Abstract:
Wireless power transfer is received using a magneto mechanical system. A magneto mechanical system may include an array of magneto-mechanical oscillators, wherein each oscillator may comprise a magnetic symmetrical part and a suspension engaged to the magnetic part. The system may further include a coil formed around the array and electromagnetically coupled to the oscillators to produce an electric current caused by electromagnetic coupling with the oscillators.

Description:
This application claims priority from provisional application No. 60/979,381, filed Oct. 11, 2007, the entire contents of which disclosure is herewith incorporated by reference. 
    
    
     BACKGROUND 
     Our previous applications have described magneto mechanical systems. Previous applications by Nigel Power LLC have described a wireless powering and/or charging system using a transmitter that sends a magnetic signal with a substantially unmodulated carrier. A receiver extracts energy from the radiated field of the transmitter. The energy that is extracted can be rectified and used to power a load or charge a battery. 
     Our previous applications describe non-radiative transfer of electrical energy using coupled magnetic resonance. Non-radiative, means that both the receive and transmit antennas are “small” compared to the wavelength, and therefore have a low radiation efficiency with respect to Hertzian waves. High efficiency can be obtained between the transmit antenna and a receive antenna that is located within the near field of the transmit antenna. 
     SUMMARY 
     The present application describes techniques for capturing wireless power based on a magnetic transmission. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the Drawings: 
         FIG. 1  shows a block diagram of induction between transmit and receive loops; 
         FIG. 2  shows an elemental torsion pendulum 
         FIG. 3  shows a dynamo receiver; 
         FIGS. 4A and 4B  show flux and field strength within a sphere; 
         FIG. 5  shows an integrated embodiment; 
         FIG. 6  shows a disc shaped array; 
         FIG. 7  illustrates how a coil can be wound around the disc shaped array. 
     
    
    
     DETAILED DESCRIPTION 
     The classical principle of non-radiative energy transfer is based on Faraday&#39;s induction law. A transmitter forms a primary and a receiver forms a secondary separated by a transmission distance. The primary represents the transmit antenna generating an alternating magnetic field. The secondary represents the receive antenna that extracts electrical power from the alternating magnetic field using Faraday&#39;s induction law. 
                 -     μ   0       ⁢       ∂     H   ⁡     (   t   )           ∂   t         =     ∇     ×     E   ⁡     (   t   )                 
where ∇×E(t) denotes curl of the electrical field generated by the alternating magnetic field
 
     The inventors recognize, however, that the weak coupling that exists between the primary and secondary may be considered as a stray inductance. This stray inductance, in turn, increases the reactance, which itself may hamper the energy transfer between primary and secondary. 
     The transfer efficiency of this kind of weakly coupled system can be improved by using capacitors that are tuned to the precise opposite of the reactance of the operating frequency. When a system is tuned in this way, it becomes a compensated transformer which is resonant at its operating frequency. The power transfer efficiency is then only limited by losses in the primary and secondary. These losses are themselves defined by their quality or Q factors. 
     Compensation of stray inductance may also be considered as part of the source and load impedance matching in order to maximize the power transfer. Impedance matching in this way can hence increase the amount of power transfer. 
       FIG. 1  illustrates impedance matching between the transmit and receive portions of a non-radiative system. 
     As the distance D between the transmitter  100  and the receiver  150  increases, the efficiency of the transmission can decrease. At increased distances, larger loops, and/or larger Q factors may be used to improve the efficiency. However, when these devices are incorporated into a portable device, the size of the loop may be limited by the parameters of the portable device. 
     Efficiency can be improved by reducing antenna losses. At low frequencies such as less than 1 MHz, losses can be attributed to imperfectly conducting materials, and eddy currents in the proximity of the loop. 
     Flux magnification materials such as ferrite materials can be used to artificially increase the size of the antenna. Eddy current losses are inherently reduced by concentrating the magnetic field. 
     Special kinds of wire can also be used to lower the resistance, such as stranded or litz wire at low frequencies to mitigate skin effect. 
     An alternative to non-radiative transfer uses a magneto mechanical system as described in our co-pending application Ser. No. 12/210,200, filed Sep. 14, 2008. This picks up energy from the magnetic field, converts it to mechanical energy, and then reconverts to electrical energy using Faraday&#39;s induction law. 
     According to an embodiment, the magneto mechanical system may be part of an energy receiving system that receives energy from an alternating magnetic field. 
     According to an embodiment, the magneto mechanical system is formed of a magnet, e.g. a permanent magnet, which is mounted in a way that allows it to oscillate under the force of an external alternating magnetic field. This transforms energy from the magnetic field into mechanical energy. 
     Assume a charged particle moving at a velocity γ and a magnetic field H.
 
