piezoline

Piezo electrical elements have a very high inner resistance, so no current is needed for static or quasi-static work. But by their nature, piezo elements are capacitors. If they work dynamically, a high current is necessary for charging and discharging. So, often in a dynamical application, the maximum current of the power supply determines the shortest rise times of the actuators.

Our team from piezosystem jena is experienced in working with piezoelements and we can give advice in solving your positioning problems. We can advise you of the parameters you should induce to reach an optimal result when working with piezo electrical elements.

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1. Piezoelectrical effect - inverse piezoelectrical effect

The result of external forces to a piezo electrical material is positive and negative electrical charges at the surface of the material. If electrodes are connected to opposite surfaces, the charges will generate a voltage U.

(1.1)

C = electrical capacitance
d = piezoelectrical module; parameter of the material (depending on the direction)

By generating forces F to the piezoelectrical material, the volume (bulk) of the material will be approximately constant.

The Curie brothers first discovered piezoelectricity in 1880. It was found by examination of the crystal TOURMALINE.

Modern applications of the piezoelectrical effect can be found in sensors for force and acceleration, musical discs, microphones, actuators and stages for nanopositioning and also in lighters.
An applied voltage to a piezoelectrical material can cause a change of the dimensions of the material, thereby generating a motion. Lippmann predicted this inverse piezoelectrical effect and the Curie brothers were the first to experimentally demonstrate it.

The first applications were in ultra sonic systems for underwater test and also underwater communications. For actuators , the inverse piezoelectrical effect was applied with the development of special ceramic materials. Materials for piezoelectrical actuators are PZT (lead-zirconium-titanate). For the electrostrictive effect the materials used are PMN (lead-magnesium-niobate).
When speaking about actuators, the phrase "piezoelectrical effect" is often used - strictly speaking, it should be called "inverse piezoelectrical effect".

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2. Design of piezoactuators

2.1. Piezostacks - stacked design

Piezostacks consist of a large number of contacted ceramic discs. The electrodes are arranged on both sides of the ceramic discs and are connected in a parallel line as shown. Piezostacks are also called actuators, piezoelectrical actuators or piezoelectrical translators.

Figure 2.1.1. construction of a piezostack

The maximum motion caused by the inverse piezoelectrical effect depends on the electrical field strength and saturation effects of the ceramic material. The breakdown voltage of the ceramic limits the maximum field strength. Normally, piezostacks work with a maximum field strength of 2 kV/mm. This strength can be reached with different voltage values if used with different thickness of the single ceramic plates.

Example number 2
An actuator consists of 20 ceramic plates. The thickness of one plate is 0.5 mm.The total length of the actuator is 10 mm. The actuator will reach a maximum expansion of approximately 10 µm for a voltage of 1000 V. For plates with a smaller thickness the maximum voltage will be less. Modern multi-layer actuators consist of ceramic laminates with a thickness of typically 100 µm.

Example number 3
A multi-layer actuator with a total length of 10 mm consists of 100 disks with a thickness of 100 µm. The stack will reach nearly the same expansion of 10 µm with a voltage of 150V. However, it should be mentioned that the capacitance of this multi-layer actuator is much higher than the capacitance of high voltage devices. This can be important for dynamical applications (see also section 3.7: Capacitance, section 5: Dynamic properties and chapter 10: Electronics).

It is more complicated to produce multi-layer piezoelectrical actuators. Because of the advantage of the lower voltage, some companies are developing so called monolithic actuators. This means, the green sheet ceramic will be laminated with the electrode material. In this way, the full actuator will be made as one system. So the actuator will have the equivalent parameters (for example a high stiffness) of a solid ceramic material.

Piezostacks with and without mechanical pre-load

Because of their construction, the compressive strength of piezostacks is more than one order of magnitude larger than its tensile strength. Mostly, the glue used to laminate the actuators determines the tensile strength. When actuators are used for dynamical applications, compressive and tensile forces occur simultaneously due to the acceleration of the ceramic material. To avoid damage to the actuators, the tensile strength can be raised by a mechanical pre-loading of the actuator. Another advantage of the pre-load is better stability of the actuators with a large ratio between the length and the diameter. Normally the mechanical pre-load will be chosen within 1/10 of the maximum possible loads. You can find more information in sections 4 and 5 of the piezoline.

Figure 2.1.2. stacks with and without pre-load

We recommend to use a pre-loaded actuator from piezosystem jena when:

tensile forces can affect the actuator
they are used in dynamical applications

Figure 2.1.3. Tilting forces

Actuators without pre-load should be mounted on the end faces. This can be done using adhesive or threads in the bottom of the housing. You should not apply shear, cross-bending or torsional forces to the actuator. Clamping around the circumference is not allowed. External forces on the top of the actuator should mainly be in the direction of expansion central to the end faces.

If you wish a detailed discussion, please contact our team or your local dealer!

