Question: For the network below, the switch has been at position b for a long ti

Question

Question:

For the network below, the switch has been at position b for a long time. At t = 0, periodic switching at a 100 times per second. Let Vg = 9 VDC, C = 1mu F. Design values for R1 and R such that The capacitor is fully charged during the charging cycle The capacitor only falls down 30% of its maximum value during the discharging time Determine Vca(t) = Voltage across the capacitor at position a Vcb(t) = Voltage across the capacitor at position b The value or R1 and R to satisfy the design requirement Hint use x(t) = xf + (x0-xf)e-(t-to)/T to find quick solution

Explanation / Answer

T1 =charging duration =1/100 =10^-2 s

T1 should be more than the time constant

   that is    R1*C << T1

                R1 << 10^4

hence   R1 sholud be less than   10 k ohm ..

let   R1   = 0 ohm ...

T2 =discharging duration ...

   0.3 =e^(-t/RC)

    1.203 = -t/(RC)

      t    = 1.203*RC

      t =T2

     1.203*RC   = 10^-2

                R   = 10^4/1.2 = 8.33 k ohm

b) Vc(t) = voltage across the capacitor when switch is at position a

              = 9   , 0<t<10^-2

     Vc(t) =voltage across the capacitor when switch is at position b

             = 9e^(-t*10^3/8.3)   , t>10^-2

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