Abstract:
A novel motor drive system has been described for use in electrostatic generator/motor systems based on the time variation of capacity of a rotating condenser comprised of segmented rotor and stator elements. It takes advantage of the fact that the motor action of such a system depends only on the rms value of the drive pulses, which therefore can be formed simply by periodically interrupting a high-frequency ac wave train. This new circuitry simplifies the drive system and takes advantage of recent developments of devices used in the art of inversion of dc voltages to high-frequency (tens of kiloHz) ac.

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0001]    The United States Government has rights in this invention pursuant to Contract No. DE-AC52-07NA27344 between the U.S. Department of Energy and Lawrence Livermore National Security, LLC, for the operation of Lawrence Livermore National Laboratory. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates to drive systems for electrostatic generators and motors, and more specifically, it relates a pulse-train drive systems for such circuits. 
         [0004]    2. Description of Related Art 
         [0005]    Electrostatic generator/motors, as described in the prior art, can be operated in either a generator or a motor mode. This invention describes an improved electronic drive system for applications requiring the motor mode. Such devices have application to flywheels for bulk energy storage, among myriad commercial and defense uses. 
         [0006]    In general, electrostatic generator/motors involve the use of an assembly of rotating and stationary elements that together comprise a condenser (or, capacitor), the capacitance of which varies periodically with the motion of the rotating elements relative to the fixed elements. An example of the prior art is shown in  FIG. 1A  (top view) and in  FIG. 1B  (side view). Turning now to  FIG. 1A , a circular array of fan-like stationary elements,  100 , is depicted. Below this stationary array of elements is a similar circular array of elements, which is allowed to rotate about a vertical axis, referred to as the “rotor.”  FIG. 1B  shows a cross-sectional side view of the overall ES structure, showing an embodiment consisting of a set of two rotors,  106 , with each respective rotor also comprised of a circular array of fan-like elements. Each respective rotor plate is bound on each circular surface by a stationary array of opposing, fan-like elements,  100 ,  104 , and  102 , respectively. The rotor plates are allowed to rotate about an axis oriented orthogonal to the plane of the fixed plate(s), as shown in  FIG. 1A . 
         [0007]    As shown in  FIG. 1B , the rotors are comprised of a set of annular fan-like elements, with each element having a thickness greater than the basic substrate of the disc. In general, the thick fan-like sections elements of the rotor can consist of metallic (conductive) material, dielectric material or combinations thereof. Each pair of fixed fan-like elements, which comprises the opposing stationary plates, forms a capacitor of a fixed gap, g, in between which, each respective rotor traverses. As the moving discs rotate, the capacitance between each pair of opposing stationary plates will vary periodically, owing to differences in the gap dimension, g, and the properties of the rotor material, as each fan-like element of the rotor passes between each respective annular capacitor element in the array. 
         [0008]    This structure can be configured either as an electrostatic (ES) generator or as an electrostatic (ES) motor, dependent on the details of an electrical circuit that includes this device. In the so-called “generator mode” of operation, rotation of the moving element results in the generation of an ac voltage arising from the basic equation for the voltage across a charged condenser when the capacitance varies with time, as indicated by Equation (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       q 
                       