 F=qμ   0 ( v×H )
 
     In an embodiment, this oscillation uses rotational moment around an axis perpendicular to the vector of the magnetic dipole moment m, and is also positioned in the center of gravity of the magnet. This allows equilibrium and thus minimizes the effect of the gravitational force. A magnetic field applied to this system produces a torque of
 
 T=μ   0 ( m×H )
 
     This torque aligns the magnetic dipole moment of the elementary magnet along the direction of the field vector. The torque accelerates the moving magnet(s), thereby transforming the oscillating magnetic energy into mechanical energy. 
     The basic system is shown in  FIG. 2 . The magnet  200  is held in place by a torsion spring  210 . This torsion spring holds the magnet in position shown as  201  when no torque from the magnetic field is applied. This no-torque position  201  is considered θ=0. 
     Magnetic torque causes the magnet  200  to move against the force of the spring, to the position  202 , against the force of the spring with spring constant K R . The movement forms an inertial moment I that creates a torsion pendulum that exhibits a resonance at a frequency proportional to the square root of the ratio K R  over I. 
     Frictional losses and electromagnetic radiation is caused by the oscillating magnetic dipole moment. 
     If this system is subjected to an alternating field H AC  at the resonance of the system, then the torsion pendulum will oscillate with an angular displacement data depending on the intensity of the applied magnetic field. 
     According to another embodiment, some or all of the torsion spring is replaced by an additional static magnetic field H DC . This static magnetic field is oriented to provide the torque
 
 T=μ   0 ( m×H   DC )
 
     Another embodiment may use both the spring and a static magnetic field to hold the device. 
     The mechanical energy is reconverted into electrical energy using ordinary Faraday induction e.g. the Dynamo principle. This can be used for example an induction coil  305  wound around the magneto electrical system  200  as shown in  FIG. 3 . A load such as  310  can be connected across the coil  305 . This load appears as a mechanical resistance. The load dampens the system and lowers the Q factor of the mechanical oscillator. In addition, when the coil has a load across it, the eddy currents in the magnets may increase. These eddy currents will also contribute to system losses. 
     In an embodiment, the Eddy currents are produced by the alternating magnetic field that results from the coil current. Smaller magnets in the magneto system may reduce the eddy currents. According to an embodiment, an array of smaller magnets is used in order to minimize this eddy current effect. 
     A magneto mechanical system will exhibit saturation if the angular displacement of the magnet reaches a peak value. This peak value can be determined from the direction of the external H field or by the presence of a displacement stopper such as  315  to protect the torsion spring against plastic deformation. This may also be limited by the packaging, such as the limited available space for a magnet element. 
     According to one embodiment, optimum matching is obtained when the loaded Q becomes half of the unloaded Q. According to an embodiment, the induction coil is designed to fulfill that condition to maximize the amount of output power. 
     When using an array of such moving magnets, there may be mutual coupling between the magnets forming the array. This mutual coupling can cause internal forces and demagnetization. According to an embodiment, the array can be radially symmetrical, e.g., spheroidal, either regular or prolate, as shown in  FIGS. 4A and 4B .  FIG. 4A  shows the parallel flux lines of a magnetized sphere. This shows the magnetic flux density B.  FIG. 4B  shows the magnetic field strength in a magnetized sphere. From these figures that can be seen that there is effectively zero displacement between magnets in a spheroid shaped three-dimensional array. 
     Therefore, the magnets are preferably in-line with the axis of the spheroid shown as  400 . This causes the internal forces to vanish for angular displacement of the magnets. This causes the resonance frequency to be solely defined by the mechanical system parameters. A sphere has these advantageous factors, but may also have a demagnetization factor is low as ⅓, where an optimum demagnetization factor is one. Assuming equal orientation of axes in all directions, a disc shaped array can also be used. Discs have a magnetization factor that is very high, for example closer to 1. 
     Magnetization factor of a disc will depend on the width to diameter ratio. The shaped elements also have a form factor that is more suitable for integration into a device, since spheroids have a flat part that may be more easily used without increasing the thickness of the structure 
     The following is a comparison of magneto-mechanical systems with classical ferrimagnetic materials (ferrites). Ferrimagnetic materials or ferrites may be modeled as a magneto-mechanical system or conversely, magneto-mechanical systems may be considered as ferrites with special properties that may not be achievable with the classical ferrite materials. This will be shown in the following: 
     In ferrimagnetic substances, the magnetic moments of adjacent atoms are aligned opposite like in antiferromagnetic materials but the moments do not fully compensate so that there is a net magnetic moment. However, this is less than in ferromagnetic materials that can be used for permanent magnets. 
     Even though there are weaker magnetic effects, some of these ferrimagnetic materials, known as ferrites, have a low electrical conductivity. This makes these materials useful in the cores of AC inductors and transformers since induced eddy currents are lower. 
     A low electrical conductivity can also be found in a magneto-mechanical system composed of a multitude of small elementary magnets that are mutually electrically isolated so that eddy currents are attenuated. 
     The crystalline Ferromagnetic and ferrimagnetic materials are typically structured in magnetic domains also called Weiss domains. Atoms in a domain are aligned so that a net magnetic moment results. These domains may be considered as the magnets of a magneto-mechanical system. 
     In many magnetic materials, to a varying degree, the domain magnetization tends to align itself along one of the main crystal directions. This direction is called the easy direction of magnetization and represents a state of minimum energy. In a ferrite material, the directions of crystal domains may be considered randomly oriented so that there is complete cancellation and the resultant net magnetic moment at macroscopic level is zero, if no external magnetic field is applied. This is in contrast to the magneto-mechanical systems where the “elementary” magnets are equally oriented. 
     To rotate the magnetic moment of a crystalline domain in another (non-easy) direction, a certain force and work is required depending on the angle of rotation. Such work is performed if the ferrimagnetic material is subjected to an external magnetic field. The underlying physical phenomenon is Lorentz force applied to the magnetic moment, as described above. 
     The torsion spring (mechanical or magnetic) of a magneto-mechanical system sets the magnetic orientation of domains back to their state of minimum energy. If the external field is removed, it may be considered as the torsion spring of a magneto-mechanical system. Since crystal domains in ferrites have different shapes and sizes, they appear as different spring constants. Another embodiment uses elementary oscillators which all have an equal spring constant. 
     Stronger external fields cause more domains to be aligned or better aligned to the direction given by the external magnetic field. This effect is called magnetic polarization. This may be mathematically expressed as
 
