1. Piezoelectrical effect – inverse piezoelectrical effect

The result of external forces to a piezoelectrical 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.


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

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, 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”.

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 2.1.1construction 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, piezo stacks 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 (high voltage actuator).

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 and works at voltages typically 130 V high.

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 130 V. 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. Such monolithic actuators are provided by piezosystem jena.

Piezostacks with and without mechanical pre-load

Because of their construction, the compressive strength of piezo stacks 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.

Bild 2.1.2 Piezostapel ohne und mit Vorspannung

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

• tensile forces can affect the actuator
• they are used in dynamical applications
• shear forces (shear strain) affect the actuator (external forces perpendicular to the direction of motion

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!

2.2 Tube design

For this actuator there is the used 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.

Figure 2.2.1 piezotube

Example number 4

Consider a tube actuator with a diameter of 10mm, a wall thickness of 1mm 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.

2.3 ring actuators with central hole

Ring actuators are build similarly according to multi layer stack actuators. But with their free hole (Figure 2.3.1) making it possible to perform radiation or small components. This type of actuator is mainly used in the field of beam manipulation and laser resonators. The mechanically prestressed versions of these actuators can be used dynamically.

Figure 2.3.1 ring actuator

2.4 Bimorph Elements

These elements are made from two thin piezoelectrical 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 100 Hz. 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).

Figure 2.4.1serial 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.

2.5. 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 2.5.1 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 piezo element 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 piezo elements 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 piezo stack with 400 µm motion would need at minimum a length of 400 mm!

3. Properties of piezo mechanical actuators

3.1 Expansion

The relative expansion S = l/L0 (without external forces) of a piezoelement is proportional to the applied electrical field strength. Typical values of the ceramic materials are S0.1 - 0.13% (field strength E = 2kV/mm).


S - relative stretch (without dimension)
d = dij - piezo module, parameter of the material
E = U/ds electrical field strength
U - applied voltage
ds - 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 - 100µm. 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.

3.2 Hysterese

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.

Bild 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 10V to 20V (for multi-layer elements). Therefore we drive our elements with voltages from -20V up to +130V.

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 systems 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 100V 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!

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 1/100nm 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 plug in PC card has a voltage noise of ? 3mV at the output. Relative to 150V maximum voltage this is a value of 2 · 10-5. Operating a piezoelement with a maximum expansion of 20µm, the mechanical noise of this system will generate oscillations in the order of 0.4mm.

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

We have several different voltage amplifiers (power supplies). A compact 3 channel supply, or power supplies in 19 inch eurosystem. A very interesting supply is our plug in PC card, which controls up to 3 different piezoelements directly from the PC.

3.4 Polarity

In general our piezoelements work with a positive polarity. A minimum reversal voltage on the order of 10 % of the maximum voltage (for example -10 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.

3.5 Stiffness

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


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 5mm2 and an active length of 9 mm has a stiffness of cT1E = 120 N/µm. With the same construction (cross section, material) but double the length (18mm), 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 18mm) is used, the stiffness will be 240 N/µm.

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 12·10-6 K-1. Imagine a cube of 10x10x10mm3 - 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 ?HV (7 - 10)·10-6 K-1.
Multi-layer stacks show a negative temperature coefficient of ?NV ? -6·10-6 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.


Δltherm = thermal expansion of the whole actuator
Lpiezo = length of the piezo stack
Lmetal = length of the metal housing
αpiezo = temperature coefficient of the piezo ceramic
αmetal = 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 150V (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 Δlactuator=0.78µm

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

In the region of liquid nitrogen (T1 ca. 77K), 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:

ΔlT1= expansion at T1
ΔlT0 = expansion at room temperature
ΔT = T0 - T1
αpiezoeffect= temperature coefficient of the piezo effect

In the range of 260K to 390K 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=185K. The estimated stroke is around 25µm.

Figure 3.6.1. Example of temperature dependence of multilayer ceramic Lpiezo=18mm at room temperature.

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

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 (4K).

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.


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, piezoelements show the best accuracy.

3.7 Capacitance

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




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 16mm, a cross section of 25mm² and a thickness of the ceramic plates of 110µm consists of approximately 144 plates. With a relative dielectricity of εr = 5400 one yields a capacitance of the actuator of approximately 1.6µF (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 (ds2 = 5 ·ds1) 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.

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.



Δl0.1 - motion length after 0.1s after ending of rise time of the voltage.

