Yesterday, in Logarithmic amplifier, I ended with the following plot:
I was bothered by the broad cloud of points, and wanted to come up with a better test circuit—one that would give me more confidence in the parameters. It was also quite difficult to get close to Vbias—the closest this could measure was one least-significant-bit of the DAC away (about 5mV). A factor of 512 from the largest to the smallest signal is 54dB, but only about the upper 40dB of that was good enough data for fitting (and very little time was spent at near the Vbias value).
I think that part of the problem with the cloud was that the input signal was changing fairly quickly, and the Arduino serializes its ADC, so that the input and output are measured about 120µsec apart. I decided to use a very simple slow-changing signal: a capacitor charging toward Vbias through a large resistor. My first attempt used a 1MΩ resistor and a 10µF capacitor, for a 10-second time constant:
The capacitor charging should be a smooth curve exponential decay to Vbias, so the log amplifier output should be a straight line with time. There were two obvious problems with this first data—the output was not a straight line and there were weird glitches about every 15–20 seconds.
The non-straight curve comes from the capacitor not charging to Vbias. Even when the capacitor was given lots of time to charge, it remained stubbornly below the desired voltage. In think that the problem is leakage current: resistance in parallel with the capacitor. The voltage was about 1% lower than expected, which would be equivalent to having a 100MΩ resistor in parallel with the capacitor. I can well believe that I have sneak paths with that sort of resistance on the breadboard as well as in the capacitor.
According to Cornell Dublier, a capacitor manufacturer, a typical parallel resistance for a 10µF aluminum electrolytic capacitor would be about 10MΩ [http://www.cde.com/catalogs/AEappGUIDE.pdf]:
Typical values are on the order of 100/C MΩ with C in μF, e.g. a 100 μF capacitor would have an Rp of about 1 MΩ.
So I may be lucky that I got as close to Vbias as I did.
The glitches had a different explanation: they were not glitches in the log amplifier circuit, but in the 5V power supply being used as a reference for the ADC on the Arduino board—I had forgotten how bad the USB power is coming out of my laptop, though I had certainly observed the 5V supply dropping for a second about every 20 seconds on previous projects. The drop in the reference for the ADC results in a bogus increase in the measured voltages. That problem was easy to fix: I plugged in the power supply for the Arduino rather than running off the USB power, so I had a very steady voltage source using the Arduino’s on-board regulator.
To solve the problem of the leakage currents, I tried going to a larger capacitor and smaller resistor to get a similar RC time constant. At that point I had not found and read the Cornell Dublier application note, though I suspected that the parallel resistance might scale inversely with the capacitor size, in which case I would be facing the same problem no matter how I chose the R-vs-C tradeoff. Only reducing the RC time constant would work for getting me closer to Vbias.
Using a 47kΩ resistor and a 470µF capacitor worked a bit better, but the time constant was so long that I got impatient:
The calibration of 9.7mV/dB seems pretty good, so the 409mV range of the recording corresponds to a 42dB range. The line is straighter, but I’m still not getting as close to Vbias as I’d like.
I then tried a smaller RC time constant (hoping that the larger current with the same capacitor would result in getting closer to Vbias, and so testing a larger dynamic range on the log amplifier). I tried 16kΩ with the 470µF capacitor:
The two models I fit to the data give me somewhat different mV/dB scales, though both fit the data fairly well. The blue curve fits better up to about 65 seconds, then has quantization problems. Using that estimate of 9.8mV/dB and the 560mV range of the output, we have a dynamic range here of 57dB. There is still some flattening of the curve—we aren’t quite getting to the Vbias value, but it is pretty straight for the first 50 seconds.
Note: the parallel resistance of the capacitors would not explain the not-quite-exponential behavior we saw in the RC time constant lab, since those measurements were discharging the capacitor to zero. A parallel resistance would just change the time constant, not the final voltage.
I was using the Duemilanove board for the log-amplifier tests. I retried with the Uno board, to see if differences in the ADC linearity make a difference in the fit:
The 625 mV range over 250 seconds corresponds to about 69dB, assuming that the 9.1 mV/dB calibration is reasonably accurate (and 64dB if the earlier 9.8mV/dB calibration is better).
My measurements of the log amplifier do not seem to yield a very consistent mV/dB parameter, with values from 9.1mV/dB to 9.8mV/dB using just the Arduino measurements (and even less consistency when a model of RC charging is used). I’m not sure how I can do more consistent measurements with the equipment I have. Anyone have any ideas?
Incidentally, my son has decided not to include a microphone in his project. The silicon MEMS mic was small enough, but the op amp chip for the analog processing was too big for the small board area he had left in his layout, and he decided that the loudness detector was not valuable enough for the board area and parts cost. I believe that his available board area shrunk a little today, because he discovered that the keep-away check had not been turned on in the Eagle design-rule checker. Turning it on indicated that he had packed the capacitors too close in places, and he had to spread them out. (At least, I think that’s what he told me—I’ve not been following his PC board layout very closely.)
I’m still interested in learning about log amplifiers and precision rectifiers, so I’m still going to breadboard the components of the design and test them out. I’m not sure when I’ll ever use the knowledge, since the Applied Circuits course does not cover the nonlinear behavior of pn junctions nor the forward-voltage drop of diodes (we don’t use diodes in the course).