SCHOOL
OF
ENGINEERING
DEPARTMENT OF CIVIL AND INDUSTRIAL ENGINEERING
Master of Science in Aerospace Engineering
Master Thesis
PIEZOELECTRIC
ACTUATORS’
EFFECTS
ON AN ACCELERATOR
GRID
OF GRIDDED ION
THRUSTER
Supervisor
Chia.mo Prof. Mario Rosario Chiarelli
Candidat
e
Guglielmo Neri
Abstract
The growing demand for space missions, in particular missions requiring the introduction into orbit around the earth of satellites such as Earth observation and GPS applications, is encouraging the development of increasingly reliable and low-cost systems. Every satellite placed in orbit, whatever the specific mission it has to perform, needs to be equipped with reliable control systems. In this sense gridded ion thruster is one of the most reliable propulsion system with higher operational efficiency. In any spacecraft installation of an ion propulsion system it is likely that there will be a need to alter the position of the thrust vector with respect to the centre of the vehicle, in order to minimize attitude and orbita perturbations during operation. Of most importance is the need to correct for the movements of the centre of mass of the spacecraft during operation. These movements are caused by the consumption of propellant, by the deployment and rotation of solar arrays, and by the varying radiation flux from the sun, hence several reasons justify the development of an ion propulsion system thrust vectoring system. Spacecrafts launched to date have used ion thrusters mounted on gimbals to control the thrust vector within a range of about ±5°. Such devices have large mass and dimensions, hence the need exists for a more compact system, preferably mounted within the thruster itself and to reduce overall thrust vectoring system mass. In the past few years, several studies have been conducted on translating the accelerator Grid in the plane perpendicular to the thrust axis with coefficient of 0.003-0.004 mm per degree of ion beam deflection and is found that this can be a best solution to obtain thrust vectoring in the range of ±8°. Use of micro electromechanical system, such as shape memory alloy actuators and piezoelectric actuators are promising. This thesis aims to study the real efficiency of the thrust vectoring system using piezoelectric actuators that move the accelerator grid and analyzing the effects that these movements have on the grid itself. The analysis is carried out using a FEM analysis software, first by analyzing only one piezoelectric actuator and then analyzing the complete system composed by the grid and the piezoelectric actuators, focusing on mechanical and thermal responde of the grid.
List of Figures i List of Figures . . . v List of Tables v List of Tables . . . vi Introduction vi 1 Piezoelctric Materials 1 1.1 Piezoelectric properties . . . 2 1.2 Governing equation . . . 10 1.3 Equation of motion . . . 15
2 Gridded Ion Thruster 17 2.1 Principles of operation . . . 19
2.2 Thrust vectoring system . . . 25
3 ACC Grid 27 3.1 ACC Grid material and properites . . . 31
3.2 ACC Grid initial geometry . . . 33
3.3 Research of equivalent material for FEM analysis . . . 35
3.4 ACC grid final geometry . . . 47
4 Piezoelectric Actuators 48 4.1 Smart Material Actuators . . . 51
4.2 Selection Criteria . . . 52
4.3 Piezoelectric Stack . . . 53
4.4 Piezoelectric Stack Analysis . . . 56
5 Complete system analysis 68 5.1 Model for FEM analysis . . . 68
5.2 Static Structural analysis . . . 71
5.4 Thermal analysis . . . 92
Conclusions 99
Bibliography 101
1.1 Direct Piezoelectric Effect . . . 1
1.2 Converse Piezoelectric Effect . . . 2
1.3 Poling . . . 3
1.4 Crystalline structure of a piezoelectric material (titanium zirconate lead) 3 1.5 Calculation of the input electrical and the output mechanical energy . . 7
2.1 Ion thruster schematic . . . 18
2.2 Illustration of a DC-discharge electron bombardment ion thruster. . . . 19
2.3 Electrical schematic of a DC-discharge ion thruster with the cathode heater, keeper and discharge power supplies. . . 19
2.4 Simplified schematic of ion thruster. . . 20
2.5 The two grid acceleration system. . . 21
2.6 Ion beam deflection using accelerator grid translation. . . 26
3.1 Non-planar sheath model approximation for a two-grid system. . . 29
3.2 Ion optics parameters. . . 29
3.3 Plasma beam divergence angle and beam current density profile. . . 30
3.4 ACC Gird 3D model. . . 33
3.5 ACC Gird 3D model thickness. . . 34
3.6 ACC Gird holes pattern. . . 34
3.7 Cut of the Grid 3D model. . . 35
3.8 Square-shaped plate 3D model. . . 36
3.9 Square-shaped plate without holes 3D model. . . 36
3.10 Tensile load applied on the model. . . 37
3.11 constrain applied on across the face that blocking the displacement in x. 37 3.12 constrain applied on an edge of the face that blocking the displacement in y and z. . . 38
3.13 Directional deformation along x axis. . . 38
3.14 Force reaction on constrain. . . 39
3.15 Force reaction on constrain for the plate model with holes. . . 40
3.16 Shear load applied on one side of the model. . . 41
3.18 constraints that block the y and z directions. . . 42
3.19 constraints that block the x and z directions. . . 42
3.20 Shear elastic strain. . . 43
3.21 Force reaction on one constrain. . . 43
3.22 Shear elastic strain. . . 44
3.23 Force reaction on one constrain. . . 44
3.24 Force reaction on constrain. . . 45
3.25 Force reaction on one constrain. . . 45
3.26 Directional deformation along x axis. . . 46
3.27 Shear elastic strain. . . 46
3.28 Equivalent ACC Gird 3D model. . . 47
4.1 Actuating Unit . . . 48
4.2 Piezo disk model . . . 54
4.3 Schematization of piezoelectric stack connected in parallel . . . 55
4.4 Radius of piezoelectric disk . . . 55
4.5 Thickness of piezoelectric disk . . . 55
4.6 Length of piezoelectric stack . . . 56
4.7 Properties of piezoelectric material . . . 56
4.8 Model material . . . 57
4.9 Bonded constraint . . . 57
4.10 Mesh of piezoelectric stack . . . 58
4.11 Fixed support . . . 58
4.12 Dielectric and electromechanical properties of the piezoelectric material 59 4.13 Dielectric and electromechanical properties of the piezoelectric material 59 4.14 Applied voltage . . . 60
4.15 Applied voltage . . . 60
4.16 Directional deformation of piezoelectric stack . . . 61
4.17 Own frequencies . . . 61
4.18 First mode of vibrating . . . 62
4.19 Second mode of vibrating . . . 62
4.20 Third mode of vibrating . . . 63
4.21 Fourth mode of vibrating . . . 63
4.22 Fifth mode of vibrating . . . 64
4.23 Sixth mode of vibrating . . . 64
4.24 Seventh mode of vibrating . . . 65
4.25 Participation factor summary . . . 65
4.26 Voltage Time dependence of voltage . . . 66
4.27 Deformation over time . . . 66
4.28 Total acceleraion . . . 67
5.4 Ring model . . . 70
5.5 Symmetric model . . . 70
5.6 T ZMeqproperties in Engineering Data . . . 71
5.7 T ZM properties in Engineering Data . . . 72
5.8 Zirconia properties in Engineering Data . . . 72
5.9 Geometry model for FEM analysis . . . 73
5.10 Material of the grid in FEM model . . . 73
5.11 Material of the Annulus in FEM model . . . 74
5.12 Material of other components in FEM model . . . 74
5.13 Material of Piezo stack in FEM model . . . 75
5.14 Bonded constrains in FEM model . . . 75
5.15 No Separation constrains in FEM model . . . 76
5.16 Reference System of the model . . . 76
5.17 Symmetry function . . . 77
5.18 Model mesh . . . 77
5.19 Fixed support constrain . . . 78
5.20 Dielectric and electromechanical properties of the piezoelectric material 78 5.21 Dielectric and electromechanical properties of the piezoelectric material 79 5.22 Applied voltage . . . 79
5.23 Applied voltage . . . 80
5.24 Pressure due to the ion flux . . . 80
5.25 Directional deformation along X axis . . . 81
5.26 Directional deformation along Z axis . . . 82
5.27 Equivalent von-Mises stress . . . 82
5.28 Normal stress along X axis . . . 83