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2.2 Tube design

For this actuator there is the transversal piezoelectrical effect. The tubes are made from a monolithic ceramic; they are metalized on the inner and outer surface. Normally, the inner surface is contacted to the positive voltage. If an electric field is applied to the tube actuator, a contraction in the direction of the cylinder's axis, as well as a contraction in the cylinder's diameter, results in a downward motion. If the outer electrodes are divided, the tube can work as a bimorph element. In this way, it is possible to reach a larger sideways motion. Piezotubes are used for mirror mounts, inchworm motors, AFM (atomic force microscopes) and STM microscopy.

Bild 2.2.1: tube actuator

Example number 4
Consider a tube actuator with a diameter of 10 mm, a wall thickness of 1 mm and a length of 20 mm. The maximum operating voltage is 1000 V. So, the applied field strength is 1 kV/mm. The transversal piezoelectrical effect shows a relative contraction of approximately 0.05 %. For the length of 20 mm, one will get an axial contraction of 10 µm. Simultaneously the circumference of 31.44 mm will be shorter by 15 µm. This is related to a radial contraction of 4.7 µm.

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2.3. Bimorph Elements

These elements are made from two thin piezo electrical ceramic plates mounted on both sides with a thin substrate. The principle is similar to thermo bimetal circuits.

Applying opposite field strength to the ceramic plates, one plate shows a contraction, the other will expand. The result is bending in the order of sub-mm up to several mm. Bimorph elements use the transversal piezoelectrical effect (see also section 4), the working piezoelectrical module is the d31 coefficient. Piezoelectric bimorph elements have a resonant frequency of several 100Hz. Because they show a large drift (creep) while doing static work (because of shear stress in the layers) they are often used in dynamic applications. Because of their construction, they have a low stiffness and they cannot make a parallel motion (almost circular).

In the following figure, two kinds of piezo electrical bimorph elements are represented.

Figure 2.3.1: serial and parallel bimorph

Serial bimorph
Both piezoelectrical plates are polarized in opposite directions. A voltage is applied to the electrodes on the ceramic plates on the outside. If a voltage is applied and the plate shows a contraction, the other will show an expansion.

Parallel bimorph
A metal plate middle electrode is between the two ceramic plates. The polarization of both ceramic plates is in the same direction. The bending of this bimorph will be reached by applying opposite voltages to the electrodes. Because of the metal plate in the middle, these bimorph elements have a higher stiffness.

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2.4. Hybrid Design

For many applications it is necessary to have a motion on the order of 50 µm - 300 µm (for example fiber coupling problems). To use stacked actuators for a motion of 300 µm, one needs a translator with a length of 300 mm, independent of whether you are using high or low voltage stacks.The high capacitance is another disadvantage of such large stacks. Because of the inhomogeneous expansions of the ceramic plates, the top plate of the stack will always show a slight tilting motion. That's why bimorph elements are not suited for parallel motion or force generation.

piezosystem jena has developed a hybrid piezoelectrical element for parallel motion with high accuracy. A lever design of the construction gives very compact dimensions. We have developed the hybrid elements for three dimensional motions. Since we use solid state hinges, mechanical play does not occur.

The working principal is shown in the figure below.

Figure 2.4.1. Parallelogram design

The flexmount points A,B,C and D are solid state hinges. piezosystem jena uses a monolithic design; the motion is achieved by bending these flexmounts. Because of the rectangular design and the thread holes, it is very simple to combine these elements with normal mechanical stages. The advantage is a much higher accuracy and an excellent resolution of the motion. Because most of these elements have an integrated pre-load, they are suited for dynamical motions (see also section 6: lever transmission!).

Please note the following advantages of piezoelectrical driven stages:
When a piezoelement is working, no manual forces are required to position the stage. Using only mechanical positioning systems, the position cannot be held if the external forces are removed. These positioning problems (for example for fiber coupling) can be avoided by using piezoelectrical elements.

Example number 5
The piezoelements MINITRITOR 38 from piezosystem jena generates a rectangular motion of 38 µm in x, y and z direction. Integrated solid state hinges with parallelogram design provide parallel motion without any mechanical play. The dimensions are 19 mm x 19 mm x 16 mm. Another element is the piezoelement PX 400. This element gives a motion of 400 µm; the dimensions are 52 mm x 48 mm x 20 mm. This element is also suited for dynamical motion. For more details please see our data sheets and section 6 of this catalog. For comparison, a piezostack with 400 µm motion would need at minimum a length of 400 mm!

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3.1. Expansion

The relative expansion S = ∆l/L0 (without external forces) of a piezoelemen t is proportional to the applied electrical field strength. Typical values of the ceramic materials are S = 0.1-0.13% (field strength E = 2 kV/mm).

(3.1)

S - relative stretch (without dimension), d = d_ij - piezo module, parameter of the material, E = U/d_s electrical field strength, U - applied voltage, d_s - thickness of a single disk.

