piezo ceramic tutorials

Definition & HistoryPiezoelectricity - Definition and HistoryPiezoelectricity (from: pressure electricity) is a property of certain classes of crystalline materials including natural crystals of Quartz, Rochelle Salt and Tourmaline plus manufactured ceramics such as Barium Titanate and Lead Zirconate Titanates (PZT).When mechanical pressure is applied to one of these materials, the crystalline structure produces a voltage proportional to the pressure. Conversely, when an electric field is applied, the structure changes shape producing dimensional changes in the material.The piezoelectric materials from Morgan Electro Ceramics use polycrystalline ceramics instead of natural piezoelectric crystals. These are more versatile with physical, chemical and piezoelectric characteristics able to be tailored to specific applications. The hard, dense ceramics can be manufactured in almost any given shape or size. They are chemically inert, and immune to moisture and other atmospheric conditions.Morgan Electroceramic, as part of a group which is a world leader in Advanced Ceramics, has piezoelectric R&D and manufacturing facilities in Southampton, England; Ruabon, Wales and Bedford, Ohio.Historical Note:The Piezoelectric effect was discovered byPierre Curie (1859-1906) in the early 1880's.

Piezoelectric ActionsPiezoelectric ActionsThe piezoelectric effect for a given item depends on the type of piezoelectric material and the mechanical and electrical axes of operation can be precisely orientated within the shape of the ceramic. These axes are set during "poling"; the process that induces piezoelectric properties in the ceramic. The orientation of the dc poling field determines the orientation of the mechanical and electrical axes.The poling field can be applied so the ceramic exhibits piezoelectric responses in various directions or combination of directions.The poling process permanently changes the dimensions of a ceramic element. The dimension between the poling electrodes increases and the dimensions parallel to the electrodes decrease. The effect is shown in the fig 1, albeit greatly exaggerated.

http://www.morganelectroceramics.com/resources/piezo-ceramic-tutorials/piezoelectric-actions/Axes

Piezoelectric materials are anisotropic - their electrical, mechanical, and electromechanical properties differ for electrical or mechanical excitation along different directions. Thus for systematic tabulation of properties, we require a standardized means for identifying directions. Where crystals are concerned, we use the orthogonal axes originally assigned by crystallographers. However, we identify the axes by numerals:1 corresponds to x axis,2 corresponds to y axis,3 corresponds to z axis.Piezoelectric ceramics are isotropic and are not piezoelectric before poling. Once they are polarized, however, they become anisotropic. The direction of the poling field is identified as the 3 direction. In the plane perpendicular to the 3 axis, the ceramics are non directional. Accordingly, the 1 and 2 axes may be arbitrarily located but, of course, they must be perpendicular to each other.

Piezoelectric VoltagePiezoelectric Voltage - Static ActionsAfter the poling process is complete, a voltage lower than the poling voltage changes the dimensions of the ceramic for as long as the voltage is applied.A voltage with the same polarity as the poling voltage causes additional expansion along the poling axis and contraction perpendicular to the poling axis (fig. 2). A voltage with the opposite polarity has the opposite effect: contraction along the poling axis, and expansion perpendicular to the poling axis. In both cases, the ceramic element returns to its poled dimensions when the voltage is removed from the electrodes. These effects are shown greatly exaggerated in fig. 2.

Piezoelectric VoltageAfter the poling process is complete, compressive and tensile forces applied to the ceramic element generate a voltage. Refer to fig. 3. A voltage with the same polarity as the poling voltage results from a compressive force (a) applied parallel to the poling axis, or from a tensile force (b) applied perpendicular to the poling axis. A voltage with the opposite polarity results from a tensile force (c) applied parallel to the poling axis, or from a compressive force (d) applied perpendicular to the poling axis.

The instances above can also be arranged to occur in the other two planes, i.e. thickness shear and face shear.

Generally two or more of these actions are present simultaneously. In some cases one type of expansion is accompanied by another type of contraction which compensate each other resulting in no change of volume. For example, the expansion of length of a plate may be compensated by a n equal contraction of width or thickness. In some materials, however, the compensating effects are not of equal magnitude and net volume change does occur. In all cases, the deformations are very small when amplification by mechanical resonance is not involved. The maximum displacements are on the order of a few microinches.http://www.morganelectroceramics.com/resources/piezo-ceramic-tutorials/piezoelectric-voltage/

Vibrations & DisplacementsPiezoelectric Ceramic Vibrations & DisplacementsPiezoelectric Ceramic Vibrations & DisplacementsWhen the crystalline structure of a material has no centre of symmetry, it is noncentrosymmetric. A single crystal with this structure has anisotropic characteristics: the properties of the material differ according to the direction of measurement. Piezoelectricity is an anisotropic characteristic. The non-centrosymmetric crystalline structure provides a net electric dipole moment within the crystal unit cell. Any dipoles aligned in the same direction will arrange themselves into regions called domains.Piezoelectric ceramics are ferroelectric materials. These materials have noncentrosymmetric unit cells below a certain temperature and a centrosymmetric structure above that temperature. That temperature is the Curie temperature. Above the Curie temperature, these ceramics have a centrosymmetric structure and have no piezoelectric characteristics. Below the Curie temperature, these ceramics have a noncentrosymmetric structure.A ceramic material is composed of many randomly oriented crystals or grains, each having one or a few domains. With the dipoles randomly oriented, the material is isotropic and does not exhibit the piezoelectric effect. By applying electrodes and a strong d.c. electric field, the dipoles will tend to align themselves parallel to the field, so that the material will have a permanent (or remanent) polarization. Refer to Figure 5. Not as many domains can align their dipoles in ceramic materials as in single crystals, but enough do so that the material will become piezoelectric. After poling, the material has a remanent polarization (Pr) and remanent stress (Sr) as shown in Figures 6A and B.Fig 5 Poling

