Calculations
In order to pull down the engineered ferritin-containing statoliths, the Hardware team aimed to construct constant current drivers that could drive a pair of Helmholtz coils at different currents. This generates a gradient magnetic field between the two coils, and the gradient strength is directly related to the strength of the force due to the magnetic field.
The magnetic field produced by the coils is derived from the Biot-Savart law, and approximated to point entirely along the axis that goes through the two coils (defined as the z-direction):
This magnetic field generates a force proportional to its gradient on ferritin, giving us the following equation of motion:
Plugging in the magnetic field from the coils, we find that the force is at its maximum in the midpoint between the two coils (z=0), which is where the plant’s roots will be. Approximating the deviations from z=0 to be small and negligible, we obtain an ordinary differential equation in the form:
Where the coefficients are:
This then becomes an initial value problem, where the goal is to make the motion of ferritin similar to that of nanoparticles involved in gravitropism. The conditions necessary are for the nanoparticles to move along an entire plant cell (10-100 μ meters) in 30 minutes or less. Thus, we can include the condition
and find a unique solution to the ODE.
Plugging in the known constants for ferritin and water:
With distance set to 15 cm and Coil N2 to 10 turns, 12 gauge wire:
We used Accel Instruments' online coil properties calculator to determine the inductance, wire length, and other statistics not resulting directly from the ODE solution.
Design
In order to select passives and devices capable of generating the desired current (9.87 A), we used a CircuitLab simulation and DC sweep. The details for the CircuitLab simulation are included on our Model page.
Once we had defined a set of devices to be used in the constant current generation, we set out to identify other components used for the control of the direction, switching speed/waveform, and amplitude of the output current. These include a full H-bridge for controlling direction of current flow, an embedded ATTiny microcontroller and discrete logic gates for controlling the switching of the H-bridge, and a digital rheostat. We followed these considerations during component selection:
- Can be powered off of a single supply (+5V)
- All interfacing over an i2c bus with at least 2 configurable addresses
- No smaller than 0805 sized passives and no QFN/BGA packages to enable hand placement and rework
- Components in-stock on DigiKey with limited design risk as determined by findchips.com
- Limited component cost where possible
We ended up using logic level P-Channel and N-Channel MOSFETs on the high and low sides of the H-bridge respectively, using MOSFET drivers. This allows us to avoid more complex driver solutions that require multiple supply voltages, at the expense of higher ON resistance and cost for P-Channel MOSFETs.
For safety purposes, we used discrete digital logic to ensure that the two current directions of the H-bridge could not be activated simultaneously.
In searching for i2c PWM generator devices, we found that many were designed for low-frequency PWM that may not be able to generate alternating magnetic fields that could be useful in other experiments, or had stocking and lead time issues as a result of the global chip shortage. We instead decided to use a more modern ATTiny series microcontroller (specifically the ATTiny412), which would be able to generate PWM and function as an i2c peripheral by running Arduino code using Spence Konde’s megaTinyCore.
This work resulted in the generation of our board schematic after many iterations:
Fabrication
We also decided to include a current sensor inline with the load to allow continuous current measurement and correction or safety shut off of the system. This is an analog hall effect current sensor, and we use an i2c analog-to-digital converter (ADC) to read this current value over i2c. A 15A fuse was used for circuit protection, and sufficient bulk capacitance was added on the board’s power input to prevent voltage rise (resulting from dissipation of Back Emf) from affecting the supply.
We designed the following board with dimensions less than 100x100 mm to reduce costs at many PCB fabricators. Due to the high currents in use, the board is a 4-layer board with considerable stitching vias to improve thermal conductivity between top and bottom signal layers. Inner layers are power and ground. Heatsinks were added to safely dissipate the up to 4.4 Watts lost as heat from the D44VH NPN power transistors.
We were able to assemble and test components on this board as well as Arduino and CircuitPython firmware on the ATTiny412 and an RP2040 acting as an i2c controller. The code, build documentation, wiring schematics, and mechanical/electrical source are posted on our GitLab, which can be accessed here.
In order to safely contain this high-power device and AC-DC converter power supply (Mean Well UHP200A), we required a true earth grounded electronics enclosure. The walls of the enclosure are 3D printable and the bottom and top are laser-cut acrylic and aluminum, respectively. The power supply and high-power components are grounded through the aluminum plate, which also satisfies the heat dissipation requirements.
