2.0 Piezo Actuator Materials
2.1 RoHS Exemption
3.0 Properties of Piezo Actuators
3.1 Displacement Performance
3.2 Hysteresis Effects
3.3 Creep and Drift
3.4 Force and Displacement
3.4.1 Force Versus Displacement Characteristics
3.4.2 Displacement With a Constant External Load
3.4.3 Displacement With an External Spring Load
3.6 Heating and Power Dissipation
The converse or inverse piezoelectric effect, or the application of an electric field to induce strain, was discovered using thermodynamic principles in 1881 by Gabriel Lippmann. It is the inverse piezoelectric effect that enables piezoelectric materials to be used in positioning applications.
The inverse piezoelectric effect can be described mathematically as:
where, xj is strain (m/m), dij is the piezoelectric charge coefficient (m/V) and is a material property, and Ei is the applied electric field (V/m). The subscripts i and j represent the strain direction and applied electric field direction, respectively. Electric field is a voltage across a distance, so large electric fields can be generated with small voltages if the charge separation distance is very small.
As a general rule, the strain (xj) for most PZT materials found on the market is around 0.1 to 0.15% for applied electric fields on the order of 2 kV/mm. For example, a 20 mm long active-length PZT actuator will generate approximately 20-30 μm of maximum displacement. One can easily see that to generate 250 μm, a PZT stack would be approximately 170 to 250 mm long. Therefore, most piezo flexure stages with >50 μm of travel use lever amplification to achieve longer travels in a more compact package size. A tradeoff is made between the final device package size and stiffness because the stiffness of the device decreases with the square of the lever amplification ratio used. Aerotech’s piezo nanopositioning stages are optimized to provide superior mechanical performance in a compact stage package.
As the electric field is cycled from positive to negative to positive, the following transformations occur in the piezo actuator:
A: Initially, strain increases with electric field and is only slightly nonlinear. As the electric field is increased, the dipoles of all the grains will eventually align to the electric field as optimally as is possible and the distortion of the grains will approach a physical limit.
B: When the field is reversed, strain decreases more slowly due to the reoriented dipoles. As the field gets smaller, the dipoles relax into less ideal orientations and strain decreases at a faster rate.
C: As the field becomes negative the dipoles are forced away from their original orientation. At a critical point they completely reverse direction and the piezo actuator becomes polarized in the opposite direction. The electric field at the point of polarization reversal is known as the coercive field (Ec).
D: After polarization reversal, the piezo expands again until it reaches its physical strain limit.
E: The electric field is reversed again and the same hysteretic behavior that occurred along curve B occurs as strain decreases.
F: The electric field is driven to the coercive limit for the opposite polarization direction and the dipoles reorient to their original polarization.
G: The piezo actuator expands with the applied electric field to its physical limit.
For positioning applications, piezo actuators are generally operated with a semi-bipolar voltage over an area of the curve (ABC) away from the saturation and coercive field limits. An example of displacement versus applied voltage for a piezo actuator stack in this region of the curve is shown in Figure 2.
Aerotech amplifiers take full advantage of the semi-bipolar operation of piezoelectric stack actuators. Our actuators are designed to operate from -30 V to +150 V with very high voltage resolutions. Over this voltage range, open-loop hysteresis values can be as large as 10-15% of the overall open-loop travel of the piezo stage. Operation of the piezo stage in closed-loop effectively eliminates hysteresis of the actuator enabling positioning repeatabilities in the single-digit nanometer range.
A few interesting characteristics become evident upon inspection of Figure 3. The piezo actuator’s force and displacement increase as the applied voltage is increased. The maximum force output of a piezo actuator, or blocking force, occurs when the rated voltage is applied across the actuator and the output of the actuator is “blocked” or not allowed to move. As the actuator expands, the force production capability reduces until the force output reaches zero at the maximum rated displacement of the actuator.
In the case of a piezo stage or actuator without any applied load (case 1), the stroke of the piezo is given as ΔL1. When a mass is applied to the piezo stage (with expansion in the direction of gravity), the initial deflection (ΔLo) is calculated as:
where kp is the stiffness of the piezo stage in the direction of motion and m is the applied mass. With mass m applied to the piezo stage, the stage is compressed a distance ΔLo but the stroke ΔL2 remains the same as the unloaded stage. That is:
In the case of a piezo stage or actuator without any applied load (case 1), the stroke of the piezo is given as ΔL1. For case 2 when driving against a spring load, the piezo stage stiffness (kp) and the external stiffness (ke) act in series and decrease the overall stroke of the actuator. The stroke in case 2 is given by:
It is evident upon inspection of Equation 4 that in order to maximize the stroke of the piezo stage, the piezo stage stiffness (kp) should be much larger than the external spring stiffness (ke).
where C is capacitance (units of F), A is the cross-sectional area of the capacitor perpendicular to the direction of the electric field (units of m2), T is the thickness of the dielectric material separating the charge (units of m), and ε is the material permittivity of the dielectric material separating the charge. The material permittivity is described as:
where ε0 is the permittivity of a vacuum (~8.85 x 10-12 F/m), and εr is the relative permittivity of the material (also called the dielectric constant).
