# Low-Voltage Plasma Generator Based on Standing Wave Voltage Magnification
## 2 Materials and Methods  
The previously presented methods could generate the required oscillations to operate the coil as a helical resonator. Usually, an air-core transformer is used to transfer the excitation signal. It acts as an ideal inductor, but its coupling coefficient varies as the operating frequency changes. A low coupling coefficient is required for the resonator that follows this study to create a medium in which voltage magnification due to standing wave is achieved. A typical high-voltage air-core transformer is presented below in Figure 1. The secondary will have a low coupling coefficient as a quarter-wave oscillator. At the same time, a low coupling coefficient makes possible a medium in which constructive interference contributes to the output voltage, forming a standing wave from the incident wave and the reflected wave summation. Figure 1: Helical resonator air-core transformer, Reprinted from [PERSON] and [PERSON] (2015) [12]. Possibility for primary side natural frequency adjustment: (**a**) L1 has a low inductance value due to winding pitch and area; (**b**) L1 has a higher inductance value compared to the previous case. For a low-loss, non-radiating, quarter-wave line with an open circuit at the load end, the voltage magnification ratio at resonance between the top and bottom of the resonator is calculated using Equation (1), where \(\alpha\) is the line damping factor, and \(h\) represents the length [13]:  
\[\frac{V_{top\_load}}{V_{initial}}=-j\frac{1}{\alpha h} \tag{1}\]  
In this equation, the left operand represents the voltage magnification ratio. Calculating the phase shift resulting from the right operand transfer function further correlates the model with the mathematical part and the experimental part. These three sections are covered in the following chapters. Having the secondary winding operating in the radio waves domain, each time the wave enters a different medium, part of the radiation will reflect into the conductor. The reflection factor can be calculated based on several constraints [14]. As the signal propagates at the speed of light through a conductor, Equation (2) can be used to match the secondary resonant frequency to one quarter wavelength multiple, considering an ideal transmission line. \[\frac{4}{(2N-1)}L=\frac{v_{p}}{f_{sw}}, \tag{2}\]  
Equation (2) is another form of Equation (3), presented below:  
\[\lambda=\frac{v_{p}}{f_{sw}}, \tag{3}\]  
providing the relation between several variables described below. These variables represent:  
1. \(N\)--number of maximum points due to standing wave;
2. \(L\)--secondary winding wire length;
3. \(v_{p}\)--propagation velocity, usually considered as the speed of light;
4. \(f_{sw}\)--transmission line switching frequency;
5. \(\lambda\) --wavelength of propagated signal. An illustration for these equations is presented below in Figure 2, where maximum and minimum points due to interference are identified. In any marked maximum points, a reflection of the illustrated incident wave will overlay the reflected wave onto the incident wave. Through repeated summation, the standing wave is formed. Figure 2: Illustration of secondary wire length versus the propagated wave for one-quarter and three-quarters wavelength resonators. A typical high-voltage air-core transformer is presented below in Figure 1. Having a low coupling coefficient, the secondary will act as a quarter-wave oscillator. At the same time, a low coupling coefficient makes possible a medium in which constructive interference contributes to the output voltage, forming a standing wave from the incident wave and the reflected wave summation. Based on the equation presented above, maximum and minimum electric field values can be identified from Figure 3.  
In Equation (2), one can choose different values for the number of maximum points. However, as shown in Figure 3 on the left side for the quarter-wave topology, the electric field vector is concentrated in one maximum point compared to the two maximum points for three-quarters of the wavelength.