US2872602A - Arbitrary function generators - Google Patents

Description

Feb. 3, 1959 n. L. HERR ARBITRARY FUNCTION GENERATORS 2 Sheets-Sheet 1 Filed June 15, 1955 /NVEN7'O,Q DONALD L. HERR Feb. 3, 1959 D. L. HERR 2,872,602

ARBITRARY FUNCTION GENERATORS Filed June 13, 1955 2 Sheets-Sheet 2 400 CONDUCTORS INVEN TOR. DONALD L HERR BY W a pn/fiuaz ATTORNEYS United States Patent ARBITRARY FUNCIIGN GENERATORS Donald L. Herr, Santa Monica, Calif. Application June 13, 1955, Serial No. 514,882

12 Claims. (Cl. 310-111) This invention relates to electromagnetic induction transfer devices and more particularly, to novel transfer devices capable of synthesizing arbitrarily chosen continuous mathematical functions.

Electromagnetic voltage and torque transfer devices for generating certain specific elementary functions are well known in the art and find wide application in analog computers and fire control systems. Basically, these devices comprise. a rotor body secured to a control shaft and concentrically positioned for rotation Within a surrounding stator body- Both the. rotor and. stator are provided with discretely distributed conductor windings. Bythis: arrangement, the degree of inductive coupling between the rotor and stator windings, when one or the other is electrically excited, is dependent upon the angular relationship. of the rotor with respect to the stator.

A typical electromagnetic transfer device of the abovev type now in general use, is the electromagnetic resolver described and claimed in my U. S. Patent No. 2,608,682, entitled Electromagnetic Resolver, issued August 26, 1952. In the resolver, the rotor and stator are each provided with two windings mechanically 90' degrees apart. An alternating input signal applied to. one of the stator windings will result in an output voltage from one of the rotor windings proportional to the product of the initial input signal and the sine or cosine of the angle of the rotor shaft or windings with respect to the stator. Theer is thus provided a device which may be used to continuously compute the sine or cosine of a varying angle.

The present invention has as its primary object the provision of an electromagnetic transfer device capable of generating any desired arbitrary function of the rotor shaft angle, providing such function is continuous, thereby resulting in a vastly more universal instrument than has heretofore been available.

More particularly, an object of the invention is to provide an electromagnetic transfer device which may be pro-designedv in accordance with the. characteristics of the desired arbitrary function to be generated, so that upon operation, the output function of the rotor shaft angle is either, (1) an electrical signal representing a synthesis of the desired arbitrary function, or (2) a restraining or opposing torque on the output shaft varying in accordance with said arbitrary function.

The basic theory underlying the present invention for attaining these and other objects and advantages, stems from the fact that any arbitrary mathematical function that is continuous over a finite period may be expanded into a Fourier series of sines and cosines. Since electromagnetic resolvers, such as described above and in the foregoing mentioned patent, are already known in the art and are individually capable of generating sinev and cosine functions, it will. be realized atonce that a series of such resolvers individually designed to generate a sine or cosine of given amplitude and shaft angle frequency, may have their outputs combined together to provide a function that is the. summation of such. sines and cosines- BY changing the amplitude and space frequency of each indi- 2,872,602 Patented Feb. 3, 1959' 2. vidual generator in accordance with the separate terms in the Fourier expansion of an arbitrary function, it is possible. to provide acombined. or'summed output which represents the arbitrary function itself.

In accordance with the present invention, a single electromagnetic transfer device. comprising a stator and rotor is provided which will immediately yield the. desired arbitrary function. This result is achieved by distributing the number of conductor. windings in. the stator and the rotor slots in a manner determined by the coefficients of the sine and cosine terms in the Fourier series expansion of the. arbitrary function. It ispossible, accordingly, to provide in a. single device, an output signal or shaft torque that represents any desired arbitrary function of shaft angle, including the elementary sine. and cosine functions.

A better understanding. of this; invention will be had by referring to the accompanying. drawings, in which:

Figure 1 is a schematic cut-away perspective view of an electromagnetic; transfer device showing stator and rotor bodies coaxially positioned;

Figure 2 is an enlarged cross-section of a portion of the stator and; rotor bodies of an. electromagnetic transfer device, such as schematically shown in Figure. 1;

Figure 3 is a schematic plan view of the. stator illustrating the conductor distribution in accordance with an illustrative example of the invention; and,

Figure 4 is a similar, schematic plan view of the rotor for the example illustrated in Figure 3.

