IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 8, …ims.unipv.it/FIRB2006/pub/Dallago10b.pdf · 2010. 7. 21. · Enrico Dallago, Member, IEEE, Marco Marchesi, and Giuseppe

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 8, AUGUST 2010 1989

Analytical Model of a Vibrating ElectromagneticHarvester Considering Nonlinear Effects

Enrico Dallago, Member, IEEE, Marco Marchesi, and Giuseppe Venchi

Abstract—In this paper, an analytical model of a vibrating elec-tromagnetic harvester, taking into account nonlinear effects, is pre-sented. Knowing the acceleration applied to the harvester system,the model is able to predict the performance of the harvester interms of induced voltage on the load. The energy transducer con-sists of four magnets, two movable and two fixed, arranged in a waysuch that both fixed magnets repel the moveable one. The modelwas implemented in Simulink and exploits the results of a finiteelement method (FEM) solver (Flux2D) to estimate the non-linearelectromagnetic repulsion force and the flux linkage by the coil. Toevaluate the effect of a load applied to the harvester, the effect ofthe Lorentz’s force, caused by the interaction of the current thatflows in the coil and the flux density of the moveable magnet, wastaken into account. As a consequence, the error in the estimate ofinduced voltage, at resonance, is reduced from about 80% to 7%.Finally, at resonance, the maximum power that could be deliveredby the harvester and dissipated on a resistive load was estimatedto be about 6 mW.

Index Terms—Electromagnetic harvester, energy harvesting,magnetic transducer.

I. INTRODUCTION

NOWADAYS, the concept of energy harvesting has gaineda large consideration in the scientific community because

of the continuous decrease in the power consumption of micro-electronic devices and systems [1]–[4]. For example, the powerconsumption of sensor network systems, wireless sensors orsmart dust is becoming so low to allow driving them with theenergy stored in a capacitor or in a super-capacitor by a harvester.Recently, different types of harvesters, both integrated [5]–[10]and macroscopic [11]–[17] have been presented. Furthermore,the possibility to merge different types of technologies has beenpresented in [18]. All these efforts are aimed at using an energyharvester to drive portable or wireless devices.

Among the various types of available environmental en-ergy, vibrations have always been focused on by a lot of re-searchers because a number of different transducers can success-fully be employed (e.g., magnetic, piezoelectric, or electrostaticdevices).

Manuscript received July 24, 2009; revised September 21, 2009 andDecember 20, 2009. Date of current version June 25, 2010. This work was sup-ported by the Italian Ministry of University under FIRB project RBAP065425“Analog and Mixed-Mode Microelettronics for Advanced Systems.” Recom-mended for publication by Associate Editor C. R. Sullivan.

The authors are with the Department of Electrical Engineering, Univer-sity of Pavia, 27100 Pavia, Italy (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2010.2044893

Integrated harvesters, i.e., systems in which not only the front-end circuitry but also the transducer itself are integrated, arerapidly improving; nevertheless, the energy they can actuallyharvest is still not enough, mainly because the efficiency of agiven transducer principle reduces as the physical dimensionsare reduced, especially, if one considers the typical frequencydistribution of vibrations in the environment and the fact thatmost transducers are resonators. On the contrary, macroscopicdevices [4], [12], [19] can be used right now to drive wear-able devices. This is why this paper focuses on a macroscopicelectromagnetic transducer.

One of the main issues of these harvesters is the necessity ofhaving an efficient front-end circuitry, in order to supply it withthe same energy the harvester stores in the intermediate storagestage (e.g., a capacitor or super-capacitor), so as to implement afully self-powered system. Furthermore, the front-end circuitryhas to load the transducer in an optimal way to maximize thegathered energy. In this scenario, the design, tuning, and opti-mization phases could largely benefit from an analytical modelof the transducer representing precisely the physical behavior ofthe transducer. This is, particularly, true when realistic workingconditions (in which all quantities, both physical and electrical,vary significantly in time) are considered.