Low-voltage, multi-layer actuators are generally used for nanopositioning because they allow for 0.1% to 0.15% nominal strains with low voltages (<200 V). The maximum applied electric field across these actuators are in the range of 1-4 kV/mm. Because these actuators are constructed from thin layers (typically 50 to 200 μm thick) separated by electrodes, the resulting applied voltages are lower (<200 V) compared to high-voltage actuators (~1000 V) where layer thickness is ~1 mm. The thickness of each layer (Tlayer) can be defined as the overall active length of the piezo actuator (Lo) divided by the number of layers (n). The piezo stack capacitance of a multi-layer actuator can then be expressed as a function of the number of layers (n) and the overall active length (Lo), as follows:
Typical capacitances of low-voltage, multi-layer piezo actuators used in nanopositioning applications are between 0.01 to 40 μF. The capacitances specified in Aerotech data sheets are measured at small signal conditions (1 Vrms at 1 kHz). For larger signal operation (100-150 V), an increase in capacitance by as much as 60% should be expected. This capacitance increase should be used when performing sizing calculations (see Section 5).
The current (i) flowing through a capacitor (C) is proportional to the change in voltage with respect to time. This is mathematically represented as:
This simple relationship will be needed to adequately size amplifiers required to drive piezoelectric stages (see Section 5).
where ESR is the equivalent series resistance of the capacitor and Xc is the capacitive reactance. The loss tangent can also be written as the ratio of active (resistive) power (P) to reactive power (Q):
The higher the loss tangent, the more energy is converted to heat (energy lost) as an alternating electric field is introduced to the material. For soft PZT materials, which are typically used for nanopositioning applications, the loss tangent generally is between .01 to .03 for lower amplitude signals (~1-10 volts) and can be as high as 0.1 to 0.25 for higher amplitude signals (~50-100 volts).
The reactive power (Q) is defined as:
For a single frequency (f) the capacitive reactance is:
Using Equations 10, 11 and 12, it can be shown that the power dissipated in a piezo actuator for a sinusoidal voltage with an amplitude of Vpp/2 and frequency f is:
Equation 13 is a very useful approximation and shows the effects of power loss in piezoelectric devices. This power loss is linearly proportional to the frequency of operation and the capacitance of the piezo actuator, and proportional to the applied time-varying voltage squared. Since voltage is proportional to position, the power loss is proportional to the square of the commanded time-varying position signal applied to the piezo stage.
Figure 6 shows an illustration of how the power loss changes as a function of frequency and applied voltage for a typical piezo actuator with a capacitance of 4 μF.
Temperature rise is proportional to the power dissipated in the actuator. To determine the temperature rise of the piezo actuator or stage requires in-depth knowledge of the exact stage characteristics and design (materials, contact area, etc.). By examining Figure 6, one can see that heating typically only becomes a concern at very large signal amplitudes (e.g., high voltage or large amplitude position) and high frequencies. For most positioning applications, the power dissipation and temperature rise in a piezo nanopositioning stage is negligible. For applications requiring large position oscillations and high frequencies, contact Aerotech’s Application Engineering Department. We will be happy to assist you in sizing the correct piezo nanopositioning device for your exact application.
Piezo actuators are well-suited for operation at extremely low temperatures, as well. The crystals in piezoelectric material remain in their piezoelectric configuration no matter how low the temperature drops. Standard commercially available stack actuators can operate down to -40ºC with no problems. The biggest issue in cold environments is not the piezo itself, but induced stress from thermally contracting mechanisms. For extremely cold environments, special design considerations are required for the actuator to survive the cooling process. Carefully chosen electrodes and extremely homogeneous ceramic must be used to prevent cracking because of unmatched thermal expansion coefficients.
Piezo ceramics do operate differently at low temperatures. At these low temperatures, the ceramic stiffens, which causes a decrease in the amount of strain generated per volt. This is offset by increased electrical stability in the crystal structure, allowing fully bi-polar operation. Other advantages of low temperature operation include lower hysteresis, better linearity, lower capacitance and smaller dielectric loss.
