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Chapter 2
Thruster Principles
Electric thrusters propel the spacecraft using the same basic principle as chemical rockets—accelerating mass and ejecting it from the vehicle. The ejected mass from electric thrusters, however, is primarily in the form of energetic charged particles. This changes the performance of the propulsion system compared to other types of thrusters and modifies the conventional way of calculating some of the thruster parameters, such as specific impulse and efficiency. Electric thrusters provide higher exhaust velocities than is available from gas jets or chemical rockets, which either improves the available change in vehicle velocity (called v or delta-v) or increases the delivered spacecraft
and payload mass for a given v. Chemical rockets generally will have exhaust velocities of 3 to 4 km/s, while the exhaust velocity of electric thrusters can approach 10 2 km/s for heavy propellant such as xenon atoms, and 10 3 km/s for light propellants such as helium.
2.1 The Rocket Equation
The mass ejected to provide thrust to the spacecraft is the propellant, which is carried onboard the vehicle and expended during thrusting. From conservation of momentum, the ejected propellant mass times its velocity is equal to the spacecraft mass times its change in velocity. The “rocket equation” describing the relationship between the spacecraft velocity and the mass of the system is derived as follows. The force on a spacecraft, and thus the thrust on the vehicle, is equal to the mass of the spacecraft, M , times its change in velocity, v :
Force = T = M
dv dt
16 Chapter 2
The thrust on the spacecraft is equal and opposite to the time rate of change of the momentum of the propellant, which is the exhaust velocity of the propellant times the time rate of change of the propellant mass:
T =
d dt ( m^ p v ex) =^ v ex
dm (^) p dt
where m (^) p is the propellant mass on the spacecraft and v ex is the propellant
exhaust velocity in the spacecraft frame of reference.
The total mass of the spacecraft at any time is the delivered mass, md , plus the
propellant mass:
M ( t ) = m (^) d + m (^) p. (2.1-3)
The mass of the spacecraft changes due to consumption of the propellant, so the time rate of change of the total mass is
dM dt
dm (^) p dt
Substituting Eq. (2.1-4) into Eq. (2.1-2) and equating with Eq. (2.1-1) gives
M
dv dt
= v ex
dM dt
which can be written as
dv = v ex
dM M
For motion in a straight line, this equation is solved by integrating from the spacecraft initial velocity, vi , to the final velocity, v (^) f , during which the mass
changes from its initial value, md + m (^) p , to its final delivered mass, md :
dv vi
v (^) f =^ ^ v ex
dM m (^) d + m (^) pM
m (^) d .^ (2.1-7)
The solution to Eq. (2.1-7) is
18 Chapter 2
Modern ion and Hall thrusters operating on xenon propellant have exhaust velocities in the range of 20–40 km/s and 10–20 km/s, respectively.
The dramatic benefits of the high exhaust velocities of electric thrusters are clearly seen from Eq. (2.1-11). For example, consider an asteroid rendezvous mission for which it is desired to deliver 500 kg of payload with a mission v of 5 km/s. A spacecraft propelled by a chemical engine with a 3-km/s exhaust velocity, corresponding to an Isp of 306 s, would require 2147 kg of propellant to accomplish the mission. In contrast, an ion thruster with a 30-km/s exhaust velocity, corresponding to an Isp of 3060 s, would accomplish the same mission using only 91 kg of propellant. High- v missions such as this are often enabled by electric propulsion, allowing either a significant reduction in the amount of required propellant that has to be launched or the ability to increase the spacecraft dry mass for a given wet mass associated with a launch vehicle or mission requirement.
2.2 Force Transfer in Ion and Hall Thrusters
The propellant ionized in ion and Hall thrusters is accelerated by the application of electric fields. However, the mechanism for transferring the thrust from the ion motion to the thruster body, and thereby the spacecraft, is different for ion thrusters and Hall thrusters.
In ion thrusters, ions are produced by a plasma source and accelerated electrostaticly by the field applied between two (or more) grids, as illustrated in Fig. 2-1. The voltage applied between the two grids creates a vacuum electric field between the grids of the voltage divided by the gap d. The ions represent additional charge in the gap between the grids that modifies the electric field. Assuming infinitely large grids, the electric field distribution between the grids can be found from the one-dimensional Poisson’s Equation:
dE ( x ) dx
( x ) o
qni ( x ) o
where o is the permittivity of free space, is the ion charge density in the
gap, q is the charge on an ion, and ni is the ion number density in the gap.
