last modified on: Mar. 25, 2005; see update below!

             – the Faraday Disk Dynamo as the original over-unity device –


[Note: Any updates made to this page will involve no non-textual changes of any kind, to preserve the integrity of the design models discussed in the Analysis sections as valid reference material. The Tesla & Back-Torque Theory feature section has been revised to include a more accurate and complete discussion of eddy current and stator coupling losses. Our thanks to the huge number of visitors this webpage receives, and to the many students and engineers who have written us about the report.]

 

Synopsis:
    As many students, engineers, and over-unity researchers may already know, it's very hard to find a single good Internet reference resource which soundly assesses the  early Faraday induction dynamo in comparison to a latter-day derivative – the "unipolar generator". Basic engineering design information on either device is often inaccurate or misleading, and sometimes erroneous.

Quite a number of determined inventors and experimenters have attempted to develop a "formal" self-sustaining over-unity variant of the disk dynamo which was 'unipolar' in that it was statorless, but virtually none have ever succeeded.
    The fact that a few have actually demonstrated a "free energy" operational power gain is important to point out in this turbulent time, when a definitive breakthrough in over-unity electrical power generation is needed so imperatively and so few people are having any real success at achieving one that is practical. Therefore, we are pleased to offer this webpage as a concise but detailed engineering analysis of these fascinating machines, one that appears to be much-needed, in order to foster a proper understanding of them and to assist students and alternative energy enthusiasts in their researches.
    A Faraday dynamo or unipolar generator doesn't lend itself well to practical commercial development because of the nature of its output, since it produces very low (even fractional) DC voltage at extremely high current. When Nikola Tesla invented polyphase alternating current near the end of the 19th century, the concentrated development of DC power systems based on Faraday's original work virtually ceased. However, Faraday dynamos and generators are well-suited to easy and precise mathematical modeling, both mechanically and electrically. If we clearly show that formal* over-unity operation can be achieved with this technology, it could be that due cause is indicated for its renewed development – given recent advances in solid-state DC-DC current conversion and regulation. And while Michael Faraday's simple "new electrical machine" may not really be suitable for commercial-scale power generation, as we will see, it's quite possible that a stand-alone residential power plant could be developed from it!
 
[ *Note: For our purposes here, we define a formal or self-sustaining over-unity device as "an electrical machine or self-contained system that, once started, will operate entirely on its own output and supply excess power to a load with no external input energy being provided thereafter by the operator." Thus, an electrical device that operates over-unity is one wherein the dynamic zero-point-energy environment contributes enough input energy that the ratio of output to operator-input ( "coefficient of performance", or COP) is greater than 1.0. Operator input is made, of course, in the form of applied torque.]

Introduction:  Before we make any detailed analyses of the Faraday disk machines, it will be useful to briefly summarize their essential mechanical and magnetic characteristics. The disk induction dynamo shown below exemplifies the following form of Faraday's Law of Induction: an electromotive force [emf, or voltage] will be induced in any conductor that is moving across or "cutting" magnetic flux lines, and the emf is proportional to the rate at which the flux lines are being cut. Thus, the rotating disk cuts the flux produced by the stationary magnet field pieces, and a voltage appears between the inner and outer edges of the disk.
 

correct interpretation of disk dynamo action, as shown, whether field pieces are conductive or not.

 
    Of course, we now know with a certainty that the rotor of such a generator is subject to a magnetic back-torque that is proportional to the output current drawn, due to the operation of Lenz's Law, which states that: a current set up by an emf induced due to the motion of a closed-circuit conductor will be in such a direction that its magnetic field will oppose the motion causing the emf. But, as it turns out, it is this same 'limiting' principle which allows this device – and nearly all common rotating power generation equipment – to also function as an electric motor: if the appropriate voltage and current are supplied, in this case between the center and edge of the conductive rotor, the disk will rotate with about the same motor torque as it requires for a corresponding level of generator action.
 
    Curiously enough, shortly after Faraday's initial experiments with a primitive version of the induction dynamo he discovered that the same voltage could be induced if the magnetic field pieces were affixed directly to the rotor, but that no voltage at all would be induced if the field pieces were rotated while the conductive disk was held stationary! He logically inferred that even if the field pieces were rotating, their magnetic fields must be stationary, and that the type of uniform magnetic field produced by a cylindrical magnet must therefore be a property of space itself (or perhaps more accurately, space-time) and was independent of the magnetic material which serves to create the field. This quality or effect is an oddity which to this day has no satisfactory explanation.
 

incorrect interpretation of statorless generator action, promulgated by certain researchers.

 
    Some theorists mistakenly assumed that a statorless "unipolar" generator (like that shown above) can no longer also serve as a motor, presumably because the magnetic field produced by an applied rotor current can't 'properly' oppose that produced by field pieces which are rotor-mounted, in "normal" accordance with Lenz's Law. The logical inference then drawn by certain researchers, perhaps most notably the late Bruce DePalma (a physics professor at MIT) and electrical engineer Paramahamsa Tewari in India, was that a statorless generator therefore would not exhibit the normal Lenz losses [as back-torque] provided that the magnets used are nonconductive – and so it should most assuredly be capable of functioning at an over-unity level of output!
    DePalma's explanation for this peculiarity was simply that the interaction of the primary magnetic field with that produced by the radial output current results in a shear torque between the conductive disk and the field pieces which is resisted by mechanical attachment, and so is wholly constrained within the rotor assembly and not reflected back to the mechanical drive! The experimental evidence seems to indicate that neither of these two predicate assumptions is true, however, as we'll see.
    DePalma's 'discovery' [circa 1977] that a unipolar generator's field pieces should be nonconductive to avoid the production of magnetic back-torque is illustrated by the "N-effect" diagram at left below – and could in fact be arrived at through simple deductive reasoning. If a rotating cylindrical magnet produces a stationary field, and the magnet itself is conductive (e.g., Alnico), then a voltage should appear between the center and outside surface by Faraday's Law of Induction! The trouble is that the act of drawing any load current from such a set-up will once again produce a classical level of back-torque, by Lenz's Law.
 
  

 


 
    Why DePalma called the device of his design shown on the right above an "N-machine" is something of a mystery, since it is in essence merely a typical unipolar generator. It is also interesting to note that although he has properly indicated the polarity of the induced voltage in the N-effect drawing on the left – according to the traditional left-hand rule** for generator action – the polarity shown in the drawing on the right is incorrect! And, as we've indicated in our preceding drawings, the preferred and logical polarity for any DC disk generator device is that whereby the negative terminal is located at the circumference – so that any "Hooper-effect" electromechanical centrifuging of mobile conduction electrons is series-aiding with respect to the induced load current flow. [In this case, it's important to realize that pure DC current has actual mechanical momentum, whereas AC current does not!]  
 
[ **The left-hand rule for generator action is as follows:  Extend the thumb, index finger, and center finger of the left hand at right angles to one another, so that the index finger points in the direction of the flux (north to south) and the thumb points in the direction of the motion; the center finger will then point in the direction of the induced electron flow (negative to positive).]


Review of Prior Art:  In an ongoing effort to develop a self-sustaining unipolar generator system that just might be able to serve as a stand-alone residential power plant, Bruce DePalma and his principal correspondent-collaborator Paramahamsa Tewari built and tested very large, unwieldy, and expensive apparatus in reliance on the inherent design scalability of the Faraday machine. A number of perceived "improvements" over the basic technology were also implemented, but the underlying logic with which these changes were selected and made was perhaps inherently flawed – with negative consequences for both the cost-effectiveness and the coefficient of performance (COP) of the equipment [examples of which may be seen at the DePalma website (
depalma.pair.com/index.html) and at www.tewari.org.
    The famous Kincheloe Report (1986) on the testing of one of DePalma's larger N-machines may be reviewed at www.totse.com/en/fringe/free_energy/dpalma5.html. Despite the fact that DePalma made a number of questionable design choices in the "Sunburst" device tested by Robert Kincheloe (Professor Emeritus, Electrical Engineering, Stanford Univ.), the data suggests that the back-torque losses were only about 20% of generated power – instead of the classical >100%. Prof. Kincheloe also states that: "while DePalma's [output] numbers were high, his basic [free energy] premise has not been disproved"; and "there is indeed a situation here whereby energy is being obtained from a previously unknown and unexplained source."  
 
    These two pioneering experimenters and many others seem to have made a primary assumption for which we have so far found little supporting evidence: that a disk dynamo-motor could be shaft-coupled to a unipolar generator, which in turn powered it, to great advantage. It might then seem that even if the dynamo-motor itself was not over-unity in nature, the external input electrical power required could be substantially reduced since the two devices have very similar [but difficult to match] voltage and current characteristics. But an under-unity machine of the original stator-and-rotor disk dynamo design would only be as efficient if used as an electric motor as it was as a generator: its motor efficiency would not significantly exceed that of today's best electric drive equipment, and the increased rolling load would tend to offset any "power gain factor" by requiring proportionally greater operator input power and cost.
    The question arises whether such a combination could become input self-sustaining if an over-unity disk induction dynamo was somehow designed and incorporated. And in fact, mathematical modeling suggests that this might be possible, in that the dynamo's power output goes up by the 4th power of increases in the rotor radius while its input power requirement goes up by the square thereof. However, in accordance with the earlier-stated relative equivalence of motor/generator action in almost all rotating electrical equipment, there is every empirical reason for believing that such a Faraday "dynamoelectric" motor could only produce as much torque as it requires as a generator under full load. Therefore, even if a given disk dynamo was able to exhibit a full-load COP of 2 (for example) when used as a generator to convert input torque to output current, it would exhibit a COP of only 1/2 or 0.5 when used as a motor to convert full-load input current into output torque!! Thus, piggybacking the two Faraday machine variants on a common shaft, in hopes of "bootstrapping" the dual device to a state of self-sustaining operation, would actually tend to be self-defeating from a practical standpoint.
    Finally, both DePalma and Tewari elected at some point to try replacing the permanent magnet fields with externally-powered electromagnetic coils, but then a special "test cabinet" source was necessary to supply proper power to these field coils – in addition to the grid power always required by the primary drive motor! This had the minor advantage of making the unipolar generator's DC output fully variable and reversible. However, a flat coil or solenoid produces a nonuniform magnetic field whose flux density falls off with increasing distance from its radial centerline, unlike the uniform field established between facing permanent magnets. With the addition of ferromagnetic cores, the relatively weaker fields of the facing electromagnets could then be greatly augmented and homogenized – and AC power output could even be achieved – but only at the expense of proportionally increased eddy current (Lenz) losses [that could only be minimized by using cores made of a very nonretentive field grade (~3% Si or other low-carbon) steel alloy, such as C1010.]
    Although the N-machine and Tewari's "Space Power Generator" have been claimed to exhibit COPs approaching 3.0 (or more), these devices were never made capable of self-sustaining operation nor were they (of course) ever cost-effectively mass-produced and marketed. And while Tewari maintains that the technology "is indeed commercially viable and should be brought to the attention of the general public", he notes that prospective manufacturers "do not see a market for a low-voltage, high-current machine."
 
