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2015 April 10

Sinusoids and impedance lecture

Filed under: Circuits course — gasstationwithoutpumps @ 20:46
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Today’s lecture in BME 101 (the Applied Electronics for Bioengineers class) was again pretty much just as I had planned.  I covered three topics:

• Sinusoids
• Capacitors
• Complex impedance (of capacitors)

The sinusoids section was a brief intro to Euler’s Formula: $e^{j \theta} = \cos(\theta) + j \sin(\theta)$, expanding to the general sinusoid we’ll use all quarter: $A e^{j \omega t + \phi}$.  I showed them what phase meant on both a time-domain sine wave and as a rotation of the unit circle.  I suspect that those who already knew Euler’s Formula had their memory refreshed, and those unfamiliar with it will either look up info about complex numbers or give up because they are math-phobic.  This year’s class doesn’t seem particularly math-phobic (so far), so I’m hopeful that they’ll refresh their memories of complex numbers, because we’ll be using sinusoids in this form a lot.

I did a lot of cold-call questioning in the capacitor section, getting students to give me the charge formula Q=CV and some descriptions of  the structure of a capacitor fairly quickly. I also mentioned the dependency of capacitance on area, insulator thickness, and dielectric constant.  I gave the relative dielectric constant of air as about 1 (I looked it up now as 1.0006), plastics as 2–4, and ceramic capacitors as around 10,000. I was wrong about the ceramics:  the class 2 Barium titanate ceramics (what we have in our cheap ceramics) have a relative dielectric constant in the range 3000–8000, and the class 1 paraelectric ceramics only 5–90.  I claimed that electrolytic capacitors relied on the thinness of the oxide and large plate area, rather than high dielectric constants, but didn’t give a value (Kemet, who makes capacitors, claims 8.5, so a little more than plastics, but nowhere near the ceramics [http://www.kemet.com/Lists/TechnicalArticles/Attachments/6/What%20is%20a%20Capacitor.pdf]).

I then got from the students that $I = dQ/dt$, and thus that $I = C dV/dt + V dC/dt$ (getting the class to apply the chain product rule took a while).  I pointed out that we would usually use examples in which C was constant, so the formula simplified to $I = C dV/dt$, but that some of our circuits would have changing capacitance (like the electret microphones that they’ll use next week and the capacitive touch sensor that they’ll design later).

I then put the two previous parts together, defining impedance as a generalization of  resistance, for sinusoidal signals: $Z = V/I$.  We then made the voltage by an arbitrary sinusoid, $V(t) = A e^{j \omega t + \phi}$, and figured out the impedance of a capacitor $Z_C = \frac{1}{j \omega C}$.  I had them give me the impedance of a resistor and capacitor in series (a couple of false starts, but quickly converging to the right answer: $R + \frac{1}{j \omega C}$). Finally, I had them give me the formulas for a couple of voltage dividers: a high-pass RC filter and a low-pass RC filter, and we simplified the formulas by multiplying top and bottom by ${j \omega C}$.

I then switched to gnuplot and showed them how to plot the magnitude of the impedance of a circuit as a function of frequency, and the gain of a high-pass filter:

j = sqrt(-1)
Z_C(w,C) = 1/ (j * w *C)
set xrange [1:10000]
plot abs(Z_C(2*pi*x, 1e-6))

set logscale xy
plot abs(Z_C(2*pi*x, 1e-6))

divider(zup,zdown) = zdown/(zup+zdown)
R=4700
plot abs(divider(Z_C(2*pi*x, 1e-6)  , R))


The gnuplot stuff was a little hurried, so I’ll spend the first part of Monday on Bode plots, corner frequencies, and the design of RC filters. They have a homework (prelab) exercise due on Monday, so they should be primed for understanding the material.

1. Did you mean product rule (for CV) or chain rule (after substituting for V)?

Either way, if you had to pry that out of them, they probably have a similar knowledge of complex numbers. I am no longer shocked to learn that students have little recall of complex numbers after I had a math prof ask me if they are ever used for anything. And they likely have never heard that sqrt(-1) is also called j or discovered that their TI calculator knows the ln(-1) when put in the correct mode.

Comment by CCPhysicist — 2015 April 12 @ 18:58

• Oops, I meant product rule. The chain rule (when differentiating the sinusoid) actually took them far less time. I’ll fix the post.

I didn’t expect students to know about the strange usage of j by electrical engineers, so I told them about it (more than once). I was a math major, but I learned about complex numbers from my dad (an engineer trained as a physicist) long before I got to them in math classes.

Comment by gasstationwithoutpumps — 2015 April 12 @ 19:18

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