 B=μ   0   H+J=μ   0 ( H+M )=μ 0 μ r   H  
 
     where J is the magnetic polarization, M is the magnetization, and μ r  the relative permeability. 
     The magnetization effect may be considered as a magnification of the magnetic flux density at the receive location by the factor μ r  using rotatable magnetic moments. This principle of local magnification of magnetic flux density is inherent to the magneto-mechanical system described above. Thus a relative permeability may be attributed to a magneto-mechanical system. In a resonant system, this relative permeability will be a function of frequency and reaches a maximum close to the resonance frequency. 
     Another mechanism for changing the domain magnetization which may occur in ferrite materials is the direction of magnetization remains the same but the volumes occupied by the individual domains may change. This process, called domain wall motion, the domains whose magnetization direction is closest to the filed direction grow larger while those that are more unfavorably oriented shrink in size. 
     This kind of magnetization process differs from that of a magneto-mechanical system as described above. If the external magnetic field is continuously increased, the ferrite material will be progressively magnetized until a point of saturation is reached. Saturation is a state where net magnetic moments of domains are maximally aligned to the external magnetic field. 
     Magneto-mechanical systems, as described above, saturate when the angular displacement of elementary magnets reaches the maximum peak angular displacement. The dynamic behavior when an alternating external magnetic field is applied is different. For this purpose the magnetization process of a bulk ferrite material can be considered. Considering a typical magnetization curve (M as a function of the external field H) of a ferrite, three major regions can be identified in which the ferrite shows different dynamic behavior. 
     At low magnetization, domain wall movements and rotations are mainly reversible. Being reversible means that the original magnetization condition can be returned when the external field is increased and then again decreased to its original field strength, other than hysteresis effects. 
     The second region of the magnetization curve is one in which the slope of magnetization (M vs. H) is greater and in which irreversible domain wall motion occurs. 
     The third section of the curve is one of irreversible domain rotations. Here the slope is very flat indicating the high field strength that is required to rotate the remaining domain magnetization in line with the external magnetic field. 
     Irreversible domain wall motion or domain rotation explains the well known hysteresis in the magnetization curve that is presented by all ferrites in a more or less pronounced manner. Hysteresis means that the magnetization or the induction B lags relative to the external magnetic field. As a consequence, the induction B at a given field H cannot be specified without knowledge of the previous magnetic history of the ferrite sample. Thus hysteresis may be considered as memory inherent to the material. 
     The area included in a hysteresis loop is a measure of the magnetic losses incurred in a cyclic magnetization process e.g. as resulting from an alternating external magnetic field. 
     With respect to the application of wireless energy transfer, there will be a requirement to drive a ferrite at least into the second region of magnetization where hysteresis losses typically become significant. This requirement is different e.g. for a communication receiver antenna. This is, however, not further shown here. 
     At higher frequencies two major loss contributors can be identified in ferrite materials: 
     hysteresis losses due to irreversible domain changes; and 
     eddy current losses due to residual conductivity in the ferrite. Hysteresis losses increase proportionally with frequency as the energy to cycle once around the hysteresis loop is independent of the speed. Eddy current losses have the effect of broadening the hysteresis loop. 
     Magneto-mechanical systems using a torsion spring as described above are largely hysteresis-free, where irreversible effects are concerned. At higher frequencies eddy current losses must be expected too. At lower frequencies (&lt;&lt;1 MHz) a magneto-mechanical system has the potential to provide high Q-factors at levels close to saturation. 
     For alternating fields, a ferrite core material may be characterized by its complex permeability
 
