Gas station without pumps

2013 January 28

Ninth day of circuit class

After the last class I wrote

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. [This did get discussed today, and students liked the idea of a T-shirt.]

Well, it will definitely be Wednesday before we get to hysteresis.  We did not get to talking much about stainless steel, though we did do some plotting of the electrode circuit.  The problem is that I started with a quick “do now” question to check that they had absorbed Friday’s material:

For each of the following circuits, give the gain at 1Hz, 1kHz, and 1MHz:

The four circuits I put on the board.  Of course, I used 0.01µF, not 10nF, and 0.56µF, not 560nF, since no one uses the milli- and nano- prefixes for capacitance, but CircuitLab is incapable of that distinction.

The four circuits I put on the board. Of course, I used 0.01µF, not 10nF, and 0.56µF, not 560nF, since no one uses the milli- and nano- prefixes for capacitance, but CircuitLab doesn’t seem to realize that.

I expected that most of the class would be able to do the resistance voltage divider, and half the class would get the low- and high-frequency gain values for the RC circuits. I expected that students would have trouble with the capacitance voltage divider, since I had suggested they look at it at the end of last Friday’s class, but with little expectation that any of them would have the curiosity to actually do it without being required to.

I was a little disappointed that no one in the class got any of the questions, even with far more time than I had planned to spend on them. I guess I’ve been guilty of pseudoteaching—something that looks like good teaching, but the students don’t actually learn. So we went over the 4 problems in class, with me extracting the answers from the students and pointing out where sanity checks are needed (like whether the corner frequency is 1000 or 0.001, based on keeping track of units). So what was intended as a 5–10-minute check on what students had retained turned into a 45-minute repeat of the previous class.

I’ve also told the students that there will be a quiz next Tuesday, covering everything we’ve done with impedance and voltage dividers. I’ll be making the quiz closed notes, since far too many students wasted time looking for magic formulas in their notes, rather than thinking about what they knew and using it. I will expect students to bring a calculator to class from now on also.

Now I have to come up with a quiz—and if it turns out that most of the class can’t learn this stuff without pages of mindless drill, then I’ll have to start assigning pages of problem sets, an approach to education that I’ve always hated.

That repeat teaching left me with less time than I had planned, so the only new material I covered was plotting impedance versus frequency for the model we had for the electrode pair:

The model the students are supposed to fit to the data from last week's lab—a standard model for a pair of polarizable electrodes.

The model the students are supposed to fit to the data from last week’s lab—a standard model for a pair of polarizable electrodes.

Before showing them how to set up gnuplot to plot the model, I first had them think about what happens at DC and at ∞ frequency.  After a bit of fishing, I finally got them to elucidate the behavior of a capacitor (open circuit at DC and short circuit at very high frequency), and figure out what that meant for the overall impedance.

I showed them the following gnuplot script (and walked them through it line-by-line, since I’m not confident that they understand function definition yet):

set xlabel "frequency (Hz)"
set ylabel "|Z| (ohms)"

set logscale xy
unset key

# voltage divider
divider(z1,z2) = z2/(z1+z2)

# two impedances (resistances) in parallel
parallel(z1,z2) = z1*z2==0? 0: z1*z2/(z1+z2)

# impedance of a capacitor at a given frequency
j = sqrt(-1)
Z_C(c,f) = 1/(j*2*pi*f*c)

# corner frequency for an RC time constant
freq(RC) = 1./(2.*pi*RC)

 R2 = 5.

 min(a,b) = a<b? a:b
 max(a,b) = a<b? b:a

set xrange [1:1e6]	# 1Hz to 1MHz
# set yrange [0.9*R2:1.1*(R1+R2)]
set yrange [*:*]

unset label
unset arrow

set label "f(R1*C1)" at freq(R1*C1),R1+R2
set arrow from freq(R1*C1),1e-12 to freq(R1*C1),1e12  nohead lw 0.5
set label "f(R2*C1)" at freq(R2*C1),R2
set arrow from freq(R2*C1),1e-12 to freq(R2*C1),1e12  nohead lw 0.5

set title sprintf("(%.3gF || %.3gohm) + %.3gohm", C1, R1,R2)
plot abs(parallel(Z_C(C1,x), R1)+ R2), \
 		R1+R2, R2

For those who don’t have gnuplot handy in another window, here’s what the script produces:

Plot of impedance of a model of polarizable electrodes, showing the asymptotes and the critical points.

