PRELAB QUESTION!! Deriving Equations

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 >> R_s (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. V_out (t). What is the maximum value of V_out(t)? What
is the minimum value of V_out(t)?

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

c. Rise time t_r of V_out(t).

d. Fall time t_f of V_out(t).

e. Delay times t_PHL and t_PLH.

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 V_out(t) during the time interval when
V_out(t) falls with time (see part 6.1.a above), write the equation
for log{ V_out(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 t.

b. Alternatively, from the equation for V_out(t) during the time
interval when V_out(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- V_out(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 t.