Abstract:
Piezoelectric harvesting devices are disclosed herein. An embodiment of a harvesting device includes a cantilever having a resonant frequency associated therewith, wherein the cantilever vibrates when in the presence of a vibration source, and wherein the harvesting device generates a current upon vibration of the cantilever. The generated current is present at an output. A bias flip circuit is used to tune the resonant frequency of the harvesting device based on measurements of the vibration source that causes the cantilever to vibrate, wherein the bias flip circuit includes a switch that connects and disconnects an inductor to the output.

Description:
[0001]    This application claims priority to U.S. provisional patent application 61/697,174 filed on Sep. 5, 2012, which is hereby incorporated for all that is disclosed therein. 
     
    
     BACKGROUND 
       [0002]    Devices for harvesting vibration energy are often mechanical resonators with high-Q, wherein Q is the quality factor of the resonator. One example of such a device is a piezoelectric harvester. The piezoelectric harvesting devices typically have a cantilever that vibrates when it is in the presence of vibration energy. The cantilever is coupled to or otherwise connected to piezoelectric material that generates electricity from the vibrations. 
         [0003]    One problem with high-Q mechanical resonators, such as piezoelectric harvesters, is that they resonate at a single frequency, whereas the sources of vibration energy are usually not monotonic, stable, and predictable. Therefore, there is a mismatch between the mechanical resonant frequency of the piezoelectric harvesters and the frequency of the vibration sources. The result is low power output from the mechanical resonators. 
       SUMMARY 
       [0004]    Piezoelectric harvesting devices are disclosed herein. An embodiment of a harvesting device includes a cantilever having a resonant frequency associated therewith, wherein the cantilever vibrates when in the presence of a vibration source, and wherein the harvesting device generates a current upon vibration of the cantilever. The generated current is present at an output. A bias flip circuit is used to tune the resonant frequency of the harvesting device based on measurements of the vibration source that causes the cantilever to vibrate, wherein the bias flip circuit includes a switch that connects and disconnects an inductor to the output. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1  is an isometric view of an embodiment of a piezoelectric harvesting device. 
           [0006]      FIG. 2A  is an equivalent circuit representative of the piezoelectric harvesting device of  FIG. 1 . 
           [0007]      FIG. 2B  is an equivalent circuit representative of the piezoelectric harvesting device of  FIG. 1  operating at the mechanical resonant frequency. 
           [0008]      FIG. 3  is a graph showing an example of the frequency response of the harvesting device of  FIG. 1 . 
           [0009]      FIG. 4  is an embodiment of a harvesting device using a tunable inductor to change the phase of the output voltage relative to the phase of the vibration source displacement. 
           [0010]      FIG. 5A  is an embodiment of a bias flip circuit connected to an equivalent circuit of a harvesting device. 
           [0011]      FIG. 5B  is an example of the gate voltage on the switch of the bias flip circuit of  FIG. 5A . 
           [0012]      FIG. 5C  is an example of the output current of the harvesting device of  FIG. 5A . 
           [0013]      FIG. 5D  is an example of the output voltage of the harvesting device of  FIG. 5A  when connected to the bias flip circuit. 
           [0014]      FIG. 6  is an embodiment of a harvesting device using a bias flip circuit to change the phase of the output voltage relative to the phase of the vibration source displacement. 
           [0015]      FIG. 7  is an embodiment of a harvesting device using a bias flip circuit to change the phase of the output voltage relative to the phase of the vibration source acceleration. 
           [0016]      FIG. 8  is an embodiment of a harvesting device using a bias flip circuit to change the phase of the output voltage based on measurements of the output current waveform. 
       
    
    