Figure 3.8.1. creep of a PU 40

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

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 0.01Torr, up to 100Torr the gases used have a very low insulating behavior. If piezoelements with vacuum options (prepared with materials 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 safety 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 5·10-9Torr, no influence of outgassing of the piezoelement was detected. The piezoelements were not heated.

Heating, baking out of piezoelements.

Piezoelements from piezosystem jena can be baked out up to 80 °C (175°F) without problems. Elements with special preparation can be baked out up to 150 °C (300°F).

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.

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 Tc/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!

4. Static behavior of piezoelectrical actuators

To generate an expansion in a piezoelectrical actuator, 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:

longitudinal effect:


transversal effect:


S - strain, relative motion, T = F/A - mechanical tension pressure (e.g. caused by external forces), sii - coefficient of elasticity (reciprocal value of the Young's modulus), Δlz - expansion of the actuator in z dimension, lz - length of piezoelectrical active part of the actuator, cF - stiffness constant of the external spring for pre-load, cT - stiffness of the actuator.

Figure 4.1. stacked actuators with longitudinal expansion and transversal compression

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

Typical coefficients are:

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.



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).

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 cTE because of the action of an external load with the pressure T, so it becomes “shorter“.


The stiffness cTE 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 cTE operates at an external force of F = 70N, using formula (4.1.1) it is easy to calculate the compression of a stack of 1µm.

4.2 No external forces, F = 0

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


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 = 635·10-12m/V
Active length L0 = 16mm

The thickness of a single plate is 100µm. The operating voltage is 150V. The field strength is E = 1.5kV/mm.

The expansion will be Δl0 = 15µm without external forces (see formula 4.2.1.).

4.3. Constant external loads, F = constant

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


However, the expansion l0 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 5mm² show depolarization effects for external loads > 1kN. 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.

Figure 4.3.1. motion under external constant force

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 = -cF·ΔL (cF stiffness of the spring) we get the following expansion of the actuator:


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


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 = 65N/µm. The motion l0 without external forces is 16 µm. This actuator is assembled in a housing with a pre-load stiffness cF = 1/10cT. In comparison with formula (4.4.2.) the motion will decrease to 16.4µm. If the stiffness of the pre-load is increased to 70% of the stiffness of the actuator cF = 0.7 cT = 46N/µm, the motion will reach only Δl = 9.4µm.

Figure 4.4.1. motion dependence of external spring forces

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


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 l0, the stiffness is cT = 65N/µm. This actuator is working against a spring with a stiffness cF = 64N/µm. In this assembly the actuator will reach an effective force of 431N. When it operates with an external spring with a stiffness of 500N/µm, it will reach an effective forces of F = 920N.

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.

4.5 Blocking forces, Delta l = 0

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


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


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 (cF = ½·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 16µm. A generated force of 320N is demanded. What motion can be reached under such conditions?


Look at the diagram, the vertical line beginning at the point of 320N 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 11µm. The same result can be calculated using (4.4.3.). For the real expansion l under external spring forces we yield from (4.4.3.) Δl = Δl0 – Feff / cT. The stiffness of the actuator is cT = 85N/µm. The result will also Δl = 11µm.

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!

5. Dynamic properties

5.1 Resonant frequency

Piezoactuators are oscillating mechanical systems, characterized by the resonant frequency fres. 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 50kHz.


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


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 300Hz up to 1.5kHz.

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 f0res = 12kHz. The effective mass can be estimated by meff = 10g. This actuator has to tilt a mirror with a mass M = 150g. Because of this mass, the resonant frequency changes to f1res = 3kHz.

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!

5.2. Rise time

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


When an actuator is given a short electrical pulse, the actuator expands within its rise time tmin. 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!

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

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

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 Fdyn occur in the same order while moving dynamically (formula given for sinusoidal oscillation).



Δl/2 - magnitude of the oscillation (Δl full motion of the actuator).
meff – effective mass

A large acceleration operates on the ceramic and electrode material.


φ - angle of the phases of the oscillation.

Example number 19

An actuator with a motion of 20µm and an operating frequency of 10kHz has an acceleration of 39500m/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.

6. Actuators with lever transmission system

Most of our elements work with an integrated lever transmission (see figure 2.5.1., page 66). 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:


The stiffness decreases quadratically with the transmission factor:


TF – lever transmission of the leverage ratio, cT - stiffness to the stack, cF - 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.


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

Figure 6.1. resonant frequency of piezoelement with lever transmission on series PU 100

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.


