Tesseract - A High-Stability, Low-Noise Fluxgate Sensor Designed for Constellation Applications

. Accurate high-precision magnetic field measurements are a significant challenge for many applications including constellation missions studying space plasmas. Instrument stability and orthogonality are essential to enable meaningful comparison between disparate satellites in a constellation without extensive cross-calibration efforts. Here we describe the design and characterization of Tesseract - a fluxgate magnetometer sensor designed for low-noise, high-stability constellation applications. 10 Tesseract’s design takes advantage of recent developments in the manufacturing of custom low noise fluxgate cores. Six of these custom racetrack fluxgate cores are securely and compactly mounted within a single solid three-axis symmetric base. Tesseract’s feedback windings are configured as a four-square Merritt coil to create a large homogenous magnetic null inside the sensor where the fluxgate cores are held in near-zero field, regardless of the ambient magnetic field, to improve the reliability of the core magnetization cycle. A Biot-Savart simulation is used to optimize the homogeneity of field generated by the feedback Merritt Coils 15 and was verified experimentally to be homogeneous within 0.42 percent along the racetrack cores’ axes; an improvement thirteen times that of the traditional ring-core sensor design. The thermal stability of the feedback windings is measured using an insulated container filled with dry ice inside a coil system. The sensitivity over temperature of the feedback windings is found to be between 13 ppm/°C and 17 ppm/°C. The sensor’s three axes maintain orthogonality to within at most 0.015 degrees over a temperature range of -45 °C to 20 °C; an improvement at least six times that of the ring-core sensor design. Tesseract’s cores achieve a magnetic 20 noise floor of 5 pT/ √ Hz at one Hz. Tesseract will be flight demonstrated on the ACES-II sounding rockets, currently scheduled to launch in late 2022 and with a thermal expansion coefficient of 16.2 ppm/°C, making it an excellent base material for linear sensitivity temperature compensation. There are materials with a lower coefficient of thermal expansion, Torlon was selected to match the thermal expansion coefficient of the copper windings (16.7 ppm/°C) and minimize stress in the sensor. Tesseract takes advantage of this property of Torlon for effective real time linear temperature compensation, 180 which should contribute to good measurement stability over changes in temperature. Torlon is also extremely resistant to shearing and skewing with temperature. We expect that this will reduce the tendency of the sensor’s axes to skew with temperature. in the sensor’s feedback windings and not the applied field. We do not measure any variation in axis alignment to within deviations of 0.015 degrees over changes in temperature from -45°C to 20°C. The sensor’s feedback windings are a less-than-ideal search coil, so the accuracy of the measurements is limited by poor Tesseract Sensor’s feedback topology for homogeneity and power consumption. The simulation results were in agreement with laboratory measurements of the prototype sensors’ feedback magnetic field. The Tesseract retains its ferromagnetic cores in a feedback magnetic field that homogeneous to within deviations of 0.42% We used a low-cost method to analyze the sensor’s gain and orthogonality over temperature. This method uses coil system measurements and a second reference magnetometer to least squares vector-vector calibration to find the intrinsic sensitivity, 425 orthogonality, and offsets of the sensor as a function of temperature. Using this method, we show that the Tesseract’s three orthogonal axes skew by no more than 0.015 degrees over changes in temperature from -45 °C to 20 °C. The Tesseract sensor outperforms the traditional 1” ring-core Sensor design in both these metrics. The Tesseract’s orthogonal axes are more stable by at least a factor of 4 and its magnetic feedback is more homogenous by a factor of 13. We surmise that Tesseract’s higher degree of feedback homogeneity and axis stability over temperature will contribute to improved fluxgate stability.


Introduction 25
Constellation satellite missions are acknowledged to have an important role in the future of space plasma science. The NASA Heliophysics Science and Technology Roadmap for 2014-2033, recognizes that one of the opportunities to drive scientific discovery will come from a constellation mission of 30 or more spacecraft (Heliophysics Roadmap 2014). Recent missions such as Cluster (Balogh et al., 2001), Swarm (Merayo et al., 2008), ST-5 (Slavin et al., 2008), THEMIS (Auster et al., 2008) and The Magnetospheric Multiscale Mission (Torbert et al., 2016) have successfully flown constellations of three to five spacecraft and 30 have made significant contributions to the understanding of solar wind and magnetospheric physics (Ganushkina et al., 2017., Nakariakov et al., 2016 and the references therein).
Recent developments in nanosatellite technology promise to provide a platform for the future of constellation missions at a low cost (Bandyopadhyay et al., 2015). Such a nanosatellite constellation mission could accurately resolve and characterize the spatial and temporal evolution of magnetic fluctuations that are indicative of larger scale magnetospheric processes, such as plasma waves, 35 https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License. field aligned currents and plasma transport at various scales. However, constructing a magnetic sensor with sufficiently small mass and volume to be accommodated on a nanosatellite, while still maintaining the instrumental accuracy and precision required for a magnetospheric constellation mission is not a trivial task. As the size of a fluxgate sensor is reduced, it becomes increasingly difficult to preserve the stability required for reliable multi-point cross-comparisons. This presents a significant challenge for the use of nanosatellites on constellation missions. The 2015 NASA Technology Roadmap has noted the importance of addressing the 40 challenge of "high measurement stability to allow inter-satellite calibration" to enable "high-stability magnetic field measurements that can be made in distributed systems" (2015 NASA Technology Roadmap 8.3.1.3).