The maximum expansion will raise with increasing voltage. The relation is not perfectly linear as predicted by equation (3.1.). The characteristic curve reflects the inherent hysteresis (see also section 3.2.). The maximum expansion that can be achieved by using normal stacks us up to 300 µm. The length of such a stack will be 300 mm! Typical piezostacks have motion of 20 - 100 µm. For greater expansion, actuators with a lever transmission are superior. It is possible to combine piezoelectrical elements with mechanically or electromechanically driven systems. So, the motion will be several cm, although the motion will show mechanical play.

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3.2. Hysteresis

Because of their ferroelectric nature, PZT ceramics show a typical hysteresis behavior. If voltage is applied in a positive direction and then in a negative direction (bipolar voltage), one can obtain the following curve.

Figure 3.2.1: Via the applied voltage, the motion of the element will follow the points ABCDEF.

If the voltage is increased, the movement increases. The maximum motion (point A) will be limited by saturation and by the voltage stability (voltage break down) of the ceramic material. If the voltage is reversed, the piezoelement shows a contraction. After removing the voltage, a permanent polarization will remain. Therefore the motion of the piezoelement is not zero (point B). If a definite negative voltage is applied (so-called coercitive voltage; point C) the motion will be zero microns.

The piezoelement will contract when the negative voltage is increased. At the same time the polarization of the dipole in the ceramic begins to change. At point D the polarization of most of the dipoles is changed, so that the element will expand again for increasing negative voltage up to point E. If the negative voltage is reversed, the piezoelement will contract according to the behavior from point A to point B, so point B is again the point which refers to the remaining polarization. By further increasing the voltage (now positive) the element contracts (up to point F) with polarization changes. By further increasing the voltage, the element expands to point A.

The butterfly curve shows that by applying bipolar voltage it is not possible to accurately determine the position of the piezoelement. For example, for the same voltage, the element can be in position G or in position F. Thus, normally one works with unipolar voltage outside the region of saturation and breakdown and outside the region of polarization changes. So piezoelements show the well-known expansion characteristics.

Figure 3.2.2: Typical hysteresis curve of a multilayer piezostack

To get a larger motion, it is possible to work with a negative voltage in the order of 10 V to 20 V (for multi-layer elements). Therefore we drive our elements with voltages from -20 V up to +130 V.

Working in that range, you find the typical expansion curve of piezoelements . The typical width of the hysteresis is 10 - 15% of the commanded motion.
Working in a small voltage range, the hysteresis is also smaller. This is also shown in the figure 3.2.2. above. Each piezoelement provided by piezosystem jena comes with the measured curve of its hysteresis.

Hysteresis closed loop

In closed loop systems the closed loop control electronics compares a given or wanted motion (e.g. through modulation input signal) with the actual position measured by the sensor system. Any deviation in both signals will be corrected. Thus closed loop systems do not show hysteresis within the accuracy of the closed loop system. For more details see chapter 8 and 9.

OEM elements for industrial applications

For piezoelements working under industrial conditions, we recommend working with voltages up to a maximum of 100 V in order to achieve the best long term reliability. This is important, especially if the piezoelement must work constantly with maximum expansion (under maximum voltage) over a long time period. Please see also chapter 11: reliability!

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3.3. Resolution

Independent of the hysteresis, the piezoelectrical effect as a solid state effect has a very high resolution. A piezoelement PX 38 from piezosystem jena was tested in an interferometer and a motion of 1/100 nm was detected.

Therefore the resolution is limited by the noise characteristic of the power supply. Our power supplies are optimized to solve this problem (please see also section 9.1. and 10.1.).

Example number 6
Our NV 40/1 CLE card has a voltage noise of < 0.3 mV at the output. Relative to 150 V maximum voltage this is a value of 2 · 10^-5. Operating a piezoelement with a maximum expansion of 20 µm, the mechanical noise of this system will generate oscillations in the order of 0.04 mm.

You are invited to speak with our team about various power supplies for their specification!

piezosystem jena offers several different voltage amplifiers (power supplies). A compact 3 channel supply, or power supplies in 19 inch eurosystem.

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3.4. Polarity

In general our piezo elements work with a positive polarity. A minimum reversal voltage on the order of 10% of the maximum voltage (for example -20 V for 130 V multi-layer elements) will increase the total expansion. A higher reversal voltage is not recommended because of depolarization effects.

On request, it is possible to construct the elements with positive or negative polarity.

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3.5. Stiffness

A piezoelectrical actuator can described by a mechanical spring with constant stiffness c_E^T.

The stiffness is an important parameter for characterization of the resonant frequency and generated forces.

The stiffness is proportional to the cross section A of the actuator. The stiffness decreases with an increasing actuator length L₀. In reality the dependence is more complicated. The stiffness is also related to other parameters, e.g. how the electrodes are connected. When the electrodes are not connected, there is no way for the energy to be dissipated; therefore in this case the stiffness has its largest value.

