Piezo Electric Ceramics | Polarization | Commonly Used Definition of Terms & Relationship | Relative Dielectric Constant
 
 
 

Piezo electricity is the property some bodies have of becoming electrically charged when a mechanical pressure is applied on it. The crystalline structure produces voltage which is proportional to the pressure applied. The same materials exhibit the reverse effect of changing shape when an electric field is applied on to it. Quartz, Rochelle salt, Tourmaline are the natural form of crystals which exhibited Piezo electric effect, and have been known to the mankind for many years.

Many contemporary applications have a requirement of very strong piezoelectric effect, which resulted in the use of polycrystalline Lead Zirconate Titanate based ferroelectric materials. These class of Piezoelectric crystals is more versatile, as their physical, chemical & Piezoelectric parameters can be tailored as per the requirement of specific applications. They are chemically inert, mechanically hard, & are not affected by the atmospheric humidity, and the ease of manufacturing into any given shape & size have resulted in their wide usage and immediate acceptance. In addition to all these, the mechanical & electrical axes of the materials could be precisely oriented in relation to the shape of ceramic.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Polarization is the most important process for the manufacture of modern day piezoelectrics. After Silver electrodes have been applied on the surface of the ceramic element, a high D.C. field at elevated temperatures is applied which results in the alignment of Dipoles in the direction in which the D.C. field is applied. Before the application of D.C. field, the dipoles are randomly oriented, with the result the net piezoelectric effect is Zero.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
After the removal of the Polarizing field and the cooling of the ceramics, the dipoles cannot easily return to their original positions, and we now have what is known as remanent polarization. The ceramic body becomes
permanently piezoelectric, and it can now convert electrical energy into mechanical & vice versa. A polarizing treatment is the final process which is carried out in its manufacture by connecting the electrodes on the ceramic to a direct High Voltage D.C. Source.
 
 
Polarisation and Charge Coefficients
 
With piezoelectric ceramics, the relationship between the applied stress and the resulting responses depend upon:
•  Piezoelectric properties of the ceramic.
•  Size and shape of the element, and
•  Direction of the electrical and mechanical vector quantities.
 
To identify directions in a piezoelectric element, three axes termed as 1,2 and 3; which are analogous to the classical three dimensional orthogonal set of axes X, Y and Z are used. Material properties along the 1 and 2 axes are identical to each other but different from those along the 3 axis. For maintaining simplicity, references are made only to the 3 and 1 directions. The poling or 3 - axis is invariably taken parallel to the direction of polarisation within the ceramic ( Fig 1(A) ). The polar axis is induced during the manufacturing process by treatment with a high voltage DC field applied between the pair of electroded faces to align the domains of the material in the direction of the field.
 
The polarisation vector P is represented by an arrow pointing from the positive to the Negative poling electrode. In shear mode operations, the poling electrodes are later Removed and replaced by a set of electrodes on the second pair of the faces. The 3- axis is not altered, but it becomes parallel to the new electrode faces as seen on the finished element ( Fig 1(B)).
 
Piezoelectric charge coefficient (d constant)
 
The piezoelectric d constant is a measure of the charge density per unit stress or the strain per unit field
dik                        =                      Coulombs/ meter2                      =                Coulombs
                                                      Newtons/ meter2                                            Newton
Dik                        =                      meter/meter                               =                   meter
                                                       volt/meter                                                       volt
 
Piezoelectric coefficients with double subscripts link electrical and mechanical quantities. The first subscript gives the direction of the electrical field associated with the voltage applied or the charge or the voltage produced. The second subscript gives the direction of mechanical stress or the strain.The piezoelectric charge coefficient d applies when the force in the 3-direction ( along the polarisation axis ) and is impressed on the same surface on which the charge is collected ( Fig 2(A) ), whereas d31 applies when the charge is collected on the same surface as before but force is applied at right angles to the poling axis (Fig 2(B)).
 
 
Piezoelectric charge coefficient (g constant)
 
The g coefficient is a measure of the field per unit stress or strain per unit charge density.
Gik                        =                       Volt/ meter                               =                Volt/ meter
                                                     Newtons/ meter2                                            Newton

gik                        =                       meter / meter                          =                 Volt/ meter
                                                     Coulombs/ meter2                                         Coulomb
 
Output voltage is applied by multiplying the calculated electric field by the thickness of the ceramic between the electrodes. The first subscript indicates the direction of the generated voltage and the second indicates the the direction of the force.A"33" subscript signifies that the electrical field generated and the mechanical stress are both along the polarisation (Fig 2(A)). A "31" subscript signifies that the pressure is applied at right angles to the polarisation axis but the voltage appears on the same electrodes as in the "33" case (Fig 2(B)).
 