μ=μ′+ jμ″ 
 
     The real and imaginary part represent the permeability with the magnetization in phase and in quadrature to the external field, respectively. 
     The two permeabilities can often be found plotted in data sheets for ferrite materials. Typically, the real component is fairly constant with frequency, rises slightly, then falls rapidly at higher frequencies. The imaginary component on the other hand first rises slowly and then increases quite abruptly where the real component is falling sharply. 
     The maximum in μ′ that occurs shortly before cut-off is ferrimagnetic resonance. Ferrimagnetic resonance is an intrinsic property of a ferrite material and may be considered as the upper frequency at which the material can be used. It is also observed that the higher the permeability μ′ of the material, the lower the frequency of the ferrimagnetic resonance. This phenomenon of resonance indicates domain rotation, a counter torque (spring), and a certain inertial moment. It can be shown that the resonance frequency depends on the so-called gyromagnetic ratio. 
     Ferrites show a resonance similar to a magneto-mechanical system however with a too low Q-factor so that this effect cannot be technically exploited to get materials with high permeability μ′ at a specified frequency. 
     Gyromagnetic resonance with high Q-factors (up to 10,000) can be observed at microwave frequencies (&gt;1 GHz) in certain ferrite materials (e.g. Yttrium Iron Garnets) if the material is subjected to strong static magnetic fields. This effect, which is based on electron spin precession, can be exploited to build microwave components such as circulators, isolators, high-Q filters and oscillators. Non-radiative energy transfer using coupled magnetic resonance in the microwave range would however be limited to extremely short range. 
     Gyromagnetic resonance may be considered as a magneto-mechanical system at the atomic level. A difference is however that magnetic moments are processing around the field lines of the static magnetic field rather than oscillating axially. In both cases there is, however, a moving magnetic moment and an angular displacement. 
     Therefore, it can be seen that the magneto mechanical systems can use ferrimagnetism and gyromagnetism as part of their energy transfer. 
     A magneto-mechanical system may be formed of a single permanent magnet or of a multitude (an array) of elementary magnets. Theoretical analyses shows that: 
     the ratio of magnetic moment-to-inertial moment increases with the number of elementary magnets. This ratio is similar to the gyromagnetic ratio known from ferromagnetism. 
     the performance of the magneto-mechanical system increases with this ratio of moments 
     A figure of merit for the performance of a magneto-mechanical system 
               k   c     =       P   av         H   AC   2     ⁢     V   S               
where P av  denotes the power that is available under the condition of optimum matching, H AC  is the external alternating magnetic field strength, and Vs the volume required by the magneto-mechanical system. This figure of merit, which is called the specific power conversion factor, is indicative of how much power per unit system volume can be extracted from an alternating magnetic field, H AC′  if penduli are perpendicularly oriented to the direction of the exciting magnetic field.
 
