an experimentally validated electromagnetic energy harvesters

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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 2314–2324

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

An experimentally validated electromagnetic energy harvester

Niell G. Elvin a,n, Alex A. Elvin b

a Department of Mechanical Engineering, The City College of New York, Steinman Hall T-228, New York, NY 10031, USAb School of Civil and Environmental Engineering, University of the Witwatersrand, Johannesburg, South Africa

a r t i c l e i n f o

Article history:

Received 10 July 2010

Received in revised form

27 October 2010

Accepted 23 November 2010

Handling Editor: M.P. Cartmellnical energy harvester using a rare-earth magnet shows good agreement with the results

Available online 15 December 2010

0X/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jsv.2010.11.024

esponding author.

ail address: [email protected] (N.G. Elv

a b s t r a c t

A relatively simple method for determining the electromechanical parameters of

electromagnetic energy harvesters are presented in this paper. The optimal power

generated through a load resistor at both off-resonance and resonance is derived

analytically. The experimentally measured performance of a rudimentary electromecha-

from the model. The parasitic generator coil resistance can have a profound effect on the

overall performance of an electromagnetic generator by essentially acting to degrade the

effective coupling coefficient. Data from the setup electromagnetic generator shows

normalized power densities of 1.7 mW/[(m/s2)2 cm3] operating at a resonance frequency

of 112.25 Hz. This power density is comparable with other electromagnetic devices of the

same volume operating at these frequencies. The power output of the presented

electromagnetic generator is comparable to equivalent piezoelectric generators.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, there has been a substantial interest in converting ambient mechanical vibration into electrical energy forpowering small-scale electronic devices such as sensor nodes. Most recent work on so called mechanical energy harvestershas focused on using piezoelectric resonators [1,2]. Other resonator architectures have also been studied (see [3,4] for areview) and include magnetostrictive, capacitive and electromagnetic based systems. A number of factors influence theconversion efficiency of resonator based harvesters including: (a) how well matched the resonator’s natural frequency is tothe dominant frequency of the ambient vibration, (b) degree of electromechanical coupling, (c) the mechanical damping ofthe resonator, and (d) the electrical impedance of the attached circuit.

Piezoelectric resonators provide some advantages over other energy harvesting methods including: few moving parts andrelatively simple circuits that can be used to store the electric energy. However, piezoelectric systems also have a fewdisadvantages, namely that they have relatively low energy conversion efficiency and that the higher energy densitymaterials (such as PZT) are stiff and brittle. High stiffness results in resonators having higher natural frequencies which makesthem difficult to implement in low-frequency vibration environments such as is typically found in large structures such asbridges and buildings [5]. Accurate theoretical models, both analytical for simple circuits [6] and numerical [7] for morecomplex circuits, have been developed for piezoelectric resonators. With only the material properties as input, these modelscan predict the behavior of piezoelectric harvesters to within a few percent.

Electromagnetic harvesting, which is the focus of the present paper, is one of the oldest techniques for energy harvesting[8,9]. When compared with piezoelectric systems the main advantages are: (a) their potentially high conversion efficiency,and (b) simple mechanical resonator structures (such as springs) that can be used for low frequency energy conversion. Themain disadvantages are: (a) extra components (such as the induction coil) which have to be aligned accurately with the

ll rights reserved.

in).

N.G. Elvin, A.A. Elvin / Journal of Sound and Vibration 330 (2011) 2314–2324 2315