Cross motion

Because of the principle of a lever transmission with parallelogram design, the motion in one direction is followed by a small motion in the other rectangular direction. Though the motion follows a parabolic curve, the end faces will make a parallel motion. As mentioned, the parallelism due to the lever transmission system is better than the parallel motion of the stack itself.

The order of this cross motion is approximately 0.2% but depends on the specific construction parameters.

For example, a TRITOR element with 40µm motion in the x direction will make a simultaneous motion of about 50nm in the y direction. For most applications this will not disturb the positioning precision. But in other cases it should be taken into account.

Because of the relation of 500:1 between the aspired motion and the cross motion, it is easy to understand that this parabola can also be described by a straight line.

Hence, cross motion can be minimized by optimizing the straightness of the piezoelement. This was done for the piezoelement PU 100. By alignment the cross motion was minimized to less than 15nm (see figure 6.2.).

Figure 6.2. optimization of the cross motion of a piezoelement PU 100

7. Simulation of dynamic properties, transformation of electrical and mechanical properties

The piezoelectrical efffect describes the electromechanical coupling behavior of ferroelectrical materials. A theoretical model of electromechanical transducers is given with an electromechanical network. This network consists of electrical and mechanical components, which are connected via a specific four-pole circuit with the coupling factor y.

Using this model makes it possible to simulate the dynamic behavior of piezoelectric actuators.
The equivalent electrical circuit of piezoelectrical actuators can be determined with the help of an electrical impedance analyzer. With the equivalent electrical circuit, it is possible to simulate the dynamic behavior of the corresponding actuator system. The electrical model can be implemented in standard simulation programs and, thus, whole systems, including the power supply and a closed loop control circuit, can be simulated.

In the case of piezoelectrical transducers, the components of the theoretical networks are: the electrical free capacitance Cb, the mechanical elements compliance n (mechanical stiffness n-1), effective mass m and the intrinsic mechanical losses h. Due to the reciprocal network characteristic with the cupling factor y, the following relation can be given:

A transformation of the mechanical components to the electrical side of the four-pole network leads to the model of the equivalent electrical circuit of piezoelectrical actuators (Figure 7.1.1.).

Figure 7.1.1. mechanical scheme and equivalent electrical network of piezoelectrical transducers.


The representative electrical circuit gives a linear approximation of the electromechanical system. The characteristic equation of this network is similar to the characteristic equation of a simple spring-mass-oscillator. If a force F is applied to a mechanical spring with the stiffness n-1, the displacement x is given with x=Fk·n. With the given coupling relation, the equivalent equation for the electrical network is


The voltage ucn is given with:



Therefore the characteristic equation for the mechanical dispplacement of piezoelectrical transducers can be determined from the equivalent electrical circuit:



The resonant frequency is given by:


As mentioned before, the model of the equivalent electrical circuit corresponds to a linear approxiamation of the real coupling behavior of electromechanical transducers. This model includes neither the piezoelectrical hysteresis nor the creep and saturation of polarisation. Further restrictions of this model are given with the special characteristic of the piezoelectrical material parameters. All specific material properties (e.g. compliance, capacitance, piezoelectric coefficient) are dependent on the the applied electrical field. This dependence is not to be considered by the linear model.

Example number 20

We tried to find a simulation model for our actuator PU90. His model should be able to calculate the dynamic behavior of this element with different additional masses.

An additional mass leads to higher inductivity Lm in the model:

meff is the effective moved mass of the actuator system without an additional load and madd is the additional mass. To calculate Lm for any loads, it is nessecary to find the values for y and meff. This can be done with two measurements on an electrical impedance analyzer (with different loads). We made four measurements to reach a higher reliability in our model.

From these measurements, we obtained the following values for the model of the actuator system PU 90:

meff=89.8 g; y=2.12 m/As; R=38.92 Ω; Cn=171 nF; Cb=1.56 µF

With this model we calculated the resonant frequency with respect to an additional load and we simulated the response of this actuator to a voltage step. We proved our model with some additional measurements which were done with the aid of an interferometer displacement sensor. The results are given in the diagrams. (Figure 7.1.2./3./4.)

Figure 7.1.2. calculated resonant frequency of the actuator PU 90 with respect to an additional load.

Figure 7.1.3. PU 90 with additional load of 51g, measured response to a voltage step

Figure 7.1.4. PU 90 with additional load of 51 g, simulated response to a voltage step.