In literature, different analytical analyses of vibrating elec-tromagnetic harvesters have been presented [1]–[4], [20]. Thosemodels are based on the solution of the second Newton’s lawof the inertial harvesting system in which the magnetic force,which could be due to the interaction of two magnets or a magnetand a yoke, is linear with respect to the position. Moreover, inthe aforesaid analytical models, the instantaneous displacementof the moving mass of the harvester is the input quantity. In thispaper, a different approach is presented: it is still based on thesolution of the second Newton’s law but it takes the nonlinear-ity of magnetic force, evaluated with a finite element method(FEM) tool (Flux2D) into account; furthermore, the input vari-able is the acceleration applied to the entire harvester, which isusually readily available. Finally, the effect of the current flow-ing in the sensing coil when loaded by a resistor is modeled.The obtained model is discussed by comparing its simulationswith experimental results.

II. STRUCTURE OF THE ELECTROMAGNETIC ENERGY

HARVESTER AND ITS ANALYTICAL MODEL

The analytical model introduced in this paper was based onthe energy transducer presented in [19], which, on its turn, canbe considered a variation of the system proposed in [21]. Theharvester, shown in Fig. 1, has been realized exploiting:

1) a 15-mm inner diameter and 56 mm long teflon tube;

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1990 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 8, AUGUST 2010

Fig. 1. Energy transducer simulated with Flux2D.

2) four magnets: two fixed magnets at the extreme (in theform of disk, with diameter of 10 mm and 1 mm thick) andtwo levitating magnet (in the form of disk with diameterof 15 mm and 8 mm thick). The levitation is obtainedby a suitable arrangement of the magnetic poles in theassembly as shown in Fig. 1, where “N” is the Northand “S” is the South pole. The magnets are commercialNdFeB, with residual magnetic induction (Br ) equal to1.3 T. The moving mass of the levitating magnets is equalto 0.0209 g;

3) a coil of 500 turns, realized with copper wire with a diam-eter of 0.11 mm, and a resulting resistance of about 60 Ωin which a voltage is induced by the motion of the twolevitating magnets.

The fixed magnets, that have been placed at the ends of theteflon tube, were selected so as to have a smaller diameter withrespect to the inner diameter: this allows the air to be easilyexpelled off the tube when the middle magnets vibrate, there-fore, reducing its contribution to the damping. Other sources ofdamping are the mechanical damping, which is due to frictionbetween magnets and the teflon tube, and the electrical damp-ing, which is due to the current that flows in the coil when avoltage is induced.

Such a system is typically modeled as a mass m, a spring ofstiffness k, and a damper system b, as shown in Fig. 2.

When an external acceleration is applied to the housing (i.e.,the teflon tube), the mass moves with respect to the housing andto the coil and an induced voltage appears at the coils’ terminals.To calculate this, the motion law of the mass has to be derived.In the following, the displacement of the mass from its restposition with respect to the housing is x(t); the position of thehousing with respect to an external reference frame (the one inwhich the applied acceleration is defined) is u(t), and that of theseismic mass is u(t) + x(t).

Fig. 2. Ideal inertial vibration based energy harvesting.

Some analysis approaches to this type of harvester have beenintroduced in [1]–[3] and further analyzed in [5], [20], and[22]–[25]. On the basis of the system analysis, the solutionof the second Newton’s law and approximating everything aslinear, the differential equation for the motion of the seismicmass m relative to the housing is

mx(t) = −kx(t) − bx(t) − mu(t) (1)

where x, x, x are the position, the velocity, and the accelerationof the mass m, respectively, while u is the external accelera-tion applied to the housing. In [2] and [3], the solution of (1)considering a sinusoidal movement for the housing, leads to thedefinition of the power dissipated in the damping element

Pd =mςU 2(w/wn )3w3

[1 − (w/wn )2 ]2 + [2ς(w/wn )]2(2)

wherew is the frequency of the external vibration;wn is the natural frequency of the system =

√k/m;

U is the amplitude of the displacement of the housing dueto the vibration;

ς is the damping ratio = b/(2mwn ).Since, up to now, no interaction with the coil has been con-

sidered, this power, which is purely mechanical, is the totalpower made available in such a system as a consequence of theapplication of an external acceleration. In case this power wascompletely converted into electric power then it would also rep-resent the electrical power ideally available from the transducer.

