113 792 THS A NEW CRETERIQN EOE SATISFACTORY COMMUTATION AND A DIGITAL COMPUTER PROGRAM FOR DESIGN Thesis for III: Degree of DH. D. MICHIGAN STATE UNIVERSITY Hiremaglur Krishnaswamy Kesavan 1959 ' THESIS This is to certifg that the thesis entitled A NEW CRITERION FOR SATISFACTORY CDMMUTATION AND A DIGITAL GJMPUTER PROGRAM FOR DESIGN presented by HIREMAGLUR KRISHNASWAMY KESAVAN has been accepted towards fulfillment of the requirements for D0 CTDR OF PHILOSOPHY degree in ELECTRIQAL ENGINEERING Major professor Date May 1, 1959 LIBRARY 1 Michigan Stan University I I I . I 0-169 A NEW CRITERION FOR SATISFACTORY COMMUTATION AND A DIGITfiL COMPUTER PROGRAM FOR DESIGN BY Hiremaglur Krishnaswamy Kesavan A THESIS Submitted to the Schbol for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1959 ,1 “I” . . I -—;-" V .. Approved var #41:; mth/ j: . ' “~._ Lax-1::- jv-Lal. .4 I a ' - :7 I 6/6438 Suzi—ea A NEW CRITERION FOR SATISFACTORY COMMUTATION AND A DIGITAL COMPUTER PROGRAM FOR DESIGN BY Hiremaglur Krishnaswamy Kesavan An Abstract Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1959 Approved -1- ABSTRACT The manifold applications of commutator-type machines in control sys- tems have placed exacting requirements on design. The chief difficulty encountered in this regard is the problem of commutation itself. The me- chanical switch, for which there is no practical substitution at the pres- ent, places several limitations on the terminal characteristics that can be realized. The 'electrical' aspects of the mechanical switching problem have long proven formidable. It is generally recognized that new concepts must be developed before a satisfactory solution could be obtained for this old and difficult problem. In recent years, the significant advances in the areas of network theory and the digital computer have had a direct impact on the study of commutation. It has not only resulted in the new formulation techniques of the problem but also has profoundly changed the concept of machine de- sign. In chapter I a brief survey of the several aspects of the problem is presented. The electrical aspects of the problem are separated from the rest and are considered under two categories. First, the inherent charac- teristics of the armature winding itself as they effect commutation. Sec- ond, the effect on commutation of the coupling between the fields and the armature circuits. In chapter II a concise survey is made of the reference [6] , for purposes of laying the mathematical foundations for both aspects of the problem. The background is set for the consideration of the problem of the thesis. The prdblem of the thesis is defined in chapter III. A criterion is -2- evolved which could serve as a basis for studying the relative commutating abilities of the various anmature windings operating in the same frame and under the same terminal conditions. This criterion is referred to as the 'commutation factor'. By making use of the commutation factor, a reactance voltage is calculated which serves as a basis for comparing the commutating ability of various armature windings operating on different frames. In chapter IV the basis for calculating the inductances which form the starting point for the calculation of the commutation factor is con- sidered. New analytical expressions for the inductances are presented and some convenient terms defined. Also, a numerical method suited for the digital computer is given for the determination of the exact locations of the coils undergoing short-circuit. Chapter V presents a complete digital computer program for the commu- tation design of the commutator-type machine. In chapter VI the effects of varying several of the design parameters on the commutation factor are discussed. The changes in parameters neces- sary for realizing the desired commutation factor are discussed with ref- erence to a flow-diagram of the computer program. The results of calcu- lations on forty machines are presented in the form of a table wherein the commutation factor is tabulated against several pertinent design parameters. From this table, it is concluded that when two windings are compared on the same frame, a lower value of the commutation factor will result in better commutation. The matrix notation is used extensively throughout the thesis. ACKNOWLEDGMENTS The author is grateful to Dr. H. E. Koenig, his thesis adviser, for his constant guidance in preparing this thesis. His fundamental work in the area of commutation study forms the background on which much of this thesis is based. The author wishes to express his thanks to Dr. M. 8. Reed and Dr. L. W. Von Tersch for their constant enc°uragement. Thanks are also due to the Reliance Electric and Engineering Company and to Mr. Lanier Greer, the chief D.C. Engineer in particular, for their cooperation and support. . .5-” CONTENTS ACKNOWLEDGMENTS (11) I. INTRODUCTION 1 II. EQUATIONS OF COMMUTATION 3 III. COMMUTATION FACTOR 13 Iv. BASIS FOR INDUCTANCB CALCULATIONS 23 V. DIGITAL COMPUTER DESIGN ON THE BASIS OFTHECOMMUTATION PACmR O'COOOOOOOOOOOOOOOOOOOO 40 VI. COmLUSION O O O O O O O O O O O O O O O O O O O O O 0 O O O O O O O O O O O O O 0 O 0 50 APPENDIX A .00...00.00.000.000.0000000000000QOOOOOOOOO. 58 APPENDIX B OOOOOOOOOOOOOOOOOOOOOOOO 00.00.00.000... .0 0.. 6o LISTOF RBFERBmBS 0....0.0.0.0...00000009000.00.0.0... 63 I. INTRODUCTION The rapid growth of the control systems area has greatly enhanced the importance of the commutator type machine. This electro-mechanical component, by virtue of its unique characteristics, renders it a very ver- satile component, suitable for application in a wide area of control sys- tems. However, the chief obstacle in its design is the problem of commu- tation itself. The mechanical switch used to realize the terminal char- acteristics presents several problems, many of which are not 'electrical' 1 points out, like many others on the in nature. In a recent paper, Tur subject, that although the differential equations of the problem can be obtained, these equations can not be solved analytically, as the voltage in a sliding contact depends upon a number of factors, eg., mechanical, thermal and chemical processes in the contact layer. 2 has studied in detail the effect of the quality of the car- Wilhite ban-brush on commutation. Korecki3 has established criteria useful in selecting commutator bar material and design. In a recent paper, Hind- marsh4 discusses the mechanical limitations of commutation. Experimental investigations of commutation are difficult because of inadequate techniques of accurately measuring the factors effecting commu- tation. Thielers5 points out that some machines which are found good commutation-wise in the first few days deteriorate with time while others improve with time. In spite of all the many difficulties associated with commutator de- sign, the compelling fact remains that there is no substitute at the pres- ent for the mechanical switch. With recent developments in transistors there is perhaps renewed hope in the possibility of replacing the mechani- -2- cal switching system by one devoid of moving part. HOwever, with the present over-voltage limitations on transistors, the problem of minimizing the terminal voltages of the armature circuits during the switching inter- val is expected to be as critical for a transistorized commutator as it is for the mechanical commutator. Moreover, the problem of replacing the commutating machine by some other device, however desirable, is truly a problem in invention and hence belongs to the realm of speculation. Although it is generally recognized that the design and construction of mechanical commutators and the associated armature circuitry is a highly developed art, it is also