UnpoledPoled

Fig 6 Hysteresis

Fig 6AFig 6B

As this "poled" ceramic is subjected to stress, the crystal lattice or grains distort causing some of the domains to grow at the expense of others. This changes the total dipole moment of the material. Within a certain range of stress, this change of dipole moment with stress gives rise to piezoelectric property (and its converse) can be used practically, because the material's response is nearly linear.

Dynamic PerformanceDynamic PerformanceDynamic PerformanceDynamic performance relates to the behaviour of a material when subjected to alternating fields or stresses at frequencies close to the mechanical resonance of a component. Piezoelectric transducers may be approximately represented by the equivalent circuit shown in Figure 7. The mechanical resonance of the device is represented by L1, C1, and R1 . Since it is a dielectric with electrodes it also has an electrical capacitance C2.

It is this parallel combination of C2 with L1, C1, and R1 that dictates the reactance of the transducer, which varies with frequency as shown in Figure 8.The graph shows the curve decreasing to a minimum impedance, at a frequency fm followed by a sharp increase to a maximum at fn These two frequencies are the resonance and anti-resonance respectively. The performance of a transducer will have a maximum response at a point which lies between these points.

Circuit ConsiderationsCircuit ConsiderationsCircuit ConsiderationsTo obtain optimum performance from a piezoelectric device, the circuit to which it is connected must have certain characteristics which are dictated by the design of the device. In discussing this subject, it is convenient to divide piezoelectric devices into two broad categories non-resonant devices and resonant devices. Non resonant devices are so named because they are designed to operate well below resonance, or over a relatively large frequency range, usually several octaves. A resonant device either operates at its mechanical resonance or over a band of less than one octave around this resonance.Circuit Considerations - Non-Resonant Mechanically Driven DevicesFor most practical circuit design purposes, a non-resonant piezoelectric generator device, such as a microphone or accelerometer, together with the load on the device, may be represented by the equivalent circuit of Fig 9A,

where Ce represents the capacitance of the piezo-electric element, Rc is the shunt leakage resistance of the device, CL is the shunt load capacitance, RL represents the load resistance and eL represents the load potential.The values of the shunt leakage resistance and the capacitance of the piezoelectric element are dependent on the dimensions of the element and, in the case of leakage resistance, also upon the dryness of the surface. Under normal operating conditions ,the value of this resistance is many times greater than the normal load resistance, ranging from several hundred megohms to many thousands of megohms. Generally, except at very low frequencies, the capacitances in the circuit have reactances very much lower than the circuit resistances. Under these conditions, the circuit reduces to the simple capacitive voltage divider circuit (Fig 9B).

It can be seen from this circuit that the shunt load capacitance (CL) of wiring, cable, and amplifier input does not bring about a relative loss of high frequency response as in the case of resistive or inductive sources shunted by capacitance loads, but instead reduces the output at all frequencies (except very low frequencies as discussed in the next paragraph).By the application of elementary network theory, the basic circuit can be reduced to the equivalent circuit of Fig 9C which is useful for analyzing the low frequency response. From this it can be seen that when the combined reactance of the piezoelectric and circuit capacitances in parallel approaches the combined resistance, the low frequency response begins to fall off. This is analogous to the situation existing in conventional RC-coupled amplifiers.

Circuit Considerations - Non-Resonant Electrically Driven DevicesIn the case of non-resonant piezoelectric devices which are electrically driven, the electrical impedance of the device may, for most practical purposes, be considered to be purely capacitive. For all frequencies well below the first mechanical resonance of the device, the electromechanical relationships are such that the displacement of the piezo electric element from its normal position, at any instant, is directly proportional to the electric charge applied at that instant.