    We feel that successfully bringing an over-unity induction dynamo system to market may actually be achievable, and that the secret to doing so is quite simple: use a back-to-basics 'systems approach' to develop a very straightforward yet sophisticated design which avoids all of the unnecessary pitfalls just cited, one which is self-sustaining once started with standard 12vdc car batteries and whose DC output is made inverter-ready using advanced solid-state current converters for output voltage pre-amplification. In keeping with this strategy, we will investigate a feasible design for a product that would hopefully be home-owner affordable (using off-the-shelf or non-exotic materials and components whenever possible) and compact (where the unit's bulk size has been minimized while its output-to-weight ratio has been absolutely optimized). The design theory we will examine and develop below clearly sugggests that, under properly controlled conditions, this is possible; it could turn out, however, that the system as a whole might not yet be cost-effective under 'normal' circumstances . . .
 
Tesla and Back-Torque Theory:  There's ample evidence to support both the view that back-torque will be produced in any Faraday disk machine and the claim that 'drag' may be artfully reduced to much less than a classically-figured level. Oddly enough, the notion that rotor back-torque might not be produced in one type of Faraday machine or another can probably be traced back to Nikola Tesla – and an obscure 1891 paper entitled " Notes on a Unipolar Generator". Taken out of context, Tesla's bald statement that "such a machine differs from ordinary dynamos in that there is no reaction between armature and field" was perhaps misconstrued as an absolute in the case of the statorless variant:  little was known about it, since no one could otherwise see any good reason to rotate the extra mass of the field pieces, and everyone knew that back-torque was normally produced in a typical fixed-stator dynamo.  

    [Significantly, perhaps, Tesla makes no mention whatever of the statorless Faraday generator variant in this paper. It should also be clarified that while some people may refer to such a statorless homopolar machine as a "unipolar" generator, it's more correct to let the latter term signify – as it did for Tesla – that such an acyclic-voltage device produces pure DC current having only one constant polarity (as in a battery), whether or not the field pieces are stationary.]

    However, careful review of the paper reveals that Tesla was referring specifically to the case where we "assume the current [is] to be taken off . . . by contacts uniformly from all points of the periphery of the disc." His analysis suggests that when a load current is drawn, eddy currents will be confined to those radial sectors of the disk which do not lie directly between the shaft and an outer pickup brush. In other words: regardless of whether we rotate the field pieces or not, or whether they're conductive or not, if we collected the load current from a continuous peripheral brush which enclosed the entire edge of the disk, no rotor eddy currents could be generated and the 'normal' level of counter-torque would not be present. [Its heavy eddy currents are of course a well-known major component of a typical disk dynamo's total magnetic losses.]
    What Tesla is really saying, then, is that the more uniformly a disk dynamo's output current is drawn through the periphery of the disk, the less "armature reaction" can be supported by eddy currents within the disk – which are induced by the stator magnets and therefore attracted to them. [This could also be seen to validate the earlier contention that a disk machine whose output had been sufficiently optimized to allow it to function as an over-unity generator would constitute a proportionally under-unity motor!]

[Note: We can further show that no significant reduction of the Lenz losses expressed in a disk dynamo can be made, even with ideal stator shielding, until the angular width of the primary (radial conduction) "brush sectors" – which is equal to the width of each pickup brush – exceeds that of the adjacent ‘neutral’ (transverse deflection) "eddy current sectors". Of course, the potential value of a practical liquid metal brush system which provided 100% rotor disk edge 'coverage' is also apparent – in that such a dynamo’s Lenz losses could theoretically then be entirely eliminated when ideal stator shielding is also employed – and is no doubt the reason why the U.S. Navy imposed secrecy and gag orders on inventor Adam Trombly.

   It's very important to realize at this point that Lenz losses in a fixed-stator disk dynamo may take two (2) forms: eddy current coupling, and stator inductive coupling. In the first case, a full classical level of rotor counter-torque will tend to be produced due to the attractive magnetic drag or 'friction' caused by direct polar coupling between the applied stator field and induced microcirculatory rotor eddy currents.
    As it turns out, though, a full classical measure of similar Lenz losses may also be produced by the stator, as a result of simple attractive polar coupling between the net rotor magnetic field [as expressed by "uncompensated" or asymmetrically-balanced primary load current distribution] and any (i) exposed (unshielded) outer stator magnet poles or (ii) incompletely field-piece-saturated stator shielding.  

[Note: An "ideal stator shielding design" is one wherein the field piece assemblies exhibit no external magnetic field outside of the flux gap (as in the "closed-path" configuration developed by Adam Trombly.]

    Thus, to the extent that stator coupling occurs, it will act to produce additional magnetic drag upon the rotor which is linearly proportional to the load current drawn - and thereby to satisfy "Lenz’s Law". We may further infer that, even with perfect (ideal) stator shielding to prevent any load current inductive coupling, a fully-proportional Lenz-loss eddy current counter-torque load will still seek to develop, to the extent possible (or allowed) unless the rotor disk is highly sectored with pickup brushes.

   [The percentage of 'normal' eddy current losses that any particular disk dynamo will exhibit is a very complex function of its geometry and that of its collector brush system. However, we have developed a viable proper method for actually calculating the reduced eddy current back-torque ratio that any given design should exhibit. Interested persons may obtain a copy of our definitive
Eddy Current & Stator Loss Analysis white paper upon request, with prior submission of a simple 1-pg. NDA.]

   We can arrive at a new and much clearer picture of the nature of reactive stator losses in this device, by means of the following unbroken chain of logic:
 
  – (i) The magnetic field produced by any disk rotor current(s) must rotate with the motion of the disk, being in opposition to the stationary field(s) applied by permanent magnets, or no motor action would result with an applied input current (instead of applying torque); and
  – (ii) this rotor current field must fully enclose the disk without intersecting it, or it would act either to generate a "free" voltage or to influence the disk's own inertia (neither of which can occur, classically).
  – (iii) Ordinarily, in the absence of an applied axial field, the rotor current field would then take up a simple symmetrical dual toroidal configuration (above and below the rotor plane) because of the disk's radial geometry. But in this case, it must establish the attenuated field configuration that's shown in the diagram on the right below, because flux lines never cross – and those of the rotating current field can't intersect those comprising the stationary axial applied field (in which nearly all of the disk is immersed).
 


  
 
  – (iv) Moreover, it can be seen that the rotor current field completely encloses not only the disk but the applied stationary fields as well. Therefore, the disk's rotating current field may inductively couple to any ferromagnetic material on or near the field pieces, which can thereby cause the stationary applied field to impart additional armature reaction (or back-torque) to the rotor, unless that material is magnetically saturated due solely to induction by the field pieces.
  – (v) Toroidal fields exhibit an inherently unfixed "radial polar tendency", and will polarize attractively to any applied axial polar field when radial torque is applied to them. Thus, any extant net rotor field due to self-generated current(s) will experience 'drag' against the stationary applied field, regardless of its own polarity (and disk current flow direction), as classically it must. In practice, end-pole keeper plates can be used to nearly eliminate any magnetic interaction between induced rotor current and the field pieces themselves – even if they're metallic and rotating.
  – (vi) Most importantly, however, it can be shown by integral calculus that the primary rotor current's net magnetic field around the disk is inversely proportional to the extent to which that current is drawn in opposite radial directions (i.e., by pairs of 180o-spaced brushes), because the net load current enclosed [by the path of integration] is then zero – due to equal currents flowing in opposite directions – thereby mathematically confirming the fundamental validity of Tesla's guiding design principle!
 
    Recalling now that a conductor must move relative to a stationary field for voltage to be induced, we may also deduce the following design first-principles: (1) to wit, the primary rotor current field will not act to induce eddy currents in metallic stator magnets, which in turn won't impose corresponding additional back-torque (as Lenz losses); but (2) however, when metallic magnets are used for the applied field in a 'unipolar generator', there is relative motion between a rotating conductor and a stationary field, so field piece eddy currents will be induced which may couple to the rotor field and contribute to back-torque!
    Taken together, the preceding material clearly shows that – with ideal stator shielding – back-torque will only be expressed in an induction dynamo (with a fixed stator) to the extent that eddy currents are allowed to circulate in "unused" radial sectors of the disk*, and this principle can now be used to figure what percentage of the classical eddy current back-torque will be produced: it is simply (but not strictly) proportional to the ratio of that portion of the disk's circumference which is not 'covered' by its collector brushes to its entire circumference!  [Note: This presumes of course that the load current is shared equally by uniformly-spaced pairs of oppositely-vectored brushes. Many different brush system configurations are possible for any given size dynamo; e.g., we will use a total of 64 brushes in the 'preferred embodiment' 18"-dia. prototype we're building.]

