Dependence of Modified Butterworth Van-Dyke Model Parameters and Magnetoimpedance on DC Magnetic Field for Magnetoelectric Composites

This study investigates the impedance curve of magnetoelectric (ME) composites (i.e., Fe80Si9B11/Pb(Zr0.3Ti0.7)O3 laminate) and extracts the modified Butterworth–Van Dyke (MBVD) model’s parameters at various direct current (DC) bias magnetic fields Hdc. It is interesting to find that both the magnetoimpedance and MBVD model’s parameters of ME composite depend on Hdc, which is primarily attributed to the dependence of FeSiB’s and neighboring PZT’s material properties on Hdc. On one hand, the delta E effect and magnetostriction of FeSiB result in the change in PZT’s dielectric permittivity, leading to the variation in impedance with Hdc. On the other hand, the magnetostriction and mechanical energy dissipation of FeSiB as a function of Hdc result in the field dependences of the MBVD model’s parameters and mechanical quality factor. Furthermore, the influences of piezoelectric and electrode materials properties on the MBVD model’s parameters are analyzed. This study plays a guiding role for ME sensor design and its application.


Introduction
Magnetoelectric (ME) materials produce strong ME effects due to the mechanical coupling between magnetostrictive and piezoelectric materials, which has been studied intensively in both theories and experiments [1][2][3][4][5][6]. Such ME effects provide a promising candidate for the highly sensitive DC magnetic field sensor due to its significant variations with external direct current (DC) magnetic field. Dong et al. [7] presented a ME laminate under a constant drive of H ac = 1 Oe, which can reach the limit of detection (LOD) for a DC magnetic field H dc of 10 −4 Oe. Sun et al. [2] reported a novel Nano-Electromechanical System (NEMS) AlN/FeGaB resonator with a high DC magnetic field sensitivity of 280 kHz/Oe and a LOD of 8 × 10 −6 Oe. Liu et al. [8] demonstrated a highly sensitive DC magnetic field sensor with a LOD of 2 × 10 −5 Oe. Martins et al. [9] showed a Metglas/poly(vinylidene fluoride)/Metglas magnetoelectric laminate with the sensitivity of 30 mV·Oe −1 and resolution of 8 µ Oe for H dc detection, and its correlation coefficient, linearity and accuracy values reached 0.995, 95.9% and 99.4%, respectively. Yao et al. [10] developed a Metglas/PMNT/Metglas laminate with the LOD of 10 × 10 −5 Oe for H dc detection. Wang et al. [11] also proposed a transformer-type magnetic sensor consisting of soft magnetostrictive alloy FeBSiC/piezoelectric ceramics Pb(Zr,Ti)O 3 /FeBSiC heterostructure wrapped with both the exciting and sensing coils, which provided the maximum magnetic field sensitivity of 2.12 V/Oe and equivalent magnetic noise of 114 × 10 −8 Oe/ √ HZ (at 1 Hz).
Meanwhile the material property and structure of ME composite have been researched intensively for the magnetic sensor application. As such, the field-dependent characteris-

Experiment
The ME sensor consists of PZT/FeSiB laminated composite, where the sizes of magnetostrictive (FeSiB, supplied by Foshan Huaxin Microlite Metal Co., Ltd., Foshan, China) layer and the piezoelectric (PZT, produced by Zibo Yuhai Ceracomp Co., Ltd., Zibo, China) layer are 12 mm × 5 mm × 0.03 mm and 12 mm × 6 mm × 0.8 mm, respectively. First, the PZT plate and FeSiB ribbon are dipped in organic impregnant to clean them. Subsequently, the soft magnetic ribbon FeSiB is bonded with PZT plate by using epoxy glue. Here the West System 105/206 resin/hardener epoxy with a good mechanical property and a low viscosity is utilized to provide strong bondings among layers. The mixture ratio for the epoxy part 'Resin' and part 'hardener' is specified as 5:1 by the supplier. Then the PZT/FeSiB laminated composite is compacted in a vacuum bag and cured for 12 h at room temperature to further guarantee the strong bonding among layers. The thickness of the epoxy layers is controlled to be less than 5 µm with vacuum bagging techniques, which has been proved to negligibly affect the ME performance, according to previous research [19]. Considering the ease of fabrication and ME performance, the PZT/FeSiB laminated composite is designed to operate in the L-T (i.e., longitudinal-transverse) mode. That is to say, the FeSiB layer is magnetized along the longitudinal direction (i.e., length direction) since the demagnetizing field is much smaller along this direction. Meanwhile the silver electrodes of piezoelectric layer are at its top and bottom surfaces, and the PZT is poled along the transverse direction (i.e., thickness direction).