Resonant DevicesCircuit Considerations - Resonant DevicesThe electrical impedance of a piezoelectric device is in reality more complicated than the simple capacitor representation generally employed in discussing non resonant devices. A more proper representation would be a capacitor representing the static capacitance of the piezo electric element, shunted by an impedance representing the mechanical vibrating system. In most non resonant devices, the latter impedance may be approximated by a capacitor. Therefore, we have a capacitor in parallel with a capacitor- hence the single capacitor representation.In devices designed for operation at resonance, the impedance representing the mechanical system may become, at resonance, a resistance of relatively low value and this is shunted by the same static capacitance.The shunt static capacitance generally is undesirable, whether the device is designed for operation at resonance or for broadband, below resonance operation. ln electrically driven devices, it shunts the driving amplifier or other signal source requiring that the source be capable of supplying extra current. In the case of mechanically driven devices, the static capacitance acts as a load on the active part of the transducer, reducing the electrical output.In non-resonant devices, not much can be done about the shunt capacitance, except choose a piezoelectric material having maximum activity. In resonant devices, however, the static capacitance may be "neutralized " by employing a shunt or series inductor chosen to resonate with the static capacitance at the operating frequency. This is illustrated in Figure 10.

fig10 resonant device with static capacitance neutralized by inductor

Typical ShapesTypical ArrangementsUnimorph

The magnitude of piezoelectric forces, actions and voltages is relatively small. The maximum extension of a single element is in the order of fractions of a micron. Amplification is often required and can be achieved by various arrangements of the piezo ceramic such as Unimorph, Bimorph and Stacks.A Unimorph is made by bonding a thin piece of piezo ceramic to an inactive substrate. Driving the piezo-ceramic will lead to deformation of the entire structure. Transducer Products Division has expertise in making various high specification Unimorphs for Hydrophones, Sensors and Actuators

Bimorphs

A Bimorph is made by bonding two pieces of piezo-ceramic together so that differential changes in length of the two pieces can produce relatively large movements. This element consists of two transverse expander plates secured together face to face in such a manner that a voltage applied to the electrodes causes the plates to deform in opposite directions, resulting in a bending action. The displacement of the Bimorph in response to an applied voltage is many times greater than the corresponding displacement of a single plate, typically in excess of 10m per volt. Thus Bimorphs have, in effect, 'built-in' levers to provide or require much greater motion but less force than single plates.

Actuator Stacks

Stacks are several piezoelectric elements connected mechanically in series and electrically in parallel. The displacement of each transducer element adds to the total displacement. The displacement of the whole stack assembly is equal to the sum of the individual displacements. Stacks are generally required for applications requiring large displacements (typically between 5 and 180m).

Sandwich Transducers

It is difficult to make single blocks of ceramic resonating below about 100 KHz. Instead a composite half wave resonator is used consisting of two or more piezo-ceramic rings sandwiched between metal layers.

This arrangement can achieve low frequencies at high drive levels.http://www.morganelectroceramics.com/resources/piezo-ceramic-tutorials/typical-shapes/

Symbols & NotationSymbols & NotationThe piezoelectric properties are described by a system of symbols and notations, identifying compliance, electromechanical coupling, etc.The tables include the MKS units used in measuring each property.Also see the later section onUnits & Symbols for a complete listing.Ceramic Property DefinitionsPropertyDefinitionMKS Units

Electro Mechanical Coupling Coefficient

kor...-

-

Piezoelectric Constants

dm / V

C / N

gV -m/ N

m2/ C

Relative Dielectric Constant

K-

Modulus of Elasticity

YNm-2

Density

kg/m3

Frequency Constant

NControlling Dimension x Resonant FrequencyHz m

Superscript and subscript notations describe the characteristics of a property. The next table illustrates and explains several examples of annotated symbols. The superscripts describe external factors (physical mounting, electrical conditions, etc.) that effect the piezoelectric property. The subscripts describe the relationship of the property to the poling axis.The characteristics of piezoelectric properties depend on their orientation to the poling axis. This orientation determines the direction of the action or response associated with the property. The subscript notations define the axes of a component in terms of orthogonal axes: 1 corresponds to the x-axis, 2 corresponds to the y-axis, and 3 corresponds to the z-axis. Conventionally, the direction of polarization is defined as the 3 axis. (See below)

(4, 5 and 6 refer to shear strains)The first subscript position identifies the direction of the action; the second identifies the direction of the response. For example, refer to the piezoelectric "d " constant in the following table. The first subscript refers to the direction of the field and the second refers to the direction of the strain. For the converse piezoelectric constant "g", the first refers to the stress and the second to the voltage.indicates that compliance is measured with electrode circuit open

indicates thatstressor strain is in direction 1

indicates thatstrainor stress is in direction 1

Compliance = strain / stress

indicates that compliance is measured with electrodes connected together

indicates thatstressor strain is in shear around axis 3

indicates thatstrainor stress is in direction 3

Compliance = strain / stress

indicates that all stresses on material are constant; for example zero external forces

indicates that electrodes are perpendicular to axis 1

Relative dielectric constant =

indicates that all stresses on material are constant; for example material completely blocked preventing deformation in any direction


Relative dielectric constant =

indicates that stress or strain is in shear around axis 2


Electromechanical coupling

Planar, used only for thin discs. It indicates electrodes perpendicular to axis 3 and stress or strain equal in all directions perpendicular to axis 3.