Stiffness
However, formula 3.5.1 does not describe the reality exactly enough. Depending on the kind of operation (static, dynamic operation) and the environment influence (load, electrical parameters of the electronic supply, small or large signal operation) the stiffness can vary up to a factor of 2 or more. Thus using formula 3.5.1 can give only a rough estimation of the expected properties of the piezoelements. Please consider, the electrical capacitance measured for piezoelements with small signals can increase up to 2 times when operated with large signals (under full motion).

Example number 7
An actuator with a cross section of 5 x 5 mm² and an active length of 9 mm has a stiffness of c_T1^E = 120 N/µm. With the same construction (cross section, material) but double the length (18 mm), the stiffness will be a half stiffness (60 N/µm). If an actuator with a cross section 4 times larger (for example 10 mm x 10 mm, length 18 mm) is used, the stiffness will be 240 N/µm.

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3.6. Thermal Effects

Temperature variation is an important factor in the accuracy of a micropositioning system. The thermal expansion coefficient of stainless steel for example, is about . Imagine a cube of 10 x 10 x 10 mm³. At temperature change of only 1K leads to an expansion of more than 0.1µm in each direction. With these relationships in mind, it is easy to understand that the calibration of piezoelements with integrated measurement systems depends on the temperature. If the operating temperature is different from the temperature during calibration, errors will occur.

When speaking about temperature coefficients of piezoelements , we must consider three effects:

a) The temperature behavior of the piezo ceramic material depends on the type of ceramic material. Piezo stacks operating with high voltages show a positive temperature coefficient on the order of α_HV ≈ (7 - 10) x 10^-6 K^-1. Multi-layer stacks show a negative temperature coefficient of α_NV ≈ -6 x 10^-6 K^-1 in the range up to 120 °C.
The thermal length variation of a whole short circuit actuator (e.g. series P, PA, PAHL) is the sum of the thermal expansions of the piezo ceramic and of the metal parts of the actuator.

(3.6.1)

Δl_therm = thermal expansion of the whole actuator
L_piezo = length of the piezo stack
L_metal = length of the metal housing
α_piezo = temperature coefficient of the piezo ceramic
α_metal = temperature coefficient of the metal housing
ΔT = temperature differential

Example number 8
If the temperature around a PA 16 actuator changes from 20°C to 30°C the length difference at a voltage of 150 V (full stroke) is

The length of the steel parts is 16 mm:

The length of the piezo is 19 mm:

So the total difference is:

b) The piezo effect itself also depends on the temperature. In the range <260 K, the effect decreases with falling temperature with a factor of approximately 0.4% per Kelvin .

In the region of liquid nitrogen (T₁; ca. 77 K), the expansion due to the piezoeffect will be around 10 - 30% of the expansion at room temperature (T₀). Assuming the relation between the change of the piezo electrical expansion with temperature is lineal, it can be expressed as :

Δl_T1= expansion at T₁ 
Δl_T0 = expansion at room temperature
ΔT = T₀ - T₁α_piezoeffect = temperature coefficient of the piezo effect

In the range of 260 K to 390 K the change of the piezoeffect can be neglected.

Example number 9
To estimate what maximum stroke by a PX 100 at -195 °C (liquid nitrogen) can be expected, the temperature difference to -10 °C should be calculated. So it is ΔT= 185 K. The estimated stroke is around 25 µm.

c) The ferroelectric hysteresis decreases with falling temperature. The hysteresis of piezoelectric actuators is a result of the ferroelectric polarization (see also chapter 3.2.). At very low temperatures of four Kelvin for example, there are almost no changes of the electrical dipoles (domain switching) and so there is very little hysteresis. In the region of room temperature, the influence of temperature variations to the hysteresis can be neglected.

Figure 3.6.1. Example of temperature dependence of multilayer ceramics L_piezo = 18 mm at room temperature

Figure 3.6.2. Hysteresis curve of a PA 25 element at room temperature and at 4 K

But please take into account:

Although the piezo effect decreases with falling temperature, piezoelectric actuators principally can work at very low temperatures - down to the temperature of liquid He (4 K).
If you want to work in a low temperature regime, please tell us about this fact, so we can prepare the actuator for this temperature region.

Stages

The temperature behavior for elements integrated into a lever design depends on both the temperature effect for the piezoelement and the behavior of the stage. It may differ from the behavior described above for the piezoelement itself. Because of the different constructions used for different stages a general rule cannot be given.

Closed loop stages

Please take care to use closed loop stages at near the temperature at which they were calibrated. Only at the temperature of calibration, piezoelementsshow the best accuracy.

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3.7. Capacitance

As mentioned a stack actuator consists of thin ceramic plates as dielectricum and electrodes. This is a system of parallel capacitors.

(3.7.1)

n - number of ceramic plates, ε₃₃ - dielectric constant, A - cross section of the actuator or the ceramic plates, ds - thickness of a ceramic plate.

