Fluxgate magnetometers do not measure the magnetic field absolutely and therefore must be calibrated in order to make accurate measurements. However, the calibration parameters sensitivity, orthogonality, and offset vary with changes in temperature or over time. Fluxgate stability is the degree to which these calibration parameters remain constant. 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 addressed by many authors in the literature. Fluxgate offsets are thought to originate primarily from the cores and driving electronics (Ripka et al., 2014), while changes in sensitivity and orthogonality are caused predominately by changes in the geometry of the sensor (Acuña et al., 1978; Miles et al., 2017). In this paper, we are concerned with the stability of the sensor. Factors suspected of degrading a fluxgate sensor's stability include an inhomogeneous magnetic null (Ripka, 1992) and skewing of the axes 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 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). Our magnetic sensors that are nulled on two axes, such as the CASSIOPE/e-POP fluxgate which maintained a sensitivity stability of approximately 18 ppm ∘C-1 (i.e., Wallis et al., 2015; Miles et al., 2013), outperform our sensors zeroed in only a single component (i.e., Narod and Bennest, 1990; Acuña et al., 1978) which achieved a stability of 26 ppm ∘C-1 over the temperature range from -20 to 40 ∘C. Thus, we hypothesize that a null on all three axes will further improve stability compared with one- or two-axis nulled sensor designs.

Recent constellation missions have increasingly focused on flying three-axis compensated sensors in the interest of maximizing instrumental stability. However, sensor designs that have flown on past constellation missions have been constrained to accommodating 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 feedback coil wound on a MACOR shell to create a very homogenous three-axis null at the locations of the cores. 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 100×100×50 mm, 500 g CSC sensor has maintained a sensitivity stability over temperature of 10 ppm ∘C-1 and a very high axis stability of 0.002∘ from -20 to 40 ∘C while achieving a noise floor of 6.6 pT √Hz-1 at 1 Hz (Merayo et al., 2008).

The THEMIS mission incorporated 70×70×45 mm, 75 g, low-mass fluxgate sensors that achieved a sensitivity stability of 22 ppm ∘C-1, an axis stability within 0.017∘ from -100 to 60 ∘C and a noise of 10 pT Hz-1 at 1 Hz (Auster et al., 2008). The 42.4×44.3×48.7 mm, 88 g Magnetospheric Multiscale Mission Direct Fluxgate instrument achieved a sensitivity stability over temperature of 30 ppm ∘C-1, an axis stability of about 0.03∘ between -50 and 30 ∘C, and a noise floor of 8 pT √Hz-1 at 1 Hz (Russell et al., 2016). Notably, the temperature-dependent sensitivity can be corrected to first order by fluxgate electronics using a compensated feedback current (Primdahl, 1970; Acuña et al., 1978; Miles et al., 2017); however, no 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 the Geospace Dynamics Constellation (Pfaff, 2016), MagneToRE (Maruca et al., 2021), and NanoSWARM (Garrick-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 in these efforts.

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 19 g fluxgate magnetometer with a noise floor of 120 pT √Hz-1 at 1 Hz (Kepko et al., 2017). The ExAlta1 CubeSat flew a 47 g miniaturized fluxgate sensor with a noise floor of 150 pT √Hz-1 at 1 Hz (Miles et al., 2016). However, these miniature instruments typically sacrifice noise and stability in the interest of small size. With a sensitivity stability poorer than 50 ppm ∘C-1, these sensors function primarily as variometers. If used in a constellation setting, measurements from disparate sensors would be far less comparable than those from a previous constellation mission without complex and time-consuming cross-calibration. One of the best small sensors to date appears to be the Small Magnetometer in Low-mass Experiment (SMILE): a 20×20×20 mm, 40 g cubic sensor based on three rod cores within a three-axis feedback winding, which achieved a thermal sensitivity stability of 11 ppm ∘C-1, an axis stability better than 0.02∘ from -30 to 45 ∘C, and 30 pT √Hz-1 at 1 Hz (Forslund et al., 2008).

Like the CSC and SMILE sensors, Tesseract's feedback windings create a three-axis “magnetic null” inside the sensor where the fluxgate cores are held in the 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 thought to contribute to degradation of the stability of a fluxgate's offset (Ripka, 1992), sensitivity (Korepanov and Marusenkov, 2012), and orthogonality (Petrucha et al., 2015).

Here, we present the design and characterization of Tesseract: a high-stability fluxgate magnetometer for constellation missions. 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 et al., 1983), allowing it to accommodate more cores (six instead of three). The idea of more complex feedback winding is not new. Designs such as Primdahl and Jensen (1982), Auster (2008), and Chulliat (2009) have used different methods of stacking concentric circular coils to create a three-axis null sensor. A study by Petrucha et al. (2015) experimented with a laboratory prototype sensor that used a Merritt coil feedback topology with ring geometry cores. Tesseract's design draws inspiration from the aforementioned SMILE sensor (Forslund et al., 2008), which used three equally spaced square feedback coils and a parallel rod sensor in each axis to create a cube-shaped, three-axis nulled sensor. Tesseract's Merritt coil feedback winding has been optimized to hold six racetrack cores within a highly homogeneous null field (deviations from the 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 (Ripka et al., 2014; Korepanov and Marusenkov, 2012) and linearity (Brauer et al., 1997) of the sensor.

