1. In the name of GOD

2. Ultrasonic Transducers and Arrays
Chapter 3
Ultrasonic Transducers and Arrays

3. To produce ultrasonic wave it is necessary to use a transducer
A transducer is a device which convert electric energy to mechanical energy
Piezoelectric effect
It is discovered by Pierre and Jacques (1880)
It is a phenomena in which a material upon the application of electric field changes its physical dimensions and vice versa
Types of piezoelectric
Natural: quartz and tourmaline
artificial: polycrystalline ferroelectric ceramic material such as lead zirconate titanate, Pb(Zr,Ti)O3 or PZT with strong piezoelectric properties; BaTiO3; PbNb2O3; LiNbO3 etc

4. Piezoelectric constitutive equation
Piezoelectric effect results from interaction of electric and mechanical properties of the materials
The stress-strain relationship in anisotropic materials is:
[K]=[C][ε]
[K], [C] and [ε] are the stress, the elastic constant and strain tensors
When the electric field is applied, the above equation changes to:
[D] electric displacement; [e] piezoelectric stress constant tensor; [CE] elastic constant tensor when the electric field [E]=0 and [Kε] dielectric constant tensor when [ε]=0 or clamped dielectric constant tensor.
The physical meaning of [CE] and [Kε] is understood by setting E and [ε]=0 ; then: [e}=-[k]/[E]
Where [e] , the piezoelectric stress constant is the resultant stress change per unit change in electric field without strain or while being clamped. The unit is Newton/(volt-meter) or coulombs per squire meter
The constitutive equation when [k] and [E] are independent are:
[K+] is free dielectric constant (when no stress present)
[d]=[ε]/[E] is transmission or piezoelectric strain constant in
Strain per unit change in electric field with a unit of coulomb per newton when no stress is present

5. [γE]=[ε]/[k] is the compliance of the material for [E]=0 and =1/[CE]
The relation between [e] and [d] when [k]=0 is [CE][ε]-[e][E]=0
Therefore [e]=[CE][ε]/[E]=[CE][d]
If [D] and [k] are assumed independent then: [E]=[αk][D]-[g][k] and [ε]=[g][D]+[γE][k]
Where [g]=-[E]/[k] is the receiving constant in volt.meter/newton
[γD]is the compliance when [D]=0 and [αk]=1/[Kk] when [k]=0
[K] depends on extent of freedom of the material. If it is clamped then strain=0 and is designated as clamped dielectric constant [Kε]
If it is freedom to move denoted [Kk] and is free electric constant
The transmitting constant and receving constant are related by [d]=[g][K+]
These parameters are in tensor form and in general for anisotropic materials with 18 constant.
In most cases there are some symmetry which reduce these constants. For example for quarts only five constant.
A plate cut with its surface perpendicular to x-axis called an x-cut and so on.
The x, y and z direction is denoted by 1, 2 and 3 direction
A pizeoelectric constant d13 represent the strain produced in the 1-direction by applying an electric field in the 3-direction when no external stress is present
The piezoelectric properties also depends on shape and boundary effects

6. The ability of a material to convert one form of energy to another is a measure of electromechanical coupling coefficient it is different from transducer efficiency

7. A few important geometries used for transduce