Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite

Abstract

An equivalent circuital model of magnetostrictive/piezoelectric laminated composite has been developed in order to predict its behavior in presence of dynamic electromagnetic fields. From magnetostrictive and piezoelectric constitutive equations, and using an equation of motion, magnetic-mechanical-electric equations are: obtained by building a symmetric adhoc equivalent circuit about the magnetoelectric (ME) coupling. The coefficients of the direct and converse effects are simulated. The circuit is further used to predict the voltage coefficients of laminated composite. The multilayer material is found to have significantly higher ME coefficients near resonance frequency. The ME coefficients and the voltage coefficients change significantly with the configuration of the multilayer, more specifically when the laminate operates in longitudinal-transverse (L-T) and transverse-transverse (T–T) modes.

1. Introduction

The Magnetoelectric (ME) effect arises in multiferroic materials that are electrically and magnetically polarizable due to coupling between electrical polarization and magnetization. These materials are interesting for technological applications, such as magnetic field sensors, transducers, resonators, and devices that provide opportunities in the area of renewable and sustainable energy through energy harvesting [1, 2].

Although the coupling can have nonlinear components, the ME effect is usually described mathematically by the linear ME coupling coefficient, which is the dominant coupling term [3]. The ME effect is characterized by an electric polarization induced by an applied magnetic field or a magnetization induced by an applied electric field called direct magnetoelectric effect (DME) and converse magnetoelectric effect (CME), respectively. The coefficients of the direct (${\alpha }_{\mathit{DME}}$) and converse (${\alpha }_{\mathit{CME}}$) effects can be expressed as:

$${\alpha }_{\mathit{DME}}=\frac{\partial P}{\partial H}$$E1

$${\alpha }_{\mathit{CME}}=\frac{\partial B}{\partial E}$$E2

being $E$ electric field intensity, $B$ magnetic flux density, $P$ electric polarization, and $H$ magnetic field intensity.

Maxwell’s equations of classical electromagnetism state that, in free space, $E$ and $B$ are not independent but are intimately linked to each other¹. Similarly, a theoretical method for the solid state that simultaneously describes relationships $P\sim H$ and $B\sim E$ (i.e. it describes the DME and CME behaviors) is developed below.

Multilayered composites can have even higher magnetoelectric coefficients than the particulate composites [4] and are therefore considered as viable alternatives [5].

2. Modes of operation and vibrational modes

The system to be modeled consists of bonded ferrite-ferroelectric layers of rectangular geometry. The dimensions are considered to be such that the width $w$ and thickness $t$ are very small relative to the length$\ell$; that is, the system is considered as one-dimensional (1D). The thicknesses of the magnetostrictive and piezoelectric layers are designated as $t_m$ and $t_p$, respectively. It is assumed that the interface (boundary between layers) is continuous.

In the modeling of ME laminate, different modes of operation or configurations need to be considered; in addition, to properly defining the coordinate systems (local and global) to be used. For historical reasons, the modes of operation find different lexicons. To prevent confusion, the notation adopted in this work is detailed below. Four basic modes of operation are distinguished and defined, according to the directions of sensed² magnetic field ($H$) and electric field ($E$). When the magnetic and electric field directions coincide with the longitudinal direction of the laminate, the mode is defined as “L-L”. Similarly, the modes “L-T,” “T-L,” and “T–T” designate longitudinal magnetic field - transverse electric field, transverse magnetic field - longitudinal electric field, and transverse magnetic field - transverse electric field, respectively. The basic modes of operation are outlined in a summary in Figure 1.

Figure 1.

Modes of operation of laminated magnetoelectric material.

When a magnetic field is applied to an ME laminate, more than one magnetostrictive vibrational mode is excited. However, the problem can be simplified, without losing correlation with reality, by considering a material with dimensions such that its width and thickness are very small compared to its length. In such case, the longitudinal axis direction of the laminate can be considered as the reference direction. Thus, in this chapter, the main direction of vibration is the longitudinal direction of the laminate³ (this justifies the selection of a one-dimensional 1D model).

Once the direction of vibration of the laminate is designated, it is then possible to carry out a consistent analysis of the system. When the material is excited with a magnetic field perpendicular to the longitudinal direction of the laminate, the fundamental mode of vibration will be the transverse mode and when it is excited with a magnetic field parallel to the longitudinal direction of the laminate, the fundamental mode of vibration will be the longitudinal mode (in both cases, as was mentioned, other modes of vibration can be considered as negligible).

3. Constitutive equations of the piezoelectric and magnetostrictive phases

If the system is considered in T–T mode of operation, the direction of polarization in the piezoelectric layer and the direction of magnetic field are along the thickness of the laminate (direction that is made to coincide with the cartesian coordinate axis z). The displacement $u=u\left(x\right)$ coincides with the longitudinal direction of the laminate (direction of the coordinate axis x). Figure 2a shows the directions of vectors $D$ (electric displacement or electric flux density), $B$ (magnetic flux density), $S$ (mechanical strain), $T$ (mechanical stress), along with the global coordinates x,y,z.

Figure 2.

(a) Schematic illustration of the laminated bilayer ME in T–T mode and local coordinates for the (b) piezoelectric and (c) piezomagnetic layer.

The ME effect of composite materials is due to the combined action of multiple intrinsic characteristics of the materials involved. In materials composed of two phases, two sets of constitutive equations are: required to describe the ME product property. Constitutive equations, in their most general form of the tensor type, are reduced to linear equations (both piezoelectric and piezomagnetic). The equations are: established from an electro/magneto-static point of view (i.e.,, for small excitation signals), and the simplifications are mainly a result of the crystal symmetry of the materials [7]. In the treatment of piezoelectric and piezomagnetic constitutive equations, the subscripts were established taking into account the reference axes used to define piezoelectric coefficients. To comply with this definition, the local coordinates shown in Figure 2b are defined in the piezoelectric layer and the local coordinates shown in Figure 2c are defined in the piezomagnetic layer (i.e., for the magnetic material has established a direction of magnetization⁴ in the direction of the axis 3).