Equation (2.2-1) can be integrated from the screen grid to the accel grid to give
(^) (^) Fig. 2-1. Schematic of ion thruster acceleration region.
Thruster Principles 19
E ( x ) =
q o
ni ( x ) d x 0
x
+^ E screen^ ,^ (2.2-2)
where E screen is the electric field at the screen grid. Assuming that the screen
grid is a perfect conductor, its surface charge density, , is
= o E screen. (2.2-3)
The surface charge is an image charge and is attracted by the ion charge in the gap. Since the field drops to zero inside the conductor, the screen grid feels a force per unit area equal to the charge density times the average field (which is half the field on the outside of the conductor):
F screen =
( E screen +^0 )
o E screen^2 , (2.2-4)
where F screen is the force on the screen grid. Correspondingly, at the
accelerator grid there is an electric field, E accel , and a surface charge density
equal to that on the screen grid but of the opposite sign. The accel grid feels a force is in the opposite direction:
F accel =
( E accel +^0 )
o E accel^2. (2.2-5)
The net thrust on the ion engine is the sum of the forces on the screen and accel grids,
T = F screen + F accel =
o ( E screen^2 E accel^2 ) , (2.2-6)
where T is the force in newtons. The force per unit area on the ions in the gap between the grids can be calculated using the fact that the force on an ion equals its charge times the local electric field, and integrating that force across the gap:
F ion = q ni ( x ) E ( x ) dx 0
d
.^ (2.2-7)
Eliminating the ion density ni ( x ) using Eq. (2.2-1), the integral can be done
directly:
Thruster Principles 21
Using quasi-neutrality and the definition of the Hall current density, J Hall = – ene v e , the force on the ions is shown to be equal to Lorentz forces on
the electrons:
F i = 2 (^) q ni E rdr dz + 2 (^) J Hall B rdr dz = 0. (2.2-13)
Solving Eq. (2.2-13), the force on the ions is then
F i = J Hall B (2.2-14)
By Newton’s second law, the Hall current force on the magnets is equal and opposite to the Hall current force on the electrons and, therefore, is also equal and opposite to the force on the ions:
T = J Hall B = F i. (2.2-15)
In Hall thrusters the thrust is transferred from the ions to the thruster body through the electromagnetic Lorentz force. These thrusters are sometimes called electromagnetic thrusters because the force is transferred through the magnetic field. However, since the ion acceleration mechanism is by the electrostatic field, we will choose to call them electrostatic thrusters.
2.3 Thrust
Thrust is the force supplied by the engine to the spacecraft. Since the spacecraft mass changes with time due to the propellant consumption, the thrust is given by the time rate of change of the momentum, which can be written as
T =
d dt
( m^ p v ex) =^
dm (^) p dt
v ex = m (^) p v ex (2.3-1)
where m ˙ (^) p is the propellant mass flow rate in kg/s. The propellant mass flow
rate is
m^ (^) p = QM , (2.3-2)
where Q is the propellant particle flow rate (in particles/s) and M is the particle mass.
The kinetic thrust power of the beam, called the jet power, is defined as
P jet =
m (^) p v ex^2. (2.3-3)
22 Chapter 2
Using Eq. (2.3-1), the jet power is then
P jet =
T^2
2 m (^) p
This expression shows that techniques that increase the thrust without increasing the propellant flow rate will result in an increase in the jet power.