    This same reduced back-torque ratio also applies, of course, to a 'unipolar' generator. Unfortunately, it can also be seen why only the much-less-powerful nonconductive (ceramic) magnets have generally been used in the statorless variant, since induced field piece eddy currents can transfer magnetic load to the rotor field in a statorless generator if the magnets used are metallic – because the load current's rotating magnetic field encloses the combined stationary fields of the permanent magnets and tends to polarize attractively to them. [An applied disk current's rotor field would polarize repulsively, of course, resulting in motor action, in the absence of an applied radial torque.]
    Finally, in this regard, it could be said that in a Faraday disk generator the motor field encloses the stator field, and not the reverse – as almost universally is done in common electromechanical practice. Perhaps the possibilities inherent in this unusual arrangement are part of the mystery and fascination these machines have always held for the engineering-inclined. And the greatest "secret" of both these disk machines (as we see it) is that it's possible in effect to trade the classical Lenz loss back-torque for a much-smaller collector brush torque load requirement [in the generator variants].
    In any event, in the Analysis sections to follow, we will calculate the input torque, output power, and efficiency (or coefficient of performance ) of both an induction dynamo and a unipolar generator of the same physical size, for both the classical case and the adjusted theoretical model developed from the refined design principles just discussed. It is then up to the reader to empirically decide which results are the more accurate . . .    

Getting Back to Basics:  In this section, we'll concentrate on deriving some essential formulas which properly govern and define the basic operating characteristics of both the Faraday disk dynamo and its unipolar generator variant (i.e., voltage, power, input torque, etc.). From these simple equations, we will be able to develop some further sound engineering guidelines and appropriate design first-principles.
   


Voltage:  While many texts will show calculus used to determine the accepted generalized formula for induced voltage in the disk machines, this relation is much easier to derive by simple algebraic means from Faraday's own general rule of induction.
 
 
  [a] To wit, the magnitude of the terminal voltage induced in a conductor depends on three factors: (i) the flux density of the applied field; (ii) the length of the conductor immersed in the field; and (iii) the velocity at which the conductor moves through the field. Taking the simple product of all three factors yields the equation  E = B l v ,  where E is in volts, B is the flux density in tesla (or webers/m2 ), length l is in meters, and v is the velocity in m/sec.
  [b] In a general sense, we may let  length l = r , where r is simply the full annular width of the rotor (in meters). Then, we can let the average angular speed of rotation  v = (2π r f) / 2 , or v = π r f , where f is the rotor's frequency of rotation in revolutions per second.
  [c] Therefore,  E =  output voltage Vo =  B r (π r f) . And thus, nonrigorously,  Vo = π r2 f B .
  [d] In reality, we must let  the length l = R, where Ra is the radial width of the rotor's net working flux gap area (in meters). Also, we'll let the average angular speed of rotation  v = 2π r f ,  where r now specifies the mean radius of the flux gap area as measured from the rotor's axis, and f is the rotor's frequency of rotation (in rps).
  [e] Then,  Vo =  B R(2π r f ) ; and, merely rearranging terms,  Vo = 2πr Rf B . This formula will now yield the necessary accuracy of voltage calculation, since that portion of the rotor disk which is actually immersed in the field is properly indexed to as-specified dimensions for the field pieces.
 
    It is important to point out that Vo as figured by the formula just derived can actually be treated as both an open-terminal and full-load value. It can be shown theoretically, and has been experimentally verified, that a disk dynamo or generator's rotor charge will be distributed in such a way that its output voltage is quite constant regardless of the load current drawn, and these machines therefore behave as if they were a regulated voltage source!  
 
Resistance:  Having developed the relative radial planar-dimensional relationships for the disk devices, so that we can properly project operating voltage, it then becomes necessary to address the issues of volume resistivity and internal rotor circuit resistance if we wish to correctly figure the output current I (according to V = I R ) and power P (according to P = V I ) for any given size machine. No other aspect of system design is more crucial to optimizing output or more responsible for unrealistic projections of system performance.
    It's easy to see that open-terminal resistance of a disk dynamo or generator has only one seemingly major component: the resistance of at least one pair of brushes. In virtually all cases, the resistance of the rotor disk itself is and should be entirely negligible, generally being measured in only the single-digit micro-ohms. And, it's true, proper brush design and material selection will probably "make or break" the system's COP in most cases. However, in addition to brush-and-rotor resistance there's another type of resistive loss that is sometimes not accounted for in design reckoning: that of the microscopically-thin field discharge contact zone between each brush and the rotor! This is actually the primary resistance in all Faraday disk machines; it's largely responsible for brush heating, and can be as much as 3 orders of magnitude larger than the actual brush resistance.
    It is essential in this technology to use the highest quality brushes, having very high conductivity and a low coefficient of friction (k). For silver-graphite brushes running on silver slip rings or plated surfaces, static k is ~0.3 and dynamic k is ~ 0.2. Each brush's contact interface resistance in such case should not exceed 0.005 ohm initially and should decrease to an average of Rz = ~ 0.003 ohm after extended open-terminal run-in.
 
  [a] The most convenient formula for volumetric resistance is:  R = ρ L / A , where R is resistance in ohms; ρ is the volume resistivity in ohm-cm; L is the length in the current direction, in cm; and A is the current's cross-sectional area in cm2. Appropriate use of this formula to figure brush and rotor resistances will be illustrated in the Analysis sections below.    
 
Output Power:  In most cases, voltage can be thought of as the primary component of electrical power, as seen in the relation given earlier above. Thus, to maximize the output power Po of a disk dynamo or generator, we absolutely must maximize the unavoidably tiny voltage it produces. Not considering the required input power (and torque) for the moment, it can easily be seen in the formula derived for output voltage (Vo) that the most effective way to do this is simply to increase the rotor radius – since a major measure of its radial width is factored in twice.
    At some point, however, the product of increasing rotor size and speed will result in an unacceptably high value for brush speed, and so the maximum OEM ft./min. rating of the brush material selected will define an upper limit for the device's output voltage and its rotating inertia. Of course, the flux density B (as the final determining factor) can be maximized to the extent permitted by the cost and availability of specialized magnetic materials which have their own concrete natural field-strength limits.
 
  [a] From the two formulas for power P given earlier above, it follows that  P = V (V / R) = V2/ R . Therefore, by substitution, the full-load output power  Po = (2πr Rf B)2/ R . Of course, if output current Io has previously been calculated (since I = V / R), Po is also conveniently equal to Io2R . In principle, however, output power should only be based on the calculated or measured value for Vo using P = VI , due to the device's constant voltage characteristic (as discussed above).
 
    It should be noted that metallic neodymium iron boron magnets are now available with flux densities approaching 1.4 tesla (or 14,000 gauss) This is the world's most powerful permanent magnet material. The strongest nonconductive magnets generally available in large sizes are made of sintered Ferrite 5 (BaO-6Fe2O3), a ceramic material with residual flux density of 0.38 tesla (3,800 gauss).
   


Input Torque & Power:  Before we derive a generic formula for no-load mechanical input power (in watts) that is strictly a function of a device's rotating inertia and variable starting time, it is important to realize that power P more fundamentally must reflect the brush and load input torque T required, according to Pi = 2π f Ti , and that torque is rotational force applied at a given distance from a central axis of rotation. Torque may be expressed in newton-meters (N-m) when the power is in watts. In the Analysis sections below, a handy formula is provided for figuring applied induction motor torque in foot-pounds (ft.-lb.) and rpm, in which case input power will be in units of horsepower (Hp) [where 1 Hp = 746 watts].
    In the induction dynamo, the primary load is the 'normal' generator back-torque Ta, whose magnitude can be simply computed (by virtue of Lenz's law) from the traditional equation for the "force on a current element in a magnetic field": F = B I L , where for present purposes I  is equal to Io (or the load current) and as before L = Ra , where Ra is the radial width of the flux gap area. The full-load back-torque due to induction may then be found by using r once again to specify the mean radius of the flux gap area in the standard relation for torque: Ta = F r , where F will be a negative (retarding) value in N-m. [It is important to note that:  1 N-m = 0.7376 ft.-lb. = 8.851 in.-lb.]
    In an ideal disk generator, the primary load is just the dynamic friction of the brushes, although an OEM-provided value for static ('starting') coefficient of friction  must be considered.  Of course, this will be a substantial source of retarding torque and secondary load in any practical disk induction machine. Recommended values for spring pressure vs. material  will allow the total brush 'drag' to be accurately computed. The negative brush(es) will contact the rotor disk's outer edge in the design analyses we'll study below, so the full 'nominal' rotor radius Ro will be used to figure outer brush speed and resultant counter-torque. For simplicity, we'll consider the positive brush(es) as running directly on a conductive rotor shaft (as shown in the drawings above). The net forward force required to keep the brushes from decelerating the rotor is:  F = p k A , where F is in pounds, p  is spring pressure in psi, k = coeffic. of friction, and A = total contact area in square inches. [OEM-suggested minimum spring pressure for silver-graphite brushes is 4 psi.]
 
  [a]  No-load mechanical power to the rotor assembly is equal to the kinetic energy stored therein at a given constant operating speed divided by the elapsed time needed to achieve that speed: So, Pr = Ek / t . Ek is in turn equal to half the product of the rotor's moment of inertia (I) and the square of its final "run" angular velocity (ω, in radians/sec), where ω = 2πf :  Pr = [(½mRo2)(4π2f 2/ 2)] / t . And thus,  Pr = mπ2R2f 2 / ts , where Pr is in watts, m is mass in kg, R in this case is the 'equivalent annular inertial radius' [(A/π)1/2] of the disk* (as figured below), and ts is "start" time in seconds.
* [It is acceptable in this case to ignore the trivial inertial moments of the shaft and two disk mounting flanges.]

