Abstract:
A motor/generator having its stationary portion, i.e., the stator, positioned concentrically within its rotatable element, i.e., the rotor, along its axis of rotation. The rotor includes a Halbach array. The stator windings are switched or commutated to provide a DC motor/generator much the same as in a conventional DC motor/generator. The voltage and power are automatically regulated by using centrifugal force to change the diameter of the rotor, and thereby vary the radial gap in between the stator and the rotating Halbach array, as a function of the angular velocity of the rotor.

Description:
[0001] The United States Government has rights in this invention pursuant to Contract No. W-7405-ENG-48 between the United States Department of Energy and The University of California for the operation of Lawrence Livermore National Laboratory. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0002]    1. Field of the Invention  
           [0003]    The present invention relates generally to generators and motors and, more particularly, to automatically regulating their voltage and power as a function of the angular velocity of a rotor concentrically rotating around a stator.  
           [0004]    2. Description of Prior Art  
           [0005]    There are numerous applications that require compact pulsed-power systems with power outputs of hundreds of megawatts, associated with energy storage capabilities of hundreds of megajoules. These range from emergency power needs and utility electric power conditioning and stabilization, to pulsed laser fusion or magnetic fusion systems. Utilities employ battery banks or large cryogenic systems including superconducting magnets for energy storage. Condenser banks are commonly used in laser and magnetic fusion applications.  
           [0006]    Flywheel energy-storage systems with rotors fabricated from high-strength fiber composite are integrated with high-power electrical generators for use during unexpected intermittent loss of network electrical power to sensitive electronic equipment, such as computers or automated production lines. One highly successful embodiment of such a system utilizes a generator/motor that employs permanent magnets arranged in a dipole version of the Halbach array configuration.  
           [0007]    Halbach arrays comprise the most efficient way to employ permanent-magnet material for the generation of dipole and higher-order pole magnetic fields within a given volume of space. They require neither “back-iron” elements nor iron pole faces in their construction, and they produce fields that approach the theoretical ideal of field uniformity (dipole arrays) or of sinusoidal variation with rotation (higher-order arrays). As such they are ideally suited for use in generators or motors constructed with air-cored stator windings; that is, windings constructed without the use of the laminated iron elements typically used in conventional generators and motors. Using air-cored stator windings avoids the hysteresis losses and limitation on the peak power caused by the magnetic saturation of laminated iron elements and the increased inductance of the stator windings in comparison to air-cored stator windings.  
           [0008]    A generator/motor that employs a dipole version of a Halbach array is described in U.S. Pat. No. 5,705,902, titled “Halbach Array DC Motor/Generator” issued to Bernard T. Merritt, Gary R. Dreifuerst, and Richard F. Post, the present inventor. A generator/motor system that employs higher-order Halbach arrays to produce its magnetic fields is described in U.S. Pat. No. 6,111,332, titled “Combined Passive Bearing Element/Generator Motor,” also issued to Richard F. Post.  
           [0009]    Flywheel energy storage systems typically operate over a range of angular velocity lying between a maximum determined by structural limitations, and one-half the maximum, at which point ¾ of the kinetic energy of the flywheel has been extracted. In the absence of voltage compensation, the output voltage will fall to half its initial value at this point. However, compensating for this great a change by electronic regulation or external circuits is expensive. An example of a means for regulation of the voltage that requires external circuitry is described in U.S. Pat. No. 5,883,499, “Method for Leveling the Power Output of an Electromechanical Battery as a Function of Speed,” issued to the present inventor.  
           [0010]    Motor/generators are useful in electric vehicles, adjustable-speed DC drives, and flywheel energy storage systems. The new invention incorporates novel features in such a way as to overcome some significant limitations of the prior art and to improve the performance of devices and systems incorporating motor/generators.  
         SUMMARY OF THE INVENTION  
         [0011]    Briefly, the present invention is a motor/generator having its stationary portion, i.e., the stator, positioned concentrically within the rotor along its axis of rotation. The rotor includes a Halbach array. The stator windings are switched or commutated to provide a DC motor/generator much the same as in a conventional DC motor. The commutation may be performed by mechanical means using brushes or by electronic means using switching circuits. Centrifugal force changes the diameter of the rotor. This varies the radial gap in between the stator and the rotating Halbach array, and thus changes the output voltage or power. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0012]    [0012]FIG. 1 is a section view taken normal to the axis of rotation of the rotor of the motor/generator of the present invention.  
         [0013]    [0013]FIG. 2 is a section view of the motor/generator of the present invention taken along line  2 - 2  of FIG. 1.  
         [0014]    [0014]FIG. 3 is a schematic drawing of a stator winding of the present invention, represented by a series circuit composed of a voltage source, two resistors, and an inductor.  
         [0015]    [0015]FIG. 4 is composed of two graphs: a function for the regulated voltage generated by the generator/motor of the present invention and the same function for the unregulated voltage generated by a generator/motor of the prior art, with both graphs plotted on the ordinate as a function of the angular velocity of the rotor shown as a percentage of its maximum angular velocity. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0016]    Turning to the drawings, FIG. 1 is a cross section taken normal to the axis of rotation  11  of rotor  13  of motor/generator  15  of the present invention. More particularly, rotor  13  includes Halbach array  17  consisting of magnets arranged in a Halbach configuration to form a cylinder about axis of rotation  11  and having a rotational degree of freedom about axis  11 . Halbach array  17  is concentric with and attached to the inner circumference of cylinder  19 . Halbach array  17  includes inner surface  21  lying at radius α with respect to axis  11 , and outer surface  23  lying at radius β with respect to axis  11 . Cylinder  19  has an inner circumference having radius β with respect to axis  11 , and an outer surface  24  lying at radius δ with respect to axis  11 .  
