Coupling Factor Between the Magnetic and Mechanical ... - IEEE Xplore

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between wire coils and miniature magnets to convert mechanical energy to electricity ... the magnetic induction field in the vicinity of the cylindrical magnet.
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Coupling Factor Between the Magnetic and Mechanical Energy Domains in Electromagnetic Power Harvesting Applications John Cannarella1 , Jerry Selvaggi2 , Sheppard Salon2 , John Tichy1 , and Diana-Andra Borca-Tasciuc1 Mechanical, Aerospace and Nuclear Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Micro-power generation is an area developing to support autonomous and battery-free wireless sensor networks and miniature electronic devices. Electromagnetic power harvesting is one of the main techniques for micro-power generation and it uses the relative motion between wire coils and miniature magnets to convert mechanical energy to electricity according to Faraday’s law of induction. Crucial for the design and analysis of these power systems is the electromechanical coupling factor , which describes the coupling between the is defined as : the product between the number of turns mechanical and electromagnetic energy domains. In current literature in the coil ( ), the average magnetic induction field ( ), and the length of a single coil turn ( ). This paper examines the validity of the current definition and presents two case studies involving cylindrical permanent magnets and circular coil geometries to demonstrate its limitations. The case studies employ a numerical method for calculating which uses the toroidal harmonics technique to determine the magnetic induction field in the vicinity of the cylindrical magnet. Index Terms—Coupling factor, electromagnetic power generation, energy harvesting, toroidal harmonics.

I. INTRODUCTION

W

IRELESS sensor networks (WSN) have numerous potential applications from controlling energy resources [1] to continuous health monitoring [2]. In these networks each sensor must be energetically autonomous, which presents a major obstacle to the widespread implementation of WSNs. Using batteries is often not a viable solution: changing them poses a major problem for sensors physically embedded into objects/tissue. Moreover, even in consumer applications such as home automation, where a comparatively small number of sensors are employed, it is undesirable to change dozens of batteries every few months to maintain system operation [3]. However, wireless nodes typically demand small amounts of continuous power in their idle state and have only short periods of high power use when active. For many systems the power consumption is consequently in the range of a few mW and continues to decrease [4]–[7]. Hence it is now feasible to harvest ambient energy to power WSN nodes. The most appealing source for harvesting energy for WSN applications is ubiquitous ambient vibration. Mechanical vibrations with frequencies of up to 200 Hz [8] are found in numerous environments including common household appliances, industrial plant equipment, motor vehicles, and human body motion. At present, several methods for conversion of mechanical vibration energy to electric energy are being explored including electromagnetic [9]–[21], piezoelectric [22]–[28], and electrostatic [29]–[31] conversion. Among these methods, the simplest and most robust is electromagnetic power conversion, which uses a system of small magnets and coils vibrating relative to each other to induce a voltage across the coils according to Faraday’s

Manuscript received August 08, 2010; revised November 19, 2010; accepted February 22, 2011. Date of publication March 03, 2011; date of current version July 27, 2011. Corresponding author: D.-A. Borca-Tasciuc (e-mail: borcad@rpi. edu). Digital Object Identifier 10.1109/TMAG.2011.2122265

Fig. 1. Schematic of the vertically (z direction) oscillating coil setup.

law. Electromagnetic harvesters are already being commercialized (e.g., Ferro Solutions from the US [32] and Perpetuum from the UK [33]). In order to optimize such systems and extract maximum power output for a given set of design constrains, it is necessary to accurately model and predict system behavior. The analysis typically proceeds as follows. The basic vibrating coil system is modeled as wire coils attached to a seismic mass, which oscillates relative to a permanent magnet as shown in Fig. 1. The general equation of motion is then (1) where is the relative position between the seismic mass and the magnet, is the mass of the seismic mass, is the damping is a time-varying coefficient, is the spring constant, and force applied to the base of the device. The damping coefficient is a combination of mechanical and electrical damping . The mechanical damping damping is assumed to be viscous and the electrical damping is assumed to be solely due to the induced current in the coil; unwanted eddy currents are neglected, which is a reasonable assumption for most practical applications. The electrical

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CANNARELLA et al.: COUPLING FACTOR BETWEEN THE MAGNETIC AND MECHANICAL ENERGY DOMAINS

damping is proportional to velocity as will be subsequently demonstrated. The electrical damping force depends on the current in the coil such that

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TABLE I NUMERICAL VALUES USED IN SIMULATION

(2) where is the electromagnetic coupling factor, which is a function of magnetic induction and setup geometry [9], [10], [16], [18], [34]–[36]. The current in the coils is a function of the voltage induced in the coils during oscillation, while the induced voltage is proportional to the coil velocity:

(3) where

is the same electromagnetic coupling factor from (2), is the resistance of the electrical load of the device, and is the resistance associated with the coils and other undesirable electrical losses in the device. The total damping coefficient associated with the system can therefore be expressed by substituting (3) into (2) and adding to arrive at the mechanical damping component

Assumed values for numerical case studies.