5.29 Normal stress along Z axis . . . 83
5.30 Shear stress in XY plane . . . 84
5.31 Voltage . . . 84
5.32 Own frequencies . . . 86
5.33 Participation factor summary . . . 87
5.34 First mode of vibrating . . . 87
5.35 Second mode of vibrating . . . 88
5.36 Third mode of vibrating . . . 88
5.37 Fourth mode of vibrating . . . 89
5.38 Fifth mode of vibrating . . . 89
5.39 Sixth mode of vibrating . . . 90
5.40 Seventh mode of vibrating . . . 90
5.41 Eighth mode of vibrating . . . 91
5.42 Ninth mode of vibrating . . . 91
5.44 Temperature of the deep space . . . 93
5.45 Temperature of the ion flux on ACC Grid . . . 93
5.46 Convection parameter of the Grid . . . 94
5.47 Convection parameter of the holder . . . 94
5.48 Convection parameter of the ring . . . 95
5.49 Convection parameter of the case . . . 95
5.50 Convection parameter of the piezo stack . . . 96
5.51 Total Temperature . . . 97
5.52 Total heat flux . . . 97
1.1 Piezoelectric properties of representative piezoelectric materials. . . 9
3.1 TZM chemical composition. . . 32
3.2 TZM Physical properties. . . 32
3.3 TZM Mechanical properties. . . 32
3.4 ACC Grid geometrical data. . . 33
3.5 T ZMeqMechanical properties. . . 47
4.1 Type of Actuators and Their Features . . . 49
4.2 Type of Actuators and Their Features . . . 50
5.1 Results summary. . . 85
5.2 Results summary. . . 85
In the recent decades, a lot of research has been carried out on finding a new types of control systems for satellites that can be reliable, low-cost and low-weight. This thesis aims to describe, analyze and study the real efficiency of one possible type of control system, in particular the thrust vectoring system using piezoelectric stack actuators that move the accelerator grid of a gridded ion thruster. Initially, the piezoelectric materials are described both in terms of their chemical-physical properties and in terms of the constitutive and motion equations that govern them. In the same way the principle of working of the gridded ion thruster is described, paying particular attention to the importance of the acceleration grid (ACC grid) and what consequences it can have on the exhaust ion flux moving it even by a few micrometers. It also describes a particular analysis carried out on the grid used to create a model of the same more simplified, from a geometric point of view, to be used in the FEM analysis. The piezoelectric actuators are then described, focusing on the piezoelectric stacks and analyzing their behavior when subjected to pre-established potential difference. Finally, a representative model of the grid - piezoelectric stack system is described, on which static structural analysis, thermal analysis and modal analysis are carried out using a FEM analysis software.
Chapter 1
Piezoelctric Materials
Piezoelectricity is the property of certain materials (such as crystals, certain ceramics, and biological matter such as bone, DNA and various proteins) that accumulates elecrict charge in response to applied mechanical stress (direct piezoelectric effect) (Figure 1.1), or to deform in response to applied electric field (reverse piezoelectric effect) (Figure 1.2). The word piezoelectricity, derived from the Greek, means electricity resulting from pressure and latent heat.
The first scientific publication describing the phenomenon, later termed as piezoelectricity, appeared in 1880. It was co-authored by Pierre and Jacques Curie, who were conducting a variety of experiments on a range of crystals at the time. In those experiments, they cataloged a number of crystals, such as tourmaline, quartz, topaz, cane sugar and Rochelle salt that displayed surface charges when they were mechanically stressed. The discovery of the direct piezoelectric effect is, therefore, credited to the Curie brothers. They did not, however, discover the converse piezoelectric effect. Rather, it was mathematically predicted from fundamental laws of thermodynamics by Lippmann in 1881. Having said this, the Curies are recognized for experimental confirmation of the converse effect following Lippmann’s work.
Figure 1.2: Converse Piezoelectric Effect.
1.1
Piezoelectric properties
The piezoelectric effect is manifested in those materials that have a non-symmetrical crystalline structure. Stress (tensile or compressive) applied to these types of crystals alters the separation between the sites containing the positive and negative charges in each elementary cell, leading to a clear polarization on the outer surfaces of the crystal. The effect is practically linear, i.e. the induced polarization varies proportionally with the stress applied, and is also dependent on the direction; according to this principle, compressive or tensile stress generate electric fields, and therefore tensions of opposite polarity. Since the phenomenon is also reciprocal, if the crystal instead of being subjected to a force is exposed to an electric field, it will undergo an elastic deformation that causes an increase or a reduction in its length, in accordance with the polarity of the applied field. Piezoelectirc materials can be natural or man-made. The natural PEM are crystal like quartz (SiO2), Rochelle salt, Topaz, Tourmaline- group minerals and some organic substances. Man-made piezoelectric materials are crystals that are quartz analog, ceramics, polymers and composites. There are 32 crystal classes which are divided into the following seven groups:
triclinic,monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. These groups are also associated with the elastic nature of the material where triclinic represents an anisotropic material, orthorhombic represents an orthotropic material and cubic are in most cases isotropic materials. Only 20 of the 32 classes alow piezoelectric properties. Ten of these classes are polar, i.e. show a spontaneous polarization without mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell. The remaining 10 classes are not polar, i.e. polarization appears only after applying a mechanical load. Piezoelectric ceramics are inherently made up of micro-domains, i.e. from small areas, in which the electric dipole moments are oriented in the same way because of the mutual interactions of electrical type between the reticle ions, which tend to align in precise directions. Due to of the random orientation of domains within the material, the resulting polarity in a ceramic is nothing. To obtain piezoelectric properties is therefore an external electric field must be applied according to a process commonly
Piezoelctric Materials
referred to as poling. The poling process occours when the material is subjected, under a predetermined temperature, to a very high electric field that orients all the dipoles in the direction of the field, Upon switching off the electric field, most dipoles do not return back to their original orientation giving rise to a total net dipole (and therefore a non-zero polarity). It is noteworthy that the material can be de-poled if it is subjected to a very high electric field oriented opposite to the poling direction or is exposed to a temperature higher than the Curie temperature of the material. (Figure 1.3) A piezoelectric ceramic is a mass of perovskite crystals. Each crystal is composed of a small, tetravalent metal ion placed inside a lattice of larger divalent metal ions and O2, (Figure 1.4). Mechanical compression or tension on the element changes the dipole moment associated with that element. This creates a voltage. Compression along the direction of polarization, or tension perpendicular to the direction of polarization, generates voltage of the same polarity as the poling voltage. Tension along the direction of polarization, or compression perpendicular to that direction, generates a voltage with polarity opposite to that of the poling voltage.
Figure 1.3: Poling.