         [0017]    Stator  25  lies concentrically within rotor  13 , and does not rotate relative to axis  11 . FIG. 2 is a cross section taken along line  2 - 2  of FIG. 1. As particularly shown therein, stator  25  includes conductive windings  27 . Electrical leads for windings  27  are not shown. Each winding  27  is a rectangle comprised of inner section  29  located at inner radius R in  with respect to axis  11 , and outer section  31  located at outer radius R out  with respect to axis  11 . Gap  33  between each outer section  31  of windings  27  and inner surface  21  is thus equal to (α-R out ). Each winding  27  is mounted on a rigid, non-extensible support  35 .  
         [0018]    Inner radius α varies as a function of centrifugal force as rotor  13  rotates at angular velocity ω 0 . R out  of windings  27  remains constant. Thus, gap  33  will vary as a function of ω 0 . As will be subsequently explained, when motor/generator  15  functions as a generator, the voltage induced in windings  27  varies as a function of gap  33 , and thus varies as a function of ω 0 . Furthermore, when motor/generator  15  functions as a motor, the back emf produced varies as a function of gap  33 , and thus also varies as a function of speed.  
         [0019]    With regard to the operation of motor/generator  15  as a generator, the upper limit to the specific power output (power output per kilogram of weight) of air-cored generators of the type of motor/generator  15  can be estimated by analyzing the so called Poynting vector, P, which defines the local value of the energy flux, in units of Watts/m 2 , carried by an electromagnetic field. The Poynting vector, P, in a vacuum is defined by the equation:  
           P= ( E×B )/μ 0  watts/m 2   (1)  
         [0020]    where: E (Volts/m.) is the value of the electric field;  
         [0021]    B (Tesla) is the magnetic field vector; and  
         [0022]    μ 0 , the permeability of free-space,=4π×10 −7  (henrys/meter).  
         [0023]    The electric field in the frame at rest (stator  25 ) arises from the relativistic transformation of the magnetic field from the rotating frame (rotor  13 ). This transformation is governed by the relativistic relationship:  
         ( E+v×B )=0  (2)  
         [0024]    where: v is the is the velocity vector in the direction of transport of the rotating magnetic field, and has only one component in this case, v φ  the azimuthal component.  
         [0025]    Solving Equation 2 for E, inserting this result into equation 1, and performing the vector product called for therein results the following relationship for P:  
           P=[B   2   v− ( v·B ) B ]/μ 0  watts/m 2   (3)  
         [0026]    Taking the component of P in the direction of v provides an expression for the rate of power flow through the circuits.  
         ( P·v )/ v=v   φ   B   2 [1−cos 2 (θ)]/μ 0   (4)  
         [0027]    where the angle θ is the angle between v and B.  
         [0028]    In the magnetic field emanating from rotating Halbach array  17 , the angle θ rotates continuously at a rate equal to the angular velocity of rotor  13 , ω 0 , multiplied by the order, N, of the pole. Taking the time average of the Poynting vector component gives a value for the average power through a surface area perpendicular to v as follows:  
         &lt;( P·v )/ v&gt;=v   φ ( B   2 /2μ 0 ) watts/m 2   (5)  
         [0029]    This equation can be interpreted as representing the result of transporting magnetic stored energy, U=(B 2 /2μ 0 ) Joules/m 3  at a velocity, v φ , through a surface perpendicular to v φ . It is assumed that v 0 &lt;&lt;δ, where δ is the velocity of light. The magnitude of the energy flux represented in a practical situation can be estimated from the following example. Assume that Neodymium-Iron-Boron permanent magnets, for which the remanent field, B r  is equal to or greater than 1.2 Tesla, are used in Halbach array  17  and that the azimuthal velocity, v φ , of the array equals 10 3  m/sec, a typical value for a fiber composite rotor. From the equations for Halbach array 17 given below, the peak surface field of the array, B 0 , can be determined. In typical cases it is approximately equal to 1.0 Tesla. Using the above velocity and a surface magnetic field B 0 =1.0 Tesla as illustrative values, the power per unit area predicted by Equation 5 is 400 mw/m 2 . As will be later discussed, generators of the type that are the subject of the present invention can achieve power outputs into matched loads that represent a substantial fraction of this calculated incident power level, a level that represents the theoretical upper limit to the power transfer. Such power flux levels are very high as compared to conventional commercial iron-cored generators, for which the corresponding power flux levels are typically less than 1.0 mw/M 2 .  
         [0030]    Equation 5 can be used to provide an estimate of the power output of motor/generator  15 , and to derive scaling laws for that power output. The magnetic field from Halbach array  17 , in cylindrical coordinates, is given by the following equations:  
               B   ρ     =           B   0          [     ρ   α     ]         N   -   1            cos        (     N                 φ     )                 (   6   )                 B   φ     =       -         B   0          [     ρ   α     ]         N   -   1              sin        (     N                 φ     )                 (   7   )                 B   2     =         B   0   2          [     ρ   α     ]           2      N     -   2               (   8   )                 where        :                     B   0       =           B   r          [     N     N   -   1       ]            [     1   -       (     α   /   β     )       N   -   1         ]            C   N               (   9   )                 C   N     =         cos   N          (     π   /   M     )            [       sin   (     N                   π   /   M           (     N                   π   /   M       )       ]               (   10   )                               
 
