PIEZOELECTRIC MATERIALS
4.7 TRANSDUCER COMPARISON
Substituting these values into f1(τ) produces
aluminum: f1(1)=0.3734 brass: f1(1)=0.3427 steel: f1(1)=0.2956.
The amount of deflection was 0.2 mm with a negligible substrate; therefore, the free deflection for the various substrates are
aluminum: δo=(0.3734)(0.2 mm)=74.5µm brass: δo=(0.3427)(0.2 mm)=68.5µm steel: δo=(0.2956)(0.2 mm)=59.1µm.
Table 4.2 Comparison of transducer properties for a piezoelectric stack and an ideal piezoelectric bimorph
Stack Cantilevered Bimorph Short-circuit stiffness Y3EAp
Ls
Y1Ewpt3p 4L3p Displacement/voltage d33
Ls
tp
3d13
L2p t2p Blocked force/voltage Y3Ed33
Ap tp
3 4Y1Ed13
wptp Lp
The expressions for the blocked force and free deflection of stacks and bimorphs are also listed in Table 4.2.
Generalizing the transducer equations also allows us to compare other aspects of transducer performance. A parameter that is often of interest in transducer design is the time response of the actuator. Piezoelectric materials are often utilized because of their fast response to changes in voltage. This allows them to be used in applications that require fast positioning. Equation (4.153) allows us to quantify the time response and compare the response speed between piezoelectric stacks and bimorphs. Assuming that the resistance force on the piezoelectric element is due to an inertial load with massm, the equation of motion for the system can be written
mu(t)¨ +kEpu(t)=kEpuov(t). (4.155) Dividing by the mass allows us to write the equation in the familiar form of a single- degree-of-freedom oscillator:
¨
u(t)+ωEn2u(t)=ωnE2uov(t), (4.156) whereωnE=kEp/mis the short-circuit natural frequency of the oscillator for a short- circuit electrical boundary condition; the superscript notation is dropped for conve- nience. Equation (4.156) represents the equation of motion for an undamped oscillator.
The simplest method of adding energy dissipation to the equations is to add a linear, velocity-dependent damping term,
¨
u(t)+2ζωnEu(t˙ )+ωEn2u(t)=ωEn2uov(t), (4.157) whereζ is thedamping ratiothat represents the energy dissipation in the transducer.
A parameter of interest in design is the speed at which the transducer will respond to a step change in the applied voltage. The inertial forces and damping forces will impede the mechanical response and produce a delay in the step response of the transducer. Writing the transducer equation as a single-degree-of-freedom damped oscillator allows us to utilize well-known results in controls and linear systems theory to quantify the delay in transducer response.
The response to a step change in potential can be solved with a variety of methods;
including Laplace transforms and the convolution integral (discussed in Chapter 2).
Using Laplace transforms, we write (assuming the initial conditions are zero) s2+2ζωEns+ωEn2
!
u(s)=ωEn2uov(s). (4.158) Solving for the ratiou(s)/v(s) yields
u(s) v(s) =uo
ωEn2
s2+2ζωnEs+ωnE2
. (4.159)
The Laplace transform of a step voltage input isv(s)=V/s. Substituting this result into equation (4.159) and finding the inverse Laplace transform produces
u(t) δo
=1−ωnE
ωdE
e−ζ ωnEtsin
ωdEt+φ
, (4.160)
whereωEd =ωnE
"
1−ζ2andφ=cos−1ζ. Equation (4.160) assumes that the damping ratio of the system is less than 1, which is typical for most applications in which damping is not specifically designed into the device.
The transducer response to a step change in potential is affected strongly by the variation in energy dissipation. Figure 4.31ais a plot of the step response for three values of the damping ratio. We see that an undamped system will exhibit a peak response that is equal to 2δo, and increasing the damping ratio will produce a decrease in the peak response. The number of oscillations that occur until the response decays to the steady-state value also decreases as the damping in the system increases.
0 5 10 15 20 25 30
0 0.5 1 1.5 2
ωn t u/δo
0 5 10 15 20
0 0.5 1 1.5 2
ωn t u/δo
tst tr
upk
(a) (b)
tpk
Figure 4.31 (a) Transducer step response for three damping ratios:ζ =0 (solid),ζ=0.05 (dashed), andζ=0.3 (dotted). (b) Representation step response.
The response of a damped oscillator to a step input is often characterized by four parameters:
1. The peak response,upk, the maximum output over all time 2. The time required to reach the peak response,tpk
3. The rise time,tr, the time required for the output to go from 10% to 90% of its final value.
4. The setting time, tst, the time required for the response to decay to within a prescribed boundary (typically,±2%) of its steady-state value
These parameters are illustrated in Figure 4.31bfor a representative step response.
Expressions for these parameters have been derived and are written as upk=1+e−ζ π/
√1−ζ2
tpk= π ωEn
"
1−ζ2
(4.161) tr ≈ 1.8
ωEn
tst= 4 ζωnE
.
These expressions are useful for estimating the time response characteristics of a piezoelectric actuator.
Example 4.13 A piezoelectric stack actuator with a square cross section is being designed using PZT-5H piezoceramic. The positioning application requires the ac- tuator to move a 300-g load with a free displacement of 30µm. The rise time for the actuator must be less than 0.2 ms. Assuming that the maximum electric field is 1 MV/m, compute the geometry required to obtain these design specifications.
Solution The free deflection of a piezoelectric stack is obtained from equation (4.82). Replacingv/tp with the maximum electric field of 1 MV/m and solving for the stack length, we have
Ls = 30×10−6m
(650×10−12m/V)(1×106V/m)
=46.2 mm.
