# Piezo Engineering – A Guide

## Table of Contents

The Direct and Inverse Piezoelectric Effect

Piezo Actuator Materials

RoHS Exemption

Properties of Piezo Actuators

Displacement Performance

Hysteresis Effects

Creep and Drift

Force and Displacement

Capacitance

Heating and Power Dissipation

Environmental Effects

Piezo Stage Properties and Nomenclature

Accuracy/Linearity

Resolution

Repeatability

Stiffness

Resonant Frequency

Load Ratings

Expected Lifetime

Amplifier Selection

## The Direct and Inverse Piezoelectric Effect

In 1880, while carrying out experiments with topaz, tourmaline, quartz, Rochelle salt crystals, and cane sugar, Pierre and Jacques Curie learnt that when mechanical stress was applied to a crystal, faint electric charges developed on the surface of that crystal. The prefix “piezo” comes from the Greek *piezein,* meaning to squeeze or press. As a result, piezoelectricity is electrical charge that is generated on specific materials when that material is subjected to an applied mechanical stress or pressure. This is called the direct piezoelectric effect.

The *converse or inverse piezoelectric effect,* or the application of an electric field for inducing strain, was discovered by using thermodynamic principles in 1881 by Gabriel Lippmann. It is the inverse piezoelectric effect that allows piezoelectric materials to be used in positioning applications.

## Piezo Actuator Materials

Despite the fact that many materials display the inverse piezoelectric effect, the most widely applicable and popular piezoelectric material by far is PZT, or lead-zirconium-titanate. The term PZT is usually used for referring to an extensive range of ceramics which display varied properties based on the grainsize and mixture ratios of their main raw materials: lead, titanium and zirconium. It is also possible to manipulate the properties of the ceramic by adding dopants and making adjustments to the manufacturing process. The recipes for specific materials are generally proprietary and differ between suppliers.

### RoHS Exemption

Despite the existence of lead as a doping material, PZTs are exempt from RoHS directive 2002/95/EC because of a lack of an ideal replacement material. Although efforts are underway to develop alternative materials, no ideal alternative is expected in the field for years to come.

## Properties of Piezo Actuators

### Displacement Performance

The response of a piezoelectric material to an applied stress or applied electric field relies on the direction of application relative to the polarization direction. Due to this, most mechanical and electrical properties that describe piezo materials are also dependent on direction. The inverse piezoelectric effect can be described mathematically as:

(Equation 1) |

where, x_{j} is strain (m/m), d_{ij} is the piezoelectric charge coefficient (m/V) and is a material property, and E_{i} 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 huge electric fields can be produced with small voltages if the charge separation distance is extremely small.

As a general rule, the strain (x_{j}) 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 instance, a 20 mm long active-length PZT actuator will produce almost 20-30 μm of maximum displacement. One can effortlessly see that to produce 250 μm, a PZT stack would be around 170 to 250 mm long. Thus, most piezo flexure stages with >50 μm of travel use lever amplification in order to obtain longer travels in a more compact package size. A tradeoff is made between the stiffness and final device package size since the stiffness of the device decreases with the square of the lever amplification ratio used. Aerotech’s piezo nanopositioning stages are optimized to provide exceptional mechanical performance in a compact stage package

### Hysteresis Effects

Piezoelectric materials are considered to be a subset of a larger class of materials called ferroelectrics. Ferroelectricity is a property of specific materials that have a spontaneous electric polarization, which can be reversed by applying an electric field. Similar to the magnetic equivalent (ferromagnetic materials), ferroelectric materials display hysteresis loops based on the applied electric field and the history of that applied electric field. Figure 1 presents an illustration of a strain (X) versus electric field (E) “butterfly” curve for a PZT material driven to its excitation limits.

**Figure 1. **Strain (displacement) behavior of a ferroelectric material like PZT with an applied electric field driven to its excitation limits.

