Prelab 9

The questions below are due on Thursday April 13, 2023; 11:00:00 PM.
 
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Last week, we saw that for the following circuit, if the input voltage V_i(t) was a sine wave, then the output voltage v_o(t) would be a sine wave at that same frequency (but possibly scaled down and shifted by some amount). And what we found was that as the frequency of the input sine wave got higher, v_o(t) got smaller in terms of amplitude.

In this week's lab, we'll explore this idea a bit further, interpreting it as a "filter" that attentuates certain frequencies but leaves others alone. Ultimately, our goal is to build a bass-boost system for music, which you can hook up to your computer or phone to play music with the low-end frequencies amplified.

In this pre-lab, we'll refresh ourselves on a bunch of the pieces that we'll use in constructing this system, most of which are based on ideas/circuits we have seen before. Then in lab, we'll put it all together, building up a little system that works on arbitrary audio input.

Audio Input/Output

The circuit we're going to build will center around processing audio, which we'll get from your computer/phone/whatever via a "TRRS" cable (tip, ring, ring, sleeve), named for the various regions of the connector, each of which carries a different signal. In order to grab those signals, we'll use a connector like this one, which will allow us to get access to those signals on our breadboard:

For this lab, we're going to ignore the "sleeve" signal, which usually corresponds to a microphone (and we won't be using a microphone here). Once we have audio playing through a cable connected here, we can model this connector like so:

where V_L(t) and V_R(t) are voltages proportional to the air pressure needed to make a particular sound.

These connectors come to us with the board and the pins separate, so we'll need to solder them together. You can take a look back at section 2 of the week 3 prelab if you want a refresher on soldering, and we'll have some additional special instructions for this particular soldering job in the lab itself.

Averager

We can compute the weighted average of two potentials e_1 and e_2 by using a voltage divider like the following:

The potential at the node in the middle (where the two resistors are connected to each other) can be expressed as \alpha e_1 + \beta e_2 for some values of \alpha and \beta. What are these values?

Enter your answers as Python expressions, using R_1 and R_2, respectively, to represent R_1 and R_2.
 

  

  

If we want the potential at the middle node to be e_1 + e2 \over 2, how should R_1 compare to R_2? Enter an expression for R_1 involving R_2.
R_1 = ~

Buffer

In previous labs, we have seen the following op-amp configuration, which we've called a "buffer" (or a "voltage follower"):

If we make the ideal op-amp assumption and ignore output limitations of the op-amp, solve for e_o in terms of e_i (e_i):
e_o =~

Check Yourself 1:

Given the relatively simple relationship between e_o and e_i, why do we even care about using this kind of circuit? When/how is it useful?

"Virtual Ground"

In the result that we computed above, we ignored the output limitations of the op-amp. However, when we're working in the lab, we won't be able to ignore that. The op-amp's output range will be limited by the supply voltage we provide to it. In lab, we're going to use a 9Volt battery to power the op-amp, which means that it will not be able to output voltages outside of the 0V-9V range (relative to the "-" terminal of the battery).

However, the voltages we'll get out from our audio source will be able to be positive or negative, so if we're going to work with them and with our op-amps, we'll need to take some care so that we don't end up losing the negative values.

For example, if we just took V_L(t) and plugged it into the input of a buffer, where the op-amp was powered by our 9V battery, we would lose all of the negative values!

In lab, we'll get around this by creating a "virtual ground" at 4.5Volts relative to the "-" side of our battery. If we then view that node as our "ground" (our 0V reference), then the two ends of the battery will be at -4.5V and +4.5V, respectively, which will allow us to make use of voltages that go positive or negative within that range and thus avoid some of the problems that can arise from the output limitations of the op-amp.

Amplifier

Let's take another look at a different circuit we've seen before as well:

Which of the following best describes this configuration?

If we make the ideal op-amp assumption and ignore output limitations of the op-amp, we can express e_o = ke_i, where k is a constant. In this formulation, what is k, in terms of e_i (e_i), R_1 (R_1), and/or R_2 (R_2)?
k =~

Amplifier with Offset

This analysis, as well, depended on ignoring the output limitations of the op-amp. However, we often need to adjust things so as to avoid those limitations. Consider the following variation of the above circuit (where we have replaced a connection to ground with a connection to an arbitrary node with potential e_2:

If we make the ideal op-amp assumption and ignore output limitations of the op-amp, we can express e_o as:

e_o = k (e_i - e_{\rm ref}) + e_{\rm ref}.

In this formulation, find k and e_{\rm ref}, in terms of e_i (e_i), e_2 (e_2), R_1 (R_1), and/or R_2 (R_2).
 

  

  

Putting It All Together

This circuit is going to be the most complicated circuit we've built this semester. But how do you eat a big pizza? One little bite at a time...As such, we're going to want to split the process into many pieces, each of which can be designed, implemented, and tested independently of the others. We'll think about several pieces here:

  • We'll want to make a stable 4.5Volt source (relative to the "-" sign of our battery) and do our work around that point, so as to avoid output limitations of the op-amp.

  • We'll want to offset V_L(t) and V_R(t) by this amount, and then average the results together (to get a single signal for the audio, which we can then manipulate). Let's call this new signal V_a(t).

  • We'll want to buffer V_a(t) so that we can make use of it in multiple places.

  • We'll want to use an RC circuit to filter V_a(t), making a version containing only the bass frequencies (let's call that new signal V_b(t)).

  • We'll want to amplify V_b(t) in such a way that we avoid output limitations of the op-amp, getting a new voltage kV_b(t).

  • We'll want to average together V_a(t) and kV_b(t) to get our final output (which should emphasize low-end frequencies while not losing high-end frequencies).

  • This new signal can then be hooked up to another TRRS connector so that we can listen to it via headphones :)

Check Yourself 2:

Try your hand at designing each of the components above, and draw a schematic for each. What should each piece look like, and how can we connect them to achieve the overall effect we want?

Bring these schematic drawings with you to lab. If you get stuck, or if you're unsure of your designs, that's OK! Just try your best, and we'll get the details sorted out during lab.