For the highest accuracy, Aerotech recommends operation at or near 20°C because that is the temperature in which the nanopositioning stages are built and calibrated. Contact an Aerotech Applications Engineer if extreme temperature environments are expected in your operation as we will assist you in selecting or customizing the proper piezo positioning stage for the highest level of performance in any environment.
The terms accuracy and linearity are sometimes used synonymously when describing the positioning capability of piezoelectric nanopositioners. However, they can have subtle differences in meaning.
Accuracy is defined as the measured peak-peak error (reported in units of micrometers, nanometers, etc.) from the nominal commanded position that results from a positioning stage as it is commanded to move bidirectionally throughout travel.
Linearity is defined as the maximum deviation from a best-fit line of the position input and position output data. Linearity is reported as a percentage of the measurement range or travel of the positioning stage.
An example of the raw measurement results from an accuracy and linearity test is shown in Figure 7. The accuracy plot is shown in Figure 8. Notice how the accuracy results have a small residual slope remaining in the data. The deviation of a best-fit line to the measurement data taken in Figure 7 is used to calculate the linearity error. The residuals from this best-fit line and an illustration of how linearity error is calculated is shown in Figure 9.
In conclusion, the term accuracy is used to quantify both sensitivity effects (slope of measured versus actual position) as well as nonlinearities in positioning and is reported as a pk-pk value. The term linearity is used to quantify the effects of nonlinearities in positioning only and is reported as a maximum error or deviation of the residuals from the best fit line through the measured versus actual position data. The positioning accuracy can be approximated from the linearity specification by doubling the linearity specification. For example, a 0.02% linearity for a 100 µm stage is a 20 nm maximum deviation. The approximated accuracy error is 2 x 20 nm or 40 nm pk-pk.
Aerotech specifies the resolution as a 1 sigma (rms) noise, or jitter, value as measured by an external sensor (either precision capacitance sensor or laser interferometer) at a measurement bandwidth of 1 kHz, unless noted. The stage servo bandwidth is set to approximately 1/3 to 1/5 of the 1st resonant frequency of the piezo nanopositioner because this is generally the highest frequency that the servo bandwidth can be increased to before servo instability occurs. Because the noise is primarily Gaussian, taking six times the 1 sigma value gives an approximation of the pk-pk noise. Unless specified, the measurement point is centered and at a height of approximately 15 mm above the output carriage. In noise critical applications, measuring at a lower servo bandwidth will result in a lower noise (jitter).
Values are specified for open-loop and closed-loop resolution. Open-loop resolution is governed only by the noise in the power electronics whereas closed-loop resolution contains feedback sensor and electronics noise as well as power amplifier noise.
The repeatability of Aerotech’s QNP piezo nanopositioning stages is specified as a 1 sigma (standard deviation) value calculated from multiple bidirectional full-travel measurements. To obtain an approximate peak-peak value for bidirectional repeatability, multiply the 1 sigma value by 6. For example, a 1 nm value specified as a 1 sigma repeatability will be approximately 6 nm peak-peak.
Unless specified, specifications are measured centered and at a height of approximately 15 mm above the output carriage. The specification applies to closed-loop feedback operation only.
As mentioned in Section 3.1, most longer-travel (>50 μm) piezo flexure stages use lever amplification to achieve longer travels in a more compact package size. Lever amplification designs cause the stiffness in the direction of travel (inversely proportional to the square of the lever amplification ratio) to be reduced when compared to a directly-coupled design. Also, most lever amplification designs cause the stiffness of the actuator to change depending on location in travel due to the non-linear nature of the amplification gain. For this reason, along with manufacturing and device tolerances, the stiffness of Aerotech’s piezo nanopositioning stages is specified at a nominal value of ±20%.
Aerotech piezo nanopositioning stages are optimized to provide both premium dynamic performance and a compact stage package.
where fn is the resonant frequency (Hz), k is the stiffness of the piezo nanopositioner (N/m) and meff is the effective mass of the stage (kg).
In a very general sense, it is typically the first (lowest) resonant frequency of the positioning system that limits the achievable servo bandwidth. The design of the flexure, supporting mechanics and piezo actuator stiffness govern the location of this resonant frequency. Aerotech has optimized the dynamics of our nanopositioning piezo stages to provide a stiff, high-resonant frequency design in an optimal stage package.
By adding an applied mass to the piezo stage, the resonant frequency will decrease by the following relationship:
where mload is the mass of the applied load.
In lever amplification designs, the stiffness can change throughout travel, as mentioned above. As a result, the resonant frequency will change by the square root of the change in stiffness. For example, if the stiffness changes by 7%, the resonant frequency will shift by approximately 3.4% throughout travel.