As the electric field is cycled from positive to negative to positive, the following transformations take place in the piezo actuator:

A: Initially, strain increases with electric field and is just lightly nonlinear. As the electric field is increased, the dipoles of all the grains will ultimately 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 because of the reoriented dipoles. As the field gets smaller, the dipoles relax into less ideal orientations and strain decreases at a rapid rate.

C: As the field becomes negative the dipoles are forced away from their original orientation. At a critical point, they totally reverse direction and the piezo actuator becomes polarized in the opposite direction. The electric field at the point of polarization reversal is called the coercive field (E_{c}).

D: Following 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 took place along curve B takes place again 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 usually operated with a semi-bipolar voltage over an area of the curve (ABC) away from the saturation and coercive field limits. Figure 2 presents an example of displacement versus applied voltage for a piezo actuator stack in this region of the curve.

**Figure 2. **Typical hysteresis curve of a piezo electric stack actuator operating from -30 V to +150 V.

Aerotech amplifiers take complete advantage of the semi-bipolar operation of piezoelectric stack actuators. The actuators are designed for operating from -30 V to +150 V with extremely 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 piezostage in closed-loop efficiently prevents hysteresis of the actuator enabling positioning repeatabilities in the single-digit nanometer range.

### Creep and Drift

The response time for a piezo ceramic subjected to an electric field is much faster than the reorientation time of the individual dipoles. This phenomenon results in undesirable behavior in open-loop position control. When an electric field is applied, the piezo stack will make a corresponding displacement in an almost instantaneous manner. If the field is then held constant, the piezo stack will continue to move slowly as the dipoles reorient, a phenomenon called creep. It can take many minutes or even hours to reach steady state, with strain increasing by as much as 1% to 5% past the initial strained position.

There is a similar effect known as zero point drift. When an electric field is removed, the dipoles will slowly relax and motion will then continue slowly until a steady state is reached. Operating the piezo actuator or stage in closed-loop control prevents this drift since the controller is compensating for this movement in real time in order to keep the output motion at the desired position.

### Force and Displacement

**Force Versus Displacement Characteristics**

The force generated by an actuator acting in the polarization direction is totally independent of the complete length of the actuator and is just a function of the cross-sectional area of the actuator and the applied electric field. An illustration of the force versus displacement output of a piezoelectric actuator with different applied voltages is shown in Figure 3.

**Figure 3. **Force vs. displacement output of a piezo actuator at various applied voltages.

A few interesting characteristics become evident upon inspection of Figure 3. The piezo actuator’s displacement and force increase as the applied voltage is increased. The maximum force output of a piezo actuator, or blocking force, takes place 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 decreases until the force output reaches zero at the maximum rated displacement of the actuator.

**Displacement with a Constant External Load**

Figure 4 illustrates the case of a piezo actuator or stage with a constant, external load applied.

In the case of a piezo stage or actuator without any applied load (case 1), the stroke of the piezo is provided as ΔL_{1}. When a mass is applied to the piezo stage (with expansion in the direction of gravity), the initial deflection (ΔL_{o}) is calculated as:

(Equation 2) |

where k_{p} 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 compresses a distance ΔL_{o} but the stroke ΔL_{2} continues to be the same as the unloaded stage. That is:

ΔL_{2} = ΔL_{1} (constant, external load) |
(Equation 3) |

**Displacement with an External Spring Load**

Figure 5 demonstrates a case where a piezo actuator or stage is driving against an external spring load.

In the case of a piezo stage or actuator without any applied load (case 1), the stroke of the piezo is provided as ΔL_{1}. For case 2 when driving against a spring load, the piezo stage stiffness (k_{p}) and the external stiffness (k_{e}) act in series and decrease the total stroke of the actuator.

**Figure 4. **Displacement of a piezo stage or actuator with a constant external load.

The stroke in case 2 is given by:

(Equation 4) |

It is evident upon inspection of Equation 4 that in order to maximize the stroke of the piezo stage, it is necessary for the piezo stage stiffness (k_{p}) to be much larger than the external spring stiffness (k_{e}).

