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Sensors and Actuators A 205 (2014) 103– 110

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical

jo ur nal homepage: www.elsev ier .com/ locate /sna

nalysis and fabrication steps for a 3D-pyramidal high density coillectromagnetic micro-generator for energy harvesting applications

ayra Peraltaa,∗, José L. Costa-Krämerb, Ernesto Medinaa, Arnaldo Donosoa

Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC), 1020A Caracas, VenezuelaInstituto de Microelectrónica de Madrid, CSIC, Tres Cantos, 28760 Madrid, Spain

r t i c l e i n f o

rticle history:eceived 9 April 2013eceived in revised form 7 September 2013ccepted 15 October 2013vailable online 6 November 2013

eywords:lectromagnetic micro-generatornergy harvestingEMS

a b s t r a c t

In the field of energy harvesting, recycling the energy of mechanical vibrations from the environmentbecomes an attractive option to use in low power applications. With this goal, mechanisms of transduc-tion have been developed with electrostatic, piezoelectric and electromagnetic principles of operation.In this work we present the analysis for an electromagnetic micro-generator using the linear generatorapproach combined with the dependence of the electromotive induced force (EMF) with geometricalparameters such as the radii of the coil and the magnet, the distance between them, and the ”slope” ofthe coil, a parameter representing the reduction of radii as a function of each turn of the helicoidal wind-ing. The latter feature introduces the possibility of pyramidal coils instead of solenoidal coils. We showthat when the distance between the coil and the magnet is of the order of the radius of the magnet, the

icro-coilithography

pyramidal geometry presents a higher induced electromotive force than a solenoidal coil of comparabledimensions. We also illustrate a method for the construction of a micro-electromagnetic generator whichcontemplates the fabrication of pyramidal coils with tridimensional electron beam lithography. Results forresonant structures and microcoils are discussed. The estimated induced EMF and power were calcu-lated for a pyramidal high density coil generator, showing a better performance for the pyramidal whencompared with the solenoidal geometry.

. Introduction

With the emergence of wireless technology and wirelessetworks, research on energy sources to supply them is critical1–3]. Despite that batteries being currently the most commonnergy source, they have limited lifetime and require constanteplacement, which can be a complicated and expensive task forarge wireless networks. Besides, used batteries are a source ofnvironmental pollution. As an alternative, devices that can takedvantage of environmental waste energy in the form of light, heatradients, and mechanical vibrations can be used to harvest usefulnergy. The field that studies those ways to reuse energy and theirssociated devices is called energy harvesting [4,5].

There are three well known mechanisms of transduction to con-ert mechanical vibrations into voltage: piezoelectric, electrostatic

nd electromagnetic [6–8]. The electromagnetic generator in theicro-scale offers lower power than the other two mechanisms,

enerally between nW and �W [5,9]. However, with the decrease

∗ Corresponding author. Tel.: +58 2125041391; fax: +58 2125041391.E-mail addresses: mayrafisucv@gmail.com (M. Peralta),

ramer@imm.cnm.csic.es (J.L. Costa-Krämer), ernestomed@gmail.com (E. Medina),rnaldoivic2008@gmail.com (A. Donoso).

924-4247/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.sna.2013.10.012

© 2013 Elsevier B.V. All rights reserved.

of power consumption of the wireless devices, it is still an interest-ing alternative, due to its ease of integration with micro-fabricationtechniques and its simple principle of operation [10,11].

One of the most important challenges in the design of electro-magnetic generators is miniaturization, given the important effectsof scaling [12,13] and due to the difficulty in the constructionof micrometric coils [14]. An example of the possible applicationof the electromagnetic generator at the micrometric scale is inthe powering of active sensors networks for building monitoring,environment control, etc. [2,3,5,13]. The parameters to design andoptimize these devices are often extracted from the linear gener-ator configuration [10,11,15] taking as the relevant ones the massof the generator and its characteristic length, in order to resonatewith a dominant frequency range of the environmental source ofvibrations; the mechanical and electrical damping ratios; and thegeometry of the magnetic field [16,17]. In this work we present ananalysis for a micro-electromagnetic generator based on a 3D pyra-midal high density coil with a single magnet configuration. For thisconfiguration we introduce a new design parameter which is theslope p of the coil and discuss also the dependence of the electro-

motive induced force on parameters such as the relation betweenthe sizes of the magnet and the coil; the distance between them; thespacing between turns in the coil and the number of turns. We alsopresent a fully MEMS compatible methodology for the construction

