RC Circuits

The questions below are due on Friday April 07, 2023; 05:00:00 PM.

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In this lab, we'll experiment with RC networks. We'll start by running some small experiments to see that the currents and voltages throughout such a network match what we expect from circuit theory; and then we'll run a couple of additional experiments to see some useful applications of RC networks (some of which we'll expand on more in future weeks).

Getting Help

Remember that labs are a learning experience, not a test, so please ask us questions as you're working through!

For this lab, we also expect that checkoff 1 will take the majority of the time for this lab. Please read everything and don't rush through it.

Although we do want everyone to work individually and build their own circuits, it is also more than OK to ask friends/neighbors for help, too.

Rise and Shine

Here is the circuit we're going to be working with today:

To start, we'll use 0.1{\rm \mu F} for the capacitance.

Given the values of R and C above, what is the (theoretical) time constant of this circuit, in seconds?
\tau =~

Check Yourself 1:

In a moment, we're going to build the circuit above, and we're going to set up the voltage source so that it produces a 10Hz square wave, alternating between 3.3 Volts and 0 Volts like so:

Given this shape of V_s(t), sketch i_C(t) and v_C(t) as functions of time (by hand, not with a graphing calculator or desmos or whatever). Be prepared to discuss these sketches and your method for generating them during your checkoff.

Given this shape of V_s(t), what are the approximate maximum and minimum values we expect i_C(t) to have? Enter your answers as a Python list \displaystyle [\max_t(i_C(t)), \min_t(i_C(t))] with the values measured in Amperes.

Now we'll build this circuit and run an experiment to verify that we actually see results that match our expectations. We should be able to verify the general shape that we're expecting, as well as the numerical values like the time constant.

Before Building

Before building, let's reset the scope to all of its defaults by:

Pushing the "Default Setup" button in the upper left, and
Under the "Wave Gen" menu (accessible by pushing the button in the bottom right), click the "Settings" button (the bottom one on the screen), then click the "Default Wave Gen" button (the bottom one again) You should get a message on screen saying that things have been reset.

Let's go ahead and build it now, by following these steps:

Set up your scope's wave generator to a square wave with a 10Hz frequency, 3.3Volt peak-to-peak amplitude, and an offset of 1.65Volts. This will act as the voltage source in the schematic diagram above.
Hook up a resistor and capacitor as shown in the schematic above, using a .1{\rm \mu F} capacitor to start. Capacitors are in the same cabinet as the resistors.
Set up channel 1 on the scope to measure V_s(t) and channel 2 to measure v_C(t).
Set the trigger to normal mode, and have it trigger on the rising edge of channel 1.
Set the vertical scale to 500{{\rm mV}\over \text{div}} for each channel and set the time scale to 5{{\rm mS}\over \text{div}}. Adjust things so that you can see the entirety of the waveform.

Check Yourself 2:

Does the shape you see on channel 2 match what you expected given the graph?

Let's also measure the rise time to see whether it matches our prediction from earlier. We can do this using the "Meas" button. Under the "Type" submenu, we can choose "rise time," which computes the measure we discussed in the prelab.

Check Yourself 3:

Use this menu to add a measurement for the rise time of v_C(t). Does the result match your prediction?

Finally, we'll add another graph so we can measure the current. Unfortunately, we don't have a great way to measure current directly (the scopes only really measure voltages). But luckily, we have a voltage elsewhere in the circuit that is proportional to i_C(t), so we can measure that and back-solve for i_C(t). In particular, we'll measure the voltage drop across the resistor.

Unfortunately, though, our scopes make measuring that value directly a little bit difficult, since the "-" sides of the scope's input channels are all internally wired to ground. So we can't just put probes across the resistor to measure this voltage. Luckily, though, the scope still gives us a way to make this measurement, using its built-in math capabilities. Click the "Math" button near the bottom left of the scope. This will bring up a menu where we can force the scope to compute various different values for us. The "Operator" submenu lets us choose a mathematical operation to perform, and we can also choose what sources are used for the calculation. The result will be plotted in pink on its own scale, which you can adjust using the knobs right near the Math button.

Check Yourself 4:

How can we compute the voltage drop across the resistor from the two signals we're already measuring?

Check Yourself 5:

Use the 'Math' menu to add a plot of the voltage drop across the resistor. Do these results match what you would expect given your predictions for i_C(t), both in terms of its general shape and in terms of its time constant and max and min values?

Check Yourself 6:

As one final test, we'll try swapping out the capacitor for one with a value of .033{\rm \mu F}. Before you do that, try to predict: how will the values on the scope change? Then try it, and verify that the results match what you expect.

Checkoff 1:
Discuss the results of your experiments so far with a staff member. Be prepared to explain your results and also to relate them to circuit theory.

Two Applications

For the remainder of the lab, we'll play around with a couple of interesting ideas related to the circuit, both of which have some practical applications.

PWM

The first application we'll look at is PWM ("Pulse-width Modulation"), which is a different way of implementing something like the DAC we built before (which allows producing analog voltages from a digital source). This approach can often be more power-efficient than the DAC we built before, and it can also give us more-finely-grained control over the voltages we can create.

To start, let's re-set our circuit to its original configuration (with the .1{\rm \mu F} capacitor and 10{\rm k}\Omega resistor) and adjust the scope so that channels 1 and 2 are shown on the same scale (and with their reference points in the same spot).

Now let's crank up the frequency of the wave gen's square wave. Instead of the 10Hz we were using before, set it to be 5kHz. Something interesting happens! Specifically, the oscillations in v_C(t) get quite small, so that it starts to resemble a (approximately) constant voltage.

Next we'll change a different property of V_s(t): the "Duty Cycle" (which is near the bottom of the wave gen menu).

Check Yourself 7:

Try adjusting the duty cycle up and down. Looking at V_s(t), what feature(s) of the input wave is this actually changing? Looking at v_C(t), how does it change as you adjust the duty cycle?

Check Yourself 8:

How does this relate to the DAC circuit we built before? Why might someone prefer this circuit over our old design, or vice versa?

Sinusoids

Finally, let's explore a different kind of input V_s(t). Instead of a square wave, let's use a sine wave; and we'll see how the output v_C(t) looks. This will turn out to have enormous practical value, as it is sort of the first step toward understanding filtering, which allows us to manipulate the frequency content in signals.

Start by resetting the wave generator (Wave Gen \to Settings \to Default Wave Gen) and then adjusting it so that it outputs a 20Hz sine wave. Take a look at the output. Notice that it also looks like a sinusoid! While this may be surprising at first, it turns out that this is not a coincidence; we'll talk about it more next week and show why this should be the case in theory, as well as some compelling practical ways that we can make use of this idea.

Check Yourself 9:

How does the frequency of the sinusoid v_C(t) relate to that of the sinusoid V_s(t)?

Check Yourself 10:

Now try adjusting the frequency of V_s(t) slowly from your 20Hz starting value to around 5kHz. How do the frequency and amplitude of v_C(t) change as you do this?

Note that in order to really see what's going on, you may need to adjust the display scales on the scope as you move the frequency through this range.

Checkoff 2:
Discuss the results of your experiments (with PWM and with sinusoidal input) with a staff member.

When you are all done, clean up by putting things back where they belong. It is OK to throw out the capacitors and resistors, as well as any cut wires you used.