sensors Article

Equivalent Circuit Model of Low-Frequency Magnetoelectric Effect in Disk-Type Terfenol-D/PZT Laminate Composites Considering a New Interface Coupling Factor Guofeng Lou, Xinjie Yu * and Shihua Lu State Key Lab of Power System, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China; [email protected] (G.L.); [email protected] (S.L.) * Correspondence: [email protected]; Tel.: +86-10-6278-0250 Received: 10 May 2017; Accepted: 13 June 2017; Published: 15 June 2017

Abstract: This paper describes the modeling of magnetoelectric (ME) effects for disk-type Terfenol-D (Tb0.3 Dy0.7 Fe1.92 )/PZT (Pb(Zr,Ti)O3 ) laminate composite at low frequency by combining the advantages of the static elastic model and the equivalent circuit model, aiming at providing a guidance for the design and fabrication of the sensors based on magnetoelectric laminate composite. Considering that the strains of the magnetostrictive and piezoelectric layers are not equal in actual operating due to the epoxy resin adhesive bonding condition, the magnetostrictive and piezoelectric layers were first modeled through the equation of motion separately, and then coupled together with a new interface coupling factor kc , which physically reflects the strain transfer between the phases. Furthermore, a theoretical expression containing kc for the transverse ME voltage coefficient αv and the optimum thickness ratio noptim to which the maximum ME voltage coefficient corresponds were derived from the modified equivalent circuit of ME laminate, where the interface coupling factor acted as an ideal transformer. To explore the influence of mechanical load on the interface coupling factor kc , two sets of weights, i.e., 100 g and 500 g, were placed on the top of the ME laminates with the same thickness ratio n in the sample fabrication. A total of 22 T-T mode disk-type ME laminate samples with different configurations were fabricated. The interface coupling factors determined from the measured αv and the DC bias magnetic field Hbias were 0.11 for 500 g pre-mechanical load and 0.08 for 100 g pre-mechanical load. Furthermore, the measured optimum thickness ratios were 0.61 for kc = 0.11 and 0.56 for kc = 0.08. Both the theoretical ME voltage coefficient αv and optimum thickness ratio noptim containing kc agreed well with the measured data, verifying the reasonability and correctness for the introduction of kc in the modified equivalent circuit model. Keywords: magnetoelectric effect; magnetoelectric laminate composite; equivalent circuit model; interface coupling factor; magnetoelectric laminate based sensor

1. Introduction The magnetoelectric (ME) effect is defined as a dielectric polarization of a material when a magnetic field is applied, or, conversely, a magnetization of a material when an electric field is applied [1–3]. The intrinsic ME effect was first observed in single-phase material by Curie in 1894 [4]. Due to its high requirement for the temperature and low inherent magnetoelectric coupling, little research has been done on single phase ME material. Since van Suchtelen proposed the concept of a product property in the composite combining magnetostrictive and piezoelectric phases in 1972 [5–7], the large extrinsic ME effects in bulk composites have drawn great attention. The development of bulk ME composites reached a milestone in 2001, when the giant ME effect was reported in the Terfenol-D/PZT laminate composite by Ryu et al. [8–10]. Sensors 2017, 17, 1399; doi:10.3390/s17061399

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In the past two decades, researchers have focused on developing analytical models to explain and predict the ME effect in ME laminate composites. The first theoretical analysis was performed by Harshe et al. in 1991 [11–13]. They proposed a simple static elastic model, in which the magnetic and piezoelectric layers were assumed to be perfectly bonded and the strains in the both layers along the transverse directions were equal. Compared to this simple model, Nan developed a rigorous Green’s function technique to solve the constitutive equations [14,15], which is universal to the ME composites with various connectivity, not only 2-2 type laminate composite. By applying this method, all effective composite properties could be derived, but it could hardly provide a simple expression for predicting the ME coefficient. The models proposed by Harshe and Nan both supposed the interface between the phases to be ideally coupled, which does not exist in the actual condition. On this basis, Bichurin et al. proposed another generalized static elastic model [16,17] to calculate the effective ME coefficient at low frequency by introducing an interface coupling k, which represents the actual bonding condition between the magnetic and piezoelectric layers. In their modeling, the laminate composite was considered as a homogeneous bilayer whose effective material parameters in the constitutive equations were estimated by an averaging method, which is similar to the Green’s function. However, the calculation and expression for the effective ME coefficient was still complicated, although much simpler than Green’s function technique. Filippov agreed the coupling between the layers was not ideal, but suggested that the laminate composite could not be regarded as a homogeneous medium [18]. The definition of coupling coefficient between the phases proposed by Bichurin was also applied in his theory of ME effect for heterogeneous composites. Martins et al. defined a similar interface coupling factor for the polymer based composites and further proposed a ME coupling model for the particulate composites [19–21]. However, the above modeling methods are all applicable for static or quasi-static magnetic field, while the energy transduction of ME laminate composite commonly undergoes dynamic drive. Dong et al. proposed an equivalent circuit method account for ME effect under dynamic magnetic field drive (quasi-static condition as well), in which the magnetostrictive and piezoelectric layers were coupled though an equation of motion [22,23]. Although the theory provides simpler calculation and ME coefficient expression than the static elastic methods, it considered the laminate composite as a homogeneous medium and assumed that the magnetostrictive and piezoelectric layers have the same displacement and strain, i.e., the interface between layers were perfect. Consequently, the ME coefficient derived from the equivalent circuit could not reflect the actual interface coupling condition. In brief, Bichurin’s generalized elastic model considered the inherent imperfect coupling interface but got a complicated expression for ME coefficient for static or quasi-static magnetic field, while Dong’s equivalent circuit model provided a simple and straightforward ME prediction for dynamic situations but ignored the actual interface coupling condition. To overcome their drawbacks and combine their advantages, a modified equivalent circuit model considering the newly proposed interface coupling factor, under quasi-static and dynamic magnetic field drive, is proposed to predict the ME coefficient of 2-2 type laminate composite in this paper. Furthermore, the previous studies on the ME laminate composite are all based on long-type configuration. It is necessary to take the other configurations, such as disk-type, into consideration. This paper focuses on the modeling of disk-type ME laminate composite and aims to provide guidance for the design, fabrication and application of the ME laminate based devices, such as current sensor, magnetic sensor, energy harvester and wireless energy transfer system. 2. Theoretical Analysis Figure 1a shows the schematic of T-T mode disk-type laminate composite, which consists of a piezoelectric PZT-5 layer poled in its thickness (or transverse) direction sandwiched between two magnetostrictive Terfenol-D layers magnetized in their thickness (or transverse) direction. The thicknesses of the magnetostrictive and piezoelectric layer are tm and tp , respectively. Their radii

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are both a. The dashed volume represents a micro sector ring shaped mass unit. For the convenience modeling, thethe diameter of of each layer is assumed much larger than itsits thickness and thethe thickness of of modeling, diameter each layer is assumed much larger than thickness and thickness thethe epoxy resin layers could be ignored, i.e., i.e., the laminate couldcould be regarded as anasin-plane, twoof epoxy resin layers could be ignored, the laminate be regarded an in-plane, dimensional disk-type laminate. two-dimensional disk-type laminate. It also alsoshould shouldbebe noticed the cylindrical coordinate system is applied in the theoretical It noticed thatthat the cylindrical coordinate system is applied in the theoretical analysis analysis due to the disk-type configuration. For a micro sector ring shaped mass unit in the disk due to the disk-type configuration. For a micro sector ring shaped mass unit in the disk shown in shown in Figure 1a,b, r axis is along the radial direction, θ axis is vertical to the edge, and z axis is Figure 1a,b, r axis is along the radial direction, θ axis is vertical to the edge, and z axis is along the alongdirection. the axial direction. axial

Figure 1. 1. (a) The schematic of the T-T mode mode disk-type disk-type ME ME laminate laminate composite composite in in the the cylindrical cylindrical Figure coordinate, which of aofpiezoelectric PZT-5 PZT-5 layer poled its transverse sandwiched coordinate, whichconsists consists a piezoelectric layerinpoled in its direction transverse direction between two magnetostrictive Terfenol-DTerfenol-D layers magnetized in theirintransverse direction. The sandwiched between two magnetostrictive layers magnetized their transverse direction. tp, respectively. Their radii are thicknesses of the andand piezoelectric layer are are tm and The thicknesses of magnetostrictive the magnetostrictive piezoelectric layer tm and tp , respectively. Their radii are both a. The dashed volume represents a microsector sectorring ringshaped shapedmass massunit. unit. (b) (b) The The schematic of aa both a. The dashed volume represents a micro micro mass unit. unit. (c) (c)The Thelocal localcoordinate coordinateininthe themagnetostrictive magnetostrictivelayer, layer,ininwhich which3 micro sector ring shaped mass 3denotes denotesthe themagnetization magnetizationdirection. direction.(d)(d) The local coordinate thepiezoelectric piezoelectriclayer, layer,ininwhich which3 The local coordinate ininthe 3denotes denotesthe thepolarization polarizationdirection. direction.

