## 2 Materials and Methods  
A brief review of the operating principles of an asymmetrical capacitor EHD thruster will be the starting point of this study. In an asymmetrical capacitor, one electrode acts as the emitter or anode, and the other as the collector or cathode. When a potential difference is applied between them, a non-uniform electric field is produced. Once the applied voltage reaches the value necessary to produce the corona effect, the gas surrounding the emitter is ionized, throwing ions towards the collector. If the displacement of the ions were to take place in an absolute vacuum, the ions would accelerate within the electric field until they are captured by the cathode.  
However, when they do so in a gas, they collide with neutral particles to which they give up kinetic energy. The accelerated neutral particles, not being affected by the electric field, freely move between the emitter and the collector, causing a reaction force on the electrodes that translates into thrust. Figure 1 shows an elementary asymmetrical capacitor.  
The equations used below are based on the references [8; 9], where a deep review of the physical processes involved in the system shown in Figure 1 is addressed. However, such a complex and thorough analysis is out of the scope of this paper. Therefore, as the authors themselves indicate, the use of only three equations is enough to understand all the involved phenomena, as well as the methodology of our study.  
\[I=C\times V(V-V_{0}) \tag{1}\]  
\[T=F_{EHD}-F_{D}=\frac{I\times d}{\mu}-F_{D} \tag{2}\]  
\[\theta\cong\sqrt{C}\times\left(\frac{d}{\mu}\right)^{\frac{3}{2}}\times\frac{ 1}{\sqrt{F_{EHD}}} \tag{3}\]  
\(I\) being the current per length unit, \(V_{0}\) the initial voltage, \(C\) a constructive constant which depends on gas properties and the electrode design, \(T\) the resultant thrust, \(F_{EHD}\) the electrohydrodynamic force, \(F_{D}\) the drag caused by the ionic wind flowing around the collector (a cylindrical electrode identical as the one used in our study), \(d\) the distance between emitter and collector, \(\mu\) the ion mobility and \(\theta\) the electromechanical effectiveness, measured as the ratio between force and electrical power.  
Figure 1: Asymmetrical capacitor.  
It can be deduced from Equation (2) that the thrust can be increased by reducing \(F_{D}\) and increasing the value of \(I\) as well as the distance between electrodes. However, the increase in \(F_{EHD}\) caused by \(I\) also decreases the electromechanical effectiveness \(\theta\).  
The drag \(F_{D}\) is proportional to the ionic wind velocity [8] and the velocity, in turn, is proportional to \(\sqrt{I}\) so thrust can be expressed as follows:  
\[T=\left(\frac{d}{\mu}-\alpha\right)\times I \tag{4}\]  
where \(\alpha\) is constant and the rest of the variables are as defined above.  
From this mathematical analysis it was concluded in [9] that the configuration that optimized the thrust had to contain 3 anodes. However, in the study to be presented below, it was decided to use only one. Although the use of a single anode reduces the efficiency of the asymmetric capacitor, this design is simpler constructively, and allows to achieve the objectives sought; to experimentally evaluate the thruster performance with different distances between collector and emitter, to quantitatively determine the influence of the electrode length, and to use a dual configuration with two identical asymmetric capacitors in parallel at different distances. All this, with uA current resolution and a reduction of the surface area occupied by using a single collector.