**2. Magnetoelastic energy harvester modeling**

This section aims to introduce different basic approaches that can be considered when a magneto-elastic energy harvester is modeled. Particular emphasis will be devoted to the material modeling, starting with the classic description under the general hypothesis of linearity (in analogy with the typical modeling of piezoelectric materials), as well as with more sophisticated approaches, taking into account the material nonlinearities and rate-independent hysteresis. Then, the whole device is modeled from the magneto-mechanical point of view. In the scientific literature, usually, a *semicoupled* modeling approach is adopted: the mechanical stimulus is *ideal* and only its effect on the magnetic characteristic is considered. On the other hand, a *fully coupled* approach can be considered and, in this case, the influence of the material response on the mechanical side can be taken into account too, if the vibration source is not ideal.

This analysis will be done employing an analogy with a circuital description by means of the two-port circuits formalism. As interesting application, the capabilities of the harvesting process to damp the vibrations of the mechanical source will be discussed.

All the analysis will be performed under the following fundamental assumptions:


The first one is necessary to have, together with an isotropic material behavior, a scalar description of the constitutive relationships. The second one allows, for instance, to have the *long solenoid* hypothesis for the coil and to treat the mechanical stress as uniform along the material. The third and the fourth assumptions allow to neglect any propagation effect into the device.

#### **2.1. Material modeling - constitutive relationships**

In the hypothesis of monodimensional operating regime (all the fields and the mechanical input along the same direction), the magneto-elastic characteristics can be written as:

$$\begin{cases} S = \mathcal{S}(H, T) \\ B = \mathcal{B}(H, T) \end{cases} \tag{1}$$

where *S* is the strain, *H* is the applied magnetic field, *T* is the compressive stress and *B* is the magnetic induction. Typical behaviors of those characteristics are shown in Fig. 1 and the following general properties can be inferred:


#### 490 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges Magnetoelastic Energy Harvesting: Modeling and Experiments <sup>5</sup> Magnetoelastic Energy Harvesting: Modeling and Experiments 491

4 Will-be-set-by-IN-TECH

characteristic is considered. On the other hand, a *fully coupled* approach can be considered and, in this case, the influence of the material response on the mechanical side can be taken

This analysis will be done employing an analogy with a circuital description by means of the two-port circuits formalism. As interesting application, the capabilities of the harvesting

(b) the length of the structure along the field axis is much larger than the other dimensions, (c) the vibrations frequencies are much lower than the mechanical resonance of the structure,

(d) the electric load is a lumped element, i.e. electromagnetic field propagation is neglected.

The first one is necessary to have, together with an isotropic material behavior, a scalar description of the constitutive relationships. The second one allows, for instance, to have the *long solenoid* hypothesis for the coil and to treat the mechanical stress as uniform along the material. The third and the fourth assumptions allow to neglect any propagation effect into

In the hypothesis of monodimensional operating regime (all the fields and the mechanical

*S* =*S*(*H*, *T*)

where *S* is the strain, *H* is the applied magnetic field, *T* is the compressive stress and *B* is the magnetic induction. Typical behaviors of those characteristics are shown in Fig. 1 and the

• *B*(*H*): if the compressive *T* increases then the *B*(*H*) cycles drop down (the material

*<sup>B</sup>* <sup>=</sup>*B*(*H*, *<sup>T</sup>*) (1)

input along the same direction), the magneto-elastic characteristics can be written as:

• *S*(*H*) and *B*(*H*) are non-linear with non-local memory, i.e. hysteresis,

• *S*(*H*): there is an optimum *T*<sup>0</sup> that makes Δ*S* largest at *H*max,

• *S*(*H*): the *S*(*H*) cycles are self-similar with respect to the stress,

becomes magnetically harder under increasing constant stress).

process to damp the vibrations of the mechanical source will be discussed.

All the analysis will be performed under the following fundamental assumptions:

(a) All the field are coaxial and directed along the magneto-elastic material axis,

into account too, if the vibration source is not ideal.

i.e. sound propagation can be neglected,

**2.1. Material modeling - constitutive relationships**

following general properties can be inferred:

• *S*(*H*) and *B*(*H*) show saturation when |*H*| → ∞, • *S* and *B* are even and odd function of H, respectively,

• *S*(*H*): if *T* > *T*<sup>0</sup> then the *S*(*H*) cycles drop down,

the device.

**Figure 1.** Typical magnetostrictive characteristics (Terfenol-D) at different constant stresses (the titles are |*T*| [MPa]).

The eqs. (1) can be obtained by suitable derivatives of the Gibbs free energy expression:

$$\text{G}(T, H) = \frac{T^2}{2E} + \frac{\mu\_0}{2}H^2 + \mathbb{1}(T, H) \to \begin{cases} S = \left. \frac{\partial G}{\partial T} \right|\_H\\ \left. B = \left. \frac{\partial G}{\partial H} \right|\_T \end{cases} \tag{2}$$

where the first and second terms are pure linear elastic and magnetic energies, respectively, and the third one is the magneto-elastic energy. *E* is the Young modulus and *μ*<sup>0</sup> is the vacuum magnetic permeability. It is worth noting that |*H*,*<sup>T</sup>* means that the derivatives are made at constant *H* and *T*, respectively.

The simplest way to model the magnetoelastic materials behaviour is the linear one. In that case the Gibbs free energy expression reads:

$$G(T, H) = \frac{T^2}{2E} + \frac{\mu\_0}{2}H^2 + \frac{\mu\_0}{2}\chi H^2 + dTH \rightarrow \begin{cases} S = dH + \frac{T}{E} \\ B = \mu H + dT \end{cases} \tag{3}$$

where *μ* = *μ*0(1 + *χ*) = *μ*0*μ<sup>r</sup>* and *d* is the, so called, piezo-magnetic coefficient. It is worth noting that in this case it is assumed that any transformation taking place in the material is *lossless*. This assumption, from the thermodynamic viewpoint, leads to the following constraint [48]:

$$
\left.\frac{\partial \mathcal{S}}{\partial H}\right|\_{T} = \left.\frac{\partial \mathcal{B}}{\partial T}\right|\_{H}.\tag{4}
$$

It can be noted that the eqs. (3) are the magnetic counterpart of the piezoelectric ones. Obviously, in a B-H plane (or in a S-H) at a fixed stresses T (or at a fixed magnetic fields H), the linear modeling predicts parallel lines. This behavior is far from the actual material response because magnetoelastic materials show saturation phenomena and also a magnetic hardening, as stated above. So, this approach can fail in predictions when large variations of

**Figure 2.** Nonlinear magnetostrictive characteristics obtained from eqs. (6). The values *Ms* = 0.8 T, *γ* = −347 T, *E* = 110 GPa match the typical behavior of Terfenol.

the inputs are considered but can be fruitfully used if the inputs have small variations around a magneto-mechanical working point.