To measure the impedance of ME composite as a function of the external DC magnetic field H dc , H dc is applied along the longitudinal direction of FeSiB layer with a pair of electromagnets driven by a SR830 Lock-In Amplifier. Here the H dc varies from 0 to 400 Oe, which is calibrated with a Gauss magnetometer (Lake Shore 455 DSP, Columbus, OH, USA). Additionally, when analyzing the dielectric characteristics of the ME sensor, an Impedance Analyzer (4194 A HP Agilent, Santa Clara, CA, USA) is used to measure the magnetoimpedance (Z) of ME composite with the excitation frequency ranged from 125 kHz to 155 kHz. Figure 1 shows the impedance Z of the ME sensor as a function of electrical excitation frequency f when the varied DC bias magnetic field is applied along the length direction. As illustrated in the inset of Figure 1, the maximum and minimum impedance as a function of excitation frequency show a strong dependence on DC bias magnetic field. which has been proved to negligibly affect the ME performance, according to prev research [19]. Considering the ease of fabrication and ME performance, the PZT/F laminated composite is designed to operate in the L-T (i.e., longitudinal-transve mode. That is to say, the FeSiB layer is magnetized along the longitudinal direction length direction) since the demagnetizing field is much smaller along this direction. M while the silver electrodes of piezoelectric layer are at its top and bottom surfaces, and PZT is poled along the transverse direction (i.e., thickness direction).

Results and Discussion
To measure the impedance of ME composite as a function of the external DC m netic field Hdc, Hdc is applied along the longitudinal direction of FeSiB layer with a pa electromagnets driven by a SR830 Lock-In Amplifier. Here the Hdc varies from 0 to 400 which is calibrated with a Gauss magnetometer (Lake Shore 455 DSP, Columbus, USA). Additionally, when analyzing the dielectric characteristics of the ME sensor, an pedance Analyzer (4194 A HP Agilent, Santa Clara, CA, USA) is used to measure the m netoimpedance (Z) of ME composite with the excitation frequency ranged from 125 to 155 kHz. Figure 1 shows the impedance Z of the ME sensor as a function of electrical excita frequency f when the varied DC bias magnetic field is applied along the length direc As illustrated in the inset of Figure 1, the maximum and minimum impedance as a f tion of excitation frequency show a strong dependence on DC bias magnetic field. Figure 1. Impedance curve of the ME sensor at various bias DC magnetic fields, and the insets s enlarged details around the maximum and minimum impedances. The maximum relative stan deviations (RSD, i.e., standard deviation/mean × 100%) of impedance with multiple measurem is 0.27%.

Results and Discussion
It is known that the impedance of the ME sensor is defined by [21] Z= where and are the effective relative permeability and permittivity, an are vacuum permeability and permittivity, respectively. The effective relative permitt can be represented as [22]. It is known that the impedance of the ME sensor is defined by [21], where µ e f f and ε e f f are the effective relative permeability and permittivity, µ 0 and ε 0 are vacuum permeability and permittivity, respectively. The effective relative permittivity ε e f f can be represented as [22].
