Digital-to-Analog Converter

The questions below are due on Friday February 27, 2026; 05:00:00 PM.

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Getting Help

As always, we have a help queue active during the lab! Go through the lab, answer the questions on this page, and ask for checkoffs on the queue! When you're all done with everything, you're done! If you get stuck on the way, let us know; we're happy to help!

Remember that labs are intended to be a learning experience, not a test, so take your time and understand things, and please ask us questions as you're working through!

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.

1) Analysis

Here's the circuit we'll be using today. This should look familiar from the prelab.

Using superposition, solve for the v_\text{out}. Be prepared to talk through your work during Checkoff 1.

In this configuration, what is the voltage v_\text{out} in terms of V_1, V_2, V_3, V_4, V_5, V_6, and/or R? Use superposition. You may find it easier to start by finding V_1's contribution, rather than the other way around).
Enter your answer as a Python expression, using values V_1, V_2, V_3, V_4, V_5, V_6, and R, respectively, to refer to these values.
v_\text{out} =~

2) The Digital Abstraction

As mentioned in the prelab, this circuit is useful as a digital-to-analog converter (DAC, often pronounced like "dack"). We're going to hook it up to our RP2040, which is only capable of producing two voltages: a "high" voltage of 3.3 Volts and a "low" voltage of 0 Volts relative to its internal ground. Programmatically, we can set any pin to "high" or "low", making the corresponding pin look like this:

By hooking various pins of the RP2040 up to a circuit like the one at the very top of the page, we'll be able to get a much wider variety of voltages from the RP2040.

Imagine hooking up the RP2040's pins so that they provide V_1, V_2, V_3, V_4, V_5, and V_6 in the circuit at the top of the page. How many distinct values of v_\text{out} will we be able to make?

In that same configuration, what is the maximum voltage we'll be able to make by turning pins "on" and "off", in units of Volts? Enter your answer accurate to within 10^{-2} Volts, as a number (not an expression).

In that same configuration, what is the smallest change we'll be able to make by turning a pin on/off, in units of Volts? Enter your answer accurate to within 10^{-4} Volts.

3) Grab Your Equipment

Now that we know what to expect, we're going to build the circuit from up above, using several of the RP2040's pins to provide the voltage sources indicated there.

Since it takes a while to boot up, turn on your scope before going to grab your parts so that it can be warming up while you're grabbing things. Then grab a RP2040, a breadboard, and a speaker. We'll also need to cut some wires, so grab yourself some clippers as well.

When using the RP2040 with a breadboard, it's best to plug it in on the very end of the board so that it's spanning the gap in the board, with its USB jack facing outward; like so:

It can take some work to get it into the breadboard. Be careful not to break things, but it's OK to use a little bit of force to get the pins to go in.

Note, you do not need to provide external power to the RP2040, your computer will provide the power through the USB cable.

Once you've done that, construct the circuit from the very top of the page, using pins D0, D1, D2, D3, D4, and D5 to provide V_1, V_2, V_3, V_4, V_5, and V_6, respectively. Use a value for of 1{\rm k}\Omega for R when building your circuit.

As we build more complicated circuits in lab, the layout of your breadboard becomes more important. In particular, the nicer you layout the circuit the easier it is to troublshoot (both for yourself and when you ask for help from the staff). So taking the time to layout your circuit clearly will save you time in the long run.

As a layout tip, it is good to try to make the structure of this circuit on your breadboard somewhat match the structure of the schematic, if possible. Specifically, we recommend having the 2R resistors cross the center gap in the breadboard, with two open holes between each one (which makes connecting them via R resistors easy); and having the v_{\rm out} side of the circuit be on the side of the RP2040 (i.e., having V_1 close to the RP2040 and V_6 farther away).

We also have some little pre-cut wires in the bottom row of the resistor cabinet; the orange ones are conveniently-sized for jumping across the center gap of the breadboard or from the middle part of the breadboard to the closer of the red/blue rails; and the yellow ones work for jumping from the middle part of the board to the farther one of the red/blue rails. You should, of course, feel free to cut your own wires; but the little pre-cut ones are often super convenient.

Lastly, it might be helpful to cut the legs of your resistors to they sit closer to the breadboard. When the legs are long it is easier to accidentally short them together causing issues in wiring.

We're happy to consult on board layout as well; just ask!

Also note that the pin labeled "GND" in the very corner of the board is the RP2040's internal reference. The "-" sides of the voltage sources in the circuit above are all connected to that spot internally by the RP2040 (as shown below), so there are some other components we'll need to connect there as well to complete the circuit.

