Half-Wave Rectifiers Notes

Half-Wave Rectifiers

Section 3: Overview

• Single-phase Half-wave Rectifiers (Sections 3.1/3.2)
• Resistive and Inductive Loads (Section 3.3)
• Free-wheeling Diode (Section 3.7)
• Half-wave Rectifiers with Capacitor Filter (Section 3.8)
• Controlled Half-wave Rectifier (Section 3.9)

Half-Wave Rectifier: Resistive Load

• Average Voltage Equation:
  V{avg} = rac{Vm}{rac{ heta}{ ext{2} imes ext{π}}}, ext{ where } heta = ext{T}
• RMS Voltage Equation:
  V{rms} = rac{Vm}{rac{1}{ ext{π}}}

Half-Wave Rectifier: Inductive Load

• Voltage and Current Relationship:
  V_{m} = L rac{di(t)}{dt} + Ri(t)
• Forced Response:
  if(t) = rac{Vm}{Z} ext{sin}( heta)
• Natural Response:
  i_n(t) = Ae^{-rac{t}{ au}}
• Total Current Expression:
  i(t) = if(t) + in(t)

Free-Wheeling Diode

• Function: Prevents output voltage, $v_o(t)$, from becoming negative.
• Diode Behavior:
  • When $V_s(t) > 0$, D1 is on.
  • When $Vs(t) < 0$, D2 takes over, and $vo(t) = 0$.

Circuit Example: Half-Wave Rectifier

• Given:
  • R = 100Ω
  • L = 0.1H
  • $V_m = 100V$
  • $ ext{ω} = 377 ext{ rad/s}$
• Current Expression: To be determined for an ideal diode.

Sample Problems (with Free-Wheeling Diode)

• Example:
  • Source: 240V rms at 60Hz, R = 8Ω
  • Find:
  • Average output current, $I_o$
  • Average power delivered
  • Power factor of the source
  • Average current in each diode
  • Inductance, L, to limit peak-to-peak ripple current (less than 10% of $I_o$)
• 1st Harmonic Approximation for ripple current: ext{Peak-to-peak ripple current} ~ 2I_1
  • Required $L = 719.9 ext{ mH}$.

Phase Control in Half-Wave Rectifiers

• Methods:
  V{avg} = rac{Vm}{2} ext{ (over angle α)}
  V{avg} = rac{Vm}{ ext{π}}( ext{cos } α - ext{cos } β)

Half-Wave Rectifier with Capacitor Filter

• Calculation for Ripple voltage:
  rac{ΔVo}{Vm} ≈ rac{I}{f RC}

Important Notes:

• Remember that for a half-wave rectifier, the output voltage will always be less than the average voltage due to its rectifying nature.
• Values of inductance/ capacitance should be specified carefully to ensure ripple current limits are respected in application.
• Utilize Fourier analysis for determining harmonic relationships in voltage/current ripple levels, where the first harmonic is often sufficient.