Fluxgate Stability
Fluxgate stability is critically important for a magnetospheric constellation mission, as it enables comparison of disparate magnetic measurements without the need for intensive cross calibration. However, it remains poorly understood and is not explicitly 45 addressed by many authors in the literature. Factors suspected of degrading fluxgate stability include an inhomogeneous magnetic null (Ripka, 1992) and skewing of the axes and cores due to mechanical and thermal strain (Primdahl, 1979).
Fluxgate magnetometers (Primdahl, 1979) measure the static and low-frequency vector magnetic field by modulating or gating the local magnetic flux and measuring the induced electromagnetic force (EMF) in a sense winding. A ferromagnetic core, periodically driven into magnetic saturation at frequency f, is used to gate the local field, thereby inducing a second harmonic (2f) signal due 50 to the nonlinear magnetic permeability of the core as it enters magnetic saturation twice per cycle. In this paper, we will be discussing only second harmonic fluxgates. In many instruments, including that presented here, magnetic feedback is used to null the magnetic field in the sensor, which linearizes and extends the measurement range of the instrument and is thought to improve overall fluxgate stability (Primdahl and Jensen, 1982). Magnetic sensors that are nulled in two axes, such as the CASSIOPE/e-POP fluxgate which maintained a sensitivity stability of approximately of 18 ppm/°C (Wallis et al., 2015), outperform sensors 55 zeroed in only a single component i.e. Narod and Bennest, (1990); Acuña et al (1978), which achieved a stability of ± 0.25% (Acuña et al., 1978) over the temperatures ranges from -20°C to 40°C which we calculate is equivalent to 26 ppm/°C. Thus, we hypothesize that a null in all three axes will further improve stability compared to one or two axis nulled sensor designs.

Current State of High Stability Fluxgate Sensors
Recent constellation missions have increasingly focused on flying three axis compensated sensors in the interest of maximizing 60 instrumental stability. However, sensor designs that have flown on past constellation missions have been constrained to accommodate a traditional one-inch diameter ring-core geometry ferromagnetic core.
Potentially the most stable magnetospheric field fluxgate measurements to date were taken with the Compact Spherical Coil (CSC) Sensor aboard Swarm which implements a nested three axis Helmholtz coil feedback wound on a MACOR shell to create a threeaxis null in a 100×100×50 mm instrument. The 500g CSC sensor has maintained a sensitivity stability over temperature of 10 65 ppm/°C and very high axis stability of 0.002 degrees from -20°C to 40°C while achieving a noise floor of 6.6 pT/√Hz at 1 Hz (Merayo et al., 2008). From Primdahl and Jenson (1982), we estimate that the CSC's feedback coils hold their ring-cores in a field that deviates from uniformity by 1.5%.
The THEMIS mission incorporated 70×70×45 mm fluxgate sensors that achieved a sensitivity stability of 22 ppm/°C, an axes' stability within 0.017 degrees and a noise of 10 pT/Hz at 1 Hz (Auster et al., 2008). The ST-5 magnetometer (MAG) has dimensions 70 of 50×50×30 mm and a total mass with electronics of approximately 600g and achieved a noise of 0.1 nT rms at 1 Hz. (Slavin et  Hz, an axis stability of about 0.03 degrees between -50°C and 30°C, and a sensitivity stability over temperature of 30 ppm/°C after compensation was applied (Russell et al., 2016). Notably, the temperature dependent sensitivity can be corrected to first order by fluxgate electronics using compensated feedback current (Primdahl 1970;Acuña et al., 1978;Miles et al., 2017); however, no 75 equivalent compensation exists for orthogonality. In general, smaller temperature dependent sensitivity is desirable but smaller orthogonality dependence is more critical.
Future constellation missions, such as Geospace Dynamics Constellation (Pfaff et al., 2016), MagneToRE (Maruca et al., 2021) and NanoSWARM (Gerrick-Bethell et al., 2021) promise to build on the success of these constellation magnetometers, and the continued development of increasingly precise, robust, magnetic field instruments remains an important means to these efforts. 80