A partial view of a model of the RP2040's internals

Check Yourself 1:

In our original schematic, we had the two 2R resistors on the sides connected to a single node shared by the - sides of all of the sources. Here, though, we only have a single wire connecting to the RP2040's ground (pin G). How can we adjust the schematic from up above to account for this fact? Where and how should we wire up the RP2040's pin G? What components should it connect to?

We're going to use a couple of different programs to test things today. You can download them all as a ZIP file: dac_code.zip.

To start, we're going to use the program called "stepper" (stepper/stepper.ino inside of that ZIP file), which you should flash to your RP2040 using a similar process to what we did last week:

Open Arduino (teensyduino on Mac) and open stepper/stepper.ino.
From the menu, under Tools\toBoard: ..., select Seeed XIAO RP2040. This tells Arduino what kind of board we're programming.
Plug your RP2040 into the computer via USB cable (we have extras if you need one).
From the mentu, under Tools\toPort: ..., select the port you are connected to (if you can't figure out which one, unplug the RP2040 and see which port option disapears.)
Click the second button from the left in the Arduino window, which looks like a rightward-pointing arrow (it should say "Upload" when you hover over it). This should compile things and send the result to the RP2040 with no orange error messages.

We're also going to make some measurements soon, so if you don't already have a probe for channel 1 of your oscilloscope, go ahead and grab one from the front of the room.

If you use channel 1 on the scope to measure v_\text{out}, you should see what looks like a little ramp repeating itself every so often (you'll likely need to zoom in/out to get a clear view of it). Make sure to turn on "BW Limit" so things aren't too fuzzy. If you zoom in closely, you can see that it's not actually a smooth ramp, though; it's got lots of little sements in it if you zoom in enough. You may also wish (for now) to set a trigger in the middle of the ramp or something like that, and switch the trigger to Normal mode (which should help hold things in place). Or you can use the Run/Stop button manually to freeze things on the screen.

Check Yourself 2:

Use the cursors on the scope to measure various properties of the wave to see whether they agree with the theory from above:

What is the maximum voltage in that waveform?
What is the spacing (in Volts) between the voltage levels?
How long (in seconds) is the voltage held at each of those individual voltage levels?
How often does the whole ramp repeat?

Given those measurements, how many distinct steps are there in the ramp?

Be prepared to talk about these measurements and how you made them during your checkoff.

Check Yourself 3:

Given the pattern you see, how is the RP2040 toggling the pins? Can you predict the pattern? Try using your scope to measure the individual pins after you have a hypotheses. Does it make sense that changing them in this way causes the pattern we see in v_\text{out}?

Checkoff 1:
Discuss your results so far with a staff member. How did you solve for the values in the circuit at the top of the page? Do your experimental results match the theory?

4) A Second Program

Now load the "song" program (song/song.ino in the ZIP file) and continue measuring the output. Hit "Default Setup" and then use the controls on the scope to get a nice view of the output wave (we want the wave to be taking up most of the screen). If you had your trigger in Normal mode for measuring the ramp, switch it back to Auto.

Now, let's listen to it by hooking up a speaker across the port labeled v_{\rm out} in the diagram. You should hear a song. But oh, no! It's really quiet, way quieter than the ~2.2 volts we were measuring should be. And indeed, look at the scope!!! What changed about your signal?

Check Yourself 4:

Try plugging and unplugging the speaker while looking at your scope; what changes about the signal when the speaker is connected?

Let's see if we can understand this change using circuit theory. But oh, no! The DAC is a complicated circuit with six independently-controllable voltage sources and thirteen resistors, so solving that seems like kind of a pain. . It wasn't too bad to analyze using series/parallel combinations and divider relationships, but still...it sure would be nice if we could model it as a simpler circuit instead...

...wouldn't it?

...

Thankfully, as we've seen over the past week or so, we do have a way to do this! We can model our whole DAC circuit (including the RP2040) with a Thévenin equivalent!

By thinking about the Thévenin equivalent of our DAC, we can model our circuit, with and without the speaker connected, as shown below; much nicer to analyze with way fewer components! Yay!


without speaker	with speaker

Check Yourself 5:

Given this model, can you see why the voltage output changed when we added the speaker?

Let's analyze this in a little bit more detail. To start, let's model the speaker as a resistor. What is its resistance, in Ohms? (Note that you can measure this directly using your multimeter).

Now, let's make some measurements and see if we can empirically determine the Thévenin resistance of our DAC circuit. Firstly, let's switch back to the stepper.ino program on our RP2040. Then, use the scope to answer the following two questions:

What is the peak voltage of the stepper program without the speaker connected?
What is the peak voltage of the stepper program with the speaker connected?

