A Night Out on the \taun

The questions below are due on Tuesday October 15, 2024; 10:00:00 PM.
 
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1) Part 1

Consider the following circuit. For this circuit, assume that at t=0_-, the capacitor has fully discharged (i.e., the voltage drop across it is 0). At time t=0, we close the switch:

If we plot v_C as a function of time, we end up with a graph like the following (where we are only showing a small range of t):

Answer the following questions about this circuit and graph:

  1. Derive a closed-form expression for v_C(t) in terms of t, V, R, and C.
  2. Use this expression to show that, for this circuit, v_C\left(R C\right) \approx 0.63212\times V
  3. Using your results from the first two questions, if R = 200{\rm k}\Omega, approximately what values of V and C gave rise to the graph above? Include units in your answer.

2) Part 2

Now consider the following (related) small circuit consisting of two switches, two constant voltage sources, a resistor, and a capacitor:

For all of the examples in this problem, we will make the following assumptions:

  • For all t < 0, the switch labeled S_1 is closed and the switch labeled S_2 is open.
  • For all t \geq 0, the switch labeled S_2 is closed and the switch labeled S_1 is open.

For all of the graphs below, estimate the values of V_1, V_2, and C so as to match the graph. In all cases, pay careful attention to the labels on the axes of the plots, and clearly label the units of the different quantities in your answer(s).


 

 


 
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