     Theoretical analysis using the assumption of rod magnets of length l em  shows that for a given system Q-factor and operating frequency, the specific power conversion factor increases inversely proportional to l em   2 ; and thus to N e   2/3  where N e  is the number of elementary oscillators fitting into the unit system volume. This equation does not hold for items in saturation, which means that the angular displacement of the torsion penduli is not limited by stoppers. This is a very interesting result indicating the advantage of an array of elementary magnets over a single oscillating magnet. 
     Higher specific power conversion factors can have lower field strengths where the system saturates. 
     As a consequence of saturation, at a given frequency there exists an upper bound for the available power per unit system volume, which depends on 
     the maximum peak angular displacement θ peak    
     the strength of the external alternating magnetic field H AC . 
     Theory shows that this upper bound linearly increases with H AC . This upper bound is an important design parameter for a magneto-mechanical system. It also shows that there exists some degree of freedom to design magneto-mechanical systems as long as the ratio 
                 Q   UL     ·     H   AC         l   em   2           
remains constant, where Q UL  is the unloaded Q-factor of the magneto-mechanical system.
 
     The above analysis shows that using an array of micro magneto-mechanical oscillators enables the design of a system with a performance better than anything achievable in practice with a single macro oscillator. A macro sized oscillator would require an extremely high Q-factor that could not be realized in a mechanical system. 
     Another embodiment uses micro-electromechanical systems (MEMS) to create the magneto mechanical systems.  FIG. 5  shows one embodiment of forming an array of magneto mechanical oscillators using MEMS technology. 
     An array  500  may be formed of a number of magnet elements such as  502 . Each magnet elements  502  is formed of two U-shaped slots  512 ,  514  that are micro-machined into a silicon substrate. A permanent rod magnet  504 ,  506  of similar size is formed within the slots. The magnet may be 10 μm or smaller. At the micrometer level, crystalline materials may behave differently than larger sizes. Hence, this system can provide considerable angular displacement e.g. as high as 10°. This may provide the ability to increase the Q factor of such a system. 
     The magnet itself may be on the order of 10 μm or smaller. These devices may be formed in a single bulk material such as silicon. The magnets  504 ,  506  can have a high magnetization e.g. higher than 1 Tesla. 
     The magnet itself is composed of two half pieces, one piece attached to the upper side and the other piece attached to the lower side. Preferably these devices are mounted so that the center of gravity coincides with the rotational axes. 
     The device may be covered with a low friction material, or may have a vacuum located in the area between the tongue and bulk material in order to reduce type the friction. 
       FIG. 6  shows a cut through area of a three-dimensional array of magnets. In one embodiment, the array itself is formed of a radial symmetric shape, such as disc shaped. The disc shaped array of  FIG. 6 ,  600  may provide a virtually constant demagnetization factor at virtually all displacement angles. In this embodiment, an induction coil may be wound around the disc to pick up the dynamic component of the oscillating induction field generated by the MEMS-magneto mechanical system. The resulting dynamic component of the system may be expressed as
 
 m   x ( t )=| m |·sin θ( t )· e   x  
 
       FIG. 7  illustrates how the induction coil can be wound around the disc. 
     Mathematical equations for the power that can be transferred through a magneto-mechanical system per unit system volume can be derived in terms of
         system parameters such as geometry (e.g. size or number of elementary oscillators)   material properties   frequency   external alternating magnetic field strength       

     Equations for the maximum available power are determined under the constraints of a limited angular displacement and Q-factor of the magneto-mechanical oscillator. 
     These equations analyze the potential of magneto-mechanical systems and to find optimum design parameters. 
     A primary system parameter is a parameter that is independent of any other parameter of the set and thus cannot be expressed as a function of another parameter. 
     To analyze the system, the following set of primary parameters have been chosen: 
     V S : Volume of magneto mechanical system [m 3 ]. 
     l em : Length of elementary rod magnet [m] 
     ρ em : Length-to-radius ratio of elementary magnet 
     ν em : Specific volume of elementary magnet in [m 3 /kg] 
     H em : Internal magnetic field strength of elementary magnet [A/m] 
     α: Fill factor (Ratio of total magnetic volume to system volume) 
     Q UL : Unloaded Q-factor of mechanical resonator(s). It includes the losses due to mechanical friction, radiation, and due to conversion from mechanical to electrical energy. 
     θ peak : Maximum peak displacement angle of magnet rod supported by the mechanical resonator [rad] ______ f 0 : Resonance frequency [Hz] 
     H AC : Externally applied alternating magnetic field [A/m] 
     P av     —     mech : Available mechanical power. (maximum power into load) 
     Secondary system parameters and physical quantities include:
         r em : Radius of elementary rod magnet [m] (=l em /ρ em )   V em : Volume of an elementary magnet=l em   3 π/ρ em      V e : Volume required by an elementary system (resonator) (=V em /α)   N e : Number of elementary magnets in system volume=V S /V e      I: Moment of inertia of elementary magnet [kg m 2 ]. It is a function of ν em , l em , and ρ em      K r : Torsion spring constant [kg m 2  s −2 ]. It is a function of Q UL , f 0 , and I   Γ s : Dynamic rotational friction (angular velocity proportional to torque) representing all system losses [kg m 2  s −1 ]. It is a function of Q UL , f 0 , and I and includes the losses due to mechanical friction, radiation, and due to conversion from mechanical to electrical energy.   Γ L : Load equivalent dynamic rotational friction [kg m 2  s −1 ].     ω : Angular velocity of oscillating elementary system   m: magnetic moment (vector) [Am 2 ]. It is a function of l em , ρ em , and H em      θ: Displacement angle [rad]   φ: Angle between magnetic moment vector at zero displacement and vector of externally applied alternating magnetic field [rad]
 
