**1. Introduction**

In the past few years, electric propulsion has received widespread attention as an alternative to chemical propulsion for spacecraft. A chemical thruster is capable of producing high thrust, but it provides a relatively low specific impulse. This limitation of chemical thrusters opens up discussion of the use of electric thrusters, which provide much higher specific impulses. A high specific impulse allows the spacecraft to reach the same speed with lower propellant consumption than chemical ones or travel faster with the same propellant mass [1–10]. In addition, the use of a high specific impulse, electric thruster permits a significant reduction in propellant mass on spacecraft, which helps reduce launching cost, extending mission lifetime or increase the payload mass on the spacecraft [2]. Therefore, the benefits that electric thrusters provide make them the best choice for in-space propulsion of spacecraft.

To date, several different types of electric propulsion thrusters have been developed, of which Hall thrusters are the most reliable and widely used. Hall thrusters are simple in design, consisting of an annular discharge channel, an anode, a cathode and a radial magnetic field across the channel [1]. A propellant, usually Xenon, is injected to channel through the hollow anode and a high potential difference is

applied between the anode and cathode. The electrons emitted from the cathode and start moving towards the anode due to the potential difference. As they enter the discharge channel, they get trapped in the radial magnetic field generated by electromagnetic coils (or permanent magnets) and start drifting in the azimuthal direction (E x B). Due to this trapping, the residence time of electrons in the discharge channel increases and they move very slowly towards the anode. Electrons, then collide with neutral propellant atoms entering the discharge channel and ionize them. Then, these generated ions are accelerated to high velocity towards the thruster exhaust by the electric field to generate thrust. On the thruster exterior, the ion beam is neutralized by electrons from the virtual cathode so that no charge builds up on the spacecraft's surface [2]. More details on Hall thruster operation and fundamentals can be found in [1–13].

Hall thrusters offer several benefits like simple design, high thrust-to-power ratio, high efficiency, improved performance, etc., which give them a clear advantage over other electric thrusters. For an input power range of 0.1 kW–20 kW, they can produce a few mN to 1 N of thrust and offers a specific impulse in the range of 1000s–3000s with more than 50% efficiency [14]. Hall thrusters can also adjust their thrust level and specific impulse by varying the discharge voltage and propellant mass flow rate, which makes them suitable for applications such as precision maneuvering, attitude control, station keeping and orbital raising [15]. In addition, plasma in a Hall thruster remains quasi-neutral, which eliminates the issue of space charge, allowing the Hall thruster to achieve higher thrust densities. Because of all these characteristics, the space community has shown great interest in Hall thrusters and they have been used successfully on many spacecraft for space missions and maneuvers.

Thrust is caused by a change in a substance's momentum as a result of a chemical reaction or an electrical principle. The thrust indicates how much force, in newtons (N), the propulsion system exert on the vehicle. Let us denote *m*\_ *<sup>p</sup>* as the mass flow rate, the exhaust velocity *U* ! *ex* and *g* is the acceleration due to gravity, then spacecraft's thrust denoted by

$$T = \dot{m}\_p \overrightarrow{U}\_{\text{ex}} \tag{1}$$

The performance of thrusters is usually determined by thrust *T*, which is the total force undergone by the rocket. Thrust also has same unit as a force in newton, which shows the movement of the propulsion system. Thrust, is generated by the burning of fuel or by the electrostatic forces. The thrust*T* ¼ *m*\_ *pU* ! *ex*, if the mass flow rate is constant. The specific impulse *Isp* is used to compare the efficiencies of different type of propulsion systems [2]. The specific impulse is expressed as*Isp* <sup>¼</sup> *<sup>T</sup> m*\_ *pg* . In general, the higher the specific impulse the less fuel is required. Therefore the specific impulse simplifies to *Isp* <sup>¼</sup> *<sup>U</sup>* ! *ex <sup>g</sup>* . The specific impulse has the dimension of time and is ameasure for the effective mission time of the thruster. The high value of the specific impulse reduces the mission time. If we denote the thrust efficiency *η* and the input power *Pt*, then these are related by

$$T = \frac{2\eta P\_t}{I\_{\eta \mathbf{g}} \mathbf{g}} \tag{2}$$

Tsiolkovsky rocket Eq. (1) can be read as

$$
\Delta \overrightarrow{\nu} = \overrightarrow{\nu}\_f - \overrightarrow{\nu}\_i = \overrightarrow{U}\_{\text{ex}} \ln \left( \frac{m\_f}{m\_f + m\_p} \right) \tag{3}
$$

*Numerical Investigations of Electromagnetic Oscillations and Turbulences in Hall… DOI: http://dx.doi.org/10.5772/intechopen.99883*

The above Rocket equation defines the change in velocity of a spacecraft. It is clear that a higher *dυ* ! demands more propellant. In terms of specific impulse, the above equation simplifies to

$$
\Delta \overrightarrow{\nu} = \overrightarrow{\nu}\_f - \overrightarrow{\nu}\_i = I\_{\text{sp}} \text{lg} \ln \left( \frac{m\_f}{m\_f + m\_p} \right) \tag{4}
$$

We can now say that Isp plays a key role in the design of a space mission propulsion system.
