# Gas station without pumps

## 2013 January 23

### Seventh day of circuits class

Filed under: Circuits course — gasstationwithoutpumps @ 19:57
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We’re now about 1/5th of the way through the course (2/10 labs done, 7/34 lectures done).  Things have smoothed out a bit since the first week, and I have a better understanding of what pace I can keep with the students.  I’ve lost two of the top students whose schedules did not permit them to stay for the whole 3-hour lab.   They were finding it too difficult to do the labs on their own time.  Part of the problem here is that seniors are over-scheduled—I had intended this course for sophomores and juniors, not seniors, but I ended up with those students who’d been putting off learning circuits as long as possible.

Today’s class went almost exactly as I had planned it (a rare luxury for me).  I had just a few concepts to get across, and fairly simple progression from each topic to the next:

• $e^{j\theta} = \cos(\theta) + j \sin(\theta)$
• Polar notation: any complex number can be represented as $z= A e^{j \theta}$ (This included the notion of amplitude.)
• sinusoids:  $e^{j\omega t}$ We talked about frequency, angular frequency, and period, reminding them of the relationships between them.
• Phasors: multiplying by a complex number does amplitude and phase change $z e^{j\omega t} = A e^{j\theta} e^{j \omega t} = A e^{j (\omega t + \theta)}$.
• From $Q=CV$, we derived $i(t) = C \frac{dV}{dt}$.  Actually, I did the full chain rule, and talked about when $\frac{dC}{dt}$ was close enough to zero to be ignored.  We plotted (crudely) voltage and current versus time, noticing the difference in phase and relating the movement of the charge to the increase or decrease of voltage.
• We then divided voltage by current to impedance of a capacitor: $Z = 1/ (j \omega C)$ and we noticed that at low frequency a capacitor is a an open circuit and at high frequency the capacitor is a short circuit.
• Finally, we analyzed a voltage divider consisting of a resistor and capacitor, rederiving the voltage-divider formula, since no one saw quickly that the formula was one they already knew, just using impedance in place of resistance.

That’s where we ran out of time, so I gave the students a homework exercise: use gnuplot to plot the magnitude of Vout/Vin for the RC voltage divider as a function of frequency for different R and C values.  I did not give them values for R and C (I should have given them 1kΩ and 1µF) nor a frequency range (I should have given them 1Hz to 1MHz), but I did tell them to use a log-log plot. They should end up with something like the following, though I don’t expect them to put in the asymptotes.

Low-pass filter made with a voltage divider having a 1kΩ resistor on top and a 1µF capacitor on the bottom.

On Friday, after tomorrow’s lab, I’ll have to remember to bring in my laptop so that we can explore the gain of simple RC filters and I can introduce the notion of the Bode plot.  I think that we’ll leave phase alone for a while, though.  I’ll probably also have to explain RMS voltage on Friday, since that is what they’ll measure in the lab tomorrow.  I’ll also want to talk a little about stainless steel, silver/silver-chloride, and other bio-relevant materials.

I’ve been thinking a bit about how to schedule the course next time, to make the first two weeks less hectic.  I think that twice a week labs (with the first seven sessions being parts kit, thermistor resistance, thermistor voltage to Arduino, microphone DC characteristics, microphone on oscilloscope, stainless steel electrodes, and Ag/AgCl electrodes) would put us just half a week behind where we are this year in terms of labs, but probably a bit ahead in terms of theory, since we would not have used a lecture on the parts kit.  The lab write-ups would probably still be weekly, even with twice-weekly labs.

Distributing the parts kit in a lab session would give us a good time to teach using the wire strippers and making clip leads with the alligator clips.  Perhaps there would even be time to learn the pattern of breadboard connections by probing with a multimeter and a couple pieces of wire.