A more advanced model, with the only further hypothesis to neglect the hysteresis, is represented by the following magnetic characteristic:

$$G(T,H) = \frac{T^2}{2E} + \frac{\mu\_0}{2}H^2 + f(T)\chi\left(\frac{H}{f(T)}\right) \to \begin{cases} S = \left(\frac{\partial G}{\partial T}\right)\_H = \frac{T}{E} - f'(T)\left[z\chi'(z) - \chi(z)\right] \\ B = \left(\frac{\partial G}{\partial H}\right)\_T = \mu\_0 H + \chi'(z) \end{cases} \tag{5}$$

where *z* = *H*/ *f*(*T*) and *f*(·) and *χ*(·) are suitable one-variable functions and can be chosen in order to model the physical behavior of the material, as saturation effect and stress dependence [19]. Indeed, if *χ*� (*z*) = *Ms* tanh(*z*) [3] and *f*(*T*) = *<sup>T</sup> <sup>γ</sup>* then the magnetic characteristic can be well approximated. In this case, *χ*(*z*) = *Ms* ln(cosh *z*) and eqs. (5) become:

$$\begin{cases} S = \frac{T}{E} - \frac{M\_s}{\gamma} \left[ z \tanh(z) - \ln(\cosh z) \right] \\\\ B = \mu\_0 H + M\_s \tanh \left( \gamma \frac{H}{T} \right) \end{cases} \tag{6}$$

where *Ms* is the magnetic polarization saturation and *γ* is a parameter modifying the approach to saturation which should be properly identified, as reported in [10]. As can be noted, it shows saturation by means of hyperbolic tangent and the magnetic hardening by means of the *z* dependence for compressive stresses.

Finally, considering that also hysteresis is shown by magnetoelastic materials another generalization can be made. A phenomenological approach for including hysteresis is to introduce a Prandtl-Ishilinskii operator [24]:

$$
\pi[\mathbf{x}] = \int\_0^{+\infty} \xi(r) \mathcal{P}\_r[\mathbf{x}] dr,\tag{7}
$$

where P*<sup>r</sup>* is a Play operator of threshold *r* and *ξ*(*r*) is a weight function that can be identified from experimental data. It is constructed by a linear superposition of Play operators, but it

cannot model saturation phenomena. In order to circumvent this limitation and to model the magneto-mechanical coupling taking place in the magnetoelastic material, the above operator can be generalized. In particular, it can be written:

$$B = G\left(\pi[H], T\right),\tag{8}$$

with

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−500

Strain [ppm]

T= −4MPa T= −16MPa T= −30MPa T= −48MPa

x 10 5

� *H f*(*T*) � →

**Figure 2.** Nonlinear magnetostrictive characteristics obtained from eqs. (6). The values *Ms* = 0.8 T,

the inputs are considered but can be fruitfully used if the inputs have small variations around

A more advanced model, with the only further hypothesis to neglect the hysteresis, is

⎧ ⎪⎪⎨

*S* =

*B* =

(*z*) = *Ms* tanh(*z*) [3] and *f*(*T*) = *<sup>T</sup>*

*<sup>γ</sup>* [*<sup>z</sup>* tanh(*z*) <sup>−</sup> ln(cosh *<sup>z</sup>*)]

� *γ H T*

� *∂G ∂T* � *H*

� *∂G ∂H* � *T*

⎪⎪⎩

where *z* = *H*/ *f*(*T*) and *f*(·) and *χ*(·) are suitable one-variable functions and can be chosen in order to model the physical behavior of the material, as saturation effect and stress

characteristic can be well approximated. In this case, *χ*(*z*) = *Ms* ln(cosh *z*) and eqs. (5)

where *Ms* is the magnetic polarization saturation and *γ* is a parameter modifying the approach to saturation which should be properly identified, as reported in [10]. As can be noted, it shows saturation by means of hyperbolic tangent and the magnetic hardening by

Finally, considering that also hysteresis is shown by magnetoelastic materials another generalization can be made. A phenomenological approach for including hysteresis is to

where P*<sup>r</sup>* is a Play operator of threshold *r* and *ξ*(*r*) is a weight function that can be identified from experimental data. It is constructed by a linear superposition of Play operators, but it

� +∞ 0

−2 −1 0 1 2

<sup>=</sup> *<sup>T</sup> <sup>E</sup>* <sup>−</sup> *<sup>f</sup>* �

= *μ*0*H* + *χ*�

(*T*) � *zχ*�

(*z*)

� (6)

*ξ*(*r*)P*r*[*x*]*dr*, (7)

Magnetic Field [A/m]

x 10 5

(*z*) − *χ*(*z*)

*<sup>γ</sup>* then the magnetic

�

(5)

−2 −1 0 1 2

T= −4MPa T= −16MPa T= −30MPa T= −48MPa

a magneto-mechanical working point.

<sup>2</sup>*<sup>E</sup>* <sup>+</sup> *<sup>μ</sup>*<sup>0</sup>

dependence [19]. Indeed, if *χ*�

Magnetic Field [A/m]

*γ* = −347 T, *E* = 110 GPa match the typical behavior of Terfenol.

represented by the following magnetic characteristic:

<sup>2</sup> *<sup>H</sup>*<sup>2</sup> <sup>+</sup> *<sup>f</sup>*(*T*)*<sup>χ</sup>*

⎧ ⎪⎪⎨ *<sup>S</sup>* <sup>=</sup> *<sup>T</sup>*

*<sup>E</sup>* <sup>−</sup> *Ms*

*B* = *μ*0*H* + *Ms* tanh

*π*[*x*] =

⎪⎪⎩

means of the *z* dependence for compressive stresses.

introduce a Prandtl-Ishilinskii operator [24]:

−1.5 −1 −0.5 0 0.5 1 1.5

*<sup>G</sup>*(*T*, *<sup>H</sup>*) = *<sup>T</sup>*<sup>2</sup>

become:

Flux Density [T]

$$G(\mathbf{x}, T) = \mu\_0 \mathbf{x} + M\_\mathbf{s} \tanh\left(\gamma \frac{\mathbf{x}}{T}\right). \tag{9}$$

The function G is the same considered in the memoryless modeling approach and allows to take into account saturation.

However, it is worth to note that when the hysteresis is introduced the method adopted for the memoryless case basing on the thermodynamic constraint (eq. (4)) cannot be considered because the process is not lossless. Moreover, this approach models only one between magnetostrictive and magnetic characteristics. This can be still valid if the vibration source can be considered ideal (*semi-coupled* approach). On the other hand, the problem of a fully coupled approach with hysteresis is still open.

#### **2.2. Device modeling**

A magnetoelastic harvester can be arranged in different configurations. They can be summarized in two main categories:


All the harvesters where the force source is in mechanical direct contact to the active material, in the so called longitudinal mode, belong to the first group. A representative device of such a harvester is shown in Fig. 3 (a). In this case, usually, the material is fabricated in the form of rods, disks or cymbals.