The objective of this paper is to inspect the validity of the current definition (6). This exercise is carried out for a configuration employing circular coils vibrating in the vicinity of a cylindrical magnet, an arrangement similar to that which has been used in practice [10], [11], [37]. Two cases are analyzed: coilloop vibration in the axial and radial direction of the magnet, respectively. For each case the coupling factor is derived from fundamental electromagnetics equations and its numerical value , where is deteris compared to that provided by mined employing both arithmetic and integral averages. The magnetic induction in the vicinity of the magnet is calculated employing the toroidal harmonics technique [38]. II. SYSTEM CONFIGURATION AND MAGNETIC INDUCTION CALCULATION PROCEDURE

(4) The parameter of main interest is the instantaneous power delivered to an electrical load (the useful harvested power), which can be calculated from

(5) is From (4) and (5) one can see that the coupling factor important to the modeling of both the mechanical and electrical behavior of the vibrating coil energy harvester. This coupling factor is widely used in electromagnetic energy harvesting literature and commonly defined as [9]–[11], [34]–[36]

(6) is the average where is the number of turns in the coil, magnetic induction, and is the length of a single turn in the coil. Equation (6) is used irrespective of setup geometry and without defining how the average field is determined. In gendefinition is appropriate for geometries in which eral, the a straight length of wire moves through a constant magnetic induction field, but not for other more complicated setups. Specifically, the coupling factor can be strongly influenced based on how one chooses to define the average magnetic induction. For many setups, an average magnetic induction computation is not always straightforward, and potential errors may arise if an invalid definition is used. Therefore, a more accurate derivation of coupling factor may be imperative in certain situations.

The dimensions of the systems considered here are shown in the Table I and are relevant to practical applications [11]. In order to calculate an exact coupling factor for the magnet-coil system, it is necessary to be able to compute the magnitude and direction of the magnetic induction of the magnet at any location in space. The magnetic induction values are calculated using toroidal harmonics and verified using elliptic integrals. Both are numerical methods which allow off-axis computation of the radial and axial magnetic induction components produced by cylindrical permanent ferrite magnets. Full details of these procedures are provided in [38]. Briefly, with this method, the magnetic induction components are expressed in terms of the , as and magnetic scalar potential . All expressions are in cylindrical coordinates. For an axially polarized cylindrical permanent magnet of and diameter , can be expressed as a sefinite length ries of hypergeometric functions [39]. The procedure is straightforward but the resulting expressions are extremely complex and symbolic integration becomes computationally prohibitive, especially as more terms are used in the series. Consequently, a midpoint Riemann sum is used in lieu of symbolic integration wherever integration is employed in the numerical portions of the following case studies. III. RESULTS AND DISCUSSION A. Vertically Oscillating System First, the case of a circular single-loop coil oscillating coaxially at a height above a permanent magnet shown in Fig. 1 is considered. The voltage induced in the circular wire can be

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found using the principle of motional EMF and is generally expressed as [40]

(7) where is the velocity vector of the loop, and is the magnetic induction vector at the location of a differential length of wire . Because the coil loop and magnet are coaxial, the radial is constant along the component of the magnetic induction should be evaluated at entire length of the loop. Note that , where is the radius of the coil. Furthermore, , , and are all orthogonal, allowing (7) to be rewritten as

Fig. 2. The average radial magnetic field B as a function of peak-to-peak oscillation amplitude, assuming a minimum height of 1 mm above the magnet surface using an integral average (dots) and an arithmetic average (daggers).

(8)

The average magnetic induction can also be approximated by taking the arithmetic average:

(9)

(13) To compare the ability of the arithmetic average to approximate the integral average for the given configuration, the average magnetic induction is plotted as a function of the , assuming a minpeak-to-peak oscillation amplitude imum height of 1 mm above the magnet surface in Fig. 2. Thus, a distance of 0.003 m on the axis represents an oscillation from 0.001 m to 0.004 m. From Fig. 2 it can be seen that the is an acceptable approximation arithmetic average for for the integral average for small oscillations. However, as oscillation amplitude increases, the arithmetic average incurs a greater amount of error due to the nonlinear nature of the magnetic induction’s decay as a function of distance from the magnet. The error is only 2% for a 1 mm peak-to-peak amplitude, but increases up to 90% for an amplitude of 1 cm. This error propagates to the coupling factor. Moreover, the simple multiplication of the single-turn coil coupling factor by to arrive at the multi-turn coil coupling factor can increase the error because each turn is actually exposed to a slightly different , which varies with . This error is minimal for thin tightly wound coils, but increases with coil size.