104 M. Peralta et al. / Sensors and Actuators A 205 (2014) 103– 110

Fig. 1. (a) Scheme of the proposed electromagnetic generator, the magnetic cylinder of radius r and magnetization M is mounted on the resonant structure, the distance zi umpe[ es aret

odt

2

ccvtecis

m

wdbmitrttib

Z

woo�grss

P

s measured from the center of the magnet to the center of the coil of radius a. (b) L15]. All the mass of the resonant structure is described by m and its elastic propertihe system respectively.

f the electromagnetic generator that allows for control over theesign parameters introduced; easy assemblage and applicabilityo large substrates adequate for serial production.

. Theoretical basis for electromagnetic generators

The configuration we used for an electromagnetic generatoronsists of a magnet mounted on a resonant structure near a fixedoil (see Fig. 1(a)). When the device is coupled with a source ofibrations, there will be a relative motion between the magnet andhe coil, and therefore by the Faraday’s law, there will be an inducedlectromotive force. Taking the lumped model shown in Fig. 1(b),onsisting of a spring mass system coupled to an external oscillat-ng force, the equation that governs the motion of the oscillatingtructure can be written [15] as

Z + (be + bm)Z + kZ = −my, (1)

here m is the mass of the resonant structure; Z is the verticalisplacement of the mass with respect to its equilibrium position;T = be + bm is the total damping coefficient given by the sum of theechanical damping produced mostly by the medium surround-

ng the mass and the electrical damping that takes into accounthe energy transformed into electricity and the dissipation by theesistance of the electrical components. The parameter k involveshe elastic properties of the resonant structure and y is the accelera-ion of the vibrating source. Taking y(t) = Y0 sin(ωt) and substitutingn Eq. (1), the time dependent position of the resonant structure cane obtained as

(t) = ωY0 sin(ωt + ı)

[(2ω0�T )2 + (ω20 − ω2)

2/ω2]

1/2, (2)

here ω and ω0 are, respectively, the frequency of oscillationsf the external force and the fundamental frequency of the res-nant structure; Y0 is the amplitude of the external oscillations;T = �m + �e = bT/(2mω0) is the total damping ratio of the systemiven by the sum of the mechanical and the electrical dampingatios. Using the Eq. (2) to obtain the velocity of the oscillatingtructure and assuming that it is in resonance with the externalource of vibrations (ω = ω0), the electrical power is

e = beZ2 = 2mω�eZ2 = m�eω3Y20

4�2T

. (3)

d model that describes the linear conversion from mechanical energy to electricity in the spring of constant k. be and bm are the electrical and mechanical damping of

From this equation two independent design parameters areobtained. For optimization of the electrical power, the followingrestrictions are imposed on the parameters

• The electrical damping �e, has to be equal to the mechanicaldamping and as small as possible [18]. This factor includes param-eters like the load and coil resistance.• The mass m, which involves the volume and the material of the

generator. Those parameters have to be calculated in order totune to the characteristic frequency of the external source, whichin the case of environmental vibrations is typically between 1 and200 Hz [18].

3. Magnets and coils

To analyze the dependence of the induced electromotive forceon the dimensions of the coil and the magnet, and the distancebetween them, the magnetic field produced by a cylinder of radiusr, length l and uniform magnetization M is calculated (see Fig. 1(a)).With the components of the magnetic field Bz(r, l, �, z) and B�(r, l,�, z), the magnetic flux through a coil of radius a, at a distancez from the center of the magnet is calculated as �m(r, l, a, z) =∫ 2�

0

∫ a

0�B(r, l, �, z) · d�A. In order to simplify the expressions pre-

sented in this article, we write the magnetic flux in terms of themagnetic permeability of the vacuum �0; the modulus of the mag-netization of the magnet M; and a function F(r, l, a, z) of thedimensions of the coil, the magnet and the distance between themas �m(r, l, a, z) = (��0M)F(r, l, a, z). The function F(r, l, a, z) is a sum ofseveral algebraic and logarithmic terms, we present the derivationmethod in Appendix.