generally recognized that a substantial improve- ment in the mathematical description of the switching system is essential to the design of commutator machines suitable for application to fast re- sponse control systems. In this thesis, attention is confined only to the electrical aspects of the problem. The electrical problem can be broadly classified into two categories. First, the inherent characteristics of the armature winding itself as they effect commutation. Second, the effect on commutation of the coupling be- tween the fields and the armature circuits. In this thesis, a mathematical foundation fOr the study of both aspects is presented but only the first is studied in detail, a criterion for the choice of the armature winding is presented, and a working digital computer program established for de- sign based on this criterion. -3- II. EQUATIONS OF COMMUTATION 2.1 History Reference is made first to the report entitled ’Commutation Study' (CS) by Koenigé. This work represents a departure from any other work found in the literature in at least two respects. First, the problem is viewed as a network switching problem with the exploitation of linear transformation theory. Second, new and different procedures are used to relate the inductance numbers to the machine geometry. These procedures, while too complex for calculation by hand, are ideally suited for digital computation. The theoretical development and results presented in that report constitute the starting point for this thesis. In view of its re- lationship to this thesis, the salient features of the report are reorgan- ized and presented as they effect the development of this thesis. 2.2 Equations of Commutation In the CS report, it is pointed out that for each interval of time during which a given number of commutator segments are contacted by the brush, a system of differential equations can be written which describe the interrelationship between the coil currents, the effects of armature and field currents, the mechanical position of the rotor and its velocity. These equations are valid for each interval of time during which a given set or commutator segments are short-circuited by the brushes. Fig. 2.1 shows a connection diagram for a 2-pole machine, for any interval in which there is no switching action. The mesh circuit equations for the network are of the form -4- A...“ Nu No. 5 .0 v 5 ~o mm 5. Nu. ma Nu- §<‘\\\ \\\\\\a a I, n . + e x .3“ $ H Law +lI mus “\Q N.\ “a” 776,05) M/‘Q(CT’ 4,067 7:2“) _777:u ('6‘) M21“: sfla 66—)! F— “ fl 777/3“) mums x4306) “mason Maw) .9409 - — .J (2.2.1) where the elements in the column matrices J1”) and J2“) represent respectively the currents in the circuits closed by the contact surfaces of brush one and two. The elements in column matrices $030) and 504“) represent cur- rents in the main circuits of the armature and stator field windings re- spectively. In particular, for a 2-pole lap wound machine, are) Jam =- “0420.9 4411?) = 43c (*7 where if(t) is the shunt field current. For any given investigation Of commutation the variables in '=JL(t) and 4(t) are considered as known functions of time consistent with terminal conditions for which the commutating characteristics are to be investigated. The effect of eddy currents in the solid frame are included in the inductance numbers used for the various circuits. That is, these induct— ance numbers are evaluated at the frequencies of time variation involved in commutation. The coefficients in submatrices 65Y11(t) and €z7é2(t) represent re- spectively the self and mutual inductance coefficients of the circuits —6- closed by brush one and two, and 55%,, represents the mutual inductance coefficients between those circuits closed by brush one and those closed by brush two. The elements of c=p3(t) and. (sx'mmszx , Mama .42., 3;" A‘Al‘ A13... (55’ m,,.6;") (S; musg'). . - '(sgmansj) 13-93, -I . . ~ ~/ sh A?" AGRSM EQ’WINSZ’) {Sh/mflasa.‘ I ’ I ' (3". mums" 1543., (2.3.6) The coefficient matrix in (2.3.6) is symmetric since the original matrix of (2.3.4) is symmetric and fslil is symmetric. Furthermore, (2.3.6) con-' tains exactly p non-trivial equations, one corresponding to each slot cur- rent variable. Having once defined the term 'slot current‘ and the form of the re- . sulting equations, these results can be obtained directly by inspection. 2.4 Necessary_Conditions for Satisfactory Commutation In the CS report, the necessary conditions for satisfactory commuta- tion are stated. In view of their importance for the further development of this thesis, these results are stated here in the form of a postulate and two theorems. Postulate 2.4.1 Let S be the set of n coils sharing the same slot and '73 and '13 th represent respectively the time at which the j coil of S enters and leaves the commutating zone. Then, for satisfactory commutation, >1 , . ' 7" 3;»; swamp = grauw- Tat/(7;) (2.4.1) where i. represents the jth J commutating coil current. The Ial(’T3) and 133(7—5) terms represent respectively the currents in the armature circuits -11- from which and to which the commutating coils are switched. The postulate (2.4.1) requires for its verification, the time-domain values of Ia1('rj) and Ia2('73) for the set of n coils during the succes- sive switching intervals, which are encountered during the commutation period. Since these currents can not be determined either by direct mea- surement or by a solution to the system of differential equations of commu- tation, (2.4.1) can hardly be verified by direct experimentation. Rather, it is based on the observed properties of switching under similar condi- tions. If the armature current per path is represented by Ia and is considered constant over the interval of time (7},“71 ), (2.4.1) reduces to 71 ) ' (N) I i 4:. T. __b: I. = an 0- 3% 3X} a} 3. (2.4.2) Theorem 2.4.1 Let m be the number of switching intervals encountered as the armature rotates one slot and.Ck be the total number of circuits closed by a brush during the kth interval, then, ‘mCK,,J_+ _hi.,_..;~. ZZE-Lr(‘tk)’ T€tl<")] " EERXTJJ 3-(75'fl (2.4.3) K:] Y=I Again, if Ia is constant over the commutation period, then m C“ . _ ' + _ Z ZE‘Y(JCK)“°YCTK")] ‘ 1771—“- (2.4.4) sz ‘Y=I Theorem 2.4.2 Let 5 and P represent respectively the slot pitch in degrees and the coils closed by all the brushes and Cslc the change in the variable number of poles and Ic represent the summation of all the currents of all as the armature rotates through an angle of 73‘ , then, \ \ -12- 7] A IQ = Z IagchD- IQ.(7}) . (2.4.5) J2! If the change in the magnitude of the armature terminal current is ‘negligible over the commutation period of one slot, then A la :. .2 71 Ia. (2.4.6) The important consequence of the theorems (2.4.1) and (2.4.2) is that it is only necessary to consider the time-variation of only one current variable, I , which represents the summation of currents of all the coils c and under all the brushes, and so, the study of commutation is finally reduced to the study of only a single differential equation instead of a system of differential equations representing all the coils undergoing commutation. Thereby, the rigid restriction of continuity on all the in- dividual coil currents is relaxed and the study of the single differential equation mentioned above, for satisfactory commutation, marks a definite departure from all the study hitherto made on that subject. -13- III. THE COMMUTATION FACTOR 3.1 Definition of the Problem The problem of this thesis can now be stated. Instead of focusing attention on realizing the commutating fields which would give satisfac- tory commutation, attention is directed towards developing a numerical measure of the relative commutating abilities of the various armature windings operating in the same frame and under the same terminal condi— tions. This numerical measure is called the 'commutation factor' (CF). In addition to establishing such a numerical measure of the relative commu- tating abilities of various armature winding configurations, a complete digital computer program is to be established for including the commuta- tion factor in the design of commutating machines. 