Half-Wave Rectifiers Notes

Half-Wave Rectifiers Overview

  • Sections Covered:
    • 3.1/3.2: Single-phase Half-wave Rectifiers
    • 3.3: Resistive and inductive load
    • 3.7: Free-wheeling diode
    • 3.8: Half-wave Rectifiers with a capacitor filter
    • 3.9: The Controlled Half-wave Rectifier

Half-Wave Rectifier with Resistive Load

  • Average Output Voltage ($V{avg}$):
    V</em>avg=VmhetaV</em>{avg} = \frac{V_m}{ heta} where $ heta$ is the angle of conduction.
  • RMS Voltage ($V{rms}$):
    V</em>rms=Vm2V</em>{rms} = \frac{V_m}{2}
  • Equations:
    • I<em>avg=V</em>avgRI<em>{avg} = \frac{V</em>{avg}}{R}
    • P<em>avg=I</em>avgimesVavgP<em>{avg} = I</em>{avg} imes V_{avg}

Half-Wave Rectifier with Inductive Load

  • Current Response:
    V<em>mdi(t)dt+Ri(t)+Li(t)=V</em>mextsin(heta)V<em>m\frac{di(t)}{dt} + Ri(t) + Li(t) = V</em>m ext{ sin }( heta)
  • Forced Response:i(t)=i<em>f(t)+i</em>n(t)i(t) = i<em>f(t) + i</em>n(t)(total current)
    • Where $Z = R + j rac{1}{ heta}$
  • Natural Response:
    i(t)=Ae−t/au+exttermfromforcedresponsei(t) = A e^{-t/ au} + ext{term from forced response}
  • Extinction Angle ($eta$):
    • Current $i(t)$ goes to zero at extinction angle $eta$.

Free-Wheeling Diode

  • Prevents back EMF from causing negative voltages.
  • Configuration:
    • When $V_s > 0$: D1 conducts;
    • When $V_s < 0$: D2 conducts.
  • Maintains output voltage ($V_o$) positive.

Example Problem from Free-Wheeling Diode Section

  • Given:
    • Source: 240 V rms at 60 Hz
    • Load Resistance ($R = 8 \, ext{Ω}$)
    • Steps to Solve:
    • (a) Calculate average current, $I_o$.
    • (b) Find the average power delivered.
    • (c) Determine the power factor of the source.
    • (d) Find the current in each diode.
    • (e) Use a 1st harmonic approximation to calculate the inductance $L$ to limit ripple.

Capacitor Filter in Half-Wave Rectifiers

  • For small changes in output voltage ($ΔVo$): rac{Vm}{R C} < f
  • Used to smoothen the output and reduces ripple.

General Notes

  • Half-wave rectifiers are simpler but less efficient than full-wave rectifiers.
  • Output voltage/current waveforms can be further analyzed using Fourier series to estimate ripple levels.
  • Phase control techniques allow for different conduction times.