3.1 Constitutive equations of the piezoelectric laminate

The constitutive equations of the piezoelectric material for the strain $S_{1p}$ and the electric displacement $D_3$, are:

$$S_{1p}=s_{11}^E{T}_{1p}+d_{31,p}{E}_3$$E3

$$D_3=d_{31,p}{T}_{1p}+{\epsilon }_{33}^T{E}_3$$E4

Being $E_3$ the electric field in the piezoelectric layer along the z direction, $T_{1p}$ the stress in the piezoelectric layer along the x direction, $s_{11}^E$ the elastic compliance at constant $E$, $d_{31,p}$ the piezoelectric constant, and ${\epsilon }_{33}^T$ the dielectric permittivity at constant stress. The subscript $p$ refers to the piezoelectric phase.

3.2 Constitutive equations of the magnetostrictive laminate

The constitutive equations of the magnetostrictive material for the strain $S_{1m}$ and the magnetic flux density $B_3$, are:

$$S_{1m}=s_{11}^H{T}_{1m}+d_{31,m}{H}_3$$E5

$$B_3=d_{31,m}{T}_{1m}+{\mu }_{33}^T{H}_3$$E6

Being $H_3$ the magnetic field intensity in the magnetostrictive layer along the z-direction, $T_{1m}$ the stress in the piezomagnetic layer along the x direction, $s_{11}^H$ the elastic compliance to constant $H$, $d_{31,m}$the piezomagnetic constant, and ${\mu }_{33}^T$ the magnetic permeability at constant stress. The subscript $m$ refers to the magnetostrictive phase.

There is a widely used notation in the design of laminated sensors in which the fundamental vibration mode is designated by two numbers: the first number indicates the direction of the excitation field and the second number indicates the direction of the longitudinal axis (both defined through local coordinates). Taking into account the Figure 2, the fundamental vibration mode of the magnetostrictive phase can be called “31-mode” and the fundamental vibration mode of the piezoelectric phase can be called “31-mode”. That is, the phases of the composite material will be modeled as magnetostrictive and piezoelectric transducers, both in mode 31. Note the correspondence that exists (for each phase) between its fundamental mode of vibration and the subscripts of its “piezo-” parameters in the constitutive equations⁵.

4. Equation of motion: general solution and boundary conditions

In order to design laminates operating under dynamic conditions, the coupling between the composite layers is considered through an equation of motion. Considering that the applied external field (electric or magnetic) is sinusoidal, the vibrational movement of the laminate will also be sinusoidal (harmonic oscillator).

The differential equation of motion for any element of mass can be obtained using Newton’s second law, written as:

$$\overline{\rho }\frac{{\partial }^{2}u\left(x,t\right)}{\partial t^2}=r_m\frac{\partial T_{1m}}{\partial x}+r_p\frac{\partial T_{1p}}{\partial x}$$E7

Where $u\left(x,t\right)$ is the displacement function of the medium and $r_m$ and $r_p$ are the relative relations of thickness of the magnetostrictive and piezoelectric layers, respectively, that is: $r_m=t_m/t$ and $r_p=t_p/t$, and it is evident that $r_m+r_p=1$. Furthermore, $\overline{\rho }=r_p{\rho }_p+r_m{\rho }_m$is the average mass density, with ${\rho }_p$ and ${\rho }_m$ as the mass densities of the piezoelectric and magnetostrictive components, respectively.

In the 1D model, mechanical movements are considered only in the longitudinal direction of the laminate (x-axis). In the direction perpendicular to it, the component layers are considered as free bodies and for the model, there are no strains or stresses between the layers of the composite.

The magnetostrictive and piezoelectric mass elements share the same displacement component $u$ and the same strain component, that is:

$$u_m=u_p=u$$E8

$$S_{1m}=S_{1p}=S=\frac{\partial u}{\partial x}$$E9

From Eq. (3) and (5), the mechanical stresses in the phases can be expressed as:

$$T_{1p}=\frac{S_{1p}}{s_{11}^E}-\frac{d_{31,p}}{s_{11}^E}{E}_3$$E10

$$T_{1m}=\frac{1}{s_{11}^H}{S}_{1m}-\frac{d_{31,m}}{s_{11}^H}{H}_3$$E11

Substituting in Eq. (7) the derivatives of Eq. (10) and (11) and considering that $\left(\partial E_3\right)/\partial x=\left(\partial H_3\right)/\partial x=0$ (since both $E_3$ and $H_3$ do not vary along the length of the material), the equation of motion is written as:

$$\frac{{\partial }^{2}u\left(x,t\right)}{\partial t^2}={\overline{\nu }}^2\frac{{\partial }^{2}u\left(x,t\right)}{\partial x^2}$$E12

where $\overline{\nu }$ is the speed of sound in the composite material:

$${\overline{\nu }}^2=\left(\frac{r_m}{s_{11}^H}+\frac{r_p}{s_{11}^E}\right)/\overline{\rho }$$E13

Eq. (12) is the classical wave equation of motion. Your resolution makes it simplified to:

$$\frac{{\partial }^{2}u\left(x\right)}{\partial x^2}+k^{2}u\left(x\right)=0$$E14

Which has the form of the equation of motion of a simple harmonic oscillator but with the spatial variable instead of temporal. In addition, the so-called dispersion relation $k=\omega /\overline{\nu }$ holds, where $k$ is the wave number and $\omega$ is the angular frequency.