Instead, the second group exploits the inertial forces of a proof mass oscillations induced by a vibrating bond. Typical example is a cantilever, where a thin sheet of the active material is bonded on a elastic structure and bounded on one side, while on the other there is a mass free to vibrate, as sketched in Fig. 3 (b). The thin sheet of active material undergoes heavy stress variations when the mass vibrate. Conceptually, the energy conversion follows the same path of the force driven harvester. But, this configuration has a resonant-like mechanical behavior with a relatively small bandwidth and a resonant frequency related to the mass (the larger is the mass, the lower is the frequency) [44]. The first group has, instead, a larger bandwidth from the vibration frequencies point of view but a lower peak specific power. In the following we will refer to a generic harvester with particular reference to the force-driven.

Despite of the arrangements, the main elements of a magnetostrictive harvester are:


8 Will-be-set-by-IN-TECH 494 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 3.** Magnetostrictive energy harvester concept devices.

Let us consider the sketch of a magnetostrictive harvester under an external ideal time-variable force F1(t), as sketched in Fig. 3 (a). The active material of such a force-driven harvesters has a rod shape. In order to achieve an easier modeling and design of the device, the rod should be a cylinder with the length *l*<sup>0</sup> far larger than the section diameter. Indeed, in this case, two hypothesis can be made: the stress and the magnetic field lines are uniform along the rod leading to, eventually to 1-D problems along the radial direction [15]. The section area *A* should be chosen such that the available force variation gives the best stress variation for harvesting (see section 3.3). The magnetic circuit is needed to provide the magnetic bias to the active material and suitable permanent magnets can be used then. But, it is worth noting that the magnets cannot be coaxial to the active material because, usually, they are sintered and cannot stand the time-variable mechanical stresses. The N-turns coil is wound around the active material where N is chosen in order to achieve the best compromise between available voltage and current over the electric load. Finally, it should be noted that the modern electronic allows the construction of suitable energy conversion circuits that helps to match the electric impedance seen by the harvester itself to the best value, improving conversion efficiency then [8, 35].

Let us recall the eqs. (3):

$$\begin{cases} \text{S} = dH + \frac{T}{E} \\ \text{B} = \mu H + dT \end{cases} \tag{10}$$

Starting from these relationships, a general model of the energy harvester of Fig. 3 can be obtained. The procedure is based on the analogy of the whole system with a two-port: the first port with *mechanical* variables and the second with *electrical* variables. Then, a two-port circuit model of the magneto-elastic device can be obtained by exploiting the following analogy: the input force *F*<sup>1</sup> corresponds to a *primary two-port voltage* (*F*<sup>1</sup> ⇔ *v*1); while the rod tip velocity *u*<sup>1</sup> = *dx*/*dt* corresponds to a *primary current* (*u*<sup>1</sup> = *l*0(*dS*/*dt*) ⇔ *i*1). Instead, the current *i*<sup>2</sup> and voltage *v*<sup>2</sup> on the actual magnetostrictive rod coil represent the current and voltage on the second port.

It is worth noting that such an approach exploits all the power of the lumped circuit theory, making very easy to treat different cases. Moreover, the two-port equivalent model can be solved by means of standard circuit simulators [10].

**Figure 4.** Two-port equivalent circuits. (a) the part inside the dashed box implements the eqs. (12). (b) The part inside the dashed box implements the eqs. (15).

In order to do so, the local quantities of eqs. (10) have to be related to the measurables one, i.e. *F*1, *u*1, *v*<sup>2</sup> and *i*2, by using the hypothesis already made: *H* = *Ni*2/*l*<sup>0</sup> (long solenoid), Φ<sup>2</sup> = *NAB*, *x* = *l*0*S*, and *F*<sup>1</sup> = *AT* (an applied uniform in the space force on the material cross section *A* is assumed), where Φ<sup>2</sup> is the magnetic flux and *x* is deformation.

So the eqs (10) can be recasted as follows:

8 Will-be-set-by-IN-TECH

Let us consider the sketch of a magnetostrictive harvester under an external ideal time-variable force F1(t), as sketched in Fig. 3 (a). The active material of such a force-driven harvesters has a rod shape. In order to achieve an easier modeling and design of the device, the rod should be a cylinder with the length *l*<sup>0</sup> far larger than the section diameter. Indeed, in this case, two hypothesis can be made: the stress and the magnetic field lines are uniform along the rod leading to, eventually to 1-D problems along the radial direction [15]. The section area *A* should be chosen such that the available force variation gives the best stress variation for harvesting (see section 3.3). The magnetic circuit is needed to provide the magnetic bias to the active material and suitable permanent magnets can be used then. But, it is worth noting that the magnets cannot be coaxial to the active material because, usually, they are sintered and cannot stand the time-variable mechanical stresses. The N-turns coil is wound around the active material where N is chosen in order to achieve the best compromise between available voltage and current over the electric load. Finally, it should be noted that the modern electronic allows the construction of suitable energy conversion circuits that helps to match the electric impedance seen by the harvester itself to the best value, improving

> ⎧ ⎨ ⎩

*S* = *dH* +

Starting from these relationships, a general model of the energy harvester of Fig. 3 can be obtained. The procedure is based on the analogy of the whole system with a two-port: the first port with *mechanical* variables and the second with *electrical* variables. Then, a two-port circuit model of the magneto-elastic device can be obtained by exploiting the following analogy: the input force *F*<sup>1</sup> corresponds to a *primary two-port voltage* (*F*<sup>1</sup> ⇔ *v*1); while the rod tip velocity *u*<sup>1</sup> = *dx*/*dt* corresponds to a *primary current* (*u*<sup>1</sup> = *l*0(*dS*/*dt*) ⇔ *i*1). Instead, the current *i*<sup>2</sup> and voltage *v*<sup>2</sup> on the actual magnetostrictive rod coil represent the current and voltage on the

It is worth noting that such an approach exploits all the power of the lumped circuit theory, making very easy to treat different cases. Moreover, the two-port equivalent model can be

*B* = *μH* + *dT*

*T E* **vibr. bond**

**x**

(10)

**u1(t)**

**i2(t)**

**v2(t)**

(b) Velocity-driven.

**electric load**

**perm. mag.**

**elastic material iron back yoke**

**active material mass**

**l0**

**i2(t)**

**<sup>x</sup> u1(t)**

**v2(t)**

**A**

(a) Force-driven.

conversion efficiency then [8, 35].

solved by means of standard circuit simulators [10].

Let us recall the eqs. (3):

second port.

**rigid bond**

**electric load**

**active material iron permanent magnets**

**N**

**Figure 3.** Magnetostrictive energy harvester concept devices.

**F1(t)**

$$\begin{cases} \mathbf{x} &= \frac{l\_0}{EA} F\_1 + dNi\_2\\ \frac{\Phi\_2}{NA} &= \frac{d}{A} F\_1 + \mu \frac{Ni\_2}{l\_0}. \end{cases} \rightarrow \begin{cases} \mathbf{x} &= \mathbf{s}^H F\_1 + dNi\_2\\ \Phi\_2 &= dNF\_1 + L\_0 i\_2. \end{cases} \tag{11}$$

where *<sup>L</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup> <sup>N</sup>*2*<sup>A</sup> <sup>l</sup>*<sup>0</sup> and *<sup>s</sup><sup>H</sup>* <sup>=</sup> *<sup>l</sup>*<sup>0</sup> *EA* have been defined, the latter being the mechanical compliance at constant magnetic field H.

