## 3 Results and Discussion  
### Sensitivity and Linearity  
The sensor was first tested with several excitation frequency values and excitation current amplitudes in order to find the optimum operating conditions. The sensitivity measurement was implemented by applying an external direct current (DC) magnetic field to the sensor by a solenoid coil (Tian Heng Control Technology Co., Ltd., Tianjin, China, amplitude of \(\pm 1.92\) mT), and the outside of the coil was covered with 10 layers of FeNi thin films to form a magnetic shield. As shown in Figure 4a, we clearly observed that a higher driving frequency resulted in greater sensitivity of the sensor. However, when the frequency value exceeded 500 kHz, a clear decrease in sensitivity was found in our sensor response. According to the operation function of the second harmonic principle, the sensitivity empirical formula for a long-rectangular-core fluxgate sensor can be simplified as follows [28]:  
\[S_{\mathrm{max}}=8 fNA\mu_{a} \tag{1}\]  
Figure 3: (**a**) Schematic illustration of test system circuit; (**b**) Photograph of the test system. where \(S_{\text{max}}\) is the maximum sensitivity of the fluxgate sensor, \(f\) is the excitation frequency, \(N\) is the number of turns of the pick-up coil, \(A\) is the cross-sectional area of the magnetic core, and \(\mu_{\text{a}}\) is the effective longitudinal apparent permeability for the rectangular prism core, which is always proportional to the saturated magnetic induction (\(B_{\text{s}}\)) of the core materials. From Equation (1), it is evident that the sensitivity of the fluxgate sensor can be linearly enhanced by increasing the drive frequency, as shown in Figure 4a. However, at an overlarge excitation frequency (over 500 kHz in this work), increasing eddy current losses and the demagnetization effect at high frequency may significantly decrease the effective permeability (\(\mu_{\text{a}}\)) of the sensor core, resulting in a decrease in sensor response [7]. Based on this, an excitation frequency of 500 kHz, which was proven to be an optimal condition for the sensitivity of the fabricated sensor, and the same excitation conditions were used in the subsequent experiments. The changes of the sensitivity with different excitation current amplitudes were also investigated and the results are shown in Figure 4b. It is apparent that the sensitivity of our sensor increased with increasing excitation current amplitude and then remained nearly the same after a current amplitude over 70 mA. This phenomenon is mainly attributed to the influence of the excitation current on the saturation extent of the sensor core: a larger current amplitude always means a deeper saturation state of the magnetic core of the sensor, resulting in higher sensitivity in the output response. However, for the excitation current of 70 mA, the sensor core may have become fully saturated, and therefore no significant difference in sensor sensitivity was observed when the current amplitude exceeded 70 mA. Thus, the optimum excitation current of 70 mA rms (root mean square) was used for subsequent experiments unless otherwise noted. A maximum sensitivity of 1985 V/T was achieved under the excitation current amplitude of 70 mA at an excitation frequency of 500 kHz. Figure 5a indicates the relation of outputs to the external magnetic field under optimum excitation conditions. A straightforward linear relationship between the magnetic field values and the output voltage of the sensor is observed. The linear regression equation is expressed as Y = \(-\)0.15972 + 1.98521X with a correlation coefficient of 0.99738 in the linear range of approximately \(\pm\)1.05 mT. The effect of excitation current magnitude and frequency on the linearity range of our sensor was also studied and the results are shown in Figure 5b,c, respectively. The results demonstrate the independence of the linear range on the excitation current amplitude and excitation frequency. Similar results were found in a previous study by [PERSON] et al.