The general solution of Eq. (14) is:

$$u\left(x\right)=Acos\mathit{kx}+Bsin\mathit{kx}$$E15

To solve the integration constants $A$ and $B$, the boundary conditions expressed in terms of displacement speeds $\dot{u}$are established: considering that the ends of the material are free of external stresses, let ${\dot{u}}_1$ be the displacement speed at $x=0$ and ${\dot{u}}_2$ the displacement speed in $x=\ell$, then:

$${\dot{u}}_1=\mathit{j\omega u}\left(0\right)$$E16

$${\dot{u}}_2=\mathit{j\omega u}\left(\ell \right)$$E17

The resolution leads to the expression:

$$u\left(x\right)=\frac{{\dot{u}}_1}{\mathit{j\omega }}cos\mathit{kx}+\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{\mathit{j\omega }sink\ell }sin\mathit{kx}$$E18

And, according to Eq. (9) and (18), the strains at the faces of the laminate are:

$$S\left(x=0\right)=\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{j\overline{\nu }sink\ell }$$E19

$$S\left(x=\ell \right)=\frac{{\dot{u}}_{2}cosk\ell -{\dot{u}}_1}{j\overline{\nu }sink\ell }$$E20

5. Potential: current coupling

Substituting Eq. (10) in Eq. (4), is obtained:

$$D_3=d_{31,p}\left(\frac{S_{1p}}{s_{11}^E}-\frac{d_{31,p}}{s_{11}^E}{E}_3\right)+{\epsilon }_{33}^T{E}_3$$E21

By definition of electric potential and taking into account the geometry of the system, it can be expressed that the piezoelectric vibrator is satisfied: $V_p=t_p{E}_3$. Furthermore, if defined:

$$\overline{{\epsilon }_{33}}={\epsilon }_{33}^T\left(1-\frac{{d^2}_{31,p}}{s_{11}^E{\epsilon }_{33}^T}\right)$$E22

Then the electrical displacement can be written as:

$$D_3=\frac{d_{31,p}}{s_{11}^E}{S}_{1p}+\overline{{\epsilon }_{33}}{E}_3$$E23

The electric charge at the surface of the piezoelectric is obtained integrating over the surface, according to:

$$Q={\int }_0^{\ell }{\int }_0^{\mathcal{w}}{D}_3\mathit{\text{dxdy}}=\mathcal{w}\frac{d_{31,p}}{s_{11}^E}\left[u\left(\ell \right)-u\left(0\right)\right]+\mathcal{wl}\overline{{\epsilon }_{33}}{E}_3$$E24

From Eq. (18), the expressions for $u\left(\ell \right)$ and $u\left(0\right)$ are obtained according to:

$$u\left(\ell \right)=\frac{{\dot{u}}_1}{\mathit{j\omega }}cosk\ell +\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{\mathit{j\omega }sink\ell }sink\ell =\frac{{\dot{u}}_2}{\mathit{j\omega }}$$E25

$$u\left(0\right)=\frac{{\dot{u}}_1}{\mathit{j\omega }}cosk0+\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{\mathit{j\omega }sink\ell }sink0=\frac{{\dot{u}}_1}{\mathit{j\omega }}$$E26

Then:

$$Q={\int }_0^{\ell }{\int }_0^{\mathcal{w}}{D}_3\mathit{\text{dxdy}}=\mathcal{w}\frac{d_{31,p}}{\mathit{j\omega }{s}_{11}^E}\left({\dot{u}}_2-{\dot{u}}_1\right)+\mathcal{wl}\overline{{\epsilon }_{33}}{E}_3$$E27

Taking into account the properties of harmonic phasor (the analysis is in the field of angular frequency, not temporal), the electric current in the piezoelectric vibrator $I_p$ can be expressed as:

$$I_p=\frac{\mathit{dQ}}{\mathit{dt}}=\mathit{j\omega Q}$$E28

Defining the static capacitance in the piezoelectric layer $C_0$ and the electro-mechanical coupling factor ${\phi }_p$ as:

$$C_0=\frac{\overline{{\epsilon }_{33}}\mathcal{wl}}{t_p}$$E29

$${\phi }_p=\mathcal{w}\frac{d_{31,p}}{s_{11}^E}$$E30

And substituting Eq. (29) and (30) in Eq. (28), the expression of coupling for piezoelectric material is:

$$I_p=\mathit{j\omega }{C}_0{V}_p+{\phi }_p\left({\dot{u}}_2-{\dot{u}}_1\right)$$E31

Similarly, from the constitutive equations of the magnetostrictive material, replacing Eq. (11) in (6), is obtained:

$$B_3=\left(\frac{d_{31,m}}{s_{11}^H}\right)S_{1m}+\overline{{\mu }_{33}^T}{H}_3$$E32

where is defined:

$$\overline{{\mu }_{33}^T}={\mu }_{33}^T\left(1-\frac{{d^2}_{31,m}}{{\mu }_{33}^T{s}_{11}^H}\right)$$E33

According to Faraday’s law of induction, on the solenoid $V_m$ can be expressed as:

$$V_m=-N\frac{d\mathrm{\Phi }}{\mathit{dt}}$$E34

where $\mathrm{\Phi }=B_3\mathcal{w}{t}_m$ is the magnetic flux. And under concepts of harmonic phasors, the differential of the magnetic flux can be expressed as:

$$V_m=-\mathit{Nj\omega }\mathrm{\Phi }=-\mathit{Nj\omega }{B}_3\mathcal{w}{t}_m$$E35

Substituting Eq. (32) in Eq. (35):

$$V_m=-\mathit{Nj\omega }\mathcal{w}{t}_m\left[\left(\frac{d_{31,m}}{s_{11}^H}\right)S_{1m}+\overline{{\mu }_{33}^T}{H}_3\right]$$E36

Considering a solenoid with $N$ turns of total length $\ell$ and through which an electric current $I_m$ circulates, the generated magnetic field in the solenoid can be expressed as:

$$H_3=\frac{N}{\ell }{I}_m$$E37

And taking into account that the relationship between deformation and displacement velocities can be expressed by the relation [8]:

$$S_{1m}=\frac{{\dot{u}}_2-{\dot{u}}_1}{\mathit{j\omega }\ell }$$E38

Then:

$$V_m=-\frac{N\mathcal{w}{t}_m}{\ell }\frac{d_{31,m}}{s_{11}^H}\left({\dot{u}}_2-{\dot{u}}_1\right)-\mathit{j\omega }\frac{\mathcal{w}{t}_m{N}^2}{\ell }\overline{{\mu }_{33}^T}{I}_m$$E39