where ε r is relative permittivity of piezoelectric material, d 31,p is the piezoelectric coefficient, f s is the resonance frequency, E pzt and E m are the Young's modulus of piezoelectric and magnetostrictive materials, respectively. n pzt and 1 − n pzt are the volume fractions of piezoelectric material PZT and magnetostrictive material FeSiB in the ME sensor, respectively. By applying a DC bias magnetic field to the magnetostrictive material FeSiB, the magnetostriction is produced by FeSiB and transferred to the PZT layer through interfacial coupling. Meanwhile the magnetostrictive stress will also change the Young's modulus E m of magnetostrictive material FeSiB and corresponding resonance frequency f s . Correspondingly from the inset of Figure 1, the electromechanical resonance frequency f s of the ME sensor shows a strong dependence on DC bias magnetic field H dc . Specifically, the resonance frequency f s of the ME sensor is determined by the geometrical dimensions and material parameters (i.e., Young's modulus and mass density) of both piezomagnetic and piezoelectric materials, and is expressed as [12], where l is the length of the ME sensor, ρ and E are the average density and equivalent Young's modulus of ME laminate, respectively. For the ME composite, E and ρ are determined by [6], where ρ pzt and ρ m are the densities of piezoelectric and magnetostrictive materials, respectively.
Here the Young's modulus of magnetostrictive material FeSiB is given by [23], where s e , σ, s me are the elastic strain, elastic stress and magnetoelastic strain, respectively. The magnetoelastic strain arises from the magnetic domain reorientation during the varied H dc [24,25], which results in the change in effective Young's modulus with H dc . As a result, the shifts in corresponding resonance frequency (Equation (3)) with H dc are observed. According to Equations (1) and (2), the variations in the Young's modulus E m of FeSiB and resonance frequency f s with H dc also lead to the changes in effective relative permittivity and corresponding impedance with H dc for ME composite. It is noted that the combination of magnetoresistance (MR) and the Maxwell-Wagner effect could also cause the magnetodielectric effect, according to the previous report [26]. However, for our asymmetric PZT/FeSiB laminate, piezoelectric material PZT is covered with the insulating epoxy glue at surface to prevent the current penetrating into the neighboring magnetic ribbon FeSiB. Hence, there is no giant magnetoresistance effect since the sensing current cannot go through the magnetic layers and, correspondingly, no spin dependent scattering phenomenon happens in the ferromagnetic layer. Furthermore, Castel et al. [22] have also reported that the magnetodielectric effect of BaTiO 3 -Ni laminated composite could reach 10% near the resonance frequency at H dc = 6 kOe and clarified that the magnetodielectric mechanism of their composites was based on the strain effect instead of the Maxwell-Wagner effect.
It is also interesting to find in Figure 2 that the maximum impedance Z m at the antiresonance frequency (f a ) increases to a maximum value at H dc = 30 Oe, and then decreases with further increasing H dc , while the minimum impedance Z n at the resonance frequency (f s ) varies in the opposite trend. Namely, Z n decreases to a minimum value, and then increases with the increasing H dc . This is mainly because the capacitance is directly proportional to dielectric permittivity, the minimum capacitance value at f a results in the maximum impedance Z m and the maximum capacitance value at f s leads to the minimum impedance Z n according to Equation (1).