         [0031]    ρ(m.) is the radius variable;  
         [0032]    N is the pole order of array  17  (the number of wavelengths around inner surface  21 );  
         [0033]    B r  is the remanent field of the permanent-magnet material; and  
         [0034]    M is the total number of magnets in array  17 . (As there are 4 magnets per azimuthal wavelength in the Halbach array shown in FIG. 1, in this case M=4N).  
         [0035]    To evaluate the Poynting vector of the power flux from the Halbach array  17 , it is necessary to insert B 2  from equation 8 into Equation 5, set v φ =ρω 0 , and integrate over the radius between 0 (axis  11 ) and α, thereby finding the maximum value of the power flux through a single winding  27  having an axial length h(m.), and lying in a radial plane. The result is:  
               P   0     =       h          ∫   0   α            [       〈     P   ·   v     〉     v     ]             ρ           =           B   0   2          α   2        h                   ω   0         4        μ   0        N                     watts        /        winding               (   11   )                               
 
         [0036]    More power can be extracted, of course, by increasing the number of windings  27  deployed azimuthally. At some point, however, the energy extracted will approach the limiting rate at which it can flow in from the electromagnetic field. In keeping with the purpose of this discussion, no attempt will be made to solve the problem of determining the ideal number of windings  27 ; instead, it is assumed that if windings  27  are spaced one half-wavelength apart azimuthally, they will be sufficiently decoupled from each other so that the following simple calculation will provide a reasonable estimate. As will be later shown, where an explicit calculation is done for the energy coupled out of windings  27  into a matched load, this assumption is useful for determining the scaling laws of the system and for estimating the maximum power output that can be expected.  
         [0037]    Assuming that the number of windings  27  equals 2N (i.e., that they are spaced one-half wavelength apart, the total power flow through the area circumscribed by windings  27  is expressed as follows:  
               P     2      N       =       [         B   0   2          α   2        h                   ω   0         2        μ   0         ]                   watts             (   12   )                               
 