3.2 Commutation Factor To define the proposed basis for comparing the commutating ability of various armature windings, consider the sketch of Fig. (3.2.1) which rep- resents the commutating coil circuit in the process of being opened. Let this circuit (and possibly one under the second brush also) open at the end of the time interval under consideration. Then the equations repre- senting the changes in commutating coil currents for this interval tn< t is obtained by merely replacing all the elements of the first row of £5 , except the (1x1) element by -1. Por a two-pole machine or a machine with an even number of segments per pole pair, (3.2.4) can be written as A'Ac, + Awe: 77%., me”. Aye, A7102. Active; 7720.1: MC}; Ale: -17.. where A We, and A‘Vc; are column matrices with an entry (AW, '*‘ AW; ) in the (1x1) position of qum and all the other variables of the col- umn matrices are the same as before the transformation. The first vari- able in Adm represents the sum of the changes in all the currents closed by both brushes. Theorem (2.4.2) is stated in terms of this var- iable. To separate this variable from the remaining, let (3.2.5) be re- partitioned as follows AA“ AW, +54%), Mm, mam. Alan A“ CO. + AWQa. — meal m Caa AJQOC (3.2.6) where AJCO» includes all current variables in (3.2.6) except the first Axle, variable. The most efficient way to solve for (3.2.6) for Avie. is to triangu- larize the coefficient matrix by multiplying both sides of the equation by .— / ’mcla. mcmoj O M L _) (3.2.7) with the result __ AAc.‘mclo.mc(La. AVLQQ. AV|+ALP°1mcm7WQaaA AW CC, + A woo. (Men mammcaa meal 0 _ _AJ.Q,- L. mam 77%an AJQ“; (3.2.8) -18.. From the top equation in (3.2.8) and with the assumption ATP, = A ‘l’ . A“1| and AWQ=0 the ratio , takes on the form Awe Mn , (Mcu' 7n... ma)... Meal MM-mm ma)... A91... A1... ' 1+ K A4... (/+f\’) where H A")? = — [Me/mmagaj A wto. (3.2.10) all the non-zero elements of the column matrix Ame are identically equal to AW. I I‘ If only one circuit is opened at the end of the interval under commu- tation, then AWQfO. Note that if the armature windings to be compared are assumed to have the same coupling to the external circuits or if the comparison is not to include the effects of the commutating field and external circuits, then the last term in (3.2.9) is not included and the ratio is a function of the inductance coefficient matrix of the winding only. Therefore, the commutation factor is defined as AW : (Men ~mc/amg’mmm.) Aim (/ + H) (3.2.11) In evaluating the above commutation factor, the assumption was made that all the contact voltages of the column matrix of AW are zero ex- cept for the coils which are about to leave the commutating zone. When . . . -I these Contact voltages are included they are multiplied by Mammal)? This triple product is very small compared to other terms and the commu— tating factor defined above is not appreciably altered. The commutating factor given in (3.2.11) is used as a basis of com- paring the commutating ability of armature windings designed for the same -19- armature punchings and commutator and operating in the same frame. Uhder these conditions the switching sequences and length of intervals are identical and changes in the current variable Aiacl, during the last interval of commutation, are essentially equal for various winding con- figurations such as lap, wave and frog-leg windings. Under these condi- tions the ratio gives a measure of the relative magnitudes of the average bar-to-bar voltage during the last interval of commutation. The calculation of the commutation factor can be further simplified in windings like the simplex-lap where the winding repeats every pole or every pair of poles. The simplification is realized by means of 'equiva- lent representations' of the armature winding discussed in appendix B. The commutation factor which gives the inherent characteristic of a given winding on a given frame, can not be used in its present form for comparing armature windings on different frames and of different current ratings. To achieve the latter purpose, the commutation factor is used to calculate the voltage between the brush and commutator bar at the time of opening. This voltage is known as the 'reactance voltage'. 3.3 Derivation of the Formula for the Reactance Voltage (Br) Definition 3.3.1 The ratio A”; is known as the reactance volt- age; thus, Z§LV R 131' (3.3.1) Substituting (3.2.11) into (3.3.1), we have Autc ER: (C F) At (3.3.2) In (3.3.2) [it represents the time required for the current vari- ~ - Asiag able Awe to change from + I, to - Ia' Thus, 15-? represents the average rate of change of the Z33Lc variable during the commutation period. Fig. (3.3.1) shows a diagrammatic representation of the time variation of the Aid variable during the time interval At . Equation (3.3.2) can be stated in a fbrm more convenient in design. AJC = 215 where Is is the steady-state value of the current in the armature circuits. .2 map where m number of sections of the winding and .4 = Tbtal amperes X) number of parallel paths of the winding A‘t gzp775t< Where 5‘ is the slot pitch in commutator segments and is equal to Kn, K. is the total number of commutator segments, 13 is the total number of poles, 715 is the revolutions per second and finally, 2 . ‘ 0 '~" 15 the rotation of the armature in segments. Fri £1 . (CF>-—— ’“° = .22.) 775 K p (CF) The commutation factor in (3.2.11) is calculated on the basis of only one turn/section. The total reactance voltage is ob- tained by multiplying this factor by (turns/section)2. ER: ZZ—L-tc (CF): (02. “LP >73 KP) (CF) (tanS/SCCfIOH)& (3.3.3) Equation (3.3.3) can be expressed in terms of the ampere turns per pole (AT?). if K (tHYHS/SQCttoh) 73 (3.3.4) ATP - -21- A—B represents the commutating period. Curve 1: represents the time varia- tion of the [5‘101 variable during the interval AB. Curves 2, 3, 4 and 5: represent the ‘rIc- time variation of the individual coil r§§;;:“\ \\ I currents during the \ \\\\ commutating period. \ \ \ \ \\ \ \ \\ \ \ \\ \\ \\ a \\ \\ f \ \\ \ \ ‘5 \ \ ———————4-t \ \ ) \\ \ 3 \ \\ \ \ \ \ \ \ \ 5}\\ \ Qish ‘Ic "Ia FIG. (3.31) Coil currents as function of time during commutating period. -22- 01', yo : ATp X’p P K X turns/sectioh (3.3.5) Substituting (3.3.5) into (3.3.3) we have, ER =a (ATP))< p“ x 7). x (t urns'/$ectn'or\)(CF) (3.3.6) and substituting the relationship 775‘;§§g1 into (3.3.6), we have fin- ally, - Hi rpm , ER - 32 x ATP x P x —-\6-)--O~X('Cuvm$/Sccfuoh) x (CF) (33.7, .23.. IV. BASIS FOR INDUCTANCB CALCULATIONS 4.1 The commutation factor that is proposed is ultimately a numeri- cal measure for the commutating ability of an armature winding on a given frame. Thus, for successful application of this criterion for design, the commutation factor should be calculated with utmost preci- sion. The crucial step in its realization is the availability of accurate Ianalytical expressions for the calculation of the inductance numbers of the commutating coils, which form the starting point of the derivation of the commutation factor. Dreyfus7 and Thielers present methods of calculating some of the required inductance numbers for a two-pole ma- chine under the assumption of a unifOrm airgap in the commutating pole region. However, these expressions do not warrant their application