Simulation models of several of piezosystem jena´s actuator systems were made. With these models it is possible to determine significant mechanical parameters for the dynamic use of piezoelectrical actuators. In this way, custom-designed actuator systems can be realized much easier and more efficiently.

7.2. FEM optimization

A lot of applications require special mechanical properties to be considered from the beginning of the development. Additional loads to be moved dynamically require a complex optimization process of the stage. Using only formula 3.5.1. or 6.2. does not allow one to develop optimized stages. The full construction should be designed using FEM calculations. With our extensive experience in FEM calculations we can optimize more accurate parameters as stiffness, minimum tilting properties and others.
The following 2 pictures show how to optimize a stage for minimum cross motion. Cross motion occurs if one axis is moved (here the y-axis) but the other axis still shows a small motion. This cross motion is a result of a non-optimized construction, material imperfections and other factors.

In figure 7.2.1. the stress inside a stage is shown while the stage moves in y axis. Occurring tensile forces in x-axis leading to a tilt of 67µrad (calculated by FEM analysis) because of the above-mentioned imperfections.

In figure 7.2.2. the holes for mounting the stage are replaced; optimized for a minimum x tilt. The result is a tilt of 6,6µrad, which is 10 times smaller than the non-optimized stage in figure 7.2.1.
piezosystem jena uses its extensive experience in FEM calculations to develop special products optimized for the very special needs of your particular application.

Figure 7.2.1. non-optimized stage          Figure 7.2.2. optimized stage

8. Position control – closed loop systems

Because of the nearly unlimited resolution of the motion, piezoelectrical actuators are excellently suited for high precision positioning in the µm range to the nm range.

However, because of the hysteresis the relation between the applied voltage and the actuator’s motion is not unique.

There are some applications in practice where the high resolution of the motion is necessary, but the absolute positioning accuracy is not. The classic example is the problem of fiber positioning. The light of one fiber has to be coupled most efficiently into a second fiber, the knowledge of the absolute position of the fiber is not important.

Another example: If it is possible to return after each positioning event (transaction) to the 0 voltage position, the hysteresis does not affect the event (see also chapter 3.8.).

Of course some applications demand a high positioning repeatability. This can be reached by combining piezoelectrical actuators with a measurement system. Because of their high dynamics, piezoelectrical actuators are well suited for a closed loop system with a measurement system.

piezosystem jena uses different measurement systems. With strain gauges/gages it is possible to reach a position accuracy of 0.1 - 0.2%. Better results can be reached with special inductive and capacitive sensors.

You have to be careful working with a measurement system! Each measurement system always measures the motion at the place were the measurement system is located.

Variations between the measurement system and the point which should positioned (such as temperature effects), cannot be detected by the system.

piezosystem jena has developed piezoelements with a measurement system and we have also developed complete electronic controllers with integrated closed loop control. The closed loop is controlled by a PI or PID regulation circuit; the actual position measured is shown on the display. (Figure 8.1.)


Figure 8.1. principle of the closed loop control

Please note:

Often it is not possible to widen a piezoelectrical element with a measurement system. Therefore it is important to investigate carefully if an integrated measurement system is really needed. Of course such a system with sensors is more complicated and more expensive than a piezoelectrical system without measurement system.

If a measurement system is used in a closed loop system, the full range of the motion will be smaller by about 10 - 20% to preserve the dynamic of the closed loop regulation.

9. Characterization of measurement systems


The influence of the ferroelectrical hysteresis and the effect of the time dependent creep limit the best mechanical parameters of piezoelectrical actuators. In a large number of applications these effects do not play an important role. For other applications it is advantageous to implement a closed loop system. In a closed loop system the motion of the actuator will be measured and any unwanted changes from the given position will be corrected by the closed loop electronics.

piezosystem jena uses different types of measurement sensors:
• strain gauge/gages measurement systems,
• inductive (LVDT) and
• capacitive sensors. Strain gauges are very compact.

They can be integrated in nearly all piezoelectrical translation stages from piezosystem jena.

Capacitive or LVDT sensors should be used for systems needing the highest accuracy and/or dynamics. In some cases it is necessary to measure the displacement somewhere outside of the actuator. To provide the best performance for special requirements it is necessary to know the fundamental properties of the different sensor systems. In all cases it must be taken into account that, for µm and sub-µm accuracy, the full system has to be optimized (actuator, sensor, electronics, environmental conditions, etc.).

