Design and characterization of a low frequency 2-dimensional magnetic levitation kinetic energy harvester

Sensors and Actuators A 236 (2015) 1–10

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Design and characterization of a low frequency 2-dimensional magnetic levitation kinetic energy harvester Manuel Gutierrez a,∗ , Amir Shahidi a , David Berdy b , Dimitrios Peroulis a a b

School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA Qualcomm Technologies, Inc., San Diego, CA 92121, USA

a r t i c l e

i n f o

Article history: Received 6 May 2015 Received in revised form 2 October 2015 Accepted 3 October 2015 Available online 22 October 2015 Keywords: Electromagnetic Low frequency Energy harvester Magnetic levitation 2-Dimensional Simulation

a b s t r a c t This paper presents the design and characterization of the first completely orientation-independent magnetic levitation energy harvester for low power, low frequency, and low g applications. A fully symmetric design is presented able to operate in any orientation in a particular plane. A MATLAB/Simulink model has been developed to aid in optimizing harvester design. Resonant frequencies in the order of 2 Hz are desirable for human walking motion energy harvesting applications. The presented harvester has been tuned to a resonant frequency of 8.2 Hz, allowing it to take advantage of harmonics available from human walking motion. The harvester may be tuned to different frequencies with different magnet configurations and harvester dimensions with the aid of the MATLAB/Simulink model. The presented energy harvester has a measured power output of 41.0 ␮W and 101 ␮W at 0.1 g and 0.2 g accelerations, respectively. © 2015 Elsevier B.V. All rights reserved.

1. Introduction With the advent of wearable electronics and sensors in recent years, the ambient energy from human motion kinetic energy has become a large area of interest. The implementation of effective energy harvesting in low power wireless applications could lead to long device operating lifetimes. Many different approaches have been taken for kinetic motion energy harvesting, the two most dominant being piezoelectric and electromagnetic [1–8]. Levitating magnet energy harvesters have shown success in utilizing ambient vibrations for harvesting energy [9,10]. The use of fixed magnets to provide spring force eliminates the use of a physical spring, one of the most likely components to fail in a mechanical spring-damper system. There are also no wires attached directly to the moving mass, further eliminating another fault condition. Levitating magnet energy harvesters utilize Faraday’s law of induction which states that a changing magnetic flux through a coil will result in an induced electromotive force. This EMF can be attached to a load and used to power an external system. These magnetic levitation harvesters have shown success in many different applications, ranging from wrist shaking [7] to vehicle suspension [3].

∗ Corresponding author. E-mail addresses: [email protected] (M. Gutierrez), [email protected] (A. Shahidi), [email protected] (D. Berdy), [email protected] (D. Peroulis). http://dx.doi.org/10.1016/j.sna.2015.10.009 0924-4247/© 2015 Elsevier B.V. All rights reserved.

One of the main difficulties with all previously presented levitating magnet energy harvesters has been their limited degrees of freedom and their high resonant frequencies. They have been limited to only one orientation for proper or ideal operation. If the harvester is tilted at an angle or rotated within the ideal orientation, harvester output can become greatly limited or eliminated [11]. Constantinuo et al. [12], Berdy et al. [4], and Mann et al. [9] have developed devices that respond in this way as a consequence of having a free magnet that is free to move in one dimension only. Devices that have achieved rotation independence have done so without the capability of harvesting low frequency vibrations less than 10 Hz, as is needed for human walking energy harvesting. Yang et al. [13] and Bartsch et al. [14] have developed rotation independent energy harvesters but with resonant frequencies of 25 Hz and 370 Hz, respectively. An energy harvester with a resonant frequency of 10 Hz was developed by Moss et al. [15], but the device is not rotation independent in its performance. The limitation of the harvester rotation presents issues when the harvester may be worn, and the wearer is not careful to attach the device in the ideal rotation. Given the desire to use these harvesters in wearable electronics, it is expected that the device will rotate and be held at different angles. Clip-on devices will likely be clipped at angles and often even upside down, rendering 1-dimensional harvesters inoperable. Furthermore, while human walking has a predominant vertical acceleration force, there are also horizontal vibrations which are not utilized in these limited harvesters.