For ion and Hall thrusters, ions are accelerated to high exhaust velocity using an electrical power source. The velocity of the ions greatly exceeds that of any unionized propellant that may escape from the thruster, so the thrust can be described as
T =
dm (^) p dt
v ex m i vi , (2.3-5)
where m ˙ i is the ion mass flow rate and vi is the ion velocity. By conservation
of energy, the ion exhaust velocity is given by
vi =
2 qVb M
where Vb is the net voltage through which the ion was accelerated, q is the
charge, and M is the ion mass. The mass flow rate of ions is related to the ion beam current, I (^) b , by
m^ ˙ i = I^ b^ M q
Substituting Eqs. (2.3-6) and (2.3-7) into Eq. (2.3-5), the thrust for a singly charged propellant ( q = e ) is
T = 2 M
e
I (^) b Vb [newtons]. (2.3-8)
The thrust is proportional to the beam current times the square root of the acceleration voltage. In the case of Hall thrusters, there is a spread in beam energies produced in the thruster, and Vb represents the effective or average
beam voltage. If the propellant is xenon, 2 M e = 1.65 10 ^3 , the thrust is
given by
24 Chapter 2
where I +^ is the singly charged ion current and I ++^ is the doubly charged ion current, the total thrust for the multiple species, Tm , is the sum of the thrust
from each species:
Tm = I +^
2 MVb e
+ I ++^
MVb e
= I +^
2 MVb e
I ++
I +
.^ (2.3-13)
The thrust correction factor, , for thrust in the presence of doubly ionized atoms is defined by the ratio of Eqs. (2.3-13) and (2.3-8), where the beam current in Eq. (2.3-8) is given by Eq. (2.3-12):
I +^ +
I ++
I +^ + I ++^
I ++
I +
I ++
I +
where I ++^ / I +^ is the fraction of double ion current in the beam. A similar correction factor can be easily derived for higher charged ions (see Problem 4), although the number of these species is typically found to be relatively small in most ion and Hall thrusters.
The total thrust correction is the product of the divergence and multiply charged species terms:
= Ft. (2.3-15)
The total corrected thrust is then given by
T = m i vi =
2 M
e
I (^) b Vb. (2.3-16)
The total thrust for xenon can be simply written as
T = 1.65 I (^) b Vb [mN]. (2.3-17)
For example, assuming an ion thruster with a 10-deg half-angle beam divergence and a 10% doubles-to-singles ratio results in = 0.958. For a
thruster producing 2 A of xenon ions at 1500 V, the thrust produced is 122.4 mN.
Thruster Principles 25
2.4 Specific Impulse
Specific impulse, termed Isp, is a measure of thrust efficiency and is defined as the ratio of the thrust to the rate of propellant consumption. Specific impulse for constant thrust and propellant flow rate is
Isp =
T
m (^) p g
where g is the acceleration of gravity, 9.807 m/s 2. For a xenon thruster, the Isp can be expressed as
Isp = 1.037 10 6
T [N]
Q [sccm]
T [N]
Q [mg/ s]
where Eq. (2.3-2) and the flow conversions in Appendix B have been used.
Using Eq. (2.3-1) for the thrust in Eq. (2.4-1), the Isp for any thruster is
Isp =
v ex g
where v ex is the effective exhaust velocity.
Defining the Isp in terms of the exhaust velocity relative to g is what gives rise to the unusual units of seconds for Isp. In electric thrusters, the thrust is due primarily to the ions. Using Eq. (2.3-5), the Isp is given by
Isp =
vi g
m i m (^) p
where vi is the exhaust velocity for unidirectional, monoenergetic ion exhaust.
The thruster mass utilization efficiency, which accounts for the ionized versus unionized propellant, is defined for singly charged ions as
m =
m i m (^) p
I (^) b e
M
m (^) p
In the event that the thruster produces a significant number of multiply charged ions, the expression for the propellant utilization efficiency must be redefined.
Thruster Principles 27
Using our previous example of a 10-deg half-angle beam divergence and a 10% doubles-to-singles ratio with a 90% propellant utilization of xenon [in Eq. (2.4-5)] at 1500 V, the Isp is 123.60.9580.9* 1500 = 4127 s.
Specific impulse is functionally equivalent to gas mileage in a car. Cars with high gas mileage typically don’t provide much acceleration, just as thrusters with high Isp don’t provide as much thrust for a given input electrical power. Of critical importance is the ratio of the thrust achieved to total power used, which depends on the electrical efficiency of the thruster (to be described in the next section).
2.5 Thruster Efficiency
The mass utilization efficiency, defined in Eq. (2.4-6), describes the fraction of the input propellant mass that is converted into ions and accelerated in the electric thruster. The electrical efficiency of the thruster is defined as the beam power, Pb , out of the thruster divided by the total input power, PT :
e =
Pb PT
I (^) bVb I (^) bVb + Po
where Po represents the other power input to the thruster required to create the
thrust beam. Other power will include the electrical cost of producing the ions, cathode heater or keeper power, grid currents in ion thrusters, etc.