When operating in this mode, the device is being used as a sensor. That is, the ceramic element converts the mechanical energy of compression or tension into electrical energy. Values for compressive stress and the voltage (or field strength) generated by applying stress to a piezoelectric ceramic element are linearly proportional, up to a specific stress, which depends on the material properties. The same is true for applied voltage and generated strain. If a voltage of the same polarity as the poling voltage is applied to a ceramic element, in the direction of the poling voltage, the element will lengthen and its diameter will become smaller. If a voltage of polarity opposite to that of the poling voltage is applied, the element will become shorter and broader. If an alternating voltage is applied to the device, the element will expand and contract cyclically, at the frequency of the applied voltage. When operated in this mode, the piezoelectric ceramic is used as an actuator. That is, electrical energy is converted into mechanical energy.
There are five important figures of merit in piezoelectrics: the piezoelectric strain constant d, the piezoelectric voltage constant g, the electromechanical coupling factor k, the mechanical quality factor QM, and the acoustic impedance Z.
Piezoelectric Strain Constant d is the magnitude of the induced strain x by an external electric field E and it is represented by this figure of merit (an important figure of merit for actuator applications):
x = dE
Piezoelectric Voltage Constant g is the induced electric field E and is related to an external stress X through the piezoelectric voltage constant g (an important figure of merit for sensor applications):
E = gX
Taking into account the relation, P = dX, we obtain an important relation between g and d:
g = d 0
where is the permittivity and 0 is the vacuum permittivity.
Electromechanical Coupling Factor k can be, electromechanical coupling factor, energy transmission coefficient, and efficiency. All are related to the conversion rate between electrical energy and mechanical energy, but their definitions are different.
Piezoelctric Materials
• Electromechanical coupling factor k:
k2 = (Stored mechanical energy / Input electrical energy)
or
k2 = (Stored electrical energy / Input mechanical energy)
Let us calculate the first equation when an electric field E is applied to a piezoelectric material.
Since the input electrical energy is (1/2)0E2 per unit volume and the stored mechanical energy per unit volume under zero external stress is given by (1/2)x2/s = (1/2)(dE)2/s, k2 can be calculated as k2 = 1 2 (dE)2 s 1 20E2 = d 2 0s
• The energy transmission coefficient λmax:
Not all the stored energy can be actually used, and the actual work done depends on the mechanical load. With zero mechanical load or a complete clamp (no strain) zero output work is done.
λmax = (Output mechanical energy / Input electrical energy)max
or
λmax = (Output electrical energy / Input mechanical energy)max
Let us consider the case where an electric field E is applied to a piezoelectric under constant external stress X (< 0, because a compressive stress is necessary to work to the outside). As shown in Figure 1.5, the output work can be calculated as
Z
(−X)dx = −(dE + sX)X
while the input electrical energy is given by
Z
EdP = (0E + dX)E
We need to choose a proper load to maximize the energy transmission coefficient. From the maximum condition of
λ = −(dE + sX)X (0E + dX)E we can obtain λmax = " 1 k − r ( 1 k2 − 1) #2 = " 1 k + r 1 k2 − 1 #−2 Notice that k2 4 < λmax < k2 2
depending on the k value. For a small k, λmax = k2/4, and for a large k,
λmax = k2/2.
It is also worth noting that the maximum condition stated above does not agree with the condition which provides the maximum output mechanical energy. The maximum output energy can be obtained when the load is half of the maximum generative stress: − dE − sdE 2s −dE 2s = dE 2 4s
Piezoelctric Materials
In this case, since the input electrical energy is given by (0E + d(−dE/2s))
λ = 1
2k22 − 1
which is close to the value lambdamax, but has a different value that is predicted theoretically.
• The efficiency η:
η =(Output mechanical energy) / (Consumed electrical energy)
or
η =(Output electrical energy) / (Consumed mechanical energy)
In a work cycle (e.g. an electric field cycle), the input electrical energy is transformed partially into mechanical energy and the remaining is stored as electrical energy (electrostatic energy like a capacitor) in an actuator. In this way, the ineffective energy can be returned to the power source, leading to near 100% efficiency, if the loss is small. Typical values of dielectric loss in PZT are about 1 - 3%.
• Mechanical Quality Factor QM:
The mechanical quality factor, QM, is a parameter that characterizes the sharpness
of the electromechanical resonance spectrum. When the motional admittance Ym
is plotted around the resonance frequency ω0, the mechanical quality factor QM is defined with respect to the full width at Ym/p2[2∆ω]as:
QM =
ω0
2∆ω
Also note that QM−1 is equal to the mechanical loss (tanδm). The QM value is very important in evaluating the magnitude of the resonant strain. The vibration amplitude at an off-resonance frequency (dEL, L: length of the sample) is amplified by a factor proportional to QM at the resonance frequency. For a longitudinally vibration rectangular plate through d31, the maximum displacement is given by
Piezoelctric Materials
• Acoustic Impedance Z:
The acoustic impedance Z is a parameter used for evaluating the acoustic energy transfer between two materials. It is defined, in general, by
Z2 = (pressure/volume velocity)
In a solid material
Z =√ρc
where rho is the density and c is the elastic stiffness of the material.
Parameters Quartz BaTiO3 PZT 4 PST5H (Pb,Sm)TiO3 PVDF-TrFE
d33(pC/N) 2.3 190 289 593 65 33 g33(10−3Vm/N) 57.8 12.6 26.1 19.7 42 380 kt 0.09 0.38 0.51 0.50 0.50 0.30 kp 0.33 0.58 0.65 0.03 3T/0 5 1700 1300 3400 175 6 Qm > 105 500 65 900 3 − 10 TC(◦C) 120 328 193 355
1.2
Governing equation
For a general piezoelectric material, the total internal energy density U is given by the sum of the mechanical and electrical work done, i.e. in differential form it is
dU = σijdsij + EmdDm
Here themechanical stress σij and strain sij are second rank tensors, Emis the vector
of electric field, Dmis the vector of electrical displacement. All indices run from 1 to 3 and the summation convention over repeated indexes is implied. The polarization vector Piis introduced to quantify the degree of polarization of the material and it is connected
with the vectors of electric field and electrical displacement by the relation: Di = 0Ei+ Pi , Pi = χijEj
In order to derive the constitutive equations of a piezoelectric material different types of thermodynamic potentials can be used as e.g. internal energy U = U (sij, Di),
the electric Gibbs energy (electric enthalpy) Ge = Ge(sij, Ei), the Helmholtz free
energy F = F (σij, Di), the elastic Gibbs energy G1(σij, Pi) and the Gibbs free energy
G = G(σij, Ei). The different thermodynamic potentials will facilitate different sets of
piezoelectric constitutive formulations. Here the constitutive equation derived by using the Gibbs electrical function (electric enthalpy) Ge(sij, Ei) is presented, assuming it is
a quadratic form of sij, Ei . The Gibbs electrical function is a thermodynamic potential
in which the independent variables are the strain deformation sij and the electrical field
Ei, and the dependent flux variables are the stress σij and electric displacement (electric flux density) Di, i.e.
dGe = ∂Ge ∂sij e dsij + ∂Ge ∂Em s dEm
The differential form of Ge = U − EiDi is:
Piezoelctric Materials
Comparing equations yields:
σij = ∂Ge ∂sij e dsij Dm = ∂Ge ∂Em s dEm
Having in mind that σij = σij(sij, Em) and Di = Di(sij, Em), the differentials of
stress and electric displacement have the form: dσij = ∂σij ∂skl E dskl+ ∂σij ∂Em s dEm dDm = ∂Dm ∂skl E dskl+ ∂Dm ∂Ek s dEk
The physical meaning of the partial derivatives is as follows: • ∂σij ∂skl E
= Cijkl is the fourth rank tensor of the elastic stiffness constants at E = const with Cijkl= Cijlk= Cjikl = Cklij;
• ∂σij ∂Em s = − ∂Dm ∂skl
E = −eijm is the third rank tensor of the piezoelectric
constants at sij = constwithekij = ekji;
•
∂Dm
∂Ek
s= mk is the second rank tensor of the dielectric permittivity constants at
sij = const with ik = ki.