Current State of Miniaturized Fluxgate Sensors
Nanosatellites as a platform for magnetospheric measurements are an emerging topic in the literature. Some tend to focus on miniaturizing traditional fluxgate magnetometer designs for CubeSat applications without overly compromising their measurement capability (e.g., Ripka, 2003 and references therein). Recent missions, such as Dellingr and ExAlta1, have implemented miniaturized, boom-deployed fluxgate magnetometers on nanosatellites. The Dellingr nanosatellite flew a very small 19g fluxgate 85 magnetometer with a noise floor of 120 pT/√Hz at 1 Hz (Kepko et al., 2017). The Ex-Alta 1 CubeSat flew a 47g miniaturized fluxgate sensor with a noise floor of 150pT/√Hz at 1 Hz (Miles et al., 2016). However, these miniature instruments typically sacrifice noise and stability in the interest of small size. With a stability poorer than 50 ppm/°C, these sensors function primarily as variometers. If used in a constellation setting, measurements from disparate sensors would be far less comparable than those from previous constellation mission without complex and time-consuming cross calibration. One of the best small sensors to date 90 appears to be the Small Magnetometer in Low-mass Experiment (SMILE) developed by Forslund et al (2007), which achieved a thermal sensitivity stability of 11 ppm/°C, an axis stability better than 0.02 degrees from -30 to 45°C, and 30pT/√Hz at 1Hz with a 40g, 20×20×20 mm cubic sensor based on three rod cores within a three-axis feedback winding (Forslund et al., 2007).

Tesseract Sensor Overview
Like the CSC and SMILE sensors, Tesseract's feedback windings create a three-axis 'magnetic null' inside the sensor where the 95 fluxgate cores are held in near-zero field. This ensures that the cores do not exceed their linear sensitivity region regardless of the magnitude of the ambient magnetic field (Primdahl and Jensen, 1982). An inhomogeneous magnetic null at the cores is known to be a primary contributor to degrading fluxgate stability (Ripka, 1992).  Here, we present the design and characterization of Tesseract: a high-stability fluxgate magnetometer for constellation missions. 105 The Tesseract sensor's three-axis feedback windings are arranged in a four-square Merritt coil, which creates a proportionally larger homogeneous region than the Helmholtz coil for the same external volume (Merritt, 1983) allowing it to accommodate more cores (six instead of three). Tesseract's Merritt coil feedback winding has been optimized to hold these cores within a highly homogeneous null field (deviations from average of less than 0.42%), which allows for a reproducible magnetization of the ferromagnetic cores. This reproducibility is designed to improve the measurement stability of the sensor. 110 Table 1: The Tesseract Sensor's specifications as measured in the laboratory compared with the specifications of a more traditional ringcore sensor design. Tesseract is marginally larger than the ring-core sensor but has much higher axis stability and magnetic feedback homogeneity, which are associated with higher instrumental stability. The Uncompensated Stability and Orthogonality angles are both 115 measured from -45℃ to 20℃ Tesseract's custom low noise racetrack geometry cores are securely and compactly mounted within a single solid three-axis symmetric base of 30% glass-filled Torlon, reducing the potential for mechanical strains due to uneven coefficients of linear expansion and reducing the tendency of the sensor to skew with temperature, thus limiting these potential sources of instrument 120 instability. Tesseract's cores achieve a magnetic noise floor of 5 pT/√Hz at one Hz, an orthogonality within 0.015 degrees and temperature stability of 13-17 ppm/°C between -45℃ and 20℃. https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.

Sensor Design
Tesseract is designed to mitigate known sources of instability such as uncompensated residual field inside the sensor and skewing of the axes and the cores due to mechanical and thermal stresses while still maintaining low noise and a small size. Here we 125 describe the design, optimization, manufacturing, and assembly processes for the Tesseract sensor.

Racetrack Core Design
The noise floor of a fluxgate is typically limited by the intrinsic magnetic noise of a permalloy core that is periodically driven into magnetic saturation to modulate the local magnetic field. Recent development in custom low noise core manufacturing by Miles (2019) enabled us to create custom low-noise miniature racetrack sensors (Miles et al., 2022). 130 Custom permalloy created using the Miles et al., 2022 process is cold rolled into 50 µm thick foil. The permalloy foil is then cut to 5 cm length, stacked, drilled, and secured in a tight bundle. A milling machine was used to machine 6.45 mm wide by 31.45 mm long racetrack foil washers. The racetrack washers are placed in the furnace, heat-treated, and stacked into a Torlon bobbin  A plastic lid closes the core and serves as a base upon which to wind a quasi-toroidal drive of AWG 32 magnet wire ( Figure 2b).
Production cores are interleaved with a polymer between the foil layers to prevent them from moving during the magnetizing drive pulses. The stacked foil washers remove the need to spot-weld, as is done in traditional spiral-wound sensors, avoiding the heat-140 affected area around the weld and its unpredictable magnetic properties. Heat-treating the foil washers individually removes the risk of undesired welding between layers that can cause unintended shorting.
The race-track geometry aligns ferromagnetic mass on one axis, producing lower noise. However, the race-track geometry cannot be double-wound like a ring-core to sample two orthogonal components. The closed flux path of the racetrack should reduce stray fields and offset error compared to traditional parallel rod sensors (e.g., Janosek 2017; Moldovanu et al., 2000). 145