Defining the static inductance in the magnetostrictive layer $L_0$ and the magneto-mechanical coupling factor ${\phi }_m$ as:

$$L_0=\overline{{\mu }_{33}^T}{N}^2\frac{\mathcal{w}{t}_m}{\ell }$$E40

$${\phi }_m=\mathcal{w}{t}_m\frac{N}{\ell }\frac{d_{31,m}}{s_{11}^H}$$E41

The expression of coupling for magnetostrictive material is:

$$V_m=-{\phi }_m\left({\dot{u}}_2-{\dot{u}}_1\right)-\mathit{j\omega }{L}_0{I}_m$$E42

6. Magneto-mechano-electric coupling

Based on the constitutive equations of both components (Section 3) and the equation of motion (via strain–stress coupling between layers; Section 4), an equivalent circuit model for laminated ME is developed. The mechanical forces are relative to the stresses. If the forces at the ends of the laminate are called $F_1$ at $x=0$ and $F_2$ at $x=\ell$, it is satisfied that:

$$-F_1=\mathcal{w}{t}_p{\left.T_{1p}\right|}_{x=0}+\mathcal{w}{t}_m{\left.T_{1m}\right|}_{x=0}$$E43

$$-F_2=\mathcal{w}{t}_p{\left.T_{1p}\right|}_{x=\ell }+\mathcal{w}{t}_m{\left.T_{1m}\right|}_{x=\ell }$$E44

Substituting Eq. (10), (11), (19), and (20) in Eq. (43) and (44) and taking into account Eq. (18), the expressions are: obtained:

$$F_1=-\left(\frac{\mathcal{w}{t}_p}{s_{11}^E}+\frac{\mathcal{w}{t}_m}{s_{11}^H}\right)\left(\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{j\overline{\nu }sink\ell }\right)+\mathcal{w}{t}_p\frac{d_{31,p}}{s_{11}^E}{E}_3+\mathcal{w}{t}_m\frac{d_{31,m}}{s_{11}^H}{H}_3$$E45

$$F_2=-\left(\frac{\mathcal{w}{t}_p}{s_{11}^E}+\frac{\mathcal{w}{t}_m}{s_{11}^H}\right)\left(\frac{{\dot{u}}_{2}cosk\ell -{\dot{u}}_1}{j\overline{\nu }sink\ell }\right)+\mathcal{w}{t}_p\frac{d_{31,p}}{s_{11}^E}{E}_3+\mathcal{w}{t}_m\frac{d_{31,m}}{s_{11}^H}{H}_3$$E46

Furthermore, with the equations obtained in Section 5, the force expressions can be simplified to:

$$F_1=-\overline{\rho }\overline{\nu }\mathcal{w}t\left(\frac{{\dot{u}}_2-{\dot{u}}_{1}cosk\ell }{jsink\ell }\right)+{\phi }_p{V}_p+{\phi }_m{I}_m$$E47

$$F_2=-\overline{\rho }\overline{\nu }\mathcal{w}t\left(\frac{{\dot{u}}_{2}cosk\ell -{\dot{u}}_1}{jsink\ell }\right)+{\phi }_p{V}_p+{\phi }_m{I}_m$$E48

If mechanical impedances are defined as:

$$Z_1=\frac{\overline{\rho }\overline{\nu }\mathcal{w}t}{jsink\ell }$$E49

$$Z_2=j\overline{\rho }\overline{\nu }\mathcal{w}ttan\frac{k\ell }{2}$$E50

Then the mechanical forces are rewritten as:

$$F_1=Z_1\left({\dot{u}}_1-{\dot{u}}_2\right)+Z_2{\dot{u}}_1+{\phi }_p{V}_p+{\phi }_m{I}_m$$E51

$$F_2=Z_1\left({\dot{u}}_1-{\dot{u}}_2\right)-Z_2{\dot{u}}_2+{\phi }_p{V}_p+{\phi }_m{I}_m$$E52

The ME equivalent circuit corresponding to the magneto-mechano-electric expressions obtained is represented in Figure 3. In this circuit, the forces $F_1$ and $F_2$ act as “mechanical voltage” and the displacements ${\dot{u}}_1$ and ${\dot{u}}_2$ as “mechanical currents”. Thus, an applied magnetic field produces a“mechanical voltage” via the magneto-mechanical coupling factor ${\phi }_m$ and due to the electro-mechanical coupling factor ${\phi }_p$, generates an electrical potential difference across the piezoelectric layer. In the circuit model of Figure 4, a transformer with ratio $1:{\phi }_p$ is used to represent the electro-mechanical coupling. Similarly, an applied electric field due to the magneto-mechanical coupling factor ${\phi }_m$, generates a magnetization in the magnetostrictive layer. In the circuit model of Figure 4, a transformer with ratio $1:{\phi }_m$ is used to represent the magneto-mechanical coupling. Note that, the equivalent circuit model considers both electro-mechanical coupling and magneto-mechanical coupling.

Figure 3.

Magneto-mechano-electric equivalent circuit.

Figure 4.

Magneto-mechano-electric equivalent circuit under condition of free edges.

Under free-free boundary conditions, that is, null forces at $x=0$ and $x=\ell$ ($F_1=F_2=0),$ the input and output terminals are grounded, and the circuit is simplified to the one shown in Figure 5 and then simplified to that shown in Figure 3.

Figure 5.

Magneto-mechano-electric equivalent circuit under free edge condition simplified.