                         C 
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       Volts 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here q (Coulombs) is the charge on the condenser and C(t) (Farads) is the time-varying value of the capacitance, the latter owing to the rotation of the element. If the capacitance varies periodically with time, then the ac output of the electrostatic generator will also be periodic, with an ac waveform that depends on the geometry of the time-varying condenser and on the charging circuitry that is employed. 
         [0009]    The variable-capacity system described above is a “reciprocal” device, in that it is capable of functioning either as a generator or a motor, depending only on the circuitry to which it is attached. As a generator, an example of which is an energy storage flywheel, the generator output is high-frequency alternating current, which can easily be converted to mains-frequency power. As an example, the high-frequency generator output can be first rectified to dc, with the resultant dc output driving an electronic inverter to produce a 60 Hz output, the latter suitable for commercially powered devices. 
         [0010]    By reciprocity, operation of the system in the so-called “motor mode” requires a drive circuitry that generates a pulse-like waveform, which is synchronized with the rotation frequency and phase of the rotating elements. 
         [0011]    The physics principle here is exemplified by a simple example, as depicted in  FIG. 2 . Consider a 1-D parallel-plate capacitor of length L, comprised of a pair of stationary plates,  200  and  204 , respectively, separated by a gap of dimension, g. Without loss of generality, assume that the system is fitted with a third plate,  206 , also of length L. The third plate  206  is assumed to be a moveable metal (or dielectric) plate, which can be inserted between the pair of stationary plates, along the length of the capacitor, as shown in  FIG. 2 . This system results in a parallel-plate capacitor (the pair of fixed plates), whose value of the capacitance can be varied as the overlap, x, varies of the moveable plate relative to the fixed parallel plates. In general, as the overlap, x, of the plates increases, there is a concomitant increase in the value of the overall device capacitance. 
         [0012]    Consider the case where, initially, there is no overlap of the moveable plate relative to the pair of stationary plates (x=0). That is, all three plates are parallel to each other; however, the central plate,  206 , is laterally displaced to the left, by a distance, L, relative to the pair of (overlapping) stationary plates,  200  and  204 . In other words, the central plate is initially positioned to be completely outside the gap between the two fixed plates. 
         [0013]    Assume now that an external potential,  210 , of magnitude V 0 , is applied across the two stationary plates. Under this condition, as the leading edge of the moveable plate enters the gap between them (i.e., x&gt;0), the moveable plate will be subject to an attractive electrostatic force, F ES , that will persist until the plate  206  has reached the point whereby the capacitance of the capacitor is a maximum (x=L). If the plate is moved beyond this maximal point (x&gt;L), so that the capacitance begins to decrease, the direction of the attractive force, F ES , will reverse and then approach zero again as the plate emerges from the far side of the capacitor (x&gt;2L). 
         [0014]    To optimally apply this electrostatic force on the moveable plate to produce motor action in the same direction, it is necessary to ensure that the driving voltage be turned off during those times when the capacity is decreasing, and vice versa. 
         [0015]    In other words, the driving voltage is to be switched “on” as the plate moves forward and enters the gap (x&gt;0)—during which time, the electrostatic force, F ES , is attractive—further drawing the plate into the gap. Then, as the moveable plate begins to emerge from the gap (x&gt;L)—during which time the attractive force would have otherwise changed direction—the driving force is switched “off,” enabling the plate to continue its forward propagation through the gap (x&gt;L), without any applied force to retard its trajectory. 
         [0016]    The physics of the linear device shown in  FIG. 2  can be extended to that of a device with circular symmetry, in the context of the ES motor geometry. Turning now to  FIG. 3  and  FIG. 4 , an analogous set of conditions apply.  FIG. 3  depicts a top view of an ES motor, and,  FIG. 4  shows a cross sectional drawing of the same, along the A-A′ cut. 
         [0017]    In the example shown in  FIG. 3  and  FIG. 4 , the ES motor consists of two stationary plates,  300  and  304 , separated by a fixed distance (or, gap), g, each with a respective set of fixed elements, arranged in an annular pattern. Between these two discs, a rotating plate,  306 , is positioned, with a corresponding set of thick fan-like elements, each of which is comprised of conductive and/or dielectric materials. For ease of viewing, the cross-sectional view, A-A′, in  FIG. 4  depicts a rotor with only a single pair of diametrically opposed thick elements. As the rotor revolves, the capacitance between the stationary plates,  300  and  304 , as formed by opposing pairs of the fixed fan-like sections, will vary periodically. 