     DETAILED DESCRIPTION 
       [0017]    Piezoelectric harvesting devices (sometimes referred to herein simply as “harvesting devices”) and circuits connected to the harvesting devices are disclosed herein. An example of a harvesting device  100  is shown in  FIG. 1 . The harvesting device  100  includes a cantilever beam  102  (sometimes referred to simply as the “cantilever  102 ”). The cantilever  102  has a length  103  extending from a support structure  104  to an end  105 . The cantilever  102  has two elements, a first element  106  and a second element  108 , which are piezoelectric materials. A mass  110  is attached to the cantilever  102  proximate the end  105 . In some embodiments, cantilever  102  does not have a separate mass  110  attached thereto, so the mass  110  represents an effective mass of the cantilever  102 . As the cantilever  102  bends, strain forces in opposite directions are generated in the elements  106 ,  108 . These strain forces generate current, which is harvested by the harvesting device  100 . In the embodiment of  FIG. 1 , the cantilever  102  is shown bending or displacing a distance  112  from a point of equilibrium. The displacement is sometimes referred to herein as X and the displacement as a function of time is referred to as X(t). 
         [0018]    The magnitude of current generated by the harvesting device  100  is proportional to the displacement of the cantilever  102 . The cantilever  102  has a resonant or mechanical frequency ω m , so when a vibration source oscillates or vibrates the cantilever  102  at the resonant frequency ω m , maximum displacement is achieved and the maximum current is generated by the harvesting device  100 . An embodiment of an equivalent circuit  130  of the harvesting device  100  is shown in  FIG. 2A . An equivalent circuit  140  of the harvesting device  100  operating at the mechanical frequency ω m  is shown in  FIG. 2B . The mechanical frequency ω m  is sometimes referred to as the resonant frequency. The circuit  140  has a current source I P , an internal capacitance C mc , and an internal or input resistance R in . The current source I P  generates current based on the vibration of the cantilever  102  of  FIG. 1 . As described in greater detail below, the circuit  130 , and thus the harvesting device  100  can be tuned to the frequency ω of a vibration source by adding a resonant tank circuit  142  to the circuit  130 . A bias flip circuit may also be connected to the harvesting device  100  in order to achieve the maximum output when the source frequency ω is different than the mechanical frequency ω m . 
         [0019]    The generation of current by the harvesting device  100  will now be described with reference to the circuit  140 . When the cantilever  102  is vibrating at the mechanical or resonant frequency ω m , the current source I P  has a value as shown by equation 1 as: 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     p 
                   
                   = 
                   
                     
                       ρ 
                        
                       
                           
                       
                        
                       
                         Q 
                         m 
                       
                        
                       
                         ω 
                         m 
                       
                        
                       
                         C 
                         mc 
                       
                        
                       
                         Z 
                         / 
                         d 
                       
                     
                     = 
                     
                       
                         Z 
                         / 
                         d 
                       
                       
                         R 
                         in 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
         [0020]    The resistance R in  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       in 
                     
                     = 
                     
                       
                         ( 
                         
                           ρ 
                            
                           
                               
                           
                            
                           
                             Q 
                             m 
                           
                            
                           
                             ω 
                             m 
                           
                            
                           
                             C 
                             mc 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     where 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     ρ 
                     = 
                     
                       
                         κ 
                         2 
                       
                       
                         1 
                         - 
                         
                           κ 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     κ 
                     2 
                   
                   = 
                   
                     
                       Y 
                       ɛ 
                     
                      
                     
                       d 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
         [0021]    where Y is Young&#39;s modulus of the material of the cantilever  102 , ∈ is the dielectric constant of the cantilever  102 , and d is the piezoelectric coupling constant. As described above, the current generated by the current source I P  of equation 1 only applies when the source of the vibration energy has the same frequency ω as the mechanical frequency ω m  of the cantilever  102 . When the two frequencies are not equal, the current output by the current source I P  decreases. 
         [0022]    The circuits and methods described herein overcome the problem of the harvesting device  100  only providing maximum power output when the cantilever  102  oscillates at the mechanical frequency ω m . More specifically, the circuits and methods described herein disclose techniques for electrically changing the mechanical frequency of the energy harvesting device  100  to frequencies different than the mechanical resonance ω m  where it would otherwise resonate. 
         [0023]    The technique for tuning the harvesting device  100  is based on the principal that the output voltage (V in  FIG. 2B ) of the harvesting device  100  results from an electric field in the piezoelectric material in the cantilever  102 , which changes the effective Young&#39;s modulus. The change in the Young&#39;s modulus changes the spring constant and the resonant frequency of the cantilever  102 . The tuning technique can also be understood in terms of coupled resonators. The output inductor L, or a bias flip inductor as described further below, forms an electrical tank circuit, together with the capacitor C mc . This electrical resonator is coupled to the mechanical resonator resulting in two coupled resonators (a 4-pole system), and the electrical resonator can “pull” the mechanical frequency ω m  of the cantilever  102  toward the frequency ω of the vibration source. The inductor L is shown in the tank circuit  142  of  FIG. 2A  and the bias-flip circuit is described further below. 
         [0024]    Based on the techniques described above, there are two electrical variables with which to optimize output power, which is directly related to the current I P . The first is by way of a load resistor R L , which is the real part of the electrical impedance in the tank circuit  142 . The load resistor R L , can shift the mechanical frequency ω m . In embodiments where the output voltage is rectified and the energy is stored, the rectified voltage V rect  has an effect similar to the load resister R L . The load inductance L, or the effective value of the bias-flip inductor that is described below, affects the reactive impedance. This inductance can be used to tune the mechanical frequency ω m  as described below. 
         [0025]    Based on the foregoing description, impedances and resistances may be connected to the harvesting device  100  to optimize the output power. A variable w will be used herein to normalize the frequencies. The variable w is equal to the frequency ω of the vibration source divided by the mechanical frequency ω m  of the cantilever  102 . At the frequency ω here w=1 (ω=ω m ), the equivalent circuit  140  shows that output of the current source I P  is optimized by impedance matching. 
         [0026]    A technique for tuning the harvesting device is achieved by changing the output voltage V. Two equations related to the operation of piezoelectric devices are shown in equations (5) and (6) as follows: 
         [0000]    
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     
                       σ 
                       Y 
                     