Referring to Figure 1, there is shown shaped stator body 10 having a central opening within which is coaxially positioned a rotor body 11. The rotor 11 is secured to a rotor shaft 12 and is adapted to be rotated to assume different. angular positions 7. with respect to the stator. As shown, the stator is provided with aplurality of radially cut stator slots 13, and the rotor is similarly provided with a plurality of radially cut rotor slots 14. The sets of slots 13 and 14 are adapted to re ceive conductor windings. whereby the stator body may establish an electromagnetic field across the air gap 6 to which the conductor windings of the rotor body are inductively coupled, or vice versa.

In Figure 2, there are shown symbols. c c

3. 4 designating the per unit. conductor density in the. respective stator slots 13, and symbols d d designating the per unit conductor density in the respective. rotor slots 14. Let c equal the per unit conductor density in the s stator slot, and ai equal the per unit conductor density in the r rotor slot. The actual number of con;- ductors in the s and r slots is. determined by multiplying the absolute values of' 0 and d,. by the total number of conductors employed in the stator and the rotor respectively.

Stated another way, the per unit conductor density, e in the s stator slot, is, in magnitude, the ratio of the actual number of physical conductors in the s stator slot to the, total number of net effective conductors in all the stator slots, comprising the particular stator distributed winding made up of a group of c with a plus sign. prefixed for one chosen (reference) direction of actual current flow in the conductors in the slots, and with a minus sign. prefixed for the opposite direction of actual current flow in the. conductors in slots having. Such cur rent flow opposite to the reference direction. The same situation obtains for the a, per unit conductor density in the r rotor slot. The factor times the per unit, conductor density is the percent, conductor density in a particular slot.

In accordance with the present invention, the conductor winding distribution among the slots of the stator and rotor; that is, the: actual values of the 0 and d l are uniquely predetermined bythe coefficients of the Fourier Series expansion of the arbitrary function to be generated.

a cylindrically' With such distribution, the variation of the ratio of the output signal to the input signal or the variation of the rotor shaft torque with changes in the rotor shaft angle may be made to represent the desired arbitrary function. .Themanner in which and d are uniquely related to the coefiicients of the Fourier series in order to determine the distribution of the conductors in the slots will now be described.

Referring to Figure 2, let the dashed line 15 represent a. reference angle line for the stator 10. Any point in the airgap 6 having a certain fiux density generated by the stator windings alone may then be located by specifying an angle 0 so many degrees from the reference line 15.

Similarly, let the dashed line 16 represent a reference angle line for the rotor. Any point in the airgap 6 having a certain fiux density generated by the rotor windings alone, maythen be located by specifying an angle 1; so many degrees from the reference line 16.

The angle of shaft rotation 7 representing the degree of rotation of the rotor with respect to the stator, will then be the angle between the reference lines 15 and 16 as shown in Figure 2.

.The arbitrary function to be generated in accordance with the invention will be represented by changes in the output of the device with variations of therefore, let f('y) represent the desired arbitrary function. v

In the case where it is desired to synthesize the arbitrary function as an electrical output signal,

no g n) (1) where E =output voltage from the device, and E =input voltage to the device.

In the case where it is desired to provide a shaft torque which varies in accordance with an arbitrary function, this torque will be proportional to the product of applied stator and rotor voltage, as well as a function of the shaft angle, or:

where T(-y) represents the arbitrary torque function, =voltage applied to the rotor, and 1 Eg =voltage applied to the stator.

f(v)=2 cos w where n=harmonic order, and A are the Fourier coefficients. For these cases:

Let f(0) equal the normalized stator distribution of flux density generated in the airgap 6 for different angles of position '6 around the stator periphery, referred to the stator. reference line 15, when the stator conductor distribution alone is excited.

Let f(1;) equal the normalized rotor distribution of flux density generated in the airgap 5 for different angles of position 1 around the rotor periphery, referred to the rotor reference line 16, when the rotor conductor distribution ,alone is excited.

f(0) and f(1 may then be represented by Fouriers Series as follows:

f(n)= /2E n i y where na nr equal the n harmonic amplitude for the stator and rotor 1 respectively.