magnet/s, and (b) output voltages tend to be low which have to be stepped-up in order to drive most circuits. Furthermore,simplified methods that accurately predict the electromagnetic harvester’s performance are unavailable. The present paperpresents and experimentally verifies some theoretical models for electromagnetic energy harvesting. Recently, there hasbeen significant work in determining the optimal power output from both piezoelectric and electromagnetic harvesters withrelatively simple electronic circuits such as purely: (1) resistive loads, (2) resistive and capacitive loads, and (3) resistive andinductive loads [10–12]. For the most studied case of an optimal resistive load operating under resonant conditions, andassuming very large electromechanical coupling the maximal power output for both electromagnetic and piezoelectricgenerators have been shown to be equivalent. Recently, models for optimal resistive loading and the associated power outputof piezoelectric generators have been extended to include operation at off-resonance conditions and performance over thefull range of electromechanical coupling coefficients [11,12]. In order to compare electromagnetic harvesters to piezoelectricharvesters, it is important to develop equivalent optimal power expressions for electromagnetic generators at off-resonanceconditions and small to large electromechanical coupling coefficients. The case of electromagnetic generators is furthercomplicated because the parasitic resistance of the induction coil in many cases cannot be ignored. In this paper, we deriveanalytical expressions for off-resonance operation which include the full range of electromechanical coupling coefficientsand non-negligible coil resistance. The power conversion of a rudimentary electromagnetic harvester is then presented andverified with experimental measurements. The power conversion of this electromagnetic harvester is compared with somepreviously studied piezoelectric energy harvesters.

2. Electromagnetic basics

A simple electromagnetic generator typically consists of a magnet, a mass M, a suspension spring K, and an induction coil Lc

attached to an electric circuit. One simple arrangement with the magnet acting as the mass is shown in Fig. 1. As the hoststructure vibrates with acceleration Ag, the magnet undergoes vibrational displacement (y) which induces an electromotiveforce (EMF) in the inductor coil. Similar configurations are often used for electromagnetic loudspeakers (and microphones)and the derivations presented in this paper follow basic loudspeaker theory.

Assuming a massless spring, the one degree of freedom force equilibrium gives

M €yþD _yþKyþFL ¼�MAg (1)

where the dots represent derivatives with respect to time, D is the structural damping coefficient, and FL is the verticalcomponents of the electromagnetic (Lorentz) force vector (FL) due to the induced current in coil L given by

FL ¼ I

ZdL � B (2)

where I is the induced current, dL is the differential length vector along the inductor coil, and B is the magnetic field vector.The Kirchoff voltage law through the circuit is given by

Lc_IþRcIþZI¼ E (3)

where Rc is the internal resistance of the inductor coil, Z is the impedance of the circuit and E is the motional EMF given by

E ¼Z

dLUðv� BÞ (4a)

where v is the velocity vector of the magnet.

K

y

B

L

Ag

Circuit

M

Fig. 1. A rudimentary electromagnetic energy harvester configuration. The magnet and mass (M) are the same.

N.G. Elvin, A.A. Elvin / Journal of Sound and Vibration 330 (2011) 2314–23242316

The EMF in Eq. (4a) can also be calculated using Faraday’s Law

E ¼�dR

BUdA

dt(4b)

where the term in the integral is the net magnetic flux (FB=B �dA) through the differential elemental area, dA. In general thecalculation involved in evaluating Eq. (4b) is somewhat more numerically expensive than Eq. (4a).

In the one dimensional case shown in Fig. 1,

v¼ ½0 _y 0�, FL ¼ IðdLzBx�dLxBzÞ, E ¼� _yðdLzBx�dLxBzÞ (5)

For a single loop induction coil j, subjected to an axisymmetric distribution of magnetic field, the Lorentz force and EMFcan be expressed as

FLj ¼ Ið2pRcBRjÞ, Ej ¼� _yð2pRcBRjÞ (6)

where Rc is the radius of the generator coil, and BRj is the radial magnetic field at location y of coil j. For a N turn induction coilthe Lorentz force and EMF are then given by

FL ¼ I2pRc

XN

j ¼ 1

BRj, E ¼� _y2pRc

XN

j ¼ 1

BRj (7)

The radial magnetic field BR needs to be evaluated for each loop of the induction coil. In general there are no closed formsolutions for the near-field of a magnet, and thus the value has to be calculated numerically. A number of standard methodsare available for evaluating the magnetic field (including finite element and boundary element techniques). One of the easiestmethods for calculating relatively simple magnetic fields is using the solenoid approximation for the permanent magnet. Inthis method, the magnet is subdivided into Nm coil loops with radius Rm each carrying a current of

Im ¼Brhm

m0Nm(8)

where Br is the residual magnetic field strength, hm is the height of the magnet (and thus the solenoid model), and m0 is themagnetic permeability of free space.