Electromechanical coupling

indicates that the piezoelectric induced strain, or the applied stress, is in direction 3


strainapplied stress=short circuit charge / electrode areaapplied stress

Indicates that stress is applied equally in 1, 2 and 3 directions (hydrostatic stress; and that electrodes are perpendicular to axis 3)

short circuit charge / electrode areaapplied stress

indicates that the applied stress or piezoelectric induced strain is in direction 1


fieldapplied stress=strainapplied charge / electrode area

indicates that the applied stress or piezoelectric induced strain is in shear form around axis 2


fieldapplied stress=strainapplied charge / electrode area

Typical PropertiesEquivalent DoD Navy Type IParameterSymbolUnit

General Material Designations4

KT331115 - 1435

Dissipation Factor0.004

Qm500 - 600

Densitykg/m37600

Curie TemperatureC320

CouplingCoefficientskp0.580

k31-0.340

PiezoelectricCharge Constantsd33x10-12C/N or m/V290

d31x10-12C/N or m/V-125

PiezoelectricVoltage Constantsg33x10-3Vm/N24.6

g31x10-3Vm/N-10.6

Frequency ConstantNpHz.m2220

Elastic Constants Short CircuitSE11SEx10-12m2/N12.4

YE11SEx10-12m2/N8.1

Elastic Constants Open CircuitSD11SDx10-12m2/N11.0

YD11SDx10-12m2/N9.1

High Field Dielectric Properties (200kV/m)KT33%100

Time Constant@ 100Cs>5

Time Constant@ 200Cs>0.07

Aging Rates & Time StabilityKT33% / time decade-4.6

kp% / time decade-1.7

d33% / time decade-3.4

Np1.0

Temperature Stability% change in KT33from 0 - 50C8.0

Equivalent DoD Navy Type IIParameterSymbolUnit


KT331770


Qm75

Densitykg/m37800


Coupling Coefficientskp0.600

k31-0.343


d31x10-12C/N or m/V-177


g31x10-3Vm/N-11.1



YE11SEx10-12m2/N6.2


YD11SDx10-12m2/N7.1

High Field Dielectric Properties (200kV/m)KT33%12.5

DF0.0238


DF0.0551

Volume Resistivity@ 25Cohm.m>1012



Time Constant@ 25Cs

Time Constant@ 100Cs





Np0.2


Equivalent DoD Navy Type IIIParameterSymbolUnit


KT33980 - 1180


Qm900 - 1600

Densitykg/m3>7500

Curie TemperatureC>300

CouplingCoefficientskp>0.520

k31-0.350


d31x10-12C/N or m/V-127


g31x10-3Vm/N-12.2



YE11SEx10-12m2/N7.8


YD11SDx10-12m2/N8.9


DF

High Field Dielectric Properties (400kV/m)KT33%

DF






Time Constant@ 200Cs>0.01




Np1.0


Equivalent DoD Navy Type VParameterSymbolUnit

General Material Designations5J

KT332650


Qm71

Densitykg/m3>7600

Curie TemperatureC>250


k31-0.375


d31x10-12C/N or m/V-230


g31x10-3Vm/N-9.8



YE11SEx10-12m2/N6.2


YD11SDx10-12m2/N7.3


DF


DF

Volume Resistivity@ 25Cohm.m



Time Constant@ 25Cs






Np0.2


Equivalent DoD Navy Type VIParameterSymbolUnit

General Material Designations5H

KT333300


Qm67

Densitykg/m37500



k31-0.375


d31x10-12C/N or m/V-264


g31x10-3Vm/N-8.9



YE11SEx10-12m2/N5.9


YD11SDx10-12m2/N6.9


DF


DF










Np0.3


Custom MaterialsParameterSymbolUnitPZT 5BPZT 5RPZT 5MPZT 5KPZT 7APZT 7DPT2 / PC6

General Material Designations5B5R5M5K7A7DPT

KT3323502000401555004101300218

Dissipation Factor0.0200.0200.0280.0230.0250.0050.022

Qm808045615806001150

Densitykg/m37900790077007900790078006900


CouplingCoefficientskp0.6400.6300.6300.6500.5100.510kt=0.510

k31-0.380-0.385-0.370-0.380-0.300-0.300-0.030


d31x10-12C/N or m/V-210-200-270-323-60-112-3

PiezoelectricVoltage Constantsg33x10-3Vm/N25.527.120.018.641.321.034.8

g31x10-3Vm/N-10.1-11.5-7.6-6.9-16.2-9.6-2.1


Elastic ConstantsShort CircuitSE11SEx10-12m2/N14.715.715.016.010.611.87.5

YE11SEx10-12m2/N6.86.46.76.39.48.513.4

Elastic ConstantsOpen CircuitSD11SDx10-12m2/N12.613.312.913.89.710.77.4

YD11SDx10-12m2/N7.97.57.87.310.49.413.5

High Field DielectricProperties (200kV/m)KT33%-0.9

DF0.016

High Field DielectricProperties (400kV/m)KT33%-1.0

DF0.016

Volume Resistivity@ 25Cohm.m>1011>109>1010

Volume Resistivity@ 100Cohm.m>1011>108

Volume Resistivity@ 200Cohm.m>1010>106.5

Time Constant@ 25Cs>2000>10

Time Constant@ 100Cs>1800>0.5

Time Constant@ 200Cs>250>0.03

Aging Rates &Time StabilityKT33% / time decade-0.3-3.60.060.03-2.3

kp% / time decade0.00.00.00.006kt=1.7

d33% / time decade-3.0-3.10.00.0

Np0.10.2-0.050.020.2

Temperature Stability% change in KT33from 0 - 50C15.616.318.111.3

Back to top...

