Power flow in a small electromagnetic energy harvesting system excited by mechanical motion
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a y Power flow in a small electromagnetic energy harvesting systemexcited by mechanical motion
L.E. Helseth
Department of Physics and Technology,University of Bergen, N-5007 Bergen, Norway
Abstract
In this study the power flow in a coupled mechanical and electromagnetic harvesting systemin presence of both positional and electrical fluctuations is analyzed. Explicit expressions forthe power into and out of the mechanical and electrical parts of the system are found in thecase of weak coupling, and it is shown how the power flows between the domains consistent withenergy conservation. The case of thermal fluctuations is considered in particular, and use of thefluctuation-dissipation theorem explicitly demonstrates that the power delivered to the mechanicalsystem from the electrical system is the same as the power delivered to the electrical system fromthe mechanical system. On the other hand, the power dissipated in the electrical circuit is notthe same as the power transferred from the mechanical domain if the electrical circuit contains itsown current fluctuations. The electrical noise power dissipated in a load resistor is calculated, andfound to consist of a component due to electromagnetic coupling in addition to the well-knownNyquist component. The component due to electromagnetic coupling scales with size, and becomesmore important than the Nyquist component only for sufficiently small systems. . INTRODUCTION Vibration energy harvesting is a method to transfer mechanical energy into electromag-netic energy with the aim of powering small-scale sensors or devices. It is likely that futuremicro and nanosystems will harvest all the energy they need from the environment. Energyharvesting based on mechanical vibrations is a subject of increasing interest, due to the factthat vibrations are present in a large number of physical systems. Pioneering experimen-tal studies on miniature vibration energy harvesting systems were reported by Yates andcoworkers[1, 2]. Since then, considerable work has been done to understand theoretically,and experimentally, the extraction of electrical power from vibration harvesting systems, seeRefs. [3–5] and references therein. In particular, considerable efforts have been concentratedon optimizing the mechanical and electrical power as well as the electromagnetic couplingcoefficient[6–9]. Studies have also shown that coupling different harvesting modes, e.g. elec-tromagnetic and piezoelectric, may enhance the power output[10]. Poulin et al. pointed outthat electromagnetic and piezoelectric vibration harvesting systems can be modeled usingthe same differential equations, thus allowing a unified description of both[7]. Arroyo et al.compared theoretical and experimental results, demonstrating a decrease in electrical powerwith an increase in the electrical loss coefficient[11]. Since the electrical loss coefficient,which is essentially given by the ratio between the real and imaginary part of the electricalimpedance, often becomes more important as the circuit decreases, electromagnetic gener-ators have not been miniaturized at the same pace as their piezoelectric and electrostaticcounterparts. In fact, although the first electromagnetic generators were miniature devicesproducing low output powers, the trend has facilitated the development of much largerdevices ( > cm ) producing energies far in excess of 1 mW[1, 3, 5]. From an engineeringperspective, piezoelectric and electrostatic harvesters are straightforward to fabricate downto micrometer scale, and can in many cases be combined with existing mass-productiontechnology solutions. On the other hand, electromagnetic harvesters are often fabricatedusing micromachining and manual tooling, which is irreconcilable with current wafer pro-duction technology. The belief that electromagnetic generators do not perform equally wellas other types of devices when size is decreased was questioned in ref. [11], where it wasfound that as long as the electromagnetic coupling coefficient remains high while the electri-cal loss parameter stays below a certain value, there is no formal reason why electromagnetic2nergy harvesting systems should perform worse than their piezoelectric and electrostaticcounterparts.Until a few years ago, most of the research in the field of vibration energy harvestingconcentrated on either impact or oscillation-based devices exhibiting a narrow frequencyresponse. In many practical systems of interest it is necessary to utilize the wide-band tuningdevices, and this was addressed by Sari et al., who