Materials 2021, 14, x FOR PEER REVIEW 5 of 12 reach 10% near the resonance frequency at Hdc = 6 kOe and clarified that the magnetodielectric mechanism of their composites was based on the strain effect instead of the Maxwell-Wagner effect. It is also interesting to find in Figure 2 that the maximum impedance Zm at the antiresonance frequency (fa) increases to a maximum value at Hdc = 30 Oe, and then decreases with further increasing Hdc, while the minimum impedance Zn at the resonance frequency (fs) varies in the opposite trend. Namely, Zn decreases to a minimum value, and then increases with the increasing Hdc. This is mainly because the capacitance is directly proportional to dielectric permittivity, the minimum capacitance value at fa results in the maximum impedance Zm and the maximum capacitance value at fs leads to the minimum impedance Zn according to Equation (1). In order to understand the trend of impedance as a function of DC magnetic field, the electromechanical (ME) sensor is characterized with a lumped-parameter equivalent circuit based on the MBVD model, as shown in Figure 3. To characterize the loss from the electrodes, the MBVD model adds two additional loss resistors (i.e., R0 and Rs) to obtain a more accurate model compared with the standard Butterworth-Van Dyke model. It consists of two network branches in parallel, where R0 represents the resistance associated with dielectric losses of the ME sensor, Rs represents the resistance associated with electrical losses of electrode, Rm denotes the resistance associated with mechanical losses, Lm and Cm denote the motional inductance and capacitance, C0 represents the static capacitance formed between top and bottom electrodes of the ME sensor. In order to understand the trend of impedance as a function of DC magnetic field, the electromechanical (ME) sensor is characterized with a lumped-parameter equivalent circuit based on the MBVD model, as shown in Figure 3. To characterize the loss from the electrodes, the MBVD model adds two additional loss resistors (i.e., R 0 and R s ) to obtain a more accurate model compared with the standard Butterworth-Van Dyke model. It consists of two network branches in parallel, where R 0 represents the resistance associated with dielectric losses of the ME sensor, R s represents the resistance associated with electrical losses of electrode, R m denotes the resistance associated with mechanical losses, L m and C m denote the motional inductance and capacitance, C 0 represents the static capacitance formed between top and bottom electrodes of the ME sensor. reach 10% near the resonance frequency at Hdc = 6 kOe and clarified that the magnetodielectric mechanism of their composites was based on the strain effect instead of the Maxwell-Wagner effect. It is also interesting to find in Figure 2 that the maximum impedance Zm at the antiresonance frequency (fa) increases to a maximum value at Hdc = 30 Oe, and then decreases with further increasing Hdc, while the minimum impedance Zn at the resonance frequency (fs) varies in the opposite trend. Namely, Zn decreases to a minimum value, and then increases with the increasing Hdc. This is mainly because the capacitance is directly proportional to dielectric permittivity, the minimum capacitance value at fa results in the maximum impedance Zm and the maximum capacitance value at fs leads to the minimum impedance Zn according to Equation (1). In order to understand the trend of impedance as a function of DC magnetic field, the electromechanical (ME) sensor is characterized with a lumped-parameter equivalent circuit based on the MBVD model, as shown in Figure 3. To characterize the loss from the electrodes, the MBVD model adds two additional loss resistors (i.e., R0 and Rs) to obtain a more accurate model compared with the standard Butterworth-Van Dyke model. It consists of two network branches in parallel, where R0 represents the resistance associated with dielectric losses of the ME sensor, Rs represents the resistance associated with electrical losses of electrode, Rm denotes the resistance associated with mechanical losses, Lm and Cm denote the motional inductance and capacitance, C0 represents the static capacitance formed between top and bottom electrodes of the ME sensor.  The analytical expression of impedance Z(ω) and the electrical admittance Y(ω) for MBVD model are given by [27,28], Y(ω) = 1 The series resonance frequency f s and antiresonance frequency f a can be expressed as [27,28], Using Equations (7), (9) and (10), the model parameter values of C 0 , R 0 , R s , L m , C m and R m are extracted from the measured Z. Table 1 lists all the extracted model parameters.
To verify the MBVD model for further design of the conditioning circuit, the simulation of the model is implemented with the electrical simulator Agilent ADS. Figure 4 presents the computed impedance Z and phase based on the extracted model parameters at H dc = 30 Oe, which shows a good agreement with the measured data.  The analytical expression of impedance ( ) and the electrical admittance ( ) for MBVD model are given by [27,28] ( ) = + ( + 1 )( + 1 + ) The series resonance frequency and antiresonance frequency can be expressed as [27,28] = 1 2 (9) Using Equations (7), (9) and (10), the model parameter values of C0, R0, Rs, Lm, Cm and Rm are extracted from the measured Z. Table 1 lists all the extracted model parameters. To verify the MBVD model for further design of the conditioning circuit, the simulation of the model is implemented with the electrical simulator Agilent ADS. Figure 4 presents the computed impedance Z and phase based on the extracted model parameters at Hdc = 30 Oe, which shows a good agreement with the measured data.   According to the measured Z with DC bias magnetic field H dc , the corresponding equivalent circuit parameters (i.e., C m , L m , C 0, Q s , R s + R m , f s and f a ) are calculated and analyzed as a function of H dc , as shown in Figures 5-8, respectively. = = (12 where d31, , , lw, lt and l are the piezoelectric coefficient, elastic compliance coefficient density, width, thickness and length of the ME sensor, respectively. = is the plate area. It is obvious that the the length l and elastic compliance coefficient have strong influences on the Cm and Lm according to Equations (11) and (12). Specifically, due to the stress-strain coupling of interlayers, the magnetostrictive strain produced by FeSiB under varying Hdc results in the change in the length and elastic compliance coefficient for piezoelectric material. As a result, the equivalent electrical parameters Cm and Lm of the ME sensor strongly depend on Hdc and vary in the opposite ways, as illustrated in Figure   5. This is due to the fact that Lm and Cm are proportional to and , respectively.