Evaluating the line integral yields

where is the circumference of the coil loop. From (8) one can for a single loop. For a coil see that the coupling factor is with turns, all exposed to the same magnetic induction, one finds . The subscript has been added to to is an instantaneous coupling factor and varies emphasize that with coil position. can be arrived at starting with the The same definition for expression for the force opposing the coil motion. The force on a current carrying wire in the presence of a magnetic field of induction is generally expressed as [40]

(10) where is the current in the wire and is a differential length vector in the direction of the current flow. For the vertically oscillating coil, the magnetic induction component of interest which is orthogonal to the is again the radial component vector and constant along the entire length of the coil loop. Thus, (10), in a similar manner as (7), reduces to (11) and as before, for turns coil. The equivalence of the two coupling factors is expected and is a consequence of energy conservation. To determine the average coupling factor for a single oscillamust be averaged over the vertical range of oscillation. tion, The most accurate way to do this is by taking the integral av: erage over the amplitude of the vibration

(12)

B. Horizontally Oscillating System Now consider the case of a circular coil oriented parallel to the flat surface of a cylindrical permanent magnet moving along the axis shown in Fig. 3. The magnetic induction of interest—the component orthogonal to the plane of coil motion—is now the . Replacing the axial component of the magnetic induction vector quantities in (7) with appropriate scalar quantities results in

(14)

In (14) is a function of radial distance from the central axis of the magnet and evaluated at a constant height above

CANNARELLA et al.: COUPLING FACTOR BETWEEN THE MAGNETIC AND MECHANICAL ENERGY DOMAINS

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Fig. 3. Schematic of coil magnet setup for a horizontally (x direction) moving coil. The z direction is out of the page along the axis of the permanent magnet.

the magnet, . To evaluate the line integral it is necessary to and terms to polar coordinates. can be convert the written as a function of by writing in terms of :

(15) where is the distance between centers of the coil and magnet along the radial coordinate and is the radius of the coil (Fig. 3). yields Substituting (15) into (14) and rewriting as

(16) From (16) one can see that the instantaneous coupling factor for a single coil moving in the direction is

(17) The same coupling factor expression can be arrived at by analyzing the force opposing the coil motion, . Referring back to (10), the expression for magnetic force on a current carrying wire, it is again necessary to replace the vector terms with the appropriate scalar quantities and then evaluate the integral using polar coordinates. The cross product in (10) can be replaced by , as only the components of the coil crossed with the magnetic induction in the direction will yield a force in the direction. When the magnetic induction and length vectors are rewritten in terms of , the force is given by

(18) It follows then that the instantaneous coupling factor is the same as (17). By inspection of (17) it is clear that for the case of the horizontally oscillating coil, the definition for instantaneous coupling factor is inappropriate and would yield erroneous

Fig. 4. Instantaneous coupling factor versus coil-magnet center-to-center spacing (dimension d in Fig. 3) using NBl definition with axial field (bold line) and definition in (18) (dots) for a single horizontally moving coil at above magnet surface. z

= 1 mm

results. Fig. 4 shows a plot of the coupling factor for a horizontally moving coil loop calculated using the parameters in Table I and the additional assumption that the loop oscillates at a height mm above the magnet. The solid line is the instantaof definition, subneous coupling factor calculated using the stituting for the axial magnetic induction at the center of the as a function of . The curve takes on the coil curve, which is expected since and are conshape of the stant. This curve starts at a maximum on the center axis of the magnet and decreases as the radial distance increases and the field lines begin to bend back on themselves, eventually going curve as given by (17) is negative. On the other hand, the clearly different in trend, starting at zero on the central magnet axis, reaching a maximum at the edge of the magnet, and then becoming negative before approaching zero as radial distance is is a result of the axial field increased. The change in sign of switching direction as the radial distance from the magnet is increased and the consequential reversal in EMF polarity. In this case, since the instantaneous coupling factor is signif, it is clear that a simple arithmetic icantly different from average for magnetic induction cannot be used to calculate the coupling factor associated with the oscillation. IV. CONCLUSION Two case studies of electromagnetic harvesters using cylindrical magnet and circular coil geometries have been analyzed. In the case of a coil oscillating perpendicular to the surface of definition the magnet, it was shown that the instantaneous of coupling factor is appropriate as long as only the radial component of the magnetic induction at the location of the coil is used for . It was further shown that for small oscillations, a simple arithmetic average of the magnetic induction would lead to relatively accurate results. In the case of a coil oscillating parallel to the magnet surface it was shown that the simple expression for instantaneous coupling factor definition breaks down because it does not take into account the vector nature of electromechanical coupling. The coupling factor in this case cannot be calculated using an average magnetic induction, and a line integral expression must be used instead. For other, more