Typical Thermal EffectsTemp CPZT4DPZT5APZT5A

1st HeatingFirst HeatingSubsequentHeatings

-196--0.02-0.02

-80-0.025-0.03-0.02

-60-0.025-0.03-0.02

0-0.025-0.04-0.02

30-0.028-0.06-0.02

60-0.026-0.07-0.02

80-0.025-0.09-0.02

100(a)-0.11-0.02

200(a)-0.17-0.04

300(a)-0.23-0.09

Pyroelectric effects, in 10-6coul/cm2(10-2coul/m2C)(a) Above about 80C the pyroelectric effect is masked by anomalous dielectric charges.

Thermal Expansion Coefficient (in 10-6/C)Poled PZT5A1st Heating1st HeatingSubsequentHeatingsSubsequentHeatings

C1313

0+1.5+2+1+4

50+1.5+2+1.4+4

100+6-6+2+3

150+6-7+2.7+1

200+5-7+3.3-1.6

250+4.2-6+3.9-4.2

Poled PZT4D1st Heating1st HeatingSubsequentHeatingsSubsequentHeatings

C1313

0+1.5+0.1+3.8+1.7

50+4.5-0.1+3.8+1.7

100+5.8-6+3.8-1

150+6.4-6+3.8-1.4

200+5.4-6.1+3.4-2.4

250

Virgin UnpoledPZT5APZT4D

C

0+2.5+2.0

50+2.1+1.8

100+2.0+1.5

150+1.8+1.1

200+1.5+1.0

250+1.0+0.3

300+0.70.0

350-3.0+6.2

400+5.0+7.8

500+8.2+8.2

As noted above, thermal expansion of PZT4D and PZT5A is extremely anisotropic only on first heating, and on first heating only above about 50CHeat CapacityPZT,approx 420 joules/kgC (138 joulesC mole)

Thermal ConductivityPZT,approx 1.8W/mC

Typical Responsesg31vs Temperature

d31Vs Temperature

Relative Dielectric Constant Vs Temperature

Mechanical Q Vs Temperature

Planar Coupling Factor Vs Temperature

Frequency Constant Vs Temperature

Back to top...Typical High Signal PropertiesPZT4DPZT5APZT8

AC depoling Field>1.00.7>1.5

AC field for tan=0.04, 25C(a)0.390.45>1.0

% increase ofT33at above electric field171110

AC field for tan=0.04, 100C0.330.045n/a

Max rated static compressive stress (maintained) PARALLEL to polar axis25C82.720.7 or 34.5(c)82.7

100C41.420.741.4

% change ofT33with stress increase to rated max compressive stress at 25C(b)+25% approx(d)-3% approx(d)+18% approx(d)

% change of d33with stress increase to rated max compressive stress at 25C(b)15% approx(d)0% at 20.7-13% at 34.5 approx+6%(d)

Max rated compressive stress (cycled) PARALLEL to polar axis25C82.720.782.7

100C41.420.741.4

Max rated static compressive stress (maintained) PERPENDICULAR to polar axis25C55.213.855.2

100C27.613.827.6

% change ofT33with stress increase to rated max compressive stress at 25C(b)+10% approx--2%

% change of d31with stress increase to rated max compressive stress at 25C(b)-10% approx(f)--10%

Maximum rated hydrostatic pressure345138345

Compressive Strength>517>517>517

Tensile Strength, Static(g)75.875.875.8

Tensile Strength, Dynamic (peak)(g)24.127.634.5

Mechanical Q at 0MPa600751000

Mechanical Q at 7MPa180 approx25 approx800

Mechanical Q at 14MPa110 approx25 approx500

% increase in sE11at 7MPa1.7 approx10.5 approx0.1

% increase in sE11at 14MPa3.7 approx17 approx0.2

Notes(a) The value of tanat a given electric field is a function of time after poling or after any major disturbance such as exposure to an elevated temperature.(b) After appropriate stabilizing treatment. This consists of a temperature stabilization plus a few minutes soak at the appropriate static stress. The temperature stabilization is, however, more important than the stress soak.(c) The higher figure applies to a receiver, the lower to a radiator. The recommended use of PZT-5A or PZT-5H is the former.(d) In range to 70 MPa.(e) In range to 35 MPa.(f) The lateral d-constant perpendicular to the stress and polar axis is up about 20%.(g) These figures are dependent upon configuration and perfection of fabrication. The static tensile strength figures were obtained from bending tests on thin "Bimorph" structures,while the dynamic tensile strength figures were obtained from measurements of high amplitude resonant vibration of rings The latter tests are more sensitive to minor flaws.Back to top...Ageing Rates and Time StabilityMost of the properties of piezoelectric ceramics change gradually with time. The changes tend to be logarithmic with time after poling. The ageing rate of various properties depends on the ceramic composition and on the way the ceramic is processed during manufacture. Because of ageing, exact values of various properties such as dielectric constant, coupling, and piezoelectric constants may only be specified for a standard time after poling. The longer the time period after poling, the more stable the material becomes. The ageing process in any ceramic can be accelerated by exposing the ceramic to one or more of the following conditions.(l ) high mechanical stress(2) strong electric depoling field(3) high temperature approaching the Curie pointMaterial selection should be based on the conditions of a given application. Some typical ageing rates of various material properties are given in the following tableTime Stability (percent change per time decade) for some common materialsPropertyMaterial