         [0018]    The basic operation of the circular device, in terms of its periodically varying capacitance during operation, can be appreciated by viewing an annular cross section B-B′, depicted by the dashed arc in  FIG. 3  and, also, by the cross-sectional cut, also labeled as B-B′, in  FIG. 4 . The linear representation of the varying capacitance, as discussed with respect to  FIG. 2 , provides a good approximation of the salient details that underlie the operation of the circular ES device. Under this approximation, the “arc-axes,” labeled by “x” in  FIG. 3 , are analogous to the linear x-axis shown in  FIG. 2 . In essence, therefore, the view seen in  FIG. 2  is essentially equivalent to the B-B′ cross-sectional cut depicted in  FIG. 3 . 
         [0019]    Returning now to  FIG. 3 , upon the application of a voltage across the pair of fixed capacitor plates,  300  and  304 , the thick segments along the rotating plate,  306 , will each experience a tangential electrostatic force, F ES , in the direction of the stator plates, resulting in a rotation of the rotor in the direction of this force. The rotor will revolve about its axis so that its thick segments will be attracted into the respective regions between the pair of opposing fixed-plate segments, in complete analogy to the linear representation shown in  FIG. 2 . Upon complete overlap of the rotor segments  306  with the respective segments of the fixed plates, the voltage across the fixed plates  300  and  304  is switched off, so that the rotor continues to freely rotate in the same direction, without any counteracting tangential force. Therefore, the rotor will continue to rotate in the same sense, until it approaches the stationary plates of a subsequent pair of opposing fixed elements. In the case of a rotor with a single-segmented device, this condition will occur after a single revolution. In analogy to motors in general, the IS motor can be scaled so that the rotor and stator consist of more than a single element. Hence, in the case of a multi-segmented device, the condition of subsequent overlap will occur after rotation by an angle of 3607°/2n, where n is equal to the number of segments about the annular circumference of the device. In general, at the point of subsequent overlap of the rotor element with that of the stationary pair of elements, the voltage is again switched on, resulting in an attractive tangential force, and, the process repeats. Thus, by proper synchronization and application of the time-dependent attractive force, continuous angular rotation in one direction is realized, thereby resulting in an electrostatic motor. 
       SUMMARY OF THE INVENTION 
       [0020]    A novel motor drive system is described for use in electrostatic generator/motor devices based on the time variation of the capacitance of a rotating condenser, comprised of segmented rotor and stator elements. The embodiments described herein take advantage of the fact that the motor action of such a system depends only on the rms value of the drive pulses, as discussed below with respect to Equation (6), which therefore can be formed by periodically interrupting and gating a high-frequency ac wave train. The drive circuitry simplifies the drive system but also takes advantage of recent developments in the art of inversion of dc voltages to high-frequency ac (e.g., tens of kHz). 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0021]    The accompanying drawings, which are incorporated into and form a part of the disclosure, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention. 
           [0022]      FIG. 1A  depicts a top view of a multi-element electrostatic motor in the prior art. 
           [0023]      FIG. 1B  depicts a cross-sectional side view of the multi-element electrostatic motor of  FIG. 1A , in the prior art. 
           [0024]      FIG. 2  shows a diagram of a fixed-plate capacitor in the presence of a moveable conductive and/or dielectric material within its gap. 
           [0025]      FIG. 3  depicts a top view of a multi-element electrostatic motor, revealing a fixed (stator) and a moveable (rotor) plate. 
           [0026]      FIG. 4  depicts a cross-sectional, side view of the multi-element electrostatic motor of  FIG. 3 . 
           [0027]      FIG. 5  shows a plot of the power output of an electrostatic motor as a function of the rotor tangential speed. 
           [0028]      FIG. 6  shows a schematic of a bipolar waveform generator circuit to drive an ES motor. 
           [0029]      FIG. 7  shows details of the ES motor bipolar drive wave train, as generated by the circuit depicted in  FIG. 6 . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0030]    From electromagnetic theory, a simple expression can be derived that provides the magnitude of the attractive force exerted on the moveable plate in terms of the respective maximum and minimum values of the capacitance, C max  and C min  (in Farads), the applied voltage, V 0  (in Volts), and a functional form factor, f(x). The form factor, f(x), is geometry dependent and corresponds to the variation of the capacitance with displacement, x (meters), where x is the direction along which the moveable plate traverses through the gap between the fixed capacitor plates. 
         [0031]    As an example, if the variation of the capacitance scales linearly with displacement, then the expression for the capacitance as a function of displacement would take the form shown in Equations (2) and (3): 
         [0000]        C ( x )= f ( x ) C   max  (Farads)  (2)
 