### Capacitance

PZT actuators can be electrically modeled as a capacitor. The principle equation that describes a capacitor in terms of material properties and geometry is:

(Equation 5) |

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 m^{2}), 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:

ε = ε_{r} • ε_{o} |
(Equation 6) |

where ε_{0} is the permittivity of a vacuum (~8.85 x 10^{-12} F/m), and ε_{r} is the relative permittivity of the material (also known as the dielectric constant).

Low-voltage, multi-layer actuators are usually employed for nanopositioning as they allow 0.1% to 0.15% nominal strains with low voltages (<200 V). The maximum applied electric field across these actuators are available in the range of 1-4 kV/mm.

**Figure 5. **Displacement of a piezo stage or actuator driving against an external spring of stiffness k_{e}.

Since these actuators are constructed from thin layers (usually 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 (T_{layer}) can be defined as the complete active length of the piezo actuator (L_{o}) divided by the number of layers (n). The piezo stack capacitance of a multi-layer actuator can then be defined as a function of the number of layers (n) and the overall active length (L_{o}), as follows:

(Equation 7) |

Typical capacitances of low-voltage, multi-layer piezo actuators used in nanopositioning applications are between 0.01 to 40 μF. The capacitances indicated in Aerotech datasheets are measured at small signal conditions (1 V_{rms} at 1 kHz). For bigger signal operation (100-150 V), an increase in capacitance by as much as 60% should be expected. This capacitance increase should be used for carrying out 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:

(Equation 8) |

This simple relationship is essential for adequately sizing amplifiers needed for driving piezoelectric stages (see Section ‘Amplifier Selection’).

### Heating and Power Dissipation

A perfect capacitor does not disperse any power in terms of heat. However, in practice, a piezo actuator does not behave as a perfect capacitor and does have some internal resistance that produces heat when current is flowing through the actuator. The dielectric loss factor, or loss tangent, is defined as:

(Equation 9) |

where ESR is the equivalent series resistance of the capacitor and X_{c} is the capacitive reactance. It is also possible to write the loss tangent as the ratio of active (resistive) power (P) to reactive power (Q):

(Equation 10) |

The higher the loss tangent, the more energy is transformed to heat (energy lost) as an alternating electric field is launched into the material. For soft PZT materials, which are usually employed for nanopositioning applications, the loss tangent usually is between 0.01 to 0.03 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:

(Equation 11) |

For a single frequency (f) the capacitive reactance is:

(Equation 12) |

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 V_{pp}/2

(Equation 13) |

Equation 13 is an extremely 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 considered to be proportional to the square of the commanded time-varying position signal applied to the piezo stage.

Figure 6 presents an illustration of how the power loss transforms as a function of frequency and applied voltage for a typical piezo actuator with a capacitance of 4 μF.

Temperature increase is proportional to the power dissipated in the actuator. To establish the temperature rise of the piezo actuator or stage needs in-depth knowledge of the exact stage characteristics and design (materials, contact area, etc.). By analyzing Figure 6, it is possible for one to see that heating usually only becomes a concern at extremely large signal amplitudes (for example, high voltage or large amplitude position) and high frequencies. For most positioning applications, the temperature rise and power dissipation in a piezo nanopositioning stage is negligible. For applications needing high frequencies and large position oscillations, customers can contact Aerotech’s Application Engineering Department, who will be happy to assist them in sizing the correct piezo nanopositioning device for their exact application.

**Figure 6. **Estimated power dissipated as a function of frequency and applied voltage for a typical piezo actuator with a 4 µF capacitance.

### Environmental Effects

**Humidity**

Protecting the piezo actuator against humidity is one of the most critical factors to ensure long life. This is the reason why Aerotech uses specially-sealed coatings on the actuators that protect the actuator from moisture. Operation at 60% or lower RH environments is always preferred because it further extends the life of the actuator.