Equations 14 and 15 will provide a first-order approximation of resonant frequency in piezo nanopositioning systems. Complex interactions of the dynamics due to damping, nonlinear stiffnesses and mass/inertia effects cause these calculations to provide only an approximation of the resonant frequency. If a more exact value is required for your application or process, please contact us and we will assist in the design and analysis of an engineered solution.
Aerotech specifies the resonant frequency of our piezo nanopositioning stages at a nominal value with a ±20% tolerance along with the given payload (unloaded, 100 grams, etc.).
Factors such as humidity, temperature and applied voltage all affect the lifetime and the performance of piezo actuators. As discussed in Section 3.7, our actuators are sealed and life-tested to ensure thousands of hours of device life. Based on empirical data developed over years of testing, we can provide lifetime estimates based on the desired move profiles and expected environmental conditions where the piezo nanopositioning system will reside.
Because the displacement of a piezo stage is proportional to the applied voltage, the basic travel is defined by the operating voltage of the amplifier. In our data sheets for open-loop operation, a voltage range is given alongside the open-loop travel. Typically, the closed-loop travel is less than the open-loop travel because closed-loop control usually requires larger voltage margins to achieve equivalent travels (due to hysteresis, dynamic operation, creep, etc.). Although the margins used for closed-loop control are stage and application dependent, it is conservative and safe to assume that closed-loop travel is achieved using the voltage range specified for open-loop control.
Most applications require some form of dynamic operation. Even if the application is positioning a sample or optic at various points in travel and dwelling for long periods of time, the piezo stage will need to move to those positions.
At operational frequencies well below the piezo stack’s lowest resonant frequency (typically 10s to 100s of kilohertz), the piezo stack acts as a capacitor. Recall Equation 8:
Since voltage is proportional to position, the piezo actuator draws current any time the position changes (e.g., during velocity of the piezo stage). This is different than a typical Lorenz-style servomotor that only draws current during acceleration and deceleration (neglecting losses).
The output of our amplifiers are rated for continuous current and peak current. The continuous and peak currents are calculated as follows:
The current requirements of the desired move profile should be compared against these specifications to determine if the amplifier is capable of sourcing the desired current to the piezo actuator.
The example curve shown in Figure 10 gives the maximum peak-peak voltage possible for an amplifier, based on the current ratings and frequency of operation for sinusoidal motion of various piezo stack capacitances.
Consider the following additional examples of voltage, power and current calculations for selecting a piezo stage:
Example 1A 100 μm pk-pk sinusoidal motion at 35 Hz is desired from a stage with a piezo capacitance of 5 μF. The selected amplifier has a semi-bipolar supply of +150 V/-30V, a 300 mA peak current rating and a 130 mA continuous current rating. Will this amplifier be able to supply enough current to perform this move?
Example 1 CalculationsAssume that to perform the 100 μm pk-pk motion, the full voltage range is used and at mid-travel, the voltage is at the mean of the rail voltages (e.g., 60 V). Therefore:
V(t) = 90 • sin(2 • π • 35 • t) + 60
Recalling that the capacitance can increase by as much as 60% for large signal conditions, the capacitance used for this calculation is assumed to be 5 μF • 1.6 = 8 μF. The current is then calculated as:
i(t) = (2 • π • 35) • 90 • 8e-6 • cos(2 • π • 35 • t) = 0.158 • cos(2 • π • 35 • t)
Therefore, ipk = 158 mA and irms = 112 mA. The voltage and current waveforms are shown in Figure 11.
In this example, the peak and continuous currents are all less than the amplifier rating. Therefore, this amplifier is capable of supplying the necessary current to perform the desired move profile.
Example 2A move from 0 to 100 μm in 4 ms, dwell for 60 ms, then move back from 100 μm to 0 in 4 ms is the desired output move profile of a stage with a piezo capacitance of 5 μF. The desired amplifier has a semi-bipolar supply of +150 V/-30 V, a 300 mA peak current rating and a 130 mA continuous current rating. Will this amplifier be able to supply enough current to perform this move?
Example 2 CalculationsThe same calculations performed in Example 1 are performed using Equations 16, 17 and 18. Again, the capacitance is assumed to increase by approximately 60% due to large signal conditions. The voltage and current waveforms are shown in Figure 12.
In this example, the continuous current is below the rating of the amplifier. However, the peak current exceeds the maximum current rating of the amplifier. Therefore, this amplifier is NOT capable of supplying the necessary current and power to perform the desired move profile.
The complete Piezo Engineering Tutorial is available HERE as a pdf.
Information on our QNP Series piezo nanopositioners is available HERE.Information on our Ensemble QLAB piezo motion controller is available HERE.