A Method of Designing Medium and High Frequency Alternators with a Special Reference to the Inductor Type Citation Oboukhoff, Nicolas Mikhailovich (1929) A Method of Designing Medium and High Frequency Alternators with a Special Reference to the Inductor Type. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/fgd1-7v75. https://resolver.caltech.edu/CaltechTHESIS:08162024-200618128 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1929 Record Number: CaltechTHESIS:08162024-200618128 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08162024-200618128 DOI: 10.7907/fgd1-7v75 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16645 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 16 Aug 2024 23:39 Last Modified: 16 Aug 2024 23:56 Thesis Files PDF - Final Version See Usage Policy. 25MB Repository Staff Only: item control page

A METHOD OF DESIGNING MEDIUM AND HIGH FREQUENCY ALTERNATORS WITH A SPECIAL REFERENC;E TO THE INDUCTOR TYPE THESIS by NICOlAS M. OBOUKHOFF rn Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California I CONTENTS Page . 2 SYNOPSIS . . . . . . INTRODUCTION. Modern Applications of Medium and W-~gh Frequency Alternators and Their Usual Types .. . . . . . . . . . . . . .. . . . . . 4 SECTION I. General Principles and Methods of Designing and Fundamental Formulas. First Method . _. . . . . . . . . . . . . . 7 SECTION II. General Methods and Fundamental Formulas (continued). Second Method . . . 23 - SECTION III. Predetermination of the Coefficient of Flux Oscillationr. Experimental Formulas for T . Their Rational Basis . . . . 40 SECTION IV. Verification of the Theory and Method Presented in the Foregoing Sections. . 60 DRAWINGS. - 5 Sheets, Figs. 1-12 Inclusive . . . REFERENCES. 70 . . . . . . . . . . . . . . . . . . . 75-76 METHOD OF DESIGNING tEDIUM AND HIGH FREQUENCY ALTERNAT-ORS With a Special Referenc~ to the Inductor Type. by N. M. O~oukhoff SYNOPSIS. The introduction to the present paper contains a recapitulation of various modern applications of medium and high frequency alternators together with a brief description of their usual types. In Section I the peculiar features of medium and high frequency alternators are considered together with some experimental data, and fundamental equations and expressions are derived in terms of the ratio of the minimum flux density in rot or slots to the maximum flux density in the Bmin. Various wave forms of Bmax the field flux and EMF are considered and numerical coeffi- air-gap, i.e. in terms of o cients dependent upon them computed. A method for the pre- determination of the principal dimensions of an inductor alternator is given. In Section II are presented alternative equa- tions, formulas and numerical coefficients in terms of the ratio of the maximum flux linking one coil of the armature winding to the minimum flux, i.e. in terms of = f max which min is regarded as one of the most fundamental characteristics of the inductor alternator. mination of A method for the experimental deter- on existing machines is given . Section III is 3 devoted to the predeterminat io~., of 1 ; for that purpose a general expression and particular formulas are obtained and discussed, their bases being partly theoretical and partly empirical. A method for the predetermination of the maximum voltage obtainable at no load is suggested and developedo Section IV contains the verification of the theory and methods presented in the foregoing sections and the discussion of the results. The following criteria are used for the verification: 1. Saturation curve; 2. 3. Optimum pole wid.th; 4. Maximum voltage obtainable at no load; Particular limiting cases. The structural and experimental data of ten inductor alternators of different types, frequencies, capacities, makes and applications are used for the verification. There is a good agreement between the computed and experimental results. Introduction Modern applications of medium a.nd high fr~uency alternators and their usual types. Besidesthe well known a pp lica~ion of medium and high frequency alternators in the realm of the radio wave transmission, which dates from long ago, the following more recent uses are to be pointed out: 1. A new system of generating high frequency currents by means of alternators of an intermediate frequency (say from 5000 to 10000 cycles) has recently been devised. It contains a special type of a frequency multiplier in the form of a transformer whose magnetic circuit is highly saturated by the primary alternating current itself witr10ut any use of direct current. The above alternators have successfully been used for the transmission of the shorter radio waves(in the band 280 - 900 - 2100m.) 1 and quite recently for broadcasting purposes. There is a good reason to expect that the system is capable of successful operation for the generation and transmission of short waves in the band below 100 m. 2. During the recent years medium and hieh frequency alterna- tors have been used in metallurgy in this country and Europe for the purpose of electric heating and melting. High frequency ironless induction furnaces which originated from that application show a steady progress as to the scope and variety of their uses. 2 r 3. The recent application of medium frequency alternators for the power control on transmission lines has proved to be successful and very promising.3 ~ 4. High and medium (audio) frequency alternators have been used with success for energizing submarine signalling systems si nee long ago. 5. There are mo4ern mining prospecting methods in which medium (audio) frequency alternators are used to energize the region to be explored. 6. In the chemical industry the applications of medium and ·high frequency alternators are constantly increasing, the produc ... tion of ozone. being one of the oldest ones. 4 7. Medium f requency alternators have successfully been used for supplying the current to high speed induction motors which drive directly, without gear, high speed machine tools for fashioning and shaping wood, centrifugal machines, pumps and etc. 8. Medium frequency current generated by alternators has found a new application in machines for testing materials. 