Overview

• Turns: 1870
• Wire Size: 16-18 AWG
• Length: 185 mm
• Resistance: 8.89 Ohms at 10 Hz
• Inductance: 72.7 mH at 10 Hz
• Sensitivity: 9.36 mT/Amp

The DC bias coil is the keystone component of the magnetometry and relaxometry modes. It enables us to characterize the particles' magnetization response up to much higher amplitudes (+/- ~80mT, but this value varies depending on amplifier selection, heating limits, and duration). The reason it enables this flexibility is that we utilize the context of the coil (high amplitude, low frequency, and high acceptable noise limits) to employ components that would be unacceptable to use for the drive coil. To mitigate heat build-up between layers, we include air-gaps every few layers.

Design

Coil Length: 185 mm

-----Inner Layers----- ID: 41 mm OD: 53 mm

5 layers 16 AWG 130 turns per layer

-----Mid Layers----- ID: 66 mm OD: 78 mm

2 Layers 16 AWG 130 turns per layer 3 Layers 18 AWG 160 turns per layer

-----Outer Layers ----- ID: 90 mm OD: 96 mm

3 Layers 18 AWG 160 turns per layer

Wire diameter

This point is discussed in every coil design so as to not belabor the discussion from the parent page on the overall MPS system design, I'll leave it at this: thinner wire is more resistive, but higher turn density. In thermally limited cases, the resistance (and thus power density) is the driving factor. The matching impedance should include any inductive impedance.

The length of the coil

The length of the coil is somewhat pre-determined by the Rx coil geometry, which in turn is loosely determined by the sample (e.g. 3 mm bulb or microcentrifuge tube). The reason is that for a solenoid there are going to be "edge effects", this is when the homogeneity of the field at the core of the solenoid breaks apart and starts wrapping around to the other pole and thus lowers the magnitude of flux density when compared to the center. An interesting technical note discussing the topic written in 1960 by NASA is linked here. For the purposes of this design, the coil should extend about one radius' length beyond the edge of the Rx coil to minimize edge effects. There is room to optimize the turn distribution to maximize efficiency and homogeneity, but no optimization was done on this design.

Turn count, N

Before getting into the design discussion it is worth noting that turn count is perhaps not the right variable to optimize rather it is the turn density that is the number of turns per axial distance along the solenoid. But as discussed before, the length has been already determined so they are one and the same (assuming even distribution).

The turn count is inseparably coupled to the wire diameter, and for all intents and purposes needs to be determined together. To do this we wrote a MATLAB script which given a current limit and length determines the "ideal" wire. What the script does is first calculates the length of wire needed to match the 8 Ohms(or whatever resistance you want to match) which the amplifier specifies is the max-power load in the datasheet, and from there, the script determines the number of turns for that length of wire, discretizes that into pairs of layers, calculates the field from a given layer, and then repeats the process for multiple wire thicknesses. In the script, there is an option to space out every two layers with an air gap to enable airflow. The gap every two layers is probably overly generous, and much higher efficiency can be achieved with making every layer directly on the previous if you are okay with occasionally allowing it to cool off during high-current operations.

By knowing the approximate cost of wire (based on ~15 USD per pound) we can also approximate the cost of the DC biasing coil. Below is a set of figures from this script.

This script does not account for inductive impedance, which if using an AC-coupled audio amplifier (as we do), is significant.

This figure shows the field contribution for each layer of the solenoid.

This figure shows what the field is at the "sample location" which can be selected within the script.

Finally, this figure shows the cost of the DC bias coil as the wire diameter, and as expected the cost dramatically rises as diameter goes up. (The cost is likely much higher if you buy in small quantities).

Fabrication

Thermal considerations

The purpose of these coils is to produce a high amplitude field so inherently the will get hot if used at high duty cycles. To mitigate this we designed in air gaps thus increasing the surface area of the coil and a fan which blows over the turns. To model this and attempt to verify this design theory I used a simplified cross-section of the coil and assumed radial symmetry. Each layer is considered a solid bar of copper with uniform power density from the MATLAB script described above. The boundaries are depicted in the illustrations below:

The rationale for these boundaries being on the surface with free convection you shouldn't expect high values for the heat transfer coefficient, and the green boundary (air-air interface) was included to ensure that the model didn't presume the surrounding air was perfectly stationary (admittedly this is probably not the best way to model it, but for a comparative analysis it should suffice). Some air in the simulation is necessary, and this is because each layer (even those without the intentional air gap) has some air in between because we are using round magnet wire. The modeling software has trouble when the wires are explicitly modeled. In the case that there are no gaps between layers (minus the 0.1mm air gap) there is no convection except on the top surface of the outermost layer and the copper tube. Here are the results:

There are profound limitations of this model, and some major assumptions that were made along the way, despite this it serves to illustrate our design's ability to passively dissipate heat. This design is far more effective dissipating heat if a fan is employed to blow air through the fin-shaped layers. In the current design we have a fan included, and this fan uses the same logic which enables the stepper motor such that they are activated and deactivated together. By disabling them during acquisition, we reduce the likelihood that they add noise to the measurements.