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The levitating magnetic energy harvester presented is capable of harvesting energy from ambient vibrations in two dimensions along the harvester’s plane of operation. This implies that the presented harvester can be rotated at any angle and will not suffer from performance degradation. The presented harvester has been tuned to an 8.2 Hz resonant frequency, making it capable of harvesting energy from the higher harmonic frequencies of human step frequencies. 2. Energy harvester structure The magnet configuration and general layout of the presented 2-dimensional energy harvester is shown in Fig. 1. A free moving, axially magnetized disk magnet lies on a 2-dimensional plane with freedom of radial movement. A circular sidewall is constructed to constrain the center magnet boundary and hold fixed magnets. Stationary axially magnetized cuboidal magnets are distributed around the circumference of the sidewall to provide a spring force which returns the free moving magnet to an equilibrium position following perturbation. While the casing can be machined out of a wide range of materials, in this device it is machined out of Teflon for its low coefficient of friction and for its softness which makes for ease of fabrication. With the harvester resting upright such that the force of gravity is towards the bottom of the harvester, the free moving center magnet will be offset from the center of the harvester, as shown in Fig. 1. Many factors are involved in the design of the energy harvester dimensions and configuration. Due to the desire for full 2-dimensional functionality, symmetry in a circular design is a requirement. A sufficient number of perimeter magnets are then required to ensure minimal variation in harvester output due to rotation. Minimal has been defined as a maximum 0.5 Hz variation in resonant frequency with harvester rotation. The magnet force also needs to be sufficient such that the center magnet does not hit sidewalls at appreciable accelerations. One coil of N turns is z-displaced above the free moving center magnet on one side of the energy harvester. Due to Faraday’s law of induction, the changing magnetic flux through this coil will result in an induced EMF. The coil output may then be attached to a load and energy may be harvested from the motion of the center magnet. The inductance of the coil can be roughly estimated using air core coil inductance calculators to be about 50 ␮H. With a frequency of 10 Hz, this is an impedance of 3, or <1% of the coil resistance. The inductance is then considered insignificant in further analysis, and a detailed computation will be excluded. In previous literature, the free moving magnet physically moves through the coil [16,12]. However, due to the 2-dimensional capabilities and axially

Fig. 2. Parameters for presented energy harvester. Table 1 Parameters for presented energy harvester. Parameter Perimeter magnets: Number lf × wf × hf rf Grade Center magnet: dl × hl y0 Mass Grade Coil: N dc Resistance Casing: dh Material

Value

Unit

12 1.59 × 1.59 × 1.59 22 Neodymium N42

mm mm

12.7 × 4.76 −6.2 4.53 Neodymium N42 1200 17 425 47 Teflon

mm mm g

mm  mm

magnetized nature of the magnet, the presented harvester’s coil must be displaced above the center magnet. A low resonant frequency is desired such that the harvester can be used in human walking kinetic energy harvesting applications. Dominant step frequency of human walking falls at around 2–3 Hz [17]. Due to the inherently inverse relationship between resonant frequency and harvester size, obtaining a resonant frequency in the 2–3 Hz range would require an obtrusively large harvester, excluding it from wearable electronics applications. For this reason, wearable energy harvesters are designed to have a resonant frequency matching one of the higher harmonics of the dominant step frequencies, around 6–8 Hz [18]. The presented harvester is characterized by Fig. 2 and Table 1. The model shown in Section 3.5 allows for optimization of the design without the need for excessive prototyping. 3. Model 3.1. Electromechanical modeling

Fig. 1. Energy harvester magnet configuration. Free moving center magnet is levitated by perimeter magnets, and offset from center of harvester due to gravity. The perimeter magnet forces return the center magnet towards equilibrium position following perturbation.

