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Unformatted text preview: 86 1 2 2 2 1 M V B R C = + and 2 2 2 2 1 M RC V B R C  = + Therefore, the forced solution to the equation is ( ) , 2 2 2 2 2 2 sin cos 1 1 M M C f V RC V v t t t R C R C = + + and the total solution is ( ) ( ) ( ) , , C C n C f v t v t v t = + ( ) 2 2 2 2 2 2 sin cos 1 1 t M M RC C V RC V v t Ae t t R C R C  = + + + The initial condition for this problem is that the voltage on the capacitor is zero at the beginning of the half cycle: ( ) 2 2 2 2 2 2 sin 0 cos 0 1 1 M M RC C V RC V v Ae R C R C  = + = + + 2 2 2 1 M RC V A R C  = + 2 2 2 1 M RC V A R C = + Therefore, the voltage across the capacitor as a function of time before the PNPN diode fires is ( ) 2 2 2 2 2 2 2 2 2 sin cos 1 1 1 t M M M RC C RC V V RC V v t e t t R C R C R C  = + + + + If we substitute the known values for R , C , , and V M , this equation becomes ( ) 83.3 35.76 7.91 sin 35.76 cos t C v t e t t  = + This equation is plotted below: 87 It reaches a voltage of 40 V at a time of 4.8 ms. Since the frequency of the waveform is 60 Hz, the waveform there are 360 in 1/60 s, and the firing angle is ( ) 360 4.8 ms 103.7 1/ 60 s = = or 1.810 radians Note : This problem could also have been solved using Laplace Transforms, if desired. (b) The rms voltage applied to the load is / 2 2 2 rms 1 ( ) sin M V v t dt V t dt T = = / 2 rms 1 1 sin2 2 4 M V V t t = ( ) ( ) g187 g188 g186 g171 g172 g170 = 2 sin 2 sin 4 1 2 1 2 rms M V V Since = 1.180 radians, the rms voltage is rms 0.1753 0.419 71.0 V M M V V V = = = 314. Figure P39 shows a threephase fullwave rectifier circuit supplying power to a dc load. The circuit uses SCRs instead of diodes as the rectifying elements. (a) What will the load voltage and ripple be if each SCR is triggered as soon as it becomes forward biased? At what phase angle should the SCRs be triggered in order to operate this way? Sketch or plot the output voltage for this case. (b) What will the rms load voltage and ripple be if each SCR is triggered at a phase angle of 90 (that is, half way through the halfcycle in which it is forward biased)? Sketch or plot the output voltage for this case. 88 S OLUTION Assume that the three voltages applied to this circuit are: ( ) sin A M v t V t = ( ) ( ) sin 2 / 3 B M v t V t = ( ) ( ) sin 2 / 3 C M v t V t = + The period of the input waveforms is T , where 2 / T = . For the purpose of the calculations in this problem, we will assume that is 377 rad/s (60 Hz). (a) The when the SCRs start to conduct as soon as they are forward biased, this circuit is just a three phase fullwave bridge, and the output voltage is identical to that in Problem 32. The sketch of output voltage is reproduced below, and the ripple is 4.2%. The following table shows which SCRs must conduct in what order to create the output voltage shown below. The times are expressed as multiples of the period...
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 Spring '05
 Shaban

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