Plot of impedance of a model of polarizable electrodes, showing the asymptotes and the critical points.

I had a little time to show them the plot of data that I had taken on the stainless steel electrodes and the somewhat poor fit I got. I also explained (a little) what the resistances and capacitance corresponded to physically: R2 is the saline solution, C1 is the insulation of the chromium oxide layer on the steel (and maybe other insulating effects), and R1 is leakage through the insulating layer.

After class, I went to the lab for office hours, and spent another 2 hours explaining the model, how to represent it in gnuplot, how to get impedance from the voltages they measured, and what plots I expected. I also talked a bit about the meaning of models—how there are no “correct” models, just ones that are more or less useful. One of the students had tried to fit a simple capacitance model to the electrode data, and the resulting plot was instructive in seeing where that simple model failed (the resistance is not infinite at DC, nor does it go to zero at very high frequency).

The students seemed to be understanding what I was talking about (I ask a lot of questions and draw most of the stuff out of the students, rather than just talking at them), but the same was true in class last Friday, but nothing was retained for today’s “do now” question. One of the students commented that he/she could do the problems with a few hints, but was lost without them. That’s actually an encouraging sign—the student has almost learned the material, and if they can just learn to give themselves the hints, they’ll be all set. One habit I want to wean them of is memorizing huge numbers of formulas. There are very few formulas in circuits worth the trouble to memorize—a few basic principles and an ability to apply them is far more versatile and less prone to stupid mistakes.

Maybe I should recommend some generic problem-solving books, like Polya’s How to Solve It, since it seems that the class has not been taught many problem-solving skills. I keep feeling that I’m teaching stuff that they should have had much earlier, but clearly haven’t, as if every teacher they’ve ever had has pushed off teaching the important stuff in order to cram in more factoids. There are times when I’m tempted to kick the can down the road myself, but these are all seniors, and it seems almost criminal to let them graduate as engineers without ever having been taught to think like engineers. I’m going to end up working them pretty hard, trying to get them up to speed in circuits, writing design reports, and how-to-think-like-an-engineer in just 7 more weeks.

After the office hours, I chatted a bit with a student in the EE circuits course who was in the lab trying to make up their first lab (which is a somewhat simpler voltage divider lab than our thermistor lab). The total set of parts for the EE circuits class is a lot smaller than for our applied circuits class and undoubtedly much cheaper (no sensors, no instrumentation amp, no PC boards, no solder sucker, only a handful of resistors in selected sizes, a 0.2W speaker instead of a 10W speaker, no transistors, …). The student was a bioengineering student who was taking the EE course because he wanted to do bioelectronics and EE won’t accept my course as a prereq for any of theirs (turf battles). He was envious of the more interesting labs that we were doing, even if they did take a lot longer. He also said that the EE course has not gotten to capacitors or impedance yet—they’ve spent all three weeks on equivalent circuits with resistors, voltage sources, and current sources.

I suspect that at the end of the quarter my students would not be able to pass the final exam for the regular circuits class (they have only a vague understanding of equivalent circuits), but could do all the labs and design exercises. I suspect that the EE circuits students would not be able to pass the final exam nor do the labs for my class.  Although both courses are intro circuits courses, we’ve chosen to emphasize very different aspects of the subject.

On Wednesday, I need to cover the following topics:

  • RMS voltage. I keep putting off a discussion of the 3 different systems for reporting AC voltage (amplitude, peak-to-peak, and RMS), so I’d better start with it.
  • hysteresis.  I have a pretty decent writeup (I think) in the lab handout, but I’m going to have to step the students through it, because I’m not sure that all of them learn well from reading.
  • hysteresis oscillator. Yet another time to talk about RC time constants.  The problem here is going to be that there is a somewhat arbitrary scaling of the RC time constant based on what the threshold voltages are, and I’m not sure I can convey that clearly, since I’m sure that developing the differential equation for charging and discharging capacitors through a resistance will just make their eyes glaze over.  They had all that in physics, and doing it again isn’t going to make it stick any better than last time.  I’ll have to think about this some for tomorrow, and see what I can come up with to make it more intuitive for them.

