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Water Theremin: Circuit Description and Performance

Theremin as Transducer. As introduced in Overview Part 4, my water surface wave pick-up is based on the theremin, a musical instrument. For playing music, the distance between a theremin's pitch-control antenna and the player's hand determines the output frequency. The instrument mixes (sums) two radio-frequency (RF) signals and outputs the audio-frequency difference (heterodyne or "beat") signal. Two separate oscillators make the RF signals: one has a fixed frequency, while the other is proximity-sensitive due to an antenna hooked to its L-C resonant network. The antenna acts as one plate of a capacitor, the player's hand the other, and air the dielectric. The distance between plates affects total capacitance in the resonant circuit, and hence RF frequency. Since the audio output is a difference signal, small capacitance changes translate into wide pitch swings (one reason a theremin is so notoriously hard to play in tune). In my Water Theremin (WT) transducer, the antenna is insulated and water serves as one capacitor "plate," so immersion depth determines output frequency. Water surface waves modulate that frequency, so the output is a FM analog of those waves. 

WT Construction. Photographs of the WT are shown in Figures 9 through 11 of Overview Part 4. The oscillators are wired point-to-point on separate 0.6-inch square ground plane-clad perfboard squares (after using a small router to clear immediate copper from holes for the non-grounded leads). I cut a standard perfboard to fit inside a 2x3x1-inch plastic project box. This "mother board" hosts the oscillator modules and all other components, and is wired point-to-point. A small perfboard glued to the outside of the box has three solder terminals for the power supply/output cable.

The WT responds to sub-millimeter depth changes, so it needs a stable physical support. Ideally, the support should not interfere with the water surface waves being recorded. I used salvaged laboratory "ring-stand" hardware for the 5-1/2-inch long, 3/8-inch diameter horizontal arm terminated with a right-angle coupler. The vertical support for bench tests was an old laboratory stand with a 12-inch post and a Bakelite base. I made a vertical support for field recordings by pushing a 30-inch length of 1/2-inch (O.D.) aluminum conduit into a 32-oz plastic funnel, then filling the funnel with about five pounds of concrete to serve as the base. The metal support hardware grounds the WT to the water, which is essential as I will describe below. My WT's enclosure is not water-proof (however, the cable is). A water-proof enclosure would be an improvement--it would allow accidental immersions and recording in all weather conditions.

Circuit Description: Oscillators. Figure 1 shows a schematic diagram of the WT (figure and component numbers are specific to this page). I based the two identical Colpitts-type RF oscillators on a design by Aurthor Harrison (within navigate to "144 Theremin"). Each uses a common-base-wired PNP transistor (Q1) with an L-C resonant network (L1, C2, and C3) in its collector circuit. Positive feedback links a capacitance "tap" (C2-C3 junction) to the emitter, which develops the output voltage across R3.

Without external capacitors, the oscillators run at about 1.2 MHz. But C4-C6 (reference oscillator) and C7-C8 (depth-sensitive oscillator) make this about 1.0 MHz. I tweaked C5's value for a null heterodyne (equal RFs) when the antenna tip is about 2 mm deep. You may need a different value. Stray coupling tends to synchronize a theremin's oscillators when their frequencies nearly match. To minimize this, I physically separated the oscillators as much as possible and included power supply decoupling (R4 and C1) in each. As discussed below (see Figure 3), however, the normal operating range avoids frequencies close to the null heterodyne. The individual oscillators are sensitive to temperature changes, but the overall WT is remarkably temperature-stable. Heating it with a hair dryer for 5 minutes caused a mere one-percent change in output frequency. Presumably this is because the identical oscillators drift the same way, rejecting common-mode temperature changes.

Figure 1: Schematic diagram of Water Theremin, a transducer for water surface waves

Figure 1.  Schematic diagram of the Water Theremin, a transducer for water surface waves.

Circuit Description: Mixer and Envelope Detector. Oscillator outputs AC-couple through C9 and C10 to two-input CMOS NOR gate U1A, which acts as a mixer. Resistor network R5 through R9 biases the signals into the gate's ersatz "linear" range. I picked R5's value empirically. You may need a different value, or better yet, design a better mixer (see "Resolution: Room for Improvement," below). The envelope of U1A's output is "beats"--bursts of pulses when input signals reinforce each other (positive interference). Threshold response by U1B helps square up the envelope.