                     + 
                     
                       d 
                        
                       
                           
                       
                        
                       E 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
             
               
                 
                   D 
                   = 
                   
                     
                       ɛ 
                        
                       
                           
                       
                        
                       E 
                     
                     + 
                     
                       d 
                        
                       
                           
                       
                        
                       σ 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0027]    where δ is the mechanical strain (displacement/length), σ is the mechanical stress (force/area), Y is Young&#39;s modulus (force/area), d is the piezoelectric coefficient (m/volt), E is the electric field (volts/meter), D is the electrical displacement (coulomb/m 2 ), and ∈ is the dielectric constant (coulomb/volt-meter). 
         [0028]    When the output is shorted, E is equal to zero and the mechanical stiffness of the cantilever  102  is determined by Young&#39;s modulus where k m =ApY/tp, and where tp is the distance between the plates. The short circuit resonant or mechanical frequency ω m  is given by equation (7) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     m 
                     2 
                   
                   = 
                   
                     
                       k 
                       m 
                     
                     m 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0029]    where k m  is the mechanical short circuit spring constant and m is the mass  110  of the cantilever  102 . 
         [0030]    When the output is in the open circuit condition, D is equal to zero and the mechanical stiffness is defined as k oc =k m (1+ρ). Based on the foregoing equations, the open circuit and closed circuit resonant frequencies ω oc  and ω m  are related by equation (8) as follows: 
         [0000]      ω oc   2 =ω m   2 (1+ρ)  Equation (8)
 
         [0031]    Two capacitances are defined, an electrical capacitance Ce and a constrained capacitance C mc . The electrical capacitance Ce is equal to ∈Ap/tp and is related to situations where there is no stress (σ=0) on the cantilever  102 . The variable Ap is the area of the plates of the piezoelectric device and the variable tp is the distance between the plates. The constrained capacitance C mc  is defined as C e /(1+ρ) for the case when δ=0. When the capacitances and frequencies are substituted into the equations related to piezoelectric devices, the output voltage V is calculated per equation (9) as follows: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       
                         - 
                         ρ 
                       
                        
                       
                           
                       
                        
                       
                         ω 
                         4 
                       
                        
                       
                         Z 
                         / 
                         d 
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     ω 
                                     m 
                                     2 
                                   
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                   + 
                                   
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ωω 
                                         m 
                                       
                                     
                                     
                                       Q 
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     mc 
                                     2 
                                   
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                   + 
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ωω 
                                       m 
                                     
                                      
                                     
                                       G 
                                       L 
                                       N 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             ρ 
                              
                             
                                 
                             
                              
                             
                               ω 
                               2 
                             
                              
                             
                               ω 
                               m 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0032]    For reference, when the vibration source frequency ω is equal to the mechanical frequency ω m , equation (9) reduces to the common form of voltage as shown in equation 10. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         = 
                           
                          
                         