9. High voltage direct current can be obtained by combined ppplication of medium frequency alternators and rectifiers which system contains apparatus of small size. 10. High and medium frequency alternators are used in laboratory work as well as for industrial measurements. Thus the manifold uses of medium and high frequency alternators Justify the attention given in the present article to their design and computation from the electrical point of view. The main features of inductor alternators are shown in Figs. u,f~ 1, la, lb, 2, 2a, 2b, 3, 4, 5, where the eurront types with differ• 6 ent arrangements of teeth and slots are represented. Figs. 1, la, lb, 2, 2a, 2b ijhow homopolar alternators with a cylindrical rotor; Fig. 3 represents a homopolar disk rotor type which is known as Alexanderson's high frequency alternator. In Figs. 4 and 5 are represented the special forms of peripheral toothed zones as devised by Latour. A similar arrangement though not identical has been previously invented by Alexanderson in connection with his alternator. Fig. 6 gives as idea of Bethenod's heteropolar inductor alternator. An improved alternator of this type has lately been described by M. C. Spencer5, whereas the modified type of a homopolar inductor alternator has quite recently been presented in a paper by Dr. K. Schmidt. 6 In particular, Figs. 1 and 2a represent the alternator type which has been designed by the author and built by the Siemens and Halske Company of Petrograd. This is a slight modification of the usual homopolar inductor alternator(see Figs. 2 and 2b). Fig. la is a drawing on a larger scale of Fig. l in a paper by K. Schmidt. 7 Of course there are other types of medium and high frequency alternators, but those referred to appear to be most widely used~ Furthermore, in the course of the discussion other papers will be mentioned which contain description of other systems. In all the Figs. referred to, A is used to indicate field winding coils (d.c. coils), B armature a.c. winding, R rotor body, S stator body, Z rotor pole or tooth, s1 laminated armature iron core, N' ands' designate north and south polarity respectively. The dotted closed lines in Fig . 1 represent t he magnetic circuit 7 in usual homopolar inductor generators. hen revolving, the rotor peripheral toothed zone makes the magnetic flux linked with the armature a.c. winding coils oscillate, inducing hereby an electromotive fo~ce. Section I General Principles and Methods of Designigg and Funda- mental Formulas. First Method. The armature winding together with the magnetic circuit in both medium and high frequency alternators, besides meeting voltage and current requirements must also possess a definite amount of inductance. This is necessitated by the fact that in almost all applications these machines are working on the resonance principle. On the one hand, the alternator inductance must not exceed a given value lest the tuning of the circuit shoula become impossiw ble, with the capacity introduced into the circuit by some special arrangement, as for instance by the antenna. On the other hand, it is desirable to limit the use of tuning coils or at least their size by concentrating all or moat of the necessary inductance in the alternator itself. Therefore a designing engineer must con- sider with care the question of providing for a definite amount of inductance in such an alternator under design and must not have to chance its value. This requirement introduces new equations into the problem of designing this type of alternators so that this method becomes different from the usual one applied to o ~ commercial generators. 8 Furthermore, the inductance of such an alternator changes with the relative position of a· stator and rotor, as the latter revolves. The coefficient of self induction has, for inst ance, different values for the relative positions I and II represented in Fig. 7. Thus the question arises as to what value of induct- ance should be taken into account in designing an alternator of that type. Both the electromotive force and the current have their maximum(amplitude or crest value) at resonance d tion II (Figs. 7 and 10). the posiw It is the inductance at this instant that must be considered in determining the condition of resonance. This statement has already been published by the author in 1925 8 ; it will now be considered more or less in detail. At resonance the ma..~imum electromagnetic energy (at the instant of the maximum current) is e~qal to the maximum electrostatic energy (at the instant of the maximum voltage acnoss the equivalent circuit condenser) so that we have 1 ,._/, - -- ........ . ... .. . ( 1) "'J where~ is the coefficient of self induction of the circuit at the relative position II J I the amplitude value of the current, Cr the equivalent capacity of the circuit, which is practically equal to the capacity of a corrlenser inserted in the resonant circuit and Vis the crest or amplitude value of the voltage across it. '7 />,, Substituting v=uh.,,-t . , where i,,J:=. 2 ,,:,;- f, f being the resonance fre- quency we get from ( 1) the well known relation tJ-:::::. 1 , which CrLr shows that the fundamental frequency is determined by the value 9 of Lr inherent to the relative position II of the stator and rotor. In medium and high frequency inductor alternators the alternating reactive field penetrating into the poles (rotor teeth) is small in comparison with the effective direct current excitation field. This is due to the fact that in these machines the ratio of the effective d.c. field ampere turns to the reactive a.c. armature ampere turns corresponding to the said armature field is considerably greater than in the alternators of the usual commercial type so that the distortion of the excitation field flux will be much less than in the latter; the position of the vector of the electromotive force at load will almost coincide with that of the corresponding vector at no load, and the reactive armature flux will practically act like the leakage armature flux, say across stator slots, constituting thus one part of the entire resonant inductance, the other part