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2018 June 3

VCO (voltage-controlled oscillator)

Filed under: Circuits course — gasstationwithoutpumps @ 11:01
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My students requested that I talk about voltage-controlled oscillators (VCO) and low-frequency oscillators (LFO) for audio work in my applied electronics course.  (We’re in the last week, and they have everything they need to know for the final EKG lab.)

I spent some time Friday night and this morning designing, building, and testing a couple of VCO circuits—one for which frequency is linear with voltage and one for which frequency is exponential with voltage.  Both can easily be turned into low-frequency oscillators by increasing the size of one capacitor.

The oscillators have two outputs: a square wave and a triangle wave. I chose a triangle-wave oscillator, because the design is simpler than for sine waves, and the students have all the concepts they need to understand the design.

The oscillator consists of two parts: an integrator to convert a constant current input into a constant slope: dV/dt = I/C, and a Schmitt trigger to change the current from positive to negative.  Rather than using a Schmitt-trigger inverter chip, I made the Schmitt trigger from an op amp (a comparator would give crisper transitions, but that is not important at the low speeds we’re dealing with).

This design has the frequency linear with the voltage.

The op amp in the upper-left corner is a unity-gain buffer to isolate the input from the rest of the circuit.  It isn’t really needed in this design, and it limits the input voltage range to the power-supply voltage (I used 3.3V).

The next op amp is the integrator, which turns an input current into a constant slope on the output voltage.  The current through the capacitor is 10nF dV/dt.

When the FET is turned off, the current flows through R6+R7, and I= (Vin/2 – Vin)/ (R6+R7), so dV/dt = – Vin/(2* 10nF * 6.6kΩ) = -Vin / 132µs.

When the FET is turned on, the current is the difference between the current through R8 and through R6+R7, and so is (Vin/2)/R8+(Vin/2 – Vin)/ (R6+R7), which simplifies to Vin/(2*6.6kΩ), and dV/dt = Vin/132µs.

The lower-right op amp is an inverting Schmitt trigger with thresholds at 1/3 and 2/3 of Vdd.  It turns on the nFET when the voltage of the triangle wave has dropped below the lower threshold and turns it off again when the voltage has risen above the upper threshold.

The upper-right op amp is just a unity-gain buffer to isolate the output from the oscillator.

My first attempt at this design used larger resistors for R6=R7=R8 and a smaller capacitor, but it had problems when the nFET turned off—the voltage continued to rise for a little while.  The problem was that the drain of the FET had to be charged through R6+R7 before the current through the capacitor was reversed, and this took too long.  Shrinking the resistors made the capacitance on the drain of the FET much less important.

This design has the frequency exponential with voltage (about one octave per 70mV).

The oscillator design is the same as for the linear one, but the diodes provide an exponential current from the input voltage. The input unity-gain buffer is now important, as it provides current limiting to prevent damage to the diodes.  To get double the current through the FET when it is turned on, the diodes are put in parallel on the lower leg (a series arrangement on the upper leg like for the resistors would not halve the current).

I tested both circuits and they seem to work ok, with duty cycles close to 50% for the square wave.  There are slight high-frequency glitches at the peaks of the triangle wave, so it may be worth replacing the output unity-gain buffer with a 40kHz low pass filter.

This is a fairly high frequency from the diode-controlled oscillator (15.431kHz) and the duty cycle is a bit off (59.36%), because the current has gotten large enough that the on-resistance of the nFET matters.

At 500mV, the frequency is 25.77Hz, the duty cycle is about 48%, and the glitches are not visible at this time scale.

I measured and plotted the frequency as a function of voltage for VCO with diodes:

The fit is done here omitting the lowest and highest points. I believe that the highest point has hit the current limits of the input unity-gain buffer, and so is not on the straight line.

The frequency scaling of the diode-based VCO could be changed by replacing the input unity-gain buffer with an amplifier with a different gain.

(2018 June 23: I feel compelled to point out that this circuit does not have any thermal compensation, and the exponential conversion of voltage to current in the diode junctions is highly temperature sensitive.  I’ll try this summer to redesign the circuit to have temperature compensation, though the use of 3 diodes here makes that difficult—a sawtooth oscillator, with only diode would be somewhat easier.)

2015 June 3

Crude theremin

On Monday I again had 2 lectures: one in Banana Slug Genomics on the Burrows-Wheeler Transform and FM index, and one in applied electronics finishing up the EKG preparation and starting into optional material.

The BWT and FM-index lecture was a bit rough though I had prepped on the material for a fair amount of time over the weekend.  I think that the written presentations I’d used to prep from (which included the Li and Durbin paper on BWA) were better than my lecture, so I ended up pointing the students to them as well, hoping that my lecture would at least make them more willing to click through and read the paper.