Example number 10
A multi-layer stack with an (active) length of 16 mm, a cross section of 25 mm² and a thickness of the ceramic plates of 110 µm consists of approximately 144 plates. With a relative dielectricity of ε_r = 5400 one yields a capacitance of the actuator of approximately 1.6 µF (see formula 3.7.1).

Capacitance of multi-layer actuators - capacitance of high voltage actuators

Let us consider the following comparison:

Example number 11
A multi-layer actuator (index 1; parameter see example number 10) should be replaced by a high voltage element with the same length (index 2). For simplicity, both stacks consist of the same material. Refer to formula 3.7.1. The thickness of the ceramic plates of the high voltage actuator is 5 times larger (d_s2 = 5 · d_s1) so the number of plates is 5 times lower (n₂ = 1/5 · n₁). Thus the capacitance of the high voltage actuator is much lower than the capacitance of the multi-layer actuator C₂ = C₁/25.

The operating voltage for the same expansion is lower for multi-layer stacks. But the capacitance is increasing quadratically.

Please note:
Because of the higher capacitance of low voltage multi-layer stacks, these actuators need much more current in dynamical applications. The current can be neglected for static and quasi-static motions.

Please note:
The piezoelectrical properties of actuators are not constant as assumed in simple descriptions. Most of the parameters depend on the strength of the internal field. Most of the values given in the literature are for low electric fields. These values can differ for high electric fields. As an example, the capacitance for high voltage operation is nearly twice that given for low voltages.

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3.8. Drift - creep (open loop systems)

Another characteristic of piezoelectrical actuators is a short dimensional stabilization known as creep. A step change in the applied voltage will produce an initial motion followed by a smaller change in a much longer time scale as shown in the figure 3.8.1.

As one can see, the creep will be within 1% to 2%, in a decade of time. The creep depends on the expansion Δl, of the ceramic material (parameter of the material γ), on the external loads, and on time. The dependence of the creep can be shown also as a logarithmic dependence of time.

(3.8.1.)

Δl_0,1 - motion length after 0,1 s after ending of rise time of the voltage

In this case we reach a value for γ≈ 0.015. The value of γ depends on the material, the construction and the environmental conditions (e.g. forces). When the motion (voltage) is stopped, after a few seconds, the creep practically stops.

Repeatability for periodical signals

When working with periodic signals, the repeatability of a position will not be deteriorated with creep. Because of the strong time dependence of the motion, creep occurs in all oscillations in the same order.

In the figure 3.8.2. we have shown a periodic oscillation of a mirror mount PSH. The power supply is a normal power supply controlled by a function generator. The full tilting angle is approximately 380 arcseconds. In the picture there is a section of only 10 arcseconds (from 302" up to 312"). It can be seen that the repeatability is better than 0.1" which is better than 0.03%.

As a result of this experiment, we have reached a high repeatability within the system without a closed loop control. For such experiments the repeatability is only determined by the quality of the power supply.

Figure 3.8.1. creep PU 40

Figure 3.8.2 repeatability of a position with periodic motion of a mirror mount

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3.9. Working under vacuum conditions

The piezoelectrical effect, in general, works also under vacuum conditions. The only problem arises from the outgassing of the materials used.

For protection of both people and the piezoelectrical actuators from electrical breakdown, the actuators are insulated using rubber materials. However, these materials exhibit bad outgassing characteristics. That is why piezoelectrical actuators for vacuum applications are produced from materials (for example, adhesives) with low outgassing characteristics. We do not use any rubber materials. Consequently the outgassing is extremely low.

We can offer most of our elements with vacuum options.

Please note: In the pressure region between 0.01 Torr, up to 100 Torr the gases used have a very low insulating behavior. If piezoelements with vacuum options (prepared with materiala with very little outgassing) are used in this pressure region, the elements can be damaged because of electrical breakdown. Piezoelectrical actuators prepared for vacuum applications should not be used in this pressure region. For sefety reasons, piezoelements with vavuum options must not be used in environments where someone can touch the contacts.

Example number 12
The piezoelectrical driven optical slit from piezosystem jena was especially prepared for vacuum applications. Up to a pressure of 5·10^-9 Torr, no influence of outgassing of the piezoelement was detected. The piezoelements were not heated.

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3.10. Curie´s Temperature

The ferroelectric nature, and so the piezoelectrical properties, will be lost if the material will be heated over the Curie point, 150 °C. So it is important to work below the Curie temperature T_c.

The Curie temperature is dependent on the material. Normally, multi-layer actuators have a Curie temperature of 150 °C. High voltage actuators have a Curie temperature of 250 °C.

In special cases it is possible to work with other ceramic materials with varied Curie temperatures. If a piezoceramic is heated (for example by dynamical motion) up to the Curie temperature, thermal depolarization will occur.