being identified with the leakage armature flux. Under the effective field ampere turns are understood only those ampere turns which correspond to Bmax-=-.9A1,n, where Bmax 2 is the maximum magnetic induction in the air gap, and Em.in is the minimum magnetic induction 1n the rotor slot. This point will be considered in detail later in the section III. The armature reactive flux is schematically represented in Fig. 10 by the paths like e d c, qr m n r q, q' r' q' etc., while the leakage flux is exemplified by the lines of forces such as x y z, h g f. In determining the contribution of the reactive flux to the resonant inductance it is necessary to take into account 10 both the rotation of the rotor and the mutual inductance between the two armature windings Band Bon either half of the stator (Figs. 1 and la). ~he usual normal arrangement of those two wind~ ings is such that the coils of one of them are in one alignment, along a parallel to the axis of the rotor, with the corresponding coils of another winding, so that the two corresponding coils have one and the same radial median plane, the poles(rotor teeth) of one half of the rotor being also in one alignment with the corresponding poles of another. Applying Kirchhoff's laws to the magnetic circuit, it is found that the effect of mutual inductance in each winding is equivalent to that of half a reactive flux represented in Fig. 10. Since the self inductance effect itself in each winding is contributed also by half a reactive flux because of the rotation of the rotor, the resultant inductance corresponds to the full amount of the reactive flux at rest plus the leakage flux. This conclusion is different from the view according to which only half a reactive flux must be taken into account in determining the value of the resonant inductance and which, for instance, has been expressed in a paper by Professor A. Blondel.9 The writer's conclusions are in agreement with the experimental results obtained by him on several inductor alternators of medium frequency. For instance, the inductive reactance Lr4' of one such alternator with the usual coil arrangement, determined by the resonance method gave the value of 10.18 ohms at a normal load current and at the frequency f = lOOO. Direct measurement by means of alternating current of the same frequency flowing 1/ through the armature wi nding of the same machine at rest and supplied by another alternator gave for Lrw the value from 10 to 10,4 ohms at the rotor position II, and from 7.35 to 8.85 10 ohms at the position I. In both cases the upper values were determined at the lower values of the field current, and the experiment was so coniucted as to reproduce load conditions of the teated alternator. In interpreting these results, it must be borne in mind that the common armature flux interlinked with both the windings Band B1 (Fig. 1 and la), as indicated, for example, in Fig. 1 by a closed dotted line X Y, practically vanishes when the experiment is made on an inductor alternator at rest, since that flux lies within the solid metal(iron or steel) parts along a considerable length and is therefore almost entirely damped at the frequency used. Consequently there remains only the flux repre- sented in Fig . 10 whose magnetic paths lie in planes perpendicular to the rotor axis, and this result has been predicted on the basis of the outlined theory, namely: that the resonance inductance is determined by the entire flux at rest at the position II of Fig. 10 in so far as the usual normal inductor alternator is concerned. In passing, it might be pointed out that the previous consideration of the mutual inductance combined with Blondel's theory of two reactances practically leads to the same result, since the computation made by the author shows that in medium and high frequency inductor alternators the values of two reactances are much closer to each other than in ordinary commercial alternators, and the subsequent use of Bethenod's d1agram 11 completes the determination of the resonance inductance, with the result pre- viously stated. Another essential feature encountered in the design of inductor alternators is the fact tha.t the polarity of the flux linked with any coil of the armature winding does not change its sign, as the flux value oscillates between the maximum and minimum values respectively. max and min, The following is the first of two methods by which the stated peculiar features of design are taken into consideration. The first method Notation Bmax denotes the maximum magnetic induction in the air gap (!Milli/ '7 to the median plane of a pole (rotor tooth) when it faces a stator tooth. Brain denotes the minimum magnetic induction in the median plane of an interpolar space (rotor slot) I\nax and Bmin have one and the same sign and are directed radially in machines with a cylindrical rotor, on the one hand, . and unidirectionally and parallel to the axis of the shaft in machines with a disk rotor (see Figs. 1 and 3 respectively), on the other hand. u denotes the linear peripheral velocity in ems per second of a rotor. m denotes the number of parallel connected circuits or branches of the armature a.c. winding. z denotes the number of conductors connected in series in one slot of the stator or the number of turns in series in one coil, two slots being used per coil. S denotes the number of coils of the armature winding connected in series (i.e. in one branch) IJ La denotes the coefficient of self induction of the armature winding. f denotes the frequency of' EM F T denotes its period so that T 1/J .1 f denotes the coefficient of self induction of one of the L parallel branches of the armature winding so that · t -= mLa 1 denotes the gross iron length in ems of the armature core in the direction perpendicular to lateral faces of a pole i.e. measured parallel to the shaft axis in alternators with a cylindrical rotor, and radially in disk type machines (see Figs. 1 and 3). > denotes the value of the flux linked with one coil of the armature winding per cm length of the iron core in a stator and per ampere turn in a stator slot. Sometimes S is termed as leakage magnetic conductance