    Finally, once we have verified a prospective drive motor's full-load torque capability, we will use the simple power formula  ts = ωI / T  to see if start time ts is acceptable for the type of motor selected. It may be of interest and value for students to know the provenance of this formula, which is derived as follows: ω = a t , where ω = 2πf (with f in rps),  a = avg. angular acceleration, and t is the elapsed (or starting) time in seconds; and  T = I a , where T = avg. torque, and I is the moment of inertia. And thus,  t = ω / a = ω / (T / I), whereby  ts = ω I / T .    


Induction Dynamo Analysis:  In this first Design Analysis section, we will consider a disk induction dynamo with a pure copper rotor 18" in diameter and 0.187" thick, which is mounted to a 1"-dia. dual-bearing drive shaft made of a beryllium/copper alloy. The disk will be secured to the shaft using press-fit CDA18135 (99%Cu;Cd/Cr) split flanges that are silver-soldered to the disk and then set-screwed both to each other and to the shaft. To allow for stator thickness, brush slip rings, bearings, and drive coupling, the rotor shaft will be 8.5" in length.
    Two field piece arrays, each composed of 173 NdFeB disk magnets that are 1" in diameter and ½" thick, will be epoxy-resin-bonded into solid stator assemblies and mounted plane-parallel to the rotor with a realistic mechanical clearance (in the flux gap) of 0.0085" on each side.  Silver-graphite brushes
(93%Ag) will then be mounted and connected as described above, with the matching rotor-edge and shaft contact surfaces silver-plated.  
 
    

 

 
 
    Since multiple brushes will be absolutely necessary, the outer brushes will be parallel-connected in sets of four (4) per inner shaft brush, each separated by 90° of rotation – and more than one such set may in turn be uniformly distributed around the rotor by equal 'sectoring'. An even multi-pole number of such sets should be used, so that the corresponding inner brushes may be installed uniformly on the rotor shaft. The inner (shaft) brushes are assumed to be positive [see the 2nd following graphic].


 
Stator dimensions & flux density:
  [i]  flux gap outside radius OR = 8.2857" + ½" = 8.7857"
  [ii]  flux gap inside radius IR = 2.2142" – ½" = 1.7142"
  [iii]  flux gap radial width = OR – IR = 7.0715" = Ra = 0.180 m
  [iv]  mean induction radius = (OR + IR)/ 2 = 5.25" = r = 0.133 m
  [v]  net flux gap area = π(OR)2 – π(IR)2 = 233.26 sq.in.
  [vi]  total magnet area = 173 (0.7854) = 135.87 sq.in.
  [vii]  gap area B-factor = (135.87 / 233.26) = 0.5825 = 58.25%
  [viii]  disk magnet residual induction = Br = 12,900 gauss
  [ix]  computed gap flux density* = 7378 gauss (see graph at right)
  [x]  net flux density = (0.5825)(7378 gauss) = B = 0.430 T
* flux density graph courtesy of  Australian Magnetic Solutions
 
Rotor mass & moment of inertia:
  [i]  disk density (pure Cu) = 0.323 lb./cu.in.
  [ii]  disk area A = π(Ro)2 – π(½")2 = 253.68 sq.in.
  [iii]  disk volume = (253.68)(0.187") = 47.44 cu.in.
  [iv]  wt. = (47.44)(0.323) = 15.323 lb., and mass md = 6.965 kg
  [v]  equivalent inertial radius = (A /π)1/2 = 8.986 in. = 0.228 m
  [vi]  moment of inertia = ½(6.965)(0.228)2 = 0.181 kg-m2
 
Shaft mass & moment of inertia:
  [i]  density (Be/Cu alloy) = 0.302 lb./cu.in.
  [ii]  volume = 8.5 [π(½")2] = 6.68 cu.in.
  [iii]  wt. = (6.68)(0.302) = 2.017 lb., and mass ms = 0.917 kg
  [iv]  moment of inertia = ½(0.917)(0.0127 m)2 = 7.40 x 10–5 kg-m2
 
    Now that we have developed the necessary physical data for an 18" dynamo model, we need only specify a few more operating parameters before beginning a concise series of definitive performance calculations. The magnets' residual induction (or Br) has already been selected, and in this case is 12,900 gauss (1.29T) for standard grade-42 NdFeB disk magnets that have reasonable availability and justifiable price. The flux gap distance is readily figured from previous data, and is 0.204" or 5.2 mm. These criteria and the specified magnet dimensions were used to generate the flux density calculator graph provided above, from which the resultant gap flux density (quoted above) was obtained.
    Next, a rotation speed f must be selected which is not only within the brushes' maximum rating but is also hopefully at or very near an industry-standard electric motor speed. Additional data (like brush spring pressure) will be furnished as needed from OEM / vendor recommendations and specifications.
 
Voltage:  For the brush material grade selected, the maximum suggested contact speed is 5,000 fpm. While higher speeds are possible, brush wear will become excessive in a continuous-duty application. Therefore, our tentative design operating speed will be 850 rpm,  with f = 14.167 rps,  using an 8-pole AC drive motor. This yields a comfortably-high brush speed of 4,006 fpm. For comparison's sake, we'll also calculate a peak output voltage based on a rotor speed of 1150 rpm (or 19.167 rps), using a 6-pole motor for a maximum brush speed of 5,419 fpm.
    Therefore, at 850 rpm, Vo = (6.283)(0.133)(0.180)(14.167)(0.430) = 0.9163 volts. And, at 1150 rpm,  peak Vo would equal 1.240 volts.    


Resistance:  From the previous discussion of resistance, it should be realized that we can quickly derive a theoretical baseline rotor circuit resistance, and a corresponding maximum possible output current, by ignoring the trivial disk, brush, and shaft resistances and considering at first just the interface resistance of a set of 4 negative brushes load-connected to one positive brush. The combined parallel resistances of a couple even-numbered multiples of such brush sets can then be easily figured, to provide a means of increasing the output current. There will then be 1/4th as many positive brushes as negative brushes, and for convenience we will consider the number of dynamo/generator poles to be equal to the number of positive brushes.    
    For optimum performance, current must be drawn from the rotor disk in as radially uniform a manner possible. Therefore, the 2-pole (2-set) brush arrangement shown in the preceding graphic is much better than the basic 1-pole (single-set) connection, and a 4-pole connection having 4 separate 1-pole systems in parallel is better yet. Based only on the ~0.003-ohm per-brush interface resistance, these connection options result in the following minimum rotor circuit resistances and corresponding maximum currents:
 
 

  – in the 1-pole circuit, Rmin  = 1 [(0.003 / 4) + 0.003] = 0.003750 ohm , and  I max = 244.35 amps;  
  – in the 2-pole circuit, Rmin = ½ [(0.003 / 4) + 0.003] = 0.001875 ohm , and  I max = 488.69 amps;  
  – in the 4-pole circuit, Rmin = Ό [(0.003 / 4) + 0.003] = 0.000938 ohm , and  I max = 977.39 amps.


      Obviously, it is decidedly to our advantage to use the largest practical number of "pickup" brushes, although extensive mathematical modeling suggests that the total outer brush contact width should not be more than 50% of the disk's circumference in a disk induction dynamo intended for use as a motor. Accordingly, the brush current density rating will serve to limit the number of outer brushes used and the maximum possible current. The current density limit of the 93%Ag brushes we've specified is 300 A/in2. In practice, total brush ampacity should be strictly matched to the highest I max figure derived that does not exceed the calculated safe ampacity of the rotor disk (as discussed further below) and total pickup brush width should be absolutely maximized,  approaching 100% of the disk's entire circumference,  in all generator (non-motoring) variants.
    All things considered, we will assume that the 4-pole/16-point pickup brush circuit just described will be used in our 'prototype' model and in the calculations to follow.
 
    To get a better figure for our model dynamo's actual total circuit resistance Rt, we will now calculate the resistances of the brushes, rotor shaft, and disk.
    Given the OEM specs for brush current density (300 A/sq.in.) and resistivity (2.0 x 10–6 ohm-cm), we may simply divide the value for I max by 16 to find the ampacity of each negative (pickup) brush and then divide the result by the current density limit to find the contact area required. Therefore, the area of each outer brush An = (977.39 / 16) / 300 = 0.2036 sq.in. or 1.314 cm2. Outer brush thickness should be > ½ and < 2/3 of the disk thickness, as the disk's outer edges should be slightly chamfered. So, the width of each pickup brush will be 0.2036 / 0.125" = 1.63". The rotor's circumference is equal to 2πRo = 56.55", the total pickup brush width is 16 (1.63) = 26.08", and so the edge-width 'coverage' ratio is 26.08 / 56.55 or 46% (which is nearly optimum for a motor variant).
    Using brushholders which are only 5/8" 'tall', a good minimum brush length L is 1.0" or 2.54 cm. So, each pickup brush's resistance will be (2.0 x 10–6)(2.54) / 1.314 = 3.866 x 10–6 ohm, or Rn = ~ 4 μohm. Applying the same method to the 4 positive shaft brushes, with an assigned thickness of 0.50" the area Ap = 4An = 0.8144 sq.in. (5.254 cm2) and width again equals 1.63" or 4.14 cm. With length once again of 1.0", each inner brush resistance Rp = ~ 1 μohm. Finally, it will be important in practice to use the heaviest and shortest brush shunts (buss bar leads) feasible.
    The rotor shaft alloy that is greatly to be preferred is CDA17200 1.9% beryllium copper, with volume resistivity of 7.733 x 10–6 ohm-cm but the highest tensile strength of any copper-base alloy. To figure a liberal resistance for the rotor shaft, we will allocate 5" or 12.7 cm as its "electrical" length by mounting the positive brushes inboard of the bearings and drive coupling. The axial end area of the shaft is equal to π(½")2 = 0.7854 sq.in. = 5.067 cm2, and the shaft resistance Rs = ~ 19 μohm.
    The rotor disk's greatest electrical resistance is expressed through the edge of the 1"-diameter center shaft hole, the circumference of which is 3.1416". Given a thickness of 0.187", this inner edge area then equals 0.5875 sq.in. or 3.790 cm2. The volume resistivity of pure copper is 1.724 x 10–6 ohm-cm, and the disk's radial conduction length L is equal to Ro – ½" = 8.5" = 0.216 m = 21.6 cm. Accordingly, we find the resulting maximum possible rotor disk resistance Rd = ~ 10 μohm.
 