Sensor Base Design
Tesseract leverages this racetrack geometry to create a compact sensor. Six of these custom racetrack geometry cores are fixed within a three-axis symmetric 30% glass filled Polyamide-Imide (Torlon 5030) base in a compact configuration of two cores per axis shown in Figure 3a. This pairing of identical cores with opposite polarities in each axis is expected to further reduce the tendency for cross-axis contamination due to stray fields. 150 These cores are then secured firmly in place using polymerics and nonmagnetic screws so that they do not shift during vibration, but without exerting any mechanical stress on the core (Figure 3b). Vacuum tolerant epoxy is used to secure the drive windings to https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License. the cylindrical walls of the bobbin, thereby locking the drive winding in place without creating a rigid bond to the racetrack core.
The foil bobbin and the sensor base are manufactured from the same glass filled Torlon which will reduce the impact of mismatched coefficients of linear thermal expansions compared to sensors using the traditional Inconel bobbin. This further reduces the 155 potential for mechanical stress.

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Torlon is lighter and much easier to machine than the traditional Inconel ringcore bobbin or MACOR sensor winding. Small channels of a predetermined width and depth are precisely and symmetrically machined into a Torlon block. This block serves as a common base upon which to wind Tesseract's three orthogonal feedback coils (3b in red) This single solid 3 axis symmetric base for all three sense coil axes is expected to further reduce the tendency of the sensor's axes to skew with temperature, thus preserving 165 the orthogonality.
Additionally, a common base for the feedback windings provides a small, linear temperature dependence that can be characterized and then compensated in electronics. Fluxgate sensor measurements have long been known to vary with temperature (e.g., Trigg et a1., 1971). Miles (2017) documents the historical technique (Acuña, 1978) of providing realtime analog temperature compensation. Temperature compensation can be provided electronically in real time by modifying the feedback current. A 170 transconductance amplifier can be modified such that the normally constant voltage-to-current transfer function is engineered to respond to the load resistance provided by RTDs which are mounted in series with the feedback. This scaling of this transfer function allows, to first order, the effect of temperature on the load resistance offered by the sense winding to compensate for the effect of temperature on the sensor geometry.
The compensation is limited to the linear scaling of this transfer function, so it requires that the sensor be highly mechanically 175 stable with temperature such that its thermal sensitivity dependence is functionally linear through the operating temperature range. 30% glass filled Torlon has a very linear response to temperature and with a thermal expansion coefficient of 16.2 ppm/°C, making it an excellent base material for linear sensitivity temperature compensation. There are materials with a lower coefficient of thermal expansion, Torlon was selected to match the thermal expansion coefficient of the copper windings (16.7 ppm/°C) and minimize stress in the sensor. Tesseract takes advantage of this property of Torlon for effective real time linear temperature compensation, 180 which should contribute to good measurement stability over changes in temperature. Torlon is also extremely resistant to shearing and skewing with temperature. We expect that this will reduce the tendency of the sensor's axes to skew with temperature. https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.

Feedback Winding Design
Tesseract's three orthogonal feedback coils are wound into channels that have been machined symmetrically into the Torlon base ( Figure 3b). A CNC winding machine is used to guide 40 AWG magnet wire into the channels so that there is no asymmetry or 185 overlap. These feedback windings create a three-axis 'magnetic null' inside the sensor where the racetrack fluxgate cores are held in near-zero field. This ensures that the cores do not become oversaturated and exceed their linear sensitivity region regardless of the magnitude of the ambient magnetic field (Primdahl and Jensen, 1982). Retaining the cores in a homogeneous region, where the magnetic field can be effectively nulled, helps to ensure a reproduceable magnetization of the ferromagnetic cores. This is thought to improve the sensor's measurement stability and linearity (Ripka, 1992). Tesseract's feedback windings are arranged as 190 a four-square Merritt Coil in three axes. This configuration creates a relatively large homogeneous region relative to the coil's volume compared to a traditional Helmholtz Coil (Merritt, 1983)