in the present study, because additional refinements have to be definitely added for more precise calculations of the inductance numbers. The ana- lytical expressions presented in the CS report serve this need and are ideally suited to the digital computer. A detailed discussion of the problem of determining the inductance numbers for rotating machines, by correlating them to the geometry of the machines, appears in reference (8). It also contains an extensive bibli- ography on the subject. The mathematical expressions for calculating the commutating coil inductances are based on the results of a series of field maps of the commutating pole region appearing in R—299. These maps were made for a relatively wide range of commutating pole airgaps and geometries under conditions corresponding to a constant excitation of the commutating pole. From these field maps the radial component of the airgap flux density per -2 4- ampere turn of'mmm.f. so determined is plotted as a function of the po- sition of the armature surface, a curve of the form shown in Fig. (4.1.1) results. The basic assumption made in the calculation of the commutating coil inductances is that the radial component of the flux density in the commutating pole region when an armature coil is excited (instead of the commutating winding) is obtainable from the results of the field maps by simply changing the direction of the flux density as shown by curve in Fig. (4.1.2). The airgap self and mutual inductances of the commutating coils are obtained by evaluating the surface integral of the flux density curve over appropriate areas. Tb systematize this calculation of the inductance coefficients, the results of the field maps have been consolidated in the fOrn of two curves. One curve, as in Fig. (4.1.3). shows the ratio of the third harmonic, (:3 , to the peak value of flux density in a Fourier series representation of the airgap flux density curve of Fig. (4.1.1). In deriving the curves of Fig. (4.1.3) all harmonics above the third are neglected. Tb obtain the value of (0' (ratio of fundamental to the peak value of commutating pole flux density) simply subtract C3 from unity. Thus, C/=/" C3 It should also be pointed out that the Fourier series expansion of the wave form in Fig. (4.1.1) is based on taking the distance ‘71:. as one half period. This quantity ‘71:, (referred to as extent of commu- tating field) is obtained from the curves shown in Fig. (4.1.4) by forming the product ax Cd = 77.2. where CL appears as the ordinate of the curves in Fig. (4.1.4). -25- commutating pole __—_--~\i\\\\\\*Armature [80' od’ an? .9; «£9 1 l F/é. (41.1.) Radial component of flux density per ampere turn, commutating pole excited K— |80 F/G. (4. /. a) Assumed radial component of flux density per ampere turn, armature coil excited -26— 25/6. (4-/. 3.) ofié. and c K are the distances from the reference slot to the kth and '-28- 4.2 Calculation of'CommutatingTCoil Inductance: for Salient Pole Machines Fig. (4.2.1) gives a cross-sectional diagram of the iron boundary surface along with some details of the commutating pole and the armature. The details of Fig. (4.2.1) which enter into the mathematical expressions for the inductance calculations are presented below: 1.) Slot 1 is placed in a position wherein its top-section is about to break contact with the brush. 2.) X’is the angle between the center of slot 1 and the center of the nearest commutating pole. 3.) All angles are measured in the clockwise direction from the reference slot. 4.) If the top-section of slot 5 goes to the bottom-section of the slot k to form a coil, their distance OCrK is obtained by0(-a'_-C£k where th (use slots respectively. 5.) G‘is the chording angle and is equal to (TT - coil pitch). 6.) C2 is the extent of the commutating field. 7.) The slot details of the armature of Fig. (4.2.1) are shown in Fig. (4.2.2). In order to evaluate the inductances of an armature coil 3-3'.. it is necessary to integrate the flux density over the surface a-b-c-d-e (Fig. 4.2.1) when the armature coil carries unit current. Let this surface of integration be divided into the following sub-surfaces: a) integral over surface a-b and d-e define slot leakage inductance. b) integral over surface b-d defines the airgap inductance. -29.. / b s 0 “A IN“ A" 3%”0 T 4- FIG (4.1.1) JuT 51 w%/// ,%/// /////3 #75 (4.2.2.) = ist ce between the wedge of the d the top coil of the slot -31- 4.3 Slot Inductance The field distribution in the slot is assumed to be normal to the sides of the slot. It is easily shown that the self inductance 3.)? x Ls: W5 L (ad,+~"73ig+da)x lo’g hem/3,! (4.3.1) and the mutual inductance between two conductors in the same slot is taken as My 3'"? X L (ash/,1” 10-8 heva}/ W5 (4.3.2) 4.4 Airgaplnductance The assumption made in the calculation of the airgap inductance is that the excitation of the field poles will not alter the relative flux density distribution at the surface of the armature, due to the excita- tion of one armature coil. The airgap inductance is considered in two parts, that which is due to the commutating-pole flux and that due to the main-pole flux. Both the components of airgap inductance are obtained by surface integration over proper areas. A detailed discussion appears in the CS report and will not be reported here. The flux density distribu- tion curves for the commutating pole appear in the Engineering report n-299. Only the final expressions for the airgap inductances are presented below. As they are obtained by the integration of different areas, de- pending upon the positions of the coils, their ranges of application are stated. The restrictions are stated in terms of the extent of the commu- tating pole field (’7/2), the location of the slots in the commutator I zones (0Q 8), and the switching angle( 7). -32- 1) Expression valid for the region 4%. + r) edge out 4 (47,.- X) M?“ = 9% {[H— sin A ((+43)][1- sin A (Th—aha] +[l+ Sin A(:Y+ 0C3: -O"][l--$'Ih A(T+ dK-Ufl} X '0'8 (4.4.1) henvy where, qjc‘ = fundamental component of commutating pole flux per ampere turn of airgap n.n.f. (A detailed derivation of this quan- tity appears in the section (5.2) A WXHYOD T’x’tfiz where 13 is the number of poles 0“ is the chording angle when j = k, (4.4.1) represents the self inductance M... = % {[I- (sin A Wag-f] +[I-(sinn “mm x Io‘8 (4.4.2) Equation (4.4.2) is applicable for calculating the self inductance terms of any coil and also for the mutual inductances of coils in slots under the same pole. 2.a) Expression valid for the region “(771+Y)$°LK€O(J6_+O‘S(WQ~’F) (IV _ Mg“: .3: fl)-smMX+ocj—a)];{[g+sma(y+«Kflix‘0 8 (4.4.3) 2°b) Expression valid for the region ‘(772 + I) S “gr-6‘ S as (”a-'3’) M‘ ‘ YE;- fl” sm A(3’+0(a'_-O')][l-S'm A(b’+ “01)" )0‘9 'a-K (4.4.4) -33.. 3.a) Expression valid for the region (T/g-¥)>0<7;2 OQK‘T?‘(77;L+F) ' -8 M' :hil s‘mA(r+O< 5). Hence, the next step is to establish mathematical ex- pressions for their determination. Out of the many possible switching intervals, the switching interval that is selected must meet the following conditions. 1.) It must serve as the criterion for comparing different machines, 2.) It must be such that it represents the worst possible condition of commutation, 3.) It must be valid for both directions of the armature rotation. When the reference slot is in a state of leaving the commutating zone, the above three conditions are satisfied. Definition 4.4.1 The zone of armature periphery in inches corre- sponding to the interval of rotation during which the first winding sec- tion in a given slot enters and the last winding section of the same slot leaves the commutating period is known as the commutating zone. A detailed discussion of the commutator zone is found in reference (11). Only the result is given here. AY 0.1)- 3 wash Thickness Commutator zone (CZ) = com 0.1). Q°§m¢ 04 ngfl Ahqle TI" Com CID: . \f k . S (1- Com. P. ‘ Earths _ mica.» tlmckness «+- No. of bus [3“:th ( dctkcscrxcy) ’ Po\cs ]} (4.4.7) -34- where the commutator pitch deficiency is given by No. 0+ mom. boa-s - (Col‘ p'ntck) $ect‘mvxs x No. 0" Po\2.