What is the RMS Voltage Calculation for a Controlled Switch Rectifier?

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Homework Statement



See figure attached.

Homework Equations





The Attempt at a Solution



See figure attached for attempt at the solution.

Can anyone verify whether or not my workings are correct?
 
Attachments
  • Homework problem text and circuit schematic: an AC voltage source vac(t) = Vp sin(ωt) in series with a controlled switch, feeding output voltage Vo. The text asks for a plot of output waveforms and the average and RMS values as a function of k, where the switch turns on when vac > kVp and off half a cycle later.
    TT2009P1Q.JPG
    22.5 KB · Views: 458
  • Handwritten solution on graph paper titled TT 2009, problem P1: a circuit with a voltage source Vp sin(wt), a switch and output Vo, a plot of Vo(wt) showing clipped half-wave pulses starting at alpha, and the integral for average output Vo equal to Vp cos(k pi/2) over pi where 0 < k < 1.
    rsz_q-_0164.jpg
    51.7 KB · Views: 506
  • Handwritten derivation on graph paper, labelled P1 and IT 2009, of the RMS output voltage of a rectifier: Vo,RMS squared equals one over 2π times the integral from α to α+π of (Vp sin(ωt)) squared dωt, which simplifies to Vp squared over 2π times π/2, giving Vo,RMS = Vp/2.
    rsz_q-_0165.jpg
    35.6 KB · Views: 551
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Discussion

You get a straight A - from me anyway!

BTW a quick way to get rms is to realize the power in a half-wave is half that of a full wave. You know the rms of a full wave is Vp/√2 so for the half-wave it must be 1/√2 of that, or Vp/2.
 
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