## 2 System Description
### Vertical Electric Field Sensor  
For electrodes placed in free space, the presence of an electric field causes a portion of charge to accumulate on the electrodes, thereby generating a certain potential. This can be equivalently modeled as a voltage source \(V_{s}\). The electrodes themselves possess a certain capacitance \(C_{s}\) relative to an infinite distance. The preamplifier can be equivalently represented as a parallel combination of input impedance \(R_{in}\) and input capacitance \(C_{in}\), as illustrated in Figure 5, which depicts the working principle of the electric field sensor.  
According to the voltage division principle:  
\[\frac{V_{in}}{V_{s}}=\frac{Z_{in}}{Z_{s}+Z_{in}}=\frac{\frac{l}{\frac{l}{Z_{in }}+j\omega C_{in}}}{\frac{l}{j\omega C_{s}}+\frac{l}{Z_{in}^{2}+j\omega C_{in}} }=\frac{j\omega C_{s}R_{in}}{l+j\omega C_{in}R_{in}+j\omega C_{SR}}, \tag{4}\]  
If \(\omega C_{s}R_{in}>>1\), then  
\[\eta=\frac{V_{in}}{V_{s}}=\frac{C_{s}}{C_{in}+C_{s}} \tag{5}\]  
Since the sensor's inherent capacitance is in the picofarad (\(pF\)) range, measuring low-frequency electric field signals (where \(\omega\) is very small) requires \(R_{in}\) to be extremely large (\(10^{10}\,\Omega\)). Under these conditions, the reception efficiency is frequency-independent  
Figure 4: Conversion factor of the inductive magnetic field sensor.  
Figure 5: Schematic diagram of the electric field sensor working principle.  
and primarily depends on the ratio of \(C_{s}\) to \(C_{in}\). It is evident from the above that a larger inherent capacitance of the electric field sensor leads to a higher input impedance of the preamplifier and a smaller input capacitance. This configuration facilitates the detection of low-frequency signals and improves coupling efficiency.  
In this work, the schematic of the parallel-plate capacitors and low-noise amplification circuit is shown in Figure 6. The electric field sensor uses a bipolar plate made of FR4 copper-clad material with a diameter of 12 cm. The electric field antenna amplifier utilizes the OPA211, with a differential instrumentation amplification structure chosen to suppress common-mode noise.  
Based on the design parameters, the inherent capacitance of the electrode plate, \(C_{s}\), is approximately 10 pF (estimated based on the size of the electrode plates) [27], and the amplifier's equivalent input impedance, \(R_{in}\), is \(10^{10}\Omega\). When \(\omega C_{s}R_{in}=100\), the condition \(\omega C_{s}R_{in}>>1\) is considered satisfied, resulting in a frequency \(\omega\) of 1.6 kHz, which corresponds to the lower limit frequency. The upper limit frequency of the electric field sensor is determined by the cutoff frequency of the low-pass filter, which is set at 500 kHz in the design. The distance between the two pole plates is 20 cm, and the diameter of each pole plate is 15 cm (electrode plate and circuit boards combined), making the sensor compact. With the help of the front matching circuit, the antenna achieves an effective length of 1.5 m. Compared to a rod antenna, the capacitance between the two plates is larger, allowing the sensor to handle high-frequency measurements while maintaining better low-frequency performance. Overall, the electric field sensor operates within a frequency range of 1.6 kHz to 500 kHz, meeting the design requirements of 10 kHz to 300 kHz.