This result has some limitations. First of all, it considersthe equivalent spring as a linear one, in which the stiffness isindependent on the position of the mass. In the presented case,this is true only when the displacement of the levitating magnetsis negligible or very small with respect to the distance betweenthe moving and the fixed magnets. When this hypothesis is notverified, significant errors can take place.

Furthermore, we are more interested in deriving the electricalpower that can be actually obtained from the transducer than inthe power that can potentially be obtained.

The approach of this paper is to rewrite (1) as follows:

mx(t) = FMAG(x) − bx(t) − mu(t) − mg (3)

where FMAG(x) is the non-linear magnetic repulsion forcegiven by the interaction of the magnets and depends on the


DALLAGO et al.: ANALYTICAL MODEL OF A VIBRATING ELECTROMAGNETIC HARVESTER CONSIDERING NONLINEAR EFFECTS 1991

Fig. 3. Magnetic repulsion force evaluated with Flux2D.

position (as we will show in the following it can be evaluatedwith an FEM tool), and g is the gravity of the earth. Includinggravity in the force balance equation is mandatory when thetransducer is placed vertically, while it could be neglected for ahorizontal one. Intermediate orientations could be addressed byreplacing the mg term with mg cos(α), where α is the angle ofthe transducer axis with respect to the vertical direction.

This equation can then be solved directly by means of suit-able numerical methods in order to calculate the instantaneousposition of the moving magnets with respect to the sensing coilwhen a given acceleration pattern is applied to the transducer.In this way, the induced voltage and, consequently, the outputpower can be evaluated.

III. SIMULINK AND FEM ANALYSIS

To solve (3), it is necessary to know the repulsion magneticforce (FMAG ) at each relative position between the magnets.This type of analysis can be performed exploiting a magnetic fi-nite element solver, such as FLUX2D. The force versus positioncharacteristic was obtained by calculating the magnetic force ata number of positions of about 130 in the range ±16.5 mm withrespect to the center of the teflon tube (see Fig. 3). The actualmaximum excursion is about±19 mm, but since the force growssignificantly and in a nonlinear way as the distance between themagnets decreases, as Fig. 3 demonstrates, it is useless to ex-tend the analysis further. A linear approximation of the magneticforce with a reasonable error could be applied only in the rangeof ±4 mm with respect to the center of the tube, as shown inFig. 4.

To solve (3) in a numerical simulator (e.g., Simulink), it isbetter to fit the curve of Fig. 3 with a sixth-order polynomial,which is also shown in Fig. 3 itself.

The resulting model was solved by implementing it inSimulink. In particular, (3) was rewritten as following:

x(t) =FMAG(x)

m− b

mx(t) − u(t) − g. (4)

Fig. 4. Magnetic repulsion force in the range of ±4 mm with respect to thestarting position of the moving magnets.

Fig. 5. Flux linkage by the coil as a function of the position of the movingmagnets.

It is important to stress that in (3) and (4) u is the accelerationapplied to the housing and measured with the accelerometer,while x is the acceleration of the moving mass.

The magnetic simulator was also used to calculate the fluxlinkage by the sensing coil as a function of the relative positionbetween the moving magnets and the coil itself. The obtainedcharacteristic is shown in Fig. 5, which also gives the expressionof its sixth-order interpolating polynomial. This graph allows tocompute the flux linkage versus time as the magnets move; itsderivative with respect to time gives the induced voltage.

IV. EXPERIMENTAL SETUP AND RESULTS

In order to solve (4), it is necessary to know the dampingfactor. The damping is a parameter that strictly depends onthe particular system and, in general, is not easy to predict. In



Fig. 6. Experimental setup and a photo of the teflon tube with the levitatingmagnets.

Fig. 7. Measured peak of induced voltage varying the frequency of the appliedacceleration.

this paper, it was obtained by estimating it on a prototype (seeFig. 6). The experimental setup consists of a shaker, a functiongenerator, and an accelerometer. The function generator drivesthe amplifier of the shaker, which is of the open loop type: itsupplies a current to the shaker which is proportional to thereference signal of the function generator and the resulting ac-celeration depends on the dynamics of the assembly. (In a closedloop type, an on-board accelerometer is exploited so as to ap-ply acceleration to the assembly, which is proportional to thereference signal). For this reason, only sinusoidal accelerationswere considered, varying the amplitude and the frequency. Thereal amplitude of the applied acceleration was monitored with aMEMS accelerometer, mounted below the assembly.