         [0038]    In the simplest terms, this result corresponds to an amount of magnetic energy flowing through windings  27  at an angular frequency ω 0 (rad./sec.) of rotor  13  and Halbach array  17 . The controlling parameters are thus the radius, α, and length, h, of the system, and the angular velocity, ω 0 , of Halbach array  17 . The radius, α, and the angular velocity, ω 0 , are, of course, interrelated, owing to centrifugal forces that limit ω 0 . The order of the array, N, does not directly enter into the above expression, although there will later appear reasons to employ high-order (N&gt;&gt;2) Halbach arrays in order to enhance the performance of motor/generator  15 .  
         [0039]    To calculate the power output that can be expected, assume:  
         [0040]    α=0.5 meter;  
         [0041]    h=1.5 meters;  
         [0042]    B 0 =1.0 Tesla; and  
         [0043]    ω 0 =2096 rad./sec. (20,000 rpm).  
         [0044]    Inserting these values into Equation 12 gives a power level of 625 megawatts.  
         [0045]    Several requirements must be satisfied to achieve such high power levels from a relatively small generator. Firstly, the angular velocity, ω 0 , of rotating Halbach array  17  must be sufficiently high. This requirement in turn implies that cylinder  19  must withstand the centrifugal force exerted on its inner surface by Halbach array  17 . A practical solution to this problem is to fabricate cylinder  19  from a high-strength fiber composite material, such as carbon fibers bonded with epoxy resins. To avoid delamination of cylinder  19  from centrifugal stress, the wall thickness of cylinder  19  is typically limited to a radius ratio, i.e., the ratio of its outer radius, δ, to its inner radius, A, of no more than 1.3. When the inertial effect of Halbach array  17  on its inner surface  21  is taken into account, the peak tensile stress produced in such a thin-walled cylinder may be approximated by the following equation:  
           S=ρ   δ [ω 0 β] 2   G  newtons/m 2   (13)                where        :                   G     =             ρ   m       ρ   δ            {     1   -       [     1   -     α   β       ]     3       }       +     {         [     δ   β     ]     3     -   1     }         3        [       δ   β     -   1     ]                 (   14   )                                 
         [0046]    ρ δ  (kg/m 3 ) is the density of the material composing cylinder  19 ; and  
         [0047]    ρ m  (kg/m 3 ) is the density of the magnets composing Halbach array  17 .  
         [0048]    For a thin Halbach array  17  and a thin composite cylinder  19 , the function G approaches the limit of 1.0. When this value for G is inserted into Equation 13, the answer corresponds to the minimum possible stress value in cylinder  19  for a given value of radius β, and angular velocity, ω 0 .  
         [0049]    The optimum thickness for Halbach array  17  corresponds to that thickness which maximizes the ratio of the generating capacity to the mass of Halbach array  17 . The generating capacity is proportional to (B 0 ) 2 , which is in turn a function of the thickness of Halbach array  17  (see Equation 9). Taking these competing variables into account, the optimum magnet thickness (for N&gt;&gt;1) turns out to be 0.20 λ, where λ(m.) is the azimuthal wavelength of Halbach array  17 , given by the equation:  
         λ=2 παN  meters  (15)  
         [0050]    Under the stress produced by centrifugal forces, the inner radius of cylinder  19 , β, and the inner radius, α, of the Halbach array  17  will expand radially, increasing gap  33  between inner surface  21  of Halbach array  17  and outer section  31  of windings  17 . This radial expansion, Δr, which is an important factor in the embodiment of the invention, is given by the equation:  
               Δ                 r     =         (     S        /        Y     )          [       β   +   δ     2     ]                     meters             (   16   )                               
 
         [0051]    where: S(Newtons/m 2 ) is the mean stress level in rotor  13 , and  
         [0052]    Y(Newtons/m 2 ) is the Young&#39;s modulus of the composite of cylinder  19 .  
         [0053]    From Equation 13 it can be seen that the stress in rotor  13  increases in proportion to the square of the angular velocity, ω 0 .  
         [0054]    In the analysis of generator/motor  15  for the purpose of optimizing its power output, the issue of maximizing the power transfer to an external load must be considered. Since there are no ferro-magnetic materials in motor/generator  15 , the elements of windings  27  are all linear and the analysis is greatly simplified. Each winding  27  and its load can therefore be electrically represented by the circuit diagram shown in FIG. 3.  
         [0055]    More particularly, the rotating Halbach array  17  of FIG. 1 induces an rms voltage, V rms , in the stator windings  27 , characterized by an inductance, L w  (henrys), and a series resistance, R w  (ohms). The output current is delivered to a load resistance, R load . In practice, R w &lt;&lt;R load  and R w  can be neglected in comparison to other quantities. In this case the maximum power that can be delivered occurs for a load resistance equal to the inductive impedance of windings  27 , ωL w , and is given by the equation:  
               P   max     =         1   2          [       V   rms   2       ω                   L   w         ]                     watts             (   17   )                               
 