In the case of general anisotropy Cijkl , eijm, mk admit 21, 18 and 6 independent components, respectively. After integration of previous equations at constant partial derivatives the following constitutive equations are obtained:
σij = Cijklskl− eijmEm
The constitutive equations for piezoelectric materials show coupling between electrical and mechanical quantities. The direct piezoelectric effect or the sensorial effect is described by the second equation. This equation shows that an electric polarization and electric field is generated by mechanical stress. The converse effect, or the actuator effect is described by the first equation which shows that a piezoelectric material undergoes a deformation under an electric field. The strain-displacement and the electric field-potential relations are given by
sij =
1
2(uij+ uji) , Ei = −Φi
where ui is the mechanical displacement and Φ is the electrical potential.
The symmetry of the stress tensor enables nine stress components to be reduced to six independent stress components. This also enables the tensor notation to be transformed into a pseudo-tensor form. Using this so-called contracted Voigt subscript notation: the fourth order tensor Cijklreduces to the matrix representation Cαβ with (ij) that became α and (kl) that became β. In the same way the third order tensor ekij reduces to the matrix
representation ekαwith (ij) that became α. For the analysis of piezoelectric problems it is advantageous to use the notation introduced by Barnett and Lothe.With this notation, the elastic displacement and electric potential, the elastic strain and electric field, the stress and electric displacement, and the elastic and electric coefficients can be grouped as:
• Generalized displacements uI =
ui , I = 1, 2, 3
Φi , I = 4
• Generalized strain, for j = 1, 2, 3 sIj =
sij , I = 1, 2, 3 −Eji , I = 4
• Generalized stresses, f ori = 1, 2, 3, σiJ =
σij , J = 1, 2, 3
−Dj , J = 4
• Generalized stiffness matrix for i, j, k, l = 1, 2, 3,
CiJ Kl = Cijkl , J, K = 1, 2, 3 elij , J = 1, 2, 3; , K = 4, eikl , J = 4; K = 1, 2, 3, −il , J = K = 4.
Piezoelctric Materials
The symmetry properties of elastic, piezoelectric and dielectric tensors Cijkl, ekij, ij
imply the following symmetry property for the extended stiffness tensor: CiJ Kl= ClKJ i
In this definition, the lowercase and uppercase subscripts take the values of 1, 2, 3 and 1, 2, 3, 4, respectively. In terms of this shorthand notation, the constitutive relations can be unified into the one single equation
σiJ = CiJ KlsKl or in matrix notation σ11 σ22 σ33 σ23 σ31 σ12 σ14 σ24 σ34 = C s11 s22 s33 2s23 2s31 2s12 −E1 −E2 −E3 where C = c11 c12 c13 c14 c15 c16 e11 e21 e31 c12 c22 c23 c24 c25 c26 e12 e22 e32 c13 c23 c33 c34 c35 c36 e13 e23 e33 c14 c24 c34 c44 c45 c46 e14 e24 e34 c15 c25 c35 c45 c55 c56 e15 e25 e35 c16 c26 c36 c46 c56 c66 e16 e26 e36 e11 e12 e13 e14 e15 e16 −11 −12 e13 e21 e22 e23 e24 e25 e26 −12 −22 −23 e31 e32 e33 e34 e35 e36 −31 −32 −33
Piezoelectric materials show in most cases a crystal structure with a symmetry of hexagonal 6 mm class. In the case that the poling axis coincides with one of the material symmetry axes these materials become transversely isotropic. Transversely isotropic elastic materials are those with an axis of symmetry such that all directions perpendicular to this axis are equivalent. In other words, any plane perpendicular to the axis is a plane of isotropy. In the case of a transversely isotropic solid, the number of the independent elastic, piezoelectric and dielectric constants is 5, 3 and 2 respectively. In this case matrix C takes the form
C = c11 c12 c13 0 0 0 0 0 e31 c12 c22 c23 0 0 0 0 0 e32 c13 c23 c33 0 0 0 0 0 e33 0 0 0 c44 0 0 0 e24 0 0 0 0 0 c55 0 e15 0 0 0 0 0 0 0 c66 0 0 0 0 0 0 0 e15 0 −11 0 0 0 0 0 e24 0 0 0 −22 0 e31 e32 e33 0 0 0 0 0 −33 where c66= 12(c11− c12)
The elasticity coefficients Cijkland the dielectric constants ijare said to be positive-definite if
cijklqijqkl> 0, ilaial > 0
for any non-zero tensor qij and any non-zero vector ai and the following reciprocal symmetries hold due to consitutive equation
cijkl = cjikl = cklij, eijk= eikj, jk = kj
Essentially, these constraints are thermodynamic constraints expressing that the internal energy density must remain positive since this energy must be minimal in a state of equilibrium. Specializing for the case of transversely isotropic solids, one obtains:
Piezoelctric Materials
1.3
Equation of motion
The governing equations are given by the equations of motion for the mechanical displacement and by the equations of electrostatic. The electric field that develops in piezoelectrics can assumed to be quasi-static because the velocity of the elasticwaves is much smaller than the velocity of electromagnetic waves. Therefore, the magnetic field due to the elasticwaves is negligible. This fact implies that the time derivative of the magnetic field B is close to zero, i.e.
∂B ∂t ≈ 0
Thus one of Maxwell’s equations of electrodynamics becomes rot(E) = ∂B
∂t ≈ 0 hence
E = −∇Φ
Consequently, a piezoelectric continuum is based on the governing equations of elastodynamics in the case of small deformations and quasi-electrostatic fields. Restricting to the case of time-harmonic motion with frequency ω and suppressing the common factor eiωt
in all terms, the equation of motion read
σij,j+ ρω2ui = −bi , Di,i = −q
Here bi is the body force, ρ is the mass density and q is free electric volume charge. In generalized notation the equation above is written as
σiJ = CiJ KlsKl
where FJ = (bi, q) and
ρJ K =
ρ , J, K = 1, 2, 3 0 , J, K = 4
The field equations are represented by sij = 12(uij + uji) , Ei = −Φi σiJ = CiJ KlsKl σiJ,i+ ρJ Kω2uK = −FJ
These group of equations in generalized notation lead to the following equation of motion in the absence of body forces (bi = 0) and free volume charges (q = 0).
Chapter 2
Gridded Ion Thruster
The gridded ion thruster due to very high specific impulse generation, with very low fuel demand, is one of the most popular thruster using for space applications. It can compete readily with chemical propulsion technologies even if the thrust generated is much smaller. The system can be used for different mission requirements such as maintaining orbit stations for geostationary satellites, controlling orbit and attitude, and multi-goal tasks. While chemical propulsion is extremely inappropriate for deep space missions, ion thrusters also make it possible to reach deeper space further. Ion thrusters employ a variety of plasma generation techniques to ionize a large fraction of the propellant. These thrusters then utilize biased grids to electrostatically extract ions from the plasma and accelerate them to high velocity at voltages up to and exceeding 10 kV. Ion thrusters feature the highest efficiency (from 60% to > 80%) and very high specific impulse (from 2000 to over 10.000 s) compared to other thruster types.
An ion thruster consists of basically three components: the plasma generator, the accelerator grids, and the neutralizer cathode. Figure2.1 shows a schematic cross section of an electronbombardment ion thruster that uses an electron discharge to generate the plasma. The discharge cathode and anode represent the plasma generator in this thruster, and ions from this region flow to the grids and are accelerated to form the thrust beam. The plasma generator is at high positive voltage compared to the spacecraft or space plasma and, therefore, is enclosed in a “plasma screen” biased near the spacecraft potential to eliminate electron collection from the space plasma to the positively biased surfaces. The neutralizer cathode is positioned outside the thruster and provides electrons at the same rate as the ions to avoid charge imbalance with the spacecraft.