**Temperature**

Piezo actuators can be designed to work at very low temperatures (cryogenic) and very high temperatures. Curie temperature of the piezo material is the extreme upper limit of operation. The piezo material loses its piezoelectric effect at this temperature. Curie temperatures of piezo actuator materials fall between 140 °C and 350 °C; however, piezoelectric properties are dependent on temperature. Due to this reason, about 80 °C is the maximum temperature that can be used by Aerotech’s piezo actuators. In the case of precision positioning applications, temperatures approaching this can have serious detrimental effects on the performance and accuracy of the piezo stage.

Piezo actuators are also suitable for operation at very low temperatures. The crystals in piezoelectric material remain in their piezoelectric configuration, regardless of how low the temperature drops. Standard stack actuators that are commercially available in the market can operate down to -40 °C without any problems. Induced stress from thermally contracting mechanisms is the biggest issue in cold environments – not the piezo itself. For very cold environments, unique design considerations are needed for the actuator to withstand the cooling process.

Extremely homogeneous ceramic and carefully chosen electrodes should be used to prevent cracking because of unmatched thermal expansion coefficients. At low temperatures, piezo ceramics operate differently. However, the ceramic stiffens at these temperatures, causing a decrease in the amount of strain produced per volt. This is compensated by increased electrical stability in the crystal structure, enabling fully bi-polar operation. Lower hysteresis, lower capacitance, better linearity, and smaller dielectric loss are other benefits of low temperature operation.

For the highest accuracy, Aerotech proposes operation at or near 20 °C as that is the temperature in which the nanopositioning stages are developed and calibrated. If extreme temperature environments are predicted in their operation, customers should contact Aerotech Applications Engineers who will assist them in choosing or customizing the right piezo positioning stage for the highest level of performance in any environment.

**Vacuum**

Low-voltage (<200 V) piezo actuators are especially suitable for vacuum operation. Piezo actuators do not need lubrication that usually requires tremendous care when selecting for ultra-high vacuum applications. Vacuum pressures from 10 to 10^{-2} Torr should be avoided because the air’s insulation resistance considerably decreases in this range (called the corona area), thus enabling easier dielectric breakdown. Piezo nanopositioning stages from Aerotech can be prepared for ultra-high vacuum operation.

## Piezo Stage Properties and Nomenclature

Piezo nanopositioning stage series from Aerotech have been specifically designed keeping the end-user in mind. Consequently, it is important that the company’s customers have an in-depth understanding of its specifications so that they can best be matched to the end-process or application. The following is a description of the nomenclature and specifications used in Aerotech’s datasheets.

### Accuracy/Linearity

As discussed in Section ‘Hysteresis Effects’, piezo actuators exhibit non-linearity and hysteresis when operated in open-loop mode. The non-repeatabilities caused by piezo actuator hysteresis are eliminated when operated in closed-loop mode. However, the piezo stage may still show hysteresis and non-linearities that impact the device’s overall positioning accuracy. The magnitude of these non-linearities is a function of the quality of electronics and closed-loop feedback sensor used in the design, and also the quality of the mechanical stage design. With Aerotech’s optimized flexure designs, advanced electronics and high-resolution capacitance sensors, linearity errors below 0.02% can be achieved. Linearity and accuracy are measured with precise laser interferometers at a distance of approximately 15 mm above the moving carriage of the piezo nanopositioner (unless otherwise noted).

The terms linearity and accuracy are sometimes used synonymously when illustrating the positioning capability of piezoelectric nanopositioners. However, accuracy and linearity can have slight differences in meaning.

Accuracy is the measured peak-peak error (reported in units of nanometers, micrometers etc.) from the nominal commanded position that results from a positioning stage as it is commanded to move bidirectionally throughout travel. Linearity is the maximum deviation from a best fit line of the position output and position input data. Linearity is reported as a percentage of the measurement travel or range of the positioning stage.