From Figure 2 it will he noted that:

Now, the convolution integral of f(0) and f(1;) for :09-1 is given, in general, by

I fine-no Upon the substitution of 0- for 1;, from Equation 6, and f(0) and fi as given in Equations 4 and 5, into this convolution integral and integrating, there results:

' nr nr A E T 005 TL/ nn A. .(s)

.whereby the arbitrary function, f('y)- of Equation 3 is given by The next step is to relate the quantities A and A, to theper unit conductor density in all stator and rotor slots as defined generally by C and d for the s and r slots respectively;

Referring again to Figure 2, assume the stator has a total of p slots distributed over 21r radians in 0 and that the rotor has a total of q slots distributed over 21r radians in 1;.v The per unit slot-conductor densities c c c c c o where c represents the per unit slot-conductor density in the last slot of the p stator slots, are distributed in these p slots respectively, adjacent slots being separated 21r/p radians. Similarly, the per unit slot-conductor densities d d d d where d represents the per unit slot-conductor density in the last of the "q rotor slots, are distributed in these q slots, respectively, adjacent slots being separated by 21r/q radians.

In the case of continuous windings on both the stator and rotor, two conditions must be met:

and,

where each c and d is given a positive or negative sign.

Additional constraints further reducing the number of a e-anus a is, a known constant for a given, 1) and n and for known axes and locations of axes of symmetry and asymmetry of the stator slot-conductor density distributed array, and varies at most, only withn;

3,, is a known constant for a given q and n and for known axes and locations of axes of symmetry and asymmetry of the rotor slot-conductor density distributed array,

Y and varies at most, only with n;

C is, as defined previously, the per unit stator slot-conductor density, with algebraic sign, in the s stator slot;

d is, as defined previously, the. per unit rotor slot-conductor density, with algebraic sign, in the. r rotor slo g (s,n,p-) denotes the stator generating function whose form is known and determined by the knownaxes and locations of axes of. symmetry and asymmetry of the stator slot-conductor distributed ordered array over 21r radians. in 0; and whichis a function of the slot number s on the stator, of the order n of the space harmonic referred to Zn radians in 0 as the fundamental or first harmonic, and of p, the number of equispaced slots in the stator; and,

g (r,n,q denotes the rotor generating function whose form. is known and determined by the known axes and locations of axes. of symmetry and asymmetry of the rotor slot-conductor distributed ordered array over 27! radians-in 1 and which is. a function of the slot number r, on the. rotor, of the order n of the space harmonic referred to 211 radians in '1 as the fundamental or first harmonic, and of q, the number of equi-spaced slots 1 in the rotor.

For known 2 and q" and for known axes and locations of axes of symmetry and asymmetry of the stator and rotor slot-conductor density distributed arrays, Equations 12 and 13 yield a set of simultaneous equations as follows:-

order, n: of harmonic amplitudes; A to approximate; the

given function to within prescribedtolerances over a prescribed range of the independent variable 7. These A s then, are given knowns in the set of simultaneous Equations 14. Further, the a s, fl s, g s and g s are known. Only the c s and d s are unknowns in the set of Equations 14 and therefore these equations may be simultaneously solved to provide the values of the c s and d,.s, there being as many equations as there are un- 4 knowns.

In other words, if the number of the highest harmonic amplitude, A,, (not the order), is denoted by N, and the least number of independent c s and dgs to synthesize the arbitrary function required, are denoted by S and R respectively, then:

and therefore, it is possible to solve for each c and each d d from Equations 14.

Conductor windings are then applied to the stator and rotor slots 13 and 14 in accordance with the solutions for c, and d,. When an input signal is then applied to the device-to excite the particular distributed winding array, the electrical output signal will vary with the shaft angle in accordance with the. initially selected arbitrary functiOn f('y).

It will be recalled that the above analysis is for generating an arbitrary function which may be expressed by a Fourier series of cosines, Equation 3. Another class. of arbitrary functions may be represented by a Fourier series of sines, thus:

The analysis for relating c and d to the Fourier coeificients, B,,, in Equation 16, is carried out in an identical manner. to that given above for Equation 3, except. for a rotation of 1r/2: radians of the 0 and referencelines.

in the most general case in which the arbitrary function is represented by both summations of the Fourier series, thus:

f('y) =2 (A,, cos 11 3,, sin 11 (17) the c s and d s determined by the B s, may be superimposed on the c s and d s determined by the A s, re-

sulting in a single, composite stator sl'otconductor density distribution on the stator, and ina single, composite rotor slot-conductor density distribution on the rotor.