For aagiven given geometrical dimension, the magnetostrictive main magnetostrictive deformation is maximum along the For geometrical dimension, the main deformation is along the maximum dimension of the sample. When an ACfield magnetic Hac is applied along the dimension direction ofdirection the sample. When an AC magnetic Hac isfield applied along the thickness thickness direction (denoted 3 in the local coordinate of magnetostrictive layer in Figure 1c), the direction (denoted 3 in the local coordinate of magnetostrictive layer in Figure 1c), the magnetostrictive magnetostrictive layers stretch and shrink along main strain (or vibration) direction, i.e., the radial layers stretch and shrink along the main strain (or the vibration) direction, i.e., the radial direction. Through direction. Through the bonding of epoxy resin, the piezoelectric layer is forced into oscillation along the bonding of epoxy resin, the piezoelectric layer is forced into oscillation along the same direction, the same direction, which excites a radial symmetric vibration mode, generating a voltage across the which excites a radial symmetric vibration mode, generating a voltage across the thickness direction thickness3direction (denoted 3 inofthe local coordinate ofFigure piezoelectric Figure 1d) due to the (denoted in the local coordinate piezoelectric layer in 1d) duelayer to theintransverse polarization transverse polarization of the piezoelectric layer. of the piezoelectric layer. Therefore,the theworking workingmechanism mechanismofof mode laminate could be physically divided Therefore, thethe T-TT-T mode MEME laminate could be physically divided into into three steps: the radial vibration of the magnetostrictive layer, the transfer of the strain, and the three steps: the radial vibration of the magnetostrictive layer, the transfer of the strain, and the radial radial vibration of the piezoelectric layer. Except Bichurin’s group, previous researchers have paid vibration of the piezoelectric layer. Except Bichurin’s group, previous researchers have paid more more attention to the magnetostrictive and piezoelectric constitutive equations, rather strain attention to the magnetostrictive and piezoelectric constitutive equations, rather than thethan strainthe transfer transfer between them, which is mostly dependent on the interface coupling condition in strainbetween them, which is mostly dependent on the interface coupling condition in strain-mediated bulk mediated bulk ME laminate Dong et al.inignored thetransfer losses and in the strain the transfer ME laminate composite. Dong composite. et al. ignored the losses the strain assumed strainsand of assumed the strains of the magnetostrictive and piezoelectric layers to be equal [22], resulting in the the magnetostrictive and piezoelectric layers to be equal [22], resulting in the discrepancy between the discrepancy between the theoretical and experimental data.was Although a rectifying factor was theoretical and experimental data. Although a rectifying factor empirically introduced to adjust empirically introduced to adjust the discrepancy [24], it lacked a realistic physical meaning. the discrepancy [24], it lacked a realistic physical meaning. Essentially, the strains of both layers are not Essentially, theworking strains ofcondition. both layers equal the in actual working condition. Inmagnetostrictive our analysis, the equal in actual In are ournot analysis, vibration and strains of the vibration and strains of the magnetostrictive and piezoelectric layers are first modeled through the equation of motion separately [25–28], and then coupled together with a new interface coupling factor

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and piezoelectric layers are first modeled through the equation of motion separately [25–28], and then coupled together with a new interface coupling factor that physically reflects the strain transfer between the phases, which is different from the definition by Bichurin et al. [17]. 2.1. Equivalent Circuit of Magnetostrictive Layer Terfenol-D is a typical ferromagnet with the cubic symmetry. When Hac is applied along the z-axis, a radial symmetric vibration is excited in the plane of Terfenol-D layer. The corresponding constitutive equations consist of nonzero terms for the cubic symmetry are H H Srm = s11 Trm + s12 Tθm + d31,m Hz ,

(1a)

H H Sθm = s12 Trm + s11 Tθm + d31,m Hz ,

(1b)

T Bz = d31,m Trm + d31,m Tθm + µ33 Hz .

(1c)

where the subscript m denotes the magnetostrictive layer. Srm and Sθm are the strain along the radial and azimuth direction, respectively. Trm and Tθm are the stress along the radial and azimuth direction, respectively. Hz and Bz are the magnetic field and magnetic flux density along the thickness H and s H are the compliance coefficients at constant H. d direction. s11 31,m is the transverse piezomagnetic 12 T is the permeability at constant stress along the axial direction. The constitutive coefficient, and µ33 equations can be rewritten in terms of the stress and magnetic flux density as Trm =

Tθm =

d31,m Hz , H s11 (1 − σm )

(2a)

d 1 (Sθm + σm Srm ) − H 31,m Hz , H (1 − σ 2 ) s11 s ( m 11 1 − σm )

(2b)

1 H (1 − σ 2 ) s11 m

(Srm + σm Sθm ) −

T Bz = d31,m ( Trm + Tθm ) + µ33 Hz .

(2c)

sH

where σm = − s12 H is the Possion ratio of Terfenol-D. According to the Newton’s second law, the radial 11

equation of motion for the Terfenol-D layer is ∂Trm Trm − Tθm ∂2 urm + = ρm , ∂r r ∂t2

(3)

where ρm and urm are the density and displacement of Terfenol-D layer, respectively. By substituting Equations (2a) and (2b) into Equation (3), and assuming that the magnetic field is evenly distributed along the radial direction, i.e., ∂Hz /∂r = 0, and using the definition of strain Srm =

urm ∂urm , Sθm = , ∂r r

(4)

the radial equation of motion can be rewritten as (urm is simplified as um for convenience) ∂2 um 1 ∂um um 1 ∂2 um + − = , 2 ∂t2 r ∂r ∂r2 r2 υm where υm =

q

1 H (1− σ 2 ) ρm s11 m

(5)

is the sound velocity of Terfenol-D along the radial direction. Under radial .

harmonic vibration, the displacement um can be represented as the phasor um and Equation (5) can be transformed into its phasor expression as .

.

.

. ∂2 um 1 ∂ um um + − 2 + k2m um = 0, 2 r ∂r ∂r r

(6)

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where km = υωm is the wave number and ω is the angular frequency of the AC magnetic field. By dividing k2m , Equation (6) is written as the standard form of the first order Bessel’s differential equation .

∂2 um ∂ ( k m r )2

+

whose general solution is

.

. ∂ um 1 ]um = 0, + [1 − 2 km r ∂(km r ) (km r )

1

(7)

.

um = AJ1 (km r ) + BY1 (km r ),

(8)

where J1 (km r ) is the first order Bessel function of the first kind, Y1 (km r ) is the first order Bessel function of the second kind, and A and B are the parameters determined by the boundary conditions. It is obvious that there is no displacement at the center r = 0, where the corresponding Y1 (km r ) tends to infinite, so the general solution is not available unless B = 0. Then, .

um = AJ1 (km r ).

(9)

According to the equilibrium of forces, by applying Equations (2a) and (4), the external force exerted on the circumferential surface of Terfenol-D satisfies . . . . Am ∂ um um Fm + H ( + σm ) − Nm H 3 = 0, (10) 2 r r=a s11 (1 − σm ) ∂r where Am = 2πatm is the circumferential surface area of Terfenol-D, Nm =

2πatm d31,m H (1− σ ) s11 m

is the

magnetoelastic coupling factor, a and tm are the radius and thickness of Terfenol-D, respectively. By applying the following recursion formula of the Bessel function of the first kind J10 ( x ) = J0 ( x ) −

J1 ( x ) , x

(11)

Equation (9) can be transformed into .

1 ∂ um = Am [km J0 (km r ) − J1 (km r )]. ∂r r

(12)

Then, the radial vibration velocity of Terfenol-D layer at the circumferential edge could be ∂um represented as vm,a = ∂t . By substituting Equation (12) into Equation (10) and applying .

r=a

.

vm,a = jω um,a , the magnetoelastic coupling equation of Terfenol-D is given as Sensors 2017, 17, 1399

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. J0 (km a) 1 − σm . + ]vm,a + Nm H 3 , Fm = −jρm υm Am [− (13) J1(km a) km a   F H  v m 3 m , a where and can be analogous to mechanical voltages in an equivalent circuit, can be . . . J ( k a ) 1 − σ m an equivalent m where Fm and H 3 can be analogous to mechanical circuit, vm,a can be Zm = jρ mυ mvoltages Am [ − 0 in ] + J0 (km aJ)1( k 1a −)σm k m a m analogoustoto mechanical current can be analogous to ] analogous mechanical current and Zand = jρ υ A [− + can be analogous to mechanical m m m m km a J1(km a) mechanical impedance. Thus, thecircuit equivalent of thelayer Terfenol-D layer is shown impedance. Thus, the equivalent of thecircuit Terfenol-D is shown in Figure 2. in Figure 2. .