    Finally, we can now make the best possible projection of the rotor circuit Rt in milli-ohms as follows:
 
 

Rt = Ό [(Rz + Rn) / 4 + (Rz + Rp + Rs + Rd)]                  
= Ό [(3 + 0.004) / 4 + (3 + 0.001 + 0.019 + 0.010)]
= 3.781 / 4 = 0.945 m ohm = Rt = 945 μohm.       

 
Thus, it's conclusively demonstrated just how little effect the electrical 'hardware' really has on the total rotor circuit resistance of a Faraday disk dynamo (if properly designed), since the figure we just derived with fair effort differs from the quick estimate we made earlier by only 7 micro-ohms! Our revised figure for the "nominal" output current is then equal to 0.9163 / 0.000945 = nom. Io = 969.6 A.
   

Output Power:  The output power of our model dynamo at the given nominal rotation speed of 850 rpm (14.167 rps) is equal to I2R = Po = 888 watts. But, at the 'peak' rotation speed of 1150 rpm (19.167 rps), the output power would be equal to V2/R = (1.24)2 / 0.000945 = Ppeak = 1,627 watts. It can therefore be seen that in raising the operating speed by 1150 / 850 or 35.3%, the output current possible would rise to peak Io = 1,312 amps and the available power would nearly double! Of course, every brush's contact area would also have to be increased by 35.3%, and in the case of the outer brushes by increasing the brush width to 2.02". Total pickup brush contact width then increases to 35.29" or a reasonable 62.4% of disk circumference.
    Unfortunately, it can be shown by rather involved numerical anlysis of existing copper wire data that the safe ampacity of the 0.187"-thick rotor disk is 'only' 1,129 amps around its inner circumference. By increasing the shaft size to 1Ό" the safe ampacity could be raised to 1,376 amps, thereby enabling the dynamo to be operated at its peak Io. Rather than effect such a substantial device redesign, though, we will continue our analysis on the assumption that the brush size and operating speed shall be adjusted such that max. Io = 1,129 amps.
    Accordingly, the pickup brush width will have to be increased by 1,129 / 969.6 or 16.44% (to 1.90"). The total pickup brush width is 16(1.90) = 30.37", and so the edge-width coverage ratio is 30.37/ 56.55 or 54%. The new value for max. Vo = 1.067 volts, and the corresponding maximum rotor speed is then 990 rpm or f = 16.497 rps. And so, the maximum allowable output power is equal to (1.067)(1,129) or max. Po = 1,205 watts, and the new maximum brush speed is 4,665 fpm.
   


Input Torque & Power:  Although it might seem that we now have good final figures for operating speed and output power, we have yet to determine if the rotation speed we just derived is acceptably close to that of an off-the-shelf electric motor (as a practical source of input torque) which will provide adequate start and run torque at a given available horsepower rating. This will not be nearly as difficult as it might sound, with the aid of a simple yet indispensable electric motor formula that relates speed (rpm), torque (T), and power (Hp):  T = (Hp x 5252) / (rpm) , where the constant 5252 is equal to 33,000 ft.lbs./min./Hp divided by 2π radians/rev., and T is the torque in ft.lbs. [The equivalent metric expression is:  T = P / ω , where ω = 2πf , P is power in watts, and T is torque in N-m.]  
 
For the 'classical' case:
  [i] primary full-load counterforce = Fa = B I Ra = (.43)(1,129)(.180) = 87.39 N
  [ii] primary back-torque = Ta = Fa (r) = (87.39)(.133) = 11.62 N-m = 8.57 ft.lb.
  [iii] neg. brush counterforce (ea.) = Fnb = pkA = (4)(0.2)[.125 x (1.1644 x 1.63)] = 0.190 lb. (run)
  [iv] pos. brush counterforce (ea.) = Fpb = pkA = (4)(0.2)[(1.1644 x .500) x 1.63] = 0.759 lb. (run)
  [v] total brush retarding torque = Tb = 16(Fnb)(Ro) + 4(Fpb)(½") = 28.88 in.lb. = 2.41 ft.lb.
  [vi] total load torque T = Ta + Tb = 8.58 + 2.41 = 10.98 ft.lb. (treating the ~ 0.5% rolling losses as negligible).
 
    Correctly matching a standard electric motor to the mechanical load in this unusual application can be a complex and challenging task. All things considered, there are a number of reasons for selecting a permanent magnet or shunt-wound (wound field) DC motor, since the required AC input converter (as the power source) is usually also an economical variable speed control. This feature is especially desirable in situations like the present case, where our tentative operating speed falls right between the 'standard' 1150 and 850 rpm motor speeds. Also, in certain cases it may be inadvisable (although less expensive) to use an AC capacitor-start motor in this application, since they're designed to start under fully-loaded conditions and can briefly draw over 300% of normal running amps to do so. [In an average-load starting situation like the present case, this will also put a huge and unnecessary strain on the rotor assembly, due to its large moment of inertia.]
    Referring to a motor selection and ordering guide (such as any recent Grainger catalog), we find that a 2 Hp 1150 rpm PM motor (with 180vdc armature; FL amps = 9.8) is available [GE 5CD125TP002B] that develops full-load torque of 9.133 ft.lb. (109.6 in.lb.). Even though this is rather less than the 10.98 ft.lb. that would be needed to operate our model dynamo at its maximum capability, it must be remembered that (according to the torque formula above) this motor's output torque will climb as its speed is reduced – and it may be that just enough torque will be available at or very near the maximum 990 rpm operating speed we desire.
    The most straightforward way of determining that point on the selected motor's output power / torque curve where its operating speed is maximized for this particular application's input torque requirement is by repeated spreadsheet calculations to assemble tabular data, from which the best 'solution' of such a complex covariable problem is obtained when the net motor torque developed (Tm) just exceeds the total load torque (Tn) required at the reduced speed.
    Starting at 985 rpm and figuring output voltages in descending 5-rpm increments at first, the following optimal resolution was found at 970 rpm (f = 16.167 rps):
 
  – net Vo = 1.0457 volts, net Io = 1,106.5 amps, and net Po = 1,157 watts ;
  – net Fa = B I Ra = (.43)(1,106.5)(.180) = 85.643 N ;
  – net Ta = Fa(r) = (85.643)(.133) = 11.39 N-m = 8.40 ft.lb. ; and
  – total load torque = Ta + Tb = 8.40 + 2.41 = Tn = 10.81 ft.lb. [treating the ~ 0.5% rolling losses as negligible].
  – net motor torque developed = Tm = (2 x 5252) / 970 = 10.83 ft.lb. > Tn (@ 10.81 ft.lb.).
 
  – nominal motor efficiency = (Hp x 746) / (V x I) = 1,492 / 1,764 = ~ 84.6%
  – nominal dynamo efficiency = Po / (Hp x 746) = 1,157 / 1,492 = ~ 77.5%
  – combined system efficiency = (0.846)(0.775) = ~ 65.6%
 
  – letting avg. brush k = ½ (0.3 + 0.2) = 0.25,  avg.Tb = (0.25 / 0.2)(2.41) = 3.01 ft.lb.
  – avg. no-load motor torque  Tnl = Tm – avg.Tb = 10.83 – 3.01 = 7.82 ft.lb.
  – no-load motor starting time  min. ts = ωI / Tnl = 2πf (.181) / 7.82 = 2.35 sec.
  – Pr = mπ2R2f 2 / ts = (6.965)(9.8696)(0.052)(261.37) / 2.35 = 934.00 / 2.35 = 397.5 watts

  – avg. full-load motor torque  Tf = Tm – (½ Ta + avg.Tb) = 10.83 – (4.20 + 3.01) = 3.62 ft.lb.
  – full-load motor starting time  max. ts = ωI / Tf = 2πf (.181) / 3.62 = 5.08 sec.
  – rotor mechanical power expended = Pr = mπ2R2f 2 / ts = 934.00 / 5.08 = 183.9 watts

    The start times just calculated are entirely acceptable for a PM-DC drive motor, providing for 'gentle' starting considering the disk's large moment of inertia. And it may indeed be permissible in cases with extended no-load start times of between about 2.25 and 3 seconds to use an AC capacitor-start motor as a drive if necessary and if it so happens that the factory speed of the motor selected is acceptably close to the preferred operating speed of the dynamo/generator. [Longer start times may overload the additional start windings in a capacitor-start motor, damaging or destroying the coils and/or tripping a circuit breaker, even though the effective start times will be lower by ~300%].
    It's interesting to note that our model dynamo's actual induction efficiency can be found by simply refiguring its torque requirement without considering brush drag. Thus, at 953 rpm, a 1.5 Hp PM-DC motor would be adequate to power the dynamo at a voltage of 1.0273, current of 1,087.1 amps, and output power of 1,116.8 watts. With just the reasonable added proviso that the stator's outside poles must be wholly keepered at saturation, and at less than the full disk diameter, real  induction efficiency   is then 1,116.8 / 1.5 (746) = 99.8%! [The foregoing criteria regarding the essential use of saturated steel end-plates to keeper the outer stator poles are derived from the innovative "closed (flux) path" homopolar generator design of Trombly & Kahn (1982). A saturated material will support no further passage of flux nor any further external magnetic induction – in the form of eddy current losses, in this case.]
    Purists may also notice that the motor's own armature inertia has not been considered above in the interests of clarity and brevity, having been treated as negligible since it corresponds to just 0.3% of its output torque (by OEM specs) in this particular case. The same consideration also applies to the rotor shaft's miniscule inertia in relation to that of the disk.
 
