# Adaptive Algorithm for Fast 3D Characterization of Magnetic Sensors
## 2 Materials and Methods
### Field Generation  
The field generation for the system is comprised of hardware components and of the software-controlled desired excitation signals. With the underlying idea of steering a known magnetic field towards the sensor under characterization from arbitrary directions, a triaxial Helmholtz coil setup was chosen and built for the excitation. A picture of this is shown in Figure 2a. There are three orthogonal pairs of aluminum enclosures that are nested within one another. These pairs each form individual Helmholtz coils, which are regularly used to produce approximately homogeneous magnetic fields in their center area [16]. Here, we chose to work with the common change in form factor from cylindrical coils to square ones, which is well established [17]. The individual coils have \(N_{\mathrm{coil}}=60\) windings of enameled copper wire, each with a conductor diameter of \(0.8\,\mathrm{mm}\), which corresponds to a \(0.5\,\mathrm{mm}^{2}\) cross-section. After assembly, we defined the inner coil as producing a magnetic field along the x-axis, the mid coil that as associated with the y-axis, and the outer coil that as associated with the z-axis. We also measured some parameters of the resulting practical setup. Regarding static quantities, we measured the resistances and the edge lengths \(\ell\) of the individual coils, the results of which are summarized in Table 1. For each individual coil of a pair of coils, the values are identical. Figure 2: Custom hardware setup components used for measurements in this contribution. (**a**) Helmholtz coils used for field generation in characterization system. (**b**) Rack-mountable composite amplifiers for coil excitation. We used a combined digital-analog and analog-digital converter (DAC and ADC, respectively; model Fireface UFX+; RME [18]) operated at sampling rate \(f_{\rm s}=48\,\)kHz in combination with a low-noise composite amplifier (based on [19]) to excite the coils over the three channels--one for each direction (x, y, and z). Each output channel was split afterwards and connected to both coils of the respective direction with coaxial cables. The supply voltage for the rack-mountable amplifier unit shown in Figure 2b is provided by a laboratory power supply (GPP-4323; Good Will Instrument Co., Ltd., Taipei, Taiwan [20]). In addition to the (real-valued) resistances of the coils, we had to consider the frequency-dependent impedances to properly generate a desired magnetic field. We derived the equalizers (EQs) for impedance consideration from noise excitations of each coil pair. Therefore, we generated and recorded uniformly distributed white noise, which is used as the DAC input and propagated to the amplifiers. With the current sensing modules of the amplifiers, we recorded the current flow over a \(100\,\)m\(\Omega\) shunt resistor. In each coil pair measurement, we computed the power spectral density (PSD) \(S_{vv}(\Omega)\) of the voltage signal as well as the cross power spectral density (cross PSD, CPSD) \(S_{vi}(\Omega)\) of the current and voltage signals. Time signals of \(90\,\)s were recorded, with \(1\,\)s discarded at the start and end to avoid the impact of possible transient behavior. The current signals were scaled by \(\frac{1}{100\,\)m\(\Omega}=10\frac{1}{\Omega}\) to account for the shunt resistance. As parameters for the PSD calculation, the size of the fast Fourier transform (FFT) was set to about a second with an even number of frequency bins for the filter design. We used a Hann window of the same length as the FFT and a \(50\,\%\) overlap between the signal sections. From these calculations, the impedance \(Z(\Omega)\) for a given coil measurement can be derived as the cross power spectral density \(S_{vi}(\Omega)\) normalized by the power spectral density \(S_{vv}(\Omega)\).