                           
                             I 
                             P 
                           
                           
                             
                               G 
                               L 
                             
                             + 
                             
                               G 
                               in 
                             
                             + 
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 m 
                               
                                
                               
                                 C 
                                 mc 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     m 
                                   
                                    
                                   L 
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ρ 
                              
                             
                                 
                             
                              
                             
                               Q 
                               m 
                             
                              
                             
                               Z 
                               / 
                               d 
                             
                           
                           
                             
                               G 
                               L 
                               N 
                             
                             + 
                             
                               G 
                               in 
                               N 
                             
                             + 
                             
                               j 
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ω 
                                       mc 
                                       2 
                                     
                                     
                                       ω 
                                       m 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0033]    where G L  is the load conductance, 1/K L ; G in  is the internal conductance; G L   N  is the normalized load conductance; and G in   N  is the normalized internal conductance. Per the above-described equations, the circuit  140  is accurate when the source frequency ω is equal to the mechanical frequency ω m . At this frequency, the current I P  is independent of the load on the output. When the source frequency ω is not equal to the mechanical frequency ω m , the current I P  changes as the load changes. The tank circuit  142  includes the inductor L that serves to cancel the reactance of the harvesting device  100  when the source frequency ω is not equal to the mechanical frequency ω m . The inductor L is used to achieve the maximum power output from the harvesting device  100 , which is given by equation (11) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     max 
                     av 
                   
                   = 
                   
                     
                       1 
                       8 
                     
                      
                     
                       
                         ( 
                         
                           Z 
                           d 
                         
                         ) 
                       
                       2 
                     
                      
                     
                       G 
                       in 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0034]    When an inductor value as described in equation (12) is selected in the tank circuit  142 , two peaks may be present in the frequency response. 
         [0000]        L =(  m   2   C   mc ) −1   Equation (12)
 
         [0035]    When the frequency of equation (12) is substituted into equation (9), variations in the mechanical frequency ω mc  are equivalent to varying the inductance of the inductor L. Based on the foregoing, maximizing the power with respect to ω mc  yields equation (13) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     mc 
                     2 
                   
                   = 
                   
                     
                       ω 
                       2 
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               ω 
                               m 
                               2 
                             
                             - 
                             
                               ω 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ω 
                           2 
                         
                          
                         
                           ω 
                           m 
                           2 
                         
                          
                         ρ 
                       
                       
                         
                           
                             ( 
                             
                               
                                 ω 
                                 m 
                                 2 
                               
                               - 
                               
                                 ω 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             
                               ω 
                               2 
                             
                              
                             
                               ω 
                               m 
                               2 
                             
                           
                           
                             Q 
                             m 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0036]    An example of the results of the normalized power are shown in the graph of  FIG. 3  for different frequency regions. The result of equation 13 and the graph of  FIG. 3  show that different techniques are required for maximizing the power depending on the frequency ω of the vibration source. The graph of  FIG. 3  is divided into three regions. Region 2 relates to frequencies ω that are near the mechanical frequency ω m , which causes equation (13) to reduce to equation (14) as follows: 
         [0000]      ω mc   2 =ω 2 +(ω m   2 −ω 2 )ρ Q   m   2   Equation (14)
 