The mechanical impedance is: $Z_{\mathit{\text{mech}}}=Z_1+Z_2/2$. Therefore, from Eq. (49) and (50), is obtained:

$$Z_{\mathit{\text{mech}}}=-\frac{j}{2}\overline{\rho }\overline{\nu }\mathcal{w}tcot\frac{k\ell }{2}$$E53

As the equivalent ME circuit obtained (Figure 3) contains symmetrical parts (magnetic and electrical), is possible to discuss both direct and converse effects. To analyze these effects, “open-circuit” and “short-circuit” conditions are: imposed:

Converse ME effect: Under open circuit condition $I_p=I_m=0$, the following relationship is derived:

$$\frac{{\phi }_{m}d{V}_m}{{\phi }_{p}d{V}_p}=\frac{\mathit{j\omega }{L}_0{{\phi }_m}^2}{\mathit{j\omega }{L}_0{{\phi }_m}^2+Z_1+Z_2/2}$$E54

Taking into account that: $V_m=-\mathit{Nj\omega }\mathcal{w}{t}_m{B}_3$ in the solenoid and $V_p=t_p{E}_3$ in the piezoelectric vibrator, and according to Eq. (2), the converse ME coefficient is expressed as:

$${{\alpha }_{\mathit{CME}}}_{\left(T-T\right)}=\frac{\partial B}{\partial E}=-\frac{\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_p}{{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{\omega \ell }{2\overline{{\mu }_{33}^T}}\overline{\rho }\overline{\nu }cot\frac{k\ell }{2}}$$E55

By indicating ${\alpha }_{\mathit{CME}}$ with the subscript T–T, it has been possible to omit the subscripts of $B$ and $E$.

Direct ME effect: Under short-circuit condition $V_p=V_m=0$, the following relationship is derived:

$$\frac{d{I}_p/{\phi }_p}{d{I}_m/{\phi }_m}=\frac{\frac{1}{{{\phi }_p}^2/\mathit{j\omega }{C}_0+Z_1+Z_2/2}}{\frac{1}{\mathit{j\omega }{L}_0{{\phi }_m}^2}+\frac{1}{{{\phi }_p}^2/\mathit{j\omega }{C}_0+Z_1+Z_2/2}}$$E56

Taking into account Eq. (28) and (37), and according to Eq. (1), the direct ME coefficient is obtained as:

$${{\alpha }_{\mathit{DME}}}_{\left(T-T\right)}=\frac{\partial P}{\partial H}=\frac{\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_m}{{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{r_p}{\overline{{\epsilon }_{33}}\overline{{\mu }_{33}^T}}{\left(\frac{d_{31,p}}{s_{11}^E}\right)}^2+\frac{\omega \ell }{2\overline{{\mu }_{33}^T}}\overline{\rho }\overline{\nu }cot\frac{k\ell }{2}}$$E57

By indicating ${\alpha }_{\mathit{DME}}$ with the subscript T–T, it has been possible to omit the subscripts of $P$ and $H$.

Thus, expressions of CME and DME coefficients with physical parameters of the phases that constitute the composite material are obtained. They are mathematical expressions that no longer involve circuit parameters but physical parameters of the ferrite and ferroelectric. Eq. (55) and (57) show the expression of the CME and DME coefficients as a function of frequency, however for frequencies much lower than the resonance frequency of the material ($\omega \ll {\omega }_r$), it can be approximated that $cot\raisebox{1ex}{$k\ell $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\sim \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$k\ell $}\right.$ and as a result the CME and DME coefficients become independent of the frequency and the length of the laminate.

7. ME laminate under resonance frequency

When the alternating field of excitation (electric or magnetic) has a frequency near to the resonance frequency of the material, Eq. (55) and (57) give an infinite value of ME coefficients, which is physically impossible. To predict the ME behavior near to its resonance frequency, it is necessary to consider the effect of mechanical dissipation in the energy conversion process. The mechanical impedance in the resonant state can be approximated as a series RLC circuit shown in Figure 6.

Figure 6.

RLC equivalent circuit under resonance frequency.

For such a model, the circuit impedance is obtained simplify Eq. (53) by using the Taylor series expansion [9]:

$$Z_{\mathit{\text{mech}}}=R_{\mathit{\text{mech}}}+\mathit{j\omega }{L}_{\mathit{\text{mech}}}+\frac{1}{\mathit{j\omega }{C}_{\mathit{\text{mech}}}}$$E58

where $R_{\mathit{\text{mech}}}$, $L_{\mathit{\text{mech}}},$ and $C_{\mathit{\text{mech}}}$ are the effective mechanical resistance, inductance and capacitance, respectively, which have the expressions:

$$R_{\mathit{\text{mech}}}=\frac{\pi Z_0}{8Q_{\mathit{mec}}}$$E59

$$L_{\mathit{\text{mech}}}=\frac{\pi Z_0}{8{\omega }_r}$$E60

$$C_{\mathit{\text{mech}}}=\frac{8}{{\mathit{\pi \omega }}_r{Z}_0}$$E61

being $Z_0=\overline{\rho }\overline{\nu }\mathcal{w}t$; ${\omega }_r=\raisebox{1ex}{$\pi \overline{\nu }$}\!\left/ \!\raisebox{-1ex}{$\ell $}\right.$ the average resonance frequency of the composite material (note that it holds: ${{\omega }_r}^2={L_{\mathit{\text{mech}}}}^{-1}{C_{\mathit{\text{mech}}}}^{-1}$) and $Q_{\mathit{\text{mech}}}$ the mechanical factor (numerically equal to the inverse of the tangent of mechanical losses). As the mechanical losses in the composite material are the result of the mechanical losses in the magnetic phase ($1/Q_m$) and piezoelectric phase ($1/Q_p$), it can be expressed:

$$\frac{1}{Q_{\mathit{\text{mech}}}}=\frac{r_m}{Q_m}+\frac{r_p}{Q_p}$$E62

Therefore considering mechanical dissipation, the expressions for the CME and DME coefficients are obtained according to:

$${{\alpha }_{\mathit{CME}}}_{\left(T-T\right)}=-\frac{\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_p}{{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}{8\overline{{\mu }_{33}^T}}\left(1-\frac{{\omega }^2}{{{\omega }_r}^2}\right)+\frac{\mathit{j\pi }\overline{\rho }\overline{\nu }\omega \ell }{8\overline{{\mu }_{33}^T}{Q}_{\mathit{mec}}}}$$E63

$${{\alpha }_{\mathit{DME}}}_{\left(T-T\right)}=\frac{\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_m}{{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{r_p}{\overline{{\epsilon }_{33}}\overline{{\mu }_{33}^T}}{\left(\frac{d_{31,p}}{s_{11}^E}\right)}^2+\frac{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}{8\overline{{\mu }_{33}^T}}\left(1-\frac{{\omega }^2}{{{\omega }_r}^2}\right)+\frac{\mathit{j\pi }\overline{\rho }\overline{\nu }\omega \ell }{8\overline{{\mu }_{33}^T}{Q}_{\mathit{mec}}}}$$E64

From Eq. (63) and (64), it can be seen that the CME and DME coefficients have complex expressions relative to frequency.