Furthermore, Cm is proportional to the length l of composite, whereas it is inverse proportional to the elastic compliance coefficient. Since the Young's modulus is the in verse of elastic compliance coefficient, Cm is determined by both the Young's modulus E and length l. On one hand due to the stress-strain coupling of the interlayers, the length increases quickly to a maximum value due to the large piezomagnetic coefficient d33,m o FeSiB and then l reaches the saturation with further increasing Hdc. On the other hand, the Young's modulus E of the magnetostrictive layer and, corresponding, ME composite de crease initially to a minimum value with the increasing Hdc, and then increases and reaches saturation at large Hdc when Hdc further increases [18]. When Hdc < 60 Oe, the mag netostriction does not attain to saturation, Cm is affected by both Young's modulus and the length l. However, the effect of length l on Cm is more obvious than that of Young's modulus due to the large d33,m of FeSiB at the small Hdc, which causes Cm to increase w Hdc and reach a positive peak in low magnetic field Hdc = 60 Oe. When Hdc further incre above 60 Oe, the magnetostriction reaches saturation quickly; however, the Young's m ulus E still varies significantly and plays a dominant role in Cm. Currently, Young's m ulus E and corresponding Cm reach the local minimum values when Hdc further incre to 100 Oe, then Cm gradually increases and reaches saturation with further increasing due to the variation in E, as shown in Figure 5.
Meanwhile the static capacitance C0 can be also expressed with the following exp sions [28]: where and are the relative dielectric permittivity and vacuum permittivity of piezoelectric material, respectively.
When Hdc is applied along the longitudinal direction of the ME sensor, the magn strictive material FeSiB expands with the increasing Hdc, which changes dielectric per tivity of piezoelectric material due to the transferred magnetostrictive stress. Correspo ingly, C0 varies with the DC magnetic field since C0 is strongly determined by the diele permittivity. Yao et al. [10] reported that the dielectric permittivity of Terfenol-D/ magnetoelectric composite at the resonant frequency decreased and then increased w increasing dc magnetic field. In this case, C0 varies in a similar trend as function o since C0 is proportional to the dielectric permittivity. Specifically, it is shown in Figu that the static capacitance C0 of the ME sensor first decreases with the increasing Hdc, then gradually increases. The Rm is used to characterize the mechanical loss, which is subject to energy los the ME sensor. It can be given as [28] = =

Conclusions
In summary, the impedance of the ME sensor (i.e., FeSiB/PZT composite) as a function of DC bias magnetic field is experimentally measured and theoretically analyzed. Meanwhile, the simulation results with the MBVD model of the ME sensor agrees with the measured impedance Z accurately. Specifically, the dependences of extracted MBVD model parameters and the magnetoimpedance effects of the ME sensor on Hdc are observed, which result from the varied magnetostriction and the mechanical energy dissipation of magnetostrictive material FeSiB with Hdc due to the corresponding delta E effect and magnetostrictive effect. Furthermore, the influences of piezoelectric materials property and electrode on the MBVD model parameters are analyzed. The analysis of MBVD model for ME composite is beneficial to the design of analog front-end circuits for the corresponding magnetic sensor, which could further improve the LOD.

