## 2 System Description
### Horizontal Magnetic Field Sensor  
Based on the system performance requirements outlined in the previous section and the design principles for magnetic field sensors, we have designed the system's magnetic field sensor as follows. First, the equivalent circuit diagram of the flux negative feedback type broadband inductive magnetic field sensor, utilizing a feedback coil, is shown in Figure 3.  
Figure 3: Equivalent circuit diagram of the flux feedback type broadband inductive magnetic field sensor.  
Figure 2: Schematic of the E-field and B-field sensors.  
The primary coil is modeled as an inductor \(L_{pc}\) in series with a resistor \(R_{sc}\), which is then paralleled with a capacitor \(C_{sc}\). The resistors \(R_{1}\) and \(R_{2}\) determine the amplification factor of the amplifier. The amplifier circuit primarily uses the instrumentation amplifier OPA128. The input voltage and current noise levels of the OPA128 are \(8\,nV/\sqrt{\mathrm{Hz}}\) and \(300 fA/\sqrt{\mathrm{Hz}}\), respectively, within a frequency range above 1 kHz. The output voltage \(V_{out}\) is connected to the feedback coil through the feedback resistor \(R_{f}\). The feedback coil has an inductance \(L_{s}\) and its resistance is negligible. The feedback coil and the primary coil are mutually coupled with a mutual inductance \(M\), where \(L_{pc}=1.7H\), \(C_{sc}=12 pF\), and \(R\_sc=51\,\Omega\).  
Let the induced voltage in the coil be denoted as \(e\):  
\[e(\omega)=-j\omega\mu_{app}NSB, \tag{1}\]  
where \(\mu_{app}\) is the magnetic permeability, \(N\) is the number of turns of the coil, \(S\) is the effective cross-sectional area of the coil, and \(B\) is the magnetic field induction strength.  
Therefore, when the gain applied to the induced voltage \(e\) is \(G\), the output \(V_{out}\) of this equivalent circuit in the frequency domain is expressed as follows:  
\[V_{out}(\omega)=\frac{e\times G}{1+j\omega(C_{sc}R_{sc}+\frac{MG}{R_{f}+j \omega L_{s}})+(j\omega)^{2}L_{pc}C_{sc}}, \tag{2}\]  
Analyzing the parameters in the denominator of equation, within the bandwidth range of 10 kHz to 300 kHz, the value of \(R\_f\) is a few thousand ohms, and \(L_{s}\) is on the order of tens of \(\mu H\). In this range, \(R_{f}\) is much larger than \(j\omega L_{s}\), so \(j\omega L_{s}\) can be neglected. Given that \(C_{sc}=12 pF\), \(R\_sc=51\,\Omega\), \(M\) is a fraction of a few henries, and \(G\) is a large value in the thousands, the value of \(\frac{MG}{R_{f}}\) is much larger than \(C_{sc}R_{sc}\), so \(C_{sc}R_{sc}\) can be ignored. After substituting these simplified results into Equation (2), the conversion factor \(FS\) for the flux negative feedback-type broadband inductive magnetic field sensor in the low-frequency range up to 1000 kHz is  
\[FS(\omega)=\frac{V_{out}}{B}=\frac{-j\omega t_{app}NS\times G}{1+j\omega\frac {MG}{R_{f}}+(j\omega)^{2}L_{pc}C_{sc}}, \tag{3}\]  
Analyzing the analytical expression of the conversion factor reveals that, in the low-frequency range, the amplitude of \(FS\) increases linearly with frequency. As the frequency increases to a certain point, the amplitude of becomes frequency-independent and remains constant. When frequency increases further, the amplitude of decreases linearly with frequency. In summary, \(FS\) forms a band-pass transfer function determined by two frequency inflection points.  
Based on the aforementioned design principles, a wideband magnetic field sensor can be realized by adjusting the parameters of the feedback coil and circuit components. By optimizing the coil-circuit matching, the sensor's operating frequency band is designed to be 10 kHz-300 kHz (within 3 dB attenuation), as shown in Figure 4.