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 instability. Tesseract's cores achieve a magnetic noise floor of 5 pT √Hz-1 at 1 Hz, an orthogonality within 0.015∘, and a temperature stability of 13–17 ppm ∘C-1 between -45 and 20∘.

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 et al. (2019) enabled us to create custom low-noise miniature racetrack sensors (Miles et al., 2022).

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 (Fig. 2a) and interleaved with insulating layers of Kapton of the same geometry.

Production cores are interleaved with a polymer between the foil layers to prevent them from moving during the magnetizing drive pulses. A plastic lid closes the core and serves as a base upon which to wind a quasi-toroidal drive of AWG 32 magnet wire (Fig. 2b). Finally, a solenoidal sense winding of AWG 34 magnet wire is wrapped around the length of the base.

The stacked foil washers remove the need to spot-weld, as is done in traditional spiral-wound sensors, avoiding the heat-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 racetrack geometry aligns ferromagnetic mass on one axis, producing lower noise. However, the racetrack 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 with traditional parallel rod sensors (e.g., Janosek, 2017; Moldovanu et al., 2000).

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 Fig. 3a. We hypothesize that pairing of identical cores with opposite polarities in each axis may further reduce the tendency for cross-axis contamination due to mutual cancellation of their stray fields (Ripka and Billingsley, 2000).

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 (Fig. 3b). Vacuum-tolerant epoxy is used to secure the drive windings to 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 with sensors using the traditional Inconel bobbin. This further reduces the potential for mechanical stress.

Torlon is lighter and much easier to machine than the traditional Inconel ring-core bobbin used in Miles et al. (2013). 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 (Fig. 3b in red). This single solid three-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 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 al., 1971). Miles et al. (2017) document the historical technique (Acuña et al., 1978) of providing real-time analog temperature compensation. Temperature compensation can be provided electronically in real time by modifying the feedback current. A 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 resistance temperature detectors (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 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 a thermal expansion coefficient of 16.2 ppm ∘C-1, 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-1) and minimize stress in the sensor. Tesseract takes advantage of this property of Torlon for effective real-time linear temperature compensation, 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.

Tesseract's three orthogonal feedback coils are wound into channels that have been machined symmetrically into the Torlon base (Fig. 3b). A computer numerical control winding machine is used to guide 40 AWG magnet wires into the channels so that there is no asymmetry or overlap. These feedback windings create a three-axis “magnetic null” inside the sensor where the racetrack fluxgate cores are held in a 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 reproducible magnetization of the ferromagnetic cores. This is thought to improve the sensor's measurement stability and linearity.

A homogeneous magnetic null at the cores improves stability over temperature by reducing the dependence of the sensitivity on the temperature dependence of the excitation current (Korepanov and Marusenkov, 2012). A study by Petrucha et al. (2015) determined that, in a three-axis null sensor, changing the inhomogeneity of the magnetic feedback by that deviation from the average along the core by 1.3 % can alter the measured alignment of the sensor's axes by as much as ±0.12∘. Brauer et al. (1997) demonstrated that inhomogeneities as low as 1 % cause 25 nT deviations from linearity when an uncompensated earth field is applied to a transverse axis. Tesseract's feedback windings are arranged as a four-square Merritt coil on three axes. This configuration creates a relatively large homogeneous region relative to the coil's volume compared with a traditional Helmholtz coil (Merritt et al., 1983).

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 Fig. 6. For our purposes, we are most interested in quantifying the homogeneity of the feedback magnetic field along the axis of the racetrack core. Each prototype sensor was placed within a single-axis solenoid within a three-layer Mumetal magnetic shield, and a constant current of 20 mA was applied to the feedback windings, creating a magnetic field of about 190 000 nT. A direct-current hall effect milligauss meter probe was placed in the center of the racetrack core's rectangular borehole 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 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 borehole axis between -15.7 and 15.7 mm (the domain of the racetrack core).

One of the main causes of fluxgate instability is the tendency of a sensor's intrinsic calibration parameters to change due to thermal variations in the sensor's base and feedback windings (Acuña et al., 1978). The fluxgate sensor's sensitivity and orthogonality have long been known to vary with sensor temperature (Trigg et al., 1971). Thus, a characterization of the sensor's sensitivity and orthogonality over the temperature range that would be expected on a 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 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 a 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. Here, we describe a test to measure the sensitivity and orthogonality of the Tesseract sensor's feedback windings over temperature by temporarily configuring it as an air-core search coil magnetometer. This allows us to assess the temperature stability of the sensor base and feedback windings without any dependence on cores or electronics.

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 et al. (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 min of data, while the racetrack sensors were inside a single-axis four-layer Mumetal magnetic shield. The noise floors for the racetrack core sensors that will be used in Tesseract were determined to be 5 pT √Hz-1 at 1 Hz (Miles et al., 2022).

The accuracy of these measurements is limited by the magnetic noise of the laboratory setup, and in an uncontrolled warming there is always the possibility of thermal gradients or transients. A more sophisticated experimental setup 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 that will be purpose-built for controlled temperature characterization of fluxgate sensor calibration parameters, which will incorporate better shielding and limit the possibility of thermal gradients or shocks. In future 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, Tesseract will be characterized again as a complete flight-ready fluxgate instrument. Tesseract will 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.