It will thus be seen that the conductor windings in the different slots are provided with weighted operating characteristics from the standpoint of the number of conductor windings in each slot and that such weighted operating characteristics are in accordance with the solution of a plurality of simultaneous equations. Each simultaneous equation involves the weighted operating characteristics of a different harmonic generated by the windings in a pair of Fourier Series. The product of the pair of Fourier Series expresses the arbitrary function to be generated with progressive displacements between the. rotor. and, the. stator.

Furthermore, each Fourier Series in the pair individ- Considering now the case of a torque transfer arbitrary :function generator as described by Equation 2, the restraining or opposing torque function is given by:

.where L ('y) ideally denotes the coefficient of mutual induction between the stator and rotor as a function lof the shaft angle 7, and I and I respectively are the current flowing in the stator and rotor windings as a result of the "applied voltages E 'and E, of Equation 2.

Assume first, that the arbitrary torque function is expressible by a Fourier Series of sines, thus:

whereLA' denote the Fourier coeifi cients of the torque function. J

With respect to the A and A of Equations 4 and 5 of the voltage transfer device, set

where K is the proportionality constant and is independent of 'y, I,,, and 1,. The arbitrary torque function of y given by Equation 18, is then given by:

I Ten-K U m) Fon in which again, 'y=0-1 resulting in:

As in the case of the voltage transfer arbitrary function generator analysis, Equations 19 and 26 can be equated term by. term, by making:

I I L-171 By dropping the primes in Equation 27, it is evident that a voltage transfer device whose normalized harmonic amplitudes are given by:

is also a torque transfer device whose normalized harmonic amplitudes are given by ns' nr and whose reference lines for 'y in the output function are space phase displaced 1r/2 radians with respect to the reference lines of the voltage transfer device (lines 15 and 16 in Figure'2).--

To determine c and d, for the torque transfer device, a set of simultaneous equations whose left hand sides are identical to those in the set of Equations 14, but whose right hand sides in the general term, are written nA instead of n A are used. Thus, the arbitrary torque functions may be reproduced by distributing the stator and rotor conductor windings in accordance with the new c s and d s.

The analysis for relating c and d to the Fourier coefiicients for a torque function expressible by:

- T(y)=ZB,', cos m 28) is executed in an identical manner, except for a rotation of 1r/ 2 radians in the 0 and 7 reference lines.

In the most general case for' the torque function,

wherein, I

T( )=E(A,', sin n-H-BI, cos n'y) (29) the c s and dfs determined by the B s, may be superimposed on the c s and d s determined by the A' s, as in the case of the voltage transfer device, resulting in a single composite c stator slot-conductor density distribution on the stator and in a single composite d, rotor slot-conductor distribution on the rotor.

Summarizing, the preceding formulas elicit the unique stator and rotor per unit slot conductor density distributions over all slots for the generation of the given arbitrary empirical or analytical function as avoltage or torque transfer function within the limitations of the total numbers of stator and rotor slots. Thus, having once determined the stator and rotor per unit slot conductor densities, they may be respectively multiplied by the total number of net efiective stator conductors and the total number of net effective rotor conductors to yield the actual number of conductors in each slot of the stator and rotor and the relative directions of current fiow therein.

The total number of actual conductors and the wire size employed on any stator or rotor winding in slots of a given area is determined, among other factors, by the total slot volume, the slot fill factor, insulation thickness, ampere turns for proper non-saturating use of the magnetic material, air gap width, maximum number of H conductors in the maximum filled slot, transformation for use in connection with functions having sine syme metry about the origin, :0. In the example under consideration, assume the arbitrary function to be generated is expressed by:

This function is very useful in modern computers and guided missile control systems and is an example of the latter above mentioned type of function having sine sym- Assume that it is desired to develop this function in a -0.636620 f(7) +0.636620 whereby one servomechanism and one resolver or potentiometer in a conventional computer network may be eliminated. V The corresponding range in "y is In order to develop the B harmonic amplitudes of the given functionwhich step is a prerequisiteuto the unique i v fi fin The usual Fourier analysis yields the following harmonic amplitudes through the 35th order, 11:35. Since the function has sine symme ry, only the odd. harmonics in the Fourier expansion need be computed:

10 possible as limited only by the number of stator and rotor slots available and the physical constraints of electromagnetic induction fields.