Conclusions
In summary, the impedance of the ME sensor (i.e., FeSiB/PZT composite) as a tion of DC bias magnetic field is experimentally measured and theoretically anal Meanwhile, the simulation results with the MBVD model of the ME sensor agrees the measured impedance Z accurately. Specifically, the dependences of extracted M model parameters and the magnetoimpedance effects of the ME sensor on Hdc ar served, which result from the varied magnetostriction and the mechanical energy pation of magnetostrictive material FeSiB with Hdc due to the corresponding delta E and magnetostrictive effect. Furthermore, the influences of piezoelectric materials p erty and electrode on the MBVD model parameters are analyzed. The analysis of M model for ME composite is beneficial to the design of analog front-end circuits fo corresponding magnetic sensor, which could further improve the LOD. Specifically, C m and L m are given by [27,28],  (12) where d 31 , s E 11 , ρ, l w , l t and l are the piezoelectric coefficient, elastic compliance coefficient, density, width, thickness and length of the ME sensor, respectively. A = ll w is the plate area.
It is obvious that the the length l and elastic compliance coefficient s E 11 have strong influences on the C m and L m according to Equations (11) and (12). Specifically, due to the stress-strain coupling of interlayers, the magnetostrictive strain produced by FeSiB under varying H dc results in the change in the length l and elastic compliance coefficient s E 11 for piezoelectric material. As a result, the equivalent electrical parameters C m and L m of the ME sensor strongly depend on H dc and vary in the opposite ways, as illustrated in Figure 5. This is due to the fact that L m and C m are proportional to Furthermore, C m is proportional to the length l of composite, whereas it is inverse proportional to the elastic compliance coefficient. Since the Young's modulus is the inverse of elastic compliance coefficient, C m is determined by both the Young's modulus E and length l. On one hand due to the stress-strain coupling of the interlayers, the length l increases quickly to a maximum value due to the large piezomagnetic coefficient d 33,m of FeSiB and then l reaches the saturation with further increasing H dc . On the other hand, the Young's modulus E of the magnetostrictive layer and, corresponding, ME composite decrease initially to a minimum value with the increasing H dc , and then increases and reaches saturation at large H dc when H dc further increases [18]. When H dc < 60 Oe, the magnetostriction does not attain to saturation, C m is affected by both Young's modulus and the length l. However, the effect of length l on C m is more obvious than that of Young's modulus due to the large d 33,m of FeSiB at the small H dc , which causes C m to increase with H dc and reach a positive peak in low magnetic field H dc = 60 Oe. When H dc further increases above 60 Oe, the magnetostriction reaches saturation quickly; however, the Young's modulus E still varies significantly and plays a dominant role in C m . Currently, Young's modulus E and corresponding C m reach the local minimum values when H dc further increases to 100 Oe, then C m gradually increases and reaches saturation with further increasing H dc due to the variation in E, as shown in Figure 5.
Meanwhile the static capacitance C 0 can be also expressed with the following expressions [28]: where ε r and ε 0 are the relative dielectric permittivity and vacuum permittivity of the piezoelectric material, respectively. When H dc is applied along the longitudinal direction of the ME sensor, the magnetostrictive material FeSiB expands with the increasing H dc , which changes dielectric permittivity of piezoelectric material due to the transferred magnetostrictive stress. Correspondingly, C 0 varies with the DC magnetic field since C 0 is strongly determined by the dielectric permittivity. Yao et al. [10] reported that the dielectric permittivity of Terfenol-D/PZT magnetoelectric composite at the resonant frequency decreased and then increased with increasing dc magnetic field. In this case, C 0 varies in a similar trend as function of H dc since C 0 is proportional to the dielectric permittivity. Specifically, it is shown in Figure 6 that the static capacitance C 0 of the ME sensor first decreases with the increasing H dc , and then gradually increases.
The R m is used to characterize the mechanical loss, which is subject to energy loss in the ME sensor. It can be given as [28], R m is primarily determined by the length l. Correspondingly, the variations in the length l due to the magnetostriction of FeSiB cause the variation in R m with H dc .