8. Simulation of the ME laminate in T: T mode

The behavior of a composite material with lead-free barium titanate (BTO) as ferroelectric component and nickel ferrite (NFO) as ferrimagnetic component is simulated. Table 1 lists the values used in the model for polycrystalline and polarized phases [10, 11, 12, 13]. Arbitrarily, a material geometry of length $\ell =20$ mm has been selected. With Eq. (63) and (64), the resonance profiles of $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ were obtained as a function of frequency. The programming codes were developed in the Matlab® software.

NFO	$s_{11}^H\left(\frac{m^2}{N}\right)·{10}^{-12}$	$d_{31}\left(\frac{m}{A}\right)·{10}^{-12}$	$d_{33}\left(\frac{m}{A}\right)·{10}^{-12}$	${\mu }_{r33}^T$	${\rho }_m\left(\frac{g}{{cm}^3}\right)$	$Q_m$
	6.5	125	−680	400	5	10
BTO	$s_{11}^E\left(\frac{m^2}{N}\right)·{10}^{-12}$	$d_{31}\left(\frac{m}{V}\right)·{10}^{-12}$	—	${\epsilon }_{r33}^T$	${\rho }_p\left(\frac{g}{{\mathit{cm}}^3}\right)$	$Q_p$
	7.3	−78	—	1345	7.6	65

Table 1.

Physical parameters of barium titanate (BTO) and nickel ferrite (NFO).

Figures 7 and 8 show the results of modules of ${\alpha }_{\mathit{CME}}$ and ${\alpha }_{\mathit{DME}}$ at frequencies from 20 kHz to 90 kHz, with $r_m$ as a parameter. From these profiles, it is observed that the $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ reach their maximum value when they operate near the resonance frequency. In addition, these results show that the ME coupling is strongly dependent on the thickness ratio between piezomagnetic and piezoelectric components, which is poorly reported with experimental data due to the difficulty in preparing the material.

Figure 7.

$\mid {\alpha }_{\mathit{CME}}\mid$ at frequencies from 20 kHz to 90 kHz, for ME laminate in T–T mode.

Figure 8.

$\mid {\alpha }_{\mathit{DME}}\mid$ at frequencies from 20 kHz to 90 kHz, for ME laminate in T–T mode.

Figure 9 shows the maxima of $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ as a function of the magnetostrictive phase content in the composite material. The cases for $r_m=0$ (without magnetostrictive layer) and $r_m=1$ (without piezoelectric layer) have been omitted since under these conditions there is no magneto-elasto-electric coupling and ${\alpha }_{\mathit{CME}}={\alpha }_{\mathit{DME}}=0$. It is observed that, in absolute values, the DME coefficient increases with the content of the piezomagnetic component, while the CME coefficient decreases.

Figure 9.

Maximum of $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ versus $r_m$ for ME laminate in T–T mode.

In addition, it is observed (Figure 9) that $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ become similar for composite materials with $r_m\sim 0.5$. This means that for the cases in which the structures have a similar phase content, the energy transition capacity in both effects is similar (the mechanical stress action is mutual).

The resonance frequency of the CME effect can be obtained when the module of ${\alpha }_{\mathit{CME}}$ reaches its maximum value, so deriving Eq. (63) with respect to the angular frequency ω, the expression is obtained:

$${\omega }_{\mathit{\text{resCME}}}={\omega }_r\sqrt{1+{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2\frac{8\overline{{\mu }_{33}^T}{r}_m}{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}-\frac{1}{2{Q_{\mathit{mec}}}^2}}$$E65

Similarly, the resonance frequency of the DME effect can be obtained when the module of ${\alpha }_{\mathit{DME}}$ reaches its maximum value, so deriving Eq. (64) with respect to the angular frequency ω, the expression is obtained:

$${\omega }_{\mathit{\text{resDME}}}={\omega }_r\sqrt{1+{\left(\frac{d_{31,m}}{s_{11}^H{\mu }_{33}^T}\right)}^2\frac{8\overline{{\mu }_{33}^T}{r}_m}{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}+{\left(\frac{d_{31,p}}{s_{11}^E}\right)}^2\frac{8r_p}{\overline{{\epsilon }_{33}}\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}-\frac{1}{2{Q_{\mathit{mec}}}^2}}$$E66

It can be noted that both the CME and DME resonance frequencies are independent of the total thickness of the compound $t$, they only depend on the thickness ratio between the phases.

Figure 10 shows the resonance frequencies (calculated using the relationship $\omega =2\mathit{\pi f}$). It is observed that the resonance frequencies (both CME and DME) increase with the ferrite content. In other words, the maxima in the profiles of Figures 7 and 8 change at higher frequencies with the magnetostrictive phase content. According to Eq. (65) and (66), the value of ${\omega }_{\mathit{\text{resDME}}}$ is always greater than ${\omega }_{\mathit{\text{resCME}}}$ due to the term ${\left(\raisebox{1ex}{$d_{31,p}$}\!\left/ \!\raisebox{-1ex}{$s_{11}^E$}\right.\right)}^2\left(\raisebox{1ex}{$8r_p$}\!\left/ \!\raisebox{-1ex}{$\overline{{\epsilon }_{33}}\pi \overline{\rho }\overline{\nu }\ell {\omega }_r$}\right.\right)$. This provides a mathematical interpretation for the profiles obtained in Figure 10, where it is observed that the DME resonance frequencies are always greater than the CME resonance frequencies, for any value of $r_m$. As seen in Figure 10, the difference between CME and DME resonance frequencies become smaller with the magnetostrictive phase content. That is, ${\omega }_{\mathit{\text{resDME}}}\sim {\omega }_{\mathit{\text{resCME}}}$ for high values of $r_m$. This allows to infer that the difference in resonance frequencies is mainly caused by the piezoelectric phase.