We calculate the induced electromotive force (open circuitvoltage) per turn computing the magnetic flux for a time vary-ing distance between the coil and the magnet h(t) = z − (l/2) + Z(t),where Z(t) is extracted from Eq. (2) and z − (l/2) is the distance atrest between the coil and the bottom of the magnet. We further con-sider the system to be at resonance and use the small oscillationslimit (z � Z(t)). For the root mean square average we obtain

|ε | = �0Mω0Y0�√ ∂F(r, l, a, z). (4)

RMS

2�T 2 ∂z

The electrical power can be written in terms of the function F(r, l,a, z), using the equation Pe = i2RT, where i is the electrical current

M. Peralta et al. / Sensors and Actuators A 205 (2014) 103– 110 105

Fig. 2. Electromotive force |εRMS| per turn as a function of (a) the distance z between the magnet and the coil for several magnet and coil dimensions r = l = a = 100, 200, 300and 400 �m, identified with the dashed thick, continuous thin, continuous thick, and dashed thin lines respectively; (b) the coil radius a for several distances between themagnet and the coil z = 1100, 1600, 2100 and 2600 �m, identified with the dashed thick, continuous thin, continuous thick, and dashed thin lines respectively; and (c) thecoil and magnet radius a and r.

F The ds and p

ic

P

FvwgTtf

pzt

ig. 3. (a) 3D schematic representation of the system including the coil geometry.hows the parameters of the pyramidal coil, where d is the distance between turns

nduced in the coil and RT = Rc + RL is the combined resistance of theoil Rc, and the load RL. We obtain

e = (�0Mω0Y0�)2

(2�T

√2)

2(Rc + RL)

(∂F(r, l, a, z)

∂z

)2

. (5)

or numerical calculations, we take the frequency of the source ofibrations to be the typical for a car engine, as reported in Ref. [18],ith f=200 Hz with an amplitude of 7.6 �m. We also assume the

enerator dimensions to be tuned up to resonate at this frequency.he mechanical damping ratio used is �m = 0.49 [19] and the elec-rical damping ratio is �e = �m. The magnetic field is that producedor a cylindric magnetic bar of nickel.

The dependence of the calculated EMF per turn on variousarameters is presented in Fig. 2. The EMF decreases rapidly with, the distance between the magnet and the coil. For this calcula-ion we keep the aspect ratio of the system by setting r = l = a = 100,

istance z and the radius a depend on the turn of the coil considered. (b) The panel is the slope of the coil, when p = 0 we have the solenoidal case.

200, 300 and 400 �m. As shown in Fig. 2(a). In Fig. 2(b) we showthe calculated EMF for a fixed magnet size with r = l = 400�m as afunction of a, the radius of the coil. In this case the EMF increaseswith a up to a maximum and then goes to zero at a rate depend-ing on the coil-magnet distance, z. The value of a for a maximumEMF increases with the coil-magnet distance as well, as shown inFig. 2(b) for different values of z. The EMF also increases for largervalues of the magnet radius, r.

The geometry of the coil is also an important factor in the opti-mization of the electromotive induced force. We consider three coilgeometries: solenoidal, pyramidal and planar coils, which includethe most common micro-coils reported in the literature. Depictionsrepresentative of the generator including the coil geometry and its

parameters are shown in Fig. 3(a) and (b). For the pyramidal coilthe radius of each turn ai depends linearly with the distance zibetween the magnet and the considered turn. This case includesthe solenoidal coil when p = 0, where p is the slope of the line.

106 M. Peralta et al. / Sensors and Actuators A 205 (2014) 103– 110

Fig. 4. Plots of the induced electromotive force |ERMS| as a function of (a) The slope p, for several distances between the magnet and the coil z = 300, 400 and 500 �m identifiedwith the continuous thin, continuous thick, and dashed thin lines respectively. (b) The distance between turns d for several slopes p = 0, 1.64, and 10; identified with thecontinuous thin, continuous thick, and dashed thin lines respectively; and a distance between the magnet and the coil of z = 110 �m and (c) z = 300 �m.