In the electronics class, I provided some feedback on the lab reports (reminding students once again how to figure out the DC bias resistor for an electret microphone).  Over the weekend I had finally caught up on the grading, so of course Monday saw a huge stack of redone work being turned in—I did the prelab grading Monday night, but I did not have the heart to start on the redone work last night or tonight.

After the feedback, and questions from the class on the EKG lab, I gave the EKG demo (which had worked fine before cycling up the hill).  Needless to say, it did not work when I hooked everything up in front of a live audience.  But I turned this into a teachable moment by explaining what I thought the problem was (bad electrode connections, because touching the reference wire produced a change in the output), and getting out three new electrodes, adding a a little electrode gel, sticking them on next to the ones I was already wearing, and moving the clips over.  Luckily, that was the problem, so I was able to demonstrate debugging, even with no lab equipment.

The EKG demo concluded the required material for the course, and so I asked students for suggestions on what I should cover next.  I suggested the electronics of the nanopore and nanopipette labs, and asked them for other ideas. One student wanted me to explain how theremins work, and a couple of others agreed that would be an interesting topic.  So I finished Monday’s lecture on the nanopore electronics (basically telling them it was just a transimpedance amplifier, but engineered for high gain and very low noise, and talking a bit about the copper boxes around the stations to reduce capacitive coupling), and told them I’d try to talk about theremins on Wednesday.

Tuesday’s lab went fairly smoothly, but I was kept busy helping students debug their designs—most often the problem was miswiring, but some students had electrodes that weren’t making good contact.  I had on a set of working electrodes, so that we could swap the leads to my electrodes to see if the problem was the electrodes or their boards.  A lot of the students had a lot more 60Hz noise pick up (even with my electrodes) than I was getting, and I could not always determine why. In some cases it was simply long wires on the breadboard.  They’ll be soldering up their EKGs on Thursday, and the wiring should be much more compact there.

Tuesday night I read up on a bunch of analog circuits relevant to Theremin design, and in today’s class I presented

• a block diagram for a theremin (two oscillators—one with an antenna, a mixer, and a low-pass filter to pass only the difference frequency)
• I showed them a rather crude theremin that they could implement from what they already knew: two relaxation oscillators using Schmitt triggers, an XOR gate, and an RC low-pass filter. (Only the XOR gate was new to them.)
• I then talked about mixers, presenting a ring modulator (which they probably didn’t understand, as we’ve not done transformers except in passing), a simple two-resistors plus a diode non-linear mixer, and a synchronous decoder using a switch to change an op amp gain from +1 to –1.  I mentioned a couple of Wikipedia articles they could look up for other mixer designs (Gilbert Cell and frequency mixer).
• I then introduced the notion of a sine-wave oscillator as a loop with gain exactly 1 and a phase change of 2πn for some integer n.
• The first oscillator I showed them was a Wien oscillator using an op amp, for which I derived that at $\omega RC = 1$ the phase change of the positive feedback was 0 and the gain was 1/3, so we could set the positive gain to 3 to satisfy the requirements.  I mentioned the first Hewlett-Packard oscillator, and the trick of using an incandescent light bulb to control the gain and avoid clipping.  I showed them a plot of the amplitude and phase change for the feedback circuit, using gnuplot functions that they are familiar with from their own modeling.
• Next I showed them an LC tank circuit, and derived the infinite impedance resonance $\omega^2 LC = 1$.
• I then got to a Colpitts oscillator, showing them a possible circuit using an op amp, and how to model it as cascaded voltage dividers using gnuplot.  I showed them that the phase change of the feedback was 180° and that the phase change moved away from 180° very fast.  I was running out of time, so I briefly mentioned the Hartley oscillator and the Clapp oscillator, but didn’t really go into them.

On Friday, I plan to cover one-transistor amplifiers (common emitter with emitter degeneration and common collector—I probably won’t do common base), as those seem to be popular in theremin designs.

Tonight, since I was not willing to face grading, I decided to try out the crude relaxation oscillator Theremin.  Here is the circuit I ended up with (not including the bypass capacitors):

I used the potentiometer on the second oscillator to do crude pitch matching for the two oscillators, just looking at the period on an oscilloscope. The emitter-follower amplifier was used to get enough current to drive the loudspeaker.