Since it has been initially recognized and noted that the function to be synthesized has sine symmetry about the origin, :0, the stator and rotor step-wise flux density distributions (mmf functions) are chosen from that class which also has this same sine symmetry but not sinusoidal shape. Thus, letting p equal the number of stator slots and q equal the number of rotor slots, the general Equations 12 and 13 for this class of functions, respectively take the forms:

The K factor in Equation 31 is the so-called per unit skewness and constitutes that fraction of one slot-to-slot width that the rotor stack skews in going from the front end of the rotor body to the rear end of the body; a is defined by c,=a- -a h(s) is a distribution function over one quadrant of the stator slots which is repeated over the other three quadrants; m(r) is another distribution function over one quadrant of the rotor slots which is repeated over the other three quadrants; s, as before, is the sequential numbering of the stator slots in one quadrant; and r is the sequential numbering of the rotor slots in one, quadrant. Specifically, for the case of 1:20 stator slots and q: 12 rotor slots, the above equations become explicitly:

The general Equations 14 for sine symmetry are then expressed by B -B =n B, and may be Written out:

sin 5:5 T r=d (28'l)'rr H 12 i =1 a sin 20 KW '21 d, 11.1 6 -B sin K 1 (28,1)31r k 1 7:3 T'n' 8 1 a. sin 20 T 1 d, $111 -9B3 Sin 3 (281)51r 12 T511 8 1 a, sin 20 7 1. d, sin 25B;

- Be cause the c s (as expressed in terms of a and d s .are normalized to unity:

l tl= ,E .=1 (35) v s=1 and, V v

r=3 -Ed,.=l (36) a; 0.102847 a -0.424190 0, a,a., 11-; 0.251114 I a 2.198569 K =0.890003 c; 0.192847 -0.617037 03 0.765304 0 1.947455 c 3.l98569 Plu 1.000000 In general, for this case of 1:20 stator slots and q=l2 rotor slots, the generated B obtained by substituting the above K, a! and c values back in to Equations 34 will be exactly equal to the original 13 calculated previously for the given function for 11:1 to 35 only over the order of n from 1 to inclusive utilized to obtain the K, d, and 0 values. However, they are the unique rotor skewness and rotor and stator conductor distributions for the generation of the given function,

1-cos 7 'Y over the specified range, for the case of 12:20 stator slots and q=12 rotor slots with maximum harmonic matching for this slot combination.

A comparison between the calculated B of the given function and the generated B of the function-generator is set forth below:

11 B11 Calculated B Generated This table estahlishes'the validity of the statement that the harmonics, of the original Fourier Series of the function and the harmonics of the generated function match exactly, and exactly only, up to and through n=15, the maximum possible and unique matching within the limits of the slot number's, 17:20, q=l2. V

It is desirable to finally compare the normalized function values developed by the function generator, with the normalized values of the given function. This is done, utilizing the first 35 harmonics, those above that having dropped to very small values and not contributing substantially to the error percentage.

i degrees fin fin generated Percent.

, rror.

Nowhere except in the last interval does the percent error exceed 0.5 percent for this rather unusual heretofore .unavailable, practical function, and the R. M. S. error is less than 0.08%. Since even this remaining error is quite periodic, it may be still further minimized by a simple auxiliary winding designed on the above principles for the main function' The configuration for the rotor per-unit slot conductor densities demonstrates a ratio of 4.451504 between the maximum loaded slot (r=1) and the least loaded slot (r=3 On the stator, the situation is somewhat difiercut, because some of the slots in a quadrant have conductors which are oppositely poled (reversed current direction) to others in the same quadrant of slots. Since summation of the absolute magnitudes of C is the base reference for the total number of physical conductors in one quadrant of the stator slots, and equals 6.631212, whereas the absolute magnitude of the summation of 0 is always equal to unity, this winding requires a total number of physical conductors equal to 6.631212 times the total number of net effective conductors. The ratio between the maximum loaded slot (s=5) and the least loaded Slot (s=1) is 16.586044.

are in the maximum loaded slot, s=5. Postulating 60% fill factor, including slot insulation, wire insulation, lay

factor, and so forth, the copper area of 400 conductors in the slot s=5 equals 0.014130 square inch. Hence, a wire size for which the copper diameter is 0.0126 inch, is specified, for example, AWG No. 28.

