Composite Piezoelectric Transducer

This example shows how to set up a piezoelectric transducer model following the work of Y. Kagawa and T. Yamabuchi (Ref. 1). The composite piezoelectric ultrasonic transducer has a cylindrical geometry that consists of a piezoceramic (NEPEC 6) layer, two aluminum layers, and two adhesive layers. The layers are organized as follows: aluminum layer–adhesive layer–piezoceramic layer–adhesive layer–aluminum layer.

The system applies an AC potential on the electrode surfaces of both sides of the piezoceramic layer. The potential in this example has a peak value of 1 V in the frequency range 20 kHz to 106 kHz. The goal is to compute the susceptance (the imaginary part of the admittance) Y = I/V, where I is the total current and V is the potential, for a frequency range around the four lowest eigenfrequencies of the structure.

The first step finds the eigenmodes, and the second step runs a frequency sweep across an interval that encompasses the first four eigenfrequencies. Both analyses are fully coupled, and COMSOL Multiphysics assembles and solves both the electric and mechanical parts of the problem simultaneously.

Although you could analyze this problem using a 2D axisymmetric model, in order to illustrate the modeling principles for more complicated problems, this example uses a 3D geometry.

When creating the model geometry, you make use of the symmetry by first making a cut along a midplane perpendicular to the central axis and then by cutting out a 10-degree wedge; doing so reduces memory requirements significantly.

Model Data

The model uses the following material data.

NEPEC 6 Material Parameters

Table 1: Elasticity Matrix cE.
128 GPa	68 GPa	66 GPa	0	0	0
	128 GPa	66 GPa	0	0	0
		110 GPa	0	0	0
			21 GPa	0	0
				21 GPa	0
					21 GPa

Table 2: Coupling Matrix e.
0	0	0
0	0	0
-6.1	-6.1	15.7

Table 3: Relative Permittivity εrS.
993.53	0	0
	993.53	0
		993.53

Aluminum Material Parameters


Parameter	Expression/Value	Description
E	70.3 GPa	Young’s modulus
nu	0.345	Poisson’s ratio
rho	2690	Density

Adhesive Material Parameters


Parameter	Expression/Value	Description
E	10 GPa	Young’s modulus
nu	0.38	Poisson’s ratio
rho	1700	Density

Results and Discussion

Figure 1 shows the lowest vibration mode of the piezoelectric transducer, while Figure 2 shows the transducer’s input susceptance as a function of the excitation frequency.

Figure 1: The lowest vibration eigenmode of the transducer.

Figure 2: Input susceptance as a function of excitation frequency.

The result is in agreement with the work in Ref. 1. A small discrepancy close to the eigenfrequencies appears because the simulation uses no damping.

Reference

1. Y. Kagawa and T. Yamabuchi, “Finite Element Simulation of a Composite Piezoelectric Ultrasonic Transducer,” IEEE Transactions on Sonics and Ultrasonics, vol. SU-26, no. 2, pp. 81–88, 1979.

MEMS_Module/Piezoelectric_Devices/composite_transducer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Work Plane 1 (wp1)

In the toolbar, click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 27.5.

In the text field, type 10.

Click

Click the

button in the toolbar.

Extrude 1 (ext1)

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Distances (mm)
5
5.275
15.275

Click

Click the

button in the toolbar.

This completes the geometry modeling stage.

Before defining material properties, select the domains where each physics applies. Proceeding in this order enables to preselect required material properties during their definition.

Solid Mechanics (solid)

Piezoelectric Material 1

In the window, under click .

In the window for , locate the section.

Click

Select Domain 1 only.

Electrostatics (es)

In the window, under click .

In the window for , locate the section.

Click

Select Domain 1 only.

Now materials can be defined.

Materials

Nepec 6

In the window, under right-click and choose .

In the window for , type Nepec 6 in the text field.

Locate the section. Click

Select Domain 1 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Elasticity matrix, Voigt notation	{cE11, cE12, cE22, cE13, cE23, cE33, cE14, cE24, cE34, cE44, cE15, cE25, cE35, cE45, cE55, cE16, cE26, cE36, cE46, cE56, cE66} ; cEij = cEji	{128[GPa], 68[GPa], 128[GPa], 66[GPa], 66[GPa], 110[GPa], 0, 0, 0, 21[GPa], 0, 0, 0, 0, 21[GPa], 0, 0, 0, 0, 0, 21[GPa]}	Pa	Stress-charge form
Coupling matrix, Voigt notation	{eES11, eES21, eES31, eES12, eES22, eES32, eES13, eES23, eES33, eES14, eES24, eES34, eES15, eES25, eES35, eES16, eES26, eES36}	{0, 0, -6.1, 0, 0, -6.1, 0, 0, 15.7, 0, 0, 0, 0, 0, 0, 0, 0, 0}	C/m²	Stress-charge form
Relative permittivity	epsilonrS_iso ; epsilonrSii = epsilonrS_iso, epsilonrSij = 0	993.53	1	Stress-charge form
Density	rho	7730	kg/m³	Basic

Alternatively, to define the symmetric elasticity matrix, cE, and the full coupling matrix, eES, you can click the button below the Output properties table under Component1>Materials>Nepec 6>Stress-Charge form in the Model builder and use the matrix input dialogs to enter the data as given in section NEPEC 6 Material Parameters.

Adhesive

Right-click and choose .

In the window for , type Adhesive in the text field.

Select Domain 2 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	10[GPa]	Pa	Basic
Poisson’s ratio	nu	0.38	1	Basic
Density	rho	1700	kg/m³	Basic

Aluminum

Right-click and choose .

In the window for , type Aluminum in the text field.

Select Domain 3 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	70.3[GPa]	Pa	Basic
Poisson’s ratio	nu	0.345	1	Basic
Density	rho	2690	kg/m³	Basic

Solid Mechanics (solid)

Now apply the boundary conditions for each physics.

In the window, under click .

Symmetry 1

In the toolbar, click

and choose .

Select Boundaries 1–5, 7, and 8 only.

Click the

button in the toolbar.

Electrostatics (es)

In the window, under click .

Terminal 1

In the toolbar, click

and choose .

Select Boundary 6 only.

In the window for , locate the section.

From the list, choose .

In the V0 text field, type 0.5.

This is half of the total peak voltage between the terminals, which accounts for modeling only the upper half of the transducer.

Ground 1

In the toolbar, click

and choose .

Select Boundary 3 only.

Definitions

Before generating the mesh, define a variable for the susceptance.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
B	imag(es.Y11)*36/2		Susceptance

In the above expression, the factor 36 compensates for the fact that the total current at the is only computed for a 10 degree wedge of the full transducer. Moreover, the factor 1/2 accounts for the fact that only the upper half of the transducer is modeled because of symmetry in the z direction and hence only half of the actual voltage is applied. Since no damping is modeled, the real part of the admittance es.Y11 will be zero. This is why it is suitable to evaluate only the imaginary part of the admittance, i.e. the susceptance.

Mesh 1