Figure 2.1: Ion thruster schematic.
The performance of the thruster depends on the plasma generator efficiency and the ion accelerator design.
The basic geometry of an ion thruster plasma generator is illustrated well by the classic DC electron discharge plasma generator. This version of the thruster plasma generator utilizes an anode potential discharge chamber with a hollow cathode electron source to generate the plasma from which ions are extracted to form the thrust beam. A simplified schematic of a DC electron bombardment ion thruster with these components coupled to a multi-grid accelerator is shown in Figure 2.2. Neutral propellant gas is injected into the discharge chamber, and a small amount is also injected through the hollow cathode. Electrons extracted from the hollow cathode enter the discharge chamber and ionize the propellant gas. To improve the efficiency of the discharge in producing ions, some form of magnetic confinement typically is employed at the anode wall. The magnetic fields provide confinement primarily of the energetic electrons, which increases the electron path length prior to loss to the anode wall and improves the ionization probability of the injected electrons. Proper design of the magnetic field is critical to providing sufficient confinement for high efficiency while maintaining adequate electron loss to the anode to produce stable discharges over the operation range of the thruster. Several power supplies are required to operate the cathode and plasma discharge. A simplified electrical schematic typically used for DC-discharge plasma generators is shown in Figure 2.3. The cathode heater supply raises the thermionic emitter to a sufficient temperature to emit electrons, and is turned off once the plasma discharge is ignited. The keeper electrode positioned around the hollow cathode tube is used to facilitate striking the hollow cathode discharge, and also protects the cathode from ion bombardment from the discharge chamber region.
Gridded Ion Thruster
Figure 2.2: Illustration of a DC-discharge electron bombardment ion thruster.
Figure 2.3: Electrical schematic of a DC-discharge ion thruster with the cathode
heater, keeper and discharge power supplies.
2.1
Principles of operation
Electrostatic thrusters accelerate heavy charged atoms (ions) by means of a purely electrostatic field. Magnetic fields are used only for auxiliary purposes in the ionization chamber. It is well known that electrostatic forces per unit area (or energies per unit volume) are of the order of120E2where E is the strength of the field (volts/m) and 0the permittivity of vacuum
0 = 8,85 × 10−12 Faradm
. Typical maximum fields, as limited by vacuum breakdown or shorting due to imperfections, are of the order of 106V/m, yielding maximum force densities of roughly 5 N/m2 = 5 × 10−5atm. This low force density is one of the major drawbacks of electrostatic engines, and can be compared to force densities of the order of 104N/m2in self-magnetic devices such as MPD thrusters, or to the typical gas pressures of 106-107N/m2 in chemical rockets. Simplicity and efficiency must therefore compensate for this disadvantage. The main elements of an electrostatic thruster are summarized in Figure 2.4 Neutral propellant is injected into an ionization chamber, which may operate on a variety of principles (electron bombardment,
contact ionization, radiofrequency ionization...). The gas contained in the chamber may only be weakly ionized in the steady state, but ions are extracted preferentially to neutrals, and so, to a first approximation, we may assume that only ions and electrons leave this chamber. The ions are accelerated by a strong potential difference Va applied between
perforated plates (grids) and this same potential keeps electrons from also leaving through these grids. The electrons from the ionization chamber are collected by an anode, and in order to prevent very rapid negative charging of the spacecraft (which has very limited electrical capacity), they must be ejected to join the ions downstream of the accelerating grid. To this end, the electrons must be forced to the large negative potential of the accelerator (which also prevails in the beam), and they must then be injected into the beam by some electron-emitting device (hot filament, plasma bridge...).
Figure 2.4: Simplified schematic of ion thruster.
The net effect is to generate a jet of randomly mixed (but not recombined) ions and electrons, which is electrically neutral on average, and is therefore a plasma beam. The reaction to the momentum flux of this beam constitutes the thrust of the device. Notice in Figure 2.4 that, when properly operating, the accelerator grid should collect no ions or electrons, and hence its power supply should consume no power, only apply a static voltage. On the other hand, the power supply connected to the neutralizer must pass an electron current equal in magnitude to the ion beam current, and must also have the full accelerating voltage across its terminals; it is therefore this power supply that consumes (ideally) all of the electrical power in the device. In summary, the main functional elements in an ion engine are the ionization chamber, the accelerating grids, the neutralizer, and the various power supplies required. Most of the efforts towards design refinement have concentrated on the ionization chamber, which controls the losses, hence the efficiency of the device, and on the power supplies, which dominate the mass and parts count. The grids are, of course, an essential element too, and much effort has been spent to reduce their erosion by stray ions and improve its collimation and extraction
Gridded Ion Thruster
capabilities. The neutralizer was at one time thought to be a critical item, but experience has shown that, with good design, no problems arise from it.
The geometry of the region around an aligned pair of screen and accelerator holes is shown schematically in Figure 2.5. The electrostatic field imposed by the strongly negative accelerator grid is seen to penetrate somewhat into the plasma through the screen grid holes. This is fortunate, in that the concavity of the plasma surface provides a focusing effect which helps reduce ion impingement on the accelerator. The result is an array of hundreds to thousands of individual ion beamlets, which are neutralized a short distance downstream, as indicated. The potential diagram in Figure 2.5 shows that the screen grid is at somewhat lower potential than the plasma in the chamber. Typically the plasma potential is near that of the anode in the chamber, while the screen is at cathode potential (some 30-60 volts lower, as we will see). This ensures that ions which wander randomly to the vicinity of the extracting grid will fall through its accelerating potential, while electrons (even those with the full energy of the cathode-anode voltage) are kept inside. The potential far downstream is essentially that of the neutralizer, if its electron-emission capacity is adequate. This potential is seen to be set above that of the accelerator grid, in order to prevent backflow of electrons from the neutralizer through the accelerating system. In addition, by making the total voltage, VT, larger than the Net
voltage, VN, the ion extraction capacity of the system is increased with no change (if VN
is fixed) on the final velocity of the accelerated ions.
To design an ion thruster it is possible idealize the multiplicity of beamlets as a single effective one-dimensional beam and consider the following three equations to derive the classical Child-Langmuir space charge limited current equation.
• Poisson’s equation in the gap:
d2φ dx2 = − eni 0 • Ion continuity: enivi = j = costant
• Electrostatic ion free-fall:
vi =
s
2e(−φ) mi
Combining these equations, we obtain a 2ndorder, nonlinear differential equation for φ(x). The boundary conditions are φ(0) = 0 , φ(x = d) = −Va.
In addition, we also impose that the field must be zero at screen grid: dφ
dx
x=0
= 0
This is because (provided the ion source produces ions at a sufficient rate), a negative screen field would extract more ions, which would increase the in transit positive space charge in the gap. This would then reduce the assumed negative screen field, and the process would stop only when this field is driven to near zero (positive fields would choke off the ion flux). At this point, the grids are automatically extracting the highest current density possible, and are said to be space charge limited.
Since three conditions were imposed, integration of the previous equations will yield the voltage profile and also the current density j.
j = 4 √ 2 9 0 e mi 12 V a32 d2 and also φ(x) = −V a x d 43 E(x) = −4 3 V a d x d 13
Gridded Ion Thruster
The last equation shows that the field is zero (as imposed) at x = 0, and is −43V ad at x = d (the accelerator grid). This allows to calculate the net electrical force per unit area on the ions in the gap as the difference of the electric pressures on both faces of the slab:
F A = 1 20 4 3 V a d 2 = 8 90 V a2 d2
and this must be also the rocket thrust.