PZT4DPZT8PZT5APZT5JPZT5HPZT7A

KT33-4.6-4.0-0.9-1.1-0.6+0.06

kp-2.0-1.5-0.1-0.3-0.20.0

d33-3.4-6.3-2.9-4.0-3.90.0

Np+1.2+0.9+0.1+0.2+0.3-0.05

Temperature StabilityThe performance characteristics of the electric and piezo electric properties are affected by temperature variations. Each piezoelectric material is affected differently by temperature changes, according to the method of manufacture and chemical composition of the material. The changes in the various material properties with temperature are shown in the following table for all PZT materials.Temperature Stability of KT33Material%KT33(%change from 0 - 50C)

PZT-4D8.8

PZT810.4

PZT5A11.3

PZT5J24.1

PZT5H30.7

PZT7A18.1

LimitationsLimitationsLimitationsEach piezoelectric material has a particular operating limit for temperature, voltage, and stress. The particular chemical composition of the material determines the limits. Operating a material outside of these limitations may cause partial or total depolarization of the material, and a diminishing or loss of piezoelectric properties.Temperature LimitationsAs the operating temperature increases, piezoelectric performance of a material decreases, until complete and permanent depolarization occurs at the material's Curie temperature.The Curie point is the absolute maximum exposure temperature for any piezoelectric ceramic. Each ceramic has its own Curie point. When the ceramic element is heated above the Curie point, all piezoelectric properties are lost. In practice, the operating temperature must be substantially below the Curie point.The material's temperature limitation decreases with greater continuous operation or exposure. At elevated temperatures, the ageing process accelerates, piezoelectric performance decreases and the maximum safe stress level is reduced.Voltage LimitationsA piezoelectric ceramic can be depolarized by a strong electric field with polarity opposite to the original poling voltage.The limit on the field strength is dependent on the type of material, the duration of the application, and the operating temperature. The typical operating limit is between 500V/mm and 1 000V/mm for continuous application.It should be noted that alternating fields can have the same affect during the half cycle which is opposite to the poling direction.Mechanical Stress LimitationsHigh mechanical stress can depolarize a piezoelectric ceramic. The limit on the applied stress is dependent on the type of ceramic material, and duration of the applied stress.For dynamic stress (impact ignition) the limit is less severe; materials with higher energy output (high g constant) can be used.For impact applications, the material behaves quasi statically (non-linear) for pulse durations of a few milliseconds or more. When the pulse duration approaches a microsecond, the piezoelectric effect becomes linear, due to the short application time compared to the relaxation time of the domains.Power Limitations

The acoustic power handling capacity of a radiating transducer is limited by the following factors.(1) Dynamic mechanical strength of the ceramic(2) Reduction in efficiency due to dielectric losses(3) Reduction in efficiency due to mechanical losses(4) Depolarization of the ceramic due to electric field(5) Depolarization of the ceramic due to temperature rise(6) Instability resulting from the positive feedback between dielectric losses and internal heating (2 and 5)In practice, power limitations are imposed by factors 2 and 5 and the feedback between them (6). depending on the composition of the ceramic. Factors 1, 3 and 4 may be neglected. Factor 1 may be reduced through mechanical bias in sonar, ultrasonic, and other similar applications. Factor 3 may be generally disregarded, since mechanical losses are negligible compared to dielectric losses. In the case of factor 4, the electric field necessary to cause sufficient depolarization will create extremely undesirable operating conditions with very high dielectric losses and resulting low efficiency.A transducer may be efficiency-limited, temperature limited, or dynamic-strength limited. Dynamic strength is significant only when the transducer is not mechanically biased and the ceramic has a high QM A low frequency, low duty transducer is efficiency-limited. A high frequency continuous duty transducer is temperature-limited. Temperature limited transducers are dependent on the efficiency of the heat removal from the ceramic. Between these two extremes, the specific limitation is dependent on the mechanical design of the transducer. An absolute value on the power limitation of the ceramic cannot be determined without knowledge of its operating conditions.The equations pertaining to the power handling capacities of the material may be readily derived from lumped equivalent circuits. It can be shown that the acoustic power density P per cubic metre is given by Formula 1.

where k Is equal to k33 for a stack of axially poled rings or plates or k31 for a radialy poled cylinder. E is the rms electric field, and f, is the resonance frequency.It is assumed that the mechanical losses in the ceramic and the housing are negligible compared to dielectric losses. This tends to hold for materials with QM>100 The power per cubic metre dissipated in the ceramic by dielectric dissipation Pd is given by Formula 2.Formula 2

where f is the operational frequency.The efficiency of the transducerconsidering only the internal losses of the material is approximated by Formula 3.Formula 3