Note that you may need to zoom way in to make this second measurement, and if the ramp looks really fuzzy, you'll want to measure in the middle of that fuzz, not on top of it (and make sure that "BW Limit" it on to reduce the fuzz in the first place!).

Given these measurements and the Thévenin model of the DAC up above, estimate the Thévenin resistance of your DAC, and enter your answer below, in units of Ohms. We're not asking you to solve for this using the original circuit, but rather to estimate it using the measurements you made and the Thévenin model of the circuit.
R_{\rm TH}\approx~

Now go ahead and determine the equivalent resistance using circuit theory. Hint: Scroll up to the original circuit that you analyzed with superposition. What should you do with the sources to find the equivalent resistance?

Be read to talk through your math during the checkoff.

Checkoff 2:
Discuss your results with a staff member. How did you calculate R_\text{TH} using the scope measurements? How did you solve it using theory? Demonstrate the song playing as well.

5) Op Amp to the Rescue

In this section we will be using your DAC with an additional component at the output. DO NOT DISASSEMBLE YOUR DAC

In the previous section we saw the impact of the speaker loading the circuit. This is because our circuits aren't modular, as we add components it effects the overall operation of the circuit.

In the coming weeks we will be working with a operational amplifiers (op amps), a type of component that can enable modularity in circuits. For now we can model the op amp as a dependent source. Today, we will use a simple voltage controlled voltage source (VCVS) to model an op amp configured as a buffer. In this case the VCVS has a gain of one, meaning that v_s = v_x.

We can use this VCVS to prevent the loading problem we saw when we connected our speaker to our DAC.

We will spend the next few weeks learning how to analyze and design circuits with op amps, but for today we will show you exactly how to connect the op amp so that it looks like a VCVS buffer that we described above.

Today we will be using a MCP6002 op amp. This device is an integrated circuit (IC) or chip, which internally has two op amps, but we will just be using one of them.

Go ahead and grab one of the MCP6002 from the front of the room and position it in your breadboard, like this. DO NOT DISASSEMBLE YOUR DAC Just put this on the breadboard too, or grab another breadboard if you are out of room.

Like with most chips, a small dot on the surface of the chip marks pin 1. Here is how we can model the internal structure of the device with the corresponding pin out (found on the device datasheet).

We are going to connect the op amp with the following configuration to make it a VCVS, where v_x = v_s as shown above.

We also need to provide power to the IC, where VCC+ should be connected to the 5V of the RP2040 and VCC- should be connected to the ground of the RP2040.

Lastly, our op amp can only provide about 30mA of current at the output. To make sure we don't exceed this current limit, we are going to add a 51 \Omega resistor in series with the speaker (higher total resistance means lower current at the same voltage).

For today only, we will show you exactly how to wire this up. In the picture below the GND and 5V should be tied to the RP2040 power rails, you do not need to provide additional power.

Make sure you still have the stepper.ino program on the RP2040. Then, use the scope to answer the following two questions:

What is the peak voltage at the output of the op amp without the speaker connected?
What is the peak voltage at the output of the op amp with the speaker connected?

Check Yourself 6:

How is this different than when you directly connected the speaker to the output of the DAC?

Now go ahead and try to play the song (switch back song.ino).

Wow! That sounds very different. You should now hear the song at much higher volume. Which means we are no longer seeing the loading effect from the speaker we saw earlier. This is why op amps are so interesting a powerful tool to help modularize circuits.

Checkoff 3:
Discuss your results with a staff member. What is a VCVS? Why does adding a VCVS fix our challenge with the speaker from the last checkoff? Where does the power for the speaker come from?

Please clean up. Return the RP2040, speaker, op amp and cables to the front. You can throw away the resistors and wires.

6) More (Optional) Fun Things To Do If You Want To

None of the stuff here has a checkoff or affects your grade or anything like that, but if you're having fun, there are some other things you might explore:

Try changing the rate at which we move through the samples in the song.ino file. What happens if you speed it up by a factor of two? What if you slow it down by a factor of 2?
Try using the RP2040's input pins to detect button presses and use those to change the frequency of a sine wave (the sine/sine.ino program can be a good starting point), making yourself a little musical instrument. Or, measure the (continuous) voltage from a potentiometer hooked up as a voltage divider and using that voltage to control the frequency of a sine wave so that you can smoothly go between different frequencies.

If you do try these things (or other experiments of your own), we'd love to see what you come up with! :)