By exploiting the analogies *F*<sup>1</sup> ⇔ *v*<sup>1</sup> e *dx*/*dt* ⇔ *i*1, the following relationships can be derived then:

$$\begin{cases} \frac{1}{s^H} \int\_0^t i\_1 d\tau &= v\_1 + \frac{dN}{s^H} i\_2\\ \frac{1}{L\_0} \int\_0^t v\_2 d\tau &= \frac{dN}{L\_0} v\_1 + i\_2. \end{cases} \tag{12}$$

These equations are implemented in the equivalent circuit of Fig. 4 (a). The first one can be interpreted as a voltage balance at the primary port: the primary voltage *v*<sup>1</sup> is equal to the voltage on a capacitor of value *s<sup>H</sup>* minus a voltage generator controlled by the secondary current *i*2. Furthermore, the second of eqs. (12) can be interpreted as a current balance at the secondary port: the secondary current *i*<sup>2</sup> is equal to the current in an inductor of value *L*<sup>0</sup> minus a current generator controlled by the primary voltage *v*1. For example, if the harvester undergoes a vibrating force generator with a certain mass *m* and it is connected to a resistor *Rl* then, as represented in Fig. 4 (a), this can be simply solved by connecting the series of a voltage generator and a inductor<sup>1</sup> to the first port and, of course, the resistor to the second port.

The proposed methodology can be also employed to derive a two-port representation in the memoryless non-linear case, (5). By The eqs. (6) can be recasted as follows:

<sup>1</sup> The equation of a rigid mass is *<sup>F</sup>* <sup>=</sup> *md*<sup>2</sup> *<sup>x</sup>*/*dt*<sup>2</sup> <sup>↔</sup> *<sup>v</sup>* <sup>=</sup> *mdi*/*dt* that is the inductor characteristic. A linear elastic effect is equivalent to a capacitor: *F* = *kx* ↔ *v* = *k* � *idt* (with the capacitance equal to 1/*k*) and, finally, a viscous friction is represented by a resistor: *F* = *rdx*/*dt* ↔ *v* = *ri*.

10 Will-be-set-by-IN-TECH 496 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

$$\begin{cases} \mathbf{x} = \frac{l\_0}{SA} F\_1 + g(\mathbf{i}\_{2'} F\_1) \, l \\\\ \Phi\_2 = \mu\_0 \frac{N^2 A}{l} \mathbf{i}\_2 + m(\mathbf{i}\_2, F\_1) \, \text{NA} . \end{cases} \tag{13}$$

where

$$\begin{cases} g(i\_2, F\_1) = -\frac{M\_8}{\gamma} \left[ (\frac{\gamma NA}{l} \frac{i\_2}{F\_1}) \tanh(\frac{\gamma NA}{l} \frac{i\_2}{F\_1}) - \ln(\cosh(\frac{\gamma NA}{l} \frac{i\_2}{F\_1})) \right] \\\ m(i\_2, F\_1) = M\_8 \tanh(\frac{\gamma NA}{l} \frac{i\_2}{F\_1}). \end{cases} \tag{14}$$

Assuming the same previous analogies *i*<sup>1</sup> = *dx*/*dt* and *v*<sup>2</sup> = *d*Φ2/*dt*, the following system describing the nonlinear dynamic two-port can be derived:

$$\begin{cases} \frac{1}{s^H} \int\_0^t i\_1 d\tau = v\_1 + \frac{l\_0}{s^H} g(i\_2, v\_1) \\ \frac{1}{L\_0} \int\_0^t v\_2 d\tau = i\_2 + \frac{NA}{L\_0} m(i\_2, v\_1), \end{cases} \tag{15}$$

and the two-port is formally unchanged, apart from the controlled generators and a DC current generator, as shown in Fig. 4 (b). The generator implements the effect of a magnetic bias over the magnetostrictive characteristics. It does not produce any dissipated power into the electric load and, therefore, does not contribute to the energy balance of the two-port.

Finally, the two-port modeling approach makes easier to consider different types of vibration sources, either force or velocity driven [8]. Indeed, the simplest model of *non ideal* mechanical source can be defined as:

$$\begin{cases} l\_0 \, d\mathcal{S}/dt = \mu\_1(t) - \frac{A}{\eta} T(t) \quad \text{for} \ \eta > 0; \\\ \quad AT(t) = F\_1(t) - \eta l\_0 \dot{\mathcal{S}} \quad \text{for} \ \eta < +\infty. \end{cases} \tag{16}$$

In the former case, *u*<sup>1</sup> is an impressed velocity (velocity-driven source), while in the latter *F*<sup>1</sup> is an impressed force (force-driven source). These two elements can be added to the two-port circuit as the classical current and voltage *non ideal* generators, respectively. The first equation leads to the ideal velocity-driven source when the mechanical resistance *η* → ∞, while *η* = 0 describes a ideal stress-driven mechanical source in the latter.

#### **2.3. Numerical results**

In this section the different aforementioned approaches for describing the material behavior are compared, highlighting the main features and limitations. To this aim, it is useful considering a semi-coupled problem where the mechanical stress is imposed. In this case, in fact, only the effect of the mechanical source on the harvesting performances can be considered and evaluated. In the following examples, a cylindrical active material sample with length *l*<sup>0</sup> = 0.1 m and area *A* = *π* cm2 wounded by a coil with *N* = 100 turns is considered. In Fig. 5, the harvested power on a resistor *R* = 10Ω is reported, for a mechanical sinusoidal stress *T* = *T*<sup>0</sup> + *Tm* cos(2*π f t*) with *T*<sup>0</sup> = −27.4 MPa and *Tm* = 13.9 MPa, at different frequencies.

**Figure 5.** Harvested Power at different magnetic biases. It has been considered *μ<sup>r</sup>* = 12, *E* = 30 GPa, *<sup>d</sup>* <sup>=</sup> <sup>2</sup> · <sup>10</sup>−<sup>8</sup> m/A in eqs. (12), while *Ms* <sup>=</sup> 0.8 T, *<sup>γ</sup>* <sup>=</sup> <sup>−</sup>347 T, *<sup>E</sup>* <sup>=</sup> 110 GPa in eqs. (6).

**Figure 6.** Harvesting periodic trajectories - varying external magnetic bias.

10 Will-be-set-by-IN-TECH

*SA <sup>F</sup>*<sup>1</sup> <sup>+</sup> *<sup>g</sup>*(*i*2, *<sup>F</sup>*1)*<sup>l</sup>*

)tanh(

Assuming the same previous analogies *i*<sup>1</sup> = *dx*/*dt* and *v*<sup>2</sup> = *d*Φ2/*dt*, the following system

*i*1*dτ* = *v*<sup>1</sup> +

*v*2*dτ* = *i*<sup>2</sup> +

and the two-port is formally unchanged, apart from the controlled generators and a DC current generator, as shown in Fig. 4 (b). The generator implements the effect of a magnetic bias over the magnetostrictive characteristics. It does not produce any dissipated power into the electric load and, therefore, does not contribute to the energy balance of the two-port.