proposed arrays of linear cantilevers eachwith different resonance frequency, in order to ensure that the system would be able to utilizethe available vibration spectrum[12]. Clearly, such a system may take up a larger volumethan a single cantilever, and it was desirable to find methods to allow further power densityoptimization. In Refs. [13, 14] it was demonstrated that nonlinear energy harvesting systemsare not inhibited by the same bandwidth limitations as linear systems. A theoretical analysisof the electrical power output of both linear and nonlinear systems excited by randomvibrations was given by Halvorsen[15]. Various approaches to improve output power, andin particular the shape of the potential well used, have been considered. In Refs. [16, 17]the influence of an end stop to limit the motion as well as the effect of nonlinear springsin an harvester driven by colored noise was studied. Deza et al. found that the tunableWoods-Saxon oscillator may provide a useful model system when optimizing the harvestingsystem driven by colored noise[18]. A survey of broadband techniques for vibration energyharvesting has been given by Twiefel and Westermann[19].Driving an energy harvesting system by noise is not trivial, and careful analysis optimiza-tion of the amplitude and power are required. The statistical mechanics is strongly relatedto the much investigated topic of optimally understanding and enhancing the efficiency ofBrownian motion excited by colored noise[20, 21]. Furthermore, thermal fluctuations havebeen measured in superconducting and mechanical systems in different frequency bands us-ing interferometric techniques[22, 23]. Theoretically, the generalized Langevin equation hasbeen found to be able to model a range of different mechanical and electrical systems[24–28],and should therefore be a very useful tool for modeling also vibration energy harvesting sys-tems. As the vibration energy harvesting systems become smaller, one would have to dealwith thermal excitations, either as a method to enhance the performance or by reducing itusing, e.g., external forces. Thermal position fluctuation reduction using controlled externaldissipative forces has been pursued experimentally in the field of atomic force microscopy[29],but similar studies have not been undertaken for vibration harvesting systems.3he power flow in an energy harvesting system exposed to well-defined oscillating me-chanical motion was analyzed in Ref. [6]. However, it appears that the power flow betweenand within the electrical and mechanical domains has not been addressed for noisy mechan-ical motion in such a manner that also thermal fluctuations can naturally be encounteredfor. Although rectification and signal conditioning of small currents is a most pressing issuein very small harvesting systems, it is also of importance to know how much power can beactually expected to flow between and in the mechanical and electrical parts of the energyharvester. In this study these questions are addressed, with particular emphasis on thepower flow in presence of thermal fluctuations. II. THE ELECTROMAGNETIC COUPLING
Let us assume a non-relativistic system of mass m which has one degree of freedom andis described by a position operator x ( t ). The generalized Langevin equation is given by m ¨ x ( t ) + Z t −∞ µ ( t − τ ) ˙ x ( τ ) dτ + dU ( x ) dx = F ( t ) + f em ( t ) , (1)where µ ( t ) is the memory function. The force − dU ( x ) /dx is acting on the system, where U ( x ) is the corresponding potential. The force F ( t ) represents a sum of the oscillating force f m ( t ) = f sin ( ωt ) and the fluctuation force f f ( t ) caused by the thermally excited moleculessurrounding the electromechanical system. Moreover, f em ( t ) is the electromagnetic forcedue to the current i ( t ) in the electric circuit. The case where f em ( t ) = 0 has been analyzedin detail in Ref. [24], where the Langevin equation was shown to follow from a very generalquantum mechanical independent oscillator model. Here we are mainly interested in thesituation where there is non-negligible coupling ( f em = 0), with the aim of investigating thegeneration of electrical power in an electric circuit coupled to the mechanical system.In order to proceed, it is useful to assume the following relationship linking current i andvelocity ˙ x , i ( t ) = i ( t ) + Z ∞ β ( τ ) ˙ x ( t − τ ) dτ , (2)where β is a coupling coefficient associated with the coupling of mechanical into electricalenergy and i ( t ) is the current existing in the circuit due to thermal fluctuations. Similarly,it is possible to define another coupling coefficient γ linking changes in position with current4uch that x ( t ) = x ( t ) + Z ∞ γ ( τ ) i ( t − τ ) dτ , (3)where x ( t ) is the time-dependent position due to fluctuations or well-defined oscillationsnot caused by the electromagnetic coupling. We will in the following assume that thethermal current fluctuations are uncorrelated with the position fluctuations such that = 0. In the spectral domain these equations become i ( t ) = i ( ω ) − jωβ ( ω ) x ( ω ) , (4)and x ( ω ) = x ( ω ) + γ ( ω ) i ( ω ) . (5)Here x ( ω ) is the Fourier transform of x ( t ), i ( ω ) is the Fourier transform of i ( t ), β ( ω ) = R ∞ β ( τ ) exp ( jωτ ) dτ and γ ( ω ) = R ∞ γ ( τ ) exp ( jωτ ) dτ . By inserting Eq. 5 into Eq. 4 it isalso possible to write the spectral current as a function of the uncoupled spectral components i ( ω ) and x ( ω ) i ( ω ) = i ( ω ) − jωβ ( ω ) x ( ω )1 + jωβ ( ω ) γ ( ω ) , (6)and x ( ω ) = x ( ω ) + γ ( ω ) x ( ω )1 + jωβ ( ω ) γ ( ω ) . (7)If ω | β ( ω ) || γ ( ω ) | ≪
1, one may let the denominator be unity, such that i ( ω ) ≈ i ( ω ) − jωβ ( ω ) x ( ω ) , (8)and x ( ω ) = x ( ω ) + γ ( ω ) x ( ω ) . (9)This means that when the coupling between the electromagnetic and mechanical domain isweak, such that eqs. 2 and 3 become i ( t ) = i ( t ) + i ( t ) ≈ i ( t ) + Z ∞ β ( τ ) ˙ x ( t − τ ) dτ , (10)and x ( t ) = x ( t ) + x ( t ) ≈ x ( t ) + Z ∞ γ ( τ ) i ( t − τ ) dτ . (11)Equations 10 and 11 can be viewed upon as a first-order approximation, wherein only theoriginal position and current fluctuations contribute to the coupling, and second-order effects5re neglected. Clearly, by keeping higher-order terms (i.e. not making the assumption ω | β ( ω ) || γ ( ω ) | ≪
1) one may also consider the case of strong coupling as well, but this isoutside the scope of the current study.Let us now link the parameter γ ( ω ) with the the spectral susceptibility of the system,since susceptibility is often used to describe a mechanical system. The susceptibility isdefined through the equation x ( t ) = R ∞ α ( τ ) F ( t − τ ) dτ , which in the frequency domainbecomes x ( ω ) = α ( ω ) F ( ω ) with α ( ω ) = R ∞ α ( τ ) exp ( jωτ ) dτ . If x = 0 and all the positionfluctuations are due to applied currents in the electrical circuit, we may write Eq. 5 as x ( ω ) = γ ( ω ) i ( ω ). On the other hand, we know that the Lorentz force acting on the mechanicalsystem can be written as f em ( t ) = − K m i ( t ), where K m represents the coupling betweenthe electromagnetic and mechanical degrees of freedom. Thus, we must require γ ( ω ) = − K m α ( ω ).Similarly, it is possible to link the parameter β ( ω ) with the impedance of the electricalsystem. If we assume that all the current fluctuations are due to movement in the mechanicaldomain, we may write Eq. 4 as i ( ω ) = − jωβ ( ω ) x ( ω ). The voltage is related to the currentthrough the impedance Z ( t ) as v ( t ) = R ∞ Z ( τ ) i ( t − τ ) dτ , which in the frequency domainbecomes v ( ω ) = Z ( ω ) i ( ω ) with Z ( ω ) = R ∞ Z ( τ ) exp ( jωτ ) dτ . Thus, if the voltage spectrum v ( ω ) = − jωβ ( ω ) Z ( ω ) x ( ω ) is to be consistent with Faraday’s law, which in the frequencydomain can be written as v ( ω ) = − jωK m x ( ω ), we will have to require that β ( ω ) = K m /Z ( ω ). III. POWER FLOW
The instantaneous mechanical power due to the force F ( t ) is defined as (in symmetrizedform) P M = 12 h v ( t ) F ( t ) + F ( t ) v ( t ) i . (12)Using Eq. 1, it is found that P M = ddt (cid:28) m ˙ x ( t ) + U ( x ) (cid:29) + P M + P Mout + P Min , (13)where the first term on the left, d/dt < / m ˙ x ( t ) + U ( x ) > , is zero due to the fact that theexpectation value does not change with time. Moreover, P M is P M = Z ∞ µ ( τ ) 12 h ˙ x ( t ) ˙ x ( t − τ ) + ˙ x ( t − τ ) ˙ x ( t ) i dτ = 2 Z ∞ ω Re [ µ ( ω )] S x ( ω ) dω . (14)6ere P M is the power dissipated by the lossy mechanical element (e.g. viscous forces orcoupling to the radiation bath) in absence of electromagnetic coupling (as described in theprevious section), and was studied in ref. [27]. On the other hand, P Mout is the powertransferred out of the mechanical and into the electrical domain, P Mout ≈ K m h ˙ x ( t ) i ( t ) + i ( t ) ˙ x ( t ) i = 2 K m Z ∞ ω Re [ β ( ω )] S x ( ω ) dω , (15)whereas P Min is the power transferred into the mechanical domain from the electrical domain P Min ≈ K m h ˙ x ( t ) i ( t ) + i ( t ) ˙ x ( t ) i = 2 K m Z ∞ ωIm [ γ ( ω )] S i ( ω ) dω . (16)In the equations above S x ( ω ) = < x ( ω ) > is the position spectral density and S i ( ω ) = is the current spectral density, both due to the position and current fluctuations notinduced by the electromagnetic coupling (hence the subscript). We have used the first-orderapproximation discussed in the previous section