Feedback Field Simulation
To analyze and optimize the design of the Tesseract sensor's feedback windings, we developed a Biot-Savart simulation of the magnetic field generated by these windings when a DC current is applied. The simulation of the magnetic field generated by the 195 Tesseract's x-axis feedback coils, rendered in Figure 4a, is as a slice at the plane z=0 ( Figure 4b). The main region of interest is the area occupied by racetrack cores (outlined in maroon), which is at z =0 mm, y = ± 12 mm, and x = ± 15 mm. For the purposes of this paper, the figure of merit for a feedback winding configuration will be the degree to which this winding holds the racetrack core axis in a homogeneous field; more explicitly, the maximum percentage that the magnetic field values along the y = 12 mm axis deviates from the average field along this axis. 200 The simulation of the Tesseract sensor's feedback design suggests that the deviation from a uniform magnetic field along the 1D axis of the racetrack core (at y =+12 mm) does not exceed 0.45% (Figure 4b). We surmise that keeping Tesseract's racetrack cores 210 immersed in this highly magnetically homogeneous region will contribute more stable, linear fluxgate performance. https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.
In this simulation, each loop of wire in the Merritt Coil is modeled as a square array with an assigned length, current and a location on a 3D grid (Figure 4a). A Biot-Savart integration is applied over each element in the square array to evaluate the magnetic field at every point on this grid. This process is repeated for every loop in the coil system, and the resulting field values are summed to create a model of the total magnetic field generated by a Merritt coil system. 215

Feedback Winding Optimization
Merritt (1983)  Merritt coil geometry places constraints on the number of wire turns in each channel. For example, the number of wire turns on the inner channel must be 0.4235 times the number of turns on the outer channel (Merritt et al., 1983).
However, given the physical constraints of manufacturing a three-axis sensor, it is necessary to violate these ratios slightly. The requirement for channel spacing must be broken such that b/d < 0.5 in order to maintain three-axis symmetry and the mechanical integrity of the Torlon. Other considerations about the winding (i.e. that the number of layers must be even to ensure that the 225 terminating leads start and end in the same place so they can be terminated in a twisted pair) places further restrictions in the model.
When all these constraints are applied, we are left with the wire gauge, number of wire turns, and the ratio of the number of inner loops to outer loops as parameters that can vary. The Biot-Savart model (Figure 4b) was used to simulate 72 possible feedback winding configurations and the results were analyzed to determine which feedback winding characteristics were optimal for generating a homogenous field along the axis of the racetrack cores. 230 The field homogeneity generated by a feedback winding was found to have a dependence on that winding's ratio of inner loops to outer loops. Figure 5 plots the percentage that the field at the racetrack core deviates from uniformity against the ratio of inner loops of wire to outer loops. A quadratic fit suggests that an inner to outer loop ratio of 0.3985 is optimal for homogeneity. This 240 differs from the classic Merritt coil ratio of 0.4236. Shifting the inner to outer loop ratio compensates the earlier violation of the channel distance requirements for b/d defined in section 2.3.1 that were necessary to construct a physical sensor. Other parameters (wire gauge and total number of wire loops of the feedback winding) were not found to have a discernable correlation with the homogeneity of the feedback magnetic field.
In addition to feedback field homogeneity, another important consideration in selecting an optimal feedback configuration for a 245 magnetospheric sensor is the power dissipation required for these feedback windings to null maximum earth field ~65000 nT. This Where N is the total number of turns of wire, d is the length of one side of the square coil (as demarcated in figure 4), and for our purposes: V = 5 volts (the maximum voltage allowed by the operational amplifier) and B = 65000 nT (the maximum field 250 experienced in orbit). This equation implies an inverse relationship between feedback power consumption at earth field and the total number of loops of wire . Thus, a feedback configuration with more loops of wire will require less power to null earth field with an upper limit of on the resistance of the coil of to ensure it can null the full field.
Of these 72 simulated feedback winding configurations, three were determined to be excellent candidates for good feedback homogeneity and power efficiency. Figure 6 shows the parameters of these configurations along with the modeled magnetic flux 255 density distribution along the axis of the racetrack core for each of these optimized feedback configurations. Figure 6: Homogeneity of the feedback magnetic field along the axis of the racetrack core for each axis of three optimized sensors. The racetrack core occupies the region between the vertical dashed black lines at -15mm and 15mm. Configuration (b) was optimized for best homogeneity while sensor while (c) was chosen for good homogeneity with very low power consumption.

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The configuration in Figure 6c with 864 turns and a wire gauge of 40 AWG was determined to be a suitable balance of good homogeneity and feedback power efficiency. The magnetic field along the racetrack core axis varies by less than 0.46% in the x axis, 0.38% in the y axis and 0.32% on the z axis. This design requires a sensor power consumption of 89.1 mW to compensate for a worst-case Earth field of 65000 nT, which is very reasonable for accommodation on a small satellite. This is the feedback winding design that we plan to implement on the flight-ready Tesseract magnetometer that will be flown on the ACES-II sounding 265 rockets and again on the TRACERS satellites mission. https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.

Sensor Characterization
The Tesseract sensor design was prototyped and subjected to a series of tests in order to evaluate the validity of our design models and to quantitatively characterize the sensor's performance. First, several prototype sensors were manufactured and tested for 270 feedback winding homogeneity and stability over temperature. From the results of these initial tests, as described in this section, we selected the prototype that exhibited the most homogeneous feedback field.
Then, a high-fidelity prototype was manufactured with the specifications shown in Table 1. A series of tests were conducted on this prototype to quantify the characteristics of the Tesseract sensor: the feedback field uniformity , the sensitivity and orthogonality over changes in temperature, and the noise floor of the core. All the tests described in this section were conducted at the University 275 of Iowa Magnetometer Calibration Facility.