The magnetic field at any location (for example at the position of the generator coil) can be calculated using the Biot–Savarat law

B¼m0I

4p

ZdLm � r (9)

where the integral is evaluated over the entire length of the model magnetic solenoid, dLm is the differential length of the magneticsolenoid model, and r is the displacement vector from the differential length dLm to the point where the magnetic field is beingcalculated.

Eqs. (1) and (3) can be rewritten using Eq. (7) into a set of coupled differential equations of the form

M €yþD _yþKyþBLðyÞI¼�MAg , Lc_IþRcIþZI�BLðyÞ _y ¼ 0 (10)

where BL(y) is the electromechanical coupling coefficient (from Eq. (7)) and is dependent on the position of the magnet y. For alinearized model, BL(y)=BL is calculated when the spring is undeformed, i.e. when the magnet is in position y=0.

Fig. 2a shows a typical distribution of magnetic field for a magnet with radius, Rm=2.5 mm, height, hm=5 mm, and withresidual magnetization Br=1 T at position r=3 mm from the magnet centerline; the vertical displacement is measured fromthe magnet’s mid-height. Fig. 2b shows the electromechanical coupling coefficient BL(y) for the same magnet with inductorgenerator coil of radius Rg=3 mm, generator coil height hg=10 mm and number of loops N=20.

Fig. 2 shows that the maximal electromechanical coupling BL occurs when one of the ends of the generator coil isapproximately in line with the midline of the magnet.

2.1. Parameter estimation

The mechanical parameters (M and K) in Eq. (10) can be estimated using standard techniques such as density calculationsfor the mass and static stiffness calculations for the spring. The damping D is usually more difficult to estimate and needs to beexperimentally measured. The ground acceleration Ag is usually taken from measured data. The inductance of the generatorcoil Lc can be estimated from [13]

Lc ¼m0pR2Nc

2

hck (11a)

where R is the radius of the inductor, Nc is the number of turns in the inductor, hc is the height of the coil and k is a geometricfactor which depends on the number of layers in the coil and the radius to length ratio. For example, when all coil dimensionsare in meters, the Wheeler approximation for the inductance of a single layer coil (in Henrys) is given by

k¼1

0:0254m0pð10þ9R=hcÞ(11b)

-0.5 0 0.5

-10.0

-5.0

0

5.0

10.0

-0.05 0 0.05BR (T) BL (T.m)

y (m

m)

y c (m

m)

Fig. 2. (a) Radial magnetic field BR at a radius of r=3 mm from the magnet’s centroid versus axial position y and (b) the electromechanical coupling

coefficient BL for various positions of a generator coil with centroid at yc. The midline position of the magnet is at y=0, the radius of the magnet is Rm=2.5 mm,

the height of the magnet is hm=5 mm and the residual magnetization Br=1 T; the inductor generator coil radius is Rg=3 mm, generator coil height hg=10 mm

and the number of loops N=20.


For an inductor coil with multiple layers, the Wheeler approximation gives

k¼0:8

0:0254m0pð9þ6R=hcþ10Dc=hcÞ(11c)

where R is now the mean diameter of the coil, i.e. half way between the outer and inner diameter, and Dc is the radial thicknessof the coil.

