Prelab: RC Circuits and Rise Time

The questions below are due on Friday October 11, 2024; 05:00:00 PM.

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In the this week's lab, we will be experimenting with RC circuits and seeing some of the ways the ways that we can make use of them in authentic circuits. Part of what we'll be exploring is a measure called the rise time of a circuit.

1) RC Circuits

As we have seen in lecture (and will see throughout this problem set), simple RC networks have a characteristic "first-order" response. For example, consider the following circuit (which is very similar to one from elsewhere in the pset):

Assume that:

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.

Find an expression for v_C(t) (in Volts) for all t \geq 0. Enter your answer as a Python expression below, using C, R, V_1, and V_2 to C, R, V_1, and V_2, respectively. You may use exp(x) or e**(x) to represent e^x.

For t \geq 0, v_C(t) =~

2) Shapes

The value RC is referred to as the time constant and is denoted with the Greek letter \tau (tau). If we plot the function above (for some values of the constants V_1, V_2, R, and C), we end up with a graph like the following:

As we saw in a previous exercise, this curve has a few properties that we can use to reason about the values in the circuit. For example, v_C(\tau) has a predictable value, approximately 63% of the way between the initial value and the final value:

Rise Time

Another common measure is the rise time of the circuit, t_r, which is defined to be the amount of time the signal takes to go from 10% of the way to its final value, to 90% of the way. t_r is shown graphically below:

The scopes in lab can be set to measure the rise time of a signal under the "measurement" menu, so it will be a convenient measure for us as we look at signals like this in the lab. But one reason that we care about this measure is not just because our scopes can measure it (they can measure lots of things), but because we can use t_r to find the time constant \tau.

Determine the time constant \tau in terms of the rise time t_r. Enter your answer as a Python expression. Use t_r to represent t_r. You may use ln(x) to refer to \ln(x), and you may use either exp(x) or x**(x) to represent e^x.

\tau =~

A common approximation of this result is that t_r \approx 2.2\tau.