In a comment on her post Student Thinking About Abstracting, Mylène says

What frustrates me and disorients my students is that those justifications are never discussed, and even the fact that this is a model is omitted. To further “simplify” (obscure) the situation, most discussions of the matter don’t distinguish between two ideas: “the model has a change in behavior at 0.7V,” vs. “they physical system has a change in behavior at 0.7V.” Finally, the chapter starts with the most abstracted model (1st diode approximation) and ends with the less abstracted (3rd diode approximation).

On getting students to understand models: I agree that this is a huge problem. I’ve been trying various techniques and can’t claim to have found a silver bullet.

One thing I tried in class yesterday (disguised as a gnuplot tutorial) was to build up a model a little at a time to match measured data. I was trying to build an equivalent-circuit model for a loudspeaker, so I started by gathering data (rms voltage measurements across the loudspeaker and across a series resistor at different frequencies) and plotting magnitude of impedance vs. frequency from the data, then building the model a component at a time. Before doing the modeling, we had spent some time looking at the behavior of building-block circuits (R+C, R||C, R+L, R||L, C||L, C||L||R) using gnuplot, so I could ask them things like “how can we model the impedance increasing with frequency above about 1kHz?” We could then immediately modify the model and plot the results. Once things were close, we could use gnuplot’s “fit” command to tweak the parameters.

We didn’t start with “loudspeakers are …”, though we did start with one of the specs—that this was an 8Ω loudspeaker—for our first model. I didn’t even point out to the students that the frequency of main resonance peak is given as a spec on the data sheet. The data sheet gives it at 191Hz, while our measured data show 148Hz (more than 22% off, while factory tolerances for the resonant frequency are usually ±15%). They also give the voice coil inductance as 0.44mH, while our model gets 35µH, a factor of 12.6 difference! And they give the Q_{es} of the resonance peak as 3.52, while our model of the R||L||C for the peak has .

Maybe the inductance difference can be explained by the standard measurement for the voice-coil inductance being made at 1kHz for the Theile-Small parameters, while I fitted for a wider frequency range and added an extra 112µH inductor in parallel with a 32Ω resistor to bump up the impedance around 10kHz. Or maybe my fitting is a really bogus way to get the inductance, since I’m only looking at the amplitude and not the phase of the signal, and non-linear resistance could throw things off. Or maybe the Parts-Express people mis-measured or had a typo—I have no idea what measurements they made to get the parameters they report, or maybe these loudspeakers were so cheap because they didn’t meet the specs, though they are certainly good enough for our lab.

I think that one could do the same sort of model-building with diodes (the part whose models Mylène’s students were confusing with reality): start by measuring the I-vs-V characteristics. The setup I used to get a lot of data points with the Arduino for characterizing the FET in an electret mic might be a good one for them to use, though the unipolar ADC in the Arduino might be more challenging for characterizing diodes. Then try fitting different curve families to the data. Forget about physics for explaining how the diodes work, but concentrate on finding simple models that fit the data. For example, the FET models we used for the mic are not quite the standard ones, since there is a clear slope in the saturation region, and it doesn’t match the channel-length modulation model—but it can be fit with some simple curves.

Of course, I gave up on some modeling before even having the students collect data themselves—the power FETs they are using are incredibly messy, having threshold voltages that shift a lot as the transistors warm up and having an undocumented negative dynamic resistance region when diode-connected.

So it is important that their attempts to build models be of phenomena that are relatively easy to model, but they should build and fit the models (with some guidance) rather than just be handed them. I made the mistake of handing them models to fit for the electret mic lab and for the electrode lab. They not only didn’t understand the models, but they didn’t understand how to do the fitting.

I’m planning next year to do the model-building/gnuplot tutorial much earlier in the quarter, before they do the electrode labs, so that they can build the electrode models with some understanding. I’ll need to rearrange some other material, to do inductors much sooner, if I plan to use the loudspeaker data again. I may want to rearrange the labs a lot next year, since all of my first three labs involved model fitting, and the students weren’t ready for it. It may be better to move the sampling lab (which is currently lab 6) into the beginning, so that students can learn to use the Arduino in a simpler lab. As currently written, though, that lab calls for designing a high-pass filter for DC level shifting and a low-pass filter for removing aliasing, neither of which are suitable for a first-week lab in circuits.

Scheduling the labs and the classes is difficult. Fitting in all the topics they need before each lab is a tricky jigsaw problem, particularly when I discover them having problems with topics that I assumed they knew or could pick up quickly. Sigh, some stuff in the first week or two of lab is probably going to have to be “magic” as they’ve learned so little in physics classes that I can’t count on them having any useful lab or modeling skills when they come into the class. I just have to decide which things I’m willing to give them, rather than having them do for themselves.

Currently, I’m leaning toward having every lab have a design component, and to have them build models for important concepts, but I’m willing to give them a model for thermistor behavior that they just have to fit the parameters for. The design in the first two labs this year is very light (selecting a resistor value), but the measuring and model fitting is pretty heavy. The electrode lab has no design currently, but a lot of measuring and model fitting. I think I underestimated the relative difficulty of model fitting and design for these students, and may need to move the model fitting later in the quarter. I don’t think I can start with RC filters in the first week though, as they need voltage dividers, complex numbers, sinusoids, and complex impedance—probably at least 4 classes worth of material. Maybe by week three, though.

Thanks for the exploration of this tricky question. I would love to have my students build mathematical models, but I haven’t even considered trying it — my students have a hard time just getting data onto a graph at all, and the concept that slope means something is an elusive grail. Linear algebra would take much longer than the two years we have available, and the idea of a “curve family” is alien. So mostly we tangle with narrative descriptions of models (for example, “red LEDs have a turn-on voltage around 1.7V, and then they almost level off around 2V, increasing only a tiny bit after that, while the current shoots up”). It’s more urgent to me that they notice, describe, and abstract. I do insist on the graphing, though. Then we usually describe the graphs in English, since my students don’t have the vocabulary or fluency to do it in math.

I absolutely agree that the place to start, with diodes, is the I vs. V curve — a precise, carefully-examined description of “what” is happening can take place long before we get to the physics of “why.” I wish textbook authors agreed too. The physics of the PN junction is usually at the front of the chapter, and the I vs. V curve is at the back — described in a way that presupposes knowledge of the physics, which makes it almost unusable to us until much later. I’d be interested to see a textbook that orders the ideas this way.

Comment by Mylene DiPenta — 2013 February 20 @ 09:25 |

I have the advantage that the students have had calculus and calculus-based physics, though their memories of both seem rather vague.

One of the nice things about using gnuplot to do the plotting is that they don’t have to do a lot of calculation or hand plotting. They just have to get a feel for things like “impedance of L goes up linearly with frequency”, “putting two impedances in series behaves like the bigger impedance (except where they are nearly equal)”, “putting two impedances in parallel behaves like the smaller impedance (except when they are nearly equal)”. This allows us to do empirical model building without needing a theoretical justification.

I agree with you that most circuits books are written ass-backwards. It is a common fault of EE departments to believe that all instruction must be bottom up (starting from quantum mechanics) with no useful results until the 3rd or 4th year. A lot can be done starting with empirical observations and creating models that are “good enough”. (That is how most of the textbook models were created, after all.)

Comment by gasstationwithoutpumps — 2013 February 20 @ 10:04 |