The inductor coil’s resistance Rc can be estimated directly from the resistivity of the coil material, rw, the cross-sectionalarea of the coil wire, Aw, and the length of the wire, Lw

Rc ¼rwLw

Aw(12a)

For a purely resistive circuit impedance, i.e. Z=RL, the linearized transfer function for the voltage over the load resistorcurrent is given by

VðsÞ

AgðsÞ¼

MRLBLs

ðMs2þDsþKÞðLcsþRcþRLÞþB2L s

(12b)

where s is the Laplace Transform variable. This transfer function is equivalent to the previously derived equation ofAmirtharajah and Chandrakasan [8].

The power (P) dissipated by the resistor is given by

P¼ V2=RL

P¼M2A2

go2B2L RL

½ðKðRLþRcÞ�Mo2ðRLþRcÞÞ�DLco2�2þ½DoðRLþRcÞþðB2LoþKLco�MLco3Þ�2

(13)

where o is the circular frequency of the ambient excitationEq. (13) can be rewritten as

P¼M2A2

gffiffiffiffiffiffiffiffiKMp

!dr2l

½ð1�r2Þ�2zr2l�2þ½ð2zrÞþð1þd�r2Þrl�2RL

RLþRc

� �(14)

With the dimensionless variables

on ¼

ffiffiffiffiffiK

M

r, z¼

C

2ffiffiffiffiffiffiffiffiKMp , r¼

oon

,

d¼B2

L

KLc, l¼

Lcon

RLþRc

(15)

whereon is the natural frequency, z is the mechanical damping ratio, r is frequency ration, d is the effective electromechanicalcoupling coefficient and l is the tuning ratio.


The form of Eq. (14) is similar to the one derived for piezoelectric energy harvesters [11].Eq. (14) can be optimized with respect to the load resistor to give the maximal power output at any operating frequency. In the

case when the internal coil resistance Rc is much smaller than the load resistance RL, the optimal tuning ratio lopt is given by [10,11]

lopt ¼1

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2zrÞ2þð1�r2Þ

2

ð2zrÞ2þð1�r2þdÞ2

s(16)

which is equivalent to the optimal load resistance Ropt

Ropt ¼oLc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDoÞ2þðK�Mo2þB2

L=LcÞ2

ðDoÞ2þðK�Mo2Þ2

s(17)

In the case of weak coupling, i.e. d is small, RoptEoLc.In the case when the parasitic coil resistance cannot be ignored, and the generator is operated at resonance, i.e. o=on,

the optimal resistance is given by

Ropt ¼oLc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDoÞ2þðB2

L=LcÞ2þð2B2

LþDRcÞDRc=Lc2

ðDoÞ2

s(18)

or in non-dimensional form

lopt_L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2zÞ2

ð2zÞ2þd2þð2z=lcÞð2dþð2z=lcÞÞ

s(19)

where

lc ¼Lcon

Rcand lopt_L ¼

Lcon

Ropt(20)

In the case of optimal load resistance (Eq. (18)), and assuming large electromechanical coupling, the maximal power thatcan be extracted at resonance is given by

Popt ¼M2A2

g

8zffiffiffiffiffiffiffiffiKMp

Ropt

RoptþRc(21)

Eq. (21) is the same expression derived by Beeby and O’Donnell [14] using a purely mechanical analog model and is similarto piezoelectric energy harvesting [11] except for the voltage divider term Ropt/(Ropt+Rc). However, as the parasitic coilresistance increases: (1) the optimal load resistance also must increase (Eq. (18)) and (2) to achieve optimal power, theelectromechanical coupling must increase proportionally (Eq. (13)). This means that if RoptbRc then

Popt-M2A2

g

8zffiffiffiffiffiffiffiffiKMp (22)

which is the same as for piezoelectric energy harvesters.When the generator is operated off-resonance, the optimal resistance is given by

Ropt ¼oLc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDoÞ2þðK�Mo2þB2

L=LcÞ2þ½ðDRcþ2B2

L Þo2DþRcðK�Mo2Þ2�Rc=ðLcoÞ2

ðDoÞ2þðK�Mo2Þ2

s(23)

In non-dimensional form this becomes

lopt_L ¼1

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2zrÞ2þð1�r2Þ

2

ð2zrÞ2þð1�r2þdÞ2þð1=ðlcrÞ2Þð1�r2Þ2þð2z=lcÞð2dþð2z=lcÞÞ

s(24)

Substituting Eqs. (23) and (24) into Eqs. (13) and (14), respectively, gives the optimal off-resonance power output in thecase when parasitic coil resistance cannot be ignored.