$ (4.4.8) Definition 4.4.2 One-half of the commutating zone when expressed in degrees is referred to as the switching angle (3’). Y e '15. [COY'VM ZOhQ. .lh AesrefiS] (4.4.9) 4.5 Determinations of the Slot Locations (0(3) The positions of all the slots in the commutating zones defined by all the brushes can be determined by more than one method. Only the numerical method which is highly suited for the digital computer will be presented here. The method is best explained through the use of two examples. These examples also serve to point out the simplifications possible in calculating the airgap inductances when certain patterns of symmetries exist. Example 4.5.1 The pertinent machine data used in the following calculations is: a) Armature details: 1) Armature winding ............. Simplex lap 2) Number of sections ........... 4 3) Coil pitch ................... 9 slots 4) Number of slots .............. 38 5) Armature outer dia. .......... 11.5 inches b) Commutator details: 1) Number of com. segments ...... 152 -35- 2) Commutator dia. ............. 9.25 inChes 3) Brush thickness ............. 0.625 inches 4) Brush angle ................. 30 degrees 5) Mica thickness .............. 0.035 inches c) Commutating pole data: 1) No. of commutating poles ....... 4 Calculations: 1) Com. zone as calculated from (4.4.7) is 2.03 inches Ci ‘.n "nckes y INLL 2) Com. zone angle (C2) = f Av 0.3). 34“: 3) Slot 'tch ( ) ' d re = p1 5 in eg es No.01: s\ot$ 4) Tabulate the slot angles and commutator zone angles as in table (4.5.1) 5) From table (4.5.1), determine which slots are in the commutating zone. From table (4.5.1), it is evident that slots 1, 2, and 3 are in the com. zone of brush one. Slots 11 and 12 are in the zone of brush two. Slots 21 and 22 are in the zone of brush three. And finally, slots 30 and 31 are in the zone of brush four. The angles locating the positions of the slots are: (X,=O 00;;15 (X3 :‘35 0(4: I05 (15;: [IS Duo:.~203 CX'j 2.2/5 49:29; -36... From table (4.5.1) slot 20 is in the same position with respect to the zone of the third brush as is slot 1 with respect to the zone of the first brush. This is always the case fer simplex lap. . Since the winding repeats every pair of poles, considerable simplification in the calculation of the inductance numbers is achieved by an equivalent two-pole representation. Such an equivalent two-pole representation is discussed in the appendix 8. Example 4.5.2 In this example, a machine with a simplexawave winding is considered. a) Armature details: l) Armature winding ................. Simplexawave 2) No. of sections .................. 3 3) Coil pitch ....................... 8 slots 4) No. of slots ..................... 33 5) Armature outer dia. .............. 9.5 inches b) Commutator details: 1) No. of Com. Segments ............. 99 2) Com. dia. ........................ 7.223 inches 3) Brush thickness .................. 0.5 inch 4) Brush angle ...................... 30 degrees 5) Mica thickness ................... 0.035 inch c) Commutator pole data: 1) No. of commutating poles ......... 4 d) Calculations: 1) Com. zone from (4.4.7) will yield ...... 1.69 inches LGQ x Ill/.(a 915? 2) Com. zone angle (CE) = = 20°.40 -37- O 3) Slot pitch (5) in degrees .......... = 3:3 ‘ IO°. 90‘“ 4) Thbulate the slot angles and commutating zone angles as in Thble (4.5.2). 5) From the table (4.5.2), the slot locations are: ofl=CD ocazlé O(3=QS d4: [05 d5:I75 Kg: W5 iq =QSJ fig: 105 From table (4.5.2) reference slot 1, is about to leave its commu- tator zone. There are no other slots in a similar position. This is typical of simplex-wave windings. There is no repetition in the magni- tude of the inductance numbers. -38- Table (4.5.1) Slot No. Eggrees from Sl°t_1, Commutating’ Span in Zone No. Degrees l 0 l 0-20.2294 2 9.4736 2 90-110.2294 3 18.9472 3 180-200.2294 4 270-290.2294 4 28.4208 5 37.8944 6 47.3680 7 56.8416 8 66.3152 9 75.7888 10 85.2624 11 94.7360 12 104.2096 13 113.6832 14 123.1568 15 132.6304 16 142.1040 17 151.5776 18 161.0512 19 170.5248 20 179.9984 (is actually 180.0; round off error of 0.0016) 21 189.4720 22 198.9456 23 208.4192 24 217.8928 25 227.3664 26 236.8400 27 246.3136 28 255.7872 29 265.2608 30 274.7344 31 284.2080 32 293.6816 33 303.1552 34 312.6288 35 322.1024 36 331.5760 37 341.0496 38 350.5232 Table (4.5.1) shows that slots 1,2 and 3 are in commutating zone 1; slots 11 and 12 are in zone 2; slots 20,21 and 22 are in zone 3; and finally," slots 30 and 31 are in zone 4. The winding repeats every pair of poles. -39... Table (4.5.2) Slot No. Degrees from Slot 1 Commutating Span in Zone No. Degrees 1 0 1 0-20.3867 2 10.9091 2 90-110.3867 3 180-200.3867 3 21.8182 4 270-290.3867 4 32.7273 5 43.6364 6 54.5455 7 65.4546 8 76.3637 9 87.2728 10 98.1819 11 109.0910 12 120.0001 13 130.9092 14 141.8183 15 152.7274 16 163.6365 17 174.5456 18 185.4547 19 196.3638 20 207.2729 21 218.1820 22 229.0911 23 240.0002 24 250.9093 25 261.8184 26 272.7275 27 283.6366 28 294.5457 29 305.4548 30 316.3639 31 327.2730 32 338.1821 33 349.0912 Teble (4.5.2) indicates that slots 1 and 2 are in commutating zone 1; slots 10 and 11 are in zone 2; l8 and 19 are in zone 3; and finally, slots 26 and 27 are in zone 4. The winding does not repeat. V. DIGITAL COMPU’EBR DESIGN OF THE D.C. MACHINE ON THE BASIS OF THE CWUI‘ATION FACTOR 5.1 The subject of electrical machinery is at least 70 years old, and the first paper in the A.I.E.E. dates back to the year 1886. Since that date, the subject has grown enormously. Hewever, many research and developmental problems of the commutator type of machine still exist. The various analyses of the commutation problem have not resulted in a practical design procedure, eliminating the necessity of adjustments on the test floor for achieving optimum commutation. The design of the commutator type of machine, like all the other machinery, have hitherto depended on mathematical methods which are con- ducive for numerical analysis. Only numerical methods can be successfully applied to take account of, to mention only a few, the nonlinearities in- troduced by the saturation effects, the inclusion of the actual field maps of the region of the commutating pole, etc. Also, the numerical calculations in the design procedures were kept at a minimum, (within the range of the slide rule) by means of.simplifying assumptions. Vienott12 in a recent paper on the subject of induction machine design, speaks of this 'arithmetic barrier'. The advent of the modern digital computer has seen tremendous changes in the area of machine design. These arithmetic barriers are definitely surmounted and consequently, the designer is no longer forced to make un- necessary simplifying assumptions so as to stay within simple mathemati- cal expressions. This fact has definitely resulted in more precise calcu- lations. Also, the speed of the computer has enabled the designer to look for several designs, and in effect, has raised the possibility of eliminating design Changes on the test floor. The digital computer pro- gram for commutation design presented in section (5.3) is established -41- with this as the objective. and the calculations are presented in a manner best suited for digital computer application. form of a flow diagram, indicating the parameters which can be varied in The main design is based on the commutation factor. a commutation factor investigation. 