In the next stage, timer LM555 (U2) acts as an all-or-none envelope detector. It is wired as a re-triggering monostable multivibrator. R10 and C12 set the hold time at 10 microseconds. Individual input pulses re-trigger the timer since they make Q2 discharge C12 (E.A. Parr [1978] "IC 555 Projects" Bernard Babani Publishing LTD, London; ISBN 0-85934-047-3). The result: U2's output is continuously "high" during input pulse bursts (beats) plus an extra 10 microseconds. Wired in parallel, gates U1C and U1D serve as an output buffer that drives a 30-foot waterproof cable leading to male XLR plug J1. This plug connects to the Interface Unit, which needs a 6-VDC power supply for the WT. With perhaps excess caution to minimize voltage sag, I used a heavy-duty lantern battery for field recordings. However, I found that a 0.1-V decrease in supply voltage changed the WT's output frequency by only 0.15 percent.

The WT's maximum output frequency is about 75 KHz. The duty cycle depends on frequency, due to mixer non-linearity and the 10-microsecond detector hold time. However, frequency division in the Interface Unit is routinely used, converting the duty cycle to 50 percent.

Antenna Trials. I bench-tested the WT with a jar of water containing a ground electrode. My first antenna was a 6-inch length of 1/8-inch diameter steel welding rod, shrink-wrapped and sealed at the bottom. The insulation did not shed water reliably; after a few minutes in water, it became "wetted" and did not respond well to small, rapid changes in depth. I cut most of it off, leaving a one-inch stub to serve as a solder terminal for other antennas.

Of several wire types I tested, enameled copper magnet wire performed best. I prepared straitened samples (about 3.5 inches long) by stripping the top 1/4-inch (for soldering) and sealing the bottom end with a tiny bead of epoxy cement. I carefully cleaned the antenna with alcohol and a soft cloth before each use. The water also needs to be clean; good results are not expected where there is pond scum or an oily film.

Figure 2: Effect of wire guage on WT performance
Figure 2. Effect of wire guage on Water Theremin performance in static tests (immersion depth versus output frequency). Guage is related to conductor diameter according to the inset table. All wire samples were enameled copper, sealed at the bottom end with a bead of epoxy cement. Immersion depth is measured from the bottom of epoxy bead (i.e., 0 depth is when bead just makes contact with water).

Figure 2 shows WT output frequency versus water depth for enameled-wire antennas of three different diameters. The thinnest wire (36 AWG) gave the largest depth range, so I used it on further experiments and all recordings. The source of this 36-AWG wire was an old-stock spool labeled "Heavy Formvar," manufactured in 1944 by SX Wire Products, Ft. Wayne, IN.

Antenna Model. For the WT's antenna to act as a water depth-dependent capacitor, the water must be grounded and contain charge carriers. I bench-tested the WT with distilled, tap, and brine (saturated salt; NaCl) water. In distilled water, the WT completely ignored depth. Instead, it was sensitive to the distance between antenna and ground electrode--like a musical theremin. In Figure 3, square and round data points are my observations for brine and tap water, respectively. I calculated the well-fitting curves using the model in Figure 4, which is equivalent to the resonant network of the WT's depth-sensitive oscillator. Cosc combines this oscillator's fixed capacitors (C2, C3, C7, C8 in Figure 1) and includes a fudge-factor within the 5% component tolerances. Losc is L1 (see Figure 1), and I ignore its minimal DC resistance. Cant is the depth-dependent capacitance due to the 36-AWG antenna, and Rant is the resistance between this antenna and ground, due to the water.

Figure 3: observed and modeled WT static performance in brine versus tap water
Figure 3. Observed (data points) and modelled (curves) Water Theremin static characteristic in brine versus tap water. The model is shown in Figure 4. The maximum practical depth range, which is the expected maximum surface wave height (crest-to-trough), is indicated for tap water. The "equilibrium depth" is the operating point: the optimum immersion depth for a quiet water surface corresponding to f0, a 32-KHz WT output which is the FM carrier frequency. The Interface Unit was thus normally set to divide the WT frequency output by 32, down-converting f0 to 1 KHz to match the nominal operating point of the Data Converter's demodulator. The 200-1800 Hz lock range of the demodulator sets the expected maximum surface wave amplitude marked in this Figure.

The model in Figure 4 (which fits the observed data; see Figure 3) suggests that each 1-mm increment of 36-AWG antenna immersion increases Cant by 1.12 pF. In saturated salt water, Rant is indistinguishable from zero ohms at all immersion depths. In tap water, as depth increases, Rant decreases by a factor of 55 K-ohms divided by the depth in mm. As long as the ground connection to the tap water has a large suface area compared to that of the antenna, the distance between the ground and antenna has a negligible effect (at least for distances relevent to this experiment). For the normal WT operating range, the antenna surface area in contact with the water is small, and apparently this is the limiting factor for Rant in tap water. Rant behaves like an array of 55-K resistors combined in parallel, in which the number of these resistors equals the depth in mm.