Figure Terfenol-D layer. layer. Figure 2. 2. The The equivalent equivalent circuit circuit of of the the Terfenol-D

The expression for the Bessel function of the first kind is x 1 x 4 1 x 6 J0 ( x) = 1 − ( )2 + ( ) − ( ) + 2 (2 !) 2 2 (3!)2 2 ,

J1 ( x ) =

x

−

1 x 3 1 x 5 ( ) + ( ) −

(14a)

(14b)

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The expression for the Bessel function of the first kind is x 2 1 x 4 1 x 6 ( ) − ( ) +··· , J0 ( x ) = 1 − ( ) + 2 2 (2!) 2 (3!)2 2 J1 ( x ) =

(14a)

x 1 x 3 1 x 5 − ( ) + ( ) −··· . 2 2! 2 2!3! 2

(14b)

When x is relative small, i.e., x tends to zero, by reserving the first several terms, the Bessel function can be approximate to x n 1 x n1 Jn ( x ) ≈ ( ) =( ) . 2 Γ ( n + 1) 2 n!

(15)

Under the low frequency AC magnetic field drive, taking f = 50 Hz as an example, km a = υωm a = q H (1 − σ 2 ) = 3.24 × 10−3 is very small for Terfenol-D disk, so J0 (km a) = 1 = 2 and the 2π f a ρm s11 m km a km a J (k a) 1

mechanical impedance of the Terfenol-D layer can be simplified to Zm = jρm υm Am (

1 + σm ). km a

m

2

(16)

2.2. Equivalent Circuit of Piezoelectric Layer The top and bottom surfaces of PZT are covered with conductive silver electrodes. When the PZT layer undergoes the forced radial symmetrical vibration, the constitutive equations consist of nonzero terms for PZT with ∞ mm symmetry are E E Srp = s11 Trp + s12 Tθp + d31,p Ez ,

(17a)

E E Sθp = s12 Trp + s11 Tθp + d31,p Ez ,

(17b)

Dz = d31,p Trp + d31,p Tθp + ε T33 Ez .

(17c)

where the subscript p denotes the piezoelectric layer, Srp and Sθp are the strain along the radial and azimuth direction, respectively. Trp and Tθp are the stress along the radial and azimuth direction, respectively. Ez and Dz are the electric field and dielectric displacement along the thickness direction. E and s E are the compliance coefficients at constant E. d s11 31,p is the transverse piezoelectric coefficient. 12 T ε 33 is the permeability at constant stress along the axial direction. The constitutive equations can be rewritten in terms of the stress and dielectric displacement as Trp =

Tθp =

1 E (1 − σ 2 ) s11

1 E (1 − σ 2 ) s11

(Srp + σp Sθp ) −

d31,p Ez , E s11 (1 − σ2 )

(18a)

(Sθp + σp Srp ) −

d31,p Ez , E s11 (1 − σ2 )

(18b)

Dz = d31,p ( Trp + Tθp ) + ε T33 Ez ,

(18c)

sE

where σp = − s12 E is the Possion ratio of PZT. According to the Newton’s second law, the radial equation 11

of motion for the PZT layer is Trp − Tθp ∂2 urp ∂Trp + = ρp 2 , ∂r r ∂t

(19)

where ρp and urp is the density and displacement of PZT layer. By substituting Equations (18a) and (18b) into Equation (19), considering the surface of the electrode is equipotential, i.e., ∂Ez /∂r = 0, and using the definition of strain

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Srp =

∂urp urp , Sθp = , ∂r r

(20)

the radial equation of motion can be rewritten as (urp is simplified as up for convenience) ∂2 up up 1 ∂up 1 ∂2 up + = , − r ∂r ∂r2 r2 υp2 ∂t2 where υp =

r

1 E (1− σ 2 ) ρp s11 p

(21)

is the sound velocity of PZT along the radial direction. Under radial .

harmonic vibration, the displacement up can be represented as the phasor up and Equation (21) can be transformed into its phasor expression as .

.

.

up ∂2 up . 1 ∂ up + − 2 + k2p up = 0, 2 r ∂r ∂r r where kp = k2p ,

ω υp

(22)

is the wave number and ω is the angular frequency of AC magnetic field. By dividing

Equation (22) is written as the standard form of the first order Bessel’s differential equation .

∂2 up ∂ ( k p r )2 whose general solution is

.

+

. 1 ∂ up 1 ]u = 0, + [1 − 2 p kp r ∂(kp r ) (kp r )

(23)

.

up = CJ1 (kp r ) + DY1 (kp r ),

(24)

where J1 (kp r ) is the first order Bessel function of the first kind, Y1 (kp r ) is the first order Bessel function of the second kind, C and D is the parameters determined by the boundary conditions. It is obvious that there is no displacement at the center r = 0, where the corresponding Y1 (km r ) tends to infinite, so the general solution is not available unless D = 0. Then .

up = CJ1 (kp r ).

(25)

According to the equilibrium of forces, by applying Equations (18a) and (20), the external force exerted on the circumferential surface of Terfenol-D satisfies . . . . Ap Y0E ∂up up Fp + ( + σp ) − Np V 3 = 0, (26) 2 r (1 − σp ) ∂r r=a

where Ap = 2πatp is the circumferential surface area of PZT layer, Np = .

.

2πad31,p E (1− σ ) s11 p

is the

electromechanical coupling factor, and V 3 = Ez tp is the induced voltage across the PZT. a and tp are the radius and thickness of PZT, respectively. By applying the following recursion formula of the Bessel function of the first kind J10 ( x ) = J0 ( x ) −

J1 ( x ) , x

(27)

Equation (25) can be transformed into ∂up 1 = Ap [kp J0 (kp r ) − J1 (kp r )]. ∂r r

(28)

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Then, the radial vibration velocity of PZT layer at the circumferential edge could be represented . . ∂u as vp,a = ∂tp . By substituting Equation (28) into Equation (26) and applying vp,a = jω up,a , the r=a

electromechanical coupling equation of PZT is given as .

− Fp = jρp υp Ap [−

. J0 (kp a) 1 − σp . + ]vp,a − Np V 3 . J1 (kp a) kp a

(29)

Furthermore, by substituting Equations (18a) and (18b) into Equation (18c) and applying Equation (20), the dielectric displacement can be solved as d31,p 2d231 ] Ez + E (Srp + Sθp ) E s11 (1 − σp ) s11 (1 − σp ) . d31,p ∂urp urp = ε T33 (1 − k2p ) Ez + E ( + ) r s11 (1 − σp ) ∂r

Dz =[ε T33 −

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(30)

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Integrating Equation (30) over the area of the electrode surface, the electric charges induced on the electrode surface canvibration, be determined as Under harmonic the electric charges Q induced by the piezoelectric layer and the R R 2π a current I flowing into the piezoelectric layer can be expressed as phasors Q and I , respectively. The Q = 0 0 Dz rdrdθ relationship between them isR 2π R a T d31,p ∂urp urp 2 = (31) 0 0 [ ε 33 (1 − k p ) Ez + s E (1−σp ) ( ∂r + r )]rdrdθ . 11 I = jω Q 2 V3 2πad31,p 2 T = πa ε 33 (1 − kp ) tp + sE (1−σ ) up,a 11 2πad31,p Vp T = jω [πa 2 ε 33 (1 − k p2 ) 3 + E u p,a ] t p s11by (1 the − σ ppiezoelectric ) Under harmonic vibration, the electric charges Q induced layer and the current . . (32) 2 T 2 I flowing into the piezoelectric layer can beπexpressed respectively. The relationship I,31,p a ε 33 (1 − k pas ) phasors Q and 2πad  V 3 + jω u p , a E = jω between them is tp . . s11 (1 − σ p ) I = jω Q .  = jω C 0 V 3 + N p v p , a . 2πad . = jω [πa2 ε T33 (1 − k2p ) Vtp3 + sE (1−31,p up,a ] σ ) p 11 . (32) T . πa2 ε T33 (1−k2p ) . 2πad31,p πa2ε 33 (1 − kp2 ) V + jω u = jω p,a 3 E t p s11 (1−σp ) C0 = . . Fp tp where is static and = capacitance jω C0 V 3 + Nof p vthe p,a PZT layer. In Equations (29) and (32),

v

2 T

.

