# Piezoelectric / ultrasound transducer analysis

## Performed calculations

If you have materials parameters and geometry of the piezo transducer, with the consent of Engineering Solutions, it is possible to order piezoelectric transducer parameter calculation. We can perform the following types of work:

### Harmonic analysis

• calculation of the amplitude-frequency characteristic (AFC) and phase-frequency characteristic (PFC) of the conductivity and the impedance of the transducer / piezoelectric element;
• calculation of AFC and PFC of the sensitivity of the transducer;
• calculation of AFC and PFC of average displacement of the radiating surface of the transducer;
• determination of oscillation modes of the transducer.
• • ### Transient analysis

• calculation of voltage oscillograms on a piezoelectric element;
• calculation of displacements of the radiating surface
• We also have a database of materials consisting of more than 100 components, including metals, polymers, composites, piezoceramic materials, etc.

## Finite element equations for piezoelectric analysis

Allik and Hughes developed the finite element method for a piezoelectric material in 1970 . For finite element analysis, ANSYS is using next form of the equation: 

• where {U} is an element structural displacement vector,
• {V} is a vector of nodal potential,
• [KUU] is an element structural stiffness matrix,
• [KVV] is an element dielectric permittivity coefficient matrix,
• [KUV] is piezoelectric coupling matrix,
• [CUU] is an element structural damping matrix,
• [CVV] is an element dielectric dissipation matrix,
• [MUU] is an element mass matrix,
• {F} is a vector of nodal and surface forces,
• {Q} is negative electric charge vector.

here 

• [KV] element electrical conductivity coefficient matrix,
• [σeff] is "effective" conductivity matrix,
• [NU] is element shape functions,
• [BU] is strain-displacement matrix,
• [BV] is the electric potential-electric field intensity matrix

The displacement field {U} and the electric potential {V} over the element can be defined in terms of the nodal displacements {Ui} and the nodal electric potentials {Vi} using corresponding shape functions defined as [NU] and [NV]:

The strain vector {S} and the electric field {E} are related to the displacement field {U} and the electric potential {V} using equations, respectively:

Here, [D] is the derivation operator defined as:

## Different equations and matrix notation of IEEE and ANSYS

In linear piezoelectricity the equations of linear elasticity are coupled to the charge equation of electrostatics by means of piezoelectric constants (IEEE Standard on Piezoelectricity):

• where {T} is 6x1 stress vector (component order x, y, z, xy, yz, xz), N/m2
• {S} is 6x1 strain vector (component order x, y, z, xy, yz, xz), m/m
• {D} is 3x1 electric displacement vector (component order x, y, z), C/m2
• {E} is 3x1 electric field vector (component order x, y, z), V/m (C/N)
• [cE] is stiffness matrix evaluated at constant electric field, N/m2
• [e] is 6x3 piezoelectric matrix relating stress and electric field, C/m2 or N/Vm
• [e]T is transpose of [e]
• S] is dielectric matrix evaluated at constant strain, F/m
• Piezoceramic manufacturer equations
(strain-charge form)
• where {T} is 6x1 stress vector (component order x, y, z, yz, xz, xy), N/m2
• {S} is 6x1 strain vector (component order x, y, z, yz, xz, xy), m/m
• {D} is 3x1 electric displacement vector (component order x, y, z), C/m2
• {E} is 3x1 electric field vector (component order x, y, z), V/m (C/N)
• [sE] is compliance matrix evaluated at constant electric field, m2/N
• [d] is 6x3 piezoelectric matrix relating strain and electric field, C/N
• [d]T is transpose of [d]
• T] is dielectric matrix evaluated at constant stress, F/m

IEEE and ANSYS are using different matrix notation.

## Piezoelectric transducer modeling

It is advisable to carry out the calculation of the transducer at the first stage for exploring different ideas without costly prototyping.
Modeling of physical processes allows to calculate the characteristics of the sensor and its output parameters with high accuracy (error not more than 5%), such as:
• amplitude of oscillation (displacement);
• resonant frequencies;
• voltage on the piezo element, etc.

The developer can be more flexible at the conceptual design stage. Thus, prototypes creating and experimental tests conducting required only after calculating the optimal design of the sensor. This makes it possible to significantly reduce costs during the R&D phase and anticipate in advance the pitfalls.

## Piezoelectric element FEA

The most difficult in the calculation of ultrasonic piezoelectric sensors is the simulation of the piezoelectric element. For a correct calculation of the piezoelectric element, it is required to know a large number of piezoceramic parameters. The graph (Fig. 1) shows the AFC of the piezoelectric element taken experimentally with help of impedance analyzer and calculated AFC of the piezoelectric element. The frequency error is 2.3% (at the first resonance) and 0.34% (at the second resonance).

Excitation frequency f=25kHz
Scale of oscillations 200000: 1  Excitation frequency f=73,6kHz
Scale of oscillations 10000:1  Excitation frequency f=280kHz
Scale of oscillations 10000:1
Figure 1 - Amplitude-frequency characteristic of the piezoelectric element

# Portfolio

## Ice sensor  Excitation frequency f=25kHz
Scale of oscillations 10000:1
• calculation of resonance frequency;
• determination of the frequency dependence on the amount of ice;
• calculation of spurious oscillations at the attachment point;
• determining the sensor sensitivity area.      Figure 2 - Amplitude-frequency characteristic of the sensor

## Densimeter

Determination of parasitic oscillations in a density meter.      ### Model of a piezo package used in a density meter    ## Dispergator

Comparison of different dispersant designs. Choosing the best construction.
 № Version Frequency of the first resonance f1, Гц Frequency of the second resonanceа f2, Гц Mechanical amplitude of tip oscillations at f1, µm at f2, µm 1 initial 18971 45295 36,29 24,55 2 with a whole piezoelectric element 19318 46189 42,46 28,08 3 with reduced length of mass 24697 40213 38,37 36,65 4 with increased mass radius 19226 39691 35,25 32,11 5 with a modified horn 20194 48475 40,33 37,03 6 with 4 whole piezo elements 18126 43230 46,85 33,94 ## Flowmeter

Flowmeter model
Oscillator  