         [0037]    It can be seen that when w is equal to or close to ω m , ω mc  becomes equal to ω. As described above, this situation is the same as matching the inductance of the inductor L to the capacitance of the capacitor C mc  of  FIG. 2B . In the other situation when the source frequency ω is greater than or less than the mechanical frequency ω m , the frequency response corresponds to regions 1 and 3. In these situations, equation 13 reduces to a pole splitting equation. 
         [0038]    In some embodiments, region 2 has a bandwidth of approximately +/−(2Q m ) −1 . In this region, power is optimized by using equation 14 for reactive admittance and G L  is substantially equal to G in . 
         [0039]    A bias flip circuit may be used to simulate an inductance that optimizes the power output of the harvesting device  100  when the frequency is in any of the regions 1, 2, or 3 as shown in  FIG. 3 . When the vibration source frequency ω is in region 1 where it is less than the mechanical frequency ω m , the maximum power transfer occurs when the output voltage V has a phase of positive ninety degrees relative to the frequency ω of the vibration source. The displacement of the vibration source is sometimes referred to as Z or Z(t). When the vibration source frequency ω is in region 3 where it is greater than the mechanical frequency ω m , the maximum power transfer occurs when the output voltage V has a phase of negative ninety degrees relative to the frequency ω of the vibration source. As shown above, when the vibration source frequency ω is equal to the mechanical frequency ω m , the maximum power transfer occurs when the phase of the output voltage V is equal to the phase of the vibration source. 
         [0040]    In some embodiments, a tunable or variable inductor may be used to adjust the phase of the output voltage V.  FIG. 4  shows an embodiment of a circuit  250  that uses a variable inductor  254  for adjusting or tuning the phase of the output voltage V. The circuit  250  is connected to the harvesting device  100  as illustrated by the equivalent circuit  130  described above. The variable inductor  254  is controlled by a control circuit  258  that determines the correct inductance value and sends a signal to the variable inductor  254 , which causes the variable inductor  254  to adjust to the correct value. The control circuit  258  is sometimes referred to as an inductance optimization circuit. A measurement circuit  260  measures the displacement Z(t) of the vibration source and outputs a signal to the control circuit  258  that is proportional to the displacement Z(t) of the source. 
         [0041]    The circuit  250  operates by monitoring the displacement Z(t) of the vibration source by way of the measuring circuit  260 . When the frequency ω of the vibration source is less than the mechanical frequency ω m , the control circuit  258  receives the signal from the measurement circuit  260  and causes the variable inductor  254  to change so that the phase of the output voltage V is positive ninety degrees relative to the vibration source. When the frequency ω of the vibration source is equal or substantially equal to the mechanical frequency ω m , the control circuit  258  causes the variable inductor  254  to change so that the phase of the vibration source is equal to the phase of the output voltage V. When the frequency ω of the vibration source is greater than the mechanical frequency ω m , the control circuit  258  tunes the variable inductor  254  so that the phase of the output voltage V is negative ninety degrees relative to the phase of the vibration source. 
         [0042]    The circuit  250  of  FIG. 4  may require a very large inductor as the variable inductor  254  and it may not be able to tune fast enough to operate with high frequency sources. The use of a bias flip circuit resolves the problem of the large inductor that may be required with the circuit  250 . An example of a circuit using a bias flip is shown by the circuit  300  of  FIG. 5A . The bias flip configuration of the circuit  300  is implemented using a relatively small inductor L that is connected within the circuit  300  by way of a switch, which in the configuration of  FIG. 5A  is a field effect transistor (FET) Q1. The bias flip configuration is able to flip the polarity of the voltage on the effective capacitance C mc  of the harvesting device  100 ,  FIG. 1 , by opening and closing the FET Q1 as described further below. When the FET Q1 closes, the LC tank circuit of the inductor L and the capacitor C mc  begins to resonate at a high frequency. In some embodiments, the tank circuit is allowed to oscillate for only half a period, at which time the FET Q1 is opened. An example of the gate voltage of the FET Q1 is shown in  FIG. 5B  wherein the gate is open when the voltage is low. The FET Q1 closes for a very short period. The result is that the bias on the capacitor C mc  flips. The flipping of the bias on the capacitor C mc  counters the effects of the capacitance. Thus, the bias flip simulates a large inductance connected in parallel with the capacitor C mc  and provides an alternative to the variable inductor  254  of  FIG. 4 . 
         [0043]    The current I P (t) is shown in  FIG. 5C  and the voltage V(t) is shown in  FIG. 5D . As shown, the bias flip circuit  300  is able to sync the phase of the voltage V(t) with the current I P (t), which yields the greatest power transfer from the harvesting device  100  to a load that is connected to the harvesting device  100 . In some embodiments, the load is a rectification circuit and/or a storage circuit. 
