## 1 Introduction  
A VLF antenna is a typical electrically small antenna [1, 2], the electrically small antenna (ESA) is the antenna whose antenna size is much smaller than the wavelength (\(\mathrm{l}\ll\lambda\)), which can be equivalent to a capacitor or inductor with a small radiation resistance, which also leads to the ESA having a high quality factor, Q. Q is inversely proportional to the antenna bandwidth, so the electrically small antenna with high Q is a narrow band antenna. [PERSON]'s theorem gives the formula for calculating the Q value of electrically small antennas as follows [1, 2, 3]. \[Q=\frac{1+3k^{2}a^{2}}{\left(ka\right)^{3}\left(1+k^{2}a^{2}\right)} \tag{1}\]  
\(k\) is the wave number, \(k=2\pi/\lambda\), and as shown in Figure 1, a is the radius of the sphere that exactly surrounds the antenna. It can be known from Equation (1) that when the antenna electrical size decreases, the \(Q\) value will increase and the antenna bandwidth will become narrow. When the antenna size decreases to less than \(\lambda/2\pi\), the \(Q\) value will increase greatly. At present, the VLF communication system mainly uses an MSK (Minimum Shift Keying) signal source [4, 5], and the traditional VLF transmitter system uses a fixed tuning network, and the resonant frequency of the antenna feeder system is at the center  
Figure 1: Schematic diagram of the ESA. frequency of the MSK signal. In order to ensure that the transmitted signal has enough power, the symbol rate of the MSK signal will be very low, which also leads to a low communication rate. \[S_{MSK}=cos(2\pi f_{c}t+\frac{\tau a_{k}}{2T_{s}}t+\varphi_{k}),(k-1)T_{s}\leq t \leq kT_{s} \tag{2}\]  
Formula (2) is the expression of MSK signal [6], where \(f_{c}\) is the carrier frequency; \(\mathrm{a}_{k}=\pm 1\) (corresponding to the input symbol is \"1\" or \"0\"); \(\mathrm{T}_{s}\) is the symbol width; \(\varphi_{k}\) is the initial phase of the kth symbol, which is invariant in one symbol width. When the input symbol is \"1\", \(a_{k}=+1\), frequency is \(f_{h}=f_{c}+1/4T_{s}\), and when the input symbol is \"0\", \(a_{k}=-1\), frequency is \(f_{l}=f_{c}-1/4T_{s}\) and symbol rate is \(R_{B}=1/T_{s}\). In this paper, \(f_{h}\) is the \"mark\" frequency and \(f_{l}\) is the \"space\" frequency. According to the Shannon formula \(\mathcal{C}=B\log_{2}(1+S/N)\)[6], under the condition that SNR is almost unchanged, as the communication rate increases, the communication capacity increases, and the bandwidth occupied by the signal also gradually increases. However, when the signal bandwidth exceeds the available bandwidth of the antenna feeder system, the signal will be greatly attenuated and the information carried by the waveform will be lost. Therefore, the reason that limits the improvement of the communication rate lies in the narrowband characteristics of the high-Q electric small antenna. In order to enhance the effective bandwidth and optimize the communication rate of the VLF system, two primary approaches can be employed. One involves augmenting the series resistance between the antenna and transmitter; however, this comes at a cost of reduced efficiency, which may impact the overall communicative capability of the system [7]. Another approach to enhance the effective bandwidth is synchronous tuning technology [8], enabling the feeder system to dynamically tune to both the \"space\" frequency \(f_{l}\) and \"transmission\" frequency \(f_{h}\) of MSK signals instead of being statically tuned at the center frequency. This technique effectively broadens the system's effective bandwidth and enhances communication speed without compromising antenna efficiency, as depicted in Figure 2[9; 10; 11; 12; 13; 14]. Presently, countries such as the United States and Australia have embraced synchronous tuning technology, achieving communication speeds ranging from 200 bps to 1000 bps or even higher [15].