The crude theremin only sort-of works.  I could get high-pitched theremin-like noises from it, but when the difference frequency dropped to 500Hz, the two oscillators phase-locked and the speaker went silent (often with very irritating on-off stuttering). The phase lock occurred even though the oscillators were on different chips at opposite ends of the breadboard, and I had 10µF ceramic bypass capacitors on each Schmitt trigger package and a 470µF electrolytic on the middle of the power bus.  Everything was powered to 5V with the USB power from my MacBook Pro.

I also often got 60Hz modulation of the signal, often as a on/off square wave, when the frequency difference was close to the point where the oscillators phase locked.

I’m not sure how the oscillators are getting coupled so that they phase lock when the difference frequency is small, so I’m not sure how to fix the problem.  The 60Hz sensitivity is almost inherent in the relaxation oscillator design, since the thresholds for the hysteresis on the Schmitt trigger remain fixed, but the hand near the antenna couples in 60Hz interference.  I suppose that over the summer I should try building a theremin with Colpitts oscillators or Clapp oscillators, to see if that works any better.

2015 April 21

Loudspeaker lab went well

Filed under: Circuits course — gasstationwithoutpumps @ 22:34
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I’m falling behind in my blogging about the applied electronics class, and so the notes on the lectures may be a bit skimpy.

Last Friday, I had several topics:

• Some metacognition talk about answer-getting vs problem solving, with an emphasis on the descaffolding (removing the detailed what-to-do instructions and worksheets) that needed to happen as undergrads, and how too many of their classes had not been doing it, so I had to descaffold them faster than was really optimal.  I warned them that this would be uncomfortable.
• Test equipment as part of the circuit. One of the main points of the lab they had just completed the day before was to make them aware that test equipment (like oscilloscopes) become part of the circuit that they are testing, and can affect that circuit. I’d not made that clear in the book (there is a chapter that needs a massive rewrite this summer), so I spent some time in class on the idea.
• Inductors.  I didn’t do much with magnetic fields, but I did give them $V= L \frac{dI}{dt}$ and derived the impedance $j \omega L$.  There was an unplanned digression about transformers, in response to a question. I did have students give me the impedance of a resistor in series with an inductor, which they had to plot vs. frequency as one of their pre-lab assignments over the weekend.
• Loudspeakers. I showed them a labeled diagram of a loudspeaker, and the magnetic field near the voice coil. I reminded them of the EMF right-hand rule, and showed them how current through the voice coil would push the voice coil up or down (in the orientation of the pictures).
• I then quickly went over measuring the magnitude of impedance with a frequency generator and a pair of  AC voltmeters. This is almost identical to the lab they just did for DC characterization of a microphone, but I would have liked to spend more time on it.

The students did a prelab exercise over the weekend, but I forgot to tell them to do the prelab exercise for Thursday’s lab as well as Tuesday’s, so I had to tell them that in class on Monday (assigning it for Wednesday).

Monday’s lecture was supposed to cover several topics, but I didn’t get to them all:

• feedback on the previous week’s design report.  This took far too long, in part because I went over some fundamentals—I’m not sure, based on their writing, that all the class understands the difference between voltage and current. I introduced the hydraulic analogy (voltage=pressure difference and current=flow), but I’m sure that there are still several who think of both voltage and current as some incomprehensible measure of “electricity”.
• Another fundamental that came up as a result of the report is what a “model” is.  Too many people had been trying to stick voltmeters into the middle of the model for the loudspeaker.  I think that the physics classes are doing a poor job of getting students to think in terms of models, but I’ll do what I can to remedy the deficit.
• I also spent some time on writing fundamentals, like who the audience is for a design report, and that the document must be written for someone who does not have the book or the assignments.  Students are so used to fill-in-the-blank answers that they often don’t get the idea of a self-contained report—but that is what they will have to create as engineers, so they need to start learning how to do it. I had a full page of notes on feedback for the students, much of it specific to the writing assignment I was returning.
• I went over how to determine the shunt resistor for the current measurements in the loudspeaker lab. I was surprised to find out that most had never heard of a geometric mean and had no idea that it was the appropriate way to average numbers when errors were measured by ratios not by differences. I’m not sure that the idea came across well, because I had not anticipated having to teach them that part of high school algebra (silly me—they’ve needed to be retaught everything else they should have learned in high-school algebra, so I should have known that they would never have heard of geometric means).
• I never did get to the main topic for the day: hysteresis.  I’ll have to cover both model fitting and hysteresis on Wednesday, though I’d only originally planned to do model fitting.  Fitting the models for the loudspeaker is a bit tricky, since the main R and L values have to be fitted away from the resonance peak, then the resonance peak needs to be fitted on just that region. I hope that there is enough time on Wednesday to cover hysteresis and relaxation oscillators as well as the model fitting.