Initially, the amplitude of the applied acceleration was fixedat ±1 g and the peak of the induced voltage was measuredwhile varying the frequency. This measurement, shown in Fig. 7,allowed evaluating the quality factor, which is about 4.24. It isworth noting that the measurement of Fig. 7 forces the systemto work in a nonlinear region, especially, around the resonance;thus the measured Q indirectly includes the contribution of this

Fig. 8. Experimental (solid line) and simulation (dashed line) induced voltage;applied acceleration of ±1 g at 9 Hz.

nonlinearity. The damping factor b was then estimated by meansof the following definition:

Q =wn

b=

√km

b. (5)

This formula is the only explicit linearization step in theevaluation of the b factor. The estimated value was found out tobe about 0.31 Ns/m.

Knowing the value of the damping factor, the complete sim-ulation can now be run, obtaining, for example, the waveformof the induced voltage in the coil in a no-load condition. Acomparison between the simulated and the measured voltage isoffered in Fig. 8. As it can be seen, the matching is satisfactory:the amplitude mismatch is lower than 0.075 V considering thatthe peak is about 2.25 V.

The shape of the induced voltage (a large peak followed bya lower peak) is due to the offset in the starting position of themoving mass with respect to the center of the tube. This off-set, of about 2.5 mm, is caused by the gravity, which slightlycompresses the lower equivalent magnetic spring. To better un-derstand the shape of the induced voltage, the position versustime and the flux linkage, whose derivative is the induced volt-age, versus time are shown in Fig. 9.

This offset should be considered also when a linear approx-imation of the magnetic force versus position is used, becauseit moves the interval of abscissa where to apply the lineariza-tion. For example, Fig. 9 shows the linear approximation in theposition range from −6 to 2 mm.

In this case, the resulting linearized k is slightly higher thanthe one obtained in Fig. 4. Nevertheless, it can be demonstratedthat in both cases the full model, including the spring nonlin-earity, outperforms the linearized ones as shown in Fig. 10,which presents a comparison between the measured inducedvoltage and those simulated by the three considered models(the full model and the two linearizations). This time-domain



Fig. 9. (a) Flux linkage versus time at frequency resonance and (b) Position of movable magnets versus time at frequency resonance.

Fig. 10. Magnetic repulsion force in the range of −6 to 2 mm.

comparison between simulation and experimental results wasperformed in the frequency range of Fig. 7, keeping the am-plitude fixed at ±1 g. The results have been summarized inTable I.

Moreover, considering the linear approximation of Fig. 10, itis also possible to estimate the resonance frequency exploitingthe definition

fn =wn

2π=

12π

√k

m. (6)

It comes out to be about 10.4 Hz, in good agreement withthe measurement of Fig. 7. Driving the system with an externalacceleration at the aforesaid frequency, the harvester should givethe maximum induced voltage, as it is confirmed in Table I andshown in Fig. 12.

To complete the study of the proposed model, a sensitivityanalysis of its behavior with respect to the damping factor wascarried out, by simulating the frequency response in terms of the

TABLE ISIMULATED AND MEASURED PEAK OF THE INDUCED VOLTAGE

AT DIFFERENT DRIVING FREQUENCY

Fig. 11. Comparison between the simulation, (with and without nonlinearforce effects), and measured induced voltage at 9 Hz.



Fig. 12. Comparison between simulated (dashed line) and measured (solidline) induced voltage at mechanical resonance.

peak of the induced voltage at various damping factors between0.21 Ns/m to 0.41 Ns/m in steps of 0.05 Ns/m. The accelerationwas kept constant at ±1 g. The results are presented in Fig. 13.