The short circuit resonance (i.e. osc) for an electromagnetic generator ignoring coil resistance (i.e. Z=0 and Rc=0) can bederived directly from Eq. (10) as

osc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKþB2

L=Lc

M

s¼ooc

ffiffiffiffiffiffiffiffiffiffiffi1þd

p(25)

where ooc is the open circuit resonance which is the same as on in Eq. (15).

3. Experimental validation

An electromagnetic generator was constructed using a neodymium rare earth magnet attached to a steel spring; thegenerator coil was hand wound from 30 gauge copper wire (Fig. 3 with parameters given in Table 1). Two generator coils were

Magnet

SpringGeneratorCoil

Coil Holder

Support Attachment

Fig. 3. The disassembled electromagnetic generator. Device properties are given in Table 1.

Table 1

Properties of the electromagnetic generator used in the experimental validation. The resistance per unit length of the 30 gauge wire is taken as 0.338 O/m.

The measured and calculated values for a single wire layer is indicated by NL=1 and 2 for the double wire layer.

Dimensions and properties Measured value Calculated value

Magnet Mass, M (g) 0.69

Damping ratio, z (%) 0.58

Natural frequency, on (rad/s) 705.3

Electromagnetic coefficient, BL (N/A) 0.030 for NL=1 0.063 for NL=2

Inductance, Lc (mH) 3.78 for NL=1 13.9 for NL=2 3.73 for NL=1 14.7 for NL=2

Coil resistance, Rc (O) 0.27 for NL=1 0.50 for NL=2 0.25 for NL=1 0.45 for NL=2

Load resistance, RL (O) 986

Magnet radius, Rm (mm) 2.41

Magnet residual field, Br (T) 1.2

Generator coil radius, Rg (mm) 4.40 for NL=1 4.20 for NL=2

Generator coil height, hg (mm) 7.0

Number of coil loops in each layer, N 23


manufactured, the first with one layer of copper windings (NL=1), and the second with two layers (NL=2). In both generatorconfigurations, the magnet was positioned within the coil so as to give the maximum radial magnetic field (i.e. the magnet’smid-height is in line with the end of the coil).

Table 1 shows the calculated values for the coil inductance and coil resistance. Mechanical damping and resonantfrequency was measured using a standard ring-down test through a relatively large load resistor (RL=986 O). The generatorwas then mounted onto a linear shaker and the base acceleration was measured by an accelerometer. The measured voltagefrequency responses for the electromagnetic generators are compared with linear theory (Eq. (12)) in Fig. 4. Theexperimentally measured effect of load resistance (RL) on the voltage output when the generators are driven at resonance(i.e. r=1) is shown in Fig. 5 together with Eq. (12). The theoretical results in Figs. 4 and 5 use the measured values (column 2 inTable 1), and the calculated value for the electromagnetic coefficient (BL). It should be noted that in Table 1, 0.05 O are addedto the calculated values of the generator coil resistance to account for the extra length of wire needed for attaching the loadresistance and oscilloscope probes. The theory shows good agreement with the experiments.

The maximal (peak) power output for the single layer coil is calculated from the measured voltage to be 8 mW per m/s2 ofbase acceleration at a load resistance of 0.33 O which is in good agreement with the predicted 7.9 mW per m/s2 of baseacceleration at a load resistance of 0.43 O from Eq. (13). This gives a normalized power (NP) of 4 mW/(m/s2)2, wherenormalized power is defined as the average power per m/s2 of base acceleration [14]. The overall volume of the device is2.3 cm3. The normalized power density (NPD) which is defined as the NP divided by the volume is 1.7 mW/[(m/s2)2 cm3]. Thisnumber is typical for electromagnetic generators of similar volumes [14].