         [0056]    where: ω is the frequency for the output voltage and is equal to Nω 0 ; and  
         [0057]    V rms  is the output voltage.  
         [0058]    V rms  may be determined using the equations for the magnetic field of Halbach array  17 , i.e., equations 6, 7, 9, and 10. The voltage in winding  27  is derived from the time-varying azimuthal flux through the area between r out  and r in  produced by the azimuthal component of the fields of Halbach array  17 . Integrating this field component over the area circumscribed by one of windings  27  results in an expression for the induced voltage as a function of time:  
               V        (   t   )       =       B   0        α                 h                       ω   0          [       r   out     α     ]       N          {     1   -       [       r   in       r   out       ]     N       }          cos        (     N                   ω   0        t     )                     volts             (   18   )                               
 
         [0059]    The square of the rms value of this expression may now be inserted into Equation 17 to determine the maximized power per winding into a matched load. The result is given by the following equation:  
               P   max     =         1   4          [         B   0   2          α   2          h   2          ω   0         N                   L   g         ]                  (       r   out     α     )       2      N            [     1   -       (       r   in       r   out       )     N       ]       2                   watts        /        winding             (   19   )                               
 
         [0060]    The inductance (self plus mutual) of one of windings  27  has been calculated using theory and may be used to evaluate the inductance term, L g , in Equation 19. The result is:  
               L   g     =           μ   0          P   δ         2                 k                   d   δ                       henrys             (   20   )                               
 
         [0061]    where: P δ (m.) is the distance around the perimeter of one of windings  27 ; that is, the length of the conductor comprising one of windings  27 .  
         [0062]    k=2π/λ=N/a is the azimuthal wavelength of Halbach array  17 ; and  
         [0063]    d δ (m.) is the center-to-center spacing (in the azimuthal direction) between the circuits of windings  27 .  
         [0064]    Substituting equation 20 for L g  in Equation 19 results in an expression for the maximized power per winding:  
               P   max     =         1   2          [         B   0   2        α                   h   2          ω   0          d   δ           μ   0          P   δ         ]                  (       r   out     α     )       2      N            [     1   -       (       r   in       r   out       )     N       ]       2                   watts        /        winding             (   21   )                               
 
         [0065]    The maximized total output of motor/generator  15  is then given by multiplying Equation 21 by the number of windings  27 , n w , given by the ratio of the circumference of windings  27  to the center-to-center azimuthal spacing in between adjacent individual windings  27 :  
           n   w =2π r   out   /d   δ   (22)  
         [0066]    The maximized total power output is thus given by the following equation:  
               ∑     P   max       =       [       π                   B   0   2          α   2          h   2          ω   0           μ   0          P   δ         ]                (       r   out     α     )       (       2      N     +   1     )            [     1   -       (       r   in       r   out       )     N       ]       2                   watts             (   23   )                               
 
         [0067]    Note that the distance around the perimeter of one of windings  27 , P c , is given by the expression:  
               P   δ     =       2        [     h   +     (       r   out     -     r   in       )       ]       =     2        h        [     1   +       (       r   out     h     )          (     1   -       r   in       r   out         )         ]                     meters               (   24   )                               
 
         [0068]    where (r out −r in ) is the radial depth of windings  27 .  
         [0069]    Equation 24 may be used to obtain a further optimization of the power output. Note that the amount of flux enclosed by an individual winding  27  depends on the area it circumscibes, which is equal to its radial depth, (r out −r in ), multiplied by its length, h. Increasing the circumscribed area therefore increases the induced voltage. However, as shown by Equations 20 and 24, increasing the circumscribed area also increases the inductance of the winding, which, as shown by Equation 17, would decrease the power output. There are thus two competing effects, the result of which is to define an optimum value for the area circumscribed by a singular winding of windings  27 . To determine the optimum radial depth of windings  27 , Equation 24 is substituted into Equation 23: 
               ∑     P   max       =       [       π                   B   0   2          α   2          h   2          ω   0         2        μ   0         ]            (       r   out     α     )       (       2      N     +   1     )            {         [     1   -       (       r   in       r   out       )     N       ]     2       1   +       (       r   out     h     )          (     1   -       r   in       r   out         )           }                   watts             (   25   )                               
 