Figure 10.

CME and DME resonance frequencies ($f_{\mathit{\text{resCME}}}$ and $f_{\mathit{\text{resDME}}}$) as a function of magnetostrictive phase content, for ME laminate in T–T mode.

Another important parameter to characterize the ME effect is the so-called voltage coefficient, defined as:

$${\alpha }_V=\frac{\partial V}{\partial H}$$E67

In the piezoelectric phase is satisfied: $P_3={\epsilon }_0{\chi }_e{E}_3$, being ${\epsilon }_0$ the dielectric permittivity of free space and ${\chi }_e$ the dielectric susceptibility of the material (defined as ${\chi }_e=\raisebox{1ex}{${\epsilon }_{33}^T$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_0$}\right.-1$). Considering that $V_p=t_p{E}_3$ (Section 5) and that $\left({\epsilon }_{33}^T-{\epsilon }_0\right)\sim {\epsilon }_{33}^T$ then:

$${\alpha }_V=\frac{\partial V_3}{\partial H_3}=\frac{t_p}{{\epsilon }_0{\chi }_e}\frac{\partial P_3}{\partial H_3}=\frac{r_{p}t}{{\epsilon }_{33}^T}\frac{\partial P_3}{\partial H_3}$$E68

According to the definition of the DME coefficient (Eq. (1)), the voltage coefficient can be written as:

$${\alpha }_V=\frac{r_{p}t}{{\epsilon }_{33}^T}{\alpha }_{\mathit{DME}}$$E69

Note that, unlike ${\alpha }_{\mathit{DME}}$, the voltage coefficient depends on the thickness $t$of the material. Figure 11 shows the results obtained from the voltage coefficients at the resonance frequency (maximum voltage coefficients) as a function of the piezomagnetic phase content (considering a total thickness of $t=2$ mm in Eq. (69)). To express ${\alpha }_V$ in units of V/Oe (cgs system), the values in SI units were multiplied by the ${10}^3/4\pi \left(\frac{V/\mathit{Oe}}{\mathit{kg}{m}^3/C^{2}s}\right)$. It is observed that there is an optimum value of magnetic phase content that corresponds to a maximum value of ${\alpha }_V$. More specifically, a maximum of ${\alpha }_V\sim 80\mathit{mV}/\mathit{Oe}$ is observed for $r_m=0.3$ and at a frequency $f\sim 116\mathit{kHz}$.

Figure 11.

${\alpha }_V$ versus $r_m$, at resonance frequency, for ME composite in T–T mode.

It is interesting to note that the trend of ${\alpha }_V$ and ${\alpha }_{\mathit{DME}}$ with the values of $r_m$ do not coincide with each other. The maximum of ${\alpha }_{\mathit{DME}}$ is obtained when $r_m=0.9$ (Figure 9), while the maximum in ${\alpha }_V$ is observed with $r_m=0.3$ (Figure 11). In the field of application, such as transducer devices, most of the time the physical quantity that is measured is the electric potential, so the voltage coefficient could be considered a more significant parameter.

9. Simulation of the ME laminate in L-T mode

If the analysis is extended to an L-T configuration (Figure 1), the model must now imply a structure, where the polarization direction of the piezoelectric phase is transverse to the longitudinal direction of the material and the applied magnetic field is longitudinal. Figure 12a shows the directions of vectors $D$ (electric displacement), $B$ (magnetic flux density), $S$ (mechanical strain), $T$ (mechanical stress), along with the global coordinates x,y,z. The local coordinates shown in Figure 12b are defined in the piezoelectric layer and the local coordinates shown in Figure 12c are defined in the piezomagnetic layer.

Figure 12.

(a) Schematic illustration of the laminated bilayer ME in L-T mode and local coordinates for the (b) piezoelectric and (c) piezomagnetic layer.

In L-T mode, the constitutive equations of the piezoelectric material are Eq. (3) and (4) and the constitutive equations of the piezomagnetic material for the strain $S_{3m}$ and the magnetic flux density $B_3$ are:

$$S_{3m}=s_{33}^H{T}_{3m}+d_{33,m}{H}_3$$E70

$$B_3=d_{33,m}{T}_{3m}+{\mu }_{33}^T{H}_3$$E71

Being $H_3$ the magnetic field intensity in the magnetostrictive layer along the z-direction, $T_{3m}$ the stress in the piezomagnetic layer along the x direction, $s_{33}^H$ the elastic compliance to constant $H$, $d_{33,m}$the piezomagnetic constant, and ${\mu }_{33}^T$ the magnetic permeability at constant stress.

Note that in the magnetostrictive layer, the local coordinate “3” (which always designates the direction of magnetization) coincides with the direction of movement, therefore all the subscripts are “3” in the constitutive equations of the magnetostrictive material. In this case, the phases operate as a magnetostrictive transducer in mode 33 and as a piezoelectric transducer in mode 31.

