Electrostatic measurements with a field mill

Anyone who has enjoyed a good thunderstorm will be familiar with one consequence of atmospheric electricity: charge separation in clouds can generate electric fields strong enough to overcome the insulating properties of air, causing lightning. Less well known is the "fair-weather field" which exists even on sunny days; walk outside and your head will be approximately 200V more positive than your feet! We don't notice this field because there aren't enough charge carriers in air to create much of an electric current. Other creatures do notice, however; it was recently discovered that spiders can exploit this field to travel great distances under electrostatic levitation.

It turns out Earth's surface is like one terminal of a giant, leaky capacitor. The other terminal is the ionosphere, a region of the upper atmosphere which is somewhat conductive because of ionization from solar radiation. (This is the same reason it can reflect shortwave radio signals, enabling round-the-world radio communication without satellites.) Between the ionosphere and the ground, the much lower ion density carries a small drift current--negative ions drifting upward, and positive ions drifting downward--which slowly drains the capacitor. Thunderstorms are one of the mechanisms charging the capacitor, because most lightning deposits negative charge on the ground. In areas with clean air and flat terrain (e.g. oceans), the fine-weather field follows a daily and seasonal pattern; this is called the Carnegie curve and is almost perfectly explained by the statistical prevalence of thunderstorms as the late-afternoon peak occurs in different parts of the world.

I wanted to build a device to monitor the outdoor electric field, especially during thunderstorms. Scientists use a device called a field mill for this purpose. With a field mill, not only is it possible to sense electrically charged clouds; on a finer timescale these instruments reveal the rapid field changes involved with lightning as charge re-distributes itself inside a thundercloud.

Theory of operation

Field mills are a diverse group of instruments, but the basic principle is shown in the figure above. Sensor plates (electrodes A and B) are alternately shielded and unshielded by a grounded, rotating shutter. The ambient electric field, represented by the downward-pointing arrows, induces surface charge on the unshielded electrodes, because the electric field must be zero inside a conductor. The shielded electrodes, on the other hand, do not see the field and so do not have a net surface charge. As the shutter rotates, the alternating induced charges create a differential alternating current in the wires connected to the amplifier.

In a traditional design the A and B electrodes are fed into a pair of charge amplifiers to create a differential AC voltage. In the example shown, the signal has a frequency of twice the motor rotation rate, because one complete rotation yields two coverings/uncoverings of the A or B electrodes. The waveform is quasi-sinusoidal, depending on the shape of the electrodes, shutter, and effects of fringing fields. It can be synchronously rectified using a reference signal from the motor drive circuitry or an optical sensor on the shaft. Passing this through a low-pass filter results in an output voltage which follows the magnitude and sign of the ambient electric field. The system bandwidth is limited by the low-pass filter, which in turn depends on the motor speed.

In US patent 6608483, John Hill proposes a field mill which circumvents the bandwidth limit of the traditional design by providing quadrature signals from the sensor electrodes. For an electric field E(t) the sensor signal would then be E(t)*exp(j*w*t) rather than E(t)*sin(w*t) in the traditional approach. If the rotating phasor exp(j*w*t) is accurately known, then it should be possible to recover E(t) without low-pass filtering.

The patent suggests a staggered patterning of the sense electrodes as shown in one of the figures:

In the field mill presented here, I obtain the quadrature signals electronically instead, by dividing the sense plate into equal sectors of 90-degree "electrical" length. My shutter has three lobes, i.e. each rotation of the motor shaft equates to three cycles in the sensor waveforms, or 3*360 = 1080 degrees. 1080/90 = 12, so the sensor needs 12 sectors:

Denoting the twelve sectors $$s_1$$ through $$s_{12}$$, the quadrature signals traditionally denoted $$I$$ and $$Q$$ can be formed as follows:

Let $$a = s_1 + s_5 + s_9$$ $$b = s_2 + s_6 + s_{10}$$ $$c = s_3 + s_7 + s_{11}$$ $$d = s_4 + s_8 + s_{12}$$ Then $$I = (a+b) - (c+d)$$ $$Q = (b+c) - (a+d)$$

This says that we can electrically connect every fourth sector. Four charge amplifiers then suffice to create the signals a, b, c, and d. Another four summing amplifiers can create the differential I and Q signals for digitization and processing in software.

Detailed design

I have a poor aptitude for anything mechanical, so the thought of fabricating a metal shutter to attach to a motor and spin at thousands of RPM was quite daunting. It seems to demand skills I do not possess, or some form of CNC machining, which is too expensive. Then I realized that a custom PCB might do the job: double-sided 1.6mm boards with ENIG plating are quite reasonable from China, and all I needed to do was supply the CAD files. Stainless steel would undoubtedly be more durable, but I hoped ENIG would survive long enough outdoors to make the project worthwhile.

Used or surplus hard-disk spindle drives provide a ready source of inexpensive brushless motors. As a bonus, many of the motors I tested had a conductive path from the rotor to the stationary frame, eliminating the need for a metal brush to ground the shutter blades. I used Inkscape to draw the outline of the shutter PCB with mounting holes to match the most common hub and bolt-circle dimensions in my random assortment of motors from Ebay. The screw holes are plated through to connect the top and bottom copper pours. The shutter simply replaces the hard-disk platter and spins just as well.