Considering the Child-Langmuir space charge limited current equatio, if the beam has a diameter D, it is possible predict a total beam current of
I = π 4 4√2 9 0 e mi 12 D d 2 vt 3 2 = P Vt 3 2
where P is the so called perveance of the extraction system. This equation shows that this perveance should scale as the dimensionless ratio D
2
d2, so that, for example the
same current can be extracted through two systems, one of which is twice the size of the other, provided diameter and grid spacing are kept in the same ratio. While the one-dimensional model is important in identifying many of the governing effects and parameters, its quantitative predictive value is limited. Three-dimensional effects, such as those of the ratio of extractor to accelerator diameter, the finite grid thicknesses, the potential variation across the beam and so on, are all left out of account. So are also the effects of varying the properties of the upstream plasma, such as its sheath thickness, which will vary depending on the intensity of the ionization discharge, for example. Also, for small values of R = VVNT, the beam potential (averaged in its cross-section) cannot be expected to approach the deep negative value of the accelerator, particularly for the very flattened hole geometry prevalent when Dd is also small. Thus, the perveance per hole can be expected to be of the functional form
P = p d Ds , Da Ds , ta Ds , ts Ds , R , VD VT
where the subscripts s and a identify the screen and accelerator respectively, t is a grid thickness, and VD is the discharge voltage, which in a bombardment ionizer
controls the state of the plasma. Some of the salient conclusions of these dependencies will be summarized here:
• Varying the screen hole diameter Dswhile keeping constant all the ratios
d Ds , Da Ds , ta Ds , ts Ds , R , VD VT
• The screen thicknesses are also relatively unimportant in the range studied (Dts ≈ 0.2 - 0.4).
• Reducing R = VVNT always reduces the perveance, although the effect tends to disappear at large ratios of spacing to diameter (Dds), where the effect of the negative accelerator grid has a better chance to be felt by the ions. The value of Dds at which R becomes insensitive is greater for the smaller R values.
• For design purposes, when VN and not VT is prescribed, a modified perveance
I VN 3 2
• The perveance generally increases as DDas increases, with the exception of cases with R near unity, when an intermediate Da
Ds ≈ 0,8 is optimum.
• Increasing VVDT, which increases the plasma density, appears to flatten the contour of the hole sheath, which reduces the focusing of the beam. This results in direct impingement on the screen, and, in turn, forces a reduction of the beam current. two limiting conditions should be mentioned:
• Direct ion impingement on screen: At low beam current, the screen collects a very small stray current, which is due to charge-exchange ion-neutral collisions in the accelerating gap: after one such collision, the newly formed low speed ion is easily accelerated into the screen. The screen current takes, however, a strong upwards swing when the beam current increases beyond some well defined limit. This is due to interception of the beam edges, and, since the high energy ions are very effective sputtering agents, results in a very destructive mode of operation.
• Electron back-streaming: For R values near unity, the barrier offered by the accelerator negative potential to the neutralizer electrons becomes weak, and beyond some threshold value of R, electrons return up the accelerator potential to the chamber. This results in screen damage, space charge distortion, and shorting of the neutralizer supply. Kaufman gave the theoretical estimate
Rmax = 1 − 0,2 Ie Da exp Da TA
Gridded Ion Thruster
2.2
Thrust vectoring system
The relative radial alignment between the screen and accelerator apertures of an ion thruster influences the path of exiting charged particles positively charged ions are attracted to the negatively charged accelerator grid, which extracts ions from the thruster. If alignment between the screen and accelerator apertures is perfectly concentric, the radial influence on the ion path is cancelled due to symmetry and particles leave the thruster approximately along the aperture axis. If the accelerator aperture is radially offset from the axis of the screen aperture, a positively charged ion will be diverted into the direction of the closest edge of the accelerator aperture, and the particle leaves the thruster at a skewed angle. The magnitude off the skewed angle is roughly proportional to the offset between the two apertures, until ions impinge directly on the accelerator aperture electrode. For circular apertures, the direction of the diversion can occur anywhere in the radial plane. By manipulating the relative alignment between the screen and accelerator apertures, the entire beam from an ion thruster can be collectively steered in a desired direction. This could effectively provide a spacecraft with a means of thrust vector control. The ability to control the thrust vector of any spacecraft propulsion system is extremely advantageous. It can be used both to improve or optimise mission performance andalso to compensate for the shift in position of the centre of mass in order to minimise attitude control requirements. In general, thrust vectoring is achievedby the use of a mechanical gimballing mechanism. However, this system usually imposes a significant mass penalty. This is particularly true in the case of electric propulsion (EP), as the gimbal mounting can be heavier than the thruster itself. Moreover, gimbals are very expensive and, like any mechanical system, susceptible to failure. There would thus be major advantages in developing thrust vector control systems dedicated to EP. However, in order to be relevant, the performance of such a device must meet the requirements of EP as main propulsion system. For a thrust of 250 mN, a thrust deflection angle of a fraction of a degree is requiredto achieve a plane change in parallel with an orbit raising transfer. Because the angle is very small, a high accuracy is required, with very small step size. Also, as in most cases of controlledd evices, a fast response wouldbe highly beneficial. Such requirements tend towards the development of a built-in thrust deflection system. The most straightforward solution to the problem of thrust vectoring is mounting the thruster on a gimbal mount. It has an important advantage of not changing the thruster design and therefore not affecting its performance. However, it need to use mechanical actuators and it has a high system mass. Through the years significant progress in mechanical actuation methods has taken place and building a reliable gimbal system is no longer a problem. Nevertheless, a large mass penalty, as well as the space occupied by the actuators remains a significant disadvantage of this system. The simplest non-mechanical methodis electrostatic deflection of the beam after it leaves the engine. This can be achieved by placing electrically charged plates around the beam. The potential needed to deflect the thrust vector by a desired 8° is about 400 V for a 5 cm diameter thruster, however much higher voltages would be needed for larger engines. Furthermore,
the danger of creating of plasma sheaths around the plates appears, leading to the need of using much higher voltages in order to achieve the desired deflection. Because of these difficulties, the electrostatic system is not considered feasible and no laboratory tests have been conducted so far. The magnetic deflection system has a similar layout, with charged plates replaced with magnet polepieces. It has been calculated that the power consumed by electromagnets needed to deflect the exhaust beam of a 5 cm thruster only in one plane can reach 271 W. Also the mass of such a system will be prohibitively large. This makes this system infeasible.In order to reduce the voltages needed for electrostatic deflection, an attempt was made to embed the vectoring electrodes into the accelerator grid. Due to the minute sizes of grid apertures, adding extra electrodes insulated from the main grid proved to be very difficult and acceptable design was achieved only after a considerable effort. Another technique that aims to deflect the beam inside of the grid system, as shown in Figure 2.6, is that of using an accel grid that can be translated within its plane. The resulting change in electrostatic field between the grids causes ions to change direction. Various theoretical and experimental analyses have shown that deflection angle is directly proportional to grid movement with coefficient of 0,03 – 0,04mm/deg. Also correlation between the vectoring angle β, grid translation and grids separation distance lghas been derived
β = −A lg
where A is a coefficient, ranging from 16,5 to 22,8, dependable on operational parameters, mainly the power level.
Figure 2.6: Ion beam deflection using accelerator grid translation.
The main limitation to increasing the vectoring angle is grid sputtering caused by ions. However, if deflection of as low as 8° is needed, sputtering poses no major threat.