2

πa ε 33 (1−kp ) to mechanical voltage and mechanical current in an electric circuit, p,a wherecan C0 be = analogous is static capacitance of the PZT layer. In Equations (29) and (32), Fp tp . J k a ( ) 1 − σto 0 p p mechanical voltage and mechanical current in an electric circuit, and Zp =vp,a jρpυcan A [be ] − analogous + . p p . J ( k a ) 1 − σ p p 0 Zp = jρp υp Ap [−JJ1 ((kk paa)) + kkppaa ] can be be analogous to mechanical impedance, and Vand andVI3 are realistic can analogous to mechanical impedance, and I are 3 1 p electrical voltage and current, Thus, the Thus, equivalent circuit of the PZToflayer is shown realistic electrical voltage andrespectively. current, respectively. the equivalent circuit the PZT layerinis Figure shown3.in Figure 3.

Figure3.3.The Theequivalent equivalentcircuit circuitof ofthe thePZT PZT(Pb(Zr,Ti)O (Pb(Zr,Ti)O)3)layer. layer. Figure 3

Under the low frequency AC magnetic field drive, taking f = 50 Hz as an example, Under the low frequency AC magnetic field drive, taking f = 50 Hz J as an example, kp a = ( kp a) 1 2 q 0 = 1 = J0 (kp a) ωk a = ω a = 2π fa Eρ s E (1 − σ 22 ) = 1.04 × 10 −3 −3 k p a =k p a2 and the J ( k a ) a = 2π f a ρs ( 1 − σ ) = 1.04 × 10 is very small for PZT-5 disk, so = 1 p kp a υp p 11 11 kp a υp is very small for PZT-5 disk, soJ1 (kp a) and the 22 mechanical impedance of the PZT layer can be simplified to mechanical impedance of the PZT layer can be simplified to

Zp = − jρ pυ p Ap (

2.3. Introducing Interface Coupling Factor

1+σp kp a

)

(33) .

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Zp = −jρp υp Ap (

1 + σp ). kp a

(33)

2.3. Introducing Interface Coupling Factor The expression for the interface coupling factor in the long-type ME laminate composite proposed by Bichurin et al. is k c = (S1p − S10,p )/(S1m − S10,p ) [17], where S1p and S1m are the longitudinal strains of the piezoelectric and magnetostrictive layer, respectively, and S10,p is the longitudinal strain of piezoelectric layer with no friction between the layers. However, the major deficiencies of this interface coupling factor are as follows: (i) S10,p could hardly be measured or calculated, and the expression for k c lacks a simple and straightforward physical meaning; and (ii) the definition of k c proposed by Bichurin et al. is only applicable for the static or quasi-static condition in the elastic model. If it is applied to couple the equivalent circuits of magnetostrictive and piezoelectric layers for the dynamic condition, there would be two unnecessary mechanical current sources in the combined equivalent circuit of the whole laminate composite due to the existence of S10,p , resulting in a much more complicated expression for the ME coefficient. Therefore, a new interface coupling factor which physically reflects the strain transfer between the phases should be proposed in the equivalent circuit model for quasi-static and dynamic conditions. Considering the radial strain is ∂ur ur ( a ) − ur (0) Sr = = = ∂r a whose phasor form is

.

.

Sr =

R

vr ( a)dt − a

R

vr (0)dt

,

(34)

.

vr ( a ) − vr (0) , jωa

(35)

without loss of generality, let .

kc =

Srp .

Srm

.

=

.

.

vrp ( a) − vrp (0)

.

.

vrm ( a) − vrm (0)

=

vp,a .

vm,a

.

(36)

When the interface coupling factor k c = 1, i.e., Srp = Srm , the strains of two phase material are equal, which means the strain of magnetostrictive layer is totally transferred to the piezoelectric layer through the epoxy resin adhesive and the interface coupling between them is ideal. When k c = 0, i.e., Srp = 0, the strain of piezoelectric layer is always zero no matter how the magnetostrictive layers shrink and stretch, which means the strain of magnetostrictive layer could hardly be transferred to the piezoelectric layer through the epoxy resin adhesive and there is no interface coupling or friction between them. When kc is between zero and one, the strain of magnetostrictive layer is partially transferred to the piezoelectric layer and the actual coupling condition is quantitatively represented by kc . . Under free boundary condition, the radial vibration velocity at the circumferential edge vm,a . and vp,a are the mechanical current on the secondary side of the ideal transformer in Figures 2 and 3. By applying Equation (36), the equivalent circuits of the two phases could be coupled through an ideal transformer with a turn ratio kc , which transfers the radial strain (or vibration) from the magnetostrictive layer to the piezoelectric layer in the direct ME effect. Meanwhile, under the mechanical boundary condition that the resultant external force exerted on . . the ME laminate composite is zero (i.e., Fm + Fp = 0) in the unclamped harmonic vibration, and under the electrical boundary condition that the piezoelectric layer could be regarded as open-circuited (i.e., the dielectric displacement Dz = 0) in measuring its output, the ME laminate composite satisfies .

.

.

.

− Zm vm,a + Nm H 3 − Zp vp,a + Np V 3 = 0, .

.

.

I = jωC0 V 3 + Np vp,a = 0,

(37a) (37b)

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.

.

vp,a = k c vm,a .

(37c)

From the equations above, the equivalent circuit of the T-T mode disk-type Terfenol-D/PZT laminate composite could be developed, as shown in Figure 4. The applied small AC magnetic field . . drive H 3 on the Terfenol-D layer is eventually transformed into the induced electrical voltage V 3 on the PZT layer through three ideal transformers whose turn ratios are the magnetoelastic coupling factor, Sensors 2017, proposed 17, 1399 11 of 19 the newly interface coupling factor and the electromechanical coupling factor, respectively. Comparing with the equivalent circuit derived by Dong [22], this is a more physical and precise model precise model by the interface factor, which in is not discussed by introducing theintroducing interface coupling factor, coupling which is not discussed Dong’s model.inIt Dong’s should model. also be It should also be mentioned that this is a version observed from the Terfenol-D layer, in which the mentioned that this is a version observed from the Terfenol-D layer, in which the mechanical current .  v m , a electromechanical mechanical current main circuit impedance, loop is , the mechanicalcoupling impedance, the in the main circuit loopinis vthe mechanical the factor and m,a , the electromechanical coupling factor and static capacitance of PZT layer is converted to the primary side static capacitance of PZT layer is converted to the primary side of the ideal transformer whose turn of theisideal ratio kc . transformer whose turn ratio is kc.

Figure 4. 4. The The equivalent equivalent circuit circuit of of the the T-T T-T mode mode disk-type disk-type Terfenol-D/PZT Terfenol-D/PZT laminate laminate composite. composite. Figure

According to the magneto-elasto-electric equivalent circuit, the ME voltage coefficient of T-T According to the magneto-elasto-electric equivalent circuit, the ME voltage coefficient of T-T mode disk-type ME laminate composite could be obtained. By applying the voltage divider rule, the mode disk-type ME laminate composite could be obtained. By applying the voltage divider rule, the .  Np. V3 ratio of the output mechanical thethe exciting mechanical voltage NmN Hm3 H is3 is exciting mechanical voltage mechanical voltage voltage Np V 3 to to 2 2 NN p kpc kc Np V3 jωC0 N V = Nm H jω C p 3 3 = Zm + k c Zp +0

Nm H3

Zm + kc Zp +

. Np2 k c jωC02

N p kc

(38) (38)

By substituting the parameters into Equation (38), the ME voltage jωC0 .coefficient of T-T mode disk-type ME laminate composite could be derived as By substituting the parameters into Equation (38), the ME voltage coefficient of T-T mode disk dV3 2n(1 − n)ttotal d31,m g31,p kc type ME laminate composite could be as αV = = derived , (39) H (1 − σ ) E dH3 T-T ns11 (1 − σp )(1 − k2p ) + (1 − n)kc s11 m

2n(1 − n)ttotal d31,m g31,p kc

dV3

= ratio, tm α V = H the thickness thickness the− ratio tm +tp is the dH σ )(1 −means nsE (1 −which k 2 ) + (1 n)k sof (1 − σ )