to arrive at Eqs. 15 and 16, which thereforeare valid for weak coupling ( ω | β ( ω ) || γ ( ω ) | ≪ P Mout + P Min = 0, which means thatpower is either transferred into or out of the mechanical domain, with corresponding im-pact on the environment. However, in the special case that the system is in equilibrium,we must require P Mout + P Min = 0 such that the power transferred out of the mechan-ical domain into the electrical domain is the same as that coming into the mechanicalfrom the electrical domain. Comparing Eqs. 15 and 16 we then have ωRe [ β ( ω )] S x ( ω ) = − Im [ γ ( ω )] S i ( ω ). That this condition is fulfilled for a system near equilibrium can beseen by applying the fluctuation-dissipation theorem of Callen and Welton, which statesthat S x ( ω ) = ~ π Im [ α ( ω )] coth (cid:16) ~ ω k B T (cid:17) and S i ( ω ) = ~ π ωRe [ Z ( ω )] | Z ( ω ) | − coth (cid:16) ~ ω k B T (cid:17) [30].Here k B is Boltzmann’s constant and h = 2 π ~ is Planck’s constant. Thus, near equilibriumthe net dissipated mechanical power is given by Eq. 14.The total power in the electrical system is composed of one component flowing into themechanical domain and another being dissipated in the electrical circuits. The componentflowing into the mechanical domain is, due to energy conservation, given by Eq. 16, andis not of particular interest here. Of more interest is it to find the power dissipated in theelectrical circuit, since that power can be utilized in the electrical energy harvesting system.The average dissipated electrical power is given by P E = (1 / < vi + iv > , where thevoltage is related to the current through the impedance Z ( t ) as v ( t ) = R ∞ Z ( τ ) i ( t − τ ) dτ ,7hich in the frequency domain becomes v ( ω ) = Z ( ω ) i ( ω ) with Z ( ω ) = R ∞ Z ( τ ) exp ( jωτ ) dτ .The total power dissipated in the electrical circuit is given by P E = Z ∞−∞ Z ( − ω ) < i ( ω ) > dω = P E + P E , (17)where P E ≈ Z ∞ Re [ Z ( ω )] S i ( ω ) dω , (18)is due to thermal current fluctuations independent of the electromagnetic coupling, and P E ≈ Z ∞ Re [ Z ( ω )] ω | β ( ω ) | S x ( ω ) dω . (19)is the power flowing into the electrical circuit from the mechanical system. To obtain Eqs.18 and 19 we have used the first order approximation discussed in the previous section.By noting that Re [ β ( ω )] = Re [ Z ( ω )] / | Z ( ω ) | − , it is seen that P E = P Mout . Thus, thepower transferred out of the mechanical domain is dissipated in the electrical circuit, andenergy conservation is ensured in the model used here. It is interesting to note that thepower dissipated in the electrical circuit is not the same as the power transferred from themechanical domain. One may define a ratio η EM = P E /P Mout , which becomes η EM ≈ R ∞ Re [ Z ( ω )] S i ( ω ) dωK m R ∞ ω Re [ β ( ω )] S x ( ω ) dω , (20)where we have used that P E = P Mout . It is seen that for a system in thermal equilibriumthat η EM >
1, where the additional (noise) power dissipated in the electrical circuit occursas a result of the thermal fluctuations. In general, one needs to know the spectral densities( S i ( ω ) and S x ( ω )), the impedance Z ( ω ) and the susceptibility α ( ω ) in order to compute thedissipated power in an electrical circuit using the formalism presented above. We will in thenext section look at a simple example which allows us to connect the theory presented herewith some special cases previously encountered in the literature. IV. ELECTRICAL POWER DISSIPATION IN PRESENCE OF A RESISTIVELOAD
Let us consider a harvesting system where there is an electromechanical coupling betweenthe electromagnetic and the mechanical systems, and where the electrical circuit has an in-ternal resistance R i connected to a resistive load R L . The friction coefficient is constant8uch that Re [ µ ( ω )] = c . An estimate suggests that in order to use the first-order approx-imation of equations 10 and 11 one needs to set K m / ( R L + R i ) c ≪
1. As pointed out byWeber, equivalent circuits are not always good models of physically reliable circuits underall conditions[31]. Deviations are in particular expected at high frequencies, i.e. when thesmallest wavelength of the electromagnetic waves becomes comparable to the size of thecircuit. However, it should be emphasized that due to the natural frequency filtering effectof the mechanical oscillator, such high frequencies are ruled out in basically all practicalsystems. We will also neglect radiation reaction forces associated with self-induced currentsas studied in Ref. [32] since these only become important at very high frequencies.For the moment we assume that the mechanical system is forced by an external force f m ≫ f em and f m ≫ f f . The system oscillates sinusoidally at the resonance frequency ω such that S x ( ω ) = ( x / δ ( ω − ω ). Here we would like to evaluate the electrical powerdissipated in the load resistor R L , which is different from the total dissipated power givenby Eq. 17. To maximize the power transfer, one requires impedance matching with R L = R i as the resistance of the load. The power to be evaluated is now given by P Eload = < v L i > ,where v L = R L i , such that the power dissipated in the load can be found to be P ELoad ≈ K m R L ω x k B T ∆ f . (21)Here ∆ f is the frequency bandwidth of the electrical circuit. It should be emphasized thatthe first part of Eq. 21 is the same as Eq. 47 in Ref. [6] for an impedance-matched load.The second part of Eq. 21 is just the noise power associated with thermal fluctuations in theelectrical circuit, as was found by Nyquist[33]. Note that both the internal resistance and theload resistor generate equal amounts of thermal noise power, such that in thermal equilibriumthere is no net power flow in the circuit. It is seen that in the simple approximationconsidered here the total power dissipated in the electrical load is just the sum of thespecial cases considered in previous studies. Only in the case that the vibration amplitudeis very small, x ∼ √ k B T ∆ f R L / ( ω x K m ), the Nyquist noise power becomes comparableto the power transferred due to electromagnetic coupling. However, when the latter termbecomes this small, it would be more correct to look at the spectral densities caused bythermal fluctuations, both for position and current.To this end, we now set f m = 0 and consider a thermally fluctuating system where S x ( ω ) = ~ π Im [ α ( ω )] coth (cid:16) ~ ω k B T (cid:17) and S i ( ω ) = ~ π ωRe [ Z ( ω )] | Z ( ω ) | − coth (cid:16) ~ ω k B T (cid:17) . Furthermore, we9ssume that ~ ω/k B T ≪ ω =2 π ∆ f . Under such circumstances the power dissipated in the load can be found as P ELoad ≈ P Etransfer + P Ethermal = K m R L k B Tm + k B T ∆ f . (22)For a macroscopic system the thermal fluctuations are negligible, and the noise powerpredicted by Eq. 22 is under normal circumstances indeed orders of magnitude smaller thanthe power generated in the energy harvesting system reported in e.g. Refs. [1, 2]. However,when the system becomes smaller, approaching the micro and nanometer scale, one expectsthe noise power to become increasingly important. Here it should be emphasized that,unlike the Nyquist noise power, the transferred power from the mechanical domain scalesinversely with the system size. To see an example of this, assume that the harvesting systemis composed of a cube of sides l p , and that a conductor of length 4 l p , width l p and thickness t (where t ≪ l p ) is wrapped around it such that it makes one turn. In order to obtainan analytical expression, we will now assume that K m ≈ Bl p . Although the exact fielddistribution and geometry is not accounted for in such a simple expression, it does give acorrect order of magnitude, and is therefore a useful first approach. The conductor acts likea thin shell covering four sides of the cube, similar to that of a colloid covered by a metallicshell. The technical details of connecting the conductor with the external load R L will notbe considered here. The mass of the system is m = ρ m l p + 4 ρ c l t , where ρ m is the density ofthe cube and ρ c the density of the conductor. The conductor, with its conductivity ρ e andsquare cross section A = l p t , exhibits an intrinsic resistance R i = 4 ρ e l p /A . Now the noisepower transferred from the mechanical to the electrical domain is expressed as P Etransfer ≈ B tk B Tρ e ( ρ m l p + 4 ρ c t ) . (23)Figure 1 shows this noise power when k B = 1 . × − J K − , T = 300 K, ρ m = 2000 kgm − , ρ c = 8960 kgm − , ρ e = 1 . × − Ω m and t = 1 nm. The solid line corresponds to B = 2 T and the dashed line to B = 0 . l p ∼ − m due to the fact that the thickness of the conductor has been assumed tobe constant. However, for l p ≫ − m the power decreases roughly as l − p . Other selectionsof which dimensions to hold constant and which to scale with system size may give rise todifferent power scaling, but the order of magnitude is not expected to change. It is seenthat for the model given here powers less than 10 − W can be expected. The dashed line10f Fig. 1 shows the Nyquist noise power ( P Ethermal ≈ k B T ∆ f ) assuming ∆ f = 1000 Hz,for comparison. Note that only for nanometer-sized systems the transferred mechanicalnoise (due to the electromagnetic coupling) power exceeds the Nyquist noise power for theparameters used here. V. CONCLUSION
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