9.1 resolution

The piezoelectrical effect is a real solid state effect. In theory there is no limitation to the resolution; an infinitely small change in the electrical field gives rise to an infinitely small mechanical displacement. The real world offers some limits in resolution, which are caused by electrical, mechanical, acoustic and thermal noise.


The mechanical resolution is determined by the design of the drive. Piezo actuators from piezosystem jena are made with flexure hinges. Due to this construction principle no mechanical play arises, whereby the mechanical resolution is unlimited.


During operation the resolution indicates how much the piezo actuator moves if no motion is indicated i.e. the output voltage of the amplifier is to remain constant. This resolution is determined by the noise of the output voltage. Here only the frequency range of the output voltage is taken into consideration, which the actuator is able to follow.

Usual piezo actuators with lever transmission of piezosystem jena have a resonant frequency between 200Hz and 750Hz, so the output voltage is measured only up to this frequency. Frequencies above this range can not be converted into a motion by an actuator. A voltage noise of 0,3mV means that the piezo actuator has a resolution of 0,2Nm, related to a total travel of e.g. 100µm @ 150V (total voltage range: -10 … +130V).

For the resolution in the closed loop mode the noise of the measuring system must be additionally considered. Therefore the noise of the output voltage in the closed loop mode is measured. This value depends on the used actuator, the measuring system, the adjusted control parameters and the amplifier. The comutation of the resolution is made according to the formular indicated above.

The data for non-linearity related to the position, repeatability and lower and upper voltage limit are provided on the calibration report sent with the system.

To measure the resolution of a closed loop system we used a PX 100 with capacitive sensor and a power supply NV 40/3 CLE. The investigations were done considering best environmental conditions as mentioned above. The element was driven with a square function of approximately 40mV amplitude.

In figure 9.1.1. and 9.1.2. we show the sensor voltage and the measurement signal of the laser beam interferometer. The measurements were done for two different filter frequencies of the sensor electronics - 10Hz and 1kHz.


Figure 9.1.1. voltage signal from the capacitive sensor (red line) in comparison with the interferometer signal (blue line). The filter frequency of the sensor was set to 1kHz.

Figure 9.1.2. voltage signal from the capacitive sensor (red line) in comparison with the interferometer signal (blue line). The filter frequency of the sensor was set to 10Hz.

can see that the sensor signal with the 10Hz filter seems to be even better than the interferometer signal. The reason is that higher frequencies do not pass the 10Hz filter of the electronics and thus they are not measured. The sensor seems to be less noisy which can result in a higher accuracy of the full system.

Please note:

The highest positioning resolution requires very stable measurement conditions. The best measurement conditions are:

• a well-grounded environment
• an area far from electromagnetic fields (use shielded cables)
• vibrationally isolated conditions (an actively damped table is recommended)
• stable temperature conditions

Otherwise the environmental conditions will determine the resolution of the experiment.

9.2. Linearity

In the ideal case, the relation between the input signal (signal defining the position of an actuator) and the output signal (realized motion) should be linear.
When speaking about systems with integrated sensors, the linearity of the sensor (plus sensor electronics) is an important quality parameter.

Absolute position calculated from sensitivity

The linearity describes the approximation of the relation between indicated and true position. With the measured voltage (MON) the reached position is to be calculated on the basis of the formula below. The current values for the following calculations are taken from the calibration protocol (e.g. see page 108).

Example number 21

minimum voltage = - 0.007V*
total stoke = 400µm*s
sensitivity = 0.0248 V/µm*
measured voltage = 3.864V

* values are given in the calibration protocol

Absolute position calculated from sensitivity with consideration of the non-linearity As already mentioned, the monitor output voltage gives the best values for the current position of the system. Taking into account the measured non-linearity of the positioning system (see calibration curve) the absolute position calculated from the sensitivity should be corrected by the non-linearity.

The deviation of the true actuator position from this linear relation is the non-linearity. This is described by a polynomial function of higher order. In order to calculate the true actuator position on the basis of the measured voltage, the non-linearity must be taken into account.

Example number 22

minimum voltage = - 0.007V*
total stroke = 400µm*
sensitivity = 0.0248V/µm*
non-linearity @157µm=0.037%
measured voltage = 3.864V

= 156,088µm + 148nm
= 156,236µm

*values are given in the measurement report

Taking into account the non-linearity of 148nm at the position of the system of 156µm (MON = 3.864V) had to be corrected to 156.236µm. We determine the linearity of a sensor system in the following way: We operate the piezoactuator with a triangular wave over the full range of motion. The motion of the system will be measured by the integrated sensor and by the laser beam interferometer.