Figure 4: Model of Water Theremin antenna which fits observed performance
Figure 4. Model of Water Theremin antenna which fits observed performance in static ("bench") tests, in context of simplified resonant network of depth-sensitive oscillator. Cosc represents the net fixed capacitances associated with the depth-dependent oscillator (C2, C3, C7 and C8 in Figure 1). Losc is the oscillator's inductor (L1 in Figure 1). Cant is the immersion depth-dependent capacitance of  the antenna. Rant is is the resistance between the antenna and the water's ground connection.

Amplitude Range and Linearity. Against its tap water static response curve in Figure 3, I marked the WT's largest practical depth range. Five stages of frequency division (25 = 32) in the Interface Unit matches this range to the lock range of the Data Converter's demodulator. It suggests a maximum input wave height of 30 mm crest-to-trough. My sidebar on the physics of water surface waves helps put this in the perspective of water surface waves found in nature.

Within this useful range, the WT's non-linearity error is about six percent. All classic-design theremins are non-linear: equal changes in capacitance do not change output frequency equally across the working range. Review the equation for L-C resonance to understand the "root" cause of this (pun intended). Since it's a known error, one could compensate for it in recording software. Or one could try a more linear water level sensor (such as the non-theremin water level-to-frequency converter described in: P.J. Ross [1983] "A Water-Level Sensor Using a Capacitance to Frequency Converter" J. Phys. E: Sci. Instrum. 16: 827-828).

Resolution: Room for Improvement.
WT resolution can be defined as the smallest change in depth that consistently causes a proportional change in output frequency. For minimum noise, it should be smaller than the Data Converter's resolution limit (12-bit, or 1/4096th of full scale in my case) after any gain is applied. My WT design falls short, as I will explain. With a care-free "parts-on-hand" approach, I used digital gates as an analog mixer--the U1A and U1B cascade. This is a "square (digital) peg for a round (analog) hole." The gates' threshold response (i.e., non-linearity) cause stepped (not smooth) frequency output versus depth. Envelope (beat) periods are quantized in one-microsecond steps (period of the 1-MHz oscillators).

After time-averaging due to frequency division (by 32), there are 4416 frequency "steps" for the 30-mm range shown in Figure 3. But their distribution is not uniform. There are 1250, 54, and 16 steps per millimeter at the minimum, equilibrium, and maximum depths, respectively. A 1/54-mm resolution at the operating point doesn't seem too bad at first glance, but note that it's equivalent to less than 11-bit resolution (referred to a 30-mm range). A better WT would be designed around a linear mixer.

Distortion: Water-Antenna Interaction. As noted above, the old-stock 36-AWG enameled wire for my antenna came from an old spool labeled, "Heavy Formvar," which is probably a trade name for the insulation. Using a microscope, I was impressed by the very smooth appearance of the insulation surface. For another way to assess smoothness, I watched light reflected from a water surface while drawing wire samples up and down. Any visible jitter of the reflection at the surface-wire interface would expose a source of noise and distortion. I saw no jitter during motion in either direction. But I noticed a different problem: The wire attracted a small positive meniscus (about 0.2 mm high). Apparently, the insulation is slightly hydrophilic (has some affinity for water). The meniscus was shallower during insertion than withdrawal. 

However slight, water adhesion to the WT's antenna should distort surface waveforms, affecting low-amplitude signals the most. This is a likely cause of the distortion highlighted in Figure 12 of Overview Part 4. In that experiment, I recorded water surface waves resonating in a sink (Figure 11 in Overview Part 4 is a photo of the set-up). I excited the quiet water with an impulse. It took nearly three minutes for the surface to settle down (damp out) completely. A minute or two into this decay, the crests of low-amplitude waves begin to appear clipped, likely due to antenna-water stickiness.

Another distortion source is flexing of the antenna by the waves themselves. Water particle motion is circular as surface waves pass a fixed point (see sidebar on physics of water surface waves). If antenna insertion depth is small compared to wavelength, the crests exert net lateral force on the antenna in the direction of wave travel, and troughs do the opposite. Thus, a flexible antenna records crests that lag, and troughs that lead, their actual timing. This effect should increase with amplitude. One possible solution might be to attach a mass (sinker) to the tip of the antenna wire using fishing line.

General comments and discussion about the Water Theremin are in Part 4 of the Overvierw, along with links to recordings made with this transducer.

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