I also need to give them generic feedback on the second lab report—they got the lab reports back with specific feedback for individual reports, but no general comments that applied to several groups.  I think I’ll do that on the class website, though, rather than taking up class time.


  1. Interesting about the voltage dividers. A few possibilities that I have noticed in my students:
    – weak visual spatial skills. They don’t recognize the circuit as a voltage divider because it’s not in a vertical line, or because it’s not in a box shape, or because there’s a T-junction (series circuits don’t have T-junctions, of course!!) One thing that helps with this is giving a single circuit and asking them to analyze the output if the load is in parallel with one component, then the other. It breaks the assumption that a load is an invisible thing to the right of the circuit.
    – incomplete understanding of the concept of the “voltage reference.” My students need a LOT of practice remembering that there’s not really any such thing as the voltage “at” C or any other point; that’s a short-hand for the voltage from C to the reference point.
    – visual spatial skills again — not realizing that the voltage from C to ground is the same as the voltage across the resistor
    – incompletely internalized ideas about phase shift, making it difficult for them to resist treating V and I as scalars instead of vectors
    – interpreting “gain” to mean that the output is bigger than the input. Finding it hard to figure out how that could be possible.
    – difficulty interpreting gain as a ratio of output to input — i.e. a tendency to interpret gain as an amount of something. I’m reminded of the study you blogged about (sorry, can’t find at the moment) about the difference in student performance depending on whether concrete values were given in a problem or whether only symbols were used. If I was giving this to my students, I’d give them a voltage value so they could find “amounts” and then use that to scaffold their solution for the ratio. I would also show the signal source, and maybe the load too (even if the load is a voltmeter). I hesitate on that because I don’t want them to get in the habit of seeing a “load” in parallel with one of the components and assuming that you can analyze a network while ignoring a component, so long as it’s in paralllel! But once they’re gotten comfortable with the voltmeter as a high-impedance device, we can work through why it’s ok to abstract it out.
    – difficulty interpreting the concept of “load.” Is R2 the load?
    – difficulty interpreting the idea of an “open” (i.e. the open from C to ground). My students will often say that there is no voltage there, since there’s no current.
    – are they doing the phasor math by hand? Are they able to judge when to use polar and rectangular form? (My advice is , add and subtract in rectangular, multiply and divide in polar. I have entirely given up on the alternatives)
    – conceptual troubles with the distinctions between voltage and current. My students often have a hard time making sense of why a capacitor with low impedance, and correspondingly high current, gives a low gain. I mean, there are lots of electrons flowing through!
    – more conceptual troubles with V vs. I: current doesn’t flow through a capacitor, so 3 of these 4 circuits are meaningless anyway. Right? (I combat this by talking explictly about electrons flowing back and forth around the circuit, and also explicitly talking to the students about why, in electronics, information is almost always encoded by modulating the voltage, not the current — so the low impedance cap drops very little voltage, and the load is in parallel with that, also “seeing” very little voltage.
    – do your students think that electrons “flow into ground” and then disappear, never to be seen again? Mine often do, and it can hamstring their sense-making.

    Comment by Mylène — 2013 January 30 @ 20:35 | Reply

    • So far, I’ve been staying away from having them think about electrons, and just using conventional current. I think that the different directions for conventional current and the movement of electrons is more likely to confuse them than to help them think about current. I don’t think that they’re confused by the horizontal vs. vertical orientation of the upper arm of the voltage divider, since we’ve used both drawings more or less interchangeably. But I’m making an assumption there that I haven’t really tested.

      I think that most of the class got as far as the voltage divider formula, but then fell apart on plugging in the impedance of the capacitor. I might have gotten better results if I’d reordered the problems so that the resistance voltage divider was part A. If they succeeded at that one, they might have been in better shape to tackle the low-pass and high-pass filters. I’ll have to think carefully about the ordering of problems for Monday’s quiz, so that the easiest problems are first, encouraging them to continue deeper into the quiz, rather than shutting down on the first problem.

      I was only asking them for the magnitude of the gain, not the phase shift. We did phasors, but only in one lecture and have not revisited phase since. I haven’t decided yet whether to come back to phasors later in the quarter or not. I may have to in the op amp circuits, if they do RC feedback circuits, since the phase change may be enough to cause oscillation. I think I’ll be happy, though, if they just have a good grasp of the amplitude Bode plots, and not the phase ones.