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complicated geometries it is desirable to employ commercial electromagnetic field simulators to correctly determine the best arrangement that will ensure the maximum coupling between mechanical and magnetic domains, and consequently maximum power output. ACKNOWLEDGMENT D.-A. Borca-Tasciuc acknowledges NSF awards 0813598 and 0925733. REFERENCES [1] G. Piat, J. Wall, P. Valencia, and J. K. Ward, “The tiny agent—Wireless sensor networks controlling energy resources,” J. Networks, vol. 3, pp. 42–50, 2008. [2] X. F. Teng, Y. T. Zhang, C. C. Y. Poon, and P. Bonato, “Wearable medical systems for p-health,” IEEE Rev. Biomed. Eng., vol. 1, pp. 62–74, 2008. [3] R. A. Quinnell, “Networking moves to home automation,” Electronic Design, Strategy, News (EDN-US Edition), vol. 14, pp. 40–50, 2007. [4] J. Polastre, R. Szewczyk, and D. Culler, “Telos: Enabling ultra-low power wireless research,” in 4th Symp. Inf. Proc. Sens. Networks, 2005, pp. 364–369. [5] W. R. Davis, N. Zhang, K. Camera, F. Chen, D. Markovic, N. Chan, B. Nikolic, and R. W. Brodersen, “A design environment for high throughput, low power dedicated signal processing systems,” IEEE J. Solid-State Circuits, vol. 37, no. 3, pp. 420–431, Mar. 2002. [6] B. Otis, Y. H. Chee, and J. Rabaey, “A 400 W-RX, 1.6 mW-TX superregenerative transceiver for wireless sensor networks,” in IEEE Dig. Tech. Papers, 2005, vol. 1, pp. 396–406. [7] S. Roundy, E. Leland, J. Baker, E. Carleton, E. Reilly, E. Lai, B. Otis, J. M. Rabaey, P. K. Wright, and V. Sundararajan, “Improving power output for vibration-based energy scavengers,” Pervasive Comput., vol. 4, pp. 28–36, 2005. [8] S. Roundy, P. K. Wright, and J. Rabaey, “A study of low level vibrations as a power source for wireless sensor nodes,” Comp. Commun., vol. 26, pp. 1131–1144, 2006. [9] S. P. Beeby, R. N. Torah, M. J. Tudor, P. Glynne-Jones, T. O’Donnell, C. R. Saha, and S. Roy, “A micro electromagnetic generator for vibration energy harvesting,” J. Micromech. Microeng., vol. 17, pp. 1257–1265, 2007. [10] C. B. Williams and R. B. Yates, “Analysis of a micro-electric generator for microsystems,” Sens. Actuators, vol. 52, pp. 8–11, 1996. [11] R. Amirtharajah and A. P. Chandrakasan, “Self-powered signal processing using vibration-based power generation,” IEEE J. Solid State Circuits, vol. 33, no. 5, pp. 687–695, May 1998. [12] W. J. Li, Z. Wen, P. K. Wong, G. M. H. Chan, and P. H. W. Leong, “A micromachined vibration-induced power generator for low power sensors of robotic systems,” in Proc. Eighth World Automation Congress, Hawaii, USA, 2000, pp. 16–21. [13] N. N. H. Ching, H. Y. Wong, W. J. Li, P. H. W. Leong, and Z. Wen, “A laser-micromachined vibrational to electrical power transducer for wireless sensing systems,” in Proc. Transducers ’01, 11th Int. Conf. Solid-State Sens. Act., Munich, Germany, 2001, vol. 1, p. 38. [14] H. Kulah and K. Najafi, “An electromagnetic micro power generator for low-frequency environmental vibrations,” in Micro Electro Mechanical Systems 2004, 17th IEEE Intl. Conf. MEMS, 2004, pp. 237–240. [15] M. Mizuno and D. G. Chetwynd, “Investigation of a resonance microgenerator,” J. Micromech. Microeng, vol. 13, pp. 209–216, 2003. [16] T. Reissman, T. J. S. Park, and E. Garcia, “Micro-solenoid electromagnetic power harvesting for vibrating systems,” Proc. SPIE—Int. Soc. Opt. Eng., vol. 6928, pp. 692806–692806-9, 2008. [17] I. Sari, T. Balkan, and H. Kulah, “An electromagnetic micro power generator for wideband environmental vibrations,” Sens. Actuators, vol. 145, pp. 405–418, 2008.

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