If temperature parameters are not given we recommend working in temperatures up to T_c/2 (normally up to 80°C). If materials become depolarized, the piezoeffect is lost. However, the application of a high electrical field to the actuator can restore it. Thus, special piezoelectrical materials can be annealed in the vacuum chambers. The heating of piezoactuators can be ignored when working under static and quasi-static conditions. It should be taken into account for dynamical applications (see section 5).

If there is a particular problem, please contact us for more information!

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4. Static behavior of piezoelectrical actuators

To generate an expansion in a actuators , the ceramic material must be pre-polarized. The majority of the dipoles must be oriented in one direction. If an electrical field is now applied in the direction of the dipoles, (here the z direction) the actuator will show an expansion in the direction of the field (longitudinal effect) and will show a contraction perpendicular to the field (transversal effect).

The motion is expressed by the equation:

(4.0.1) longitudinal effect

(4.0.2) transversal effect

S - strain, relativ motion,
T = F/A - mechanical tension pressure (e.g. caused by external forces)
S₁₁ - coefficient of elasticity (reciprocal value of the young modulus)
Δl_z - expansion of the actuator in z-dimension
l_z - length of piezoelectrical active part of the actuator,
E=U/d_s - electrical field strength
U - applied voltage

Piezoceramics are pre-polarized ferroelectric materials; their parameters are anisotropic and depend on the direction. The first subscript in the d_ij constant indicates the direction of the applied electric field and the second is the direction of the induced strain.

Typical coefficients are:

coefficient	dimension	PZT
d₃₃	(m/v)	700 * 10^-12
d₃₁	(m/v)	-275 * 10^-12
S^E₃₃	(m²/v)	20 * 10^-12
S^E₁₁	(m²/v)	15 * 10^-12
tanδ	-	3-5 %
k	-	0,65

The negative sign represents the contraction perpendicular to the field. Typically, high voltage actuators are made from "hard" PZT ceramics and multi-layer low voltage actuators are made from "soft" PZT ceramics.
For the sake of simplicity, if not otherwise mentioned, from now on we will refer to the longitudinal piezoelectrical effect, however all relations can be written in the same manner for the transversal effect.

(4.0.3)

The first term of the equation (4.0.3) describes the mechanical quality of an actuator as a spring with a stiffness cT. The second term describes the expansion in an electrical field E.

The static behavior can be stated using formula (4.0.3).

Figure 4.1. stacked actuators with longitudianl expansion and transversal compression

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4.1 No voltage is applied to the actuator, E = 0

The actuator is short-circuited. Formula (4.0.3) becomes S = Δl/L0 = s₃₃ · T. The deformation of the actuator Δl is determined by the stiffness of the actuator c_T^E because of the action of an external load with the pressure T, so it becomes "shorter".

(4.1.1)

L₀ - length of the actuator.

The stiffness of an actuator can be calculated by taking into account the stiffness of the ceramic plates. This approximation assumes that the adhesive between plates is infinitely thin.

Monolithical multi-layer actuators perform well in this respect, giving stiffness on the order of 85% 90% of the stiffness of the pure ceramic material. Especially for high voltage actuators, the stiffness of the metallic electrodes and the adhesive have a large influence on the stiffness of the stack.

Example number 13
On a stack with a stiffness of a given operates at an external force of F = 70 N, using formula (4.1.1) it is easy to calculate the compression of a stack of 1 µm.

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4.2. No external forces, F = 0

The motion of a stack without any pre-load and without external forces can be expressd by:

(4.2.1)

The maximum expansion depends on the length of the stack, on the stack, on the ceramic material and on the applied field strength.

Example number 14
Let us consider a multi-layer stack with the following parameters:

Piezoelectrical constant
d₃₃ = 635·10^-12 m/V
Active length
L₀ = 16 mm

The thickness of a single plate is 100 µm. The operating voltage is 150 V. The field strength is E = 1.5 kV/mm. The expansion will be Δl₀ = 15 µm without external forces (see formula 4.2.1.).

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4.3 Constant external loads, F = constant

Operating with constant force F or weight, the actuators will be compressed (see Figure 4.3.1.).

(4.3.1)

However, the expansion Δl₀ due to the applied voltage will be the same as when an external force is not applied (see formula 4.2.1.).

In cases where excessively high external forces are applied, depolarization may occur if there is no applied electrical field. This effect depends on the type of ceramic materials used. This polarization may be reversed if an electrical field is applied.

However the depolarization can be irreversible if the external forces have exceeded the threshold limit for that material. Damage to the internal ceramic plates may also occur. Therefore it is important to respect the given data for the relevant materials.

Standard actuators from piezosystem jena with a cross section of 5 x 5 mm² show depolarization effects for external loads > 1 kN. Please see the given parameters in our data sheets!

If your problem needs additional clarification, do not hesitate to contact our team from piezosystem jena.