Fig. 5. (a) Plot of the coil radius a as a function of the distance between the magnet and the coil, z, for several magnet sizes and coil initial radius r = l = 200, 400, 600, and8 shed

f ar (d(

Tc

|

a

|

00 �m identified with squared and triangles. The continuous thin, dashed thin, daorce |ERMS| as a function of the number of turns N for several coil geometries: plancontinuous thick line); and a distance of z = 300�m and (c) z = 110 �m.

he induced electromotive force for a particular coil geometry isalculated as the sum over turns as shown in Eqs. (6) and (7):

ERMS |pyr = �0Mω0Y0�

2�T

√2

N∑i=0

[∂F(r, l, a, z)

∂z

]zi

, (6)

nd

ERMS |pl =�0Mω0Y0�

2�T

√2

N∑i=0

[∂F(r, l, a, z)

∂z

]ai

. (7)

thick, and continuous thick lines represent linear fits. (b) Plot of the electromotiveashed thick line), p = 0 (continuous thin line), p = 10 (dashed thin line) and p = 1.64

For the pyramidal coil, the induced electromotive force |ERMS|pyr, isa sum over the number of turns placed at a distance zi from the mag-net. The variation of the coil’s radius is taken into account throughthe linear relation with the distance z between the coil and themagnet. For a given magnet size the EMF depends on the slope ofthe line p, the vertical distance between turns d, the distance z, andthe number of turns N. In the case of the planar coil, the induced

electromotive force |ERMS|pl, is a sum over the radius ai for eachturn, and depends of the horizontal distance d between turns, thenumber of turns N, and the distance z, for a given magnet size. Forsimplicity, we neglect the wire thickness.

M. Peralta et al. / Sensors and Actuators A 205 (2014) 103– 110 107

Fig. 6. Proposed procedure for the manufacture of the electromagnetic generator. Two equal substrates are used for the fabrication of the vibrating magnetic structure (Subs.1) and the coil (Subs. 2). The procedure starts with UV lithography in both cases (step 1), the masks are designed in order to create structures that fit one over the other.

F ) Subt

tetfii

ig. 7. (a) Substrate with membranes of different areas having 240nm thickness. (bhickness.

The induced electromotive force was computed as a function ofhe geometrical parameters of the coil looking for the maximumfficiency, which corresponds to maximum change of flux through

he coil. The behavior of the EMF with the parameters extractedrom the geometry of the coil is presented in Figs. 4 and 5. Theres a value for the slope p = popt for which the EMF reaches a max-mum. For distances between the coil and the magnet such that

strate with plateau structures of approximately 1.2 mm×1.2 mm area and 350 �m

z − (l/2) � r (where l is the length of the magnet), the maximum isreached at p ≈ 0, but when z − (l/2) ≈ r then the EMF reaches a max-imum at p > 0. For this calculation we take z = 300, 400 and 500 �m

and r = l = a0 = 200 �m. As shown in Fig. 4(a). This dependence is alsoshown in Fig. 5(a) where is plotted the radius of the coil (for whichthe EMF reaches a maximum) as a function of the distance betweenthe center of the coil and the magnet z, for several aspect ratios of

108 M. Peralta et al. / Sensors and Actuators A 205 (2014) 103– 110

Fig. 8. (a) Optical microscope image of a 200 �m×200 �m×100nm magnetic membrane

case is a nickel square deposited by e-beam evaporation, of 20 �m × 20 �m and 50 nm thfabricated on the top with 3D e-beam lithography. (c) SEM image of the plateau showing

Table 1Geometrical parameters for the optimization of the induced electromotive forceconsidering the slope p of the coil, the coil-magnet gap z − (l/2), and the length ofthe coil N × d.

Parameter z − (l/2) and N × d � r, l z − (l/2) and N × d ≈ r, lMagnet’s radius r ≈a ≈a

tiEitiiwwdvEtscpauqf

ttN

TQpntt

Coil’s radius a ≈r a0 ≈ r and ai linearly dep. of zSlope p ≈0 = popt

he magnet r = l = 200, 400, 600, and 800 �m. In Fig. 5(a) we alsonclude fits which relate linearly the radius of the coil (for which theMF reaches a maximum) with the coil-magnet distance. This plotndicates that the efficiency of the pyramidal coil increases abovehat of the solenoidal configuration when the distance z − (l/2) isncreased. The EMF decreases as the distance between turns d isncreased at a rate which is almost the same for all coil slopes