It should be noted that since the spring in the generator has little lateral stiffness, thus to prevent the magnet fromdisplacing laterally and touching or mechanically interfering with the coil, the magnet diameter in the experiment was keptsignificantly smaller than the coil diameter. A larger magnet diameter would naturally increase the electromagnetic field andwould increase the coupling coefficient of the generator. The electromagnetic model of a 8 mm diameter magnet, which is theprobably the largest magnet size that can be used with the present coil configuration, gives an average power output (NP) for asingle coil generator of 9.1 mW/(m/s2)2 and a normalized power density NPD=4 mW/[(m/s2)2 cm3].

10RL (Ω)

0 2 4 6 80

1

2

3

4

5

6

7

8

9

|V/A

g| [m

V/(m

/s2 )

]

Theory Experiment

2 Layer Generator Coil


Fig. 5. Voltage generated by the electromagnetic generators in Table 1 at resonance for various load resistances.

108 109 110 111 112 113 114 115 1160

1

2

3

4

5

6

7

8

9

Theory Experiment

Frequency (Hz)

|V/A

g| [m

V/(m

/s2 )]



Fig. 4. Comparison of experimentally measured and theoretically derived voltage to base acceleration frequency response of an electromagnetic generator

for a single and double layer generator coil (physical properties summarized in Table 1).


4. Parameters effecting generator performance

For a given magnet size and strength (and thus radial magnetic field), electromechanical coupling can be increased byproviding a greater length of generator coil, for example by winding more layers. However, as wire length increases, so doescoil resistance. In this section, we calculate the effect of adding more layers (NL) to the optimal power conversion of theelectromagnetic generator. The experimentally validated case of a single coil layer NL=1 (Table 1) is used as a basis for thesecalculations, i.e. the mechanical properties are as measured and presented in Table 1, while the electromagnetic and electricalproperties (BL, Lc and Rc, i.e. Eqs. (9), (11) and (12)) are calculated for an increasing number of generator coil layers, NL.

Fig. 6 shows that for the specific configuration studied in this section, the optimal number of layers is NL=12, which givesapproximately 80% of the theoretical maximal output power (i.e. when coil resistance is zero, Eq. (22)). At maximalpower output the average normalized power would be NP=8 mW/(m/s2)2 and average normalized power densityNPD=3.5 mW/[(m/s2)2 cm3]. Less than 10% of the generator volume is taken up by the generator coil, magnet and spring;the rest of the space is used to mechanically support these generator components. Saving this space would increase the NPDby a factor of 10, making it as approximately three times less efficient than the best reported generator operating at 100 Hzand approximately 30 times less efficient than the best reported electromagnetic generator operating at any frequency [14].

10 15 20 25 30Number of Generator Coil Layers NL

0 50

5

10

15

20

25

|p/A

g| [μ

W/(m

/s2 )

2 ]

Theoretical Maximum (Eq. 22)

Fig. 6. Maximum generator power as a function of the number of generator coil layers. The dashed line shows the theoretical maximum output power

(Eq. (22)). Electromagnetic generator properties are given in Table 1.

0 5 10 15 20 25 300

5

10

15

20

25

|p/A

g| [μ

W/(m

/s2 )

2 ]

Number of Generator Coil Layers NL

Theoretical Maximum (Equation 22)

30 Gauge32 Gauge

28 Gauge

Fig. 7. Maximum generator power as a function of the number of generator coil layers and wire gauge.


Realistically, the volume of the present generator could be readily reduced by a factor of 2.5 by decreasing the diameter of thecoil holder around the spring and optimizing the mechanical attachment to the support (Fig. 3).