per cm of axial length of a stator iron core, the whole length being 1 (see Figs. 1 and 3). Eomax denotes the maximum or crest value of a complex wave of EMF of an alternator. E0 or Eeff denotes the RMS or effective value of a complex E M F wave. denotes the maximum or crest value of the fundamental sine wave of EMF. denotes the R M S of same. .J Ei - E lmax - l[2 11 ~ax denotes the maximum or crest value of a sine E M F which is -f:o le_, ~ .,u·'tl to the working circuit. E denotes the RMS value of same - ,EmaJt E _ ,-- 2 Pr denotes the pole or rotor tooth pitch. denotes the stator slot pitch ( or stator tooth pitch) w.u.., denotesthe peripheral width of a pole (rotor tooth) , a' r ' denotes the width of a pole at its root. denotes the peripheral width of a rotor slot. denotes the peripheral width of a stator tooth. denotes the peripheral width of a stator slot (the opening at the air-gap} a' s denotes the least width of a stator tooth. b denotes the width of a stator slot which corresponds to a' 8 (See Figs. 7, 10 and 11) B d.= min ~ax ¢ denotes the flux linked with one coil of the armature winding. m denotes its maximum value. 'r_max ¢ min denotes its minimum value. ½0 :s;\.~~~ max l denotes the coefficient of flux oscillation d denotes the single air-gap length, radial (see Fig. 1) or axial (see Fig. 3) .' Br denotes the maximum flux density in poles (rotor teeth). B denotes the maximum flux density in stator teeth. s denotes the armature core iron space factor, i.e. the ratio of the net iron length to the gross iron core length 1. In Figs. 7, 10 and 11 are shown the coils of a.c. or armature winding of a stator, the indicating the left side with direct con:luctors and the points denoting the right side with return conductors of a coil. The field flux diagram of Bin Fig. 7 shows that, while the magnetic field is unidirectional along the rotor periphery, one side of each of the coils is subject to the action of a pole flux which is greater and the other side is under the influence of a slot flux. It follows therefrom that (Bmax w Bmin): 1 Su 10 Eomax E Q - I 0 -8 . volts ..........• (2) (Bmax - ~in)~ 1 6 u 10...Svolts ........... (3) Two limiting forms of a complex EMF wave, between which all complex wave forms in an alternator considered can be ranged * are the rectangle, on the one hand, and the sine curve, on the other, and the corresponding values of tively, so that actually --1.... 1 /J.., ~2• a.re 1 and respeca . JI • /" These two limiting forms of the complex EMF wave are dependent upon the two limiting forms of the magnetic field which are represented in the Fig. 8 by the broken line a'b'x'1•e•u•v•h', on tne one hand, and the sine curve b'k'l'e'o'q'h', on the other, /6 while a real form of the magnetic field is given with sufficient accuracy by the trapezoidal diagram b'c'd'e'f'g'h' to which a trapezoidal complex EMF wave corresponds. The reader will be referred to Fig. 9 in ,rhich different trapezoidal complex EMF waves are represented (the right hand half of the figure). Furthermore, as an alternator will have to feed a resonant circuit which is to be tuned to the fundamental frequency, it follows that the fundamental EMF sine wave should be taken into account and regarded as a working factor of the system. Therefore, it is necessary to introduce the values Jbax and E1 into equations and final results. For the two limiting cases we have Elmax= = 4 :1.27 (Rectangular complex EMF wave) Eomax and : 1 (Sine cdlmplex EMF wave) so that ,- 1 <' " actually ·/ 'I ;TT Finally we get the following limiting values for El _, Eomax .. / --+:-= =- 0.108 (sine complex EMF wave) " = _4~'!:... ~ 0.905 (recta.ngular complex E M F wave) V ~tr;- 2 so that actually Sine wave 0.708 Rectangular wave 0.905 ... . . . . . . ....... (3 bis) In Fig. 9 different complex EMF waves of trapezoidal shape which suits the real wave form the best ara represented, and respective values o~ determined by the method Just exemplified are given in Table. I. The trapezoidal EMF waves are to be 17 represented each by Fourier aeries; then the computation of / can easily be carried out for each case. As an average suitable for most cases the value from o.8 to 9,0L say o.85, can be taken for I . If the pole pitch Pr is very small involving also a very small width of the rotor pole fls elf or a pole has a shape like that represented in t he Fig. la, the complex EMF wave will . not differ much from a sine curve; then it is safer to take for the value from 0.708 to o.80, say 0.75. Now we have E1- ',; Eomax' and using ( 3) we get since E is the R M S value of the sine E M F "hich an alternator has to supply to a resonant circuit. Furthermore, as it has already been proved, such an alternator should possess a given or required value of inductive reactance .La. which requirement leads to the following expressions LW - mLaW where w Combining equations (5) and (6) and solving them, we get the following formulas for determining the characteristic data of the armature a.c. winding and therefore the size of an inductor generator /J' z- " / . .. J where Bmax can be computed in a usual manner, since it corres• ponds to the median plane of a pole with ~ as an equivalent. air-gap length, or, if ~ax given, field ampere-turns can be calculated as well. o(_- Bmin which ratio can be either computed Bznax or determined by an experimental or graphical method, as it will be shown later in Sect1oris II and III. Investigating several medium frequency inductor alternators, it has been found by the author that ; had almost the same value, namely: from 13 to 15, averagely 14, for closed slots of a stator(Figs. la, 2, 2b, 3) and about 10 for open slots(Figs.l, 2a) 5 can be computed or determined experimentally by Anyhow means of a model of the magnetic circuit and armature winding. The value of 0\.