hen the coil-magnet distance is z = 110 �m (see Fig. 4(b)), buthen this distance is increased as in Fig. 4(c) for z = 300 �m, theecay is slower for the value of the slope at which the maximumalue of the EMF is reached in Fig. 4(a). The dependence of theMF with the number of turns N is shown in Fig. 5(b) and (c). Inhe case z − (l/2) � r, the slopes p = 0 and p /= 0 present almost theame behavior, but when z − (l/2) ≈ r the slope p /= 0 is more effi-ient than p = 0 when p = popt. Finally, for all the above cases, thelanar coil presents the highest open circuit voltage (see Fig. 5(b)nd (c)). This kind of geometry displays a very low inductance, lowniformity of the magnetic field, high parasitic capacitance, and lowuality factor, therefore, we only consider the pyramidal geometryor further analysis.

Table 1 summarizes the analysis of the parameters presented in

his paper, considering not only the gap between the magnet andhe coil z − (l/2), but also the vertical length of the coil given by

× d.

able 2ualitative comparison between the geometries with slope p = 0 (solenoidal) and

/= 0 (pyramidal). The parameters used in this table are: the gap between the mag-et and the coil z − (l/2), the compatibility with the micro-fabrication techniques,he density of turns (d is the separation between turns), and the the magnet-coil gapaking into account the density of turns.

Geometry p = 0 p /= 0

EMF for z − (l/2) � r |ERMS|pyr|p=0 = |ERMS|pyr|p /= 0 ←−EMF for z − (l/2) ≈ r |ERMS|pyr|p=0 < |ERMS|pyr|p /= 0 ←−MEMS compatibility Bad GoodDensity of turns 1/(d ≥ 25 �m) [14] 1/(d ≥ 1 �m) hereEMF for z − (l/2) ≈ r taking

the density of turns|ERMS|pyr|p=0� |ERMS|pyr|p /= 0 ←−

(see Table 3)

fabricated with the procedure explained in the previous section. The magnet in thisick. (b) SEM image of the plateau like structure with a 10 turns pyramidal structure

the height of the structure. (d) SEM image of a 10 turn pyramidal structure.

Considering the above analysis, we construct Table 2, where wecompare qualitatively the geometries p = 0 (solenoidal) and p /= 0(we discarded the planar geometry for reasons explained above).

4. Methods of fabrication

The procedure presented in this work for the construction ofan electromagnetic generator, starts with two silicon substratescoated with two thin films of Si3N4, one for each face. The magneticresonant structure is constructed in substrate 1 and the coil in sub-strate 2. We propose a four step fabrication process, illustrated inFig. 6 and explained in more detail below:

• Optical lithography: In both cases (Subs. 1 and 2) the first stepconsists of UV lithography to expose a square area of Si3N4.• Reactive Ion Beam Etching (RIBE): The area of Si3N4 exposed in

the first step is etched with a beam of ions of N2 and CHF3.• Etching of Si and magnet and coil fabrication: In this step the

silicon is etched through the depth of the substrate, as is shownin the third step of Fig. 6. The result is a membrane in the case ofSubs. 1 and a plateau like structure in the case of Subs. 2. Finally,the magnet can be fabricated over the membrane using opticallithography and depositing a magnetic material. Over the plateau,a pyramidal coil is fabricated using 3D Electron Beam Lithographytechnique [20] (details of the fabrication of micro-coils with 3De-beam lithography will appear elsewhere).• Packaging: The last step consists in covering the base of Subs. 2

with resin in order to place over Subs. 1, as is shown in Fig. 6.

An advantage of the process explained here is that it allows con-trol of all the geometrical parameters considered in the previoussection: the size of the magnet can be controlled by changing thesize of the mask, the size of the coil and its geometrical parameterscan be controlled through the pattern and dose applied during thee-beam lithography. The distance between the coil and the magnetcan be varied with the thickness of silicon removed in the KOHetching of Subs. 2 (see Fig. 6). Another benefit, which is crucialfor applications of this kind, is the simple scalability of the pro-cess for serial production of the generators. The techniques usedare reproducible, fully compatible with MEMS and easily scalableto large substrates, because the elements (coils and magnets) aremicromachined in a way that facilitates assembly.