(39) where n = t = of Terfenol-D total 3 T-T 11 p p c 11 m , layers tm to the total thickness of the ME laminate ttotal . The numerator of Equation (39) is a quadratic function of n, so it could be concluded that there is an optimum thickness ratio noptim corresponds to t tm n = mME = voltage the maximum coefficient. By letting δαV /δn = 0, the value of the optimum thickness ratio ttotal tm + tp could wherebe solved as is the thickness ratio, which means the ratio of the thickness of Terfenol-D 1 s noptim = t t o t a l . The numerator of to the total thickness of the sE (ME layers t m 1−σp ) laminate 11 1+ (1−k2p ) H k s ( 1 − σ ) Equation (39) is a quadratic function of n, so it could that there is an optimum thickness c 11be concluded m 1 s = n d2 s E (voltage 1−σp ) (40) 31,p ratio optim corresponds to the maximum By letting δα V δ n = 0 , the value of 2 11 1+ ME (1−coefficient. 1−σp ε T s E ) k c s H (1−σm ) 33 11 11 the optimum thickness ratio could be solved as = s E1 S tm

1+

noptim =

1+ =

=

s (1−σp ) 11 k c s H (1−σm ) 11 E 11

ε

1εT33 33

s (1 − σ p )

k s (1 − σ m ) H c 11

(1 − kp2 )

1 s (1 − σ p )

2 2 d31,p 1+ (1 − ) H T E 1 − σ p ε 33 (1 − σ m ) kc s11 s11 E 11

1

(40)

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3. Sample Fabrication and Experimental Setup The Terfenol-D disks (Central Iron and Steel Research Institute, Beijing, China) oriented along the thickness direction were machined to the dimension of φ20 × 2 mm. In order to explore the relationship between the ME voltage coefficient and thickness ratio and verify the existence of the optimum thickness ratio, the transversely polarized PZT-5 disks (Sunnytec Company, Suzhou, China) covered by silver electrodes on the top and bottom surfaces were machined to eleven sets of dimensions and the diameters were always kept as 20 mm. The parameters of Terfenol-D and PZT-5 are listed in Table 1. Table 1. The parameters of Terfenol-D and PZT-5. d31,m or g31,p

Material Terfenol-D PZT-5 2 1

1

10−10

2× Wb/N 10 × 10−3 Vm/N

E sH 11 or s11

σ

µr

kp

10−12

0.35 0.36

5 1

0.62

m2 /N

125 × 13.5 × 10−12 m2 /N 2

Cited from Central Iron and Steel Research Institute, China; Cited from Sunnytec Company, China.

Two Terfenol-D disks and one PZT-5 disk were laminated as a sandwich structure by epoxy resin adhesive (Eccobond 45 BLK, Emerson & Cuming, Germantown, WI, USA) and cured at 85 ◦ C for 6 h under mechanical load for a better bonding. The interface coupling factor, i.e., the strain transfer efficiency between the layers, could be influenced by the adhesive bonding condition, which is related to the temperature and pressure. Therefore in the sample fabrication (not in the test), two sets of weights, 100 g and 500 g were placed on the top of ME laminates with the same thickness ratio to provide different mechanical pressure for comparison. All the fabricated samples were tagged with numbers for convenience and their configurations are listed in Tables 2 and 3. Table 2. The configuration of ME laminate sample under 500 g weight mechanical load. Sample Number

1

2

3

4

Terfenol-D (mm) PZT-5 (mm) Thickness ratio n Pre-mechanical load

2 0.5 0.89

2 0.8 0.83

2 1 0.8

2 1.2 0.77

5

6

7

8

2 2 2 2 1.5 2 2.5 3 0.73 0.67 0.615 0.57 500 g weight

9

10

11

2 3.5 0.53

2 4 0.5

2 5 0.44

Table 3. The configuration of ME laminate sample under 100 g weight mechanical load. Sample Number

12

13

14

15

16

17

18

19

Terfenol-D (mm) PZT-5 (mm) Thickness ratio n Pre-mechanical load

2 0.5 0.89

2 0.8 0.83

2 1 0.8

2 1.2 0.77

2 2 2 2 1.5 2 2.5 3 0.73 0.67 0.615 0.57 100 g weight

20

21

22

2 3.5 0.53

2 4 0.5

2 5 0.44

Figure 5 shows the experimental setup for measuring the ME voltage coefficient. An electromagnet (EM5, East Changing Technologies, Inc., Beijing, China) is used to supply a DC bias magnetic field Hbias up to 5000 Oe for the ME laminate. A digital function generator (SP 1212B) and a power amplifier (HVP-1070B, Foneng Tech, Inc., Nanjing, China) supply the drive signal to a pair of Helmholtz coils to generate a small AC magnetic field Hac in parallel superimposed on Hbias , ranging from 0 to 10 Oe at f = 50 Hz, which is probed by a Gauss meter (Model 905, Honor Top Magnetic Tech Co., Ltd., Qingdao, China). The induced voltage of PZT layer is observed by an oscilloscope (Tektronix TPS 2014). Because the Terfenol-D layers can be regarded as conductive, the output voltage between the top and bottom surfaces of the laminate composite equals the induced voltage between the surfaces of the PZT layer because of its transverse polarization. The outlet conducting wires are soldered on the top and bottom surfaces of whole ME laminate for convenience.

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Figure Figure 5. 5. The The experimental experimental setup setup for for ME ME measurement measurement system. system.

4. Results and Discussions 4. Results and Discussions 4.1. 4.1. Magnetostriction Magnetostriction and and Piezomagnetic Piezomagnetic Coefficient Coefficient From seen that that the the ME ME voltage voltage coefficient coefficient ααv From the the theoretical theoretical expression expression Equation Equation (39), (39), it it can can be be seen v for T-T mode is proportional to the piezomagnetic coefficient d 31,m, which is a function of the DC bias for T-T mode is proportional to the piezomagnetic coefficient d31,m , which is a function of the DC bias magnetic field. Considering Considering dd31,m isisthe with respect to the DC bias magnetic field, magnetic field. the derivative derivative of of λλ31,m 31,m 31,m with respect to the DC bias magnetic field, the the magnetostriction magnetostriction of of Terfenol-D Terfenol-D should should be be discussed discussed firstly. firstly. The The magnetostriction magnetostriction of of Terfenol-D Terfenol-D is is measured by a resistance strain gauge and a Wheatstone bridge (YJZ-8, TRX Co., Ltd., Beijing, measured by a resistance strain gauge and a Wheatstone bridge (YJZ-8, TRX Co., Ltd., Beijing,China). China). For the convenience convenienceofofmeasurement, measurement, a longitudinally magnetized long-type Terfenol-D machined For the a longitudinally magnetized long-type Terfenol-D machined from from the commercial Terfenol-D rod is applied, instead of a transversely magnetized disk-type the commercial Terfenol-D rod is applied, instead of a transversely magnetized disk-type Terfenol-D. Terfenol-D. Figure 6 shows the longitudinal magnetostriction λ33 and transverse magnetostriction λ31 of Figure theoflongitudinal magnetostriction λ33atand magnetostriction 31 of Terfenol-D as6 ashows function the DC bias magnetic field Hbias roomtransverse temperature measured by a λ strain Terfenol-D as a function of the DC bias magnetic field H bias at room temperature measured by a strain gauge method. By calculating the slope of the measured λ-Hbias curves using a six-order polynomial gauge method. calculating and the slope of thepiezomagnetic measured λ-Hcoefficient bias curves using a six-order polynomial curve fitting, theBy longitudinal transverse of Terfenol-D as a function of curve fitting, the longitudinal and transverse piezomagnetic coefficient of Terfenol-D as a function of the DC bias magnetic field Hbias is illustrated in Figure 7. the DC bias magnetic field H bias is illustrated in Figure 7. It should be noticed that the transverse magnetostriction is negative when Hbias is applied It should be noticed that of thethe transverse magnetostriction is negative Hbias applied along along the thickness direction Terfenol-D bar, but it is illustrated as when positive foriscomparison. It the direction thethe Terfenol-D bar, magnetostriction but it is illustratedisasmore positive comparison. It can be can thickness be obviously seen of that longitudinal thanfor four times greater than obviously that the magnetostriction longitudinal magnetostriction is more than field four times greater of the that of theseen transverse with DC bias magnetic ranging fromthan 0 tothat 5000 Oe. transverse magnetostriction with DC bias magnetic field ranging from 0 to 5000 Oe. Upon increasing Upon increasing Hbias , the longitudinal magnetostriction almost reaches its saturation near 900 H bias, the longitudinal magnetostriction almost reaches its saturation near 900 ppm at 3000 Oe, ppm at 3000 Oe, whereas the transverse magnetostriction increases steadily without showing the whereas the transverse increases steadily showing could the sign saturation. sign of saturation. Themagnetostriction differences in the magnitude and without curve tendency be of attributed to The differences in the magnitude and curve tendency could be attributed to the demagnetization the demagnetization field. When Hbias is applied along the longitudinal direction in which the field. When Hbias isfactor applied the longitudinal in which the demagnetization of demagnetization of along the Terfernol-D bar isdirection small, the net magnetic flux is large factor and the the Terfernol-D bar is small, the net magnetic flux is large and the induced strain is saturated at lower induced strain is saturated at lower magnetic field while along the thickness direction in which magnetic field whilefactor alongisthe thickness in which theis demagnetization factorstrain is relative the demagnetization relative large,direction the net magnetic flux small and the induced could large, the net magnetic flux is small and the induced strain could hardly reach saturation. Although hardly reach saturation. Although the longitudinal magnetostriction is much higher than the transverse the magnetostriction is much higheristhan transversedone, reaches their corresponding one,longitudinal their corresponding piezomagnetic coefficient morethe comparable. its maximum 33,m piezomagnetic coefficient is more comparable. d 33,m reaches its maximum value higher than × 10−8 − 8 value higher than 2 × 10 Wb/N under 100 Oe, but drops rapidly to lower than 2.5 × 10−9 2Wb/N Wb/N under Oe, but drops rapidly to lower 2.5 ×exceeds 10−9 Wb/N Oe or more. under 1000 Oe100 or more. However, d31,m keeps rising than and even d33,munder under 1000 the magnetic field However, d 31,m keeps rising and even exceeds d33,m under the magnetic field higher than 2000 Oe, higher than 2000 Oe, which reveals an advantage of T-T mode ME laminate. Considering the ME which an advantage of T-T ME laminate.coefficient, Considering MEdisk-type voltage ME coefficient is voltagereveals coefficient is proportional to mode the piezomagnetic T-T the mode laminate proportional to the piezomagnetic coefficient, T-T mode disk-type ME laminate shows a potential for shows a potential for providing higher ME coefficient in the applications as sensors and actuators providing ME coefficient in the applications as sensors under large magnetic under largehigher DC magnetic field environment in comparison to and otheractuators configurations, suchDC as L-T mode field environment in comparison to other configurations, such as L-T mode or L-L mode long-type or L-L mode long-type ME laminate. ME laminate.