Figure 7 shows an example of the raw measurement results from an accuracy and linearity test. Figure 8 shows the accuracy plot. It is seen how the accuracy results have a small residual slope remaining in the data. Linearity error is calculated using the deviation of a best-fit line to the measurement data taken in Figure 7. Figure 9 shows the residuals from this best-fit line and an illustration of how linearity error is calculated.

In brief, the term accuracy is used for measuring sensitivity effects (slope of measured versus actual position) as well as nonlinearities in positioning. It is reported as a pk-pk value. The term linearity is used to measure the effects of nonlinearities in positioning only and is reported as a deviation or maximum error of the residuals from the best fit line through the measured vs. actual position data. From the linearity specification, the positioning accuracy can be approximated by doubling the linearity specification. For instance, a 0.02% linearity for a 100 µm stage is a 20 nm maximum deviation. The approximated accuracy error is 40 nm pk-pk or 2 x 20 nm.

### Resolution

Resolution is defined as the smallest detectable mechanical displacement of a piezo nanopositioning stage. Most piezostage manufacturers will state that the resolution of a piezo actuator is hypothetically unlimited because even the smallest change in electric field will cause some mechanical expansion (or contraction) of the piezo stack. While this fact is theoretically true, it is largely impractical because all piezo stages and actuators are used with sensors and electronics that generate some amount of noise.

In these devices, the noise normally rises with increasing measurement sensor bandwidth. Therefore, the resolution (or noise) of a piezo nanopositioner is a function of the sensor bandwidth of the feedback device. Piezo amplifiers and feedback electronics from Aerotech have been optimized to provide high resolution and low noise making them suited for some of the most demanding performance applications.

**Figure 7. **Raw measurement results from an accuracy and linearity test of a 100 µm piezo stage.

Aerotech specifies the resolution as a 1 sigma (rms) jitter, or noise, value as measured by an external sensor (either laser interferometer or precision capacitance sensor) at a measurement bandwidth of 1 kHz, unless noted. The stage servo bandwidth is set to approximately 1/3 to 1/5 of the 1^{st} resonant frequency of the piezo nanopositioner, because this is usually the highest frequency that the servo bandwidth can be increased to before servo instability occurs. Since the noise is mainly Gaussian, taking six times the 1 sigma value provides an approximation of the pk-pk noise. Unless specified, the measurement point is centered and at a height of about 15 mm above the output carriage. Measuring at a lower servo bandwidth will result in a lower noise (jitter) in noise critical application.

Values are specified for closed-loop and open-loop resolution. Only the noise in the power electronics governs open-loop resolution, while closed-loop resolution contains power amplifier noise as well as feedback sensor and electronics noise.

### Repeatability

The repeatability of QNP piezo nanopositioning stages from Aerotech is specified as a 1 sigma (standard deviation) value measured from multiple bidirectional full-travel measurements. In order to obtain an approximate peak-peak value for bidirectional repeatability, the 1 sigma value can be multiplied by 6. For instance, a 1 nm value given as a 1 sigma repeatability will be about 6 nm peak-peak. Unless specified, specifications are measured centered and at a height of about 15 mm above the output carriage. The specification applies to closed-loop feedback operation only.

### Stiffness

The stiffness of a nanopositioner or piezo actuator is specified in the direction of travel of the output carriage. In essence, the stiffness is a function of the stage flexure, piezo stack and amplification mechanism(s) employed in the design. Higher stiffness piezo stages enable higher dynamics in positioning, for example better dynamic tracking performance and faster move and settle times.