- Bmin depends upon the form of the toothed Bmax peripheral zones of the rotor and stator, in general, and upon the ratio - , in particular. Ivi.. The same reasonings which have just been developed can also be applied to heteropolar a.c. generators of non-inductor type, IJ if they are to satisfy certain conditions concerning inductance. This will lead to the following formulas analogous to the foregoing ones: /0 I I v i A heteropolar alternator 1s equivalent to an inductor generator in so far as, ceteris paribus, the maximum magnetic induction I\:iax' regardless of its polarity in the former, is equal to Bmax - Bmin in the latter. (7), In fact, if in the set I of formulas fa) and (9) (Bmax - Bmin) is replaced by 2B~ax, the group II is obtained, and vice ,v-ersa , by substituting Bmax - Bxnin for 2 ~ax, we can transform the second set of formula first one I. II into the This mutual transformation may be considered as verification of the presented formulas. rd~ Now it would be in ~ to give some suggestionsAhow to use the foregoing equations. First of all, it is necessary to carry out some preliminary calculations. The pole pitch Pr is to be determined from the admitted peripheral velocity u of a rotor and frequency f : Pr _Y._, f and then the width Sor of a pole will be chosen as a fraction of Pr; for the most part~ -;::::::, 0.5 Pr approximately. In modern high frequency machines of the type considered u is equal to from 100 to 150 m per second, whereas in some excepw tional cases it reaches 200 m per second and even more. In medium frequency generators u is considerably lower. Choosing for -fu. air- gap J' a certain value which in modern high and medium frequency alternators is only a fraction of l mm, it will be possible to determine experimentally or by computation the ratio o(:= Bmin Bmax Furthermore, taking as a basis the admitted peripheral velocity u and the number of revolutions per second, the diameter D of a rotor is directly determined; thereupon knowing pole pitch Pr and choosing a certain number m of parallel branches or circuits for armature winding, we are able to determine the number of coils Sin series contained in each branch. The value of is approximately constant for a given type of an inductor alternator to be designed; hence its value obtained by experimental investigations of other machines of the same type may be taken as a tentative value and substituted for ~ in the formulas of either of two groups I and II. In this way we obtain from equations (7) or ( 10 ) respectively, as the case may be, a tentative value for the number z of conductors in a stator slot, and from the other equations or the same set also the length 1 of the armature core. As z can be only an integer it should be rounded out accordingly. ~ must be determined for the position II (Figs. 7 a.nd 10) as it has previously been shown in this paper. The next step to be taken is the determination of the area of the cross section of conductors of the armature winding. As the value of the current supplied by an alternator is given, and the number m of the parallel branches of the armature winding has already been given, cross section of conductor ~n be easily determined. In middle frequency alternator the current density can be admitted much higher than in generator of commercial type. The writer can point out to a case when he admitted 12 amp. per 1 mm 2 , and no over-heating occurred (see also a paper by Karl Schmidt 12 ). Thus it will be possible to determine the shape and the cross section area of a stator slot for a selected z and a determined pole pitch. Of course, in outlining a stator slot allowance for insulation must be made. Following the slot selection the peripheral forms of both a,,,,£ a statorArotor can be selected, and consequently it becomes possible to make a second and more accurate determination of the value of ; In its turn, an accurate value of ~ brings about a more precise determination of z and 1 by means of the sets of equations referred to (I or II). It is seen that all the characteristic data of an alternator discussed{ D, -1, z, Pr, Sr, S, m, i; 1 ol the form and size of the slots, can be determined by means of this method, in general, and by the sets of equations I, II, in particular, so that an alternator designed should meet the requirements and conditions imposed. It should be noted that contents of this chapter are not limited to the inductor type of alternator; the set II of equations makes the method applicable also to heteropolar non-inductor generators. The writer has applied this method w i t h ~ satis- factory results. Section II General Methods and Fundamental Formulas ( continued) Second Method The foregoing formulas and equations contain the ratio = Bmin. They will now be transformed in such a way as to int~ce/\ = $ max where max denotes the value of the min flux passing through a pole and corresponding to one half of the pole pitch 1. e. to ~• while </J min is the value of the flux across a rotor slot corresponding to another half of the pole pitch, although the pole width Br and that of a slot br may differ somewhat from ~- Thus <j5 max represent the total amount of the flux passing through the air- gap across that recw tangular tangent strip area 1 x Pr which is situated along the 2 outer cylindrical surface of the pole symmetrically with respect to its median line (see Figs. 7 and 8). The ratio A • ~ is an important factor in the operation of an ind.uctor~nator; its reciprocal O =_Q,._M will henceforth be called the coeffic lent of the flu~llation. We shall at first consider two limiting cases, i.e. a rectangular field flux distribution and a sine flux oscillation or distribution. Referring to Fig. 8 where different forms of field distri• bution are represented we get in the first case (rectangular distribution) . /3?n4X + /J?'MY& 4~~ l3n~ - /Jmp,v £,6/l, £ :?, ~ ma.x ( 1 - r !(_ y-- ) :Bma.x ( 1 .. oc ) 1 Pr, a.nd as . 2 Bmax 1 Pr: O.S max' we get finally for a rectangular field . 2 T ) ............. ( 1 3 ) ) In the second case (sine wave flux oscillation or distribuw tion) we have from Fig. 8 (/) =: /~ - /Jfr,ax, - ; c2 Jg d~,- . ., Substituting for max its expression from (14) . we get 1 - and after obvious reductions we have ........... ( 15) . .- + .~{ /- ':I from which equation we obtain 1 - where ::k ( 1 .. 1 2 • . • • ( 16 ) I ) k _ Jr ~ ) ............ ( 16 bis) · l o. 285 ( 1 .. / ) and A = 1;0 In applying the expressions (16) and (16 bis) we shall discriminate between medium and high frequency alternators: while in the former the coefficient of the flux oscillation can be considered as having an average value equal to about 3.5--4 and - - 0.25 - o.285, in the latter : 1.75 - 2.25 and A= o.45 - 0.57. The reason for the dis- crimination lies in different values of Pr in medium and high f~equency alternators: 7 in the former Pr is the greater. Consequently, if we take we get o=. ., - r.2 k .. :r 2 k ... 1 w 1 1 0.75 X 0.285 1 0.43 X 0.285 4 and 0 • 1. 75 respectively, -- 1.29 for medium frequency -- 1.40 for high frequency and k ~ 1.345 for intermediate cases between medium and high Irequency alternators. It follows t hat we have generally k ( 1 ... where k ) ............. (17) = 1 for a rectangular distribution of the field flux i.e. for a rectangular wave of the EMF, and k 1.29 • 1.40 in the case of the sine EMF (sine field flux oscillation or distribution). Since the actual form of the field flux distribution will be of an intermediate shape, k will assume values between those limiting figures. The intermediate forms of the field flux as well as of the EMF can be represented by trapezoids. be done with sufficiently good approximation. This can In Fig. 9 four . trapezoids I, II, III, I.V show a gradual transition from a rec.. tangular shape to the sine wave of both the field flux and the EMF, the trapezoid IV being very close to the sine curve. Now the following computation will be applied to the wave shapes intermediate between rectangular and sine form. With the notation of Fig. 9 we have ·~ 9tm = /J, _,_;. ,m;,_ 9 + ~:i /Jm»- /(0fl 1-~;) ~ raj/ £ whence we get or where /4~ I It is seen that the relationship between 1 - ol and 1 is of the same simple form for· intermediate wave shapes as for the limiting forms of the field flux and EMF, the coefficient k being determined as equal to 1 (l,t... or from the equations (i i ) (i6GuJ ror the rectangular, trapezoidal and sinusoidal wave form respecti~ely. z lJ Substituting in the set I of equations (7), (8) and (9) k ( 1 ... A ) for 1 ... c<., and kr 1/o for A , and putting ............... ( 22) ==- S- we get the third set of equations E u .2./JJ., Jnttt .This set of equations makes possible the computation of the principal dimensions of alternator in terms of rather than oc , which is an advantage because it is easier to predetermine o( • than The method of computing Q by means of formulas presented by the author will be given in the next section. The verification of the whole method by checking computed results against experimental data obtained from observations made on various medium and high frequency alternators will be presented later in a special section. The numerical values of ~ , k and O are grouped together in Table I for different values of Q and various wave forms corres- ponding to the trapezoids of Fig. 9 with the rectangular and sine waves of the field flux and EMF as the limiting cases. I o.a95 Trapezoid I v 1 1.061 1.13 1.21 fJ. 0.905 0.895 o.865 o.a12 0.745 0.707 Rectangle Trapezoid I Trapezoid II Trape mid III Trapezoid IV Sine wave I 0.93 1.31 o. 707 0.934 1.32 0.707 0.941 1.33 0.101 0.947 1.34 0.954 1.28 o.745 0.957 lo285 o.745 0.965 1.294 o.745 0.972 0.972 1.193 0.945 o.a12 1.303 r;0.905 0.975 ' 1.2 0.812 0.98 1.200 l I k _I 3.25 0.97 0.812 (3 0.905 _ -----~ 0.955 0.982 1.122 0.947 , h Q 1.35 - I 0.707 I 0.992 0.985 0.951 0.905 f> Q.865 0.984 {;°"' 0.905 - 1.22 1.317 0.812 1.138 O.745 I 0.865 1.063 I 2.25 0.973 1.125 0.865 0.978 -·-· 0.973 0.993 1 0.988 0.895 1 ll - 1.056 0.975 0.895 o.95 I 0.905 0.905 0.953 /o 1 o.895 l k 1.128 l (1 0.905 0.865 I I o.905 I I (, 2.00 - r 3.oo 1.058 0 ~ 05 1 l 0.095 I I k -. 2.75 1.375 1.333 0.948 0 : 05 (continued) 1 T - r: - - -- 0.707 0.745 1.23 1 .142 1.06 -..---- T -= 2.50 - I -- ---- Wave fonn of Field flux and EMF TABLE I -- - 0.992 - 1.40 - I 0.707 1.05 1.485 0.707 Sine wave ---L- 1.005 1.35 0.745 1.06 1.425 0.745 Trapezoid IV 0.812 1.007 1.24 0.812 1.047 1.29 0.812. Trapezoid II o.a65 1.015 1.175 1.07 0.865 1 0.905 1.065 J-C l AND 0 -=(3 o.995 I I 1.15 0.905 6' 0.895 ~ 1.75 0.957 1 k 0.895 0.905 7 o.967 0.905 ~ - 1.oa 1 6' -- 0- - o.865 I 0.905 k 1.25 - -- NUMERICAL VALUES OF Trapezoid II Rectangle I ~ r -- ~- L . Wave form of Field flux and E MF TA:BLE I - •· ( o . 895 o . 865 0 . 812 o . 745 Trapezoid I Trapezoid II Trapezoid III Trapezoid IV I 0 . 707 0 . 905 Rectangle Sine wave /3 = 3. 50 1. 30 1 . 272. 1. 19 1. 12 1 . 055 l o . 92 o . 948 0 . 967 o . 968 o.945 0 . 905 J . 1 - "· - 1 . 29 0 0812 0 . 745 1 I o . 865 0 . 895 00905 1. 25 1 . 175 1 . 11 1. 05 l o . 932 0.957 o . 96 o . 94 0 . 905 = 5. 00 _ _ _ ___,__ _ _ _ __ 0 . 913 ,I __ - o . 94 0 . 962 I 1. 183 1 . 261 o . 965 0.94 I 1 . 05 1. 115 0 . 905 4 . 00 I- . ~71--r-/ _J -= (continued) l -...! 0 . 707 o . 745 0 . 812 o . 865 o . 895 0 . 905 l \ --·, • - - (1 - - -h-- . !!: ! ~~ Wave form" of 1 TABLE I _, :. . ' J.' In general, is to be computed from Fourier expansion in trigonometric series corresponding to a wave form considered (see Fig, 9) and k to be determined by means of equation (16 bis) or ( 21) according to the case, with reference to Fig. 9., 6" being equal to k Examining the values of (;"' given in Table I for 1.25 L L 5 and for different wave forms corresponding to Figs. 8 and 9 it is seen that for every practically poss~b1e case O. 905 L f;' L 1 . 05 .................. ( 26) so that, if S- is given an a,verage value 0.95, the e.rror will not exceed 5.5 per cent in any practical case. But in general, it is quite possible to assume for 6""" a value that would be more suitable for a given individual case than the above average value, taking into account the value of and a probable form of the field flux. For instance, in a medium frequency alternator it is quite reasonable to assume the wave form to be close to the rectangular distribution of the field, since the ratio Pr 7 is considerably larger than in high frequency alternators and, taken in itself, may be very great, reaching sometimes the value of about one hundred. In this case the toothed peripheries of both the stator and rotor are supposed to be normal (see Figs. 7 and 10) with ar ...