Referring now to Figure 3, the physical stator conductor distribution in the stator slots is shown by the numerals which indicate the actual physical number of conductors in each slot. The dots indicate that the direction of current flow in certain slots is coming out of the drawing, while the crosses indicate that the direction of current flow in certain slots is passing into the drawing. The, actual windings may be toroidaL latitude, or effected in any other convenient manner. The step-wise air-gap flux density distribution (mmf. function) generated by this Winding over 360 of the inside periphery of the stator, directly corresponds to the step-wise poled conductor distribution given in Figure 3.

Calculation of the required self-inductance of the rotor winding in a practical size 23 case, for an air-gap 6 of .00554 inch and for an existing rotor slot area of .02461 square inch necessitates a total not effective number of conductors per rotor Winding of 2000. This number corresponds to the actual total number of physical conductors in one quadrant, since the direction of current how in any one quadrant is uniform. Thus, there are 500 conductors per rotor quadrant.

The corresponding poled conductor distribution is illustrated in Figure 4 wherein a 60% fill factor is used and No. 28 AWG copper wire is employed. As in the case of Figure 3, the numerals indicate the actual physical number of conductors in each slot and the dots and crosses indicate the corresponding direction of current flow.

The above specific example thus illustrates the manner in which the slot conductor density distributions for the stator and rotor are determined in accordance with the characteristics of the arbitrary function whereby the output of the generator represents the arbitrary function.

It will be seen from the foregoing description that the present invention provides an electromagnetic transfer device which is far more. versatile than any heretofore known in the art, in that it may be employed to generate any arbitrary continuous function, reproducing such function as either an electrical voltage signal or as a shaft torque.

What is claimed is:

1. An arbitrary function generator comprising: a first body; a second body co-axially positioned in concentric relationship to said first body for rotation relative to said first body through a variable angle 7 measured from a radial: reference line on said first body, said second body and first body defining an annular air gap therebetween; said first body having a plurality of discrete radially cut slots ciitcumferentially spaced about its axis; electrical conductors positioned in said slots; each slot, in sequential order from said reference. line containing a discrete number of said electrical conductors such that the ratio of the number of conductors in any one slot to the total number of conductors in all of said slots defines a unique absolute per unit conductor density in said any one slot; means for passing current through said conductors in specified directions to establish a first body slot step-wise flux density distribution in the adjacent air-gap varying over different circumferential points measured from said reference line; and conductor means on said second body adapted to be inductively coupled to said first body step-wise slot fiux density distribution upon rotation of said second body through various values of the angle y, the effective per unit i slot conductor densities in said first body slots as derived from the absolute per unit conductor densities and said specified directions of current flow, having values, such asto: provide said step-wise flux density distribution with harmonic amplitudes and signs equal to the term by term co-efiicients of the Fourier Series expansion of said arbitrary function, whereby the signal induced in said conductor means of said second body represents a function of body for rotation relative to said stator body through a variable angle 7 measured from a radial. reference line.

on said. stator, said rotor and stator defining'an annular air-gap therebetween; said rotor and stator bodies each having a plurality of discrete radially cut slots circumferentially spaced about their respective axes; electrical conductors positioned in said rotor and stator slots; each slot in said rotor and stator containing a discrete number of said electrical conductors such that the ratio of the number of conductors in any one rotor slot to the total number of conductors in all of said rotor slots defines the absolute per unit conductor density in said any one rotor slot and the ratio of the number of conductors in any one stator slot to the total number of conductors in all of said Q 0 d S th absolute per unit conductor density in said any one. stator slot; and means for passing current through said conductors in each of said rotor and stator slots in specified directions to establish a rotor and stator slot step-wise flux density distribution in the adjacent airgap varying over different circumferential points measured from. said reference line; the respective electrical conductors associated, with said rotor and stator being inductively coupled upon rotation of said rotor through various values of the angle 7, the products of a first function of the effective per unit conductor density distribution in each of said rotor slots with a second function of the eifective per unit conductor density distribution in each of said stator slots, as derived from said rotor and stator absolute per unit conductor densities and said rotor and stator slot specified directions of current flow, being term by term respectively proportional to and of the same sign as the term by term coeificients of the Fourier Series expansion of said arbitrary function, whereby the output signal of said generator represents a function of said angle 7 corresponding to said arbitrary function.