Chapter 3
ACC Grid
Ion thrusters are characterized by the electrostatic acceleration of ions extracted from the plasma generator. The ion thruster’s main feature is that the propellant ionization process and the ion acceleration process are physically separated. The ionization process takes place in the discharge chamber of the thruster and can be achieved by different principles namely electron bombardment (Kaufman thruster), RadioFrequency waves (RF thruster) or Electron Cyclotron Resonance (ECR thruster). The ion accelerator consists of electrically biased multi-aperture grids, and this assembly is often called the ion optics. The design of the grids is critical to the ion thruster operation and is a trade between performance, life, and size. Since ion thrusters need to operate for years in most applications, life is often a major design driver. However, performance and size are always important in order to satisfy the mission requirements for thrust and specific impulse (Isp). Alongside the plasma generator and the neutralizer, ion engine grids are the components that decide the thruster’s geometry, so their study is essential for the scaling of the engine. The purpose of the grids is extract the ions from the discharge plasma and focus them through the downstream accelerator grid (ACC Grid). This focusing has to be accomplished over the range of ion densities produced by the discharge chamber plasma profile that is in contact with the screen grid, and also over the throttle range of different power levels that the thruster must provide for the mission. The ACC grids must minimize ion impingement on the screen grid and extract the maximum number of the ions that are delivered by the plasma discharge to the screen grid surface. In addition, the grids must minimize neutral atom loss out of the discharge chamber to maximize the mass utilization efficiency of the thruster. High ion transparency and low neutral transparency drives the grid design toward larger screen grid holes and smaller accel grid holes, which impacts the optical focusing of the ions and the beam divergence. The beam divergence also should be minimized to reduce thrust loss and plume impact on the spacecraft or solar arrays, although some amount of beam divergence can usually be accommodated. Finally, grid life is of critical importance and often drives thruster designers to compromises in performance or alternative grid materials.
The thruster ion optics assembly serves three main purposes:
• Extract ions from the discharge chamber;
• Accelerate ions to generate thrust;
• Prevent electron backstreaming.
The ideal grid assembly would extract and accelerate all the ions that approach the grids from the plasma while blocking the neutral gas outflow, accelerate beams with long life and with high current densities, and produce ion trajectories with no divergence under various thermal conditions associated with changing power levels in the thruster. Grids have finite transparency; thus, some of the discharge chamber ions hit the upstream screen grid and are not available to become part of the beam. The screen grid transparency, Ts , is the ratio of the beam current, Ib , to the total ion current, Ii , from the discharge
chamber that approaches the screen grid:
Ts =
Ib
Ii
The goal for screen grid design is to maximize the grid transparency to ions by minimizing the screen thickness and the webbing between screen grid holes to that required for structural rigidity. The maximum beam current density is limited by the ion space charge in the gap between the screen and accelerator grids with respect to the perveance that was specified by the Child–Langmuir equation in which the sheath was considered essentially planar. The problem is that the sheath shape in the screen aperture is not planar, as seen in Figure 3.1, and the exact shape and subsequent ion trajectories have to be solved by 2-D axi-symmetric codes. However, a modified sheath thickness can be used in the Child–Langmuir equation to approximately account for this effect, which is written as Jmax = 40 9 r 2e M VT 3 2 le2
ACC Grid
Figure 3.1: Non-planar sheath model approximation for a two-grid system.
where VT is the total voltage across the sheath between the two grids and the sheath
thickness leis given by le = s lg+ ts 2 + ds 2 4
Ion optics are characterized by a series of geometric and electrical parameters, shown in Figure 3.2, like as, grid outer diameter (D), similar to the discharge chamber diameter, grid active diameter or beam diameter (Dac), screen grid thickness (ts), accel grid thickness (ta),decel grid thickness (td), screen grid aperture diameter (ds), accel
grid aperture diameter (da), decel grid aperture diameter (dd), screen-accel gap (lsa),
accel-decel gap (lad), screen grid potential (Vs), accel grid potential (Va) and decel grid potential (Vd), where the deceleration grid is present in case of a three grid thruster.
There are also some parameters that can be used to assess the quality of the grids, that is, how well they can accomplish their functions:
The Normalized Perveance per Hole(NPH) is the amount of current that a single aperture can extract and focus. It is defined as
N P H = Jb Vt1.5
le ds
2
where Jb is the beam current per hole and Vtis the total voltage applied between the accel and screen grids:
Vt= Vs+ Va
N P H measures the extraction capability of a given aperture. Beam divergence is how much the plasma beam expands after it has been focused, as shown in Figure 3.3 and can be characterized by the divergence angle (α) which is defined as
α = arctanRα− rα L
Figure 3.3: Plasma beam divergence angle and beam current density profile. where Rαis the radius normal to the beam axis of the cone that encloses 95% of the
total beam current, rα is the ion beam radius at the thruster exit and L is the distance
from the thruster exit to the region where Rα is measured. The divergence angle can also be defined for a beamlet focused by a single aperture. Beam divergence should be minimized to reduce thrust losses and avoid plume impact on the spacecraft. For two grid acceleration systems, different experiments yielded divergence angle values between 10° and 25° approximately. Ion optics problems can be divided into three categories: launch problems, operation issues and sputter erosion. Potential launch problems relate to launch vibrations, which can generate plastic deformation, grid-to-grid contact and aperture misalignment. Displacement of the accel grid aperture relative to the screen grid centerline causes an offaxis deflection of the ion trajectories, commonly called beam
ACC Grid
steering. In order to prevent these issues, ion optics are subjected to random vibration tests at protoflight levels in three axes, both isolated and mounted on the thruster, to validate the design and prove that vibration-induced stresses will not exceed the elastic limit stress and that grid gap and holes alignment will not be altered. Ion thruster optics are operated at high voltages and elevated temperatures, which leads to thermal response, electrostatic pressure, and interference with the discharge chamber magnetic circuit. Although electrostatic pressure is not a real problem (for a potential difference of 1500 V, which is a worst-case condition, 0.254 mm grid-to-grid separation, electrostatic pressure was determined to be 152 Pa and the magnetic interference is solved by building the grids with diamagnetic or weakly paramagnetic materials (metals such as molybdenum and titanium and carbon based material), grids thermal expansion can generate problems, especially in thrusters that utilize refractory metal optics. For a 30 cm thruster operating at a 350 W discharge power, temperatures at the centers of the screen and accel grids are of the order of 350◦C and 290◦C.Temperature difference induces thermal expansion that changes size of the acceleration gap between the screen and accel grids, which will directly affect the ion trajectories and the perveance of the ion optics. Additionally, thermal expansion results in buckling and plastic deformation, which prevents proper thruster operation.