- Pr and a s :::::. ar approximately. 2 In another case represented in Fig. la where a special form of rotor poles is selected it is advisable to consider the flux wave as approaching t he sine flux oscillation curve and accordingly to choose an appropriate value of (;" from Table I or compute it Jt from the foregoing formulas. In most cases it will be possible to determine 6" within 2 per cent of error. As in medium and high frequency alternators only one stator slot is used per effective pole, in general, and only one conductor per slot is to be found in high frequency alternators, in particular, the EMF wave as a function of time may be considered identical with the corresponding field flux wave regarded as a function of peripheral arc length. Therefore, the graph of Fig. 9 can be used to represent both the waves each with its own appropriate scale. In the sets of formulas I and III Eis equal to the RMS value of the fundamental EMF wave or, in other words, denotes an assigned RMS value of a sine EMF which is to be supplied to a resonant circuit. Very often it is, however, desirable and even necessary to have relations between E0 , i.e. that RMS value of a complex EMF wave which is directly measured by a hot wire voltmeter, on the one hand, and electrical and mechanical data of an alternator, on the other. For this purpose the following formula is given E _ V2 E .................. ( 27) o- - where for a sinef E MF complex wave complex wave f :1'2 and for a rectangular f =44- =1. 27, since in this case Eo = Eomax ,E:: ~~ax on the one hand, and Eroax= _!- E0 , on the other; consequently Eo=. q , 'JC For the wave forms intermediate between a rectangular one and a sine curve we have assumed the trapezoids in the Fig. 9. Therefore E 0 i.e. the RMS of EMF of the complex wave is JJ determined by the relation ., (see Fig. 9) I -, tJ On integration we get a'p E - E \ o - omax ..... ( 28) where _L + _g_ - 3 o- 3 Furthermore, we have, talclng into account the second of the equations (28) Emax :; (2' E ~1max -- ti' = g Eomax ,~ \12 E1 : ...........• ( 29) whence it follows Eo : -1f.L : (2 E1 .....................• ( 3 0) {f;1t;;Alii; or putting 0 = f .................. (31) we fall back into the formula (27) Now taking, as intermediate wave forms, the trapezoids represented in Fig. 9, it ls possible to compu~e third equation of the group (28) and pa from the from the expansion of the trapezoids in Fourier series, and thus to determine f from {31) for each of the trapezoida~I, II, III, IV of the figure 9. The following Table II is obtained Table- II Numerical- values of Wave form r Rectangle -{/ 2~ Trapezoid I I 1.33 Trapezoid II 1.375 Trapezoid III 1.4 Trapezoid IV 1.405 Sine wave 1-~ J From equation ( 5) it follows E ~ E1 =~ Bmax(l - ) zlSu lOQS and on account of (20) and (22) it is obtained E = E1 = 6' ( 1 - 1/ )BmaxzlSu 10-8 ............. ( 32) Substituting in (27) for E (or E1 ) the foregoing expression it obtains ( 1 ~ 1/Q )BmaxzlSu 10-8 ............ (33), ~ ~ ~b • and putting E0 : . . . . . . . . . . . . ( 34) , tr/ ( 1 ... 1/ )~axzlSu 10-8 ................ ( 35) In Table III are grouped the values of corresponding to different wave forms of the field flux and EMF as well as to the numerical values of which actually occur in medium and high frequency alternators. The formula (35) will be used later for the verification of the theory. From the equations (23) and (25) Ir Sine Wa ve t II 1 . 06 II Trapezoid I V 1 . 05 /f 1 . 05 ,, Trapezoid I II 1 . 015 -- ,, r o . 992 i - ~ - ~._.-, r, I/ " I /, ,, '"' ... - •• o . 973 ,• 0 . 993 , 1 . 00 7 1 . 013 1 . 01 ~-::. 1 . 00 1 . 005 I 1. 01 ,, 1.0 18 1 1 . 04 ,, Trapezoid II ~ = 1 . 00 - II I/ ·.•, ·- -,, -- 0 . 955 " o . 982 ' I ,, 1 . 00 -~ ,, ,, Ir ,, I ,, 1 . 01 ,, ! - ~--· ·- o . 947 0 . 9 72 0 . 993 1 . 003 1 . 006 7= 1 . 00 ,, 1 . 008 1 r =- 1 . 00 '. I ,, II ,, ,, ,, - I j - l -- -- ·1 o . 983 Ir Q. 944 ,, 0 . 977 1 . 003 1 ,, 1 . 002 I 0 . 999 , II 0 . 997 ,, 0 . 957 ,, " " 1 . 00 1 -- ·- ~ ,, 0 . 92 ,, 0 . 948 " 0 . 975 ,, 0 . 994 II ~-:. 1, 00 j-,, ::::.1 . 00 . 'h :: 1 . 00 l.. I l 0 . 985 I , o . 962 •t -·- :• •---..si ,, 0 . 913 ,, o . 94 ' ,, o.932 •• o . 97 •• o . 99 I }) -.:: 1. 00 ,, o . 998 j ,, o . 997 1, :=. l . OO 3:.°_?_1"": I ~ 3_::.~_ J ..:, 3~0 _:.J: ~· 001 1= 5 . 00 - ,, 0 . 93 o . 941 " 0 . 929 ......... • - - -4--- , , _ o . 965 0 . 988 1 . 00 1. 004 ~= 1 . 00 1~2:-1.:..~I : _'.::00 - _"{ ~ _:__'.25 -~ J: 2 . ~o L 2.751 e Trapezoid II i-= 1 . 00 _ l . 25 - 1 . 025 ~ // h -,-- UUMERICAL VKL UES OF , 'A·~-~------···1·~~~-~-·---·--,r~--~•• --·-·--·-· .. - · - · - - • - · - I Rectangle E MF Wave form of TABLE I II ~ '-->. Jo For the fund.a.mental component of the EM F wave El = E = 6' ( 1 - 1,/. )LW uBmax ~ E1 : E : ...... ( 36) z ( l ... 1,/. {¾~ (siuBmax 10-8 ••••• (37) Furthermore, in view of the equations (27) and (34) we have ......... ( 38) - , c 1 .... 1,1- ) Lw w3max z Eo = ( 1 - 1ft ) '....hl!. '(51 uBmax io-3 ..... (39) (LJ For the complex Eo EM F wave ~'4> The equations ( 35), (38) and (39) establish certain relations between the electrical and mechanical features of an alternator of the inductor _type and the observed EMF at no load. Examination of Table III shows that for high frequency alternators, usually char•cterized by lower values of co?fficient i , the has almost constant value for all practically possible wave forms of · the flux, namely ~ ,::z l. This same result holds for normal medium frequency alternators, since it has been f/tf'p(_ shown that the wave form of the flux in them approaches the rectangular distribution gf ~he fie~e:-. In general, it is easy in any case to select from Table III an appropriate value of • The expression (35) makes it possible to determine experi~ r. mentally the coefficient of flux oscillation Since can be evaluated within a few per cent of error, it is quite permissa~ ble to use the equation (35) solving it with respect to • The solution is found to be r = - V-Bm ______ ax ___z_l_S_u_______ ........