**Figure 8. **Example accuracy error calculated from the measurement data shown in Figure 7.

As discussed in ‘Displacement Performance’, lever amplification is used by most longer-travel (>50 μm) piezo flexure stages to realize longer travels in a more compact package size. When compared to a directly-coupled design, lever amplification designs cause the stiffness in the direction of travel (inversely proportional to the square of the lever amplification ratio) to be reduced. In addition, most lever amplification designs changes the stiffness of the actuator based on location in travel owing to the non-linear nature of the amplification gain.

Due to this reason, together with manufacturing and device tolerances, the stiffness of Aerotech’s piezo nanopositioning stages is specified at a nominal value of ±20%. Piezo nanopositioning stages from Aerotech are optimized to provide a compact stage package and premium dynamic performance.

### Resonant Frequency

The resonant frequency of a nanopositioning stage can be estimated as follows:

(Equation 14) |

where f_{n} is the resonant frequency (Hz), k is the stiffness of the piezo nanopositioner (N/m) and m_{eff} is the effective mass of the stage (kg).

From a general standpoint, it is usually the first (lowest) resonant frequency of the positioning system that restricts the achievable servo bandwidth. The location of this resonant frequency is governed by the design of the flexure, piezo actuator stiffness, and supporting mechanics. Aerotech has optimized the dynamics of the nanopositioning piezo stages in order to provide a stiff, high-resonant frequency design in an optimal stage package.

**Figure 9. **Illustration showing how linearity is calculated based on measurement data taken from a 100 µm piezo nanopositioning stage.

When an applied mass is added to the piezo stage, the resonant frequency will decrease by the following relationship:

(Equation 15) |

where m_{load }is the mass of the applied load.

As mentioned above, the stiffness can change throughout travel in the case of lever amplification designs. Consequently, the resonant frequency will change by the square root of the change in stiffness. For instance, if the stiffness changes by 7%, the resonant frequency will shift by about 3.4% throughout travel.

A first-order approximation of resonant frequency in piezo nanopositioning systems is provided by Equations 14 and 15. These calculations provide only an approximation of the resonant frequency due to complex interactions of the dynamics caused by nonlinear stiffnesses, damping and mass/inertia effects. If a more accurate value is needed for a process or application, customers can contact Aerotech who will assist them in the design and analysis of an engineered solution. The company specifies the resonant frequency of its piezo nanopositioning stages at a nominal value with a ±20% tolerance together with the specified payload (unloaded, 100 grams, etc.)

### Load Ratings

Piezo actuators are brittle ceramic materials. Similar to most ceramics, PZTs have a higher compressive strength than tensile strength. The actuators used in Aerotech’s stage designs are preloaded, so that a compressive load state can always be maintained during standard operational limits. On its data sheets, Aerotech specifies push and pull load limits that refer to loading applied in the direction of travel. For certain stages, the load rating may be different based on the direction of the applied load. All Aerotech load ratings are a maximum value. If customers require larger load ratings than what is provided in Aerotech’s data sheets, they can contact an Aerotech Applications Engineer who will assist them in easily modifying or customizing a design to meet their exact requirements.

### Expected Lifetime

Using FEA and analytical techniques, the key guidance elements in Aerotech piezo actuator flexure stages are sized to ensure long and reliable operation. The dimensions and materials selected for these flexure elements guarantee elastic bending and stresses in critical areas that are well below the endurance limit.

The performance and lifetime of piezo actuators are affected by factors such as temperature, humidity and applied voltage. As mentioned in ‘Environmental Effects’, Aerotech’s actuators are sealed and life-tested to guarantee thousands of hours of device life. Based on empirical data developed over years of testing, Aerotech can provide lifetime estimates depending on the preferred move profiles and predicted environmental conditions where the piezo nanopositioning system will reside.

## Amplifier Selection

This section provides a basic outline for choosing a piezo amplifier based on a specified piezo actuator and move profile.