Figure 9.2.1. sensor output signal and signal from laser beam interferometer

Figure 9.2.2. linearity of a PX100 with strain gauge

9.3. Repetition accuracy (repeatability) ISO 5725

The repetition accuracy designates the error which arises if the same position from the same direction is approached again and again. In order to achieve a certain position repeatedly the same modulation voltage must be applied. The difference between modulation voltage and monitor voltage is regulated to zero by the electronic controller. The deviation of the different reached positions is indicated by repeatability. In the provided calibration protocol the maximum value of this error is indicated.

Figure 9.3.1. typ. repeatability for a PX100 with capacitive measurement system


The exact position of piezoelements cannot be accurately represented by an amplifier display due to its resolution. For highly exact positioning requirements it is recommended to supervise the position over the monitor voltage. For this an appropriate digital voltmeter is necessary.

9.4. Dynamic properties of a closed loop system

As stated, all single parts of a closed loop system influence the dynamic properties. This includes the properties of the actuator, of the sensor and the electronic system.

Please note:

When speaking about the properties of the actuator, it means the actuator as integrated into the real experiment. Additional masses or any forces from outside can influence the dynamic properties dramatically. Since we do not know how an individual might make use of our actuators, we did not load additional masses onto the actuators in our experiments. You will find further details about dynamical properties in chapter 5.

The closed loop electronics, utilizing control algorithms (P, PI, PID etc.) also affect the dynamic behavior. Each control system has to be calibrated with the special actuator. Do not change any modules or actuators of a control system!

Do not hesitate to ask us if you have any questions!

For a correct analysis of the dynamical properties, the damping curve and the phase shift over the frequency variation has to be measured. The dynamic function for operating the element should be investigated with respect to the containing frequency.
The driving frequency must be smaller than the maximum frequency of the full system. To ensure this, any curve differing from a sine wave form should be analyzed so as not to exceed the containing frequencies. Therefore, a Fourier transform must be made.

Of course this is not very practical.

An approximation in control theory says that the maximum system frequency of a feedback controlled system should be ten times less than the lowest characteristic frequency of the open loop system.

To give a simple impression of what we achieve in closed loop, we did some pure tests with the elements PX100 with strain gauge sensor. The element was driven with a rectangular function of 10Hz with an amplitude of approximately 50% of the full motion. We determined the time in which the controlled system reached an accuracy of 99% and 99,9% of the final position. (see Figure 9.4.1)

Figure 9.4.1. response time of a closed loop system PX100 with a strain gauge measurement system

9.5. Calibration protocol for closed loop system

Each closed loop system to be delivered to our customers is calibrated to reach the optimum values in linearity and repeatability.
These data are shown in calibration protocol coming with the system.


meaning of parameters

a) total range of motion
b) maximum corresponding voltage (voltage at the monitor output, system switched into closed loop operation)
c) minimum voltage (closed loop) at monitor output
d) sensitivity (range of voltage related to the full range of motion) (see also formula 9.2.1.)
e) nonlinearity@200µm = 0.0447% = 179nm

How to calculate the nonlinearity in nm from the data of the calibration protocol:

nonlinearity = 179nm

10. Electronics supplied for piezoactuators

10.1. Noise

In chapter 3.3. we mentioned that the sensitivity of piezoelectrical actuators is only limited by the voltage noise of the power supply. If a power supply has a noise given by ?U, the mechanical motion ?X, determined by this noise will be:


Where U is the current voltage to the piezoelement, Δl is the expansion for the voltage U.

Power supplies from piezosystem jena are developed and optimized especially for piezoelectrical actuators. So they have excellent noise characteristics, which allows positioning in the nm range. (see Figure 10.1.1.)

Figure 10.1.1. Noise of the power supply NV 40/3 CLE

Example Number 23

The power supply NV 40/3 CLE is suited for 3 channels (e.g. a TRITOR element for 3D positioning). This device has a voltage noise of < 0.3mV. For a maximum output voltage of 150V this is a dynamic range of < 2·10-6. For a piezoelement with a motion of 50µm we yield a mechanical noise of 1nm.

