Twenty-eighth day of circuits class

Yesterday’s lecture was a mish-mash of odds and ends that we hadn’t covered well previously, and questions that had come up.

I started with talking about where the EKG signal comes from.  The guest lecturer on Wednesday had done a good job of covering action potentials, but I pointed out that we were not sticking electrodes into their heart muscle cells: we had access to only the outside of the cells.  So where was the differential signal we were measuring coming from?  This had been one of my first puzzles in trying to understand how an EKG works, and they were mostly just as clueless about it as I had been.  The explanation that I settled on was looking at each of the cells as a little capacitor with a battery that charged it and a switch that discharged it, with one side accessible to us.  There is a resistance from each cell to each of the differential electrodes, and from the differential electrodes to the body reference electrode, so that the measurement electrodes act like they are in a voltage divider between the cell and the reference electrode.  The voltage difference between the differential electrodes depends on the difference in resistance from the cell to the electrodes, and the signal we see depends on the change in position of the discharge wave.  If everything depolarized at the same time, we’d see almost no difference voltage, but as the wave sweeps from left-to-right (or right-to left) we see difference voltages.    It’s not a perfect explanation of where the EKG signal comes from, but it is better than leaving them thinking that they are seeing the action potential directly.

After that a question came up about the 60Hz noise in the EKG signal, and where it came from.  I talked about the loop formed by the LA and RA wires and the body between them as an electromagnetic pickup, and how we could reduce the electromagnetic pickup by twisting the wires together more to reduce the area of the loop.  I also discussed capacitive coupling of 60Hz into the wires, reminding them of the capacitive touch sensor they had made earlier in the quarter.  We talked a bit about shielding cables and Faraday cages.  While on the subject of noise, I also mentioned the problem of microphonics in the nanopore equipment.  We have not discussed thermal noise or other problems of designing for very small signals.

Students asked where the 60Hz hum in their class-D power amplifiers came from, and I talked about ground loops and noise pickup in their power lines.  The op amps they were using had excellent power-supply noise rejection, but the voltage reference they were using (a pair of resistors as a voltage divider and a unity-gain buffer) provides an excellent path for noise in the power supply to be coupled into the amplifier.  I talked about two ways to reduce the hum: using twisted cables for the DC power, to reduce the electromagnetic pickup of 60Hz noise, and using a Zener diode reference instead of a simple voltage divider for the Vref signal.  I’m wondering whether I should add Zener diodes to the parts kit next year, or even whether I should get an adjustable voltage reference like the TL431ILP.  Either one is about 20¢ each in the small quantities that we would need.  The Zener diode is easier to explain, but the adjustable voltage reference is more versatile (the voltage is set by 2 external resistors as a voltage divider with the output of the voltage divider held at 2.5V) and has less variation in voltage with current (output impedance of 0.2Ω instead of 5–30Ω).  One minor problem with the TL431LP is that the lowest reference voltage it provides is 2.5V (using a wire from the “cathode” to the reference feedback input).  Since we are doing everything with power supplies of 5V or more, this shouldn’t be a problem for us.

After talking about noise, a question came up about the fat wires used for loudspeakers in stereo systems and whether they were shielded.  I managed to get the class to come up with the correct explanation: that the wires are fat to reduce resistance and avoid I²R power losses.  Currents to loudspeakers tend to be pretty big, since the resistance of the speaker itself is typically only 8Ω.  We talked about microphone cables being shielded (nowadays, I think that they are mostly twisted pairs inside a foil or metalized plastic shield, rather than coaxial cable) but that the tiny voltages and currents that speaker wires would pick up not mattering, since the signals were not amplified.  I also mentioned that solar panels were generally wired with fat wires also, to reduce the I²R power losses, since the voltages of the solar panels were fairly low (12V or 24V).

The students did not come up with any questions, so I pulled out one that had been asked weeks ago in lab: how sine-wave oscillators work.  The students had build square-wave oscillators with Schmitt triggers, and had used function generators, but had not seen any sine-wave oscillators.  I decided to do a classic oscillator: the Wien-bridge oscillator, since it uses the building blocks and concepts they are familiar with: a differential amplifier and two voltage dividers as a bridge circuit.  I started out with a generic bridge (just an impedance on each arm) and we got 3 formulas relating the nodes of the amplifier: , , and , from which we derived the stability condition .  This was all review for them, as they had had bridge nulling on a quiz.

The Wien bridge oscillator circuit. I initially gave it with just a resistor, not a light bulb, for R1, since the analysis is easier that way. The neat thing about the light bulb is that it provides an automatic gain control to set its resistance to  R2/2.

I then gave them the circuits for each arm of the bridge (just resistors on the negative feedback divider, and a series RC and a parallel RC for the arms of the positive feedback divider).  Rather than do complex impedance calculations, we just did Bode plots of the impedance of each of the RC arms, from which we could see that the voltage divider had zero output at DC and ∞ frequency, with a maximum at .  We were running out of time, so I did not derive with them that the gain of the positive voltage divider was 1/2 at that frequency, but jumped immediately to describing the use of an incandescent bulb in the negative feedback circuit to provide automatic gain adjustment (though I just waved my hands at it, not really showing how the thermal feedback mechanism worked).  I also managed to mention the historical importance of this oscillator design as the first product of Hewlett-Packard, and the start of “Silicon Valley”.

The range over which the automatic gain control with the light bulb works is determined by the range of resistance for the bulb filament. When the bulb is cold, its resistance must be less than R2/2. When the output is a sine wave with amplitude equal to the power supply, the resistance of the bulb filament must be larger than R2/2.  When the circuit is stable, the RMS voltage on the  bulb will be 1/3 the RMS voltage of the output, and the bulb filament resistance will be R2/2.  Nowadays other non-linear components are used rather than bulbs for the gain control, since bulbs suffer from microphonics and (for low frequencies) insufficient low-pass filtering (they are relying on the thermal mass of the filament to provide the low-pass filter of the automatic gain control).

On Monday, I plan to answer other questions students have, if they can come up with anything that confused them over the quarter.  If they can’t come up with any questions for me and send them to me this weekend, then maybe I’ll have to come up with some questions for them as an impromptu quiz.