The generator coil resistance can be changed by varying the diameter of the winding coil wire. Fig. 7 shows the effect ofincreasing and decreasing the diameter of the winding coils by selecting 28 gauge and 32 gauge wire. The diameter of the 30gauge wire is dw=0.25 mm giving a packing ratio in Table 1 of 85%. Assuming the same packing ratio for the 28 gauge(dw=0.32 mm) and 32 gauge (dw=0.20 mm) coils, gives the number of loops in each generator layer of N=18 and 30,respectively. The resistances per unit length for the various gauges are: 0.338 O/m for the 30 gauge wire, 0.538 O/m for the 32gauge wire and 0.213 O/m for the 28 gauge wire. Fig. 7 shows that there is little effect on maximal output power with wiregauge. The maximal power remains approximately 80% of the theoretical maximum. Slightly higher output power levels canbe achieved at a lower number of generator coils for the bigger gauge wire, which makes it more cost efficient.

5. Comparison of piezoelectric and electromagnetic generators

A standard method for comparing various electromechanical generators is by analyzing [4]: (1) the maximum poweroutput (Pmax), (2) the effective coupling coefficient (d), and (3) maximum energy density (pmax).


The maximal power output is given by [4]

Pmax ¼d

4�2doUinp

d4�2d

(26)

whereo is the driving frequency, and Uin is the energy of the input vibration. When comparing various energy generators, theinput vibration energy and driving frequency are assumed to be the same. For electromagnetic generators the effect of thecoupling coefficient is given by Eq. (15) and for piezoelectric generators by [11,15]

d¼y2

CpKp¼o2

oc�o2sc

o2oc

(27)

where y is the piezoelectric coupling coefficient, Cp is the piezoelectric capacitance, Kp is the piezoelectric generator stiffness,ooc and osc are the open-circuit and short-circuit natural frequency of the generator. It should be noted that the expressiontypically used [4] for piezoelectric coupling, i.e.

d¼d2Y

ep(28)

is purely a piezoelectric material property constant and does not take into account the geometric configuration of thegenerator and thus cannot be used directly for comparison purposes. In Eq. (28), d is the piezoelectric coupling coefficient, Y isthe modulus of elasticity and ep is the permittivity of the piezoelectric material. Most vibration based piezoelectric generatorsoperate as bending cantilevers and thus the geometric layout and structural support layer (shim) are critical for calculatingthe effective electromechanical coupling coefficient.

The maximum energy density (pmax) under weak electromagnetic coupling, (i.e. d approaching zero) is given by [4]

pmax ¼drðQAgÞ

2

4o pdQ2 (29)

Table 2Comparison of piezoelectric energy generators and the present electromagnetic generator. All piezoelectric generator properties are taken from the

experimental measurements of Erturk et al. [16]. The PZT-5H bimorph marked with (~) is taken from the experimental measurement of Roundy and

Wright [15].

Generator type d d/(4�2d) Q dQ2

PZT-5A bimorph 0.07 0.019 82 487

PZT-5H bimorph 0.09 0.023 68 397

PZT-5H unimorph 0.04 0.010 91 338

PMN-PZT unimorph 0.18 0.049 23 93

PZT-5H bimorph~ 0.02 0.005 33 22

Current electromagnetic generator 0.70 0.27 86 5224

10-3 1000

0.2

0.4

0.6

0.8

1

Pn

10-2 10-1

Fig. 8. The effect of coupling coefficient (d) and parasitic tuning ratio lc on the normalized power output of an electromagnetic generator with characteristics

given in Table 1. The black dot represents the maximal normalized power output for the single layer experimental coil.


where r is the density of the moving mass material, and Q=1/(2z) is the mechanical quality factor. For comparison purposesthe density and amplitude of the base vibration acceleration are assumed to be the same for piezoelectric andelectromagnetic harvesters.