Lab today went fairly smoothly.  Everyone managed to get good data from 3Hz to 1MHz, with lots of data around the resonance peak, so they have good data for fitting models to.  There were three problem loudspeakers: Two had bad insulation-displacement connectors on the ends of cables to the loudspeakers.  They didn’t fail as open circuits, but as about 70Ω resistances that remained fairly stable until the wire was jostled.  This meant that the students had inconsistent results that were hard to debug. I think that the cheap connectors were slightly oversized for the wire, so that there was not a gas-tight connection and some corrosion had formed between the wire and the connector.  I finally managed to figure out what was going on for one of the groups, and a little while later applied the same thinking to another group.  Unfortunately, it was getting rather late by then, so that group only got one of their two loudspeakers characterized—they’ll probably try to finish up on Thursday, after doing the next lab.

The third problem loudspeaker behaved normally as far as resistance was concerned, but its resonant peak was nowhere near where it was supposed to be.  Instead of a resonance at 137Hz (as reported on the data sheet), it was out around 700Hz—way too high for a mid-range loudspeaker.

Incidentally, I was wrong about which loudspeakers the staff had bought this year—they got the 15W midrange speaker rather than the slightly cheaper 5W one, so we don’t need to worry about burning out loudspeakers on the power-amp lab.

2014 June 22

Loudspeaker relaxation oscillator

Filed under: Circuits course — gasstationwithoutpumps @ 18:56
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Yesterday I played around a little with a hysteresis oscillator (relaxation oscillator) using a loudspeaker as the feedback device, rather than a resistor, as shown in the schematic below:

Hysteresis oscillator with loudspeaker for feedback. Changing the value of C1 should produce different frequencies. I used 3.3v from the KL25Z board as my power supply.

I was hoping that I could get the oscillator to resonate at a predictable frequency, based on the gain or the phase shift of the voltage divider consisting of the loudspeaker and the capacitor. Of course, a relaxation oscillator that relies on the hysteresis of a Schmitt trigger is emphatically not a linear circuit, and there is no reason to think that using a loudspeaker instead of a resistor for the feedback element will simplify anything.

I did have a few questions:

• Would it oscillate? (I saw no reason it shouldn’t.)
• At what frequency would it oscillate? Was there any relationship to the gain or phase shift of the loudspeaker and C1 voltage divider?
• How stable was the frequency?
• Would it be loud enough to hear on the loudspeaker?

Before even wiring up the circuit, I ran some gnuplot scripts with my model of the loudspeaker to see what the gain and phase of the voltage divider would be:

Gain from the voltage divider, with different C1 values. Note that I’ve also added a line for an RC low-pass voltage divider, for comparison.

Phase change from the voltage divider, with different C1 values. Again I’ve added a line for an RC low-pass voltage divider, for comparison.

Note that the loudspeaker+capacitor provides a bigger phase change at high frequency and a faster rolloff in gain than an RC filter would. I was hoping that the resonant peaks at high frequency might capture the oscillator and provide a moderately stable output.

To avoid load on the oscillator from a multimeter (used to measure frequency) or oscilloscope, I buffered the output with a couple more Schmitt triggers.

C1 frequency
electrolytic measured with Fluke 8060A
470µF  27Hz
220µF  50–55Hz
47µF  265–285Hz
33µF  375–406Hz
4.7µF  3077–3401Hz
ceramic measured with Fluke 8060A
4.7µF  2.56–2.66kHz
0.1µF  116–122kHz
ceramic measured with Bitscope oscilloscope
47nF  245kHz
22nF  286kHz
10nF  587kHz
2.2nF  1.1MHz

The frequencies were not stable, but shifted unpredictably within a fairly wide range. The Fluke 8060A meter is not capable of frequency measurements above about 100kHz, so I used the FFT on the Bitscope USB oscilloscope for the higher frequencies. For the Bitscope measurements, I took whichever frequency seemed to come up most often—usually the lowest frequency.

The audio-range frequencies 27Hz to 3.4kHz were clearly audible on the loudspeaker (even annoying at the higher frequencies), and the fluctuations in frequency were audible also—they are not an artifact of the measuring equipment.

Looking at the input and output of the oscillating Schmitt trigger for a moderately large capacitor is instructive:

The green trace is the input to the Schmitt trigger—it seems to be a fairly clean triangle wave.
The yellow trace is the output—it jumps when the inverter switches, but the current limitation of the Schmitt trigger seems to keep it from being a square wave.
C1 was 33µF here, and the timebase is 2ms/div.