And in the best-case theoretical model: The following computations are based on what we believe is a justifiable application of the 'Tesla' reduced back-torque ratio to our model dynamo in actual operation. As we developed earlier above, in this case that ratio is equal to 1 – 0.56 = 0.46.
 
  [i] primary full-load counterforce = Fa = 46% [B I Ra] = 0.46[(.43)(1,129)(.180)] = 40.20 N
  [ii] primary back-torque = Ta = Fa (r) = (40.20)(.133) = 5.35 N-m = 3.94 ft.lb.
  [iii] total brush retarding torque = Tb = 16(Fnb)(Ro) + 4(Fpb)(½") = 2.41 ft.lb. (same as before)
  [iv] total load torque T = Ta + Tb = 3.94 + 2.41 = 6.35 ft.lb. (treating the ~ 0.5% rolling losses as negligible).
 
    Referring again to the motor selection and ordering guide, we find that a 1.5 Hp 1150 rpm PM motor (with 180vdc armature; FL amps = 7.2) is available [GE 5CD125TP001B] that develops full-load torque of 6.85 ft.lb. (82.2 in.lb.). In this case, the motor will have more than enough torque for the dynamo to be operated at the maximum allowable rotor speed (and disk ampacity limit) of 990 rpm (f = 16.5 rps):
 
  – max. Vo = 1.067 volts, max. Io = 1,129 amps, and max. Po = 1,205 watts (from preceding subsection);
  – net Ta = Fa(r) = 3.94 ft.lb. (from above); and
  – total load torque Tn = 6.35 ft.lb. [treating the ~ 0.5% rolling losses as negligible].
  – net motor torque developed = Tm = (1.5 x 5252) / 990 = 7.96 ft.lb. > Tn (@ 6.35 ft.lb.).
 
  – nominal motor efficiency = (Hp x 746) / (V x I) = 1,119 / 1,296 = ~ 86.3%
  – nominal dynamo efficiency = Po / (Hp x 746) = 1,205 / 1,119 = ~ 107.7% (or COP = 1.077)
  – combined system efficiency = (0.863)(1.077) = ~ 92.9%
 
  – avg. starting brush retarding torque avg.Tb = 3.01 ft.lb. (same as before)
  – avg. no-load motor torque  Tnl = Tm – avg.Tb = 7.96 – 3.01 = 4.95 ft.lb.
  – no-load motor starting time  min. ts = ωI / Tnl = 2πf (.181) / 4.95 = 3.79 sec.
  – Pr = mπ2R2f 2 / ts = (6.965)(9.8696)(0.052)(272.25) / 3.79 = 973.18 / 3.79 = 256.8 watts

  – avg. full-load motor torque  Tf = Tm – (½ Ta + avg.Tb) = 7.96 – (1.97 + 3.01) = 2.98 ft.lb.
  – full-load motor starting time  max. ts = ωI / Tf = 2πf (.181) / 2.98 = 6.30 sec.
  – rotor mechanical power expended = Pr = mπ2R2f 2 / ts = 973.18 / 6.30 = 154.5 watts

Conclusions:
    We interpret the dynamo efficiency calculated above to be definitive and exciting proof [within the framework of this rigorous treatment] that our 18" Faraday disk dynamo model could in fact be built to exhibit bona fide over-unity operation, if not yet necessarily in a self-sustaining manner. As mentioned early in the course of this study, it would require a separate bank of solid-state DC-DC step-up current converters to pre-amplify the output voltage of such a device before that output can be accepted by the vast majority of available AC inverters. Most inverters today have a reliable nominal efficiency of  93%, and there are now several extant types of suitable pre-amplifying converters having similar efficiency. So, the combined 'in-line' efficiency for our model would at best be only (1.077)(.93)(.93) = ~ 93.15%, and the integrated 'system loop' – including drive motor – would obviously not be self-sustaining.
    [For an excellent and quite readable pdf technical paper (Starzyk et al.; 1999) on just one such DC current converter methodology, entitled "A DC-DC Charge Pump Design Based on Voltage Doublers", just click here.
    However, there are a number of things that can yet be done to significantly enhance the performance of our model 18"-diameter dynamo, not the least of which is to thicken the rotor disk slightly (for added rotor ampacity). We've also designed a more refined dynamo model (using many more brushes) whose inherent COP would approach 1.50, and which will in fact – if reality meets the best theoretical model – be demonstrably self-sustaining, given the typical converter, inverter, and drive motor efficiencies cited.*
    We therefore feel that an imperative course of research and development is plainly indicated.
  * [We've also drawn up an "integrated system loop flow chart for a self-sustaining Faraday generator" having a feasible 20% net back-torque ratio and COP = 1.5, whereby it can be seen that the integrated system's net over-unity COP will be 1.12 – and thus it's more than inverter-self-running – but have been advised against its publication.]
   


Statorless ('Unipolar') Generator Analysis:  In the preceding Induction Dynamo Design Analysis, we discovered undeniable evidence that the traditional stator-and-rotor Faraday disk dynamo has inherent over-unity potential – at least when ultra-high-strength NdFeB field magnets (of over 1.0 Tesla residual induction) are used. However, it could be shown from the preceding Conclusions that any such disk generator designed according to the specific design principles discussed herein would have to exhibit an over-unity COP of at least ~1.20 before it could 'drive' a self-sustaining output system of the type described.  

    And what of the homopolar disk dynamo's 'unipolar' statorless variant(s)? Wherein, so many people have tried to use Ferrite magnets which are attached unnecessarily to the rotor disk – and which have only about 30–35% of the residual induction of the available (albeit expensive) NdFeB magnets! It will obviously be a fair challenge to design such a generator which is in any way competitive with the prior model, but the process involved may serve to further illuminate the engineering methods and principles used so successfully in the preceding case.
    In that first Design Analysis section, we considered a disk induction dynamo with a pure copper rotor 18" in diameter and 0.187" thick, and in the interests of making a true and fair performance comparison we will do so again here. However, in this case the disk will be mounted to a 1½"-dia. dual-bearing drive shaft made of the same Cu/Be alloy, which will raise the rotor ampacity to 1,579 amps at the disk/shaft joint. As before, the disk will be secured to the shaft using press-fit Cu/Cd/Cr-alloy split flanges that are silver-soldered to the disk and then set-screwed both to each other and to the shaft, with the latter once again being 8.5" in length.
    Two field piece arrays, each comprising 173 Ferrite-5 disk magnets that are 1" in diameter and 5/8" thick, will be epoxy-resin-bonded into solid pole-set assemblies and mounted directly on the rotor disk (in this case) with an electrical clearance of 0.0085" on each side (as the thickness of each intervening adhesive layer). [These Ferrite magnets are equal in number and diameter to the NdFeB magnets used in the previous Analysis, although their thickness is 25% greater to enhance the much-lower gap flux density produced.] Silver-graphite brushes (93%Ag) will then be mounted and connected as before (with the negative brushes contacting the disk's outer edge and the positive brushes running directly on the rotor shaft).  
 
    
 
    The primary practical engineering problem inherent in the statorless generator design is, of course, achieving adequate dynamic balancing of each 'composite' pole-set assembly. Not only must the "one-spot" radial symmetry of magnet layout depicted in the diagram on the left above be uniformly broken, so that the magnet mass incorporated within any given-size radial sector of the rotor area is as equal as possible (across the concentric rings of magnets), but a saturated steel band which can be spot-drilled as needed to obtain maximum possible dynamic balance must encircle each pole-set assembly. We'll assume that such is the case here, although this would be difficult to manage in practice.
    Also, the outer brushes in this model will be parallel-connected in double sets of three (3) per inner brush, each set being separated by 60° of rotation, and once again an even multi-pole number of such double brush sets should be uniformly distributed around the rotor by equal sectoring to maximize rotor current and to allow the corresponding inner brushes to be installed uniformly on the shaft. As before, these inner brushes are assumed to be positive [see the 2nd following graphic].


