Hysteresis is observable in open loop operation; it can be reduced by charge control (Figure 2) and virtually eliminated by closed loop operation.

There are fundamentally two different driving methods for piezoelectric actuators:

() charge driving,

() voltage driving.

Piezoelectric stack displacement depends on the charge stored on it; so using a charge feedback amplifier, it is possible to correctly drive the actuator. Considering the piezoelectric actuator as a 4-port element, for charge driving techniques, inputs are charge and external force (if present) and the outputs are tip displacement and voltage drops on it (see Figure 3). Using instead voltage driving techniques, inputs are voltage and external force and outputs are charge and tip displacement. For piezoelectric stack, we can create an FEM model that is useful to study the displacements of the free end of the stack and the system dynamics.

This model does not consider damping in a first stage: this effect is introduced in a second stage taking into account the experimental response of the piezoelectric actuator in resonance conditions. Including the mechanical dynamic behavior, this equivalent model can be easily implemented in electronics simulation software.

For a piezoelectric material, the equations we need are clearly standardized in the document ANSI/IEEE STD 176-1987, which expresses piezoelectric equations and physical constants in SI units.

The constitutive equations of a piezoelectric material at a microscopically level are the following:

or in an equivalent form:

The meaning of each symbol conventionally used in previous expressions is the following:

(i) strain of the material,

(ii) Young's modulus

(iii) mechanical stress

(iv) charge constant

(v): electric field applied

(vi) electric displacement

(vii) permittivity

(viii) elastic stiffness constant

(ix) piezoelectric constant

(x) impermittivity constant

The relations that regulate the behavior of the entire piezoelectric actuator are derived from a classic FEM approach based on the standard defined equations.

The core of the actuator is a stack of piezoelectric layers. Electrically speaking, the layers are in parallel, while from a mechanical point of view they are in series. Applying a voltage to the electrical leads of the stack will cause a displacement of the mechanical sides and a force on the faces that are respectively the sum of displacements and forces of each layer.

Once defined the potential and kinetic energy associated to the element, it is possible to obtain the two electromechanical equations that regulate the behavior of the piezoelectric element. If we consider charge driving, the equations of the stack are

While using voltage driving, the equations are

Mechanical and electrical dissipations are not considered during this step. The meaning of each symbol conventionally used in previous expressions is the following:

(i) is the mass matrix,

(ii) is the stiffness matrix (short circuit ),

(iii) is the stiffness matrix (open circuit ),

(iv) is the electromechanical coupling matrix,

(v) is the nodal displacement,

(vi) is the external force on stack tip,

(vii) is the voltage drops on piezo stack,

(viii) is the equivalent piezo capacitance,

(ix),

(x) is the charge stored on piezo stack.

It is possible to transform the nodal equations into modal equations in order to consider each mode independent from the others. Considering only the resonances in the frequency range between 10 Hz and 100 kHz, the model has been reduced to the first five modes. The goal is to obtain electrical impedance in order to know the load to be applied to the power driver.

Reducing the mechanical model, it is possible to convert it into a frequency domain as shown in (5) as equivalent admittance.

Performing a low-frequency analysis (), the inertial properties of each mechanical parallel branches can be neglected, while elastic contributes are summed to the electrical capacitance:

Introducing the residue term , the static capacitance (at ) can be written as

So, the quantity corresponds to the equivalent low-frequency admittance. Doing then a first-mode analysis, we should detract the first-mode quantity from equivalent low-frequency capacitance:

Impedance poles are admittance zeros; so in order to find impedance natural frequencies, admittance's numerator is imposed equal to zero.

Keeping apart the parameter in the gradient of the curve, it is possible to impose equal to zero just on the part in brackets:

First zero's pulsation is

where are pulsations of admittance antiresonances, which are in correspondence to the impedance resonances. From experimental tests on the piezoelectric stack or from FEM analysis, it is possible to obtain the value of the first antiresonance pulsation (which also is the first admittance resonance value) . Substituting this value in the last expression, the first residue is

In this way, the admittance associated to the first natural frequency is

Keeping then in consideration all the successive modes, it is possible to extend the matter in order to determinate all the successive residues. For the first modes, impedance can be written as

Residues can be written as