Finally, the two-port modeling approach makes easier to consider different types of vibration sources, either force or velocity driven [8]. Indeed, the simplest model of *non ideal* mechanical

*η*

*AT*(*t*) = *<sup>F</sup>*1(*t*) <sup>−</sup> *<sup>η</sup>l*0*S*˙ for *<sup>η</sup>* <sup>&</sup>lt; <sup>+</sup>∞.

In the former case, *u*<sup>1</sup> is an impressed velocity (velocity-driven source), while in the latter *F*<sup>1</sup> is an impressed force (force-driven source). These two elements can be added to the two-port circuit as the classical current and voltage *non ideal* generators, respectively. The first equation leads to the ideal velocity-driven source when the mechanical resistance *η* → ∞, while *η* = 0

In this section the different aforementioned approaches for describing the material behavior are compared, highlighting the main features and limitations. To this aim, it is useful considering a semi-coupled problem where the mechanical stress is imposed. In this case, in fact, only the effect of the mechanical source on the harvesting performances can be considered and evaluated. In the following examples, a cylindrical active material sample with length *l*<sup>0</sup> = 0.1 m and area *A* = *π* cm2 wounded by a coil with *N* = 100 turns is considered. In Fig. 5, the harvested power on a resistor *R* = 10Ω is reported, for a mechanical sinusoidal stress *T* = *T*<sup>0</sup> + *Tm* cos(2*π f t*) with *T*<sup>0</sup> = −27.4 MPa and *Tm* = 13.9 MPa, at different frequencies.

*<sup>l</sup>*<sup>0</sup> *dS*/*dt* <sup>=</sup> *<sup>u</sup>*1(*t*) <sup>−</sup> *<sup>A</sup>*

*<sup>l</sup> <sup>i</sup>*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*(*i*2, *<sup>F</sup>*1) *NA*.

*i*2 *F*1

) − ln(cosh(

*γNA l*

*i*2 *F*1 )) �

*γNA l*

> *l*0 *<sup>s</sup><sup>H</sup> <sup>g</sup>*(*i*2, *<sup>v</sup>*1)

*NA L*0

*m*(*i*2, *v*1),

*T*(*t*) for *η* > 0;

(13)

(14)

(15)

(16)

*N*2*A*

*i*2 *F*1

> *i*2 *F*1 ).

*γNA l*

⎧ ⎪⎪⎨

*<sup>x</sup>* <sup>=</sup> *<sup>l</sup>*<sup>0</sup>

Φ<sup>2</sup> = *μ*<sup>0</sup>

⎪⎪⎩

*γ* � ( *γNA l*

describing the nonlinear dynamic two-port can be derived:

⎧ ⎪⎪⎨

1 *sH*

1 *L*0 � *t* 0

� *t* 0

⎪⎪⎩

⎧ ⎪⎨

⎪⎩

describes a ideal stress-driven mechanical source in the latter.

*<sup>g</sup>*(*i*2, *<sup>F</sup>*1) = <sup>−</sup> *Ms*

*m*(*i*2, *F*1) = *Ms* tanh(

where

⎧ ⎪⎪⎨

⎪⎪⎩

source can be defined as:

**2.3. Numerical results**

A first important difference between the linear and the nonlinear memoryless approach can be noted. In fact, in the linear case the harvested power is independent on the magnetic bias, while in the nonlinear case this strongly influences the harvester performances. This phenomenon can be easily understood referring to Fig. 6 where the periodic working trajectories are reported in the B-H plane, for a fixed frequency *f* = 168.5 Hz. It is worth to note that the area of this loops is directly proportional to the harvested energy, i.e. the averaged harvested power [9]. In the linear case (Fig. 6 (a)), the described loops are equal independently on the magnetic bias. Instead, in the non-linear case (Fig. 6 (b)), the loops area changes on this parameter. In the same figures the static characteristics of the two modeling approaches are reported. As it can be seen, these loops are within the static characteristics: in the linear case the distance is equal (parallel lines), in the nonlinear one this relative distance depends on the magnetic bias level. To support this interpretation, another test has been performed comparing again the linear and the nonlinear memoryless models, but varying the mechanical prestress level, keeping the magnetic bias constant, see Fig. 7. In this case *T*<sup>0</sup> = −20.45 MPa or *T*<sup>0</sup> = −34.35 MPa have been considered, while the stress amplitude *Tm* = 6.95 MPa is fixed. Again the harvested energies in the linear case are the same, while in the nonlinear case are strongly influenced by the prestress level. To explain this other

**Figure 7.** Harvested Power at different mechanical prestresses. The models parameters are the same of Fig. 5.

**Figure 8.** Harvesting periodic trajectories - varying mechanical prestress. The models parameters are the same of Fig. 5.

behavior, let us consider the Figs. 8 with loops in the B-H plane. In Fig. 8 (a), the linear static characteristics are also reported and as it can be noted the relative distance is still constant while in the nonlinear case it is not. So the described loops by the linear model are equal (same power level) independently on the prestress level. So summarizing, the linear model cannot take into account the magnetic bias and the mechanical prestress in the description of the harvester performances.

In Fig. 9, a comparison between the memoryless non-linear and hysteretic models is reported. In Fig. 9 (a), the static characteristics of the two models for two different mechanical stresses are drawn. For a mechanical input with a prestress *T*<sup>0</sup> = −27.4 MPa and amplitude *Tm* = 13.9 MPa, the harvested powers on a resistor *R* = 10Ω are reported in Fig. 9 (b). In this case, as additional phenomenon related to the hysteresis, it can be noted that the harvested power levels changes with the initial state (in this case, the demagnetized initial state and the saturation). This behavior can be also analyzed on the B-H plane, as reported in Figs. 10 (a) and (b). While in the memoryless nonlinear case the periodic loops are within the static characteristics, when the hysteresis is considered the periodic trajectories are within the

**Figure 9.** Comparison between Nonlinear Memoryless and Hysteretic modeling.

12 Will-be-set-by-IN-TECH

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

Frequency [Hz]

0.05 0.1 0.15 0.2 0.25 0.3

Flux Density [T]

1.2 1.3 1.4 1.5 1.6 1.7 1.8

T= −13.5MPa T= −27.4MPa T= −41.3MPa

Magnetic Field [A/m]

(b) Nonlinear Memoryless Modeling.

x 10 4

**Figure 7.** Harvested Power at different mechanical prestresses. The models parameters are the same of

**Figure 8.** Harvesting periodic trajectories - varying mechanical prestress. The models parameters are the

behavior, let us consider the Figs. 8 with loops in the B-H plane. In Fig. 8 (a), the linear static characteristics are also reported and as it can be noted the relative distance is still constant while in the nonlinear case it is not. So the described loops by the linear model are equal (same power level) independently on the prestress level. So summarizing, the linear model cannot take into account the magnetic bias and the mechanical prestress in the description of