By analyzing the variation in equivalent circuit parameters (i.e., C m , L m , C 0, R m etc.) with H dc , it is found that the varying magnetostrictive strain of FeSiB with H dc is the main reason for the H dc dependences of equivalent circuit parameters. Furthermore, it is also noted that these equivalent circuit parameters depend on the actively vibrating area A, since R m and L m decrease with the enlarged area, whereas the capacitances C 0 and C m increase with the enlarged area. Such relations are important for designing ME sensors.
Furthermore, the mechanical Quality factor (Q-factor) reflects the capability of the ME sensor to reserve mechanical energy and the corresponding loss of resonant circuit. Q is defined as the ratio of the stored energy to the dissipated energy per cycle during oscillation. According to Lakin's method, the Q s at series resonance frequency f s and Q p at antiresonance frequency f a can be defined as [27,28], On one hand, the smaller value of R S is desired to improve the effective mechanical Quality factor Q s of the ME sensor, according to Equation (15). Since R S represents the electrical loss of electrode, the type and quality of the electrode materials directly affect the Q s of the ME sensor. In this case, utilizing the electrode material with high acoustic impedance and low resistivity can reduce R S and improve the effective electromechanical coupling coefficient of the ME sensor. On the other hand, Equation (16) predicts that the high Q p value can be obtained when the ME sensor possesses a low R 0 . Since the dielectric losses R 0 of the ME sensor is mainly determined by the dielectric loss of piezoelectric material, it means the smaller dielectric loss results in the larger effective mechanical Quality factor Q p of the ME sensor. Additionally, from Equations (15) and (16), it is found that both Q s at resonance frequency and Q p at antiresonance frequency decrease with the increasing mechanical loss R m . Hence, the mechanical loss R m plays a primary role in the energy dissipations of the ME sensor.
Subsequently, the Q s , Q p and corresponding loss as a function of H dc are experimentally investigated to verify and further understand the above theoretical analysis. It is known that R m depends on the mechanical energy dissipation tanδ mech of the ME sensor [28], while Q s is inversely proportional to tanδ mech [29]. When the DC magnetic field is applied, the mechanical energy dissipation R s + R m of magnetostrictive material FeSiB changes dramatically owing to the non-180 • domain wall motions. This results in the varied mechanical quality factor Q s of the ME sensor with H dc , as shown in Figure 7a. Specifically, the quality factor (Q s ) at the series resonance frequency f s decreases from 182 to the minimum value of 160 at the H dc = 200 Oe and then gradually increases according to the MBVD model. Obviously, the variation in Q s is mainly attributed to the magnetic mechanical loss associated with magnetic domain wall movement and material damping of FeSiB.
Furthermore, the trends of Q p and R 0 + R m as a function of H dc are similar to that of Q s and R s + R m , as shown in Figure 7b. However, the magnitude of antiresonance mechanical quality factor Q p ranges from 234 to 245.6, which is higher than the resonance mechanical quality factor Q s . The differences between Q p and Q s were also reported by in previous literature [30,31]. Finally, the resonance frequency f s and antiresonance frequency f a of ME laminated sensor as a function of varied H dc are investigated, as shown in Figure 8. Both f s and f a exhibit similar trends with H dc , which increases with the increasing DC bias field. The obvious shifts of resonance frequency f s and antiresonance frequency f a with H dc indicate that f s and f a of the ME sensor are adjustable by varying the DC bias magnetic field.

Conclusions
In summary, the impedance of the ME sensor (i.e., FeSiB/PZT composite) as a function of DC bias magnetic field is experimentally measured and theoretically analyzed. Meanwhile, the simulation results with the MBVD model of the ME sensor agrees with the measured impedance Z accurately. Specifically, the dependences of extracted MBVD model parameters and the magnetoimpedance effects of the ME sensor on H dc are observed, which result from the varied magnetostriction and the mechanical energy dissipation of magnetostrictive material FeSiB with H dc due to the corresponding delta E effect and magnetostrictive effect. Furthermore, the influences of piezoelectric materials property and electrode on the MBVD model parameters are analyzed. The analysis of MBVD model for ME composite is beneficial to the design of analog front-end circuits for the corresponding magnetic sensor, which could further improve the LOD.