         [0000]      with the ansatz 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           C 
                           min 
                         
                         
                           C 
                           max 
                         
                       
                       ] 
                     
                     + 
                     
                       α 
                        
                       
                           
                       
                        
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This result holds for the condition 0&lt;x&lt;L, in which case, the capacitance increases linearly, from C min  to C max , as the moveable plate enters the fixed set of capacitor plates. By symmetry, for L&lt;x&lt;2L, the capacitance begins to decrease linearly, from C max  back to C min . For the purposes of this discussion, we need only consider the first case, namely, that whereby the capacitance increases, with the condition that 0&lt;x&lt;L. In Equation (2) and Equation (3), fringe-field effects have been neglected, which, in the context of this discussion, provides a very good approximation for most practical device configurations and scale sizes. With reference to  FIG. 2 , assuming that the dimension of the parallel-plate capacitor in the x direction is L (meters), the constant α is given by the equation: 
         [0000]    
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             C 
                             min 
                           
                           
                             C 
                             max 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         1 
                         / 
                         L 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Inserting Equation (4) for a into Equation (3) results in the following equation for f(x) for this geometry (again, subject to the condition that 0&lt;x&lt;L): 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         C 
                         min 
                       
                       
                         C 
                         max 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               C 
                               min 
                             
                             
                               C 
                               max 
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           x 
                           L 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since F=dW/dx, where W=CV 2 /2 is the energy stored in the electric field of the capacitor, the magnitude of the electrostatic force exerted on the moveable plate for any f(x) can be shown to be the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     ES 
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         2 
                       
                       ) 
                     
                      
                     
                       
                          
                         
                           f 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                          
                         x 
                       
                     
                      
                     
                       C 
                       max 
                     
                      
                     
                       V 
                       0 
                       2 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       Newtons 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    As can be seen from Equation (6), an electrostatic force is exerted on the moveable plate, in the positive direction, when the capacitance is increasing with displacement (i.e., df(x)/dx&gt;0). Conversely, the electrostatic force reverses direction when the capacitance is decreasing with displacement (i.e., df(x)/dx&lt;0). This means that the voltage V 0  should be applied only while the capacitance is increasing with displacement, and should be turned off when it is decreasing. This result, in turn, implies that the duty cycle, d, of the drive system will be limited to a maximum value of 50 percent, depending on the details of the geometry of the capacitor. For the special case of a linear variation of capacitance with displacement, as given by Equation (5), the expression for the positive force, Equation (6), becomes, after differentiation: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     ES 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           2 
                         
                         ) 
                       
                       [ 
                       
                         
                           1 
                           - 
                           
                             
                               C 
                               min 
                             
                             
                               C 
                               max 
                             
                           
                         
                         L 
                       
                       ] 
                     
                      
                     
                       C 
                       max 
                     
                      
                     
