Technical Publication TP-224
Piezoelectric Coupling
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coupling. However, one always defines the above
ratio as the square of the coupling coefficient, K.
On The Meaning of Piezoelectric Coupling
by Robert Gerson
K2
Coupling is a concept which is useful wherever
energy is to be converted from one form into another
with only small losses. The coupling coefficient is a
number representing the extent of the conversion. Let
us consider the simplest case of energy storage: a
block, B, slides on a frictionless plane, P. It is held to a
wall, W, by a spring, S. An ant, A, pulls B from W,
stretching S and storing energy in it.
Using the above definition a coupling may be defined
whenever a substance accepts energy and stores it in
more than one form. For example, when a material
expands on heating a thermomechanical coupling
may be defined, based on the relative energies stored
in expansion and in internal heat. A steam engine
represents an application of thermomechanical coupling.
W
S
A
B
P
All of the energy delivered by A is used to stretch S.
Clearly A is perfectly coupled to S. If one measures the
‘coupling”, the only logical choice for numbers to
represent it is unity (one).
In converting energy one is rarely so fortunate as in
the case pictured above. It develops that usually the
best one can do in storing energy is to partially store it
ln a desired form and partially in some undesired way
S’
Piezoelectric coupling (the coupling of electric to
stress energy) occurs in materials in which an electric
field caused a dimensional change proportional to the
field. The electric energy is stored both as electric field
energy and as expansional energy in the piezoelectric
materials. In piezoelectric materials, it is also found
that a dimensional change produces electric charge or
field.
There are thus two cases of piezoelectric coupling:
1. Electrical energy is supplied to a piezoelectric
device with the object of causing a mechanical deformation.
(Switch Closed)
S
No electric Field on
Piezoelectric
A
K2
A is now trying to store energy in S by stretching S’. If
S is stiff, A is strongly coupled to S. If S’ is ductile, S’ is
stretched considerably and S is relatively undisturbed.
This is weak coupling.
Consider the ratio:
Energy stored in S
Energy stored in S + Energy stored in S’
This could logically be the numerical value of the
=
Energy stored in S
Energy stored in S + Energy stored in S’
=
Electric Field on
Piezoelectric
Energy stored as mechanical deformation
Electrical energy supplied to crystal
The question of what portion of the energy was used
for mechanical deformation is not a trivial one and can
only be answered by another measurement. This is
made by holding the crystal rigidly clamped and
measuring the energy to charge it as before. Only
electric field energy is supplied, since the piezoelectric
has not deformed. Subtract this from the total electrical energy supplied when deformation is allowed, and
one has the energy stored in mechanical deformation.
1
It is assumed that energy losses are negligible or these
definitions do not apply.
(The dimensional change of the ceramic is, of course
tremendously exaggerated.)
2. Mechanical energy is supplied to a piezoelectric
device with the object of creating an electric field.
Now, if one couples, mechanically to the expansion ln
the longitudinal direction, one measures the k33. If one
couples to one of the lateral directions, one measures
the k31, which is invariably lower than the k33 . Partially, this is because there are two 1-directions available for coupling, but only one 3-direction.
+++++++++
--------Piezoelectric unstressed
& uncharged
Piezoelectric stressed
& charged
There were cases where the electric field caused a
mechanical deformation. Note that the reverse processes correspond to the same coupling values: By
changing the longitudinal dimensions, one generates
an electric field in the 3-direction governed by k33. By
changing the lateral dimension, again the electric field
produced in in the 3-Direction, but it is governed by
k31. Remember that in both cases, the electrodes are 3
electrodes, perpendicular to the polarization.
If the electrodes are in the 1-location,
k2 =
Electrical energy stored in piezoelectric
Mechanical energy supplied to piezoelectric
The electrical energy stored is the mechanical energy
supplied, less the mechanical energy to cause the same
deformation if the electrodes were short-circuited (no
electric field energy is built up in the latter case).
It is not obvious, but the two piezoelectric coupling
coefficients defined above are equal. Referring back to
Ant A, if he is using an electric “force” to pull the two
springs, then S’ represents the electrical storage of
energy while S is the desired mechanical energy. If, on
the other hand, A is supplying mechanical energy,
then S’ is the stored mechanical energy and S is the
desired electrical energy.
It is also conceivable that our friend ant A will have
more than one spring S’ which he can attach to the
block. With each spring there goes a different coupling
coefficient. In piezoelectric ceramics there are three
modes of coupling (three different springs) which are
of special interest. These may be understood through a
consideration of where one may apply electrodes to
the ceramic. The electrodes may be put on either
perpendicular to the electrical polarization (in the
same location as the poling electrodes) or parallel to
the electrical polarization (perpendicular to the
original polling electrodes). The first case is called the
3-location, while the latter is the 1-location. If voltage
is applied to “3” electrodes, the ceramic changes both
its longitudinal and lateral dimensions.
- - - - - -
3–Electrodes
+
(Polarization)
Electric
Voltage
Applied
+
3–Electrodes
+
+
-
(Polarization)
-
+
-
+
Electric
Voltage
Applied
+
-
-
+
+
the only response to electric field is a shear The
coupling is k15. Note that a properly applied shear
stress produces a field in the 1 direction, also governed by kl5. Only a shear can produce a field in
the 1-direction.
Another type of coupling which is convenient for
laboratory measurement is the planar coupling, kp. It
corresponds approximately to the electrodes in the 3direction, and mechanical coupling to both 1-directions simultaneously. For piezoelectric ceramics,
kp ≈ 1.7 k31.
Typical values of the various k’s,and of k , are given
by the table on the following page.
It is seen that the really strong transducing arrangements are the 33 and 15 modes for PZT.
The thoughtful reader may now reflect on the following:
Suppose the transducer is driven by an AC signal and
one approaches its frequency of mechanical resonance.
What happens for strong coupling? For weak coupling? When frictional forces are present?
+ + + + +
2
PZT-4
PZT-5A
PZT-5J
PZT-5H
PZT-8
k33
0.70
0.71
0.69
0.75
0.64
k332
0.49
0.50
0.48
0.56
0.41
k31
0.33
0.34
0.36
0.39
0.30
k312
0.11
0.11
0.13
0.15
0.09
k15
0.71
0.69
0.63
0.68
0.55
k152
0.50
0.48
0.40
0.46
0.30
kp
0.58
0.60
0.60
0.65
0.51
kp2
0.34
0.36
0.36
0.42
0.26
3