Feedback Field Homogeneity
In order to experimentally characterize the magnetic field generated by the feedback windings and test the results of the Biot-Savart model, three prototype sensors were manufactured with the optimized feedback winding configurations modeled in Figure   6. For our purposes, we are most interested quantifying the homogeneity of the feedback magnetic field along the axis of the 280 racetrack core. Each prototype sensor was placed within a single axis solenoid within a three-layer mumetal magnetic shield and constant current of 20 mA was applied to the feedback windings. A DC hall effect milligauss meter probe was placed in the center of the racetrack core's rectangular bore hole in the prototype bobbin and measurements of the magnetic field generated by this current were taken along the bore axis at 2.5 mm intervals. The final value for each measurement was found as an average of that value over four separate tests. The uncertainty was determined by the range over which each measured magnetic field value varied 285 over the four tests. Figure 7 plots the measured magnetic flux distribution along the longest axis of the racetrack core for each prototype sensor as a percentage deviation from the average field value along the bore hole axis between -15mm and 15mm (the domain of the racetrack core). 290 Figure 7: (a) The magnetic field generated by the Tesseract prototype #3 sensors' feedback windings was measured in 2.5 mm intervals along the axis of the racetrack bore hole. The magnetic field is plotted as the percentage deviation from the average value in the racetrack core region (between -14.5 mm and 14.5 mm). The feedback magnetic field does not deviate by more than 0.42% along the core axis and the feedback coils consume 29.7 mW to null earth field. (b) The measured percentage deviation of the magnetic field for each position along the axis of the ring-core hole. The magnetic field deviates as much as 5.62% in the domain of the one-inch ring-core (between -12.7 295 mm and 12.7 mm). https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.

Comparison with Ring-core Feedback Field
We performed the same test again, this time using our prototype feedback winding for our 1" ring-core sensor (Figure 1b) which has similar geometry to its heritage e-POP design (Wallis et al., 2016) and many historical missions (e.g. Acuña, 1980). The ringcore sensor uses solenoidal windings which have similar dimensions for all three axes, so we expect the feedback field to 300 be roughly the same for each axis. Figure 9 plots the measured magnetic flux distribution along the radial axis of the ring-core as a percentage deviation from the average field value in this region. Distance is measured from the center of the ringcore and the ring-core itself extends from 12.7 mm to -12.7 mm.
The maximum that the measured feedback field deviates from uniformity along the x axis of the ring-core sensor is 5.62%. The ring-core feedback windings generate a field the along the center axis of the ring-core that is ten times more inhomogeneous than 305 that generated by the Tesseract sensor.

Comparison with Biot-Savart Model
The feedback configuration of Prototype 3 (Figure 7c), with 864 turns of 40 AWG wire on each axis, was chosen to be the current high-fidelity sensor ( Figure 1a) and will be flown on future missions. This feedback winding generates a magnetic field along the axis of its cores that deviates from average by a maximum of 0.42% in the x axis, 0.39% in the y axis and 0.26% in the z axis. To 310 assess the effectiveness of our Biot-Savart simulation experimentally, we directly compared these data to the model output for the field values along the axis of the racetrack bore hole. The results of the model are plotted for each axis as solid lines (Figure 8).

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Slight differences between the field in the three axes are due to Tesseract's different sized feedback loops that are necessary to nest the physical coils (Figure 1a). The simulation does a reasonable job of modeling the data within the error bars. To quantify the goodness of the model's fit to the data, the square root of the variance of the residuals (RMSE) and the adjusted R-Squared values were calculated for each axis and for each of the three prototypes. The Biot-Savart model fits the data to 320 well above an adjusted R-Squared value of 0.95 and an RMSE below 5.0 for each axis of each prototype sensor. The quality of https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License. these fits implies a very strong agreement between the data and the model, and we take this as an experimental validation of our Biot-Savart simulation.

Calibration over Temperature
One of the main causes of fluxgate instability is the tendency of sensor's intrinsic calibration parameters to change due to thermal 325 variations in the sensor. The fluxgate sensor's sensitivity and orthogonality have long been known to vary with sensor temperature (Trigg et a1 1971). Thus, a characterization of the sensor's calibration parameters over the temperature range that would be expected on magnetospheric, or planetary mission is critical for the validation of a space-based fluxgate sensor.
The dominant effect of changes in temperature on a three-axis null sensor is the change in sensitivity due to variations in the geometry of the coils of wire used to provide magnetic feedback (Primdahl and Jensen, 1982) which is in turn dependent on the 330 rate of thermal expansion or contraction of the material of the bobbin upon which the feedback coils are wound. (Acuña et al., 1978). The temperature dependent sensitivity can be corrected to first order by fluxgate electronics using compensated feedback current (Primdahl, 1970;Acuña et al., 1978;Miles et al., 2017). Another cause of instability is the tendency of the sensor's three orthogonal feedback coils to skew over changes in sensor temperature (Ripka, 1992); however, no equivalent compensation exists for orthogonality so while smaller temperature dependent sensitivity is desirable, smaller orthogonality dependence is more critical. 335