As expected, increasing the value of the damping factor thepeak amplitude of the induced voltage decreases, in accordancewith the definition of (4) and (6). This effect is maximizedat the resonance, resulting in a large variation of the inducedvoltage for a little variation of the damping factor. Moreover,the increased damping factor causes a variation of the resonancefrequency, in accordance with the following definition:

wmax = wn

√(1 − b2

2km

)(7)

where wmax corresponds to the frequency where the maximuminduced voltage occurs.

V. EFFECT OF THE LOAD ON THE PERFORMANCE

OF THE HARVESTER

The analysis presented so far is based on the hypothesis thatthe harvester works under no-load condition. When a load isconnected to the coil and, as a consequence, a current flows init, another effect takes place. With a resistive load, the actualload voltage not only decreases because of the resistive dividerbetween the coil series resistance and the load resistance, butalso it is further reduced by the interaction between the coilcurrent and the moving magnets. This interaction consists ofvarious effects. A first effect can be modeled, given the currentin the coil, computing the magnetic field generated by the coilitself by means of its self-inductance. This field interacts withthe field of the moving magnet and gives rise to a damping of thevibration. Another effect is the resulting Lorentz’s force on thecoil due to the interaction between the current and the magneticfield of the movable magnet. This effect is the predominant

Fig. 13. Simulated induced voltage varying the frequency and the dampingfactor.

effect in the proposed magnetic configuration and is the one thatwas implemented. In order to include the Lorentz’s force, (4) ismodified as follows:

x(t) =FMAG(x)

m− b

mx(t) − u(t) − g

− (2πrmean-coil)NBradial(x)m

Vind

R(8)

where N is the number of turns of the coil, Bradial is the radialcomponent of the levitating magnets’ magnetic field, Vind/R isthe current that flows in the coil (expressed in terms of inducedvoltage divided by the total resistance R, i.e., the sum of the in-ternal resistance of the coil and the load resistance), 2πrmean-coilis the mean length of a turn of the coil. Inductive effects on thevoltage are neglected due to the small self-inductance and lowfrequency of operation, which is a small reactance with respectto the resistance of the coil.

In order to solve the (8), an estimate of the radial componentof the levitating magnets’ magnetic field (Bradial), on a lineat the mean radius of the coil and along its cross section (seeFig. 1), is required. It can, once again, be obtained by usingFlux2D. The results are shown in Fig. 14. Finally, the Simulinkmodel was modified to take into account the Lorentz’s force,giving rise to the complete model shown in Fig. 15.

To demonstrate the weight of this phenomenon on the globalperformances of the transducer, an analysis was carried out, eval-uating the peak load voltage for a given acceleration amplitudeand frequency while varying the load resistance. Fig. 16 showsthe obtained results when an external acceleration of ±1 g, at9 Hz is considered, while Fig. 17 refers to ±1 g accelerationwhen the frequency is equal to the resonance frequency of theharvester. As expected, the difference between the three curvesvanishes as the load resistance is high enough to approach theno-load condition.

On the contrary, the effect of the Lorentz’s force becomes ev-ident for lower load values, and it is emphasized at resonance:



Fig. 14. Magnetic field, on the middle length of the coil, in the useful direction for the Lorentz’s force generation.

Fig. 15. Simulink model that takes the reaction to the induced voltage intoaccount.

Fig. 16. Peak of the output voltage at various resistive load at 9 Hz. Red line(+): Lorentz’s force is neglected; black line (∆): Lorentz’s force is included inthe model; blue line (o): measured voltage.

Fig. 17. Peak of the output voltage at various resistive load at 10.4 Hz. Redline (+): Lorentz’s force is neglected; black line (∆): Lorentz’s force is includedin the model; blue line (o): measured voltage.

the maximum error in Fig. 16 between the measurement and thesimulations without the Lorentz’s force is about 20%, while inFig. 17, it is about 80%. Including the reaction of the inducedvoltage the model drops the error to 6% and 7% at resonanceand at 9 Hz, respectively. All these considerations are importantsince one of the most relevant parameters for energy harvestingtransducers is the “optimum load resistance,” that is the resis-tance that allows the maximum power to be drawn from thetransducer itself. When the reaction of the induced voltage isneglected the harvester can be modeled as a simple Theveninequivalent circuit and the optimum load resistance is equal to theoutput resistance, i.e., the coil series resistance. In the real case,the optimum load cannot be easily calculated starting from theelectrical parameters of the transducer. Furthermore, only the