The energy harvester may be modeled mechanically as a massspring-damper system, as illustrated in Fig. 3. An external source displaces the housing by q(t), causing a mass m, attached to springs k and dampers d, to displace by p(t) relative to the housing. Due to the 2-dimensional nature of the harvester, q(t) and p(t) can be in any two dimensions. The convention in this paper will be with directions x and y as shown in Fig. 3(b). The parameters k and d are simplifications, since the perimeter magnet forces are nonlinear,

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Fig. 3. (a) Mass-spring-damper system. A mass m is attached to encircling springs k and dampers d, simulating the free center magnet and the perimeter magnet forces and the frictional damping. (b) Housing is displaced by external q(t), Center magnet is displaced within harvester by p(t). f (t) is the absolute displacement of the magnet.

and the damping forces incorporate mechanical, aerodynamical, and electromagnetic losses. As Figs. 1 and 3 show, under vertical excitation, vibration always occurs in the y direction, which is also the direction of gravity. Therefore, further analysis can assume that all motion will occur in the y direction only, and 1-dimensional variables can be used. The functions f(t), q(t), and p(t) are then the y direction components of the respective vectors. With this simplification in mind, a general expression for y directional motion for the free moving center magnet of mass m is given by

Fmag (p(t)) − Ff

|p (t)| − cp p (t) − Fem (p(t), p (t)) p (t)

− mq (t) − mg = mp (t)

The relative displacement within the harvester p(t) is the value of interest since it determines the change of flux through the coil, and therefore the voltage produced. The fixed magnet force Fmag is a nonlinear function of magnet position and is calculated analytically in Section 3.2. The dry friction force Ff is constant and always acts against motion. The aerodynamic viscous damping acts against motion and is the product of cp and magnet speed. The electromagnetic damping force Fem is the force that acts to resist the magnet’s motion near the coil. It is caused by the magnetic field that the coil generates as current is induced by the changing flux of the moving magnet. This force is responsible for the energy from the magnet’s motion that is being harvested. The instantaneous power delivered to the coil and load as a function of coil open circuit voltage Voc , coil flux B , load resistance Rload , and coil resistance Rcoil is

 Pout

dB

2



dB dp

2 (2a)

The instantaneous power input from Fem acting against the magnet’s motion is Pin = Fem p (t)

Fem =

(2b)

dB dp dp dt

2

Rload + Rcoil

p (t)

Rload + Rcoil

(2d)

The fixed perimeter cuboidal magnets and the center free moving disk magnet impart a repellant force upon one another. Many different methods and solutions for determining the force between different types and shapes of permanent magnets exist [19–23]. Some of these methods are semi-analytical or completely numerical and could be used interchangeably, but an analytical solution was sought after in this work. The analytical solution exists for the force between cuboidal magnets [24]. For this reason, the center magnet was modeled as a cuboidal magnet with volume equal to that of the cylindrical magnet used in measurements. This simplification introduces error, but good agreement with the model and experimental measurements shows that the error is minimal. The force on the center magnet due to fixed perimeter magnets was calculated iteratively and then summed. In each case, the center free moving magnet and the fixed perimeter magnet were modeled as a pair of “magnetically charged” parallel plates with  and   charge, respectively. Since both magnets are the same grade (N42 neodymium),  =   . The coordinates for the pair of cuboidal magnets is demonstrated in Fig. 4. The first magnet sits centered at the origin with dimensions of 2a × 2b × 2c. The second magnet is centered in space at (˛, ˇ, ) with dimensions of 2A × 2B × 2C. The magnets are both axially S–N magnetized in the z direction to produce repulsion, which is represented by the fact that  and   have the same sign. The interaction energy between the two sets of parallel charged plates is found by using





+a

W=



+b

dx −a



2

The derivative dB /dp is the displacement-derivative of magnetic flux through the coil, a parameter calculated in Section 3.3. It depends on p(t) and is a function determined by the coil placement and magnet characteristics (strength and dimensions). Therefore, Fem is a function of p(t) and p (t) as shown in Eq. (1). The quantity −mq (t) is the pseudo-force felt by the magnet due to the acceleration of the casing. Finally, −mg is the force of gravity.

By conserving energy,

Pin = Pout ⇒ Fem p (t) =

p (t)(Rload + Rcoil )

=

dB dp

3.2. Magnetic force calculations (1)

2 dt dp dt Voc = = = Rload + Rcoil Rload + Rcoil Rload + Rcoil

 2 Voc

−b



+A

dy

+B

dX −A

dY −B

  4o r

(3a)

where 0 is the permeability of free space, and



(2c)

r=

2

(˛ + X − x)2 + (ˇ + Y − y) +  2

(3b)

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z

a L

y z

x Fig. 4. Location and coordinates of cuboidal magnets for magnetic force calculations.

