MPS Rx
The detected particle signal is always superimposed by drive field feedthrough. Therefore, to minimize the dynamic range requirement on the pre-amp, it’s good to use a gradiometric receive coil to cancel out the unwanted signal. While there have been a few proposed designs (all which have pros and cons and discussed further below) we have elected to use a two-part gradiometer of 450 turns per side (900 total). The gradiometer has a resistance of 27.0 Ohms (@10kHz) and an inductance of 2.30mH.
The parameters of this coil are chosen based on a few assumptions: The length of each coil (receive and cancellation part) should be bigger than its radius to maximize the homogeneity of the sensitive region. A certain distance between both coils is also needed to prevent signal in one coil from significantly inducing a signal in the opposite one. Both coils should also be located in the most homogeneous part of the drive field to detect the same field and enable more accurate Tx decoupling. We decided that both coils should be located in 95% of the drive field semi-arbitrarily.
To get a high induced voltage through the particles, a high turn density is necessary based on the law of reciprocity. To reach that, we are using thin magnet wire (34 AWG, 0.16mm diameter) and determined the number of turns with noise matching to the preamplifier (more turns, higher resistance, higher noise -> match with preamp noise). For now, we are using 450 turns on each part of the gradiometer, though in theory more would be better as the preamplifier states an ideal input impedance of ~200 Ohms.
As mentioned, there are many different designs for "gradiometers", but first it is worth defining what a "gradiometer" is and why it is called that. As the name implies, the gradiometer meters a gradient, specifically the magnetic gradient/differential induced due to particles disrupting the otherwise homogeneous drive field. Now, this principle can be manifested in many forms--below are three illustrations of designs that have been used.
First, looking at the top gradiometer example design below, the one used on this system, the large Tx coil creates a field around both Rx coil A and B (in blue), coils A and B are identically wound in opposite directions, therefore the voltage induced due to the drive field (H) cancel each other out. When particles are moved within coil A (or B) the flux density increases locally as seen in the equation:
B = mu0*(H+M)
Where H is the external homogeneous magnetic field and M is the magnetization. Thus in the coil with particles, the "M" contribution is proportional to the signal/voltage induced because:
emf(volts) = -N*dB/dt
In coil A (with particles) the induced voltage is -Nmu0(dH/dt+dM/dt) whereas coil B is +Nmu0(dH/dT) with the sign change due to opposite handedness of the winding. The summation is, therefore -Nmu0*(dM/dt) in an ideal world. Yet, in reality, the signal is closer to -Nmu0(dM/dt+ K*dH/dt) with K being a constant less than 1 which describes the "feedthrough", in our case it was measured at -71dB.
In each design, the functional principle is the same, with the goal to be to minimize K.
Design A: The design currently employed in the MPS system.
Design B: A gradiometer design discussed in many papers, one notable example being this paper by M. Graeser et al. [1].
Design C: A gradiometric drive coil arrangement presented in this paper [2]
In these designs, there are pros and cons associated with each. Here are my personal thoughts on each of these selected designs and why we elected for the top one (design A)
A: Pros: Simple to implement, easy to adjust mechanically, robust design Cons: Low efficiency because the sensitive regions (coils A/B) are near the edges of the drive/ bias field they need to be longer to create the required fields within the Rx coils (i.e. the drive/bias field is strongest in the center of the coil, yet there is no receive coil there so that energy is wasted.
B: Pros: More efficient because the strongest field (bias/drive) is at the center where the Rx coil with particles are. Cons: A bit more mechanically complex to adjust and implement-- despite this, other groups have had success and high feedthrough attenuation
C: Pros: very simple Rx coil and it can probably have a very narrow diameter allowing for a more narrow Tx coil which would then improve efficiency by limiting edge effects. It also allows for the possibility to electrically tune the gradiometer by differentially powering the two drive coils (though I haven't seen this implemented in literature yet). Cons: There may be destructive interference of the Tx coils lowering efficiency depending on the exact geometry.
All in all, we went with design A because of its reliable and simple nature. Design B would be another great candidate for this project which we may use on future iterations given its high efficiency. Design C poses an exciting opportunity to electrically tune the coupling, which if implemented would be a novel innovation in its own right (to the extent of my knowledge).
As stated above, we are currently using a 900 turn Rx coil (450 per side of the gradiometer) with results in both a high resistance (27 Ohms) and high inductance (2.3mH) -- These high values are sub-optimal in a few ways (high noise, and bandwidth limiting) yet by having such a high turn-count we are able to take some burden off of the need for an exquisite and extremely low-noise amplifier. As shown above the signal (to a limit) is proportional to turn count, so with N times more turns you get N times more signal. The sacrifice comes in when this results in more resistance and inductance. First resistance.
The issue with the real resistive load is that it manifests both in more thermal voltage noise, it increases the voltage stemming from current noise in the amplifier (a helpful discussion here from Analog Device) as well as lowering the Q of a resonant circuit (which we don't use to enable flexibility on the MPS but many systems do, such as the small-bore imager). If we take the measured input-referred noise (noise measured at the output and then divided by the gain) from our INA217 preamp, ~5nV/sqrt(Hz) that would be the equivalent of approximately 1kOhm of thermal noise, a value far greater than what we use. But if we were to increase our turn count to match the resistive noise to the amplifier voltage noise, our inductance would be on the order of 1 Henry which raises the next issue, bandwidth
The Bandwith issue associated with high inductances stems from the inductors' inherent property of acting as a low pass filter. For magnetometry and relaxometry where the primary signal is in the lower harmonics, this is less of an issue, especially when driving at 10-25kHz. But for spectroscopy when the primary goal is to see the harmonic distortion created by the nonlinearity in the particles' magnetization, this limitation can be profound. Perhaps one solution would be to set the cutoff frequency of the system to the expected sampling rate of the DAQ console used. Below are two examples that illustrate this bandwidth limiting when the input inductance is raised from 0.3mH to 3mH. These amplifiers are not implemented in the system (though ones using these components may be included later on).
A high inductance amplifier simulation shows the bandwidth is reduced when compared to the above amplifier which as 10x lower inductance but otherwise is identical.
In order to manufacture the coil with the right number of turns, we rotated the Rx coil holder with a stepper motor. After the first part, we rotated the coil holder 180 degrees to wind the cancellation coil in the opposite direction. After winding, we soldered the ends of the wire to a coaxial cable ("coax"). The coax enables some protection from external interfering fields, but to bolster this protection we use a third braided copper shell around the coil (not connected to either line) which then is connected to the copper tube as well as our star ground. The 34AWG wire is quite fragile and sensitive to tugs, so it is a good idea to have a strain relief. Further physical movement during acquisition will result in electrical voltage being induced, to solve both of these problems we potted the junction of the magnet wire and the coax cable in epoxy (liberally).
[1] M. Graeser et al. Towards Picogram Detection of Superparamagnetic Iron-Oxide Particles Using a Gradiometric Receive Coil
[2] D. Pantke et al. Multifrequency magnetic particle imaging enabled by a combined passive and active drive field feed‐through compensation approach