10.2. Current

For dynamical motions, all power supplies from piezosystem jena have a modulation input for each channel. So it is possible to generate oscillations given by a function generator via the modulation input. The electrical properties of piezoelectrical actuators are such that they act as capacitors with a high inner resistance of typically 1010Ω. electrical capacitance of:

high voltage actuators: 60nF
multi-layer low voltage actuators: 1800nF

For static and quasi-static applications the current needed does not play any role. Because of their high resistance piezoelectrical actuators practically do not need current to hold a position. They can also hold a position after separation from the power supply (please consider the safety instructions!). For dynamical applications we should consider the problem of charge and discharge the large capacitances C. The maximum current imax that would be needed for the actuator is:


dU/dt - rise time of the voltage.

The amount of the current needed can be very high for dynamical operations, so the rise time of the voltage and so on of the motion is often determined by the maximum current of the power supply. For a dynamical operation with a sinusoidal function, the maximum current imax is determined by:


imax – peak current required for sinusoidal operation (in A)
Upp – peak-peak drive voltage (in V)
f – frequency (in Hz)
C – capacitance of the actuator (in F)

The average current for this operation is:


10.3. Electrical power

Because of the high current that is needed for dynamical applications, the electrical power needed is high as well. The power Pmax for sinusoidal oscillation (with frequency f) of a piezoelement with capacitance C will be:

Pmax – peak power (in W)
Upp – peak-peakdrive voltage (in V)
Umax – maximum output voltage of the amplifier (in V)
f – operating frequency (in Hz)

The average current for this operation is


Example Number 24

The piezoelement PAHL 18/20 is suited for high loads and it has a capacitance of 7µF (for small field strength). This value can rise up to 14µF for large operating electrical fields. For an oscillation of 1kHz an actuator with 7µF capacitance requires a current of 3.3A. For a capacitance of 14µF one needs a current of nearly 7A. The output power will also increase 2 times and it will reach 1000W. For such electrical power, heating of the actuator should be considered (see also section 10.6. power loss).

10.4. Switched regime – oscillations with rectangular form

Due their properties piezoelements can work in a switched regime (e.g. for valve applications). For the voltage supply we use an electronic switch. If a short pulse is given to an actuator, the output voltage (also the voltage at the actuator) U(t) will rise depending on the time t, the capacitance of the actuator C and the inner resistance of the power supply Ri.


U0 - maximum output voltage of the supply.

If the inner resistance of the power supply Ri is small enough, the output voltage increases very quickly. This can be realized faster than the minimum rise time of the actuator, which is determined by the resonant frequency (see also section 5.1). That´s why the actuator is not able to expand faster. In such case the actuator will expand corresponding to the given electrical charges and so the expansion will reach an intermediate state smaller than the maximum output voltage U0 of the power supply.
In this way it is possible to generate a continuing signal form with an electrical switch by a series of charging and discharging pulses.

10.5. Coupling factor

The mechanical energy Wmech stored in the piezoelectrical material is created as a consequence of applied electrical energy. The electromechanical coupling factor k33 describes the efficiency of the conversion of the electrical energy Welectr into stored mechanical energy Wmech.


It can also be seen that the coupling factor depends on the direction and the parameters of the material. The formula is given here for the longitudinal effect.

The above mentioned formula is valid only for static and quasi-static conditions; power losses (e.g. by warming) are not included. The electrical power, which is not converted into mechanical energy (as expressed by the coupling factor), is given in form of electrical charges. These charges are returned to the power supply while unloading the actuator’s capacitance.

For piezoelectrical materials, the coupling factor k33 reaches values up to k33 = 0.68.

10.6. Power losses – dissipation factor

In the static regime the actuator stores energy W = ½ · C · U2. While unloading the piezoelements most of the electrical energy returns to the power supply. Only a small part will be converted into heating the actuator. These dissipation losses are expressed by the dissipation factor, the tangent of the loss angle δ.


Example Number 25

Let us consider the data given in example 24 (section 10.3). For the modulation of the piezoelement PAHL 18/20, an electrical power of approximately 1000W is necessary. The dissipated energy will be in the order of 50W, concentrated to a volume of 2ccm. After a short time, heating will bring the actuator in the region of the Curie temperature and the piezoelement will stop working. In such cases effective cooling will be necessary!

Power optimization

In some cases the choice of power supplies and piezoelements can be optimized for minimum power requirements. It might be better to use a longer stack with a lower operating voltage.

