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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. top of page
[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 = Ra , 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 Ra (2π r f ) ; and, merely rearranging terms, Vo = 2πr Ra f 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! top of page
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. preceding subsection top of page
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 Ra f 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 1.42 tesla (or 14,200 gauss) This is the world's most powerful permanent magnet material. On the other hand, the strongest nonconductive magnets generally available in large sizes are made of sintered Ferrite 5 (BaO-6Fe2O3), a ceramic material with flux density of 0.38 tesla (3,800 gauss).
preceding subsection top of page
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 .
preceding subsection top of page
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