Experimental Apparatus
Accurately measuring sensitivity and orthogonality over temperature is notoriously challenging, notably due to the difficulty of keeping the excitation/calibration coils thermally isolated from the sensor to ensure that measured changes result from the sensor under test rather than changes in the experimental apparatus. This test usually requires sophisticated equipment. simpler, low-cost methods have been devised (i.e., Brauer et al., 1999;Miles et al., 2017) to calibrate a magnetometer using an insulated cooler filled 340 with dry ice placed within some form of calibration coil.

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To characterize the thermal stability of the Tesseract sensor's design, we temporarily configured it as an air-core search coil magnetometer to directly access the attributes of the sensor base and feedback windings without any dependence on cores or electronics. The sensor is placed in a thermally insulated box made from 10 cm thick polystyrene to create a controlled temperature environment for the sensor. The polystyrene box is then placed within the two-meter Merritt coil  that Tesseract's axes are aligned with the coil systems axes as shown in Figure 9a. The coil system was used to generate a 60,000 nT, AC magnetic field at 23 Hz in the x axis for 2 minutes. The same field is then applied on the Y and Z axes, and the coil system is set to cycle through X, Y and Z, so that a complete measurement of sensitivity and orthogonality is taken every 6 minutes. A reference vector magnetometer is placed outside of the box, aligned with the coil system, and wrapped in insulating material to monitor the applied AC field and ensure that the magnitude does not vary. 355 Figure 10: The temperature of the Tesseract sensor (blue) is measured using a platinum RTD as the dry ice sublimates. The temperature of the reference magnetometer (red) was also monitored and found to vary less than 0.1 C over the course of the test.
Ten pounds of dry ice is placed inside the box to chill the sensor, and measurements are taken after the dry ice has sublimated and 360 the sensor is slowly warming. A platinum RTD temperature sensor is attached to the sensor and recorded the change in temperature as the sensor returned to room temperature ( Figure 10). As the Tesseract sensor temperature slowly increased, the response of the Tesseract prototype sensor's feedback windings, the reference magnetometer and the temperature sensors are digitized and recorded using a common 18-bit National Instruments DAQ.

Feedback Sensitivity and Orthogonality over Temperature 365
To evaluate the feedback sensitivity and orthogonality, a finite Fourier transform is performed for each 2-minute segment of data where the 23 Hz field is applied in X then Y then Z. The measured amplitudes of these signals at 23 Hz are recorded for each axis.
This yields nine values that define the calibration of the sensor. The sensitivity or gain on each axis is simply the output of the feedback winding when the 60,000 nT field is applied in that direction. Figure 11a plots the measured change in sensitivity measured in the X Y and Z feedback windings in parts per million over changes in temperature. The Tesseract sensor's three 370 orthogonality angles; the angles between the X and Y axes, X and Z axes and Y and Z axes, can be found using simple trigonometry and are defined as: Where Yy is the field measured in the Y axis when the coil system applies a field in Y, Zx is the field measured in X when a field is applied in Z, and Zy is the field measured in Y when a field is applied in Z. A is the total magnitude of the applied field. Figure   11b plots each of these angles over temperature. 380 Figure 11: (a) The measured sensitivity of the Tesseract Sensor's three feedback windings plotted against temperature. Each circle is a two-minute measurement of the amplitude of the 60,000nT, 23 Hz signal. The dotted lines are a robust linear fit to the data. The fits estimate that the Tesseracts sensitivity over temperature (b) The Tesseract sensor's three orthogonality angles, XY, XZ and YZ, plotted as a function of sensor temperature. None of the angles are found to unambiguously skew more than 0.015 degrees. (c) The measured sensitivity of the ring-core sensor's three feedback winding plotted against temperature. (d) The ring-core sensors' orthogonality angles 385 are found to skew over temperature by as much as 0.09 degrees Notwithstanding this modest experimental setup, we are able to measure the sensor's change in sensitivity and orthogonality over temperature. A robust linear fit to the data gives an estimate of the linear dependence of sensitivity on temperature. In all three axes, the sensitivities to temperature are very linear with R squared values of 0.995. We expect that this will make temperature 390 compensation very effective, since the compensation method used by Acuña (1978) is strictly linear. The sensitivity to temperature in the X and Y axes is 16.6 ppm/°C and 17.1 ppm/°C respectively; very comparable to the thermal coefficient of expansion of copper (16.7 ppm/°C) and of the Torlon base material (16.2 ppm/°C) as predicted by Acuña et al., 1978. The Z axis, which is aligned with the direction that Torlon extrusion was injected into the mold during manufacturing, differs slightly (13.3 ppm/°C).
Over the course of the test, the signal measured by the reference magnetometer does not vary more than 20 ppm in each axis which 395 confirms that these changes in sensitivity are due to changes in the sensor's feedback windings and not the applied field.
We do not measure any variation in axis alignment to within deviations of 0.015 degrees over changes in temperature from -45°C to 20°C. The sensor's feedback windings are a less-than-ideal search coil, so the accuracy of the measurements is limited by poor https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License.
signal. The background magnetic noise of the laboratory also limits the accuracy of this test. This test will be repeated once the sensor is integrated into a functioning fluxgate instrument in a magnetically quite facility in order to verify its flight calibrations. 400

