Yesterday I wrote my plans for today:
Tomorrow in class I plan to look at RC voltage divider circuits, plotting their amplitude response versus frequency using gnuplot. I’ll introduce the notion of Bode plots and RC time constants, to make it easier for students to visualize the plots without needing to run gnuplot every time. We’ll also talk some about stainless steel (what makes it a good material for implants but a poor one for electrodes) and what electrodes are popular for bioelectronics. Oh, and I need to explain RMS voltage to them, since that is what they measured with the multimeters, but we’ve only talked about amplitude, not peak-to-peak voltage or RMS voltage.
I’ll tell you how that went in a moment, but I need to set the stage a bit first.
Last night I had a bit of a panic, because I had missed the deadline for submitting a fellowship nomination for one of our prospective students for next year (I got the “due today” note when I finally had time to read my e-mail around 11pm), and our faculty had not yet made any decisions about who we are admitting—and we have to admit the student in order to nominate. Normally, our extremely capable grad staff adviser keeps me on track for these deadlines, but she’s seriously ill in the hospital, and I forgot that no one was keeping track of things for me.
I sent panicked e-mail last night asking if we could get an extension. Early this morning I got word that we had an extension until noon.
So this morning I sent another panic e-mail to all our faculty, and ran around buttonholing the ones in the same building, to find out whether we should admit the student and fill out the application form. Of course, everyone was busy interviewing grad student applicants today, and so had no time to look at a different student’s file. I started filling out the form anyway (I thought the student looked good enough to admit), and had the fellowship application done by the time I got confirmation from 3 other faculty that we should go for it. I got it into the grad division with seconds to spare before noon.
While doing the application, I realized that I had cycled up to campus without my laptop, which I needed for my 2:00 class. So I raced back down the hill, grabbed my laptop, and struggled back up the hill on my bike (this was a 47-minute round trip). I even had time to grab a burrito from the taco truck (luckily there was one pre-made one left, so I did not have to wait in the 25-minute line) and chat briefly with some grad students over lunch before class.
But I started class feeling very stressed and tired—not the best circumstances for a smooth lecture. Things went ok despite that, though I did not cover as much as I had hoped.
I started out with a simple “do now” problem, but one that I expected to cause problems for the students. I gave a simple voltage divider with a 1kΩ resistor on top and a 1μF capacitor on the bottom and asked for Vout/Vin at 1 Hz and 1 MHz. I also asked for the value of RC in international standard units. I gave the students 8 minutes from the beginning of class to work on this, though many came in late so only had a minute or two.
As expected, the students got no further than plugging the values into the formula for the voltage divider and getting stumped about what to do with it. I promised them shortcuts, so that by the end of class they could do the problems in seconds.
I used gnuplot to plot the frequency response of the circuit using the same formula they were comfortable with: .
I showed them the straight lines on the log-log plot at low and high frequencies, and we did some algebra to see where these came from. First we cleaned up the messy fraction to get . As ω goes to zero, this gives us the low frequency gain of 1 and as ω goes to ∞, this gives us the high-frequency response of . (I introduced them to the notion of “gain” also, since we have been doing a lot with voltage ratios.) They could then figure out the gain at 1MHz.
It took them quite a while to derive the formula for where the two lines intersected—the math skills of the bioengineers seem quite weak for engineers—but they were able to do it, so they probably once had the skills, but have let them atrophy for lack of use. We introduced the names “cutoff frequency” and “corner frequency” for this point. We then looked numerically at the gain at the corner frequency, which they saw was 0.7, and one student recognized this (after some prompting) as probably . I went through the derivation of that also.
Afterwards, I gave them the name “low pass filter” for this circuit, and started fishing for other things we could do with a voltage divider. After a little rejection of complicated ideas, we got the capacitor-on-top voltage divider, and the two-capacitor voltage divider. I flipped the order of the arguments in my gnuplot script to put the capacitor on top, and we saw the high-pass filter. Again we derived the formulas for the asymptotes, and students quickly realized that the corner frequency was the same, but it took them a long time to get the formula right for the line. Simple algebra seems to be difficult for them to do quickly in class. I’m wondering if I need to give a quiz or a homework set that practices some of the algebra relevant to the course, as the design exercises in the lab don’t seem to be enough to bring back their algebra.
Throughout the lecture I stressed the students doing sanity checks (like units: Hz or seconds?) and not memorizing piles of formulas but just a few simple principles that let you derive everything else quickly. Some of the concepts I stressed:
- RC time constant (I had to go back to fundamental definitions to convince them that 1 Ω F = 1 second)
- corner frequency where ω=1/(RC) (rather than trying to keep track of the 2 π all the time)
- slopes are 1, f, and 1/f for a simple RC circuit with only one resistor and capacitor
- looking at extreme cases (0 and ∞) to make approximations
We did introduce the notion of decibels (which most had seen before) as , and talk about the “rolloff” of 20 dB/decade as the frequency raised to the ±1 power.
So we have the basics now of Bode plots (just for gain, as I’m not sure we want to do much with phase in this class, at least not for a while). I did not get to any electrochemistry or why stainless steel is a good mechanically and chemically, but not electrically, for implants.
I think that Monday will see more gnuplot plotting, looking at the impedance of more complicated circuits, so that they can better understand the behavior of polarizable electrodes like the stainless steel electrodes they were characterizing. If I can show them how the Bode plots help think about and sketch the behavior quickly, I hope that they’ll have a better appreciation of this shortcut. Given that plotting with gnuplot is easy, I’ll have to convince them of the usefulness of the Bode plots for thinking about circuits, rather than the classical approach of using them to do quick sketches of behavior.
It may be Wednesday before we get to hysteresis and the hysteresis oscillator that they will build on Thursday.
One other thing I wanted to do today but spaced: I wore my banana slug genomics t-shirt so that we could discuss the possibility of designing a t-shirt for this course, but then I forgot to discuss it. I think I want to use the same basic “slug-dreaming” design, but put something different in the thought balloon. I don’t have any good ideas yet for the thought balloon. Given how much we’ll be doing with voltage dividers, doing something like an RC low-pass filter with the appropriate gain equation is not too bad an idea.