While I was on a roll, I decided to use a PCB for the sensor electrodes too. In fact, this project goes even further, placing all of the electronics on the underside of the sensor PCB. This board requires four layers; the inner layers are mostly ground fill to guard the sensors from the electronic signals on the back side. The cheap PCB process does not support blind or buried vias, so no vias are permitted on the sensor board except for sparing use of ground vias, otherwise they would interfere with the sensors. This strategy was successful but required the use of some zero-ohm resistors to bridge over crossing traces; in the schematic these are marked as "hop" jumpers.

Each group of three sensor sectors is connected to a charge amplifier:

Ignoring the protection components R3 and D3, the charge amp has exactly the same form as a transimpedance (current to voltage) amplifier, with gain set by R7 and a low-pass corner at 1/(R7*C9). Viewed as a charge amplifier, however, it converts a quantity of charge, Q, into a voltage, V, with gain set by C9 (1/C = -V/Q). The smaller the capacitor, the higher the charge-to-voltage gain. R7 then sets a high-pass corner at 1/(R7*C9). (This all makes sense in light of current being the time derivative of charge.) The sensor charge is e0*E*A where e0 = 8.85x10^-12 s^4 A^2 / kg m^3 is the permittivity of free space, E is the electric field in V/m, and A is the sensor area in m^2. This works out to approximately 1.4x10^-14 coulombs per volt-per-meter of field strength; assuming a sinusoidal variation of the charge and a motor speed of 3600 RPM, the peak current amplitude into and out of the charge amplifier is 2*pi*f*q = 2*pi*(3600/60)*3*1.4e-14 or about 16 pA per volt-per-meter of field strength. The voltage amplitude at the output of the charge amplifier, with C9 = 220 pF, is about 64 uV per volt-per-meter of field strength. This is roughly 1 LSB of a 16-bit A/D converter, so the instrument can measure fields of tens of kilovolts per meter which may be found beneath thunderstorms.

A low-leakage diode pair (D3) provides some input protection for the op amp inputs. R3 is intended to limit diode current to a safe value in the event of a large static discharge. The OPA4197 has a typical bias current of 5 pA, so the voltage drop across R3 is inconsequential.

After the charge amplifiers U1, the four sensor components are combined into the four terms derived above (a+b, c+d, b+c, a+d) by the inverting/summing amplifiers U2. This quad op-amp has a lower supply voltage to match the A/D inputs on the microcontroller, U4. Differential A/D inputs provide the final subtraction of terms to create the quadrature signals I and Q.

A dedicated brushless motor driver U11 (DRV10975) interfaces the hard-disk motor to the MCU via level shifter U7 (Si8605). The driver is configurable in the extreme and possibly overkill for the these motors, but because I did not know the motor parameters and have little experience in this area, it seemed like a safe choice. TI provdes a nice app note with instructions on how to tune the driver parameters.

The shutter angle is available to the MCU via two methods (which one to use is still TBD). The motor driver provides a tachometer feedback signal, and an optical reflection sensor (U6, VCNT2020) provides a periodic index pulse from a marking on the rotor.

Data transport to an external system is still undefined. I am using the MCU's built-in USB port during development, but for outdoor use during a thunderstorm, a wireless (or at least non-conductive!) channel is needed. The original, ambitious plan was to develop a low-data-rate, free-space optical system using a high-power red LED and pulse position modulation. As of 2019 this is bogged down and I haven't decided whether to pursue it further or punt and use a bluetooth module or fiber-optic transceiver or something else. In any case, the data will come out on J3, marked "optical comms" on the schematic.

The instrument uses battery powered for portability; twin lithium-ion 18650 cells provide a split-rail supply which simplifies the analog signal conditioning. The MCU is centered over +/- 1.8V rails, automatically satisfying the A/D common-mode input requirements and keeping the A/D reference voltage (and thus the gain) constant over battery life. The main drawback of this scheme is the need for an I2C level shifter (U7) between the MCU and the motor driver. It also wastes twice the power necessary on the MCU, but motor power is expected to dominate the power budget.

Digital signal processing

The STM32L4 microcontroller offers dual A/D converters with adequate SNR (about 11 effective bits at 5 MSPS) and some nice digital-filtering hardware for efficient downsampling. For intial testing, the two A/Ds sample simultaneously at 2.4 MHz. A combination of hardware and software stages downsample by a factor of 1600 for an output rate of 1500 Hz. If the quantization noise is white, this should boost the SNR past 16 bits.

The remainder of the processing is largely TBD. Here is a typical plot of the raw I/Q data:

The I and Q signals are not exactly sinusoidal, so the path traced by the phasor in the complex plane is not quite circular. To correctly demodulate this signal, we need to know the shutter angle at every input sample, and have a calibrated template signal (at constant E field), to de-rotate the phase and remove the residual amplitude modulation.

Results

Although much work remains to be done, I have verified basic operation of the mill indoors. Here is a video of a data collection run using a statically charged PVC pipe; the raw data were streamed to a PC and processed in MATLAB with the following [uncalibrated] result:

For eventual calibration, a high-voltage power supply can be connected to a conductive plane (aluminum foil) held a set distance above the instrument. In early experiments with a square of foil hand-held a couple of inches above the mill, it was easy to see the signal from my low-voltage bench supply.