      They didn’t need to do any complex math for these problems, as we had already derived the gains of 1 and corner/f for low-pass, f/corner and 1 for high-pass, and the sqrt(2)/2 gain at the corner frequency. I expect them to be able to handle complex numbers symbolically or with gnuplot, but not on their pocket calculators.

      I’ve not been using the term “load” much. They’ve only looked at unloaded voltage dividers, where there is nothing else connected. We talked about Thévenin equivalents and load lines in one lecture, but I don’t think that the ideas really stuck, since we’ve not used them again since. The EE circuits course spends about a third of the quarter drilling students on Thévenin and Norton equivalents, but I don’t see that as particularly useful for these students—I’d rather they spent their time learning how to pick component values for a design problem, only using the equivalent circuits if they are a useful shortcut (which they rarely are for real circuits, which don’t have huge numbers of resistors, voltage sources, and current sources all connected together). I’m taking the voltage divider and variants of it as the recurring theme throughout the course, as it is a fundamental building block of almost every circuit they’ll use.

      Comment by gasstationwithoutpumps — 2013 January 30 @ 21:01 | Reply

  2. I agree about the usefulness of voltage dividers as a sense-making tool (and the limited usefulness of Thevenin). When I glanced at these problems, I wonder if I fell into the same trap as your students? To figure out the gain, particularly if VDR is the most-emphasized tool, I would solve for Zr1/Zt (or whatever is appropriate for the problem). It didn’t even occur to me to check whether 1KHz happened to be the corner frequency, although the fact that you had recently emphasized that would of course make it more salient to your students.

    If the students aren’t imagining a load from C to ground, I wouldn’t be surprised if they can’t tell what the meaning of C is. Does it refer to voltage across R1? C1? Neither? Drawing a voltmeter in parallel with one of the components (or both!) might help. I can’t help but think that asking for gain is harder than asking for Vout — it requires an extra step of abstraction. Are they conceiving of gain as related to the relationship between input and output? If so, it may be hard to do any sanity checks when there is neither an input nor an output (even an imaginary load is helpful if they can be sure it’s in parallel with a given component).

    I don’t blame you for staying away from direction of current, but I wonder if your students are reasoning based (at least) on magnitude of current. My experience is limited to the engineering students who were in the MITx course I took, but “which direction is the current really flowing!” was by far the most commonly-asked question, generating the most frustration. It seemed that students who had taken a physics course were using that as the foundation of their knowledge, and if they couldn’t think straight about current, many found it hard to move on with a problem.

    I wasn’t suggesting having students solve for the phase shift — just that recognizing that phase shift happens can help them make sense of why, at the resonant frequency, a 10Vpp supply will not have 5Vpp across the output (corollary: 7Vpp + 7Vpp = 10Vpp).

    To check whether visual spatial skills are part of the issue, try giving the same problems rotated 90 degrees. Or flipped horizontally (input right, output left). Or, of course, give one in a straight, vertical line.

    One possible strategy: demonstrate the hardest problem. Then ask them to do the easier ones without a demo. This gets them tackling problems that are different from the demo. I’ve also been experimenting with some peer instruction-inspired ideas that have helped students ask better questions and figure out what they don’t know. The instruction is something like, “take 1 minute and solve as much as you can.” “In groups of 3, come to agreement about everything you can, and write down a question about anything you can’t agree on.” Then ask the groups to report back. Advantages: they have to grapple with the problem alone, confronting their actual level of proficiency; the groups can fill in some of the gaps (less for you), and they seem more willing to articulate their misconceptions and confusions when they are reporting on behalf of a group. I do this in a group of 12 — not because I don’t have time for them to contribute individually, but because the quality of their work and their questions goes up.

    Comment by Mylène — 2013 January 30 @ 21:30 | Reply

    • I might try adding in the voltage source and the voltmeter for exercises like this one, and repeat the definition of gain (asking for gain=Vout/Vin). Those changes to the problem may remove some conceptual barriers, without changing the problems in a way that affects what I wanted to learn about student progress.

      Thanks, these thoughts will help me craft a quiz that provides me with better feedback.

      I’ll try some group stuff next week, when we start struggling with op amp design (and using voltage dividers everywhere).

      Comment by gasstationwithoutpumps — 2013 January 31 @ 08:36 | Reply

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