Considering mechanical dissipation, the expressions for the CME and DME coefficients near their resonance frequency became:

$${{\alpha }_{\mathit{CME}}}_{\left(L-T\right)}=-\frac{\frac{d_{33,m}}{s_{33}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_p}{{\left(\frac{d_{33,m}}{s_{33}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}{8\overline{{\mu }_{33}^T}}\left(1-\frac{{\omega }^2}{{{\omega }_r}^2}\right)+\frac{\mathit{j\pi }\overline{\rho }\overline{\nu }\omega \ell }{8\overline{{\mu }_{33}^T}{Q}_{\mathit{mec}}}}$$E72

$${{\alpha }_{\mathit{DME}}}_{\left(L-T\right)}=\frac{\frac{d_{33,m}}{s_{33}^H{\mu }_{33}^T}\frac{d_{31,p}}{s_{11}^E}{r}_m}{{\left(\frac{d_{33,m}}{s_{33}^H{\mu }_{33}^T}\right)}^2{r}_m+\frac{r_p}{\overline{{\epsilon }_{33}}\overline{{\mu }_{33}^T}}{\left(\frac{d_{31,p}}{s_{11}^E}\right)}^2+\frac{\pi \overline{\rho }\overline{\nu }\ell {\omega }_r}{8\overline{{\mu }_{33}^T}}\left(1-\frac{{\omega }^2}{{{\omega }_r}^2}\right)+\frac{\mathit{j\pi }\overline{\rho }\overline{\nu }\omega \ell }{8\overline{{\mu }_{33}^T}{Q}_{\mathit{mec}}}}$$E73

And the expression for the voltage coefficient in the ME laminate in L-T mode is calculated by means of Eq. (69).

Using Matlab® software to develop the programming codes for the simulation in L-T mode, the results obtained are shown in Figures 13–17. The approximation $s_{11}^H\sim 3s_{33}^H$ was considered in the magnetostrictive layer [8].

Figure 13.

$\mid {\alpha }_{\mathit{CME}}\mid$ at frequencies from 20 kHz to 90 kHz, for ME laminate in L-T mode.

Figure 14.

$\mid {\alpha }_{\mathit{DME}}\mid$ at frequencies from 20 kHz to 90 kHz, for ME laminate in L-T mode.

Figure 15.

Maximum of $\mid {\alpha }_{\mathit{CME}}\mid$ and $\mid {\alpha }_{\mathit{DME}}\mid$ versus $r_m$, for ME laminate in L-T mode.

Figure 16.

CME and DME resonance frequencies ($f_{\mathit{\text{resCME}}}$ and $f_{\mathit{\text{resDME}}}$) as a function of magnetostrictive phase content, for ME laminate in L-T mode.

Figure 17.

${\alpha }_V$ versus magnetic phase content, in $f_{\mathit{\text{resDME}}}$, for ME laminate in L-T mode.

From the results obtained, it is highlighted that the range of resonance frequencies, both of CME and DME, present a shift towards higher frequencies for the laminate with L-T configuration: the range of resonance frequencies in L-T mode is $∆f\sim 120-226\mathit{kHz}$ (Figure 16), while for a T–T configuration it is $∆f\sim 108-135\mathit{kHz}$ (Figure 10).

Regarding the ME response analyzed through the voltage coefficient, for the L-T mode a maximum value of ${\alpha }_V\sim 880\mathit{mV}/\mathit{Oe}$ is found corresponding to an optimal piezomagnetic phase content of $r_m=0.2$ at a frequency $f\sim 135\mathit{kHz}$ (Figure 17).

For a better comparison between the results obtained in the L-T and T–T modes, Figure 18 shows the voltage coefficients ${\alpha }_V$ (near to resonance frequencies) with the magnetic phase content $r_m$. It is observed that the responses of the material operating in L-T mode are greater for any magnetic phase content, with respect to the responses in T–T mode. This behavior may be due, at least in part, to energy issues of the ferrite phase. Due to the geometry of the material, the energy component of shape anisotropy presents a lower demagnetization factor in the L-T case, where the magnetization is parallel to the length of the material (axis of greatest dimension). This agrees with the trend of the magnetic hysteresis cycles of pure ferrites in bulk, where with measurements in parallel configuration the magnetic responses are greater than those obtained in perpendicular configuration.

Figure 18.

Maximum ${\alpha }_V$ versus $r_m$ of the ME laminates, in L-T and T–T modes.

The results obtained by applying the circuit model to the barium titanate – nickel ferrite system (Sections 8 and 9) have similar trends to profiles reported by other researchers in analogous systems [14, 15, 16]. Additionally, the model was applied in a PZT–Terfenol system [9], observing similar trends to previously published results [17, 18, 19]. These similarities in trends allow us to infer that the designed circuit model is useful for modeling the behavior of ME laminated composites. It should be noted that the values of the CME and DME coefficients obtained in the simulation are expected to be somewhat higher than the real values, due to for example the fact that interface effects have not been considered in the model.

10. Conclusions

From the development of magneto-mechano-electric equations, in this chapter it was possible to simulate the direct and converse ME effects of composite laminates by developing an equivalent circuit model, thus predicting the behavior of materials in the presence of a dynamic electromagnetic field.

It is found that the ME material has significantly higher CME and DME coefficients near the resonance frequency. The resonance frequency depends on mode of operation and the phase thicknesses.

For the T–T and L-T modes, the coefficients of the direct and converse effects become similar for materials in which the structures have similar phase content ($r_m\sim 0.5$) suggesting that both effects share the same energy transmission capacity and are relative to the thickness ratio.

We found that the resonant frequency range of CME and DME exhibits a shift towards higher frequencies for the laminate with L-T configuration ($∆f\sim 120-226\mathit{kHz}$) compared to T–T configuration ($∆f\sim 108-135\mathit{kHz}$).

From the results obtained, it is observed that for the barium titanate – nickel ferrite system, the variation of T–T to an L-T mode generates notably higher values of ${\alpha }_V$ coefficients, in addition to a shift towards higher values of resonance frequency. The maximum voltage coefficient for the T–T mode has a maximum value of ${\alpha }_V\sim 80\mathit{mV}/\mathit{Oe}$ with an optimal piezomagnetic phase content of $r_m=0.3$ at $f\sim 116\mathit{kHz}$. In the L-T mode, the voltage coefficient reaches a maximum value of ${\alpha }_V\sim 880\mathit{mV}/\mathit{Oe}$ with an optimal piezomagnetic phase content of $r_m=0.2$ at $f\sim 135\mathit{kHz}$.

The results obtained allow us to infer that, at least in the frequency region studied, the tuning of these frequencies is an important step in the design of these materials for use as transducer devices.