         [0070]    Letting x=r in /r out &lt;1 the expression in braces can be written as:  
               F        (   x   )       =         [     1   -     x   N       ]     2       [     1   +       (       r   out     h     )          (     1   -   x     )         ]               (   26   )                               
 
         [0071]    The function F(x) has a maximum value, F max , as a function of x for given values of N and (r out /h). Inserting this value into Equation 25 there results the final maximized expression for the power output:  
               ∑     P   max       =       π        [         B   0   2          α   2        h                   ω   0         2        μ   0         ]              (       r   out     α     )       (       2      N     +   1     )            F   max                   watts             (   27   )                               
 
         [0072]    Note that the term in square brackets is identical to the expression in Equation 12. This equation was obtained by using Poynting&#39;s Theorem to estimate the maximum possible power output from motor/generator  15  and to determine the scaling laws for that output in terms of the magnetic field and dimensional parameters of the generator.  
         [0073]    As an example of the use of Equation 27 for the design of a generator, consider the following parameters for a physically small generator, but one with a relatively high power output.  
         [0074]    α=0.5 m.  
         [0075]    h=1.5 m.  
         [0076]    N=64  
         [0077]    r out /a =0.98  
         [0078]    B 0 =magnetic field of Halbach array  17 =1.0 Tesla  
         [0079]    ω 0 =2090 radians/sec. (20,000 rpm)  
         [0080]    The maximum value for F(x) in Equation 26, F max , is the optimal value for the radial depth of windings  27 : F max =0.981 at x=0.949. Inserting this value for F max  and the value of the other parameters into Equation 27 gives a maximized power output of 145 megawatts. This power level is to be compared with the theoretical upper limit to the power transfer, 625 megawatts, obtained from Equation 12. This example demonstrates the assertion made earlier that output powers that are a substantial fraction of the theoretical maximum can be achieved with generators of the present invention.  
         [0081]    The foregoing derivations and discussion provide a basis for describing the present invention. Consider first the means for regulating and controlling the power output of motor/generator  15 . The means proposed can be understood by examination of Equation 27 for the power output. This equation contains a term, (r out /α) 2 N, that expresses the variation in output with the outer radius, rout of windings  27  relative to the inner radius, α, of Halbach array  17 . If N&gt;&gt;1 this term becomes very sensitive to the ratio of the two radii, i.e., to gap  33  between the inner surface  21  of Halbach array  17  and the outer sections  31  of windings  27 . Using the parameters of the previous example, N=64, α=0.5 m. and (r out /α)=0.98; gap  33  equals 0.02*α=0.01 m. From Equation 27, a decrease in gap  33  by 1.0 mm would result in an increase in the power output from 145 megawatts to 188 megawatts. This example shows the power produced by motor/generator  15  can be regulated by varying gap  33 , e.g., by designing rotor  13  so that radius a changes by a predicted amount as a function of centrifugal force, i.e., as a function of the angular velocity, coo, of rotor  13 .  
         [0082]    Accordingly, motor/generator  15  provides for automatically holding its voltage output approximately constant as the angular velocity, ω 0 , decreases below its maximum value, down to an operational minimum. Such a situation would be encountered if motor/generator  15  is incorporated into a flywheel for use as an integral part of an energy storage device. The foregoing embodiment takes advantage of the naturally occurring variation of the circumference of rotor  13  in proportion to its angular velocity, ω 0 , owing to the changing centrifugal force acting radially outward against rotor  13 . That is, as ω 0  increases, rotor  13  radially expands and gap  33  increases, resulting in a corresponding decrease in the voltage output. It follows that as ω 0  slows down, gap  33  decreases and the output voltage increases. The increased output voltage occurring with a decrease in gap  33  can be used to compensate for the decrease in output voltage of the generator associated with a decrease in ω 0  (see Equation 18).  
         [0083]    Equation 13 indicates that the mean stress level in the composite rotor  13  will vary as the square of the rotational angular velocity, ω 0 . Equation 16 indicates that the inner radius, α, of the composite rotor  13  will increase linearly with the mean stress level. The magnets comprising Halbach array  17 , being restrained by cylinder  19 , will move outward along with cylinder  19  as it expands radially. It follows that the relationship between the inner radius of the Halbach array  17  at angular velocity α(ω 0 ), and that same inner radius at zero angular velocity, α 0 , can be expressed by the following equation:  
               α        (     ω   0     )       =       α   0          [     1   +     f        (       ω   0   2       ω   max   2       )         ]               (   28   )                               
 