Using the approximations for a second-order oscillator, equation (4.161), the natural frequency required to obtain a 0.2-ms response time is
ωnE= 1.8
0.0002 =9000 rad/s. (4.162)
The short-circuit actuator stiffness that is required to obtain this natural frequency is kEp =(9000 rad/s)2(0.3 kg)
=24,300,000 N/m.
The cross-sectional area that produces this stiffness is obtained from the expression in Table 4.2:
Ap = LskEp
Y3E = (46.2×10−3m)(24.3×106N/m) 62.1×109N/m2
=1.808×10−5m2.
Since the cross-sectional geometry is square, the side length of the actuator is wp ="
1.808×10−5m2=4.3 mm.
The actuator geometry that meets the specifications has a side length of 4.3 mm and a length of 46.2 mm.
4.7.1 Energy Comparisons
In addition to having differences in the time response to step changes in voltage, piezoelectric stacks and bimorphs exhibit important differences in the energy output of the transducers. Recall that the energy, or work, of a device is defined as the product of force and displacement. One of the primary results of this chapter is that actuator geometry can be used to vary the force–deflection trade-offs in a piezoelectric device.
A useful comparison of the transducers is to compare the amount of energy or work that can be performed as a function of actuator configuration and actuator geometry.
A useful metric for actuator comparison is the peak energy or work that can be performed as a function of voltage applied:
Epk= 1
2 fblδo. (4.163)
The volumetric energy density is the peak energy normalized with respect to the actuator volume:
Ev= fblδo/2
volume. (4.164)
The units of volumetric energy density are J/m3.
Using the values for free displacement and blocked force listed in Table 4.2, we can write the volumetric energy density of a piezoelectric stack as
Ev =d332Y3EApLs/2 ApLs
v2
t2p. (4.165)
Noting thatv/tp=E3, we can write Ev =1
2d332Y3EE32= 1 2
d33Y3EE3
(d33E3). (4.166)
Equation (4.166) is identical to the expression for the energy density of the material in the 33 operating mode. An important attribute of an ideal piezoelectric stack is that there is no reduction in energy density by amplifying the strain through parallel arrangement of the individual piezoelectric layers.
Performing the same analysis for a cantilevered piezoelectric bimorph, we obtain Ev= 9
8
d132Y1EwpLp
wpLp
v2
t2p. (4.167)
Recalling that for our definition of the bimorph geometry,E3=2v/tp, we can rewrite equation (4.167) in the form
Ev= 9 16
1
2d132Y1EE32 = 9 16
1 2
d13Y1EE3
(d13E3)
. (4.168)
Equation (4.168) demonstrates that in the case of a piezoelectric bimorph, the energy density is equal to only 9/16, or approximately 56%, of the energy density of the piezoelectric material operated in the 31 mode. The reduction in volumetric energy density is due to the fact that amplifying the displacement through bending actuation is equivalent to a compliant mechanical amplifier. The compliance in the amplifier reduces the achievable energy density of the device. Comparing the results for a stack actuator to those for a cantilevered bimorph, we note that the energy density of a bimorph is reduced further by the fact thatd13 is usually a factor of 2 or 3 lower thand33. Accounting for the reduction in the strain coefficient, we see that the energy density of a piezoelectric bimorph might only be 10 to 20% of the energy density of a stack actuator. The reduction in strain coefficient in the 13 direction is offset somewhat by the increase in elastic modulus in the 1 direction.
The energy density of stacks and bimorphs fabricated from various types of piezo- electric material can be computed using equations (4.166) and (4.168). Table 4.3 lists the extensional and bending energy density values for various types of common piezoelectric material. In all cases the energy density of the stack is approximately five to eight times greater than the energy density of a bimorph fabricated from identical material at the same electric field. All results listed in Table 4.3 are for an electric field
Table 4.3 Energy density of different types of piezoelectric materials in extensional and bending mode at an electric field of 1 MV/m
Ev: Ev: d33 d13 Y3E Y1E Extensional Bending
Company (pm/V) (pm/V) (GPa) (GPa) (kJ/m3) (kJ/m3)
Piezo Systems PSI-5A4E 390 190 52 66 4.0 0.7
PSI-5H4E 650 320 50 62 10.6 1.8
American Piezo APC 840 290 125 68 80 2.9 0.4
APC 850 400 175 54 63 4.3 0.5
APC 856 620 260 45 58 8.6 1.1
Kinetic Ceramics PZWT100 370 170 48 62 3.3 0.5
TRS Ceramics PMN-PT 2250 1050 12 17 30.4 5.3
TRSHK1 HD 750 360 57 65 16.0 2.4
of 1 MV/m. The energy density at other electric fields can be obtained by multiplying the value listed in the table by the square of the applied electric field in MV/m.
One material type stands out in Table 4.3, due to its high piezoelectric strain coefficients. The material, PMN-PT, is a single-crystal piezoelectricthat exhibits large piezoelectric strain coefficients and large coupling coefficients (>90%). The large strain coefficient is offset somewhat by the fact that single-crystal ceramics are softer than their polycrystalline counterparts. The energy density of single-crystal materials is generally three to five times larger than that of a conventional ceramic.
As of the writing of this book, single-crystal materials were also more expensive than conventional materials and were generally thought of as a good solution for high-end applications of piezoelectric materials where large strain (>0.5%) and good coupling properties were required.
The values listed in Table 4.3 are ideal values that do not account for certain limitations in the fabrication or operation of the material. For example, the values listed for piezoelectric stacks do not incorporate nonideal behavior introduced by inactive electrodes or insulating material. More important, these values do not reflect the inactive mass associated with important components such as the housing or preload springs. Adding the mass of inactive components can reduce the actual energy density by a factor of 3 to 5 compared to the energy density of the material itself. These issues are less important for bimorph actuators, which in many types of operation do not require preloading or casing.