In Fig. 9, a comparison between the memoryless non-linear and hysteretic models is reported. In Fig. 9 (a), the static characteristics of the two models for two different mechanical stresses are drawn. For a mechanical input with a prestress *T*<sup>0</sup> = −27.4 MPa and amplitude *Tm* = 13.9 MPa, the harvested powers on a resistor *R* = 10Ω are reported in Fig. 9 (b). In this case, as additional phenomenon related to the hysteresis, it can be noted that the harvested power levels changes with the initial state (in this case, the demagnetized initial state and the saturation). This behavior can be also analyzed on the B-H plane, as reported in Figs. 10 (a) and (b). While in the memoryless nonlinear case the periodic loops are within the static characteristics, when the hysteresis is considered the periodic trajectories are within the

0.5

1

1.5

Power [W]

1.2 1.3 1.4 1.5 1.6 1.7 1.8

T= −13.5MPa T= −27.4MPa T= −41.3MPa

Magnetic Field [A/m]

(a) Linear Modeling.

x 10 4

Fig. 5.

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

same of Fig. 5.

the harvester performances.

Flux Density [T]

2

Linear

Nonlinear − 20.45MPa Nonlinear − 34.35MPa

2.5

**Figure 10.** Harvesting periodic trajectories: comparison between non-linear and hysteretic models, starting from different initial internal states.

hysteresis loops but their position and area are strongly influenced by the initial state of the hysteresis model.

In order to analyze the effect of the harvesting process on the mechanical side, a fully coupled problem is also addressed. In this case, the first port of the equivalent two-port model is closed with a lumped mass *m* = 0.5 kg and a viscous friction *r* = 500 Ns/m. In Fig. 11, the free response of the model with an initial velocity *v* = 0.1 m/s is shown, for the linear and the nonlinear memoryless models. As it can be noted, the external magnetic bias strongly influences also the strain time evolution (see Fig. 11 (a)) as done on the electrical side (see Fig. 11 (b)). The linear modeling approach cannot take into account this dependence, however it can be considered for a preliminary analysis of the magneto-elastic harvesting device.

#### **3. Experimental characterization and results**

The good design of a magnetoelastic energy harvester needs the characterization of the raw active materials, by knowing their magnetomechanical characteristics. In order to do so, a rather complex experimental setup is needed because both magnetic and mechanical

**Figure 11.** Impulse response - Comparison

variables must be measured and/or enforced at the same time. Morever, the constitutive relationships of magnetostrictive materials, like Terfenol-D or Galfenol, show nonlinear and rate-independent memory effect (hysteresis) and the mechanical (*S*, *T*) and magnetic variables (*H*, *B*) are cross-coupled [19]. The cross-coupling terms are usually named piezo-magnetic coefficients and, in the energy harvesting framework, a deep knowledge of them is crucial because they are related to the material ability to convert energy [19]. For example, the accurate knowledge or modeling of their behavior, with respect to the magnetic field and prestress biases, can be useful to choose the working point (*H*0, *T*0) that guarantees the best performance in a energy harvesting device [10] or in other applications like a sensor. Now, in the case of characteristics without hysteresis, the piezo-magnetic parameters have to satisfy the thermodynamic constraint of eq. (4).

Several papers have tackled the experimental problem of measuring the piezo-magnetic coefficients of magnetostrictive material, as well other parameters as mechanical compliance, Young modulus, etc. This study is mandatory to explore the performance of new magnetostrictive compounds [5, 6, 37]. In a early paper, the problem of hysteresis of those coefficients started to be considered [34]. Also the variability of the piezo-magnetic parameter with the applied prestress has been considered [40], leading to the conclusion that magnetic and mechanical bias must be chosen with attention in order to get the best performance [49]. This result have been confirmed by a later paper on non-linear modeling [10].

The magnetostriction and the magnetic characteristics have to be measured in different stress and magnetic field conditions in order to explore the previous conjecture.

In order to measure the magnetostrictive characteristics, a combination of different standard and custom instruments have to be used, as sketched in Fig. 12. The mechanical compressive load is applied through a computer-controlled test-machine while the magnetic field is imposed through an electromagnet and measured by using a digital gaussmeter. The material magnetic induction is obtained after a numerical time-integration of a voltage, measured over a pick-up coil with 500 turns. The demagnetizing effect must be taken into account. The strain is directly measured with a strain gauges bridge, configured to have temperature self-compensation, while the applied stress is measured by a load cell.

#### 500 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges Magnetoelastic Energy Harvesting: Modeling and Experiments <sup>15</sup> Magnetoelastic Energy Harvesting: Modeling and Experiments 501

**Figure 12.** Sketch of a possible setup to measure the magnetostrictive characteristics.

**Figure 13.** Magnetostrictive cycles. (a) legend: 4 MPa–fuchsia, 6 MPa–green, 12 MPa–red, 40 MPa–black, 60 MPa–blue. (b) legend: 0 kA/m–blue, 10 kA/m–black, 20 kA/m–red, 40 kA/m–green, 100 kA/m–fuchsia.

#### **3.1. Magnetostrictive characteristics**

14 Will-be-set-by-IN-TECH

variables must be measured and/or enforced at the same time. Morever, the constitutive relationships of magnetostrictive materials, like Terfenol-D or Galfenol, show nonlinear and rate-independent memory effect (hysteresis) and the mechanical (*S*, *T*) and magnetic variables (*H*, *B*) are cross-coupled [19]. The cross-coupling terms are usually named piezo-magnetic coefficients and, in the energy harvesting framework, a deep knowledge of them is crucial because they are related to the material ability to convert energy [19]. For example, the accurate knowledge or modeling of their behavior, with respect to the magnetic field and prestress biases, can be useful to choose the working point (*H*0, *T*0) that guarantees the best performance in a energy harvesting device [10] or in other applications like a sensor. Now, in the case of characteristics without hysteresis, the piezo-magnetic parameters have to satisfy

Several papers have tackled the experimental problem of measuring the piezo-magnetic coefficients of magnetostrictive material, as well other parameters as mechanical compliance, Young modulus, etc. This study is mandatory to explore the performance of new magnetostrictive compounds [5, 6, 37]. In a early paper, the problem of hysteresis of those coefficients started to be considered [34]. Also the variability of the piezo-magnetic parameter with the applied prestress has been considered [40], leading to the conclusion that magnetic and mechanical bias must be chosen with attention in order to get the best performance [49].

The magnetostriction and the magnetic characteristics have to be measured in different stress

In order to measure the magnetostrictive characteristics, a combination of different standard and custom instruments have to be used, as sketched in Fig. 12. The mechanical compressive load is applied through a computer-controlled test-machine while the magnetic field is imposed through an electromagnet and measured by using a digital gaussmeter. The material magnetic induction is obtained after a numerical time-integration of a voltage, measured over a pick-up coil with 500 turns. The demagnetizing effect must be taken into account. The strain is directly measured with a strain gauges bridge, configured to have temperature

This result have been confirmed by a later paper on non-linear modeling [10].

and magnetic field conditions in order to explore the previous conjecture.

self-compensation, while the applied stress is measured by a load cell.

