Electrical generator or motor structure – Non-dynamoelectric – Piezoelectric elements and devices
Reexamination Certificate
1999-08-25
2001-05-22
Dougherty, Thomas M. (Department: 2834)
Electrical generator or motor structure
Non-dynamoelectric
Piezoelectric elements and devices
C310S328000
Reexamination Certificate
active
06236143
ABSTRACT:
FIELD OF THE INVENTION
This invention relates to transducer devices and, more particularly, to piezoelectric devices which exhibit enhanced coupling coefficients.
BACKGROUND OF THE INVENTION
Many mechanical transducers employ shape-changing materials as an integral part of their construction. An example of a material exhibiting such behavior is a piezoelectric ceramic. In the case of actuators, shape changes or strains, are the result of the application of an imposed external signal, such as an electric field. Device performance depends intimately on the ability of these materials to convert energy from one form to another. One measure of the effectiveness with which a material or device converts the energy in an imposed signal to useful mechanical energy is the coupling coefficient.
One definition of a coupling coefficient is the following: the ratio of the energy converted to that imposed is equal to the square of the coupling coefficient, k. Thus, no material coupling coefficient can be greater than 1.0, as this represents the limit of 100% conversion of imposed energy to mechanical energy. In addition, as the result of the ability to impose signals in different ways, as well as the ability of a material to strain in different ways, any material has multiple coupling coefficients corresponding to different modes of excitation and response. The largest coupling coefficients for piezoelectric ceramic materials are on the order of 0.7, corresponding to energy conversion factors of about 50%. Considerable research has addressed the development of new material compositions that might exhibit higher electromechanical coupling. See: Cross, et al., “Piezoelectric and Electrostrictive Materials for Transducer Applications”, 1991 Annual Report, ONR Contract No. N00014-89-J-1689.
Devices made using such active materials are also said to have coupling coefficients. These are properties of the device and, although related to the material coupling coefficients, are generally different from them. Various device coupling coefficients can also be defined, corresponding to specific modes of excitation and response. Accepted design guidelines suggest two ways to maximize device (and composite material) coupling coefficients: 1) use a material with high inherent coupling; and 2) configure the device so as to best use the available material coupling. See: Wallace et al., “The Key Design Principle for Piezoelectric Ceramic/Polymer Composites,” Recent Advances in Adaptive and Sensory Materials and Their Applications, pp. 825-838, Apr. 27-29, 1992; and
Smith et al., “Maximal Electromechanical Coupling in Piezoelectric Ceramics-Its Effective Exploitation in Acoustic Transducers,” Ferroelectrics, 134, pp. 145-150, 1992.
Considerable research has addressed ways to exploit material coupling, resulting in devices such as the “moonie”. see U.S. Pat. No. 4,999,819. It is commonly held that no device coupling coefficient can be greater than the largest coupling coefficient of the active material used in the device.
Piezoelectric Coupling Coefficients
Piezoelectric Material Coupling
The behavior of piezoelectric materials involves coupled mechanical and electrical response. The constitutive equations of a linear piezoelectric material can be expressed in terms of various combination of mechanical and electrical quantities (stress or strain, electric field or electric displacement). In light of the popularity of the modern displacement-based finite element method, the constitutive equations used herein employ the strain and electric fields. (Strain is related to the gradient of the mechanical displacement field, while electric field is the gradient of the electric potential field.) In condensed matrix notation, the nine constitutive equations for a typical piezoelectric ceramic material are:
{
T
D
}
=
[
c
E
-
e
T
e
ϵ
S
]
{
S
E
}
(
1
)
where
T is the stress vector; S is the strain vector (6 components each);
D is the electric displacement vector;
E is the electric field vector (3 components each);
c
E
is a matrix of elastic coefficients (at constant electric field);
e is a matrix of piezoelectric coefficients; and
&egr;
S
is a matrix of dielectric permittivities (at constant strain).
Simple Strain/Electric Field Patterns
In engineering analysis, materials may sometimes be assumed to experience a state in which only a single stress or strain component is non-zero, and in which only a single electric field or electric displacement component is non-zero. In that event, the nine constitutive equations may be reduced to two, so that the matrices of coefficients become scalars. The corresponding single coupling coefficient may be found from either:
The difference between the open-circuit (constant electric displacement) stiffness (c
D
) and the short circuit (constant electric field) stiffness (c
E
):
k
2
=
c
D
-
c
E
c
D
=
(
c
E
+
e
2
ϵ
S
)
-
c
E
(
c
E
+
e
2
ϵ
S
)
=
e
2
(
c
E
ϵ
S
+
e
2
)
(
2
)
The difference between the free (constant stress) permittivity (&egr;
T
) and the blocked (constant strain) permittivity (&egr;
S
):
k
2
=
ϵ
T
-
ϵ
S
ϵ
T
=
(
ϵ
S
+
e
2
C
E
)
-
ϵ
S
(
ϵ
S
+
e
2
C
E
)
=
e
2
(
c
E
ϵ
S
+
e
2
)
(
3
)
Eigen Strain/Electric Field Patterns
Eigenanalysis of the constitutive equations for a typical piezoelectric ceramic material reveals that only three characteristic strain/electric field patterns exhibit electromechanical coupling. Because each stress/electric displacement pattern is related to the corresponding strain/electric field pattern by a scalar (the eigenvalue), individual patterns may be considered to be effectively one-dimensional; the total electromechanical system may then be considered as a set of parallel one-dimensional systems. When the conventional coordinate system is used (“3”) the poling direction, and “1-2” the plane of isotropy), the three patterns which exhibit electromechanical coupling involve the three components of the electric field vector individually; the first two involve shears in planes normal to the plane of isotropy, and the third involves a combination of all three normal strains. For many materials, the coupling coefficient associated with each of these three eigen patterns is about 0.70.
Arbitrary Strain/Electric Field Patterns
An effective coupling coefficient may be defined for an arbitrary quasistatic electromechanical state of the material from energy considerations. For the selected form of the constitutive equations (block skew symmetric), the total energy density is the sum of the mechanical (strain) energy density and the electrical (dielectric) energy density:
U
tot
=
U
mech
+
U
elec
where
U
mech
=
1
2
{
S
}
T
[
c
E
]
{
S
}
and
U
elec
=
1
2
{
E
}
T
[
ϵ
S
]
{
E
}
(
4
)
Although with this form of the constitutive equations there is no “mutual” energy density, a “one-way coupled” energy density may be defined as:
U
coup
=
1
2
{
E
}
T
[
e
]
{
S
}
=
1
2
{
S
}
T
[
e
T
]
{
E
}
(
5
)
With these definitions, an effective coupling coefficient for an arbitrary electromechanical state may be defined as:
k
2
=
U
coup
U
tot
=
U
coup
U
mech
+
U
elec
(
6
)
Of course, this relation is most meaningful when the state considered corresponds to a quasistatic equilibrium attained as the result of some electromechanical loading process starting from zero initial conditions. Also, since any electromechanical state of the material can be expressed as a linear combination of the eigen patterns discussed in the preceding, the coupling coefficient associated with an arbitrary state cannot be greater than the largest eigen coupling coefficient.
When the electromechanical loading process corresponds to purely electrical or purely mechanical loading, special cases of Eq. 6 may be developed. In that case, the total energy is equal to the work done by the loading system, and the transduced energy
Davis Christopher L.
Lesieutre George A.
Dougherty Thomas M.
Ohlandt Greeley Ruggiero & Perle L.L.P.
The Penn State Research Foundation
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