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4.4 Changing external loads and forces, F = f (Δl)

As an example of changing external forces, consider attaching an external spring. Because of the spring's nature, the forces F, operating to the actuator , increase with the increasing displacement. If the external forces can be expressed as F = -c_F ΔL (c_F stiffness of the spring) we get the following expansion of the actuator:

(4.4.1)

e.g. the motion given in relation to the motion without external forces:

(4.4.2)

A part of the motion will be needed to compensate the external forces, therefore the final motion becomes smaller (see also figure 4.4.1.).

If the stiffness of the actuator and the stiffness of the external spring are equal, the actuator will reach only half of its normal motion.

Example number 15
The actuator PA 16/12 has a stiffness of cT = 65 N/µm. The motion Δl₀ without external forces is 16 µm. This actuator is assembled in a housing with a pre-load stiffness c_F = 1/10 c_T. In comparison with formula (4.4.2.) the motion will decrease to 14.5 µm. If the stiffness of the pre-load is increased to 70% of the stiffness of the actuator c_F = 0.7 cT = 46 N/µm, the motion will reach only Δl = 9.4 µm.

Using equation (4.4.2.) we can calculate the effective forces, which can be reached with an actuator operating against an external spring.

(4.4.3)

Δl0 - motion without external loads (µm),
Δl - motion under external loads (µm).

Example number 16
Again, we will use the actuator PA 16/12. For motion without external load Δl₀, the stiffness is c_T = 65 N/µm. This actuator is working against a spring with a stiffness c_F = 64 N/µm. In this assembly the actuator will reach an effective force of 516 N. When it operates with an external spring with a stiffness of 500 N/µm, it will reach an effective forces of F = 920 N.

An external variable force operating with an actuator will decrease the full motion.

Integrated pre-loads of piezoelectrical actuators are external forces. The value of the integrated pre-load often reaches 1/10 of the maximum possible load of the actuator. That is why the shorter expansion of pre-loaded actuators is very low. But pre-loaded actuators can work under tensile forces. They are well suited for dynamical applications.

Figure 4.4.1: motion dependence of external spring forces

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4.5 Blocking forces, Δl = 0

The actuator is located between two walls (with an infinitively large stiffness). So it cannot expand (see formula 4.2.1.):

(4.5.1)

In such a situation the actuator can generate the highest forces F_max.

(4.5.2)

This force is called blocking force of an actuator.

Operating against external spring forces, actuators show the following behavior of the generated forces in dependence on the expansion. This stress diagram is valid for typical actuators used by piezosystem jena.

Figure 4.5.1. stress strain diagram of piezoelectrical actuators

The cross over with the x-axis indicates the blocking force. The cross over with the y-axis shows maximum expansion without external forces. Also shown is the curve of an external spring. The cross over of this spring load line with the curve of the actuator gives the actual parameters, which can be reached with this actuator operating against a defined spring.

An actuator can generate the maximum mechanical energy if it is operating to an external spring with a stiffness of half the actuator stiffness (c_F = Ω cT). In this case the actuator reaches only 67% of its normal (without external forces) expansion.

Example number 17
An actuator of the type PA 16/12 operates to an external spring, without loads the actuator reaches a motion of 16 µm. A generated force of 320 N is demanded. What motion can be reached under such conditions?
Answers: Look at the diagram, the vertical line beginning at the point of 320 N crosses over to the actuator´s PA 16/12 curve. The horizontal line, beginning at this point of the cross over will end in the value of the possible motion, approximately 11 µm. The same result can be calculated using (4.4.3.). For the real expansion Δl under external spring forces we yield from (4.4.3.) Δl = Δl₀ - F_eff / c_T. The stiffness of the actuator is c_T = 85 N/µm. The result will also Δl = 11 µm.

Please note: In practice an infinitely stiff wall or clamping to the actuator cannot be realised. For this reason an actuator will not reach its maximum theoretical force in reality. Please note also that if the actuator should generate its blocking forces it will not show any motion!

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5. Dynamic properties

5.1. Resonant frequency

Piezo actuators are oscillating mechanical systems, characterized by the resonant frequency f_res. The resonant frequency is determined by the stiffness and the mass distribution (effective moved mass) within the actuator. Actuators from piezosystem jena reach resonant frequencies of up to 50 kHz.

(5.1.1)

An additional mass M loaded to the actuator decreases the resonant frequency of this system.

(5.1.2)

That is why the resonant frequency of a complete system can deviate considerably from the resonant frequency of the single actuator. This is an important fact for example when using the mirror for fast tilting. Actuators using a lever transmission for larger motions, get resonant frequencies typically within the range of 300 Hz up to 1.5 kHz.

In our data sheets for some elements not only the resonant frequency is given, but also the effective mass. Knowing the effective mass it is possible to estimate the resonant frequency with an additional mass (using formula 5.1.2.).

You will find more information about the simulation of dynamic properties in chapter 7.