5.2 Begin Input Data Rating: RPM Run no. H.P./K.W. rating Line volts Revolutions per minute Amperes Armature dimen 5 ion 5 : 13. 14. 15. Ar 0.D. Armature outer diameter Armature inner diameter Gross armature length No. of slots Slot depth Slot opening Widest slot width Slot width at bottom Distance between the wedge of the slot and the top coil of the slot Height of the coil in the top section of the slot Both the data The program details are then summarized in the -42- 16. d3 Insulation between the coil sections in the slot Armature‘windinggdetails: 17. Coil width less insulation 18. Coil depth less insulation 19. WPS Conductors/slot 20. 771 No. of sections 21. Type of winding 22. Coil pitch in slots 23. No. of parallel paths in the winding -“------------~~---- --“-----“---- -»----------~------------..-------- Commutator and brush: 24. K No. of commutator bars 25. Commutator outer dia. 26. Mica thickness 27. Brush width 28. Brush thickness 29. No. of brushes 30. Brush angle Main pole: 31. Width of pole 32. Length of pole 33. Height of pole 34. % pole face chord 35. ‘ No. of main poles 36. Shunt field volts -43- Frame and commutating pole: 37. 38. 39. 40. 41. 42. 43. 45. 46. 47. 48. 49. 50. No. of commutating poles Pole length Pole width Minimum inter-polar gap Frame outer dia. Frame inner dia. Effective frame length Calculations cl OI (DI Embrasure IParc MParc IPGinax END OF INPUT DATA CALCULATIONS BEGIN No. of poles x Sin" [7.7. Po\e Face and] 180 0.5 X Av 0.D. (35) x Sin" _i_x (34) 0.5 x (6) pxIl’Gmin TTxAr OD Awidth of com. pole x No. of;poles 7T x Ar OD x e’ (39)x(35) Trx (6)x(44) (35) 247' x(45) 7Tx(44)x(46) (35) m [:2 (La 5);. ws'c H WY] /+ (47) [3018)"— @4604 - 7’; MW] 13x1me in (48)x(39) 51. 52. -44- Effective airgap of commutating pole (40) +3 [(So) -(4o)] Carter constant .: wddbst slot width Effective airgap of com. pole _ (12) (51) From (45), (46) and (49) and with the aid of a sub-routine, the nmerical procedure of which is presented in appendix A, the values of C, and 772 are calculated and located in (53) and (54). 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. Cl ”’2; Slot component of self- / -5 (y) 2- inductance of a coil (L5) 3' ’9 X 0 x {/95 3(’4)+3(15)+(/6) Slot component of mutual -3 (15) inductance between two coil- 3, /9 X /0 X 111R,” + T] sides in the same slot. (1Msl) (In) A up X _I_ 17’ x a.) P 77a (35) x (54) 3./9 X (40) B (5.1) Fundamental component of com. 02(53)(59) (4,) (9) pole flux per amp. turn of airgap m.m.f. (We, ) (3.5) (.57) “I’m/5, (50/4 Commutator pitch deficiency. (————ao)(:¥()35) —- (:2 a) Commutator zone in inches [ft—652) gfi _ (.26) + 3&5.) [(30) (I * (6’))” -%] 63. 64. 65. 66. 67. 68. 69. Commutator zone angle in degrees Switching angle in elec- trical degrees Pole-pitch in degrees Distances of the pole centers from the reference pole Limits of the zones of commutation under each pole Commutating zones under each pole Slot pitch in degrees ~45- £5352 x //¢,(p (63)(35) 5‘ 360 (35) ox(45)~l-~((.(.,o) /x/(.s).. .. . .(cc./) .1 was). . . . . .(u...'L) (as) (at p- -I) ;/)x where (pl) (35)-I (cc. 0) +(43) ...... (m. o) (u... I) +0.3) ------ (c7. /) (ca.a)+(¢3).. . . . .(ma) (ALF/MM). . damp—.7) CZ, =(ac.0) To (67.0) : (QC, /) To (67.» as}: (u. a) 70 (4.7 a) : (4.4.,317) roam) o) 6\ Or 70. 71. 73. 74. 75. ~46- Angles of the slots measured from the O x (a?) , . . . . (70.0) reference slot 1. / x (‘9) . ' ‘ ‘ , (70. /) 2.x (6?) . . - . . (70n2) (NV/MM)" . . .{7o./vs-/) Search for the number of slots, 0,1,2..... (IVs-n 5' which lie in the regions 3.1 of 021 thru CZp. Renumber , them in a sequence. . 5n Distance of the slots within ’: S, x (‘9). ' . , , , (74],) all the com. zones from slot I 1 measured in mechanical 0(3) 5;, x (U1)- : . ° ' ' (702-4) degrees. . (d/’)Ol:1) ’ ' ' 'dlifi) 0“,: th(69)c' ....(7.2.n) Test each (X' for the following conditions and store them in a sequence after multiplying each angle by 4%31 to convert them into electrical degrees. If (f) 05 05$ (65) , do not change the angle (’I’7'{‘~5)$0('$2(65) subtract (65) from 0L. {’77)°2((’5)‘°('S3(65) subtract 2(65) from CC, (I‘vQW/oflgg's [9045) subtract (p-l) (65) from o<’ The distances of the slots from the reference slot 1 d/ = [74./) 'n electrical de rees - 1 g 09;,- (74,2) db: (74a )1) (.21) x (6.5) x (35) $1 Coil-pitch in electrical degrees -47- .76. Chording angle 0" 77" - (75) 77. Airgap component of self- inductance (Go) x lo‘8 {I -(5m [(57) f(6¥)+ a .33} 3‘ n. aw+ag 79. (78) -(7(.) so, 517‘) {(57)(75’)} 81. sm gamma} 82. A' t f 1f- ' - 1....” ° 5° (60>E['-<2o>“]+[~m>“]}x'0 3 Equations in (82) are used to calculate all the self inductance of all coils in (74). 83. Total self inductance of the (39.) + (5 5) coil 84. (57) [(64)+ dK] 85. (5'7) [ow-(74)] 86. 3"“ (3“) 87. sun (95) The expressions for the airgap component of mutual inductances, Mjk' depend on the regions. Therefore a decision must be made to the region of application and the expression to be used. 3' . 88. If -(/_.L+>r)>; OC?S ng (”r/4‘7) Mix : (to) x Io’8 {[I-(fioflh +(86)] + [HM/)MI 7487):]? -43- 89. J)“ —(7‘/;+y)é ngcx} + <5'<4'/.':"r M1“ = (60) x lo“? {ll-(80):”) +(6’lo)]} 90- ‘23: ' (fly/2.1”?) Sd}-O”SO(K$ (7.1-?) M“ = (60) X ”’3 {[’*(8°>]D'(8(’)]} 91. J; (’09-‘3030Cd12dk‘o- 7,-(472‘4- 3') M}, : (6.0) x w a I—(so) [l+(%’7)]§ 3 92. =9} (Vi-3') 2 “K ‘0' >/ d} 7/ (fr/«1+ Y) M... =80» ..~2 u. «8815-881; 93. For the calculations of Mjk, from (88) thru (92), test whether there is the slot component of mutual inductance or not; if 'yes', add (56). 94. From the calculations (77) thru (92), form the inductance coeffi- cient matrix (7n). 95. Form the transformation matrix S of the same order as the m matrix of (94). 96. Form the 3 matrix. Since the form of the S" matrix is known, a routine for the inverse in going from (95) to (96) is not required. 97. We : 37775" : (95) (94) (96) 98. Separate the (1x1) element of (97) MC” 99. Form row matrix containing all the elements in the first row ~yy2€4a of (97) except (98) 100. Form column matrix containing all elements in the first ”lea. column of (97) except (98) 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. Form the square obtained from -49- (97) by deleting (98). (99) and >qYCcta. (100) Inverse of 77’] C Ga Porm triple matrix pro- duct 7978,. my“ mom Subtract matrices N\QH - 'I 7who. 7)?“ mean Q. Form the summing matrix of the same order as 773caq' Form triple product 7720HL 7770La :9. Commutation factor turns/section amperes/path = «if. amp. turns/pole (ATP) Reactance voltage between bar and brush (ER ) (101)"1 obtained by an inverse sub-routine. (99) (102) (100) product obtained by a triple matrix product sUb-routine. (98)-(103) 3‘ (99) (102) (105) product obtained by a triple matrix product sub-routine. /o‘# /-(/o(.) conductors/slot 2x no. of sections (l9) (20)x2 (4) (23) turns/Sect. x amps x total no. of com. segments no. of parallel paths x poles (108) (109) (24) (35) ZxATP xpz x turns/sect. at CB x RPM 10 x 60 2x(110)x(35)2x(108)x(4)x(107) 107 x (61) --~_--“m-_—-----—mu——--_oo—----~----—-----------—--—--------.m--—----------- END OF CAICULATIONS. -50- VI. CONCLUSION 6.1 The digital computer program presented in section V is used for calculating the commutation factors of forty different machines. These machines have been built and their test performance is known. They are classified into 2-pole simplex-lap, 4-pole simplexdwave, 4-pole simplexb lap and 6-pole simplex-lap respectively. These results are presented in table (6.1.1). Table (6.1.1) includes a few pertinent details such as the length of the armature, the outer diameter of the armature, the volume (394-), the type of winding, the width of the slot, the commu- tating zone, the reactance voltage and the percentage buck and boost amperes for the "black-band" test along with the run numbers identifying the particular machine. The following conclusions are drawn from the table (6.1.1). 