Comparison with the Ring-core Sensor
The sensitivity and orthogonality of the ring-core's feedback windings over temperature were measured with the same method described above. The ring-core design sensor (Figure 1b) was also temporarily configured as an air-core search coil magnetometer so that attributes of the sensor base and feedback windings could be directly measured. The measured sensitivity and orthogonality over temperature are plotted in Figure 11b and Figure 11d. 405 While the Tesseract and ring-core sensors have comparable thermal sensitivities, the Tesseract's sensitivity over temperature is more consistent in each axis, presumably due to the greater symmetry of the sensor's feedback windings. The ring-core sensors' orthogonality angles skew by as much as 0.09 degrees (Figure 11d). The YZ Angle (blue) changes the most, presumably because the ring-core sensor base is most asymmetric between the Y and Z Axes (Figure 1b).

Sensor Noise 410
The noise floor of a fluxgate is typically limited by the intrinsic magnetic noise of a ferromagnetic core that is periodically driven into magnetic saturation to modulate the local magnetic field. The noise floor on Tesseract's custom miniature racetrack cores was characterized in Miles (2022). A single-axis electronics board was used to drive and sample each fluxgate core. The power spectral density noise floor of the instrument was estimated by using 20 minutes of data while the racetrack sensors were inside a single axis four-layer mumetal magnetic shield. The noise floor for the racetrack core sensors that will be used in Tesseract were 415 determined to be 5 pT/√Hz at 1Hz (Miles et al., 2022).

Conclusions
The novel fluxgate sensor called Tesseract has been designed and prototyped. It is a low mass, low noise sensor that mitigates several known causes of instability in the 1" ring-core design. We modeled the sensor's feedback winding using a Biot-Savart 420 simulation and used it to optimize the Tesseract Sensor's feedback topology for homogeneity and power consumption. The simulation results were in agreement with laboratory measurements of the prototype sensors' feedback magnetic field. The Tesseract retains its ferromagnetic cores in a feedback magnetic field that homogeneous to within deviations of 0.42% We used a low-cost method to analyze the sensor's gain and orthogonality over temperature. This method uses coil system measurements and a second reference magnetometer to least squares vector-vector calibration to find the intrinsic sensitivity, 425 orthogonality, and offsets of the sensor as a function of temperature. Using this method, we show that the Tesseract's three orthogonal axes skew by no more than 0.015 degrees over changes in temperature from -45 °C to 20 °C. The Tesseract sensor outperforms the traditional 1" ring-core Sensor design in both these metrics. The Tesseract's orthogonal axes are more stable by at least a factor of 4 and its magnetic feedback is more homogenous by a factor of 13. We surmise that Tesseract's higher degree of feedback homogeneity and axis stability over temperature will contribute to improved fluxgate stability. 430

Future Work
The accuracy of these measurements is limited by the magnetic noise of the laboratory setup. A more sophisticated experimental set up will be required to characterize the stability of Tesseract's calibration parameters to an accuracy acceptable for most space science applications (greater than ± 0.1 nT). We are currently developing a new calibration facility at the University of Iowa 435 calibration facility that will be purpose built for characterizing fluxgate sensor's calibration parameters over temperature. In future https://doi.org/10.5194/egusphere-2022-220 Preprint. Discussion started: 25 April 2022 c Author(s) 2022. CC BY 4.0 License. studies, we hope to use the specialized, environmentally controlled Helmholtz coil facility at Goddard Space Flight Center that would be capable of resolving Tesseract's long-term stability to a precision of ± 200 pT. (Vernier et al., 2004). Fluxgate electronics are currently being developed for the Tesseract Sensor in preparation for upcoming flights. Once the sensor has been integrated with cores and electronics, the Tesseract will be characterized again as a complete flight-ready fluxgate instrument. Tesseract will 440 be flight demonstrated on the ACES-II sounding rockets, currently scheduled to launch in December 2022 and again aboard the TRACERS satellite mission as part of the MAGIC technology demonstration which is scheduled to launch in 2023.

Code and Data Availability
Data and source code used in the creation of this paper can be accessed by contacting the authors.