The pair of magnets have magnetizations J and related to the magnetic charges densities by

J ,

Fig. 5. Cylindrical coordinate system and magnet dimensions convention for cylindrical magnet magnetic field calculation. Magnet has radius a and thickness L.

and they are

  = J · n

(4a)

 are parallel leaves Assuming that J and n =J

and   = J 

(4b)

The gradient of the interaction energy is taken to determine the force between the cuboidal magnets [24]. This expression is given by F = −W =

JJ  4o



(−1)

i+j+k+l+p+q

(u, v, w, r)

(5)

{i,j,k,l,p,q}={0,1}6

for force vector components Fx , Fy , and Fz , respectively:

(6a)

1 2 uv 1 (u − w2 ) ln(r − v) + uv ln(r − u) + uw arctan + rv 2 rw 2 (6b)

z = −uw ln(r − u) − vw ln(r − v) + uv arctan

uv − rw rw

(6c)

with u(i, j) = ˛ + (−1)j A − (−1)i a l

k

(7a)

v(k, l) = ˇ + (−1) B − (−1) b

(7b)

w(p, q) =  + (−1)q C − (−1)p c

(7c)

r(u, v, w) =



u2 + v2 + w2



dFy  dy 

kequilibrium =

fres

1 = 2

(8a) y=y0

kequilibrium

(8b)

m

3.3. Magnetic flux calculations

1 uv 1 x = (v2 − w2 ) ln(r − u) + uv ln(r − v) + vw arctan + ru 2 rw 2

y =

When the device is held upright, gravity pulls the center magnet to an equilibrium point y0 where mg = Fy . At this point, the displacement-derivative of Fy can approximate the resonant frequency of the center magnet by Eq. (8b).

(7d)

For this analysis, only the y direction force Fy is needed due to symmetry. This force is calculated at discrete y coordinates reachable by the free moving center magnet (x = 0). This is done for each perimeter magnet and the Fy forces are then summed, giving the total vertical force on the center magnet as a function of y coordinate. Given that all the magnets are centered on the same x − y plane, and no z displacement is possible, Fz is zero.

A cylindrical magnet with axial magnetization may be modeled as a cylindrical coil of equal diameter and height with azimuthal surface current equal to J, and its magnetic field can then be determined analytically [25]. The coordinate system for the cylindrical magnetic field calculations is shown in Fig. 5. The magnet has radius a and thickness L. Due to the symmetric nature of the cylindrical magnet around the origin, the magnetic field strength will vary only with and z. The parameter  does not affect the magnetic field strength. The magnet moves at a z-offset beneath the coil. As the change in magnetic field through the coil results in the induced EMF, the only magnetic field component of interest is the Bz . B runs tangential to the coil surface and B is zero. The expression for Bz above the magnet is split into two regions of interest; the region above the magnet within the radius of the magnet, and the region above the magnet outside the radius of the magnet. These are respectively given by

Bz ( < a, z > L) B∞ =−

∞  (−1)n (2n + 1)! n=0

−

22n+2 (n!)2

×

 2  2n  a

 a 2  2n  z−L

z−L

F

2

z

n + 1, n +

F

3 a2 n + 1, n + ; 2; − 2 2 z

3 a2 ; 2; − 2 (z − L)2



(9a)

M. Gutierrez et al. / Sensors and Actuators A 236 (2015) 1–10

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Fig. 6. (a) Modeled magnet force Fy , gravity, and spring constant k. Resonant frequency is determined using spring constant at the equilibrium position y0 . (b) Flux through coil as a function of magnet y position (x = 0). Maximum flux through the coil when magnet is centered beneath the coil.

Bz ( > a, z > L) B∞ ∞  (−1)n (2n + 1)!