 
Rotor dimensions & flux density:
  [i]  flux gap radial width = Ra = 0.180 m  (from Analysis 1 above)
  [ii]  mean induction radius = r = 0.133 m  (from Analysis 1 above)
  [iii]  net flux gap area = 233.26 sq.in. (from Analysis 1 above)
  [iv]  vol. of each pole-set assy. = (233.26)(0.6335) = 147.77 in3
  [v]  total magnet area = 173 (0.7854) = 135.87 sq.in.
  [vi]  vol. of each pole-set = (135.87)(.625) = 84.92 in3
  [vii]  gap area B-factor = (135.87 / 233.26) = 0.5825 = 58.25%
  [viii]  disk magnet residual induction = Br = 3,800 gauss
  [ix]  computed gap flux density* = 2375 gauss (see graph at right)
  [x]  net flux density = (0.5825)(2375 gauss) = B = 0.1383 T
* flux density graph courtesy of Australian Magnetic Solutions


 
Rotor mass & moment of inertia:
  [i]  disk mass md = 6.965 kg  (from Analysis 1 above)
  [ii]  disk equiv. inertial radius = (A /π)1/2 = 8.986 in. = 0.228 m
  [iii]  disk moment of inertia = ½(6.965)(0.228)2 = 0.181 kg-m2
  [iv]  volume of epoxy = 147.77 – 84.92 = 62.85 in3
  [v]  density of epoxy = 0.0715 lb/in3
  [vi]  wt. = (62.85)(0.0715) = 4.494 lb., and mass me = 2.043 kg
  [vii]  density of Ferrite magnets = 0.177 lb/in3
  [viii]  wt. = (84.92)(0.177) = 15.031 lb., and mass mps = 6.832 kg
  [ix]  pole-set inertial radius = (233.26 /π)1/2 = 8.617" = 0.219 m
  [x]  moment of inertia (ea.) = ½(8.875)(0.219 m)2 = 0.213 kg-m2
 
Shaft mass & moment of inertia:
  [i]  shaft mass ms = 0.917 kg  (from Analysis 1 above)
  [ii]  moment of inertia = ½(0.917)(0.0127 m)2 = 7.40 x 10–5 kg-m2
 
    Now that we have developed the necessary physical data for an 18"-dia. statorless generator model, we need only specify a few more operating parameters before beginning a concise series of definitive performance calculations. The magnets' residual induction (Br) has already been selected, and in this case is 3,800 gauss (0.38T) for standard grade-5 Ferrite disk magnets that are readily available and very reasonably priced. The flux gap distance is readily figured from previous data, and is 0.204" or 5.2 mm. These criteria and the specified magnet dimensions were used to generate the flux density calculator graph provided above, from which the resultant gap flux density (quoted above) was obtained.
    Next, a rotation speed f must be selected which is not only within the brushes' maximum rating but is also hopefully at or very near an industry-standard electric motor speed. Additional data (like brush spring pressure) will be furnished as needed from OEM / vendor recommendations and specifications.
 
Voltage:  For the brush material grade selected, the maximum suggested contact speed is 5,000 fpm. While speeds as high as 6,500 fpm are possible, brush wear may become excessive in a continuous-duty application. Therefore, our tentative design operating speed will be 1150 rpm,  with f = 19.167 rps, again using a permanent magnet (PM) DC drive motor. This yields an acceptably-high brush speed of 5,419 fpm.
    Therefore, at 1150 rpm, Vo = (6.283)(0.133)(0.180)(19.167)(0.1383) = 0.3987 volts.  


Resistance:  Similarly to the procedure used in the preceding Analysis section, we'll ignore the trivial disk, brush, and shaft resistances for now and consider just the interface resistance of two sets of 3 negative brushes, each connected to a single positive brush. The combined parallel resistances of several even-numbered multiples of such brush sets can then be easily figured, to provide a means of increasing the output current. There will then be 1/3rd as many positive brushes as negative brushes, and for convenience we will consider the number of dynamo/generator poles to be equal to the number of positive brushes (same as before).    
    For optimum performance, current must be drawn from the rotor disk in as radially uniform a manner possible. Therefore, a 4-pole (4-set) brush arrangement is much better than the basic 2-pole (double-set) connection shown in the preceding graphic, and an 8-pole connection having 4 separate 2-pole systems in parallel is better yet. Based only on the ~0.003-ohm per-brush interface resistance, these connection options result in the following minimum rotor circuit resistances and corresponding maximum currents:
 
 

  – in the 4-pole circuit, Rmin = 1/4 [(0.003 / 3) + 0.003] = 0.001000 ohm , and  I max = 398.7 amps;     
  – in the 8-pole circuit, Rmin = 1/8 [(0.003 / 3) + 0.003] = 0.000500 ohm , and  I max = 797.4 amps;     
  – in a 16-pole circuit, Rmin = 1/16 [(0.003 / 3) + 0.003] = 0.000250 ohm , and  I max = 1,594.8 amps.


      Again, the brush current density rating will serve to limit the number of outer brushes used and the maximum possible current. The current density limit of the 93%Ag brushes we've specified is 300 A/in2. In practice, total brush ampacity should be strictly matched to the highest I max figure derived that does not exceed the calculated safe ampacity of the rotor disk (as discussed further below) and total pickup brush width should be absolutely maximized,  approaching 100% of the disk's entire circumference,  in all generator (non-motoring) variants.
    Since the base 16-pole circuit resistance yields a figure for I max that is so close to the above-stated ampacity of the rotor disk, we will assume that the tentative 1150-rpm operating speed selected will be very slightly reduced to match that figure and that a 16-pole/48-point pickup brush circuit can therefore be used in this 'prototype' model and in the calculations to follow.
 
    To get a better figure for our model dynamo's actual total circuit resistance Rt, we will now calculate the resistances of the brushes, rotor shaft, and disk.
    Given the OEM specs for brush current density (300 A/sq.in.) and resistivity (2.0 x 10–6 ohm-cm), we may simply divide the value for I max by 48 to find the ampacity of each negative (pickup) brush and then divide the result by the current density limit to find the contact area required. Therefore, the area of each outer brush An = (1,579 / 48) / 300 = 0.1097 sq.in. or 0.708 cm2. Outer brush thickness should be > ½ and < 2/3 of the disk thickness, as the disk's outer edges should be slightly chamfered. So, the width of each pickup brush will be 0.1097 / 0.125" = 0.88". The rotor's circumference is equal to 2πRo = 56.55", the total pickup brush width is 48 (0.88) = 42.24", and so the edge-width 'coverage' ratio is 42.24 / 56.55 or 74.7%.
    Using brushholders which are only 5/8" 'tall', a good minimum brush length L is 1.0" or 2.54 cm. So, each pickup brush's resistance will be (2.0 x 10–6)(2.54) / 0.708 = 7.175 x 10–6 ohm, or Rn = ~ 7 μohm. Applying the same method to the 3 positive shaft brushes, at an assigned thickness of 0.375" the area Ap = 3An = 0.3291 sq.in. (2.123 cm2) and width again equals 0.88" or 2.24 cm. With length once again of 1.0", each inner brush resistance Rp = ~ 2 μohm. [As before, it will be important in practice to use the heaviest and shortest brush shunts (buss bar leads) feasible.]
    The rotor shaft alloy that is greatly to be preferred is CDA17200 1.9% beryllium copper, with volume resistivity of 7.733 x 10–6 ohm-cm but the highest tensile strength of any copper-base alloy. To figure a liberal resistance for the rotor shaft, we'll once again allocate 5" or 12.7 cm as its "electrical" length by mounting the positive brushes inboard of the bearings and drive coupling. The axial end area of the shaft is equal to π( Ύ")2 = 1.767 sq.in. = 11.4 cm2, and the shaft resistance Rs = ~ 9 μohm.
    The rotor disk's greatest electrical resistance is expressed through the edge of the 1½"-dia. central shaft hole, the circumference of which is 4.7124". Given a thickness of 0.187", this inner edge area then equals 0.8812 sq.in. or 5.685 cm2. The volume resistivity of pure copper is 1.724 x 10–6 ohm-cm, and the disk's radial conduction length L is equal to Ro – Ύ" = 8.25" = 0.210 m = 21.0 cm. Accordingly, we find the resulting maximum possible rotor disk resistance Rd = ~ 6 μohm.
 
    Finally, we can now make the best possible projection of the rotor circuit Rt in milli-ohms as follows:
 
 

Rt = 1/16 [(Rz + Rn) / 3 + (Rz + Rp + Rs + Rd)]                  
= 1/16 [(3 + 0.007) / 3 + (3 + 0.002 + 0.009 + 0.006)]
= 4.019 / 16 = 0.251 m ohm = Rt = 251 μohm.       

 
    As we saw before, the electrical 'hardware' really has little impact on the total rotor circuit resistance of a Faraday disk generator – if properly designed – since the figure we just derived with fair effort differs from the quick estimate we made earlier by only 1 micro-ohm! The rotor current developed at a preferred operating voltage of 0.3978 (as derived above) would then be equal to 0.3978 / 0.000251 or 1,588 amps, since I = V/R. Unfortunately, it can be shown using rather involved numerical anlysis of existing copper wire data that the safe ampacity of the 0.187"-thick rotor disk is 'only' 1,579 amps (as mentioned above) around its inner circumference. Our revised figure for the "nominal" output current will then be equal to the calculated safe rotor ampacity = nom. Io = 1,579 A.
   

Output Power:  The new value for max. Vo = 0.3963 volts, and the corresponding maximum rotor speed is then 1143 rpm or f = 19.05 rps (since f = Vo / 2πrRaB). And so, the maximum allowable output power is equal to (0.3963)(1,579) or max. Po = 625.8 watts, and the new maximum brush speed is 5,386 fpm. The pickup brush width will not have to be increased, and so the edge-width coverage ratio is still 74.7%.
   


Input Torque & Power:  Although it might seem that we now have good final figures for operating speed and output power, we have yet to determine if the rotation speed we just derived is acceptably close to that of an off-the-shelf electric motor (as a practical source of input torque) which will provide adequate start and run torque at a given available horsepower rating. This will not be nearly as difficult as it might sound, with the aid of a simple yet indispensable electric motor formula that relates speed (rpm), torque (T), and power (Hp):  T = (Hp x 5252) / (rpm) , where the constant 5252 is equal to 33,000 ft.lbs./min./Hp divided by 2π radians/rev., and T is the torque in ft.lbs. [The equivalent metric expression is:  T = P / ω , where ω = 2πf , P is power in watts, and T is torque in N-m.]  
 
For the 'classical' case:
  [i] primary full-load counterforce = Fa = B I Ra = (.1383)(1,579)(.180) = 39.31 N
  [ii] primary back-torque = Ta = Fa (r) = (39.31)(.133) = 5.23 N-m = 3.86 ft.lb.
  [iii] neg. brush counterforce (ea.) = Fnb = pkA = (4)(0.2)(.125 x .88) = 0.088 lb. (run)
  [iv] pos. brush counterforce (ea.) = Fpb = pkA = (4)(0.2)(.375 x .88) = 0.264 lb. (run)
  [v] total brush retarding torque = Tb = 48(Fnb)(Ro) + 16(Fpb)(Ύ") = 41.18 in.lb. = 3.43 ft.lb.
  [vi] total load torque T = Ta + Tb = 3.86 + 3.43 = 7.29 ft.lb. (treating the ~ 0.5% rolling losses as negligible).
 