Fig. 8 shows results of magnetic membranes, pyramidalstructures constructed by 3D e-beam lithography, and plateau

structures, successfully fabricated with the procedure presentedin this work. Fig. 7 shows membranes and plateau structuresfabricated in large substrates, illustrating the potential of themethod for be applied for large substrates. An experiment for

M. Peralta et al. / Sensors and Actua

Table 3Theoretical estimations of the EMF and the power for the geometries with slopesp = 0 (solenoidal) and p /= 0. popt is the value of the slope for which the electromo-tive force (EMF) reaches a maximum in Fig. 4(a), N is the number of turns of thecoil, d is the distance between them and Pe is the electrical power. For the casesz − (l/2) = 10 �m, and z − (l/2) = 250 �m, the first two rows were constructed assum-ing the same separation between turns, d, for both coils, in the third row the typicald for the solenoidal coil was used. The last column corresponds to the improve-ment of the pyramidal coil with respect to the solenoidal for the two densities ofthe solenoidal coil considered.

Pe EMF % of improvement

Geometry z − (l/2) = 10 � m � rPyramidal

popt = 0.47, N = 1000, d = 1 �m 6.6 �W 181.2 mV 0.5% and 89%Solenoidal

p = 0, N = 1000, d = 1 �m 6.5 �W 180.3 mVSolenoidal

p = 0, N = 1000, d = 25 �m 81.1 nW 20.1 mV

Geometry z − (l/2) = 250 �m≈rPyramidal

popt = 1.04, N = 1000, d = 1 �m 5.8 nW 5.5 mV 20% and 96%Solenoidal

tgoawttrttawWgs

Frf

p = 0, N = 1000, d = 1 �m 3.9 nW 4.4 mVSolenoidal

p = 0, N = 1000, d = 25 �m 7.9 pW 198.2 �V

he fabrication and testing of the pyramidal coil electromagneticenerator is currently under development. Taking the parametersf r = l = 500 �m, a0 = 500 �m, RT = Rc + RL = 5000, z − (l/2) = 10 �m,nd z − (l/2) = 250 �m, the expected maximum power and voltage,here calculated for coils with p = 0 and p = popt using the equa-

ions Eqs. (5) and (6), (all the calculations have been done takinghe optimal values and the restrictions explained in this paper). Theesults are presented in Table 3. We consider two cases for p = 0,he case in which the spacing between turns is d = 1 �m (the samehat for p = popt), and a more realistic case d = 25 �m, that takes intoccount the minimum spacing between turns that can be achieved

ith the techniques developed until now (see for example [14]).ith these estimates we show a scenario in which the pyramidal

eometry, p /= 0, becomes an attractive option due to its high den-ity of turns compared with the density experimentally achieved

ig. A.9. (a) Coordinate system used to calculate the magnetic field. (b) Coordinate Bz of = l = 500�m. (c) Coordinate Bz of the magnetic field as a function of the � coordinate for zunction of the z coordinate for � = 500�m for a magnet of r = l = 500�m.

tors A 205 (2014) 103– 110 109

with the solenoidal geometry, showing an increment in electromo-tive force of 89% and 96% depending on the gap considered. Also wecan see that for the case of the gap z − (l/2) = 250�m ≈ r, the pyrami-dal coil shows an increment of the EMF of 20% with respect to thesolenoidal case, when the same density of turns have been takenfor the two geometries.

5. Conclusions

An analysis of an electromagnetic generator including the geom-etry of the coil has been presented in this work. For the case inwhich the gap z − (l/2), between the magnet and the coil and thevertical length of the coil is of the order of the magnet’s radius, ithas been found that the pyramidal coil represents the best option,and the parameter of the slope, p, of the coil has been included inthe analysis.

A procedure fully compatible with MEMS for the fabrication ofelectromagnetic generators considering the parameters studied inthis work has been illustrated, showing results for resonant struc-tures of Si3N4 from 200 �m×200 �m to ≈2 mm × 2 mm and 10turns pyramidal structures fabricated with 3D Electron Beam Lithog-raphy. The induced electromotive force and power estimation wereobtained for the pyramidal p /= 0 and solenoidal geometries, show-ing a favorable scenario if we consider the high density of turns forthe pyramidal geometry due to its high compatibility with MEMStechniques.

Acknowledgements

We would like to thank Dr. Iván Fernández, Raquel Alvaro,Francisco Espinoza and Carmen Robles from the Instituto de Micro-electrónica de Madrid, for providing access to EBL, RIBE and e-beamevaporation facilities.