3.1
ACC Grid material and properites
Basic ion extraction systems usually follow the same pattern: screen grid is designed with a minimum thickness to increase the effective transparency and accel grid tends to be thicker for better erosion resistance. Decel grid, when included, is normally thinner than screen grid. Ion optics apertures are typically circular, and packed in a hexagonal array to produce a high transparency to the ions from the plasma source. To accelerate ions to high energy, it is necessary to reduce the dimension of an aperture at the plasma boundary to the order of the Child–Langmuir distance to establish a sheath that will accelerate the ions with good focusing and reflect the electrons from the plasma. The gap between the screen grid and the accel grid affects the thruster performance and service life. It is preset during ion optics assembly (cold gap) and changes during engine operation due to the grid thermal expansion (hot gap). In order to maximize the perveance of the accelerator, it is desirable to make the grid gap smaller than the aperture diameters and keep it during the operation. Ion engine grids need materials capable of working at elevated temperatures, but given that their main problems come from thermal expansion and sputter erosion, grid materials are selected based on their thermal and erosion properties. Molybdenum is the traditional material for the grids due to its low sputter erosion rate, ability to be chemically etched to form the aperture array, and good thermal and structural properties. Although optics made of other metals, especially titanium, have been fabricated and tested, carbon-based materials have developed as a better alternative for ion optics, particularly graphite and carbon–carbon composite material, because of their lower coefficient of thermal
expansion and lower sputtering yield. For the scope of this thesis a molybdenum alloy has been selected. The molybdenum alloy selected is the titanium-zirconium-molybdenum alloy (TZM) that is typically manufactured by powder metallurgy or arc-casting processes. It is an alloy that has a higher recrystallization temperature, higher creep strength, and higher tensile strength than pure, unalloyed molybdenum. While incredibly versatile, TZM is best utilized between 700 and 1400◦C in a non-oxidizing environment. The chemical composition of TZM is reported in Table 3.1
Element Content (%)
Molybdenum, Mo 99,4 Titanium, Ti 0,5 Zirconium, Zr 0,08
Carbon, C 0,02
Table 3.1: TZM chemical composition.
While the physical and mechanical properties are reported, respectively, in Table 3.2 and Table 3.3
Property Value U.M.
Density 10220 mKg3
Melting Point 2623 ◦C Boiling Point 5560 ◦C Linear Coefficient of Thermal Expansion 4,9x10−6 ◦1C
Thermal Conductivity 0,48 cmCal◦Cs
Specific Heat 0,073 gCal◦C
Electrical Conductivity 30% IACS Electrical Resistivity 6,85 µ hom cm Temperature Coefficient of Electrical Resistivity 0,0046 ◦C
Spectral Emissivity (0.65 µ wavelength) 0,37 Total Emissivity (at 1500 °C) 0,19 Total Emissivity (at 2000 °C) 0,24
Working Temperature 1600 and less ◦C Recrystallization Temperature 900 − 1200 ◦C Stress Relieving Temperature 800 ◦C
Table 3.2: TZM Physical properties.
Property Value U.M.
Tensile Strength (20◦C) 760 MPa Tensile Strength (500◦C) 448 MPa Tensile Strength (1000◦C) 206 MPa Modulus of Elasticity (20◦C) 317 GPa Modulus of Elasticity (500◦C) 282 GPa Modulus of Elasticity (1000◦C) 268 GPa
Poisson’s Ratio 0,321
ACC Grid
3.2
ACC Grid initial geometry
For this analysis a typical Ion thruster’s ACC Grid was carried out, with a characteritic pattern of holes, the data of which are listed in Table 3.4
Parameter Value U.M.
Diameter 350 mm Thickness 1,5 mm Hole Diameter 3,55 mm Hole Spacing 5,4 mm
Table 3.4: ACC Grid geometrical data.
For realize the 3D model a CAD software has been used, the model with geometrical dimensions can be seen in Figure 3.4 and Figure 3.5, while the holes pattern can be seen in Figure 3.6.
Figure 3.5: ACC Gird 3D model thickness.
ACC Grid
3.3
Research of equivalent material for FEM analysis
Due to the complexity of the holes pattern, in order to make certain operations easier during the FEM analysis phase, such as meshing, which with all the holes would become very complicated, weighing down the model, increasing the calculation time by a lot and if not done correctly risking returning incorrect results, it was decided to make an eqivalent material that allows to obtain the usual results using a grid model without holes. In order to do this, the 3D model has been modified, which will be described in the section 3.4. In order to be able to describe the behaviour of the grid using an equivalent material, the Young’s Modulus and the Shear Modulus of the new material must be found. The first thing to do is to make a representative model of the grid with the pattern of holes. To do this a square-shaped plate, with smallest dimensions than the grid, has been realized so that the necessary mechanical properties can be calculated using known theories. As shown in Figure 3.7, the square-shaped plate model was obtained directly from the complete 3D model of the grid by making a cut that would return the 3D model of the desired shape and size without altering the hole pattern,
Figure 3.7: Cut of the Grid 3D model.
The final model obtained is a square-shaped plate with sides length 100 mm and thickness 1.5 mm as shown in Figure 3.8.
Figure 3.8: Square-shaped plate 3D model.
The second step was to make the usual square-shaped plate model this time without holes, see Figure 3.9, to verify, using materials (in this case aluminum) with known properties, the real functionality of the model.
Figure 3.9: Square-shaped plate without holes 3D model.
As first load case, a tensile load was used, in which a uniform pressure was applied on a face equal to 100 Pa, as can be seen in Figure 3.10, and two constraints was applied on the opposite face, the first one a constrain applied on across the face that blocking the displacement in x while the second one a constrain applied on an edge of the face
ACC Grid
that blocking the displacement in y and z, as can be seen in Figure 3.11 and Figure 3.12 respectively.
Figure 3.10: Tensile load applied on the model.
Figure 3.11: constrain applied on across the face that blocking the displacement in x.
Figure 3.12: constrain applied on an edge of the face that blocking the displacement
in y and z.
Plotted some results like as directional deformation, Figure 3.13, and force reaction, Figure 3.14,
ACC Grid
Figure 3.14: Force reaction on constrain.
using the constitutive equation σ = E with simple physical and mathematical considerations one comes to the known form of Hooke’s law
F = E
Alx
Using the inverse formula we arrive at the useful form for checking the validity of the model for this load case
E = F l Ax
Where F is given by the reaction force, x is given by the directional deformation alog x axis, l is the lenght of the side and A is the area of the face on which the force is applied, in this case A = lt, where t is the thickness of the plate model. So entering the numerical data inside the formula the results is:
E = F l Ax = 1,5−2N 0.1 m 0.000 15 m2 1,4085−10m = 7,1 10Pa = 71 GPa
Which as known is the value of the Young’s module of the aluminium. So the model is verified. The next step was to repeat the same load case using the model with the holes, always using aluminum as the material, this time, being E known by the properties of aluminum will be used the same relationship in a form that allows us to calculate the area A subjected to the force, this data will be useful later to calculate the Young’s modulus
of the equivalent material. In this case the well-known relationship between F , A and σ can be used for simplicity
A = F σ
where F is given by the reaction force, as shown in Figure 3.15, σ is the applied pressure, and A is the equivalent area searched. This is how you get
A = F σ = 9,4102−3N 100 Pa = 9,4102 −5 m2
ACC Grid
The same procedure was utilized with the second load case, useful to calculate the shear modulus. As can be seen in Figures 3.16, 3.17 and in Figures 3.18, 3.19
Figure 3.16: Shear load applied on one side of the model.
Figure 3.18: constraints that block the y and z directions.
Figure 3.19: constraints that block the x and z directions.
The Square-shaped plate was subjected to two shear pressures, applied on two contiguous faces, respectively in the directions x and y with module equal to 100 Pa and constrained on the opposite faces with constraints that block the y and z directions and the x and z directions respectively. Plotted some results like as Shear elastic strain, Figure 3.20, and force reaction, Figure 3.21, and using the constitutive equation F = AGγ, where γ is directly given by the Shear elastic strain, it is possible calculate the shear modulus G of the aluminium to verify the load case. So using the inverse formula is obtained
ACC Grid G = F Aγ = 1,5−2N 0.000 15 m2 3,7465−9 = 2,67 10Pa = 26.7 GPa
Which as known is the value of the Shear modulus of the aluminium. So the model is verified.
Figure 3.20: Shear elastic strain.
Figure 3.21: Force reaction on one constrain.
As done for the previous load case, it is necessary to calculate the equivalent area of the plate model with holes. Using the inverse equation, it is obtained