1.) For the same frame-size, a simplexawave winding has a higher commutation factor than the simplex-lap winding. It is borne out by experience that satisfactory commutation is dif- ficult to achieve in large frame sizes using wave windings. As a matter of normal design practice, the wave-windings are avoided if possible in .these machines. D 2.) All other variables held constant, increase in the length of the armature results in an increase in the commutation factor. The "Erie Sparking Factor" commonly used as a "rule of thumb" in commutation design is also directly proportional to the length of the armature. 3.) The variation of the commutation factor with volume is plotted in the form of two curves in Fig. (6.1.1) --- one for the simplex-lap and the other for the simplex-wave. -51- These curves show a direct correlation between the commutation factor and volume of the armature for the relative commutation pole region geometries, brush width and slot geometry used in the forty de- signs. 4.) All the other variables held constant, increase in slot width results in a decrease in the commutation factor. 5.) Decrease in the commutating zone results in an increase of the commutation factor, when all the other parameters are the same. This conclusion is also related to experience in that a decrease in the width of the commutating period (with the number of coils shorted by the brushes remaining unchanged) increases the rate of change of com- mutating coil current. The switching angle chosen is exactly equal to half of the commu- tating zone. Hence, the variation of the commutation factor with switch- ing angle is the same as that already discussed under 5). 6.) The changes in the width of the commutating pole also effect the commutation factor. However, since there is very little change in the width of the commutating pole for the samples chosen, this effect does not show in the tabulation. 7.) The total number of slots within the commutating zone also effect the commutation factor. However, again in the samples chosen, this factor remains unchanged within each group. 8.) The reactance voltage tabulated gives a numerical measure for comparing two windings in different frames designs. For the machine chosen, these numbers, in general, agree with experience. 9.) The commutation factors do not correlate with the percentage "buck and boost" obtained by the black-band tests. This is as expected, u N U 5 l 666% one». 66.3 66m. 66% 66$ , ~r » x. - u w . ‘. cow. -52- s 4 _ n a _ T _ oxuqxswfitbx 1.... m.» ox N. OTTTF. w. .m. v. n. N. e. -53.. since the commutation factor which as calculated does not include the effects of the commutating field. The results in table (6.1.1) show that for the same frame. an arma- ture winding with a lower commutation factor results in a better commu- tation. There are a number of design parameters that can be varied in the process of design. The digital computer program presented in section (5) is designed to include the possibility of varying these parameters. The exact stages in the program wherein these factors can be varied are eXplained by means of a 'flow diagram' of the computer program given in Pig. (60201) 0 Box 1 represents the stage of calculations from 44 to 60 as presented in section (5.3). These calculations are used to establish the funda- mental component of commutating pole flux per ampere turn of airgap m.m.f. (qua, ). The factors effecting the commutation factor at this stage are: (l) the minimum inter-polar gap and (2) the width of the commutating pole. Box 2 represents the calculations involved in determining the slot inductances, ranging from 55 to 56. The factors effecting the commuta- tion factor at this stage are: (l) the gross length of the armature and (2) the width of the slot. The decision after box 2, whether to Choose a lap winding or a wave winding, is crucial in the realization of satisfactory commutation char- acteristics. Boxes 3, 4, and 5 represent the calculations ranging from 65 to 74. These calculations determine the number of slots in the commutating zones and their respective locations with respect to the reference slot. The number of slots in the commutating zones effect the commutation factor -54- and can be varied to an advantage. Boxes 6 and 7 represent the calculations ranging from 61 to 64 and 75 to 93 respectively. This is the stage where the airgap inductances are calculated. Here, the factors effecting the commutation factor are: (1) the commutating zone angle and (2) the coil pitch or the chording angle. Boxes 8 through 12 represent the calculations from 94 to 107. There is no variable parameter in this stage of the program. 6.3 unsolved Problem Closely associated with the study of the armature windings, is a problem frequently referred to as the problem of 'field weakening'. When the main field is weakened beyond a particular value, very bad commuta- tion results. The only method presently available is to make adjustments on the test floor to establish the number of turns and airgap of the com- mutating pole. These adjustments are made to give optimum commutation characteristics at rated conditions. This problem is included in the second phase of the commutation problem mentioned in the introduction. The work of Koenig brings out this problem in all its details. To achieve some concrete results from the mathematical foundations that have already been laid out, additional research of a long-term nature is needed. For the present it is sufficient to point out that the A 7\ term in (3.2.9) represents the effect of the commutating field and the external conditions on commutation. However, how this term is related to ‘field weakening' is yet unknown. -55.. Box 1 can vary IPGmin and width of con- nutating pole. CALCULATION OF (1)] 1 IE Low mm imam. “ armature length (2) and width of slot Decision wave or lap Decision Eq l-p or Z-p If wave i I ”G 11’ If BcL 2-n or 2p If Bq l l-p CAI£ULATION 0P CALCULATION OF CAIEULATION 0F (3) (4) (5) ,- CALCULATION 01‘ t (6) inductance or not can vary total number CALCULATION OF of slots undergoing AIRGAP INDUCTANCB commutation. (7) Porn matrix (tri- angular) (8) Box 6 t Can vary commutating Form total zone and chording symmetric (9) matrix angle. 7 ‘ Linear Transforma- I Triangularization tion (10) (11) J: Con. factor & BR NO. Run No. pi Type of 012. No. Poles 1434 1429 856 2 2 'Winding Lap Lap Lap 5.0 5.0 5.0 5.0 5.0 -56- Width IPG of slot 0.432 0.432 0.432 0.432 0.432 0.039 0.048 0.059 0.049 0.048 Com. zone 1.859 1.921 1.859 1.859 1.859 11 30 14 4.5 18 %BB Com. {5 5.00 0.00 0.02 6.26 ——----—-----——-——----— ——--—-—n—fican-oc---------------—---—----------- -----—-----——--- 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1392 1258 1393 1248 1301 1256 428 1243 1543 1544 1545 1551 1548 1549 1554 1546 1552 4 4 4 wave Wave- Dead Coil Wave wave wave- Dead Coil wave Wave Wave Wave Nave Wave wave wave Lap Lap Lap Lap 6.0 6.75 6.0 6.75 6.75 6.75 6.75 6.75 9.50 9.50 9.50 11.50 11.50 11.50 13.00 9.50 11.50 Length 051 Eiléii 3.375 84.5 4.50 112.5 4.50 112.5 4.50 112.5 4.50 112.5 4.50 112.5 3.025 130.5 4.50 102.0 4.50 102.0 4.25 193.0 4.50 102.0 4.25 193.0 5.75 202.0. 5.75 202.0 5.75 202.0 5.75 202.0 7.125 045.0 7.125 '045.0 9.50 800 11.50 1520 8.25 1090 8.25 1090 9.00 1520 9.50 800 11.50 1520 Table (6.601) 0.190 0.210 0.190 0.210 0.210 0.210 0.260 0.260 0.260 0.329 0.329 0.329 0.337 0.260 0.329 0.058 0.060 0.112 0.105 0.172 0.140 0.165 0.180 0.159 0.128 0.165 1.691 1.691 1.583 2.036 1.907 2.040 2.242 1.691 2.040 17 6.5 15.0 13 8.5 20 14 5.5 5.5 18.5 15.5 14.5 10.0 21.0 22.0 22.0 6.70 7.62 8.02 8.19 9.97 10.4 10.6 10.7 11.3 11.6 14.0 16.1 32.9 9.82 11.40 13.6 13.76 8.45 10.5 19.3 18.7 11.6 16.9 9.2 12.8 9.15 11.9 8.5 9.3 28.8 14.8 12.8 12.2 10.30 No. 20 27 h 28 29 30 BR 12.30 19.0 16.9 ~-------———_ ----~----—------——---------~---— —--------~-——~--————-------—-———------- 34 35 36 37 38 ------—- .