PLEASE HELP!!!! The input signal to the circuit in Figure 1 is a perfect square

Question

PLEASE HELP!!!!

The input signal to the circuit in Figure 1 is a perfect square wave with amplitude A (from 0 V to A), and period T where T >> RC. You may also assume that R >> Rs (the internal resistance of an arbitrary waveform generator). Using only symbolic parameters (e.g. R, C, A; not numerical values), derive the equations for the following quantities:

a. Vout (t). What is the maximum value of Vout(t)? What is the minimum value of Vout(t)?

b. The time values when the output reaches 10%, 50%, and 90% of its final value.

c. Rise time tr of Vout(t).

d. Fall time tf of Vout(t).

e. Delay times tPHL and tPLH.

HELP!! PLEASE HELP!

6.2 Parameter extraction via linear least-square-fit technique

Either technique below can be used to extract the time constant.

a. From the equation for Vout(t) during the time interval when Vout(t) falls with time (see part 6.1.a above), write the equation for log{ Vout(t)} as function of t. This equation should be linear in terms of t. Derive the equation for the slope of this line in terms of the time constant .

b. Alternatively, from the equation for Vout(t) during the time interval when Vout(t) rises with time (see part 6.2.a above), manipulate this equation so that the final form looks like:

              1- Vout(t) / A = e (-t/T)

where A is the amplitude of the step. Now you can write the equation for ln{1- Vout(t)/A} as function of t. This equation should be linear in terms of t. Derive the equation for the slope of this line in terms of the time constant .

Explanation / Answer

Vs = A
apply nodal analysis

(Vs-Vout)/(Rs+R) = C dVout/dt

or dt/C(Rs+R) =dVout/(Vs-Vout)

dVout/(Vout- Vs)   = -dt/C(Rs+R)

after integrating

ln[(Vout- Vs)/M]   = -t/C(Rs+R)        , where M =integration constant

(Vout- Vs)/M        = e-t/C(Rs+R)

               Vout =   Vs + Me-t/C(Rs+R)

now    at t= 0 ,    Vout   =0 , hence    M =-Vs

    so     Vout =   Vs -    Vse-t/C(Rs+R)

   Vout(max)      =     Vs               

Vout(minimum)      =     0    at t =0

rising time     Tr      =C(RS+R)

falll time Tf   = C(RS+R)

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