         [0084]    where: f is a constant determined from the Young&#39;s modulus of the composite of cylinder  19 , taking into account that, as noted, the magnets comprising Halbach array  17 , since they are being supported by the inner circumference of cylinder is  19 , will move radially with the inner circumference of cylinder  19  as it expands from centrifugal force; for example, f=0.015 (i.e., 1.5% expansion between ω 0 =0 and ω max ) for a typical carbon fiber composite material under tensile stress; and  
         [0085]    ω max  is the maximum angular velocity of rotor  13 .  
         [0086]    The second equation needed, derived from Equation 18, is the expression for the rms voltage output of motor/generator  15 :  
               V   rms     =       1   2          B   0        α                 h                       ω   0          [       r   out     α     ]       N          {     1   -       (     [       r   in       r   out       ]     )     N       }                   volts             (   29   )                               
 
         [0087]    Equations 28 and 29 can be combined to give the ratio of the output voltage at angular velocity ω 0 , to that at the maximum operating speed, ω max . By defining the variable y as the ratio of the operating speed, ω 0 , to the maximum operating speed, ω max , i.e.,  
             y   =       ω   0       ω   max               (   30   )                               
 
         [0088]    the ratio of the voltage at a given ω 0  to its maximum value is given by the equation:  
                 V   rms         V   rms          (   max   )         =       H        (   y   )       =       y        [       1   +   f       1   +     f                   y   2           ]         N   -   1                 (   31   )                               
 
         [0089]    To achieve voltage compensation, that is, to require that the voltage should not begin to decrease as the rotor slows down, it is required that the function H(y) should have a maximum for y&lt;1. This condition can be found by differentiating H(y) with respect to y and setting the derivative equal to zero.  
         [0090]    The result is given by the equation:  
             y   =       1     f        (       2      N     -   3     )                   (   32   )                               
 
         [0091]    Requiring that a maximum should occur for a frequency below the maximum operating speed implies a lower limit on the value of N. From Equation 32 this condition is given by the inequality:  
             N   ≥       1   2          (       1   f     +   3     )               (   33   )                               
 
         [0092]    Taking f=0.015 as before, the inequality requires N&gt;35. Larger values of N are required if regulation is to be effective over a large range of ω 0 .  
         [0093]    The equations given above can be used to determine the value of N required to perform automatic voltage regulation for, as an example, an angular velocity, ω 0 , ranging over a factor of two, that is, to require that the voltage at half speed, ω 0 =(½)ω max , should be the same as that at full speed, ω 0 =ω max  (with a modest voltage rise in between). This voltage regulation requirement is given by equating H(y=1) and H(y=½), yielding an equation for N.  
             N   =     1   +         log   e          (   2   )           log   e          [       1   +   f       1   +     (     f   /   4     )         ]                   (   34   )                               
 
         [0094]    For f=0.015, the nearest-integer value of N found from Equation 34 is N=64. The value of the voltage regulation function, H(y) for N=64 and f=0.015 is plotted in FIG. 4 over the range ω 0 =(½)ω max  to ω max . As shown therein, the regulated voltage output remains constant within a few percent over the entire operating range of rotor  13 . For comparison, the unregulated voltage is also shown. As the graph indicates, the unregulated voltage output would vary by a factor of 2 over this same operating range.  
         [0095]    The foregoing description of the automatic regulation of the voltage output of motor/generator  15  when operated as a generator is also applicable to its operation as a motor. More particularly, the variation in gap  33  that occurs as a function of variable angular velocity, ω 0 , of rotor  13 , will increase the torque at a constant input current as the motor speed drops. This action will therefore compensate for the loss of power capability at a constant input current that would otherwise occur as the motor speed drops. This enhances the versatility of motor/generator  15  operating in a motor mode.  
         [0096]    It is to be understood, of course, that the foregoing description relates only to embodiments of the invention, and that modification to these embodiments may be made without departing from the spirit and scope of the invention as set forth in the following claims.