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1000 1000

1000 1000

800 800

800 800

600 600

600 600

Longitudinal LongitudinalMagnetostriction Magnetostriction λλ33 33 Transverse Transverse Magnetostriction Magnetostriction λλ31 31

400 400

400 400

200 200

200 200

00 00

0.5 0.5

11

1.5 1.5

22

2.5 2.5

33

3.5 3.5

44

4.5 4.5

Transverse Magnetostriction Magnetostrictionλλ 31 (ppm) (ppm) Transverse 31

Longitudinal Magnetostriction Magnetostriction λλ 33 (ppm) (ppm) Longitudinal 33

13of of19 18 14 14 of 19

00 55

DC (kOe) Hbias DC bias bias magnetic magnetic field field H (kOe) bias

Figure λλ33 33 magnetostriction 31 Terfenol-D as Figure 6. The longitudinal magnetostriction 33 and and transverse transverse magnetostriction magnetostrictionλλλ31 31 of of Terfenol-D Terfenol-Das asa Figure6. 6.The Thelongitudinal longitudinal magnetostriction magnetostriction λ transverse aafunction function of the DC bias magnetic field H bias at room temperature. functionofofthe theDC DCbias biasmagnetic magneticfield fieldHH bias at roomtemperature. temperature. bias -8 -8

-8 -8

xx 10 10

xx 10 10 2.25 2.25

22

Longitudinal LongitudinalPiezomagnetic Piezomagnetic Coefficient Coefficient dd33,m 33,m Transverse Transverse Piezomagnetic Piezomagnetic Coefficient Coefficient dd31,m 31,m

1.75 1.75

22 1.75 1.75

1.5 1.5

1.5 1.5

1.25 1.25

1.25 1.25

11

11

0.75 0.75

0.75 0.75

0.5 0.5

0.5 0.5

0.25 0.25

0.25 0.25

00 00

0.5 0.5

11

1.5 1.5

22

2.5 2.5

33 3.5 3.5 DC DC bias bias magnetic magnetic field field H (kOe) Hbias (kOe)

44

4.5 4.5

Transverse Piezomagnetic Piezomagnetic Coefficient Coefficient dd 31,m (Wb/N) (Wb/N) Transverse 31,m

Longitudinal Piezomagnetic Piezomagnetic Coefficient Coefficientdd 33,m (Wb/N) (Wb/N) Longitudinal 33,m

2.25 2.25

00 55

bias

Figure 7. The longitudinal piezomagnetic coefficient dd33,m transverse piezomagnetic coefficient Figure 33,m and and Figure 7. 7. The The longitudinal longitudinal piezomagnetic piezomagnetic coefficient coefficient d33,m andtransverse transverse piezomagnetic piezomagneticcoefficient coefficient of Terfenol-D as a function of the DC bias magnetic field H bias , which are the respective dd31,m 31,m of Terfenol-D as a function of the DC bias magnetic field Hbias, which are the respective slopes d31,m of Terfenol-D as a function of the DC bias magnetic field Hbias , which are the respectiveslopes slopes calculated from the corresponding magnetostriction curves in Figure 6. calculated calculatedfrom from the the corresponding corresponding magnetostriction magnetostriction curves curves in in Figure Figure 6. 6.

4.2. 4.2. ME ME Voltage Voltage Coefficient Coefficient 4.2. ME Voltage Coefficient Figure Figure 88 shows shows the the induced induced transverse transverse ME ME voltage voltage as as aa function function of of the the AC AC magnetic magnetic field field H Hacac for for Figure 8 shows the induced transverse ME voltage as a function of the AC magnetic field Hac sample sample No. No. 66 at at H Hbias bias = = 2498 2498 Oe Oe and and ff =50 =50 Hz. Hz. The The scattered scattered points points represent represent the the measured measured data data and and for sample No. 6 at Hbias = 2498 Oe and f =50 Hz. The scattered points represent the measured data the the dash-dot dash-dot line line is is the the trend trend line line derived derived from from the the linear linear regression regression analysis analysis based based on on them. them. The The and the dash-dot line is the trend line derived from the linear regression analysis based on them. The linear linear regression regression equation equation and and the the corresponding corresponding correlation correlation coefficient coefficient R R beside beside the the line line illustrate illustrate linear regression equation and the corresponding correlation coefficient R beside the line illustrate the the the degree degree of of linearity. linearity. It It can can be be clearly clearly seen seen that that the the transverse transverse ME ME voltage voltage varies varies linearly linearly with with the the degree of linearity. It can be clearly seen that the transverse ME voltage varies linearly with the AC AC AC magnetic magnetic field field over over the the range range 0.5–3.5 0.5–3.5 Oe. Oe. Therefore, Therefore, the the slope slope of of the the trend trend line line is is just just the the transverse transverse magnetic field over the range 0.5–3.5 Oe. Therefore, the slope of the trend line is just the transverse ME ME voltage voltage coefficient, coefficient, i.e., i.e., ααvv == 0.182 0.182 V/Oe V/Oe in in this this condition. condition. It It also also should should be be noticed noticed that that the the linear linear ME voltage coefficient, i.e., αv = 0.182 V/Oe in this condition. It also should be noticed that the linear regression equation contains an intercept term, which indicates the ME laminate induces voltage regression equation contains an intercept term, which indicates the ME laminate induces voltage even even regression equation contains an intercept term, which indicates the ME laminate induces voltage even under under H Hacac == 0. 0. This This interference interference could could be be attributed attributed to to the the constant constant ambient ambient stray stray magnetic magnetic field. field. under Hac = 0. This interference could be attributed to the constant ambient stray magnetic field. By By varying varying the the DC DC bias bias magnetic magnetic field, field, the the variation variation of of the the transverse transverse ME ME voltage voltage coefficient coefficient as as By varying the DC bias magnetic field, the variation of the transverse ME voltage coefficient as aa function of the DC bias magnetic field for sample No. 6 could be measured, as shown in Figure function of the DC bias magnetic field for sample No. 6 could be measured, as shown in Figure 9. 9. a function of the DC bias magnetic field for sample No. 6 could be measured, as shown in Figure 9. The The transverse transverse ME ME voltage voltage coefficient coefficient increases increases with with the the DC DC bias bias magnetic magnetic field field steadily steadily and and reaches reaches The transverse ME voltage coefficient increases with the DC bias magnetic field steadily and reaches to to 250 250 mV/Oe mV/Oe in in the the vicinity vicinity of of H Hbias bias = = 3000 3000 Oe. Oe. Through Through the the comparison comparison with with Figure Figure 7, 7, itit indicates indicates to 250 mV/Oe in the vicinity of Hbias = 3000 Oe. Through the comparison with Figure 7, it indicates that that the the ααvv-H -Hbias bias curve curve approximately approximately follows follows the the tendency tendency of of dd31,m 31,m-H -Hbias bias curve curve over over the the range range from from 00 that the αv -Hbias curve approximately follows the tendency of d31,m -Hbias curve over the range from 0 to to 3000 3000 Oe. Oe. This This experimentally experimentally reflects reflects the the fact fact that that the the transverse transverse ME ME voltage voltage ααvv is is proportional proportional to to dd31,m while the other parameters in Equation (39), especially the thickness ratio n and the interface 31,m while the other parameters in Equation (39), especially the thickness ratio n and the interface