The cost of producing ions is described by an ion production efficiency term, sometimes called the discharge loss:
d =
Power to produce the ions Current of ions produced
Pd I (^) b
where d has units of watts per ampere (W/A) or equivalently electron-volts
per ion (eV/ion). Contrary to most efficiency terms, it is desirable to have d
as small as possible since this represents a power loss. For example, if an ion thruster requires a 20-A, 25-V discharge to produce 2 A of ions in the beam, the discharge loss is then 20*25/2 = 250 eV/ion.
The performance of a plasma generator is usually characterized by plotting the discharge loss versus the propellant utilization efficiency. An example of this is shown in Fig. 2-3. At low propellant efficiencies, the neutral pressure in the thruster is high and the performance curves are relatively flat. As the propellant efficiency is increased, the neutral pressure in the thruster decreases, the
28 Chapter 2
!!"
# $ % &
'()*
+, - # $ % !,
Fig. 2-3. Ion thruster performance curves consisting of discharge loss versus propellant utilization efficiency.
electron temperature increases, and the loss mechanisms in the thruster become larger. Thrusters are normally operated near the knee of this curve such that high mass utilization efficiency is achieved without excessive discharge loss. Optimized thruster designs result in lower discharge losses and low loss at high propellant efficiency.
The total efficiency of an electrically powered thruster is defined as the jet power divided by the total electrical power into the thruster:
T =
P jet P in
Using Eq. (2.3-4) for the jet power, the efficiency of any electric propulsion thruster is
T =
T^2
2 m (^) p P in
Measurements made of the thruster’s input electrical power, input mass flow rate, and thrust output (measured in the vacuum system by a thrust stand) during testing can be used to calculate the total efficiency of the thruster using Eq. (2.5-4). This is the preferred technique for determining the efficiency of Hall thrusters because the beam parameters (current and velocity) are not
30 Chapter 2
Thrusters with high exhaust velocities, and thus high Isp’s, are desirable to maximize a mission payload mass. It was shown in Eq. (2.4-9) that to achieve high Isp, it is necessary to operate at a high ion acceleration voltage and high mass utilization efficiency. Reductions in ion mass also increase the Isp, but at the cost of thrust at the same power level. This is seen by examining the thrust- to-total input power ratio. The total power is just the beam power divided by the electrical efficiency, so the thrust-to-power ratio using Eq. (2.5.1) is
T PT
T e Pb
The beam power is the beam current times the beam voltage. Using Eq. (2.3-16) for the thrust and Eq. (2.4-8) to put this in terms of Isp, the thrust per unit input power is
T PT
2 2 m e g Isp
g
T
Isp
Equation (2.5-9) shows that for a given input power and total thruster efficiency, increasing the Isp reduces the thrust available from the electric engine. This trade of thrust for Isp at a constant input power can only be improved if higher efficiency ion thrusters are employed.
2.6 Power Dissipation
The power into a thruster that does not result in thrust must be dissipated primarily by radiating the unused power into space. If the thruster electrical efficiency is accurately known, the dissipated power is
P dissipated = P in ( 1 e ). (2.6-1)
If the electrical efficiency is not well known, alternative techniques can be used to determine the dissipated power. For example, in an ion thruster, the power in the beam is well known, and a simple difference between the total input power and the beam power represents the dissipated power. The various input powers can be measured externally to the thruster on the power supplies. For example, assuming the heaters have been turned off and the hollow cathodes are self- heating, the power into the ion thruster is given by
P in = I (^) bVb + I (^) d Vd + I ck V ck + I nk V nk +I (^) A 1 ( V (^) b + Va ) + I (^) A 2 Va + I DE1 Vb + I DE 2 VG ,
Thruster Principles 31
where the subscript “ b ” represents the beam current and voltage, “ d ” is the discharge current and voltage, “ck” is the cathode keeper current and voltage, “nk” is the neutralizer keeper current and voltage, “ A 1” represents beam ions incident on the accel grid, “ A 2” represents charge exchange ions at the accel grid potential Va , “ I DE1 ” represents the decel grid (if present) current from
beam ions, and “ I DE 2 ” represents the decel grid current from backstreaming
ions from the beam plume. In reality, the accel and decel grid power are very small compared to the other power levels in the thruster.
The power that must be dissipated by the thruster is Eq. (2.6.2) minus the beam power:
P in = I (^) d Vd + I ck V ck + I nk V nk + I (^) A 1 ( V (^) b + Va ) +I (^) A 2 Va + I DE1 Vb + I DE 2 VG.