                       V 
                       0 
                       2 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       Newtons 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0032]    As an illustrative example of the force exerted in an electrostatic generator/motor with parameters appropriate to a flywheel energy storage module, consider a case where the value of C min /C max  is 0.25, C max  is 0.01 μf, L is 25 mm, and V 0  is 40 kV. For this case, using Equation (7), the electrostatic force becomes F ES =240 Newtons. 
         [0033]    Equation (7) may also be used to calculate the motor power, given by the product of the force, F ES  (Newtons), the velocity, ν (m/sec) and the duty cycle, d.  FIG. 5  is a plot of the motor power (in Watts) as a function of the tangential speed (dx/dt, in meters/second) for a duty cycle, d=0.5, given the above-stated device parameters of the capacitance and voltage. The speed range shown is appropriate to a flywheel rotor comprised of a carbon-fiber/epoxy material. 
         [0034]    As can be seen from  FIG. 5 , the power outputs are quite large, considering that the capacity value assumed could be achieved in flywheel modules storing a few tens of kilowatt-hours of energy. 
         [0035]    This invention takes advantage of an unusual functional dependence of an electrostatic motor, which differs significantly from that of other classes of motor, such as those employing forces exerted by magnetic fields, as derived by electromagnets and/or permanent magnets. This salient feature is that, as indicated by Equation (6), the motoring force depends on the square of the applied voltage (i.e., F˜V o   2 ), and is thus independent of its sign. The implication is that a burst of pulses, consisting of alternating amplitude pulses, and, synchronized with the rotation of the generator/rotor electrodes, are equally as effective in producing motor action as would be the case of relatively long, quasi-dc pulse-waveforms having the same rms value of voltage and duration. Given this equivalence, one can employ drivers that deliver high-frequency (tens of kHz) power such as is produced by modern solid-state inverters. This flexibility greatly reduces the size and cost of components, such as step-up transformers, while at the same time, operating at high efficiency. Moreover, the pulse parameters can be chosen to be within the resonant conditions of the structure for improved parametric performance and device efficiency. 
         [0036]      FIG. 6  illustrates a simple inverter circuit,  600 , which can perform the pertinent drive functions suggested above. The circuit consists of a DC power supply  610 , followed by a switching network having switches SwA 1 , SwA 2 , SwB 1  and SwB 2 , and further includes a step-up transformer  630  which provides the necessary waveform to drive the ES motor, the latter of which is represented by a capacitive load  640 . In this embodiment, the DC power supply, in conjunction with high frequency operation of the switches, generates a continuous high frequency, square-wave output, V(t), depicted in  FIG. 7  by waveform  700 . This signal drives the primary of a step-up transformer  630  the secondary of which, in turn, drives the ES motor capacitance  640 . 
         [0037]    Referring again to  FIG. 6 , as the switches (SwA 1 , SwA 2  and SwB 1 , SwB 2 ) change state from “on to off” and “off to on” respectively, the polarity of the drive signal to the variable capacitor concomitantly changes sign. This results in a bipolar, high-voltage output waveform that drives the ES motor. 
         [0038]    Details of the ES motor drive signal, are shown in  FIG. 7 . In this embodiment, the overall drive signal,  700 , is comprised of an envelope waveform of period, T, and of amplitude, V 0 . Within each period, T, a high frequency, burst signal is gated “on” for a duration, τ, followed by a zero voltage state  730  of duration T−τ. Hence, the duty cycle of this burst-mode drive signal is given by d=τ/T. 
         [0039]    Within the envelope of duration, T, a short burst of pulses is gated on. The burst mode consists of a bipolar square-wave sequence of pulses. Each pulse, in turn, is characterized by a period δτ, comprised of a positive-going  710  and negative-going  720  voltage feature, which, in this case, consists of square waves. 
         [0040]    In this example, the overall waveform is depicted as a symmetric signal, namely, the duty cycle of the overall signal, d, is 50%; and, in addition, the duty cycle of the pulses is also 50%. In general, the choice of these parameters will be a function of the ES motor configuration as well as the driving load, which is highly application dependent. Referring back to  FIG. 6 , the positive [negative] voltage component of each pulse within the waveform  700  corresponds to the switch set to position “A” [“B”]; and, the zero-state voltage level corresponds to the state where all switches are set to not allow current to flow through either switch position. During operation, the state of the switch is dynamically controlled, via open-loop or closed-loop system architectures. 
         [0041]    In another operational embodiment, the servo-system, with proper pulse timing, can operate in such a way as to automatically recover the residual stored energy stored by the ES generator/motor at the termination of each pulse train. Under these conditions, the overall efficiency of the system should be very high, especially when one considers that the electrical-to-mechanical energy conversion efficiency of a motor based on electrostatic principles is nearly 100 percent, there being no hysteresis, eddy-current, or conductor-resistance losses such as occur in conventional electric motors. 
         [0042]    The foregoing description of the invention has been presented for purposes of illustration and description and is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. The embodiments disclosed were meant only to explain the principles of the invention and its practical application to thereby enable others skilled in the art to best use the invention in various embodiments and with various modifications suited to the particular use contemplated. The scope of the invention is to be defined by the following claims.