=−

n=0

22n+2 n!(n + 1)!

 a 2n+2

−

z−L

 F

×

 2n+2  a 2

F

n + 1, n +

3 2 n + 1, n + ; 1; − 2 (z − L)2

3 2 ; 1; − 2 2 z



(9b)

where F(a ; b ; c ; d) is the Gaussian hypergeometric function, and B∞ is the B-field intensity that would be inside a solenoid of equal diameter and surface current, but of infinite length.

2.5 mm past the end of the center magnet, and contains 1200 turns of 42 AWG magnet wire. The magnetic flux through the coil was calculated numerically. First, Bz was calculated at discrete values as a function of and z. Then, the magnetic flux B through the coil’s 1200 turns of varying radii and heights was found by numerical integration to approximate Eq. (11), as a function of the y coordinate of the center magnet. 3.4. MATLAB/Simulink model

In computer modeling, the summations are finite, and an adequate upper bound for n must be used. Such an adequate value can be determined by comparing the discontinuous values of Bz /B∞ for both expressions at their joining point = a. The discrepancy is large with small n upper bounds less than 5 (at n = 5, 2.3% discrepancy). An n upper bound of 10 is sufficient to close this discrepancy to less than 0.1%. The simulation in this work uses an n upper bound of 50, to ensure accuracy with reasonable simulation times. Once the magnetic field z-component Bz can be known for any point above the magnet, where the coil will be, then the magnetic flux B can be calculated with

A MATLAB/Simulink model has been developed to aid in optimizing energy harvester design. The model implements the equations in the previous subsections to simulate the center magnet’s motion in response to an external excitation of choice. This in turn allows the model to simulate power delivery to a load of choice. The model for the presented harvester was constructed using the derived parameters shown in Fig. 6. The equilibrium point y0 is determined in the model to be at y = −6.2 mm. Using Eq. (8b), this yields a resonant frequency of 8.15 Hz. A Simulink time step model simulates the energy harvester operation by solving Eqs. (1), (2d), and (12). Through this time step model, several values are calculated. Of critical importance is the induced voltage in the coil, named hereafter as the open circuit voltage,

B =

Voc =

B∞ = 0 J

(10)



Bz dA

(11)

S

where S is the total surface enclosed by each loop of the coil. This general form of the equation accounts for the multiple loops of wire, and is necessary since Bz will vary along the different heights and diameters of the turns of the coil. The magnetic flux B will be a function of magnet position p(t), and the coil voltage will be the derivative V=

dB dt

(12)

This voltage will be distributed across the coil resistance and the load resistance in series. The radius of the coil in the presented harvester was chosen so that the edge of the coil is at the same level as the equilibrium point of the center magnet. This is due to there being maximum change in flux through the coil when the magnet is crossing the coil edge. Therefore, the inner and outer radii of the coil were chosen to be 8.5 mm and 12.5 mm, respectively. With the coordinate system shown in Fig. 2, the coil extends in the z direction from 1.0 mm to

dB dt

(13)

where B is the total flux through the coil (the number of turns has been accounted for). From here, load voltage can be determined with coil resistance Rcoil , and load resistance Rload by Vload = Voc

Rload Rload + Rcoil

(14)

The instantaneous and average power delivered to the load, respectively, are P=

2 Vload

Rload

P =

2  Vload

Rload

(15a)

(15b)

The model finds the average power delivered to the load for the given parameters of the simulation, such as vibration profile, center magnet starting position, and harvester rotation.

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Fig. 7. Modeled and measured ringdown tests conducted at two starting heights above center. Open circuit load. There is maximum voltage when the magnet is passing the coil edge, and zero voltage when the magnet is changing direction (zero velocity) or is at the center of the coil (zero change in flux).

An initial modeling test to perform is the ringdown test. With the harvester held upright and with no external excitation, the center magnet is brought to a specific height, and then quickly released. The result is a damped, undriven oscillator. The test is shown in Fig. 7, performed at starting heights of y = 0, and y = 3 mm above center. This test becomes useful in experimentally determining the unknown frictional parameters from Eq. (1), Ff and cp . To determine the best combination of Ff and cp , one can calculate the error between modeled and measured results of the discrete data sets of the two ringdown tests above. error =



(Vmeas − Vmod )2 +

0 mmtest



(Vmeas − Vmod )2

(16)

3 mmtest

The logarithm is taken twice in order to facilitate locating the minimum value of error on a contour plot. error as a function of Ff and cp is shown in Fig. 8 and it shows that the best model-measurement agreement in the ringdown test is found with Ff = 0.0007 N, and cp = 0.015 Ns m−1 . Validation of the model is still necessary, and is demonstrated in Section 4. However, a look at the good agreement in Fig. 9 of the ringdown test for the case of a 47  load shows an example of model validation. Modeling with a light load demonstrates that the model accurately simulates the electromagnetic damping force Fem from Eq. (2d) which equals zero in the open circuit case.