The current limitation on the Schmitt trigger output seems to be the controlling factor here, so we should be modeling this as a current source and capacitor, with the feedback impedance being irrelevant until it gets large enough that the Schmitt trigger output is voltage-limited instead of current-limited.

I tested this by replacing the loudspeaker by a 22Ω resistor, and got a very similar result (though without the voltage spikes from switching the loudspeaker inductance):

The nearly constant difference between the input and the output (across the 22Ω resistor) indicates nearly constant current flow.

If we look at the voltage across the resistor, we can measure the current:

The current through the 22Ω resistor is about ±18mA, though it isn’t quite as constant as I had thought from the previous plot. (This plot is 200mV/division, so about 9.11mA/division.)

With a 3.9Ω feedback resistor, I again see waveforms consistent with a ±18.5mA current limit (dropping gradually to ±15mA before it switches to the other phase). I can get the same triangle wave at the input if I use a wire for the feedback, with the output only a few mV different from the input (for a wire resistance equivalent to about 130mΩ). The period is about the same also, indicating that the current is what is determining the charging time of the capacitor, not the feedback resistor.

With a 200Ω feedback resistor, I see a little current limitation just as the output switches, but it becomes voltage-limited soon after. The period increases, as we would expect when the voltage limitation reduces the current.

So the loudspeaker in the feedback loop was not very interesting—it was behaving much like any other low-impedance feedback. At least, that is what is happening with low frequencies, where the loudspeaker is reasonably modeled as an 8Ω resistor.

With a 10nF capacitor and a 22Ω feedback resistor the oscillator oscillates at 1.27MHz, but with the loudspeaker only at 589kHz, so the loudspeaker is behaving more like a large impedance at these frequencies.  The output with the 22Ω resistor is a triangle wave, but with the loudspeaker is a square wave (with a little high frequency ringing).  I can’t see these frequencies well on the Bitscope oscilloscope, so I had to switch to my analog Kikusui COS5060 60MHz oscilloscope.  With 47Ω and no oscilloscope load, I get a frequency around 965kHz, with 22Ω around 1.13MHz, with 81Ω around 768kHz,  with 100Ω 687kHz, with 120Ω around 613kHz, with 150Ω around 526kHz, and with the loudspeaker around 590kHz.  So the loudspeaker is behaving like a 125–130Ω impedance around 590kHz, which is consistent with the measurements made of the loudspeaker impedance around that frequency. Similarly with a 3.3nF capacitor, I get 150Ω 1.23MHz, 180Ω 1.1MHz, 200Ω 1.03MHz, 220Ω 972 kHz, and loudspeaker 987kHz.  The output waveforms with the 220Ω or loudspeaker as feedback are fairly good square waves, so the oscillator feedback is voltage limited, not current limited.

So it looks like the simple analysis of the oscillator as being either a current-limited or a voltage limited output suffices for modeling the relaxation oscillator—the phase changes don’t seem to matter much, just how fast the capacitor charges or discharges, which depends mainly on the magnitude of the impedance.

Incidentally, the fluctuations in frequency for the oscillator occur with the resistors and the loudspeaker alike, and so don’t seem to be due to any physical or electrical properties of the loudspeaker.  They also occur with both electrolytic and ceramic capacitors.

2013 March 23

Triangle-wave oscillator

Filed under: Circuits course — gasstationwithoutpumps @ 23:09
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One of the student suggestions from the “bar exam” was to design a protoboard for the class D amplifier, so that they could keep it as a permanent object (a more useful one than the pressure-sensor amplifier, since they don’t have any strain gauges).

I’ve been thinking about that and see a few problems:

• The block diagram the students came up with this fall (with my guidance) calls for 3 power supplies, which makes the amplifier a bit impractical to power from a battery or wall-wart power supply.
• The students used an external triangle-wave generator.
• The protoboard would have to be fairly large to accommodate the FETs and inductor and enough wiring space, which would probably result in a larger board than 5cm × 5cm, and so $2/board rather than$1/board.

I’ve not given much thought yet to the power-supply problem, but I did think a little about the triangle-wave generator.  Since they only need to generate triangle waves at a single frequency (in the 60kHZ–100kHz range), it should be possible to use a hysteresis oscillator with an integrator rather than a simple RC timing circuit.  Something like the following might work:

The slew rate of the MCP6004 op amps is only 0.6 V/µs, but the triangle wave only swings from VIL to VIH, or about 1V, so the minimum period is about 3.3µs, for a maximum frequency of 300kHz. If we use another op amp to amplify the signal to make it swing almost 5v, instead of 1V, the maximum frequency would be only 60kHz.

I might want to try this circuit out this week, to see if it is worth having the students play with next year.