Please note!
Because of the complexity of this field, such calculations give only approximate values. These values should be experimentally verified by tests.

Example number 18
The resonant frequency of the actuator PA 25/12 is f⁰_res = 12 kHz. The effective mass can be estimated by m_eff = 10 g. This actuator has to tilt a mirror with a mass M = 150 g. Because of this mass, the resonant frequency changes to f¹_res = 3 kHz.
Moving with the resonant frequency, the amplitude of the actuator is much higher as in the non-resonant case. Actuators with a lever transmission show super-elevations up to 30 times and higher in comparison to the non-resonant case. When working with frequencies near the resonant frequency, one needs a much lower voltage for the same result. But please be careful! This advantage can damage your actuator if the motion exceeds the motion for maximum voltage in the non-resonant case!

We strongly recommend: Actuators should be used with frequencies of approximately 80% of the resonant frequency. Please consider also the heating of piezoelectrical elements while in dynamic motion.

Do not hesitate to contact us for solving your special problem!

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5.2. Rise time

Because of their high resonant frequency, piezo actuators are well suited for fast motions. Applications have been in valve technology and for fast shutters. The shortest rise time t_min, which an actuator needs for expansion, is determined by its resonant frequency.

(5.2.1)

When an actuator is given a short electrical pulse, the actuator expands within its rise time t_min. Simultaneously, the actuator´s resonant frequency will be excited. So it begins to oscillate with a damped oscillation relative to its position. A shorter electrical pulse can result in a higher super-elevation but not in shorter rise times!

The figure 5.2.1 shows a typical answer to a short electrical excitation of a piezo actuator PAHL 40/20 from piezosystem jena. Although the excitation pulse has a duration of approximately 8 µs the rise time of the actuator is only 20 µs. This value agrees with the resonant frequency of 16 kHz.

Figure 5.2.1: answer of a piezoelement series PAHL to an excitation voltage step of 20V

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5.3 Dynamic forces

While working in the dynamical regime, compressive stress and tensile forces act on piezoelectrical actuators . The compressive strength of piezoactuators is very high, but they are very sensitive to tensile strength. But both forces F_dyn occur in the same order while moving dynamically (formula given for sinusoidal oscillation).

(5.3.1)

Δl/2 - magnitude of the oscillation (Δl full motion of the actuator).

A large acceleration operates on the ceramic and electrode material.

(5.3.2)

ф - angle of the phases of the oscillation.

Example number 19
An actuator with a motion of 20 µm and an operating frequency of 10 kHz has an acceleration of 39500 m/s2. This value exceeds the acceleration of the earth by 4000 times.

Please consider dynamical forces while in dynamical motion. They also appear without external loads! That is why it is necessary to use pre-loaded actuators for dynamic applications. PA or PAHL signify pre-loaded actuators from piezosystem jena. Actuators without pre-load can only be used for small motions in special cases!

Please note
When working under dynamical conditions, the current, which will be needed for the motion, can reach large and critical values. For calculation of the required current, see also section 10, especially section 10.2. and 10.3.

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6. Actuators with lever transmission system

Most of our elements work with an integrated lever transmission (see figure 2.4.1.).
This construction has some advantages:

The motion can be much higher than the motion of the stack type actuator .
Because of using a parallelogram design, the parallelism of the motion is much better than the parallelism of the motion of a simple stack.
Because of solid state hinges, mechanical play does not occur. The fineness of the motion will be similar to that of actuators without lever transmission.
Solid state hinges work without wear for a long time.
Because of the lever transmission the capacitance of the whole system is much lower than the capacitance of an equivalent stack (with the same motion). This can be advantageous for dynamic applications because of the lower electrical current requirements (see also section 10.2. current).

As an approximation, piezoactuators with an integrated lever transmission can be seen as an actuator with a new stiffness and a new resonant frequency. In our data sheets these values are given for our elements.

Piezoelectrical actuators with lever transmission have the electrical capacitance of a stack and they have a high inner resistance. The essential changes to "normal" stack type actuators are:

The motion will be transmitted by the transmission factor TF:

(6.1)

The stiffness decreases quadratically with the transmission factor:

(6.2)

c_T - stiffness to the stack, c_F - stiffness of the lever transmission construction.

Because of the lower stiffness the superelevation will be higher (up to 100 times and more in relation to the motion in the non-resonant frequency range).

The resonant frequency decreases linearly with the transmission factor TF.

(6.3)

While the resonant frequency of a one-sided fixed piezoelectrical stack reaches frequency values up to 50 kHz, the resonant frequency of systems with integrated lever transmission will reach values of 30 Hz up to 1.5 kHz.

If the chosen experimental equipment is unfavorable, additional subordinate (cross) resonant frequencies may occur. The values of these frequencies can only be lower than the actuator´s main resonant frequency.

The blocking force (see also section 4.5.) decreases linearly with the transmission factor.

(6.4)

Piezo Syteme Jena