The finite element method (FEM) is a numerical technique that provides approximate
solutions to the governing differential or integral equations of a complicated system through a discretization process [57] and [58]. FEM is used to solve the problems of engineering disciplines such as the areas of stress/strain analysis of solid structures, heat conduction analysis and fluid dynamics. The domain of the system (physical / mathematical) can be defined or subject to frequent changes (moving boundary problems such as transient-free surface water flow, large deformation problems, etc.)[58]. The boundary conditions can be well defined regarding prescribed loads and displacements, or sometimes less well defined as in fluid–structure interactions or
x3
x1
x2
Piezoelectric solid
Figure 2. 10: Axes for surface wave solution.
contact problems. The propagation of SAW governed by differential equations given in equations (2.27) and (2.28) must be solved along with complexities in the geometry of the device, materials used in the device, and proper boundary conditions [59]. This section gives some basic procedures and techniques used in finite element method formulations.
2.3.1 Input and output information in FEM
The following information needs to be input about the structure for which analysis in FEM to be carried out.
1. Coordinate systems according to the geometry is an important parameter to be input either in 3D or 2D.
2. The desired geometry of the proposed device to be solved. Assigning contact pair between slider and stator in the case of the SAW motor. Assigning master for stator and slave for the stator in contact pair between two solid bodies.
3. Material properties of the model: for example, in SAW devices material properties of the piezoelectric substrate such as elasticity matrix, piezoelectric constant, permittivity constant, and density are to be furnished and for IDT electrodes anisotropic material properties such as Young’s modulus, density, Poisson ratio, etc.
are to be provided.
For SAW motor the material properties of the slider, such as the type of material and consequence properties.
4. Boundary conditions at the boundaries of the structure are to be provided. For example in the case of stress/strain analysis for piezoelectric material, the boundary conditions can be displacement constraints and electrical boundary conditions such as electric potential, electric displacement, charge, etc.
For SAW motors the flexibility of motion of the slider, freedom in which direction of displacement of the slider. Boundary load to the slider along with body load. Defining the contact friction between slider and stator, and assigning coefficient of friction either static or dynamic to the contact body.
Further, the geometry domains are to be divided into smaller domains called elements which are connected at specific points called nodes as shown in Figure 2.11. The elements created in the geometry can be in the shape of triangular, quadrilateral, tetrahedral, or brick depending on the dimensionality. The output information in case of stress/strain analysis in piezoelectric material involves nodal and elemental
information, the solution to the primary unknown quantities, the displacements in all directions, and voltage are determined at nodes. These unknowns are called as
degrees of freedom (DOF).
In this thesis, the simulations and analysis of SAW motors are carried out by FEM using piezoelectric module of COMSOL Multiphysics software. The software has well- developed solvers, graphical user interface (GUI) and post-processing capabilities.
More details of the software can be found in the user guide of the software [60].
2.3.2 Finite element simulation of SAW devices
FEM was a common practice to simulate the conventional SAW devices and have been reported by many researchers. Lerch et al.[61] presented a FEM scheme to calculate Eigenmodes and dynamic response to mechanical and electrical excitations of 2D and 3D piezoelectric transducer for any geometry. Rahman et al. [62] performed FEM simulation to study SAW based quantum devices. Atashbar et al. [63] simulated the mass loading effect of palladium sensing film in the presence of hydrogen gas. Ippolito et al. [64] performed the FEM based analysis of acoustic waves propagating on layered SAW devices. Wang et al. [65] used FEM carry out a study on the effects of change in film properties on the mass sensitivity of SAW sensors. Tikka et al.[66] reported the finite element modelling of SAW correlator. Subramanian et al. In most of the earlier works, the number of DOF to solve is significant due to the large size of the SAW structures considered in the model. To solve DOF for an entire SAW device, it requires
Elements
Boundary constraints Applied Voltage (V)
Nodes
Figure 2. 11: Illustration showing a solid body discretized into a finite number of elements.
high computing facility and extensive computation time. Thus a valid approximation and simplified model have an obligation to perform FE simulation of SAW devices. In this chapter, the conventional SAW devices such as one-port SAW resonators and SAW delay line devices are simulated with metallic IDT on the piezo substrate structure using COMSOL Multiphysics.
2.3.3 The mathematical model for a SAW delay line
The IDT converts electrical energy to mechanical energy and vice versa as explained in IDT modelling. The mathematical model of a SAW delay line given by Feng et al. [67] is adopted in formulating the simplified simulation methodology. The transmitting IDT can be considered as a force generator that converts the electrical voltage to mechanical forces.
These forces travel as SAW on the piezoelectric substrate. Consider a transmitting IDT as shown in Figure 2. 12, and with the number of force generators (fingers) of NM equal to 5. If the force generated at each finger of transmitter IDT is proportional to the applied voltage, the force at nth finger pair can be written as Fn(t) = K1Vin(t)
Where K1 is proportionality constant of the transducer. Vin is the input voltage. The displacement of the wave u at the edge of IDT of NM finger pair can be written as
1
( ) 1 2
NM n n
t F t n
v v
u (2.37)
The displacement of the wave at receiver IDT of the SAW delay line device with delay line of length L can be written as
p
Vin and Vo
x3
x1 λ
L
n =1 n =2 n =3 n =4 n =5
Figure 2. 12: A Typical IDT with five pairs of electrodes.
1
( ) 1 2
NM n R n
n N L
t F t
v v
u (2.38)
Where NR is the number of finger pair at the receiver IDT. Let qout denote the electric charge produced when the electrodes are subjected to local strain. Assuming that the charge is the sum of the charges produced at each finger and the charge from each finger is proportional to displacement,
2 1
( ) NR ( )
out l
l
q t K u t
(2.39)Accordingly assuming the output voltage is proportional to charge,
( ) 3 ( )
out out
v t K q t (2.40)
K2 and K3 are proportionality constants. Thus combining the above equations, the output voltage can be expressed as
1 2 3 1 1
( ) 2
R T
N N
out in
l n
K K K n l L
v t v t
v v
(2.41)If an IDT of 100 finger pair and displacement at edge of point fifth pair from equation (34) can be rewritten as
100 100
1 6
1 1
( ) 2 2
n n
n n
n n
n n
t F t F t
v v v v
u (2.42)
( )t a( )t b( )t
u u u (2.43)
The above equation can be further realised regarding time t as below
100
1
( ) 1 2
n
a n
n
t F t n
v v
u for all t (2.44)
94
1
( ) 1 2
n
b n
n
t F t n
v v
u for t >= nλ/v (2.45)
In general, equation (40) can be realised as a one-port SAW resonator consisting of single IDT with 100 finger pairs. As explained in the simulation of one-port SAW resonator it is convenient to model equation (40) and (41) by applying infinite periodic boundary conditions to a section of SAW resonator of length nλ consisting of single or a pair of IDT.
The simulation procedure adopted in COMSOL Multiphysics to solve the SAW delay line model is explained in the following section.