load + Freewheeling Diode: Improved performance, continuous current

  • RE Load: Battery charging, natural conduction constraints

  • RLE Load: DC motor control, combined effects of R, L, and E

Performance Enhancement

  • Freewheeling diode significantly improves converter performance

  • Proper firing angle selection minimizes transients

  • Load characteristics determine operating constraints