Fig. 8. Error between modeled and measured results of the ringdown test for both the y = 0 and y = 3 mm above center starting positions, as model friction parameters vary. The minimum error designates the optimal Ff and cp to use in the model.

Fig. 9. Modeled and measured ringdown tests conducting at a starting height of y = 0 mm above center. 47  load. Significant electromagnetic damping from Fem .

3.5. Optimization techniques: coil and harvester dimensions and placement With a versatile model in place, a few important optimization simulations can be produced. These simulations provide an important insight in what direction to take towards an optimal design, while reducing the need for excessive prototyping. The following simulations rely on a couple important parameters that are both modeled and measured, and are shown in Section 4. These include the optimum load for power transfer, 1700 , and the optimum excitation frequency, 8.2 Hz. It should also be noted that while the model is capable of simulating with a pure sinusoidal external excitation, better agreement was observed by inputting the accelerometer waveform from the shaker that was used to conduct measurements on the presented harvester. These simulations therefore use the accelerometer data from said shaker as excitation input. The following examples show some of the optimization capabilities that the model can realize. Using the presented harvester’s dimensions and perimeter magnet count of 12, a simulation to optimize the diameter and location of the coil can be generated. This simulation assumes the number of turns in the coil remains 1200, and that the coil is only moved up or down, along the y axis, on the harvester casing. Additionally, this optimization is while the harvester is under an 8.2 Hz, 0.2 g excitation. Different optimizations may be reached for different excitations. The simulation from Fig. 10 shows that optimizing the design would involve increasing the coil diameter as well as raising the

Fig. 10. Power delivered to a 1700  load as a function of coil radius and location on the harvester casing. Presented harvester is identified and generates 101 ␮W.

M. Gutierrez et al. / Sensors and Actuators A 236 (2015) 1–10

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Fig. 11. Power output to a 1700  load as a function of radius to perimeter magnets, and number of perimeter magnets, for external excitation of frequency (a) 8.2 Hz and (b) 9.0 Hz. The presented harvester has 12 perimeter magnets at a radius of 22 mm. The presented harvester is clearly much more optimized for 8.2 Hz than 9.0 Hz excitations.

location of the coil on the casing. Optimal power output is modeled to be over 120 ␮W. If the coil is instead held constant, so that its parameters are as those of the presented harvester, then a simulation to optimize the dimensions of the harvester, and the number of perimeter magnets can be generated. This is shown for an excitation frequency of 8.2 Hz in Fig. 11(a). This simulation is more so used to tune the harvester to a specific frequency. As such, Fig. 11(b) shows what parameters would instead be optimal if a frequency of 9.0 Hz was desired, instead of 8.2 Hz. This analysis depends on using a constant load, so the optimal load for the presented harvester (1700 ) is used. If say an energy harvester tuned to 9 Hz were to be built using this tool, then the optimum load would have to be determined again. The model can be further configured to optimize more than two parameters at a time, however, figures can’t clearly demonstrate this. Mainly, the optimization techniques shown are to display the versatile nature of the model, and that an optimization process can be created to design a device that is best suited for a particular purpose. This is done by identifying a few desired characteristics for the device, such as resonant frequency, size constraint, optimum load size, etc. Then, an optimization model can be used to determine the remaining design parameters, such as number of perimeter magnets, radius to perimeter magnets, coil location and size, even center and perimeter magnet strength and mass.

Fig. 12. Test setup for collecting measurements.

