**2. The Tsiolkovsky's equation**

The rocket equation is used in propulsion systems to find out the different parameters. Therefore high specific impulse related to better efficiency for a propellant. If we denote Δ*υ* ! ¼ *υ* ! *<sup>f</sup>* � *υ* ! *<sup>i</sup>* as the change of velocity of the rocket, then, the rocket's equation was derived by scientist Tsiolkovsky (1857–1935) and given as

$$\frac{m\_f}{m\_i} = e^{\frac{-\Delta \vec{v}}{k^{l\_p}}} \tag{1}$$

Here *m <sup>f</sup>* is the final mass and *mi* is the initial mass of the rocket respectively. Taking natural logarithm on both sides, we get

$$
\Delta \overrightarrow{v} = \mathcal{g} I\_{sp} \ln \left( \frac{m\_i}{m\_f} \right) \tag{2}
$$

With this relation, the change in velocity of the rocket can be found out in terms of specific impulse or force. This equation is called Tsiolkovsky's equation. In term of exhaust velocity it turns out to be

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

Here, the mass of propellant *mp* ¼ *mi* � *m <sup>f</sup>* and *m <sup>f</sup>* is the dry mass of the rocket. It can be seen from Eq. (3) that the higher *dυ* ! requires more propellant. Therefore to achieve higher *dυ* !, the exhaust velocity *U* ! *ex* of the propellant needs to be of the order of *dυ* !. To achieve higher Δ*υ* !, the electric propulsion play a key role in the current time. Various space mission including GEO communication satellite requires a ΔV of approximately 0.6 km/s for a 10-year period.

#### **2.1 Relation between thrust efficiency and input power**

If we denote the Thrust efficiency *η* and the input power *Pt* then these are related by

$$T = \frac{2\eta P\_t}{I\_{\text{pp}}\text{g}}\tag{4}$$

### **3. Main classes of electric thrusters**

Many types of plasma thrusters have been developed over the last 70 years. Mitsubishi electric corporation developed Kaufman ion thrusters for the Japanese


*Introduction to Plasma Based Propulsion System: Hall Thrusters DOI: http://dx.doi.org/10.5772/intechopen.96916*

#### **Table 1.**

*Classification of some electric thrusters.*

engineering test satellite in 1994, which had produced 20 mN of thrust (specific impulse of about 2400 s) [2, 3]. Another ion thruster (for commercial station keeping Applications) called Hughes- 13-cm Xenon Ion Propulsion System was launched into orbit in 1997 on the Hughes PAS-5 satellite [4]. The Hughes thrusters produced 18 mN of thrust at specific impulse of 2500 s (efficiency of about 50%).

Based on the acceleration of gases for propulsion, electrical thrusters have been classified into three main categories namely electro thermal, electrostatic and electromagnetic thrusters. In chemical thruster, the exhaust velocity depends to thermal heating, which cannot reach very high magnitude. In a chemical thruster, the propellant is burned and the hot gas is expelled from the thruster with the help of a nozzle but in plasma thrusters the plasma expels without an explosion taking place [2, 3, 5–18]. The performances of different types of electric thrusters have been discussed in **Table 1**.

### **4. Electrostatic Hall thrusters**

In Electrostatic thrusters only ions are accelerated by applying direct electric field at the exit side of the thruster to produce thrust. Hall thrusters were originally invented in United States and Russia 70 years ago. After that they have been widely researched in Europe, Japan, and the China. Hall thrusters have emerged as an integral part of propulsion technology. Unlike chemicals and electric rockets (solid rocket motors, liquid rocket engines and hypergolic engines), the propulsion thrust in a Hall thruster is achieved by a propellant (usually Xenon). Typical chemical thruster specific impulses range around 200–500 sec, though electric thrusters can have specific impulses up to 3000 sec or greater [1, 5–7, 19]. The pressure inside the channel is on the order of 0.1 Pa. Now a days, most of the countries are using the

Hall thruster technology in their space mission. Unlike chemicals and electric rockets, the propulsive thrust in a Hall thruster is achieved by an ionized inert gas (Xenon) which has high atomic number and low ionization potential. For this Xenon is mostly used. In a Hall thruster, the propellant is ionized and then accelerated by electrostatic forces.

**Figure 1** shows the internal parts of a plasma Hall thruster. Generally, the discharge channel is cylindrical shape made up with metallic material. The magnetic field of the order of 150 Gauss is applied to produce closed drift of electrons inside the channel. The applied magnetic field, which is strong enough so that the electrons get magnetized, i.e. they are able to gyrate within the discharge channel, but the ions remain unaffected due to their Larmor radius much larger than the dimension of the thruster. The magnetic structure of a conventional HET is constituted of a magnetic circuit with two pole pieces, cores and two magnetic screens, one internal coil and four external coils to achieve a maximum of radial magnetic field in the channel exit.

Thus the electrons remain effectively trapped in azimuthally *E* ! � *B* ! drifts around the annular channel and slowly diffuse towards the anode. This azimuthal drift current of the electrons is referred to as the Hall current. The propellant enters from the left side of the channel via anode and gets ionized through hollow cathode of the device. Hall thrusters can be classified into two categories. One of them is a stationary plasma thruster (has an extended acceleration zone) and second is a thruster with anode layer (has a more narrow acceleration zone). The electric field of strength �1000 V/m gets generated inside the discharge channel along the axial direction of the device [5]. ISRO (India) used Hall effect ion propulsion thrusters in GSAT-4 back in 2010, carried by GSLV Mk2 D3. It had four Xenon powered thrusters for North– South station keeping. Two of them were Russian and the other two were indigenous.