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to 3000 Oe.17, This Sensors 2017, 17, 1399experimentally Sensors 2017, 1399

reflects the fact that the transverse ME voltage αv is proportional to 15 of of 19 19 15 d31,m while the other parameters in Equation (39), especially the thickness ratio n and the interface coupling factor factor kkccc,, keeps keeps constant. constant. Another Another significant conclusion from from Figure Figure 99 is is that that the the expression expression Another significant significant conclusion coupling for ααvv-H -Hbias bias curve is a monotone function in a wide range of H bias from 0 to 3000 Oe, the curve is a monotone function in a from 0 to providing for curve is a monotone function in a wide range of H bias from 0 to 3000 Oe, providing the bias bias potential for T-T mode disk-type ME laminate composite to measure a large DC current or magnetic field under under aa constant constant AC AC magnetic magnetic field field drive drive [29]. [29]. The The related are discussed our future future related details details are discussed in in our field paper. research paper.

function of of the the AC AC magnetic magnetic field field H Hac ac for sample Figure 8. The induced transverse transverse ME voltage as aa function magnetic field H ac Hbias bias = 2498 Oe and f = 50 Hz. The green scattered points represent the measured data under under No. 6 at H H = 2498 Oe and f = 50 Hz. The green scattered points represent the measured bias= 2498 Oe and f = 50 Hz. The green scattered points represent the measured data pre-mechanicalload load and the blue dash-dot line is the theline trend line from derived from the the linear linear 500 gggpre-mechanical and the the blueblue dash-dot line isline the trend derived the linear regression 500 pre-mechanical load and dash-dot is trend line derived from analysis. The linear regression andequation the corresponding correlation coefficient R beside the line regression analysis. The linear linearequation regression equation and the the corresponding corresponding correlation coefficient R regression analysis. The regression and correlation coefficient R illustrate the degree of linearity. beside the line illustrate the degree of linearity. beside the line illustrate the degree of linearity.

Figure 9. The The induced induced transverse transverse ME ME voltage voltage coefficient coefficient as as a function function of of the DC DC bias bias magnetic magnetic field field Figure Figure 9. 9. The induced transverse ME voltage coefficient as a afunction of thethe DC bias magnetic field for for sample No. 6 at f = 50 Hz. for sample No. 6 at f = 50 Hz. sample No. 6 at f = 50 Hz.

4.3. Interface Interface Coupling Factor 4.3. 4.3. Interface Coupling Coupling Factor Factor The interface interface coupling coupling factor factor kkkcc could could be be determined determined by by substituting substituting the the measured measured ME ME voltage voltage The The interface coupling factor c could be determined by substituting the measured ME voltage coefficient α the piezomagnetic coefficient d 31,m,, the the thickness ratio n and other material parameters coefficient α vv,, the piezomagnetic coefficient d 31,m thickness ratio n and other material parameters coefficient αv , the piezomagnetic coefficient d31,m , the thickness ratio n and other material parameters into Equation Equation (39). (39). The The calculated of sample No. is listed in Table 4. By By ignoring the data into 4. into Equation (39). The calculated calculated kkkccc of of sample sample No. No. 666 is is listed listed in in Table Table 4. By ignoring ignoring the the data data corresponds to to the the too too large large or or small small Hbias bias,, it it can can be be seen seen that that the the calculated calculated interface interface coupling coupling factor factor corresponds corresponds to the too large or small HHbias , it can be seen that the calculated interface coupling factor kc k c is approximate to 0.10–0.13. However, the result for one sample may not reflect the common kisc approximate is approximate to 0.10–0.13. However, the result one sample notthe reflect the condition common to 0.10–0.13. However, the result for onefor sample may notmay reflect common condition for all all the the samples samples in in the the experiment. experiment. There There are are two two steps steps to to find find the the interface interface coupling coupling condition for all the for samples in the experiment. There are two steps to find the interface coupling factor that factor that that is is suitable suitable for all all the the samples: samples: (i) plot plot all all the the measured measured data data in in the the three-dimensional three-dimensional space space factor is suitable for all the for samples: (i) plot all(i)the measured data in the three-dimensional space formed formed by H Hbias bias,, n n and ααvv;; and and (ii) (ii) based based on on the the least least mean square square error analysis, analysis, fit the the measured measured data data formed by Hbiasby , n and αvand ; and (ii) based on the least mean mean square error error analysis, fit thefitmeasured data with with aa surface surface formed formed by by the the theoretical theoretical ME voltage voltage coefficient coefficient in in which which the the interface interface coupling coupling factor with a surface formed by the theoretical MEME voltage coefficient in which the interface coupling factor factor varies in in the the range range from from 0.10 0.10 to to 0.13. 0.13. The The surface surface with with the the least least mean mean square square error error corresponds corresponds to to the the varies most suitable k c . Figure 10 shows the measured data and surface fitting result in H bias -n-α v three most suitable kc. Figure 10 shows the measured data and surface fitting result in Hbias-n-αv three dimensional space space for for samples samples No. No. 1–No. 1–No. 11, 11, where where the the blue blue scattered scattered points points represent represent the the measured measured dimensional data and the surface corresponds to k c = 0.11. It could be clearly seen that the surface fits the scattered data and the surface corresponds to kc = 0.11. It could be clearly seen that the surface fits the scattered

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varies in the range from 0.10 to 0.13. The surface with the least mean square error corresponds to Sensors 2017, 17, 1399 16 of 19 the most suitable kc . Figure 10 shows the measured data and surface fitting result in Hbias -n-αv three dimensional space for samples No. 1–No. 11, where the blue scattered points represent the measured points quite well, indicating the correctness of kc = 0.11. In other words, it verifies the reasonability data and the surface corresponds to kc = 0.11. It could be clearly seen that the surface fits the scattered for the introduction of the new defined interface coupling factor in Equation (36). Similarly, the points quite well, indicating the correctness of kc = 0.11. In other words, it verifies the reasonability for measured data of samples No. 12–No. 22 are dealt with using the same process and the corresponding the introduction of the new defined interface coupling factor in Equation (36). Similarly, the measured kc = 0.08. data of samples No. 12–No. 22 are dealt with using the same process and the corresponding kc = 0.08. There is a qualitative physical explanation for the fact that the interface coupling factor varies There is a qualitative physical explanation for the fact that the interface coupling factor varies with pre-mechanical load in the sample fabrication. Under 500 g weight pressure, the bonding with pre-mechanical load in the sample fabrication. Under 500 g weight pressure, the bonding condition between the layers is relatively rigid, resulting in the relatively much strains transferred condition between the layers is relatively rigid, resulting in the relatively much strains transferred from magnetostrictive layer to piezoelectric layer, so the interface coupling factor is larger. Under 100 from magnetostrictive layer to piezoelectric layer, so the interface coupling factor is larger. Under 100 g g weight pressure, the bonding between the layers is weak, the strain transfer is reduced, so the weight pressure, the bonding between the layers is weak, the strain transfer is reduced, so the interface interface coupling factor is relatively small. coupling factor is relatively small. However, an interface coupling factor as high as 0.6 has been achieved by Silva et al. [20]. Three However, an interface coupling factor as high as 0.6 has been achieved by Silva et al. [20]. epoxy resins, Devcon, M-Bond and Stycast, and three thickness of 28, 52 and 110 μm for the Three epoxy resins, Devcon, M-Bond and Stycast, and three thickness of 28, 52 and 110 µm for the piezoelectric layer were studied. It is their conclusion that an epoxy resin with smaller Young’s piezoelectric layer were studied. It is their conclusion that an epoxy resin with smaller Young’s modulus and a piezoelectric layer with larger thickness result in a larger ME coefficient, i.e., a larger modulus and a piezoelectric layer with larger thickness result in a larger ME coefficient, i.e., a larger kc . kc. It should also be noticed that an optimum thickness for the piezoelectric layer exists at which the It should also be noticed that an optimum thickness for the piezoelectric layer exists at which the ME ME coefficient reaches its maximum value, since a much larger thickness will lead to the coefficient reaches its maximum value, since a much larger thickness will lead to the inhomogeneous inhomogeneous deformations of the piezoelectric layer, with more deformation at the boundary deformations of the piezoelectric layer, with more deformation at the boundary region with the epoxy region with the epoxy resin and lower deformation at the center region of the layer. In our experiment, resin and lower deformation at the center region of the layer. In our experiment, the Young’s modulus the Young’s modulus of Eccobond 45 is 0.5 GPa [30], larger than 0.3 GPa of M-Bond 600 and the of Eccobond 45 is 0.5 GPa [30], larger than 0.3 GPa of M-Bond 600 and the thickness of piezoelectric thickness of piezoelectric layer is over an order of magnitude larger than 110 μm. Therefore, it should layer is over an order of magnitude larger than 110 µm. Therefore, it should be qualitatively reasonable be qualitatively reasonable that the interface coupling factor obtained in our experiment is lower than that the interface coupling factor obtained in our experiment is lower than the best 0.6 for the M-Bond the best 0.6 for the M-Bond bonded ME laminate in [20]. Furthermore, our interface coupling factor bonded ME laminate in [20]. Furthermore, our interface coupling factor is comparable with 0.07 for is comparable with 0.07 for the Stycast bonded ME laminate. the Stycast bonded ME laminate.