         [0044]    The bias flip circuit  300  may be self tuning by monitoring a parameter associated with the harvesting device  100  and flipping or triggering the bias when the parameter changes or reaches a predetermined value. An example of a harvesting device using the bias flip circuit  300  is shown by the circuit  350  of  FIG. 6 . The circuit  350  includes a bias flip circuit  352  as described above with reference to  FIG. 5A . In addition, the circuit  350  includes a rectifier  354  connected to the output of the harvesting device  100 . An energy management circuit  360  and a capacitor C RECT  serve to store the energy generated by the harvesting device  100 . The circuit  350  includes a source monitor  362  that monitors the displacement Z of the vibration source (not shown), which is referred to as Z(t) as a function of time. The source monitor  362  generates a signal that is input to a timing calculator  364 . The timing calculator  364  monitors the displacement signal from the source monitor  362  and determines when to open and close the switch Q1,  FIG. 5 , in the bias flip circuit  352 . It is noted that the energy management circuit  360  and/or the capacitor C RECT  may serve as the resistive load R L  described above. 
         [0045]    The circuit  350  operates by monitoring the displacement Z(t) of the vibration source. In some embodiments, an accelerometer may be associated with the vibration source so that the displacement Z(t) may be determined by calculating the second integral of the acceleration. When the source is vibrating at a frequency ω in regions 1 or 3,  FIG. 3 , the timing calculator  364  triggers the bias flip circuit  352  to change state when the magnitude of the displacement Z(t) of the vibration source is at a maximum as measured by the source monitor  362 . When the vibration source is vibrating at a frequency ω in region 2, the timing calculator  364  triggers the bias flip circuit  352  to change state when the source displacement equals zero or is at a minimum. The triggering of the bias flip circuit  300  typically causes the switch Q1 to change state for a very brief period as shown in  FIG. 5B . The signal driving the switch Q2 is a function with a very short period. For example, if the vibration source frequency ω is 100 Hz, the period of the vibration source is 10 msec, but the time in which the switch Q1 is closed is on the order of 10 microseconds. By changing state, the source velocity and the force exerted on the vibration source remain in phase, which causes the maximum power transfer to the energy management circuit  360 . 
         [0046]    Another embodiment of using the bias flip circuit  300  with the harvesting device  100  is shown by the circuit  400  of  FIG. 7 . The circuit  400  includes an accelerometer  404  that measures the acceleration of the vibration source. The accelerometer  404  outputs a signal to a timing circuit  406  that is indicative of the acceleration of the vibration source. The timing circuit  406  functions in a very similar manner as the timing circuit  364  of  FIG. 6  by causing a bias flip circuit  402  to trigger. 
         [0047]    The circuit  400  functions in a very similar manner as the circuit  350  of  FIG. 6 . The accelerometer  404  measures the acceleration of the vibration source. When the vibration source is operating in either the first or third regions and the magnitude of the acceleration of the vibration source is at a maximum as measured by the accelerometer  404 , the timing circuit  406  triggers the bias flip circuit  402 . When the vibration source is operating in region 2, the timing circuit  406  triggers the bias flip circuit  3402  when the magnitude of the acceleration is zero. 
         [0048]    Another embodiment of using the bias flip circuit  300  is shown by the circuit  450  in  FIG. 8 . The circuit  450  includes a current sensor  452  that measures the current I AC  at the output of the harvesting device  100 . The energy management circuit  360  may be the load resistance R L  in some embodiments where an energy storage device is used to store the energy generated by the harvesting device  100 . The timing circuit  454  triggers the bias flip circuit  456  when the frequency ω of the vibration source is in regions 1 and 3 and the current I AC  is at its maximum or the magnitude of the current I AC  is at its maximum. When the frequency ω of the vibration source is in region 2, the timing circuit  454  triggers the bias flip circuit  456  when the current I AC  is zero or at a minimum. 
         [0049]    In some embodiments, an effective current I eff  is calculated. The effective current I eff  is calculated by measuring the voltage V of the harvesting device  100 . Details of this calculation depend on the specifics of the energy management circuit  360 . 
         [0050]    In some embodiments, the load resistance R L ,  FIGS. 2A and 2B , is variable. When the source frequency ω is equal to the mechanical frequency ω M , the load resistance R L  is set to the value of the input resistance R P . When the frequency ω of the vibration source is in either the first or third regions where it is less than or greater than the mechanical frequency ω m , the load resistance R L  is set to be greater than the input resistance R in . The change in the load resistance R L  results in greater power transfer to the output of the harvesting device  100 ,  FIG. 1 . 
         [0051]    While illustrative and presently preferred embodiments of harvesting devices have been described in detail herein, it is to be understood that the inventive concepts may be otherwise variously embodied and employed and that the appended claims are intended to be construed to include such variations except insofar as limited by the prior art.