With high values of QMpower handling capacity of the material is limited at times by the dynamic tensile strength, even though a bias compressive stress as high as about 80 MPa is used with PZT-4D. In this case, the acoustic power is given by Formula 4.Formula 4whereis the rms stress

These equations may be simplified for the specific case of a matched transducer. Matching is the term applied to the process of adjusting the acoustic load so that it corresponds to the image impedance of the transducer, which is treated as a bandpass filter. In this case, an inductor equal to:

is connected across the transducer. The impedance of the driving electric generator is set equal to the image impedance in order to maximize the transducer bandwidth, where the generator resistance, RG and the mechanical load impedance, RT are given by Formula 5; the bandwidth is given by Formula 6; and the acoustic power and efficiency are given by Formula 7.Formula 5

Formula 6where f1and f2are the lower and upper cut-off frequencies

Formula 7

Table 9 lists the relative power for PZT-4D and PZT-5A at resonance for the same acoustic load for a given volume of material, assuming that the material is limited by the dielectric losses with Tan = 0.04.Relative Power for PZT-4D and PZT-5AMaterialModeTemp CRelative Power

PZT-4DParallel25100

PZT-4DParallel10065

PZT-4DTransverse2523

PZT-4DTransverse10012.5

PZT-5AParallel252.7

PZT-5AParallel1003.2

PZT-5ATransverse250.5

PZT-5ATransverse1000.6

Useful RelationshipsUseful RelationshipsUseful RelationshipsPiezoelectric Equations and ConstantsTo a good approximation, the interaction between the electrical and mechanical behaviour of the piezoelectric medium can be described by the following relationships:S = sET + dED = dT +TEE = -gT + (T)-1DS = sDT + gDE = field (Vm-1)T = Stress (Nm-2)S = Strain (dimensionless)D = Dielectric displacement (Cm-2)and the superscripted permittivityand compliance s denotes the quantity kept constant under boundary conditions (e.gTis the permittivity under constant stress)."d" and "g" are piezoelectric constants, related by the general expression:d =rogwhere:r= relative permittivity (or dielectric constant)o= permittivity of free space ( 8.85x10-12Fm-1)The piezoelectric constants are defined as follows:direct effectreverse effect

d=charge density developedCN-1d =strain developedmV-1

applied mechanical stressapplied field

g=electric field developedVmN-1g=strain developedm2C-1

applied mechanical stressapplied charge density

As well as the above there are other parameters to be considered which characterise a piezoelectric material; of prime importance are the coupling coefficient, loss factor and the mechanical quality factor.The Coupling CoefficientThis parameter determines the efficiency of energy conversion in the component (but not the overall efficiency of the ceramic as a transducer) and is defined as follows:(i) For an electrically stressed componentk2=stored mechanical energy total stored energy(ii) For a mechanically stressed componentk2=stored electrical energy total stored energy The derivation of k from critical frequencies is complex and graphical solutions are generally used to facilitate calculations of k from (fn - fm)/fm. (see IRE Standards on Piezoelectric Crystals: Measurements of Piezoelectric Ceramics, 1961.)An approximate solution which depends on the shape of the piece, the mode of vibration as well as the material and is useful in design is given by:

This expression is often used for thick (1Ot > d) discs and is then called kD.Dielectric LossThe efficiency of a transducer depends on the mechanical and dielectric loss as well as the coupling coefficient. The dielectric loss is usually the most significant factor and is the ratio of the effective series resistance to the effective reactance, or as in the diagram to the right. It is the tangent of the loss angle tan = series resistance series reactance Ceramics with a low tanshould be employed for transducers which are to be run continuously at high power levels.

Mechanical/ Quality Factor Qmis defined as the ratio of the energy supplied per cycle to the energy dissipated per cycle and can be calculated from:

where C is the low frequency (1 kHz) capacitance and Zm the minimum impedance. QM can also be determined approximately from the frequency response curve as right:The frequency difference fz - f, is the frequency bandwidth at about 3dB where the amplitude is 1 /SQR(2) of its maximum value.QM = fr

f2 - f1(only where Q>3)

Direction DependenceBecause poled piezoelectric ceramics are anisotropic and the direction of polarising may be freely chosen, a method of identifying the axes of a component is necessary in order to specify its parameters.The direction of polarisation is conventionally taken as the 3 axis, with axes 1 and 2 perpendicular to this. The terms 4, 5 and 6 refer to shear stains associated with the 1, 2 and 3 directions.This axis notation is used when specifying mast of the piezoelectric parameters discussed above.

Permittivity:iji - direction of dielectric displacement.j - direction of electric field.E.g.11Tis the permittivity for a material whose dielectric displacement and field are in the 1 direction under conditions of constant stress.

Compliance:siii - direction of strain.j - direction of stress.E.g. s55Dis the shear strain to shear stress ratio at constant electric displacement, for shear about an axis perpendicular to the poling direction.