Appendix A. Derivation of the function F(r, l, a, z)

The magnetic field outside of a cylindrical bar magnet ofradius r, length l and uniform magnetization M (see Fig. A.9(a))

the magnetic field as a function of the � coordinate for z = 260�m for a magnet of = 500�m for a magnet of r = l = 500�m. (d) Coordinate Bz of the magnetic field as a

1 Actua

cV

wdptlfwpapmT

B

dit

[

[

[

[

[

[

[

[

[

[

[20] G. Piaszenski, U. Barth, A. Rudzinski, A. Rampe, A. Fuchs, M. Bender, U. Pla-chetka, 3d structures for uv-nil template fabrication with grayscale e-beamlithography, Microelect. Eng. 84 (2007) 945–948.

[21] M.A. Heald, J.B. Marion, Classical Electromagnetic Radiation, Brooks, Cole,1995.

10 M. Peralta et al. / Sensors and

an be obtained as the gradient of a magnetostatic potentialm [21]:

B(P) = −∇Vm = −�0

4�∇[

∫M · PP3

d3r ′]

= −�0M

4�∇[

∫ r

0

∫ 2�

0

∫ +l/2

−l/2

(z − Z)RdRd�dZ

(�2 + R2 − 2�R cos(� − �) + (z − Z)2)3/2

], (8)

here P is the position of the field point identified with the cylin-rical coordinates �, � and z; and r′ is the position of the sourceoint identified with the coordinates R, � and Z. From this equa-ion, the components of the magnetic field can be obtained as: Bz(r,, �, z) = − ∂Vm/∂z and B�(r, l, �, z) = − ∂Vm/∂�, the exact expressionsor these fields are a combination of algebraic and elliptic functionshich are cumbersome to integrate and analyze. Fig. A.9 showslots of the magnetic field integrated numerically with dashed linesnd a fit for this field with the continuous line. We can see in theselots that the numerically integrated magnetic field is approxi-ately modeled by the fit in the cases presented in this article.

he approximate coordinate Bz that results from the fit is:

z(r, l, �, z) = �0M

2[ −

√(l − 2z)2 + 4(r − �)2

l − 2z

−√

(l + 2z)2 + 4(r − �)2

l + 2z− 4r�

(l − 2z)√

(l − 2z)2 + 4�2

+√

(l − 2z)2 + 4�2

l − 2z− 4r�

(l + 2z)√

(l + 2z)2 + 4�2

+√

(l + 2z)2 + 4�2

l + 2z], (9)

The magnetic flux �m(r, l, a, z) through a coil of radius a at aistance z from the center of the magnet (see Fig. 1(a)), is obtained

ntegrating the component Bz(r, l, �, z), over the area enclosed byhe coil:

�m(r, l, a, z) = �0M

2

∫ a

0

∫ 2�

0

Bz(r, l, �, z)�d�d�

= �0M�[(l2 − 4lz + 4z2 − 6r� + 4�2)

√l2 − 4lz + 4(z2 + �2)

3(l − 2z)

+(l2 + 4lz + 4z2 − 6r� + 4�2)

√l2 + 4lz + 4(z2 + �2)

3(l + 2z)

+

√l2 − 4lz + 4(r2 + z2 − 2r� + �2)(−l2 + 4lz + 2(r2 + r� − 2(z2 + �2)))

3(l − 2z)

−

√l2 + 4lz + 4(r2 + z2 − 2r� + �2)(l2 + 4lz − 2(r2 + r� − 2(z2 + �2)))

3(l + 2z)

+ r(l − 2z) log(2� +√

l2 − 4lz + 4(z2 + �2)) + r(l + 2z) log(2�

+√

l2 + 4lz + 4(z2 + �2)) − r(l − 2z) log(−2r + 2�√

+ 2 − 4lz + 4(r2 + z2 − 2r� + �2)) − r(l + 2z)

× log(−2r + 2� +√

l2 + 4lz + 4(r2 + z2 − 2r� + �2))]a0, (10)

tors A 205 (2014) 103– 110

where � is the azimuthal angle for the coil. The expression betweenbig squared brackets in the above equation, evaluated in the limitsindicated, is the function F(r, l, a, z). It is important to note that theuse of the function F(r, l, a, z), not only facilitates the numerical cal-culations but also the analysis for the design of an electromagneticgenerator.

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