3. In apparatus for generating any desired arbitrary function having only a single value at each position along one of av pair of transverse axes, a first member, a second member movable v relative to the first member, a first plurality of generating means disposed at spaced positions along the first member and a second plurality of generating means disposed at spaced positions along the second member, the generating means in the first plurality being interrelated and being provided with, characteristics. to generate, upon progressive displacements between the first and second members, the desired arbitrary function as indicated by a convolution product integral which represents the integral of the product of a pair of functions, one representing the characteristics of the signals generated by the first generating means upon progressive relative orientations of the first member and the other representing the characteristics of the signals generated by the second generating means upon progressive relative orientations of the second memher.

4. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member provided with a plurality of apertures at spaced positions. along one periphery, a second member provided with a plurality of apertures at spaced positions along the; periphery facing the apertures in the first member, a first plurality of generating means disposed in co-operative relationship with the apertures in the first member and av second plurality of generating means disposed in co-operative relationship with the apertures in the second member, the generating means in the first plurality and the generating means in the second plurality being interrelated and being provided with weighted operating characteristics to produce in the different apertures field densities dependent upon the polarities and magnitudes of the coefiicients in a pair of Fourier Series which together represent the arbitrary function to be generated upon progressive displacements between the first and second. members and which individually represent arbi 15 trary functions to be generated upon progressive relative orientations of the first and second members.

5. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member,

a second member movable relative to the first member, a first plurality of generating means disposed at spaced positions along the first member and a second plurality of generating means disposed at spaced positions along the second member, the generating means in the first plurality and the generating means in the second plurality being interrelated and being provided with weighted operating characteristics in accordance with the solution of a plurality of simultaneous equations each involving the weighted operating characteristics of a different harmonic generated by the generating means in the first and second pluralities in relationship to the product of the coefiicients of the particular harmonics. in a pair of Fourier Series the product of which expresses the arbitrary function to be generated with progressive displacements between the first and second members and which individually express arbitrary functions to be generated with progressive relative orientations along the first member or along the second member.

6. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member made from magnetic material and provided with a plurality of slots at spaced intervals along one periphery, a sec ond member made from magnetic material and 'disposed for movement relative to the first member and provided with a plurality of slots at spaced intervals along a periphery facing the first member, and a first plurality of windings disposed in the different slots in the first member and a second plurality of windings disposed in the different slots in the second member and provided with conductor densities and wound with polarities in accordance with the solution of a plurality of simultaneous equations each involving the relationship between the conductor densities contributed by the different terms toward a particular harmonic and the coefficient of that harmonic in a Fourier Series which represents the arbitrary function to be generated upon progressive displacements between the first and second members.

7. In apparatus for generating any'desired arbitrary function having only a single value at each position along along one of a pair of transverse axes, a first member made from magnetic material and provided with a plurality of slots at spaced positions along one of its peripheries, a second member made from magnetic material and disposed for'movement along the first member and provided with a plurality of slots at Spaced positions along a periphery facing the slots in the first member, the number of slots in the second member being ditferent from the number of slots in the first member for corresponding units of distance, and a first plurality of windings disposed within the difierentslots in the first member and a second plurality of windings disposed within the difierent slots in the second member, the windings in the first plurality being disposed and interconnected and the windings in the second plurality being disposed and interconnected to provide in the different slots conductor densities representing the solutions of a pair of Fourier Series the product of which represents the arbitrary function to be generated upon progressive displacements between the first and second members and which individually represent the functions to be generated by the windings in the first plurality and the windings in the second plurality upon such progressive displacements.

8. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member made from magnetic material and provided with first and sec- 1% ond peripheries and provided with a plurality of slots at spaced intervals along one periphery and provided with a continuous portion at its other periphery for the flow of magnetic flux, a second member made from mag netic material and provided with first and second peripheries with the first periphery facing the slots in the first member and provided with a plurality of slots at spaced intervals along the first periphery and with a continuous portion at its other periphery for the flow of magnetic flux, and a first plurality of windings disposed in the different slots in the first member and a second plurality of windings disposed in the different slots in the second 'member and provided with conductor densities and wound with polarities in accordance with the convolution product integral of a pair of functions which individually represent the relationship to be generated for progressive orientations along the first and second members and the integrated product of which represents the arbitrary function to be generated for progressive displacements between the first and second members.

9. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member made from magnetic material and provided with a hollow annularconfiguration and with a plurality of slots at spaced positions along one of its annular peripheries, a second member made from magnetic material and provided with a hollow annular configuration and disposed in concentric relationship to the first member for movement along the first member and provided with a plurality of slots at spaced positions along the annular periphery facing the slots in the first member, the number of slots in the second member being different from the number of slots in the first member, and a first plurality of windings disposed within the different slots in the first member and a second plurality of windings disposed within the difierent slots in the second member, the windings in the first plurality being disposed and interconnected and the windings in the second plurality being disposed and interconnected to provide in the different slots conductor densities representing the convolution product integral of a pair of functions each of which represents the particular relationship generated for progressive orientations of a different one of the first and second members and the integrated product of which represents the desired arbitrary function to be generated for progressive displacements between the first and second members.

10. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first hollow annular member made from magnetically receptive material and provided with slots disposed at spaced positions along the inner periphery of the member and with bars between the slots, a second hollow annular member made from magnetically receptive material and provided with slots disposed at spaced positions along one annular periphery and with bars between the slots, a first plurality of windings disposed in the difierent slots in the first member and connected in a continuous circuit with the other windings in the plurality, and a second plurality of windings disposed in the different slots in the second member and connected in a continuous circuit with the other windings in the plurality, the first and second pluralities of windings being constructed to provide conductor densities in the different slots in the first and second members for the generation of the desired arbitrary function upon progressive displacements between the first and second members and in accordance with the solution of a plurality of simultaneous equations involving the conductor densities in the clifierent slots and the coeflicients of the successive terms in a first Fourier series expressing a particular function to be generated with progressive displacements along the first member and the coefiicients of the successive terms in a second Fourier series expressing a particular function to be generated with pro gressive displacements along the second member wherein the first and second Fourier series are provided with interrelated characteristics to have their product represent the desired arbitrary function.

11. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member made from magnetically receptive material and provided with an annular configuration having an opening at the center and provided with a plurality of slots spaced annularly around the periphery of the first member, a second member made from magnetically receptive material and provided with an annular configuration having an opening at the center and disposed in concentric relationship to the first member for rotation relative to the first member and provided with a plurality of slots spaced annularly around the periphery of the second member at the periphery facing the first member, and a first plurality of windings disposed in the slots in the first member and a second plurality of windings disposed in the slots in the second member, the windings being connected and being disposed in the different slots in particular patterns to produce in the different slots conductor densities dependent upon the polarities and magnitudes of the coefiicients in a Fourier Series representing the arbitrary function to be generated upon progressive displacements between the first and second members, the windings in the first plurality being connected and disposed to produce a resultant value of zero for all of the conductor densities in the diflerent slots in the first member and the windings in the second plurality being connected and disposed to produce a resultant value of zero for all of the conductor densities in the difierent slots in the second member.

12. In apparatus for generating any desired arbitrary function having only a single value at each position along one of a pair of transverse axes, a first member made from magnetic material and provided with a hollow annular configuration and with a plurality of slots at spaced positions along one of the annular peripheries, a second member made from magnetic material and provided with a hollow annular configuration and disposed in concentric relationship to the first member for rotation relative to the first member and provided with a plurality of slots at spaced positions along the annular periphery facing the slots in the first member, the number of slots in the second member being different from the number of slots in the first member, a first plurality of windings disposed in the different slots in the first member and a second plurality of windings disposed in the difierent slots in the second member, the windings in the first and second pluralities being constructed to provide the slots with conductor densities dependent upon the magnitudes and polarities of the coefiicients in successive terms of a pair of Fourier Series which individually represent progressive relative orientations along the first and second members and which together represent the arbitrary function to be generated for progressive displacements between the first and second members.

References Cited in the file of this patent UNITED STATES PATENTS 618,578 Newcomb Jan. 31, 1899 1,308,041 Chubb July 1, 1919 1,807,218 Langewiesche May 26, 1931 2,488,771 Glass Nov. 22, 1949 2,590,845 Curry Apr. 1, 1952 2,719,930 Lehde Oct. 4, 1955 2,731,574 Soredal Jan. 17, 1956 OTHER REFERENCES Book: Alternating Current Circuits, by Kerchner and Corcoran, 3rd edition; John Wiley & Sons, New York, 1951, pp. 163-188.

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