Since the displacement of a piezo stage is proportional to the applied voltage, the operating voltage of the amplifier defines the basic travel. In Aerotech’s data sheets meant for open-loop operation, a voltage range is given together with the open-loop travel. The closed-loop travel is typically less than the open-loop travel, because closed-loop control often needs larger voltage margins in order to achieve equivalent travels (due to creep, dynamic operation, hysteresis, etc.). Although the margins applies for closed-loop control are dependent on stage and application, it is safe and conservative to assume that closed-loop travel is obtained using the voltage range given for open-loop control.

Some form of dynamic operation is required by most applications. Even if the application involves positioning an optic or sample at various points in travel and dwelling for an extended period of time, the piezo stage will need to move to those positions. At operational frequencies well below the lowest resonant frequency (typically 10 seconds to 100 seconds of kilo hertz) of the piezo stack, the piezo stack serves as a capacitor. Recall Equation 8:

(Equation 16) |

As voltage is directly proportional to position, the piezo actuator draws current whenever the position changes (e.g., during velocity of the piezo stage). This is completely different from a typical Lorenz-style servomotor that only draws current during deceleration (neglecting losses) and acceleration.

**Figure 10. **Maximum sinusoidal peak-peak voltage of a given amplifier with various piezo stack capacitances.

The output of Aerotech’s amplifiers are rated for peak current and continuous current. The peak and continuous currents are calculated as follows:

(Equation 17) |

i = max[i(t)]_{pk} |
(Equation 18) |

It is important that the present requirements of the preferred move profile are compared against these specifications to establish if the amplifier can source the required current to the piezo actuator.

Figure 10 shows the example curve, which provides the maximum peak-peak voltage possible for an amplifier, depending on the frequency of operation and current ratings for sinusoidal motion of different piezo stack capacitances. For choosing a piezo stage, consider the following additional examples of power, voltage¸ and current calculations:

**Example 1**

A 100 μm pk-pk sinusoidal motion at 35 Hz is preferred from a stage with a piezo capacitance of 5 μF. The chosen amplifier has a 300 mA peak current rating, a semi-bipolar supply of +150 V/-30 V, and a 130 mA continuous current rating. However, will this amplifier be able to supply sufficient current to perform this move?

**Example 1 Calculations**

Assume that to conduct the 100 μm pk-pk motion, the full voltage range is employed and the voltage is at the mean of the rail voltages at mid-travel (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 employed for this calculation is assumed to be 5 µF • 1.6 = 8 µF. The current is subsequently calculated as:

i(t) = (2 • π • 35) • 90 • 8e^{-6} • cos(2 • π • 35 • t) = 0.158 • cos(2 • π • 35 • t) |

Therefore, i_{pk} = 158 mA and i_{rms} = 112 mA. Figure 11 shows the voltage and current waveforms. In this case, the continuous current and peak current are all less than the amplifier rating. Hence, this amplifier can supply the required current to carry out the desired move profile.

**Figure 11. **Voltage and current waveforms for the profile commanded in Example 1.

**Example 2**

The desired output move profile of a stage with a piezo capacitance of 5 μF is, a move from 0 to 100 μm in 4 ms, dwell for 60 ms, then move back from 100 μm to 0 in 4 ms. The desired amplifier has a 300 mA peak current rating, a semi-bipolar supply of +150 V/-30 V, and a 130 mA continuous current rating. Conversely, will this amplifier be able to supply sufficient current to perform this move?

**Example 2 Calculations**

Equations 16, 17 and 18 are used to perform the same calculations carried out in Example 1. Again, the capacitance is assumed to increase by about 60% because of large signal conditions. Figure 12 shows the voltage and current waveforms.

In this case, the continuous current is below the rating of the amplifier, but the peak current surpasses the maximum current rating of the amplifier. As a result, this amplifier cannot supply the required power and current to perform the desired move profile.

**Figure 12. **Voltage and current waveforms for the profile commanded in Example 2.

This information has been sourced, reviewed and adapted from materials provided by Aerotech, Inc.

For more information on this source, please visit Aerotech, Inc.