 
 
 
 
KT33 expresses the relative dielectric constant of the material relative to that of vacuum in the 3 direction. Multiplying this by ε0 , the dielectric constant of free space yields the absolute dielectric constant ( ε0 = 8.85 x 10-12 farads/meter). The superscript T applies to the mechanically free condition. KT33 therefore, expresses the relative dielectric constant measured in the polar direction under mechanically free condition. It is generally measured at 1 kHz, well below the mechanical resonance of the specimen.
 
Relationship Between g and d Coefficients
 
At frequencies far from resonance effects, piezoelectric ceramic transducers are fundamentally capacitors. Consequently, the voltage coefficient gik are related to the charge coefficient dik by the dielectric constant Ki ,as in a capacitor the voltage V is related to charge Q by the capacitance C.
Q        =             C . V
d33 = KT33 . ε0. g33
d31 = KT33 . ε0. g31
 
Coupling Coefficients
 
Sometimes also referred as electromechanical coupling coefficients, these describe the conversion of energy by the ceramic element from electrical to mechanical form or viceversa.
k                   =                              Mechanical Energy stored
                                                      Electrical Energy applied
K                   =                               Electrical Energy stored  
                                                    Mechanical Energy applied
Subscripts denote the relative directions of the electrical and mechanical quantities and the kind of motion involved. Kp signifies the coupling in a thin round disc polarised in radial expansion and contraction, whereas K33 is appropriate for a long thin bar or rod, electroded on the ends, poled lengthwise and vibrating in simple length expansion or contraction. K31 relates to a thin long bar, electroded on a pair of long faces, poled in thickness and vibrating in the longitudinal dimension. Since these coefficients are energy ratios, they are dimensionless.
 
Young's Modulus
 
The mechanical stiffness property of a piezoelectric ceramic material is expressed as the ratio of stress to strain. Because mechanical stressing of the ceramic produces an electrical response which opposes the resultant strain, the effective Young's modulus with electrodes short circuited is lower than the electrodes open circuited. Furthermore, the stiffness is different in the 3 - direction from that in the 1 or 2 direction. Therefore, in expressing mechanical quantities both direction and electrical conditions must be specified.
YD33 is the equivalent with the electrodes open circuited in the 3 direction whereas YD11 is the modulus in the 1 or 2 direction. The superscript D points out the open circuit condition. The inverse of Young's modulus Y is the elastic compliance 's'.
 
Dissipation Factor or tan δ
 
This is also frequently called loss tangent and is a measure of the dielectric losses in the material, expressed as the tangent of the loss angle or the ratio of resistance to reactance of a parallel equivalent circuit of the ceramic element. It is measured directly at 1 kHz using LCR Bridge.
 
Curie Point
 
It is the temperature at which the crystal structure of the material changes from a piezoelectric to nonpiezoelectric state. It is also the temperature at which the dielectric constant peaks. Each ceramic composition has its characteristic Curie Point and in use the operating temperature must be kept substantially
below the Curie Point.
 
Aging Rate
 
The change or aging of the material parameters that occurs after the poling of the ceramic is called aging rate. The aging is a logarithmic function of time. The aging rate defines change in the relevant parameters per decade of time, for example 1 - 10 days, 10 - 100 days etc. The most parameters that age with time are: KT33,
Kp Freq. constant, Qm, etc.
 
Mechanical Quality Factor (Qm)
 
The mechanical Q is a dimensionless number which gives the quality of the ceramic as a harmonic oscillator. It is the reciprocal of the damping factor. The electrical analogue ( in an equivalent electric series circuit, representing the mechanical vibrating resonance system) is the ratio of reactance to resistance. The shape of the part affects the value.
 
Frequency Constants
 
The frequency constant, N, is the product of the resonance frequency and the linear dimension governing the resonance. It is also equal to half the sound velocity in the same direction. The constant can be used to calculate the resonant frequency at which an element would operate.
 
Np   :                Planar mode of thin disc
Nt    :                Thickness mode of thin plate.
 
To calculate the resonant frequency of a given element in kHz, divide the frequency
constant by the controlling dimensions as shown below.
fr = Frequency Constant (Hz-m)
    Dim. (L,T,D) in m.