Table 2 shows that the effective coupling coefficient and energy density for the electromagnetic generator is significantlyhigher than for the piezoelectric generators. However, the actual power output of electromagnetic generators is stronglydependent on the generator coil resistance. Fig. 8 shows the effect of the coupling coefficient (d) and parasitic tuning ratio (lc)on the normalized power output (Pn) for an electromagnetic generator with damping ratio z=0.58%. The normalized poweroutput is calculated as the ratio of the optimal power output at resonance (i.e. Eq. (14) with r=1 andl=lopt) and the maximumpower output (Eq. (22)). The dot in Fig. 8 shows the optimal normalized power of the experimentally measured single layercoil (Table 1). Notice that the actual effective coupling coefficient (i.e. assuming Rc=0 and thus lc=N) is dA=2.8e�3 for theexperimentally measured single layer coil. This is significantly smaller than the theoretical effective coupling coefficient ofd=0.7, and thus it is critical to take into account the coil resistance (or parasitic tuning ratio lc) when calculating theperformance of electromagnetic generators. The apparent coupling coefficient increases to dAE0.01 when multiplegenerator coils are used under optimal conditions (i.e. maximal power in Figs. 6 and 7), giving approximately 80%normalized power as previously shown.

It should also be noted that both Eqs. (25) and (28) are approximations. For example, Eq. (28) shows that maximal energydensity is a linear function of effective coupling coefficient. As can be seen from Fig. 8 the power output is in fact nonlinear,especially for relatively high coupling coefficients.

6. Conclusion

Electromagnetic energy harvesting remains an active research area with various generator configurations having beenpreviously researched and commercially produced [14]. In this paper, a relatively simple computational approach forcalculating the electromechanical properties of a single mechanical degree of freedom electromagnetic generator has beenpresented. The single degree of freedom model was used to derive the optimal electric load resistor for maximal generatorpower at both resonance and off-resonance operating conditions. The parasitic effect of the generator coil resistance was alsoincluded in this derivation.

It was shown that for very large electromechanical coupling at resonance, the optimal power output is slightly less for anelectromagnetic generator than for a piezoelectric generator by the ratio of the load resistance to the total resistance (i.e. loadresistance and generator coil resistance). The derived models were compared to experimental results for a simpleelectromagnetic generator and showed good agreement. For moderate electromechanical coupling, it is critically importantto take into account the effect of parasitic coil resistance. The parasitic coil resistance acts to decrease the electromechanicalcoupling of the system. Adding additional layers of generator coil wire only increases the maximal power output to a certainpoint, beyond which increasing parasitic coil resistance begins to dominate the output power of the generator.

It should be noted that the simple generator tested in this paper might not be practical since it requires careful alignmentof the magnet within the coil and with the earth’s magnetic field to prevent the magnet (or spring) from touching or rubbingagainst the generator coil during operation and thus hinder or fully prevent the magnet from moving. The linear springprovides very little lateral stiffness, so horizontal forces, for example from ambient vibration, can have the same effect on themagnet. Due to the parasitic resistance of the coil in the experimental electromagnetic generator, the equivalent effectiveelectromechanical coupling of the current device was less than for previously measured piezoelectric energy harvesters[15,16]. However, electromagnetic generators can still be of interest for at least two reasons. First their cost is significantlyless than for equivalent ceramic based piezoelectric generators. The second reason is that their relatively low dampingcoefficients can lead to significantly greater overall power conversion even with significant parasitic coil resistance. In fact, forthe piezoelectric generators in Table 2, only the PZT-5A bimorph and PZT-5H unimorph would have better power outputsthan the multi-layer electromagnetic generator due to their low damping ratio and high electromechanical couplingcoefficients.

Future work will concentrate on electromagnetic energy harvesters with better quality factors, i.e. less mechanicaldamping. For example, by reducing the damping ratio of the single layer electromagnetic generator presented by a factor oftwo, the output of a single layer coil generator would exceed the best piezoelectric generator in Table 2.

References

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an experimentally validated electromagnetic energy harvesters

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