There is an analogy between linear electrical systems composed of inductances, capacitance, and resistances; and a rotational mechanical system formed of a torsion spring, inertial moment, and dynamic friction (angular velocity proportional to torque). This analogy is shown in Table 1.
       

     
       
         
               
               
               
             
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Electrical system 
                   
                 Rotational mechanical system 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Physical quantities: 
               
             
          
           
               
                 I (current) 
                 
                           
                 
                 T (torque) 
               
               
                 U (voltage) 
                 
                           
                 
                   ω  (angular velocity) 
               
               
                 Component parameters: 
               
               
                 L 
                 
                           
                 
                 1/K r   
               
               
                 C 
                 
                           
                 
                 I 
               
               
                 R p   
                 
                           
                 
                 1/Γ 
               
               
                 Resonance freguency: 
               
               
                 
                   
                     
                       
                         
                           f 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                           
                         
                         = 
                         
                           1 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               LC 
                             
                           
                         
                       
                     
                   
                 
                 
                           
                 
                 
                   
                     
                       
                         
                           f 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                           
                         
                         = 
                         
                           
                             1 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                               ⁢ 
                               
                                   
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 K 
                                 r 
                               
                               I 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Unloaded Q (parallel circuit): 
               
               
                 
                   
                     
                       
                         
                           Q 
                           UL 
                         
                         = 
                         
                           
                             R 
                             p 
                           
                           ⁢ 
                           
                             
                               C 
                               L 
                             
                           
                         
                       
                     
                   
                 
                 
                           
                 
                 
                   
                     
                       
                         
                           Q 
                           UL 
                         
                         = 
                         
                           
                             1 
                             Γ 
                           
                           ⁢ 
                           
                             
                               
                                 K 
                                 r 
                               
                               ⁢ 
                               I 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
     Derivations of equations are shown below. From the resonance condition one gets the torsion spring constant:
 
 K   r =(2π f   0 ) 2   I  
 
     It is assumed that the optimum matching condition
 
Γ L =Γ S  
 
     can be achieved with the magneto-electrical transducer (induction coil plus±load). From Q-factor equation (see Table 1), the dynamic frictions become: 
     
       
         
           
             
               Γ 
               s 
             
             = 
             
               
                 Γ 
                 L 
               
               = 
               
                 
                   
                     
                       
                         K 
                         r 
                       
                       ⁢ 
                       I 
                     
                   
                   
                     Q 
                     UL 
                   
                 
                 = 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       0 
                     
                     ⁢ 
                     I 
                   
                   
                     Q 
                     UL 
                   
                 
               
             
           
         
       
     
     Using above defined parameters, the magnetic moment of an elementary magnet may be expressed as:
 
 m=V   em   ·H   em  
 
     and the moment of inertia: 
     
       
         
           
             I 
             = 
             
               
                 
                   V 
                   em 
                 
                 · 
                 
                   l 
                   em 
                   2 
                 
               
               
                 12 
                 ⁢ 
                 
                   υ 
                   em 
                 
               
             
           
         
       
     
     Based on the well-known torque equation above, the RMS value of the driving torque becomes:
 
 T=m·μ   0   H   AC ·sin(φ)
 
     Applying Kirchhoffs node law provides the following relation between the torques in the circuit.
 