Figure 10. The measured data and surface fitting result in Hbias -n-αv three-dimensional space Figure 10. The measured data and surface fitting result in Hbias-n-αv three-dimensional space for for samples No. 1–No. 11. The blue scattered points represent the measured data under 500 g samples No. 1–No. 11. The blue scattered points represent the measured data under 500 g prepre-mechanical load and the surface represents the transverse ME voltage coefficient corresponds to mechanical load and the surface represents the transverse ME voltage coefficient corresponds to kc = kc = 0.11. 0.11. Table 4. The calculated interface coupling factor kc of sample No. 6. Table 4. The calculated interface coupling factor kc of sample No. 6. Parameters Parameters H (Oe) Hbias bias (Oe) −10 31,m (10 A/m) dd31,m (10−10 A/m) ααvv(mV/Oe) (mV/Oe) Calculatedkkc c Calculated

541541 0.90.9 46.57 46.57 0.224 0.224

748 748 857 857 1059 1059 1.41.4 1.7 1.92 1.7 1.92 53.32 53.32 69.03 69.03 79.51 79.51 0.113 0.129 0.129 0.134 0.134 0.113

1294 1294 2.05 84.22 84.22 0.131 0.131

1569 1569 2.60 102.52 0.122

Values Values 1787 2095 1787 2095 3.70 2.90 140.39 118.21 0.129 0.113

2288 2288 4.54 165.58 0.106

2498 2498 4.85 182.43 0.111

2754 2754 5.70 5.70 209.43 209.43 0.106 0.106

2998 2998 6.60 6.60 232.11 232.11 0.099 0.099

3259 3259 7.40 7.40 239.10 239.10 0.086 0.086

4.4. The Optimum Thickness Ratio According to Equation (40), when kc = 0.11, the optimum thickness ratio for T-T mode disk-type ME laminate is 0.57, while kc = 0.08, the optimum thickness ratio is 0.52. Considering the DC bias magnetic field has no influences on the optimum thickness ratio, the experimental data measured under Hbias = 2700 Oe are applied to verify the prediction on the optimum thickness ratio, as shown

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4.4. The Optimum Thickness Ratio According to Equation (40), when kc = 0.11, the optimum thickness ratio for T-T mode disk-type ME laminate is 0.57, while kc = 0.08, the optimum thickness ratio is 0.52. Considering the DC bias magnetic field has no influences on the optimum thickness ratio, the experimental data measured Sensors 2017, 17, 1399 17 of 19 under Hbias = 2700 Oe are applied to verify the prediction on the optimum thickness ratio, as shown in Figure Figure 11, 0.11, the the scattered scattered in 11, where where the the scattered scattered blue blue points points represent represent the the measured measured data data for for kkcc = = 0.11, green points 0.08, the the red red dash dash line line represents represents the the theoretical green points represent represent the the measured measured data data for for kkcc == 0.08, theoretical prediction from Equation (39) for k = 0.11 and the red dotted line represents the theoretical prediction c prediction from Equation (39) for kc = 0.11 and the red dotted line represents the theoretical prediction from Equation (39) for k = 0.08. It can be seen that the scattered measured points essentially fits the the from Equation (39) for kcc = 0.08. It can be seen that the scattered measured points essentially fits theoretical prediction curve. The measured optimum thickness is 0.61 for ratio kc = 0.11 0.56 for for theoretical prediction curve. The measured optimumratio thickness is and 0.61 k = 0.08. The measured optimum thickness ratios are both slightly larger than the corresponding kcc = 0.11 and 0.56 for kc = 0.08. The measured optimum thickness ratios are both slightly larger than theoretical one, buttheoretical the relationship between them remains unchanged, i.e., the measured optimum the corresponding one, but the relationship between them remains unchanged, i.e., the thickness ratio for k = 0.11 is still larger than the one for k = 0.08. c c measured optimum thickness ratio for kc = 0.11 is still larger than the one for kc = 0.08.

Figure 11. The Thetheoretical theoretical and measured transverse ME voltage coefficient as a function of the Figure 11. and measured transverse ME voltage coefficient as a function of the thickness thickness ratio under 2700 Oe DC bias magnetic field. The scattered blue points represent the ratio under 2700 Oe DC bias magnetic field. The scattered blue points represent the measured data measured data under 500 g pre-mechanical load, the scattered green points represent the measured under 500 g pre-mechanical load, the scattered green points represent the measured data under 100 g = 0.11 data under 100 g pre-mechanical the red dash represents thefor theoretical curve pre-mechanical load, the red dash load, line represents the line theoretical curve kc = 0.11 and thefor redkcdotted and the red dotted line represents line represents the theoretical curvethe fortheoretical kc = 0.08. curve for kc = 0.08.

5. 5. Conclusions Conclusions The The proposed proposed interface interface coupling coupling factor factor between between the the magnetostrictive magnetostrictive layer layer and and the the piezoelectric piezoelectric layer can effectively represent the coupling extend in a simpler but clearer way. layer can effectively represent the coupling extend in a simpler but clearer way. The magnetostrictive and The respective respective equivalent equivalent circuit circuit models models of of magnetostrictive and piezoelectric piezoelectric layers layers can can be be combined with an ideal transformer whose turn-ratio is just the interface coupling factor. In this combined with an ideal transformer whose turn-ratio is just the interface coupling factor. In thisway, way, the can represent represent the the physical physical coupling coupling effect effect between between the the two two phases. phases. the equivalent equivalent circuit circuit can The precise expression expression for for the the transverse transverse ME ME voltage voltage coefficient coefficient ααv and and the the optimum optimum The more more precise v thickness ratio n optim can be easily derived from the equivalent circuit mentioned above. A total of 22 thickness ratio noptim can be easily derived from the equivalent circuit mentioned above. A total of sample experiments with different configurations verify the correctness of the circuit and 22 sample experiments with different configurations verify the correctness of equivalent the equivalent circuit the expression. andderived the derived expression. The model in this this paper The model proposed proposed in paper is is only only suitable suitable for for the the ME ME laminate laminate composites, composites, in in which which the the magnetostrictive and piezoelectric layers are coupled together by epoxy resin adhesive or other glues. magnetostrictive and piezoelectric layers are coupled together by epoxy resin adhesive or other glues. It may not not essentially essentially explain explain the magnetoelectric effects physical aspect the particulate particulate It may the magnetoelectric effects from from aa physical aspect in in the composites, the thin films and other ME composites. However, if we regard the interface coupling composites, the thin films and other ME composites. However, if we regard the interface coupling factor factor as a rectify coefficient, it may also explain the discrepancy between the theoretical and as a rectify coefficient, it may also explain the discrepancy between the theoretical and experimental experimental results for these composites mentioned above [20,21]. results for these composites mentioned above [20,21]. Overall, this modified equivalent circuit model can provide guidance for the design, fabrication and application of the ME laminate based devices, such as current sensor [31], magnetic sensor, energy harvester and wireless energy transfer system [32]. Acknowledgments: This work is supported by the National Natural Science Foundation of China under Project 51377087.

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Overall, this modified equivalent circuit model can provide guidance for the design, fabrication and application of the ME laminate based devices, such as current sensor [31], magnetic sensor, energy harvester and wireless energy transfer system [32]. Acknowledgments: This work is supported by the National Natural Science Foundation of China under Project 51377087. Author Contributions: G.L. performed the theoretical modeling; G.L. and S.L. designed and performed the experiments; X.Y. led the research; and G.L. wrote the paper. Conflicts of Interest: The authors declare no conflicts of interest.

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