Using the same ion thruster example used previously in this chapter, producing a 2-A beam at 1500 V as an example, Table 2-1 shows some example electrical parameters for a generic ion thruster. Assuming 10% of the grid currents are due to direct interception, using the table parameters in Eq. (2.5-12) gives a dissipated power of 528.3 W. Since the discharge power in this example is
Table 2-1. Example of ion thruster parameters used for power dissipation calculation. Parameter Term Nominal Discharge voltage (^) Vd 25
Discharge current (A) (^) I (^) d 20
Beam voltage (^) VB 1500
Beam current (A) (^) I (^) B 2
Discharge keeper voltage (^) V ck 10
Discharge keeper current (A) (^) I ck 1
Neutralizer keeper voltage (^) V nk 10
Neutralizer keeper current (A) (^) I nk 1
Accel current (mA) I (^) A 20 Accel voltage (^) VA 250
Decel current (mA) (^) I DE 2
Coupling voltage (^) VG 20
Thruster Principles 33
n =
PT [torr] * 133_._ 32 [pascal/ torr] 1_._ 38 10 ^23 [J / K]* T [K]
= 9_._ 66 10 24 *
PT
T
particles m^3
where PT is the pressure in the vacuum system in torrs and T is the gas
temperature in kelvins. It should be noted that the pressure must be corrected for the gas type in whatever measurement system is used to obtain the actual pressure data. As an example, for a pressure of 10 –6^ torr and a temperature of 290 K, the density of gas atoms is 3.3 10 16 per cubic meter.
The pressure in a vacuum system [3] in which a thruster is being tested is determined by the gas flow rate and the pumping speed
P =
Q
S
where Q is the total propellant throughput and S is the pumping speed. The most common units for pumping speed are liters per second, so utilizing a throughput in torr-l/s directly provides the pressure in the vacuum system in torr. The conversions of different flow units to torr-l/s can be obtained from Appendix B.
The finite pressure in the test vacuum system causes a backflow of neutral gas into the thruster that may artificially improve the performance. This ingestion of facility gas by the thruster can be calculated if the pressure in the chamber is known by evaluating the flux of neutral gas from the chamber into the thruster ionization region. The equivalent flow into the thruster is then the injected flow Q plus the equivalent ingested flow. The ingested flow (in particles per second) is given by
Q ingested =
nc 4
A* c , (2.7-4)
where n is the neutral density in the chamber, c is the gas thermal velocity, A is the total open area fraction of the thruster to the vacuum system, and c is a
correction factor related to the conductance into the thruster from the vacuum system. The neutral gas density is given by Eq. (2.7-2), and the gas thermal velocity is given by
c =
8 kT M
34 Chapter 2
where M is the atom mass in kg. The conductance correction factor is sometimes called the Clausing factor [4] and describes the conductance reduction due to the finite axial length of the effective entrance aperture(s) to the thruster. This factor is generally negligible for Hall thrusters but appreciable for the apertured grids of ion thrusters. Due to the large diameter-to-length ratio of the accelerator grid apertures in ion thrusters, the Clausing factor is usually calculated by Monte-Carlo gas flow codes. An example of a simple spreadsheet Monte-Carlo code for calculating the Clausing factor for ion thruster grids is given in Appendix G.
The ingested flow of gas from the finite pressure in the vacuum system is then
Q ingested =
133.2 P
4 k T
8 k T M
A c 4.479 1017
[sccm]. (2.7-6)
This expression for the ingested flow can be rewritten as
Q ingested = 7.82 10 8
P A c T M (^) a
[sccm] , (2.7-7)
where P is the vacuum chamber pressure in torr, T is the backflowing neutral gas temperature in K, M (^) a is the gas mass in AMU, and A is the open area in
m^2. The total flow rate into the thruster is then
Q total = Q injected + Q ingested. (2.7-8)
References
[1] J. M. Lafferty, Foundations of Vacuum Science and Technology , New York: John Wiley and Sons, 1998.
[2] A. Ross, Vacuum Technology , Amsterdam, Holland: Elsevier, 1990.
[3] G. Lewin, Fundamentals of Vacuum Science and Technology , New York: McGraw-Hill, 1965.
[4] P. Clausing, “The Flow of Highly Rarefied Gases Through Tubes of Arbitrary Length,” Journal of Vacuum Science and Technology , vol. 8, pp. 636–646, 1971.