    As stated earlier, there are a number of reasons for selecting a permanent magnet or shunt-wound [wound field] DC drive motor in this application, since they will operate on normal AC line power and an economical variable speed control is usually also available. This feature is especially desirable in cases where the tentative operating speed falls somewhere between standard electric motor speeds. However, in the case of a statorless Faraday generator with a large moment of inertia, it may be permissible (and less expensive) to use an AC capacitor-start motor if it so happens that a standard-speed motor can be used – since they are designed to start under heavily-loaded conditions. Although they may briefly draw over 300% of normal running amps to do so, a capacitor-start motor can be used in certain instances to effect a corresponding reduction in generator starting time.
    Referring again to the motor selection and ordering guide, we find that a 1.5 Hp 1150 rpm PM motor (with 180vdc armature; FL amps = 7.2) is available [GE 5CD125TP001B] that develops full-load torque of 6.85 ft.lb. (82.2 in.lb.). Even though this is somewhat less than the 7.29 ft.lb. that would be needed to operate our model dynamo at its maximum capability, it must be remembered that the motor's output torque will climb as its speed is reduced – and it may be that just enough torque will be available very near the maximum 1143 rpm operating speed we desire.
    The most straightforward way of determining that point on the selected motor's output power / torque curve where its operating speed is maximized for this particular application's input torque requirement is by repeated spreadsheet calculations to assemble tabular data, from which the best 'solution' of such a complex covariable problem is obtained when the net motor torque developed (Tm) just exceeds the total load torque (Tn) required at the reduced speed.
    Starting at 1125 rpm and calculating output voltages in descending 5-rpm increments, the following optimal resolution was found at 1100 rpm (f = 18.333 rps):
 
  – net Vo = 0.3814 volts, net Io = 1,519.5 amps, and net Po = 579.5 watts ;
  – net Fa = B I Ra = (.1383)(1,519.5)(.180) = 37.826 N ;
  – net Ta = Fa(r) = (37.826)(.133) = 5.031 N-m = 3.71 ft.lb. ; and
  – total load torque = Ta + Tb = 3.71 + 3.43 = Tn = 7.14 ft.lb. [treating the ~ 0.5% rolling losses as negligible].
  – net motor torque developed = Tm = (1.5 x 5252) / 1100 = 7.16 ft.lb. > Tn (@ 7.14 ft.lb.).
 
  – nominal motor efficiency = (Hp x 746) / (V x I) = 1,119 / 1,296 = ~ 86.3%
  – nominal dynamo efficiency = Po / (Hp x 746) = 579.5 / 1,119 = ~ 51.8%
  – combined system efficiency = (0.863)(0.518) = ~ 44.2%
 
  – letting avg. brush k = ½ (0.3 + 0.2) = 0.25,  avg.Tb = (0.25 / 0.2)(3.43) = 4.29 ft.lb.
  – avg. no-load motor torque  Tnl = Tm – avg.Tb = 7.16 – 4.29 = 2.87 ft.lb.
  – no-load motor starting time  min. ts = ωI / Tnl = 2πf [.181 + 2(0.213)] / 2.87 = 24.36 sec.
  – Pr = mπ2R2f 2 / ts = (24.715)(9.8696)(0.0491)(336.1) / 24.36 = 4,025.4 / 24.36 = 165.2 watts

  – avg. full-load motor torque  Tf = Tm – (½ Ta + avg.Tb) = 7.16 – (1.86 + 4.29) = 1.01 ft.lb.
  – full-load motor starting time  max. ts = ωI / Tf = 2πf (.607) / 1.01 = 69.23 sec.
  – rotor mechanical power expended = Pr = mπ2R2f 2 / ts = 4,025.4 / 69.23 = 58.1 watts

    These starting times are of questionable acceptability even for a PM-DC drive motor, and are far too extended for a capacitor-start motor to be used [since no-load start times longer than about 3 seconds may overload the additional start windings in a capacitor-start motor, damaging or destroying the coils and/or tripping a circuit breaker, even though the effective start times will be lower by ~300%]. Also, it can be seen that the much lower gap flux density in the Ferrite-rotor generator has caused an almost 50% reduction in output power and a 33% drop in efficiency, despite the greatly increased rotor current, as compared to the induction dynamo in the previous example. And, the additional brush drag and far-higher rotor mass have served to cause what might be considered an unacceptably long starting time.
    It should be noted that this statorless generator's actual induction efficiency can be found by simply figuring its torque requirement without considering brush drag. Thus, at 1075 rpm, a Ύ-Hp PM-DC motor would be adequate to power the generator at rotor voltage of 0.3727, current of 1,485 amps, and output power of 553.5 watts. Provided both field pieces' outside poles will be wholly keepered at saturation, and at less than the disk's full diameter, real induction efficiency would be 553.5 / 0.75 (746) = 98.9%! [These criteria regarding the essential use of saturated steel end-plates to 'keeper' the outer stator poles derive from the innovative prior-art "closed (flux) path" homopolar generator design of Trombly & Kahn (~1982). A saturated material will support no further passage of flux nor any further external magnetic induction – in the form of eddy current losses, in this case.]
    Purists may also notice that the motor's own armature inertia has not been considered above in the interests of clarity and brevity, having been treated as negligible since it corresponds to just 0.4% of its output torque (by OEM specs) in this particular case. The same consideration also applies to the rotor shaft's miniscule inertia in relation to that of the disk. The non-trivial inertia of the steel end-plates has also not been considered, however, which would act to further extend the start times figured.
 
And in the best-case theoretical model: The following computations are based on the application of the 'Tesla' reduced back-torque ratio to the model 'unipolar' generator in question. As we developed earlier above, in this case that ratio is equal to 1 – 0.747 = 0.253.
 
  [i] primary full-load counterforce = Fa = 25.3% [B I Ra] = 0.253[(.1383)(1,519.5)(.180)] = 9.57 N
  [ii] primary back-torque = net Ta = Fa (r) = (9.57)(.133) = 1.273 N-m = 0.94 ft.lb.
  [iii] total brush retarding torque = Tb = 48(Fnb)(Ro) + 16(Fpb)(Ύ") = 3.43 ft.lb. (same as before)
  [iv] total load torque Tn = Ta + Tb = 0.94 + 3.43 = 4.37 ft.lb. (treating the ~ 0.5% rolling losses as negligible).
 
    Referring again to the motor selection and ordering guide, we find that a 1 Hp 1150 rpm PM motor (with 180vdc armature; FL amps = 5.0) is available [GE 5CD123GP001B] that develops full-load torque of 4.57 ft.lb. (54.8 in.lb.). In this case, the 1-Hp motor seems to have enough torque for the generator to be operated at the same rotor speed as it was with the 1.5-Hp motor in the classical case, at 1100 rpm (with f = 18.333 rps), thereby nicely illustrating the effective proportional reduction in input power:
 
  – net Vo = 0.3814 volts, net Io = 1,519.5 amps, and net Po = 579.5 watts (from classical case above);
  – net Ta = Fa(r) = 0.94 ft.lb. (from above); and
  – total load torque Tn = 4.37 ft.lb. [treating the ~ 0.5% rolling losses as negligible].
  – net motor torque developed = Tm = (1.0 x 5252) / 1100 = 4.77 ft.lb. > Tn (@ 4.37 ft.lb.).
 
  – nominal motor efficiency = (Hp x 746) / (V x I) = 746 / 900 = ~ 82.9%
  – nominal generator efficiency = Po / (Hp x 746) = 579.5 / 746 = ~ 77.7% (or COP = 0.777)
  – combined system efficiency = (0.829)(0.777) = ~ 64.4%
 
  – avg. starting brush retarding torque avg.Tb = 4.29 ft.lb. (same as before)
  – avg. no-load motor torque  Tnl = Tm – avg.Tb = 4.77 – 4.29 = 0.48 ft.lb.
  – no-load motor starting time  min. ts = ωI / Tnl = 2πf [.181 + 2(0.213)] / 0.48 = 145.7 sec.
  – Pr = mπ2R2f 2 / ts = (6.965)(9.8696)(0.0491)(336.1) / 145.7 = 4,025.4 / 145.7 = 27.6 watts

  – avg. full-load motor torque  Tf = Tm – (½ Ta + avg.Tb) = 4.77 – (0.47 + 4.29) = 0.01 ft.lb.
  – full-load motor starting time  max. ts = ωI / Tf = 2πf (.607) / 0.01 = 6,992 sec.
  – rotor mechanical power expended = Pr = mπ2R2f 2 / ts = 4,025.4 / 6,992 = 0.58 watts.

Conclusions:
    As should perhaps have been apparent from the outset of this example, there would be no reason not to use the far-stronger NdFeB magnets in this statorless generator model, since the generator's output power must be proportionally lower if the Ferrite magnets are used – and by all appearances achieving a COP > 1.0 would simply not be possible, unless perhaps a non-statorless dynamoelectric motor stage was very carefully employed [with far greater system size, complexity, and cost].
    Also, in much the same way, there would be no point in rotating the additional mass of the pole-sets (even if using metallic NdFeB magnets!) when the result must be an undesirable and perhaps untenable increase in the starting time and in the net mechanical torque requirement, which works against a best efficiency in disk generator design. With so much misinformation disseminated about the statorless Faraday dynamo variant, it might be tempting to speculate how much of it is truly inadvertent.  
   

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