# Suppression of Instabilities in a Plasma Voltage Stabilizer
## 2 Materials and Methods  
The design of the experimental setup, the design of the triode, and the method for studying plasma are described in detail in [20; 21]. Next, we will only describe their key features. Figure 1 shows the diagram of the experimental three-electrode device. The cathode of the device 1 was a tungsten disk impregnated with barium with a diameter of 10 mm. The main anode (MA) 2 and the control electrode 3 were polycrystalline molybdenum discs with a diameter of 30 mm. In the center of the main anode, there was a hole 2 mm in diameter. The control electrode was located at a distance of 1 mm from the main anode outside the discharge zone. The distance between the cathode and the MA was 8 mm. The main element of the heater 4 was a tantalum niobium wire. The plasma discharge channel was limited by a metal conical shield 6, whose potential correlated with that of the cathode. LVBD investigations were carried out with a method involving a flat, one-sided probe [21]. Since the plasma under the considered experimental conditions demonstrated axial symmetry, the EVDF did not depend on the azimuthal angle (Figure 2):  
\[f\left(\overset{\rightarrow}{r},\overset{\rightarrow}{v}\right)=f\left( \overset{\rightarrow}{r},v,\theta\right) \tag{1}\]  
where \(v=\left|\overset{\rightarrow}{v}\right|\), and \(\theta\) is the polar angle. The current to a flat Langmuir probe was calculated as  
\[I=qS\int v_{n}f\left(\overset{\rightarrow}{v}\right)d\overset{\rightarrow}{ v}=\frac{2 qS}{m^{2}}\int_{0}^{2\pi}d\phi^{\prime}\int_{qU}^{\infty}\epsilon d \varepsilon\int_{0}^{\theta^{\prime}_{max}}f\left(\varepsilon,\theta^{\prime },\phi^{\prime}\right)cos\theta^{\prime}sin\theta^{\prime}d\theta^{\prime} \tag{2}\]  
Figure 1: The device with a narrowed discharge channel: cathode (1), main anode (2), external control electrode (3), heater (4), alundum insulators (5), and conductive shield (6). where \(v_{n}\) is the component of the electron-velocity vector \(\overset{\rightarrow}{v}\), normal to the probe surface, \(U\) is the probe potential, \(\varepsilon=mv^{2}/2\), and \(\phi^{\prime}\) and \(\theta^{\prime}\) are the azimuthal and polar angles of the vector \(\overset{\rightarrow}{v}\), respectively. The second derivative of the probe current (2) with respect to the probe potential \(U\) is  
\[I_{U}^{\prime\prime}=\frac{q^{3}S}{m^{2}}\bigg{[}\int_{0}^{2\pi}f\big{(}qU, \theta^{\prime}=0,\phi^{\prime}\big{)}d\phi^{\prime}-\int_{0}^{2\pi}d\phi^{ \prime}\int_{qU}^{\infty}\frac{\partial}{\partial(qU)}f\big{(}\varepsilon, \theta^{\prime}_{max^{\prime}},\phi^{\prime}\big{)}d\varepsilon\bigg{]}. \tag{3}\]  
Expression (3) can be rewritten as  
\[I_{U}^{\prime\prime}(qU,\alpha)=\frac{2\pi q^{3}S}{m^{2}}\bigg{[}f(qU,\alpha)- \frac{1}{2\pi}\int_{0}^{2\pi}d\phi^{\prime}\int_{qU}^{\infty}\frac{\partial}{ \partial(qU)}f\big{(}\varepsilon,\theta\big{)}d\varepsilon\bigg{]}. \tag{4}\]  
To find the EVDF, we represent \(f(\varepsilon,\theta)\) and \(I_{U}^{\prime\prime}(qU,\alpha)\) in the form of a series in Legendre polynomials  
\[f(\varepsilon,\theta)=\sum_{j=0}^{\infty}f_{j}(\varepsilon)P_{j}(cos\theta), \tag{5}\]  
\[I_{U}^{\prime\prime}(qU,\alpha)=\frac{2\pi q^{3}S}{m^{2}}\sum_{j=0}^{\infty}F_ {j}(qU)P_{j}(cos\alpha). \tag{6}\]  
After substituting (5) and (6) into (4), we obtain the relationship between the components \(f_{j}\) and \(F_{j}\)  
\[f(qU)=F_{j}(qU)+\int_{qU}^{\infty}f_{j}(\varepsilon)\frac{\partial}{\partial( qU)}P_{j}\bigg{(}\sqrt{\frac{qU}{\varepsilon}}\bigg{)}d\varepsilon. \tag{7}\]  
Expression (7) is a Volterra integral equation of the second kind.