11. Lifetime - reliability

When speaking about lifetime of piezoelements 2 factors must be considered:

1. working under dynamic conditions
2. lifetime of stages with solid state hinges

11.1. Working under dynamic conditions

Working under static conditions, electric break can occur because of the migration of the electrode material into the ceramic layers. The reason may be high humidity and a constant electric field under high voltage. Working dynamically the reliability is much higher. A changing electric field converts the direction of the electrodes, migration is much less than under static conditions. There is no formula describing reliability under dynamic conditions because of the large number of technical parameters.
Piezoelements of piezosystem jena have been used for over 24 years in many different applications in research and development as well as in many industrial applications. Year by year we deliver many thousands of actuators to many customers. Proper handling during the construction of piezoelectrical stages and working together with our costumers ensures a long term reliability of our products without any significant failure rates.

11.2. Lifetime of stages with solid state hinges

Piezoelements consisting of a piezo stack integrated into a metal stage with solid state hinges are stable over a long period of time, if designed properly. The solid state hinges result in a lever transmission of the motion of the stack, which is amplified several times. If care is taken that the bending always keeps in its elastic range, a long lifetime exceeding billions of (109) of cycles can be guaranteed.

Because of our many years of experience, we know how to develop stages with a long and reliable lifetime. When operating piezoelements avoid excitation in the range of the resonant frequency of the stage. When this resonant frequency is excited, ringing and overshooting can lead to oscillations much higher than the resonant frequency, finally leading to a break of the solid state hinges.

Most failures of piezoelements occur because of improper mechanical handling and use of the elements. For proper handling and use please see also chapter 2.

12. Piezoelectrical, electrostrictive and magnetostrictive actuators

When using a solid state effect for generating a motion, piezoelectrical actuators are the most commonly used actuators. But a motion can also be generated by using other effects such as the electrostriction and magnetostriction.

We will give you an overview about these other principles.

Electrostriction The electrostriction effect basically exists parallel to the piezoelectrical effect.

The electrostrictive effect can be used above the Curie temperature. So electrostrictive materials are made from ceramics with a low Curie temperature. Electrostrictive actuators are also built as stack type actuators. The expansion of electrostrictive materials is nearly the same as for piezoelectrical materials.

In a small temperature regime of a few degrees electrostrictive materials show a small hysteresis (2-3%). But outside of this temperature range the hysteresis is larger than the hysteresis of piezoelectrical materials. So electrostrictive materials can be used only in a small temperature region (ΔT ~ 10 K). That is why these actuators do not find such a wide range of applications like piezoelectrical actuators.


A ferromagnetic material shows expansion under an applied external magnetic field. This effect is called the magnetostrictive effect and can be used for the construction of actuators. The material used is Terfenol.

Compared with the piezoelectrical and electrostrictive effect, magnetostrictive actuators show similar properties. Magnetostrictive actuators have a higher Curie temperature and there is the possibility of thermal separation of the cooling system and the magnetostrictive material.

13. Guidelines for using piezoelectrical actuators

Piezoceramics are relatively brittle materials. This should be noted when handling piezoelectrical actuators. All piezoelements (also elements with pre-load) are sensitive to shock forces.

Piezoelements without pre-load (e.g. series P, elements with lever transmission) should not be used under tensile forces (see also drawing in section 2).
Applications in which tensile forces or shear forces occur, should be realized by pre-loaded elements. On request we can optimize the integrated or external pre-loads for special applications.

During dynamical uses there can occur internal tensile forces due to the acceleration of the ceramic element itself (see also section 5 dynamic properties).

Pre-loaded piezoelements have a top plate with threads. Please note the depth of the treads. Do not apply large forces for fixing screws at the piezoelements!

Actuators are capacitive loads. Do not discharge actuators by short circuiting the leads. Ensure dielectric strength of your power supplies, wiring and connectors to prevent accidental arcing.

Abrupt discharging may cause damage to the stacks.

Piezoelectrical actuators such as stacks or various piezoelements with lever transmission work as capacitors. These elements are able to store electrical energy over a long time and the stored energy may be dangerous.

Connect and disconnect the elements only when the power supply is switched off. Because of the piezoelectrical and pyroelectrical effects, piezoactuators can generate electric charges if there are changes in the external mechanical loads or the temperature of the actuator.

Before you begin to work with any piezoelectrical actuating system note:

Switch off the power supply and discharge the actuator properly by setting the supplies to zero. If the actuator is disconnected, use a resistor for discharging the actuator. Do not switch on the power supply when the actuators are disconnected. Be sure that the electrical contact of the operator to the output connectors of the power supply is not possible when the supply is switched on!

Power supplies for piezoelements are developed especially for these elements. Do not use these supplies for other applications.

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