10−10

10−5

Istantaneous Power [W]

100

105

0 0.5 1 1.5 2 2.5

Linear Nonlinear − 5kA/m Nonlinear − 15kA/m Nonlinear − 30kA/m

Time [ms]

(b) Instantaneous power.

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 −300

Time [ms]

(a) Strain.

**Figure 11.** Impulse response - Comparison

the thermodynamic constraint of eq. (4).

Linear Nonlinear − 5kA/m Nonlinear − 15kA/m Nonlinear − 30kA/m

−200

−100

Strain [ppm]

0

100

200

The magnetostrictive characteristics *S*(*H*) under variable magnetic field and constant compressive stress are shown in Fig. 13 (a). The magnetostrictive characteristics does not include the elastic effect, i.e. the mechanical strain induced by the constant stress. It is worth noting that the dependence from the applied constant stress is somewhat complex. Indeed, the cycles change in shape and maximum amplitude with the stress.

Finally, the Fig. 13 (b) shows the magnetic induction induced by a cycled stress. These measurements are very important to infer the harvesting potentialities of the material. Indeed, the larger is the Δ*B* at a certain Δ*T* the better would be the energy conversion. The measurements show wider cycles, i.e. more hysteresis, at low-intermediate magnetic field. Moreover, the maximum Δ*B* is reached at intermediate field too.

#### **3.2. Piezo-magnetic coefficients**

The definition of the piezo-magnetic parameters considered here are the following [12, 34]:

$$\begin{cases} d\_{33} = \left. \frac{\partial S}{\partial H} \right|\_{T} \\ d\_{33}^{\*} = \left. \frac{\partial B}{\partial T} \right|\_{H} \end{cases} \tag{17}$$

As stated in eq. (4), they are equal if the hypothesis of lossless material is considered while, in general, they are not. These parameters are usually considered constant in datasheets and in linear models of the material [34]. Such an approximation holds if the material is employed in actuators where the mechanical prestress is higher than the external stress applied to the device. On the other hand, the approximation cannot be applied if general dynamic conditions are considered, as the case of energy harvesting applications.

Indeed, the piezo-magnetic parameters are nonlinear function of the considered (*H*, *T*) couple. Moreover, they show hysteresis. Nevertheless, the piezo-magnetic parameters can be obtained by a numerical derivative of the experimental hysteresis loops. The two branches of the loops are considered as *up* and *down* curves in the following.

In order to compare the two parameters, they have been sampled in a same matrix of (*H*, *T*) points. The corresponding surfaces are shown in Figs. 14,15. It is apparent that the orders of magnitude and the shapes are the same, but the parameters are not exactly equal for each (*H*, *T*) couple. The maximum for each set of curves (up and down) moves at higher *H* field for higher *T* and this confirms nonlinear models results in literature (see [8], Fig. 3 or [48]). Nevertheless, for all of them, the maximum values are achieved at relatively low stress and magnetic field, that are the ranges *T* ∈ (−10, 0) MPa and *H* ∈ (10, 20) kA/m. Those values are useful to choose the best magnetic bias and mechanical prestress.

**Figure 14.** The piezo-magnetic parameters (up curves) sampled at different magnetic field and stress.

#### **3.3. Harvesting results - Terfenol**

Let us consider now the specific powers generated by a laboratory harvesting device with a Terfenol-D rod 18 mm long and with 9 mm2 square section. The harvester has a pick-up coil

**Figure 15.** The piezo-magnetic parameters parameters (down curves) sampled at different magnetic field and stress.

**Figure 16.** Harvesting results of a force-driven device with a terfenol rod.

with 800 turns. The Fig. 16 (a) shows the peak voltage and average power with vibration frequency and *H*<sup>0</sup> = 18 kA/m, ΔF= 300 N, corresponding to Δ*T* = 33.3 MPa. It is worth noting that, at 50Hz, a relevant specific power of 6.17 mW/cm3 is reached, confirming the potentialities of vibration harvesting when high frequencies are concerned. The Fig. 16 (b) shows the average power at 50Hz by varying the resistance load, at different prestresses. It is apparent that the choices of the correct prestress and resistance are crucial to harvest the maximum power.

#### **3.4. Harvesting results - Galfenol**

16 Will-be-set-by-IN-TECH

The definition of the piezo-magnetic parameters considered here are the following [12, 34]:

*<sup>d</sup>*<sup>33</sup> <sup>=</sup> *<sup>∂</sup><sup>S</sup> ∂H* � � � � *T*

As stated in eq. (4), they are equal if the hypothesis of lossless material is considered while, in general, they are not. These parameters are usually considered constant in datasheets and in linear models of the material [34]. Such an approximation holds if the material is employed in actuators where the mechanical prestress is higher than the external stress applied to the device. On the other hand, the approximation cannot be applied if general dynamic conditions

Indeed, the piezo-magnetic parameters are nonlinear function of the considered (*H*, *T*) couple. Moreover, they show hysteresis. Nevertheless, the piezo-magnetic parameters can be obtained by a numerical derivative of the experimental hysteresis loops. The two branches

In order to compare the two parameters, they have been sampled in a same matrix of (*H*, *T*) points. The corresponding surfaces are shown in Figs. 14,15. It is apparent that the orders of magnitude and the shapes are the same, but the parameters are not exactly equal for each (*H*, *T*) couple. The maximum for each set of curves (up and down) moves at higher *H* field for higher *T* and this confirms nonlinear models results in literature (see [8], Fig. 3 or [48]). Nevertheless, for all of them, the maximum values are achieved at relatively low stress and magnetic field, that are the ranges *T* ∈ (−10, 0) MPa and *H* ∈ (10, 20) kA/m. Those values

(17)

−40 −30 −20 −10 <sup>0</sup>

−30

σ H [kA/m] [MPa]

(b) *d*∗ 33.

−20

−10 0 0

20

d\*33 [nm/A] (up curve)

40

60

⎧ ⎪⎪⎨

⎪⎪⎩

are considered, as the case of energy harvesting applications.

of the loops are considered as *up* and *down* curves in the following.

are useful to choose the best magnetic bias and mechanical prestress.

−40 −30 −20 −10 <sup>0</sup>

**Figure 14.** The piezo-magnetic parameters (up curves) sampled at different magnetic field and stress.

Let us consider now the specific powers generated by a laboratory harvesting device with a Terfenol-D rod 18 mm long and with 9 mm2 square section. The harvester has a pick-up coil

−30

**3.3. Harvesting results - Terfenol**

σ H [kA/m] [MPa]

(a) *d*33.