 T−T   K     r     −T   l   −T   Γ     S     −T   Γ     L   =0
 
     At resonance frequency, we get by definition:
 
 T   K     r     =−T   l  
 
hence
 
 T=T   Γ     S     +T   Γ     L    
 
     and from matching condition: 
     
       
         
           
             
               T 
               
                 Γ 
                 t 
               
             
             = 
             
               T 
               2 
             
           
         
       
     
     The available mechanical power per elementary system may now be simply expressed as: 
     
       
         
           
             
               T 
               
                 Γ 
                 L 
               
             
             = 
             
               T 
               2 
             
           
         
       
     
     Using the above equations, the following relation on the total power available from the entire magneto mechanical system can be obtained 
     
       
         
           
             
               P 
               av_mech 
             
             = 
             
               
                 3 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                 
               
               · 
               
                 
                   
                     μ 
                     0 
                     2 
                   
                   ⁢ 
                   
                     H 
                     em 
                     2 
                   
                   ⁢ 
                   
                     Q 
                     UL 
                   
                   ⁢ 
                   
                     υ 
                     em 
                   
                   ⁢ 
                   
                     V 
                     s 
                   
                   ⁢ 
                   α 
                 
                 
                   
                     f 
                     0 
                   
                   ⁢ 
                   
                     l 
                     em 
                     2 
                   
                 
               
               · 
               
                 
                   ( 
                   
                     
                       H 
                       AC 
                     
                     · 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         φ 
                         ) 
                       
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     This equation indicates that for given Q UL  and frequency, the available power increases inversely proportionally to the length of an elementary rod magnet, disregarding the resulting angular displacement. For the peak angular displacement of an elementary oscillator we get: 
     
       
         
           
             
               θ 
               peak 
             
             = 
             
               
                 
                   3 
                   ⁢ 
                   
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     π 
                     2 
                   
                 
               
               · 
               
                 
                   
                     μ 
                     0 
                   
                   ⁢ 
                   
                     H 
                     em 
                   
                   ⁢ 
                   
                     Q 
                     UL 
                   
                   ⁢ 
                   
                     υ 
                     em 
                   
                 
                 
                   
                     f 
                     0 
                     2 
                   
                   ⁢ 
                   
                     l 
                     em 
                     2 
                   
                 
               
               · 
               
                 H 
                 AC 
               
               · 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   φ 
                   ) 
                 
               
             
           
         
       
     
     indicating that the peak angular displacement at given Q-factor and frequency increases inversely proportional to the length of an elementary rod magnet thus setting some constraints on the external magnetic field strength H AC  and therefore also on the power that can be extracted from the external magnetic field. Introducing a maximum angular displacement constraint leads to a relation for the frequency-magnet length product: 
     
       
         
           
             
               
                 f 
                 0 
               
               · 
               
                 l 
                 em 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       3 
                       ⁢ 
                       
                         2 
                       
                     
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         π 
                         2 
                       
                     
                   
                   · 
                   
                     
                       
                         μ 
                         0 
                       
                       ⁢ 
                       
                         H 
                         em 
                       
                       ⁢ 
                       
                         Q 
                         UL 
                       
                       ⁢ 
                       
                         υ 
                         em 
                       
                     
                     
                       θ 
                       peak 
                     
                   
                   · 
                   
                     H 
                     AC 
                   
                   · 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       φ 
                       ) 
                     
                   
                 
                 ) 
               
               
                 1 
                 2 
               
             
           
         
       
     
     Using the constraint on the peak angular displacement (saturation), an interesting equation on the maximum available power can be obtained: 
     
       
         
           
             
               P 
               av_mech 
             
             = 
             
               
                 
                   π 
                   
                     2 
                   
                 
                 · 
                 
                   μ 
                   0 
                 
               
               ⁢ 
               
                 H 
                 em 
               
               ⁢ 
               
                 V 
                 s 
               
               ⁢ 
               α 
               ⁢ 
               
                   
               
               ⁢ 
               
                 f 
                 0 
               
               ⁢ 
               
                 
                   θ 
                   peak 
                 
                 · 
                 
                   H 
                   AC 
                 
                 · 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
             
           
         
       
     
     This equation may also be expressed in terms of the total magnetic moment m lot  of the magneto-mechanical system and the external magnetic induction B AC  as follows: 
     
       
         
           
             
               P 
               av_mech 
             
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               
                 
                   ( 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       0 
                     
                   
                   ) 
                 
                 · 
                 
                    
                   
                     
                       m 
                       tot 
                     
                     × 
                     
                       B 
                       AC 
                     
                   
                    
                 
                 · 
                 
                   
                     θ 
                     peak 
                   
                   
                     2 
                   
                 
               
             
           
         
       
     
     This equation is not anymore dependent on Q-factor, and length of rod magnet, which indicates a certain degree of freedom in the design of magneto-mechanical systems. These parameters however are hidden or implicit to the peak angular displacement θ peak . 
     The maximum available power linearly increases with frequency. This behavior can also be found in systems that are directly based on Faraday&#39;s Induction law. 
     A useful definition to quantify performance of a magneto-mechanical system is the specific power conversion factor that has already been described. 
     