4. Measurements Testing was conducted at Purdue University using the test setup shown in Fig. 12. A customizable shaker is fashioned out of a large 22 cm diameter speaker and a function generator. A styrofoam spacer minimizes electromagnetic interference from the speaker coil and magnet. A digital oscilloscope is used to collect voltage waveforms. A few important tests are done using the presented harvester in order to determine its power output, as well as its agreement with the model for model validation. A load sweep test determines the optimum load for maximum power transfer when the harvester is subjected to an external excitation of 8.2 Hz and 9.0 Hz, with a maximum acceleration of 0.1 g. The test, shown in Fig. 13, determines that a load of 1700  is optimum for maximum power transfer. It also shows that 8.2 Hz excitation near the modeled resonant frequency does indeed outperform the 9.0 Hz excitation. Also, the agreement in both scenarios shows that the model is valid for this variation in external excitation. A frequency sweep determines what frequency excitation leads to the greatest power delivery to the load. This test is done at

Fig. 13. Power delivered to varying load sizes, under constant excitation. Used to determine an optimum load of 1700  for maximum power transfer.

0.1 g and 0.2 g maximum acceleration to introduce measurement variation. The test, shown in Fig. 13, shows that that the optimum frequency, 8.2 Hz, matches the resonant frequency, 8.15 Hz, predicted by the model as shown in Fig. 6(a). This test also serves to produce the power rating for the device. This is stated to be 41.0 ␮W and 101 ␮W, at input excitations of 8.2 Hz with 0.1 g and 0.2 g accelerations, respectively.

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Fig. 14. (a) Power transfer to 1700  load as a function of frequency, at 0.1 g and 0.2 g maximum acceleration. (b) Power transfer to 1700  load as a function of maximum excitation acceleration, at 8.2 Hz frequency.

Fig. 15. (a) Convention for rotation of device. As pictured is defined as 0 degree rotation. (b) Modeled periodic change in resonant frequency due to harvester rotation. With 12 evenly spaced, symmetric perimeter magnets, harvester response repeats every 30◦ of rotation.

Lastly, an acceleration sweep test is shown in Fig. 14(b). This test reveals the predictably positive correlation between maximum excitation acceleration and power output. The frequency and acceleration sweeps again serve to validate the model by demonstrating good model-measurement agreement, though the model does slightly overestimate power output in these tests. However, this validation is necessary in order to use the model for design optimization as discussed in Section 3.5. It is worth demonstrating the device’s rotation independence to power output, which was stated in the Introduction. While the harvester exhibits nearly uniform response as a function of rotation in the x–y plane, the discrete nature of the perimeter magnets results in a periodic change in response as the harvester is rotated. Due to the symmetric arrangement of the perimeter magnets, the response is repeated periodically every 360/n◦ of rotation, for n equally spaced and symmetrically positioned perimeter magnets. For the 12 perimeter magnet configuration presented, the response of the harvester is periodic every 30◦ of rotation. The resonant frequencies for a full rotation of the harvester are shown in Fig. 15. It is apparent that the variation of resonant frequency based on angle of rotation is minimal and only about 0.2 Hz, or about 2.5%. The measured power output can also be shown to remain fairly constant as a function of device rotation. This is shown in Fig. 16, where the device is rotated in increments of 30◦ and power output is measured. It can be observed that power output is greater when

Fig. 16. Power transfer to 1700  load as a function of device rotation.

a perimeter magnet is at the bottom, vs a gap in perimeter magnets, as shown in Fig. 15(a). This can likely be attributed to the fact that the resonant frequency increases slightly by adding 30◦ of rotation, and the new resonant frequency is higher than the excitation frequency of 8.2 Hz. The consistency of the device as it is rotated is still demonstrated, with power varying by about 7 ␮W, or about 17%.

