# Expansion and Force generation of Piezo Stacks

## Non-Linear Expansion Behavior

The real piezo actuator deviates significantly from a linear behavior in large-signal operation and the expansion shows a pronounced hysteresis behavior. There are also time and temperature-dependent effects. All these non-linear effects can be "controlled away" by suitable electronics.

A plot of the expansion as a function of applied voltage for a piezo actuator is shown in the next figure. Also depicted is how effective control reduces hysteresis (dashed line).

**Active Expansion of the Stack - Stroke Capacity**

The active stroke expansion of standard actuators is approximately 0.1% (stroke per length of the actuator in m/m) in unipolar operating mode (the voltage applied between the - and + terminals varies between zero and the maximum positive value of 150 V) and 0.14% in bipolar mode (the voltage applied between the - and + terminals ramps up from the allowable negative voltage value of -30 V to the maximum positive value of 150 V). The stroke expansion is dependent on the mechanical preload. A suitable preload enhances the performance of the piezo actuator.Die aktive Hubdehnung von Standard-Aktoren beträgt etwa 0,1% (Hub pro Länge des Aktuators in m/m) im unipolaren Betriebsmodus (die an den - und + Klemmen angelegte Spannung wechselt zwischen Null bis zum maximalen positiven Wert von 150 V) und 0,14% im bipolaren Modus (die zwischen dem - und + Terminal angelegte Spannung wird von dem zulässigen negativen Spannungswert von -30 V bis zum maximal positivsten Wert 150 V hochgefahren). Die Hubdehnung ist abhängig von der mechanischen Druckvorspannung. Eine geeignete Vorspannung erhöht die Leistung des Piezoantriebs.

**Transverse Contraction on Electrical Activation**

The Poisson's ratio of the piezo actuator material is approximately 0.3. Thus, the expansion of a piezo in the field direction (the stack axis) is associated with a 30% contraction in the perpendicular direction. The phenomenon of transverse contraction, in turn, causes tensile stresses at the interfaces of the stack. These typically require special attention in the design of mechanical integration into the application system. Piezotechnology actuators are specially designed to minimize the detrimental effects of these mechanisms. One approach is to increase the layer thicknesses near the ends of the stack. This measure reduces the internal electric field (=U/h; h = layer thickness) at the stack end, effectively reducing the stress at the interfaces.

**Modulus of Elasticity of Piezo Materials**

Der Modul der Piezomaterialien liegt im Bereich von 30 bis 60 GPa. Der Elastizitätsmodul hängt von der elektrischen Randbedingungen ab, nämlich wie der Piezoaktuator elektrisch beschaltet ist. Der effektive Elastizitätsmodul ist in einem elektrisch offenen Zustand höher ist als die in einem kurzgeschlossenen Zustand.

**Blocking Force Capacity of Piezo Stacks**

The generated mechanical stress sigma is equal to the modulus of elasticity Y multiplied by the active strain S

sigma = Y S

The force generation capability of the piezo stack is in the order of 3,500 N (unipolar) and 5,000 N (semi-bipolar operation) per cm2 of the stack's base area. This is an extraordinarily high value! The high force generation is very valuable for the construction and application of actuators. The actuators are mechanically stiff elements. This ability to generate large forces can only be utilized when the load introduction is mechanically rigid. In particular, transmission elements must be free of play.

**Force Generation of Piezo Stacks**

As described, an applied electric field induces a mechanical stress, causing deformation of the actuator body. The generated force Fg is the product of the actuator area A and the piezoelectric pressure Sigma:
F
g
=
A
⋅
σ
F
g
= A⋅σ
Where:
F
g
F
g
is the generated force,
A
A is the actuator area, and
σ
is the piezoelectric pressure.

F = A Sigma

The generated stress s is proportional to the piezoelectric constant d, which is a material property, and the applied electrical voltage. The maximum generated force is listed in the specifications of the stacks. The appropriate stack size can be selected by specifying the blocking force or the base area.

It is worth noting that piezo actuators have much greater mechanical stiffness than any other actuator of comparable size. In mechanical equilibrium, the generated force equals the sum of the usable load force and the internal elastic force associated with the deformation of the piezo body. This means that the usable force decreases linearly with displacement X. X0 denotes the free displacement (case of no applied load).

FLOAD = Fg (1-X) / X0

A representation of the force-displacement relationship of a piezoelectric actuator with different applied voltages is shown in the following figure.Eine Darstellung der Kraft-Weg-Relation eines piezoelektrischen Aktuators mit verschiedenen angelegten Spannungen ist in der nachfolgenden Abbildung dargestellt.

## Creep and Drift ofPiezo Stacks

The linear piezoelectric effect is superimposed by the time-dependent phenomenon of dipole orientation movement in the electric field. The immediate expansion of the stack is caused by the force-generating interaction of the electric field with the dipoles of the piezoelectric material. Analogous to magnetic materials, the microscopic structure of the piezoelectric material comprises domains of similarly oriented electric dipoles that are coupled together through the field. These domains of similarly oriented dipoles are called domains. The alignment of the domain dipoles is influenced by both the electric field and the mechanical stress. These couplings cause further time-dependent effects. After applying an electric field, the actuator immediately expands. Then, the dipoles begin to reorient, and the stroke movement of the stack increases after the initial immediate response until a steady-state value is reached. Naturally, these time-dependent effects are much smaller than those of the initial immediate reaction. Creep can be effectively corrected by feedback of position in a closed-loop control system.

## Unipolar and Bipolarer Operation of Piezo Stacks

PIEZOTECNICS piezo actuators are designed for unipolar and bipolar voltage operation. The unipolar normal operation ranges from 0 to 150 V. The quasi-bipolar operation extends from -30 V to 150 V, providing the advantage of up to a 40% higher stroke.

Bipolar operation is an excellent way to increase actuator performance. PIEZOTECHNICS actuators are designed for this purpose. However, it is important to note that this mode of operation increases the specific mechanical and electrical stresses on the material. The losses associated with power conversion increase significantly, and the maximum temperature limit must be observed. Especially in dynamic continuous operation, adequate cooling of the piezo stack must be ensured.