Fig. 18. Mean power delivered to the load as a function of the load resistance.Solid line: model without the Lorentz’s force; dashed line: model with theLorentz’s force.

accurate model can correctly predict the actual available power.Fig. 18 demonstrates the concept, showing the comparison, atthe resonance frequency, between the output mean power calcu-lated with and without the Lorentz’s force. Taking the Lorentz’sforce into consideration, reduces the available power by ap-proximately 54%, while the optimum load changes from 60 to140 Ω. This last data could be a crucial point in the realizationof a front-end circuitry.

VI. CONCLUSION

In this paper, a vibration energy transducer for energy har-vesting application was presented; the focus is on the model thatwas developed to simulate its performance. The model, imple-mented in Simulink, exploits an FEM solver (Flux2D) to takeinto account nonlinear effects, making it able to predict the per-formance of the harvester when different external accelerationare applied to the system. The same model has been modifiedto include the effect of a resistive load connected directly to thecoil of the harvester. Considering an external acceleration at afrequency equal to the resonance frequency of the mechanicalsystem, the maximum error in terms of peak of the output volt-age was reduced from 80% to about 7%. The modified modelproved to be effective also in predicting the available outputpower and the optimum load.

The proposed model could be used to analyze the effect of acircuitry connected to the harvester. In fact, to evaluate the besttopology for a front-end circuitry, e.g., to store the energy ona tank, an analytical model of the scavenger which accuratelysimulates its physics could be very helpful.

ACKNOWLEDGMENT

The author would like to thank A. Fanzio, Department ofElectronic Engineering of the University of Pavia, for the helpin the realization of the prototype and Cedrat, Grenoble France,for the use of its software tools.

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Enrico Dallago (M’87) was born in Bolzano, Italy,in 1949. He received the Dr.Eng. degree in electricalengineering from the University of Pavia, Pavia, Italy,in 1974.

In 1974, he joined the Department of Electrical En-gineering, University of Pavia, where he is currentlya Full Professor of power electronics, and foundedthe Power Electronics Research Group. His researchinterests include circuit simulation, high-frequencyswitching power supplies, power integrated circuits,thermal analysis of electronic systems, magnetic sen-

sors, and energy-harvesting systems.Prof. Dallago is a member of the Italian Electrical and Electronic

Association.

Marco Marchesi was born in Castelsangiovanni,Italy, in 1975. He received the Dr.Eng. degree in elec-tronics engineering and the Ph.D. degree in electricalengineering from the University of Pavia, Pavia, Italy,in 2002 and 2006, respectively.

In 2006, he joined the Research and Develop-ment Division of STMicroelectronics, Cornaredo,Italy. Since 2009, he has been with the Departmentof Electrical Engineering, University of Pavia. Hisresearch interests include magnetic sensors for lowmagnetic field detection, magnetic and mechanical

finite-element method simulations for microelectromechanical systems devicesand energy harvesting transducers. He is the author or coauthor of more than fivepapers published in international journals, more than 15 presentations presentedat international conferences (with published proceedings).

Giuseppe Venchi was born in Pavia, Italy, in 1970.He received the Dr.Eng. degree in electronics engi-neering and the Ph.D. degree in electrical engineeringfrom the University of Pavia, Pavia, in 1996 and 2000,respectively.

He is currently a Researcher in the Departmentof Electrical Engineering, University of Pavia. Since2004, he has been cooperating with the NationalCenter for Oncological Hadrontherapy, Milan, Italy(CNAO), where he is currently in charge of the powersupplies of the CNAO synchrotron. His research in-

terests include device and packaging thermal analysis, circuit simulation, de-velopment of integrated smart-power circuits, magnetic field sensors, energyharvesting, and power supplies for particle accelerators.


Documents

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 8, …ims.unipv.it/FIRB2006/pub/Dallago10b.pdf · 2010. 7. 21. · Enrico Dallago, Member, IEEE, Marco Marchesi, and Giuseppe