# Towards the Feasibility of Long Range Wireless Power Transfer over an Ocean Surface
## 2 Methodology and Materials
### Experimental Setup
#### 2.4.1 Transmitting End  
At the transmitting end, the power source that drove the circuit was a 100 kV periodic pulse generator (See Figure 2). Its base frequency was 83 kHz but the spark gap released a mixture of harmonics of different resonant frequencies. The actual frequency of peak harmonics was found to be 44 MHz (See Figure 3b). Driven by the 100 kV periodic pulsating source, the spark gap together with the monopole antenna formed a transmitter very similar to [PERSON]'s version of the spark gap transmitter [1; 2; 3; 4; 5; 6] (See Figure 2). The left end of the spark gap was connected to the lower end of a vertical monopole antenna, while the right end of the spark gap was connected to a metal electrode submerged in water. Figure 3: Results of the basic functionality test. (**a**) Waveforms of the currents harvested at the transmitting end and the receiving end. (**b**) Photo showing how the surface power was received by multiple loads. Figure 2: Experimental setup on wireless power transfer. The Earth's surface is covered with an electric field of 100 volts/m pointing downwards [37]. Suppose the atmospheric potential associated with the height \(H\) is \(V_{0}\). Since the antenna top has reached the height \(H\), the body of the monopole antenna forms a conductive pathway to allow the positive ions at height \(H\) to flow towards the left end of the spark gap, thereby forming a current \(I_{0}\). Current \(I_{0}\) is a conductive current which carries charges. Whilst the current density of \(I_{0}\) is undeniably small, the high voltage mixing effect of the spark gap will amplify the harmonics enormously. Let us assume that the current flowing through the spark gap is \(I_{g}\). Let us also assume that the inductance of the spark gap terminal and the inductance of the antenna are equal. The spark gap itself is a highly nonlinear device which can be modeled using the following polynomial equation:  
\[I_{g}=a_{0}+a_{1}V^{1}+a_{2}V^{2}+a_{3}V^{3}+\ldots+a_{n}V^{n} \tag{17}\]  
where the \(V\) is the voltage across the spark gap. The voltage across the spark gap can be modeled as  
\[V=\frac{1}{2}(V_{0}+V_{1}cos(\omega t)) \tag{18}\]  
where \(V_{1}cos(\omega t)\) is the input voltage of the spark gap. By substituting \(V\) from Equation (18) back to Equation (17), we obtain:  
\[I_{g}=\sum_{n=1}^{n}\frac{a^{n}}{2^{n}}(V_{0}+V_{1}cos(\omega t))^{n} \tag{19}\]  
Equation (19) can be broken down into multiple harmonics as well:  
\[I_{g}=I_{0}+\sum_{m=1}^{\infty}I_{m}cos(m\omega t+\phi_{m}) \tag{20}\]  
where \(I_{0}\) and all the harmonics become a function of \(V_{0}\) and \(V_{1}\). For simplicity, let us say the value of \(n\) in Equation (19) is 3 as an example. By going through Equations (17)-(20), we obtain the total current  
\[I_{g}=I_{0}+\sum_{m=1}^{3}I_{m}cos(m\omega t) \tag{21}\]  
where the DC current \(I_{0}\) as well as the currents of the harmonics become:  
\[I_{0}=a_{0}+\frac{a_{1}V_{0}}{2}+\frac{a_{2}V_{0}{}^{2}}{4}+\frac{a_{3}V_{0}{ }^{3}}{8}+\frac{a_{2}V_{1}{}^{2}}{8}+\frac{3a_{3}V_{0}V_{1}{}^{2}}{16} \tag{22}\]  
\[I_{1}=\frac{a_{1}V_{1}}{2}+\frac{a_{2}V_{0}V_{1}}{2}+\frac{3a_{3}V_{0}{}^{2}V _{1}}{8}+\frac{3a_{3}V_{1}{}^{3}}{32} \tag{23}\]  
\[I_{2}=\frac{a_{2}V_{1}{}^{2}}{8}+\frac{3a_{3}V_{0}V_{1}{}^{2}}{16} \tag{24}\]  
Figure 2:0  
In Equations (21)-(25), \(I_{0}\) is the current along the body of the monopole antenna. \(I_{1}\) is the magnitude of the input current feeding the spark gap. \(I_{2}\) and \(I_{3}\) are the magnitudes of the harmonic waves being transmitted from the transmitting end to the receiving end. In the absence of any monopole antenna, the surface wave along the interface was just an evanescent wave without any charge movement. According to our experimental outcome, this evanescent wave is not detectable at the receiving end. However, after mounting the monopole antennas to both transmitting end and receiving end, the monopole antenna provides a conductive pathway for the positive ionsat a higher altitude to flow towards medium 2.