Units & SymbolsUnits & SymbolsValues are SI metric.=Permittivity

0=Permittivity of free space (8.85x10-12Fm-1)

KT33=relative dielectric constant, free

KS33=relative dielectric constant, clamped

tan=1/QE=dissipation factor at 1kHz, low electric field

kp=planar coupling factor

k31=transverse or lateral coupling factor

k33=longitudinal coupling factor

k15=shear coupling factor

kt=thickness coupling factor (laterally clamped)

d=piezoelectric constant, strain/field at constant stress or charge density/stress at constant electric field, 10-12m/V

g=piezoelectric constant, electric field / applied stress at constant charge or strain/charge density at constant stress, 10-3Volt metres/Newton

SE=elastic compliance at constant electric field, 10-12m2/N

SD=elastic compliance at constant charge density, 10-12m2/N

QM=mechanical Q. This is dependent upon configuration, and is given here for a thin disc.

N1=frequency constant of a thin bar, Hz m.

N33=frequency constant of a disc or plate poled through thickness resonating in thickness mode

=density, 103kg/m3

T=temperature, C

=thermal expansion

Static OperationStatic and Quasi-Static OperationUnder static or quasi-static (below resonance) conditions, the magnitude of the piezoelectric effect is given by piezoelectric "d" and "g" constants. For the case of the direct piezoelectric effect where the material develops an electric charge from an applied stress, the definitions for "d" for constant field and "g" for constant dielectric displacement should be used. Refer to the table insection 9, Ceramic Property Definitions. For the converse effect where the material develops a strain from an applied electric field, the definitions for "d" and "g" for constant stress should be used. These "d" and "g" coefficients are related by Formula 8 for plates and discs, and Formula 9 for rods.

Formula 8(Plates & Discs)d31= g31T31

Formula 9(Rods)d33= g33T33

whereT33 is the permittivity of the material

The permittivity of the material is related to both the permittivity of free space and the dielectric constant of the material according to Formula 10.Formula 10kT33=T31 /0

where kT33is the relative dielectric constant of the material and0 is thepermittivity of free space (8.85x10-12farad/meter).

At frequencies far below the mechanical resonance frequency, the electro-mechanical coupling factor, K, can be calculated by Formula 11 for plates, Formula 12 for discs, Formula 13 for rods, and Formula 14 for shear plates.Formula 11(Plates)

Formula 12(Discs)

Formula 13(Rods)

Formula 14(Shear Plates)where s is the compliance of the material

The coupling factor is a useful expression relating the amount of energy that can be changed from the electrical form to the mechanical form, or visa versa, for the different operational modes. The coupling factor can be expressed as Formula 15.Formula 15k2=Stored energy convertedStored input energy

This value, although related, should not be considered the overall efficiency of the electromechanical transduction, since it does not take into account electrical and mechanical dissipation or losses. When a transducer is not operating at resonance or if it is not properly tuned and matched, the efficiency can be quite low. A properly designed transducer can operate at well over 90% efficiency. The pressure P which a ceramic driver can impart is given approximately by Formula 16.Formula 16P =dEYE11

Dynamic OperationDynamic OperationUnder dynamic conditions, the behaviour of the piezoelectric material is much more complex. It can be characterized in terms of an equivalent electrical circuit which exhibits the conditions of parallel and series resonance frequencies. To approximate these frequencies, measure the frequency of the minimum impedance (f,) and maximum impedance (fa) for the component, since they differ by a very small amount (3)

The frequency difference f2- f1is the frequency bandwidth at about 3dB where the amplitude is 1/SQR(2) of its maximum value.Of these losses, the dielectric losses are usually the most significant. Therefore it is recommended that materials with a low dissipation factor be used for high power applications, particularly since these losses increase with power. For high intensity transducers, the overall electroacoustical efficiencyis given approximately by Formula 22.Formula 22

Where QAis the mechanical quality factor due to the acoustic load alone.

It should be noted that at high drive levels QE and QM are not constants. They are usually lower than the low drive level values.The dielectric permittivity of the material. and therefore the dielectric constant and capacitance, decreases as the applied frequency (mechanical or electrical) exceeds each resonant frequency of the particular ceramic part. For static operation, well below the first resonance frequency, the dielectric permittivity isT33(free).For dynamic operation well above all resonance frequencies of the ceramic part, the material behaves as if it was clamped (strain = 0), and the electric permittivity isS33(clamped). Between each, the permittivity is the product of the static permittivity and a loss term based on the coupling of the resonance mode each resonance point the applied frequency has exceeded, as described in Formula 23 (above first resonance), Formula 24 (above second resonance), and Formula 25 (above third resonance).Formula 23(above first resonance)T33(1 - k12)

Formula 24(above second resonance)T33(1 - k12) (1 - k22)

Formula 25(above third resonance)T33(1 - k12) (1 - k22)(1 - k32)

where k1, kZand k3represent the coupling factors for the particular resonance For a thin plate, k1and k2are k31and k'31(length and width, respectively), and k3is kt(thickness) For a thin disc, k1is kp(radial), k2is Kt (thickness), and there is no third resonance. For a rod, k1is k33(length), k2is k'p, and there is no third resonance.In addition to FA and fr(series and parallel resonance frequencies), there is a frequency, fmat which the transducer's electromechanical transduction is maximized This frequency represents the maximum sensitivity for receivers or the maximum output for drivers This frequency, the bandwidth, and the output are all dependent on the external resistive load, ReX.When k

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