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5. Conclusion The energy harvester presented is capable of harvesting vibrational movements that are of relatively low frequency, on the order of human walking gait frequencies of 2–3 Hz. The frequency is low enough to be relevant to the dominant harmonic frequencies of the walking gait. In this way, the energy harvester can be worn by a person to generate power from walking, or other personal movement. The device presented does not suffer from performance degradation as the device is rotated, so long as the device remains upright. This makes the device the first energy harvester optimized for low power, low frequency, and low g applications to be capable of harvesting motion along two dimensions of motion, and to not suffer from decreased performance when rotated. The device presented is modeled using a MATLAB/Simulink model. This was used to provide a tool to optimize the energy harvester design and tune it to specific uses. Good agreement between the model and experimental device measurements validates the model. An optimum load of 1700  was determined and used for all tests. Testing on the device included ringdown open circuit testing to determine frictional values for the model, frequency sweep testing to determine optimum excitation frequencies, acceleration sweep testing, and rotation testing to demonstrate the device’s imperviousness to rotation. Maximum power to the optimum load was 41.0 ␮W with an external excitation of frequency 8.2 Hz and maximum acceleration of 0.1 g, and was 101 ␮W with an excitation of 8.2 Hz, 0.2 g. Acknowledgements The authors are grateful to Dr. Nithin Raghunathan and Dr. Sean Scott for their valuable suggestions and ideas that supported this work.

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Biographies

Manuel Gutierrez, Purdue University, West Lafayette, Indiana 47906, USA, Manuel received his B.S. in electrical engineering from Purdue University, West Lafeyette, IN in 2015. He became involved in energy harvester research in 2013 through the Purdue Summer Undergraduate Research Fellowship, where his focus was on prototyping, modeling, and optimization of energy harvesters, and was invited to present at the Semiconductor Research Corporation’s TECHCON conference in Austin, TX. He will pursue an SM/PhD at the Massachusetts Institute of Technology, Cambridge, MA. Amir Shahidi, Ford Motor Company, Dearborn, MI 48124, USA, Amir received both his B.S. and M.S. in electrical and computer engineering from Purdue University, West Lafayette, IN in 2011 and 2014, respectively. Graduate research focused on condition monitoring sensors for mechanical bearings as well as energy harvesting for wearable electronics applications. He currently works for Ford Motor Company’s hybrid department, centered on hybrid and electric vehicle’s inverters and electric motors.

David Berdy, Qualcomm Technologies, Inc., San Diego, California 92121, USA, David received his B.S. degree in computer engineering from Rose-Hulman Institute of Technology, Terre Haute, IN, in 2008 and his Ph.D. in Electrical Engineering from Purdue University, West Lafayette, IN in 2013. In 2012, David received the IEEE Ultrasonics, Ferroelectrics and Frequency Control Society’s outstanding paper award. He is currently a Senior Engineer at Qualcomm Technologies, Inc. His research interests include energy harvesting, MEMS transducers, RF filters, wireless front-end architectures, wireless systems and their applications.

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Dimitrios Peroulis, Purdue University, West Lafayette, IN 47906, USA, Dimitrios Peroulis received his Ph.D. in Electrical Engineering from the University of Michigan at Ann Arbor in 2003. He has been with Purdue University since August 2003 where he is currently leading a group of graduate students on a variety of research projects in the areas of RF MEMS, sensing and power harvesting applications as well as RFID sensors for the health monitoring of sensitive equipment. He has been a PI or a co-PI in numerous projects funded by government agencies and industry in these areas. He is currently a key contributor in two DARPA projects at Purdue focusing on (1) very high quality (Q>1000) RF tunable filters in mobile form factors (DARPA Analog Spectral Processing Program, Phases I, II and III) and on (2) developing comprehensive characterization methods and models for

understanding the viscoelasticity/creep phenomena in high-power RF MEMS devices (DARPA M/NEMS S&T Fundamentals Program, Phases I and II). Furthermore, he is leading the experimental program on the Center for the Prediction of Reliability, Integrity and Survivability of Microsystems (PRISM) funded by the National Nuclear Security Administration. In addition, he is heading the development of the MEMS technology in a U.S. Navy project (Marines) funded under the Technology Insertion Program for Savings (TIPS) program focused on harsh-environment wireless micro-sensors for the health monitoring of aircraft engines. He has over 130 refereed journal and conference publications in the areas of microwave integrated circuits, sensors and antennas. He received the National Science Foundation CAREER award in 2008. His students have received numerous student paper awards and other student research-based scholarships. He is a Purdue University Faculty Scholar and has also received eight teaching awards including the 2010 HKN C. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B. Murphy award, which is Purdue University’s highest undergraduate teaching honor.