— p- '—---H~—----_--——--—- ---------- Run No. of Type of Dia. No. Poles Winding 1556 4 Lap 13.0 1557 4 Lap 13.0 1561 4 Lap 15.5 1558 4 Lap 13.0 1564 4 Lap 15.50 1563 4 Lap 15.50 1598 6 Lap 30.099 1588 6 Lap 30.099 1599 6 Lap 30.099 1592 6 Lap 30.099 1590 6 Lap 30.099 1591 6 Lap 30.099 1589 6 Lap 30.099 . 1593 6 Lap 30.099 1594 6 Lap 30.099 Length DZL Width IPG' Com. %BB Com. of Ar. of slot Eggs fa 12.00 2015 0.337 0.160 2.242 34.0 15.18 12.0 2015 0.337 0.140 2.160 12.0 15.36 11.0 3370 0.308 0.180 2.444 16.5 16.68 12.0 2015 0.308 0.143 2.000 14.5 17.56 14.0 3360 0.327 0.172 2.264 17.0 23.0 14.0 3360 0.332 0.172 2.342 13.5 24.4 19.25 17,450 0.510 0.432 2.794 22.32 19.25 17,450 0.510 0.300 2.794 22.71 19.25 17,450 0.562 0.302 2.769 26.25 19.25 17,450 0.472 0.363 2.648 30.30 19.25 17,450 0.472 0.370 2.640 30.30 19.25 17,450 0.432 0.328 2.346 36.6 19.25 17,450 0.432 0.330 2.346 37.8 19.25 17,450 0.357 0.333 2.497 42.3 31.50 28,500 0.370 0.328 2.292 69.0 Continued Table (6.1.1) -58.. Appendix A Numerical Procedure for the Calculation of Q1 and ”72. The numerical procedure for calculating items (53) and (54) in the computer program of section (5.3) is given below in nine discrete steps. 1.) Input the family of curves in Fig. (4.1.3), by the method of polynomial approximation using the method of '1east squares'. Curve- fitting as the basis of this numerical method forms an auxiliary sub- routine. All the curves within the family are approximated by a poly- nomial of the same degree. 2.) Form another polynomial of the same degree \whose whose coeffi- cients are a function of 8. For a value of 8, for which a curve is already drawn such as in Pig. (4.1.3) and for which the polynomial approximation is already made in step 1, the new polynomial of step 2) coincides with it. 3.) For a value of 8 different from the 8's of Fig. (4.1.3), the coefficients of the polynomial are calculated by interpolation formulas. 4.) Given the value of K, with the aid of the polynomial in 3), calculate C3. 5.) Value of Cl is given by the relationship, C1 = l-C3. 6.) Steps 1) and 2) are applied to the family of curves of Pig. (4.1.4). 7.) For a given value of ale , the coefficients of the polynomial relating ata'r/aw and 8 are calculated by similar interpolation formulas as in step 3). 8.) Given the value of 8 from the expression resulting from step 7), calculate 0. . -59- 9.) Calculate 172. by multiplying a. by cu = width of the commu- tating pole. -59- 9.) Calculate 172. by multiplying a. by cu = width of the commu- tating pole. -60- Appendix.B Two-pole Equivalent Representations For a machine with more than two poles, an 'equivalent two—pole' representation is possible for certain windings such as the simplexblap. Such 'equivalent two-pole' representations are not possible in simplex- wave windings since the winding does not repeat every pair of poles. The properties of the coefficient matrix and the transformations necessary to realize the equivalent two-pole representation are discussed for a six-pole machine with a simplex-lap winding in which three slots are short-circuited by each brush. The system of equations are of the form: r. - __ A (11b, 77?” ml; ‘ :— _ 74:3 WW :7703 mm ‘AJ, 1 I 9.02:: ‘ 272.2, 777., : m, >40. :Maa 7722.4 7.81, 81:13 _ 721,3 7724—: 78,, mm :70} m M. A742,, _ 77225 777.14: 771;“ 77419.: 77723 777-74 43’“ M; 22,, 7722:7331 m; m, m. M. 1- Ave! _77723 77124: 7’70; “01.24" 771217": _A": (3.1) where £>QP and [3:9 represent the same quantities as in eq. (3.2.5). The subscripts 1 through 6 refer to the respective brushes and the order of each 771 matrix is equal to the number of slots closed by the corre- sponding brush. Following the indicated lines of partitioning equation (8.1) can be written as: —A (IPA- Tn/(ML. 7(flab mab A (WE Male 7710.8 77740 £3.95 LA 19% L700. 77140 7710.3 Mo._ (0.2) _’_ 11:913- -61... For windings that repeat every pair of poles, equation (3.2) has the following properties. First, the entire matrix is symmetric. Sec- ondly, the submatrix in the (1,3) position is the same as that in the (1,2) Position and the submatrix in the (2,3) position is the same as that in the (3,1) position. In other words, the entire coefficient ma- trix is cyclic with respect to the six submatrices. It is to be empha— sized here, that it is this cyclic character of the coefficient matrix that makes an 'equivalent two-pole’ representation possible. The coefficient matrix is diagonalized with respect to the sub- matrices by the following linear transformation of variables. Let At“): A WA A “P; : S A 1P6 .9 we: LA 1%. ' (3.3) and _' ' " ‘T S A .1211 ‘ A 20A 5 A09: A «=9 Q l— _ h... .—a (8.4) where the symmetrical component transformation matrix Va 04 LP 3 ; a 61.2.46 0.13% (A 0145!. au 1 (8.5) ' jaw/3 Where U. is the unit matrix and 0.: e . The order of the unit matrix ‘LL is the same as the order of the square submatrices in (8.2). In this particular example, it is (6x6) since 3 slots are short-circuited by each -62- brush. The equations resulting from the change of variables are: F23 a”)? _(77’(o.0. + 27720.!» O O ’ (”Asa/2f A V; 2 O (maa‘yhab) 0 A-oé 5 5 _.£1 #QL L. C) (3 {)72an‘ ”nail Lgxgigg) When .AleA=-I§"DB==Z>q¥A==£5q¥g,‘ [3 ”a: , we have 5 A11); :3 AWA , AJA =3A¢QA and the top equation in (8.6) reads 5 A «pf: (m... + a‘m 0431100.: (B 7) Equation (B.7) represents the equations for the two pole equivalent of the machine. In terms of the submatrix notation in (B.l) the equivalent two-pole representation appears as _. .1 a. .1 .. _ ACLPPI 771p” WP]; JP 11%., L771.“ mp J .72. [— P61 ‘ _ l H (8.8) where P 0 (ml! *- ;mla) and 79(p‘2’ :(7}?.2’ 7“ £77,143) 707p]: = (NZ/2 + QM”) th 773P5a‘l77/5a + 2777,.) The subscripts P1 and P2 refer to the equivalent two poles. Equation (3.8) has the same form as equation (3.2.5). Note that the coefficients in the two-pole equivalent are a simple linear sum of the coefficients appearing in the first two rows of (B.l). l M 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) -63.. LI ST OP REFBRBBCBS The Problem of Commutation of D.C. Machines, by B. K. Tur, Blektrichestvo, 1956, No. 11, 30-33. Carbon-Brush Capacity Calculations, by J. G. Wilhite, Trans. Amer. Inst. Elec. Engrs., Vol. 75, 1367-9 (1956). Criteria for the Choice of the Bar Structure of the DuC. Commutator, by A. Korecki, Arch. Elektrotech, Vol. 6, No. 4, 619-29. Volts per Bar Limit on Large D.C. Machines, Methods of Limiting its Effects, by J. Hindmarsh, Elect. Times, Vol. 134, 381-5 (Sept. 11), and 461-3 (Sept. 25, 1958). A New Aspect of Commutation, by G. Thielers, Technical Publications, ASEA, Vasteras, Sweden. Commutation Study, by H. E. Koenig, Michigan State university, Reliance Electric and Engineering Co., Cleveland, Ohio. Die Stromwendung Grosser Gleichstrommaschinen Theorie Der Kommutierungen, by Ludwig Arthur Dreyfus. A Circuit Approach to the Analysis of Electrical Machinery, a Ph.D. Thesis by H. E. Koenig, University of Illinois, Urbana, 1953. 'R-29'; a technical report by the Reliance Electric and Engineering Company of Cleveland, Ohio. 'E-ll6'; a technical report by the Reliance Electric and Engineering Company of Cleveland, Ohio. The D.C. Machine Design manual of the Reliance Electric and Engi- neering Company of Cleveland, Ohio. Induction Machinery Design being Revolutionized by the Digital Com- puter, by C. G. Veinott, Power Apparatus and Systems, Vol. 28, 1509-15. “71111111914[110111111111“