−20

−10 0 0

20

d33 [nm/A] (up curve)

40

60

*d*∗ <sup>33</sup> <sup>=</sup> *<sup>∂</sup><sup>B</sup> ∂T* � � � � *H*

**3.2. Piezo-magnetic coefficients**

Let us consider now the specific powers generated by a laboratory harvesting device with a Galfenol rod 30 mm long and with 5 mm diameter. The Fig. 17 shows the average power densities (left) and the peak to peak voltage with a 3000 turns pick-up coil (parasitic resistance of 1 kΩ) and 1 kΩ termination resistor. The compressive stress variation is 30 MPa and

**Figure 17.** Average power densities (left) and peak to peak voltage (right) at *f* = 0.35 Hz, with a Δ*T* = 30 MPa stress variation.

*f* = 0.35 Hz, a very low frequency that could be found in structural vibrations of civil constructions. It is worth noting that the peaks are reached at relatively low magnetic field bias, confirming one of the advantages to use Galfenol for harvesting applications. Indeed, those magnetic biases could be easily obtained with a permanent magnet, in a engineered harvesting device. The peak specific power is about 1 *μ*W/cm3, reached at 10 kA/m magnetic bias and 16 MPa prestress. Those values are reasonable and in agreement with literature values [33]. The Fig. 18 (a) shows the average power densities with respect to different termination resistors and with a 800 turns coil (parasitic resistance of 19Ω). Moreover, the resistor value near the parasitic coil resistance gives the maximum power. The Fig. 18 (b) shows the average power density behavior with respect to the stress variation frequency with 20Ω resistance, 15kA/m magnetic bias and with a 1 − 80 MPa stress variation. As expected, the power increases with frequency.

#### **3.5. Harvesting loops behavior in the B-H plane**

It is a fundamental result the fact that magnetic loops in the B-H plane represent losses if the loops are passed through counterclockwise. On the other hand, if passed clockwise, those loops area represent *generated* energy. Then, it is apparent that a good design of the device should aim to enlarge as much as possible that area. Now, as it was shown in the previous sections, magnetic bias, prestress and other parameters can be used to do so because if in steady-state vibrations the power is increased then the loop area in the B-H plane is increased too. The Fig. 19 (a) shows the magnetic characteristics at 1.58 and 48.2 MPa constant stresses. The blue lines within the cycles represent the loops due to harvesting tests at different magnetic bias (8.7, 14.8 and 23 kA/m), 1 MΩ resistance and *f* = 0.8 Hz. It is noticeable that the loops are contained within the magnetic characteristics, as it was theoretically foreseen in [9]. Moreover, this is still true if different resistors are employed, as shown in Fig. 19 (b). The loops are wider for smaller resistor because a larger current circulates in the pick-up

**Figure 18.** Harvesting results of a force-driven device with a a galfenol rod.

coil and a magnetic field variation is introduced then. This result is important because it is evident that the effort to improve the material harvesting behavior should concentrate on the *static* magnetic characteristics at different stresses, trying to improve as much as possible the *available area* among limit cycles at different stresses.

**Figure 19.** (a) Magnetic characteristics with converted energy loops at different magnetic bias (8.7, 14.8 and 23kA/m), 1 MΩ resistance and *f* = 0.8 Hz. (b) Converted energy loops zoom with different resistance termination at 14.8kA/m magnetic bias.

#### **4. Conclusions and open problems**

#### **Materials**

18 Will-be-set-by-IN-TECH

0 20 40 60

16 MPa 35 MPa 55 MPa 75 MPa

Magnetic Field [kA/m]

Peak to Peak Voltage [mV]

16 MPa 35 MPa 55 MPa 75 MPa

0 20 40

Magnetic Field [kA/m]

**Figure 17.** Average power densities (left) and peak to peak voltage (right) at *f* = 0.35 Hz, with a

*f* = 0.35 Hz, a very low frequency that could be found in structural vibrations of civil constructions. It is worth noting that the peaks are reached at relatively low magnetic field bias, confirming one of the advantages to use Galfenol for harvesting applications. Indeed, those magnetic biases could be easily obtained with a permanent magnet, in a engineered harvesting device. The peak specific power is about 1 *μ*W/cm3, reached at 10 kA/m magnetic bias and 16 MPa prestress. Those values are reasonable and in agreement with literature values [33]. The Fig. 18 (a) shows the average power densities with respect to different termination resistors and with a 800 turns coil (parasitic resistance of 19Ω). Moreover, the resistor value near the parasitic coil resistance gives the maximum power. The Fig. 18 (b) shows the average power density behavior with respect to the stress variation frequency with 20Ω resistance, 15kA/m magnetic bias and with a 1 − 80 MPa stress variation. As expected,

It is a fundamental result the fact that magnetic loops in the B-H plane represent losses if the loops are passed through counterclockwise. On the other hand, if passed clockwise, those loops area represent *generated* energy. Then, it is apparent that a good design of the device should aim to enlarge as much as possible that area. Now, as it was shown in the previous sections, magnetic bias, prestress and other parameters can be used to do so because if in steady-state vibrations the power is increased then the loop area in the B-H plane is increased too. The Fig. 19 (a) shows the magnetic characteristics at 1.58 and 48.2 MPa constant stresses. The blue lines within the cycles represent the loops due to harvesting tests at different magnetic bias (8.7, 14.8 and 23 kA/m), 1 MΩ resistance and *f* = 0.8 Hz. It is noticeable that the loops are contained within the magnetic characteristics, as it was theoretically foreseen in [9]. Moreover, this is still true if different resistors are employed, as shown in Fig. 19 (b). The loops are wider for smaller resistor because a larger current circulates in the pick-up

0

Δ*T* = 30 MPa stress variation.

the power increases with frequency.

**3.5. Harvesting loops behavior in the B-H plane**

0.2

0.4

0.6

Specific average power [μ W/ cm3

]

0.8

1

It is evident that the research on magnetoelastic material synthesis can give a strong advancement to the energy harvesting applications. Indeed, the research should focus to maximize the piezo-magnetic coefficients. This, as explained in section 3.5, would means to have more space for harvesting loops within static limit cycles. In other words, to have a larger Δ*B* for a certain Δ*T*.

#### 20 Will-be-set-by-IN-TECH 506 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Another topic about materials synthesis concerns the *mechanical impedance* matching of the harvesters. Indeed, materials like terfenol or galfenol are quite *rigid* with a mechanical behavior near the one of bulk iron. In that case, the ideal vibrations have high stresses and low strains, in the 0 − 1000ppm range. If a softer magnetoelastic materials would exist, it would be possible to have vibrations harvesting with lower stresses and higher strains, in the 0.1 − 1% range, with a rubber-like behavior.

Recently the use of another magnetoelastic material (Metglass) has been proposed for energy harvesting [45]. The main advantage respect to the others magnetostrictive material is that it can be laminated achieving a higher harvester compactness. The material is a Fe-based amorphous ribbons with excellent magnetic softness and elastic response and it is cheaper than Fe-Ga, Fe-Tb-Dy alloys. A recent new application design reached 20 *μ*W/cm<sup>3</sup> at 100 Hz [53].