       
         
           
             
               k 
               c 
             
             = 
             
               
                 
                   P 
                   av_mech 
                 
                 
                   
                     H 
                     AC 
                     2 
                   
                   ⁢ 
                   
                     V 
                     S 
                   
                 
               
               = 
               
                 
                   
                     3 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                     
                   
                   · 
                   
                     
                       
                         μ 
                         0 
                         2 
                       
                       ⁢ 
                       
                         H 
                         em 
                         2 
                       
                       ⁢ 
                       
                         Q 
                         UL 
                       
                       ⁢ 
                       
                         υ 
                         em 
                       
                       ⁢ 
                       α 
                     
                     
                       
                         f 
                         0 
                       
                       ⁢ 
                       
                         l 
                         em 
                         2 
                       
                     
                   
                   · 
                   sin 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 φ 
               
             
           
         
       
     
     as well as the saturation field strength: 
     
       
         
           
             
               H 
               AC_sat 
             
             = 
             
               
                 
                   θ 
                   peak 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         3 
                         ⁢ 
                         
                           2 
                         
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           π 
                           2 
                         
                       
                     
                     · 
                     
                       
                         
                           μ 
                           0 
                         
                         ⁢ 
                         
                           H 
                           em 
                         
                         ⁢ 
                         
                           Q 
                           UL 
                         
                         ⁢ 
                         
                           υ 
                           em 
                         
                       
                       
                         
                           f 
                           0 
                           2 
                         
                         ⁢ 
                         
                           l 
                           em 
                           2 
                         
                       
                     
                     · 
                     
                       sin 
                       ( 
                       
                           
                       
                       ⁢ 
                       φ 
                       ) 
                     
                   
                   ) 
                 
               
               
                 - 
                 1 
               
             
           
         
       
     
     A system may be designed for a high k c , compromising with a lower saturation level. 
     Conversely, a system may be designed for a higher saturation level compromising with a lower k c . 
     Numerical Example 
     For a numerical example, the following parameters are assumed:
         V S =4·10 −6  m 3  (=4 cm 3  equivalent to a disk with a diameter of 4 cm and thickness of 3.1 mm)   ν em =131.6·10 −6  m 3 /kg   H em =1T/μ 0  A/m   α=0.25   Q UL =1000   θ peak =0.175 rad (−10°)   φ=0       

     The frequency of major interest is f=135 kHz. 
     The field strength of major interest is H AC =5 A/m 
     The power theoretically linearly increases with frequency. It must be noticed however that at higher frequencies power may be additionally limited by other factors such as maximum stored oscillatory energy in the system, mechanical strain, etc. This is not considered in this analysis. 
     The available power as a function of the external alternating magnetic field strength can be computed for different length of the elementary magnets. 
     A system using rod magnets of 20 μm length saturates at approximately 2.5 W while a system using 10 μm rod length saturates at a lower value of about 600 mW. The 10 μm system however is more sensitive (higher specific power conversion factor) than the one that uses 20 μm rods. This can be checked at a field strength of 5 A/m. 
     Based on this example, one can see that a disc shaped system with 4 cm diameter 3 mm thickness can extract up to 260 mW from a magnetic field of 5 amps per meter at 135 kHz. 
     Although only a few embodiments have been disclosed in detail above, other embodiments are possible and the inventors intend these to be encompassed within this specification. The specification describes specific examples to accomplish˜more general goal that may be accomplished in another way. This disclosure is intended to be exemplary, and the claims are intended to cover any modification or alternative which might be predictable to a person having ordinary skill in the art. For example, other sizes, materials and connections can be used. Other structures can be used to receive the magnetic field. In general, an electric field can be used in place of the magnetic field, as the primary coupling mechanism. Other kinds of magnets and other shapes of arrays can be used. 
     Also, the inventors intend that only those claims which use the-words “means for” are intended to be interpreted under 35 USC 112, sixth paragraph. Moreover, no limitations from the specification are intended to be read into any claims, unless those limitations are expressly included in the claims. 
     Where a specific numerical value is mentioned herein, it should be considered that the value may be increased or decreased by 20%, while still staying within the teachings of the present application, unless some different range is specifically mentioned. Where a specified logical sense is used, the opposite logical sense is also intended to be encompassed.