• ( 40) BmaxzlSu - 108 Eo All the quantities in the right hand side of the last equation are known for a given alternator: some of them are capable of being computed, for instance Bmaxi some constitute structural features of the machine, as z, 1 and S, and others are observed, as for example E0 • It is noteworthy that a definite wave form of the field flux is connected with a definite upper limiting value of • This may be seen from the equation (20) and (19) aw well as from (16) and (16 bis). In fact, the lowest possible value of is zero, since in inductor alternators the field flux is unidirectional at any given point on the stator periphery, Ol being equal to Bmin; Bmax therefore it follows from the above mentioned equations that the . ~x limiting value of is to be determined from the equation k (1 ~ l f ) : 1 ............... (41) Referring to the equations (16) and (16 bis) it is seen that. 1 1 - 14 1 + o. 2as = 3T 2 ...... ( 42) c1 - 11 > 1 for the sine wave. Solving (42) for l it is found that Qmax = 4.55 for the sine flux oscillation wave ••• (43) By the same method it will be found that for the trapezoid IV tt It " It l max = 5 III f max • 7 II r I o max = 23 max = 11 But these upper limiting values of .................. ( 44) (See Fig. 9) T depending upon the wave form of the field flux may, only in exceptional cases, interfere with the design of an inductor alternator, since usually 1 is less than the lowest of those permissible upper limits of r~ In those exceptional cases care must be taken of an appropriate form of the field wave so that it should be compatible with the value of Q . As seen from (43) and (44), the closer the wave form of the field flux is to the rectangle, the higher is the upper limiting value of Q. ~he foregoing theory has revealed the extraordinary importance of the coefficient of flux oscillation T . It must be a keystone of any method of designing the inductor alternator. In one other respect 1 plays an important role, namely: in determing the ratio of the effective field d.c. amperewturns to the reactive a.c. ampere-turns of an armature per pole. This ratio is exceedingly important in the operation of a.c. generators and synchronous motors, since it partly determines the relative reaction of the armature. The determination of this ratio in inductor alternators is complicated by the fact that only a part of the field d.c. ampereturns must be taken into account, which part may be termed as effective field ampere-turns. but only to Bmax - Bmin. These correspond not to Bmax Furthermore, in view of (20) we have 2 Bmax 2 Bmin :: Bmax & 1 .: o<. ) = ~ax k ( 1 - 1/l ) . ...... ( 45) The effective field ampere-turns are defined and determined as such field ampere-turns that are capable of producing the field whose magnetic induction or flux density is equal to B~ax k(l-~Q) . Even under this restriction the ratio of the field ampere' turns in inductor medium and high frequency generators to the reactive a.c. ampere-turns per pole is much . greater than in ordinary low frequency alternators. This is accounted for by the fact that in the former the a,c. ampere-turns per pole are unusually small in comparison with the field ampere-turns. That property of medium and high frequency inductor alternators has already been used in determining the resonant inductance; it is due to the fact that there is only one stator slot per effective pole and the cross-sectioral area of a slot is small. The suggestions at the end of Section I about how to use the fundamental equations apply also to those of Section II with the only modification that we have to deal with the coefficients A or instead of The results of Sections I and II are independent of the met hod by which is determined. SECTION III Predetermination of the Coefficient of Flux Oscillation T . Experimental Formulas for i , Their Rational Basis. The saturation curve of inductor alternators shows quite peculiar features which are different from those observed on ordinary a.c. generators of heteropolar type. As the excitation field increases, the EMF at no load increases up to a certain limit beyond which a further increase in the field brings about a decrease in the EMF at no load. The importance of that fact is seen at once. Furthermore, it has been shown by several research workers, for instance by Latour, Osnoe, Bethenod,..a.nd othei?& that the point at which an inductor alternator operates with maximum efficiency corresponds to the greatest EMF obtainable at no load. In making a design for an inductor alternator it is of advantage to assume that its normal operation conditions correspond to those in the neighborhood of the maximum EMF at no load; in other words, the normal excitation field must be close to that at which the EMF reaches its maximum at no load. The maximum value of the EMF at no load is determined by the maximum of the product (1 w )8max as it is seen from the expressions (3~) , (37) and (39). As 8max is increased, decreases because of the increase in the reluctance of the magnetic path lying within the pole and stator tooth. The product (1 w Jl')Bmax is de pendent upon two factors, one of which, namely (1 - l ) decreases as another is incr eased, with the result that finally Bmax reaches the value beyond which tµe product (1 w lA)B begins to decrease with a further increase / I max in Bma.x • Thus the expressions (35J, (37) and (39) show in the most evident manner the characteristic property of the saturation curve of the inductor alternator. If it is known how r varies with Bma.x 1 it is possible to predetermine the saturation curve, in general, and both Em.ax at which the maximum EMF at no load occurs and the value of that maximum itself, in particular. min which is the flux passing through the rotor slot (interpolar space) is in parallel with max• 1::::e:. the flux through the pole face linked with an a.c. coil; therefore, if the reluctance along paths within the pole and stator tooth increases as ~a.x assumes higher values, the ratio is decreased. Now the problem arises - max min as to how to determine as a function of Bmax• is superimposed upon another one; ma.mely, how This problem should depend upon the ratios: _ , ar, ~, (See Notation and Figs. 10 and 11). Pr Pr Ps The whole problem of determining is very complicated and has not yet been solved in a satisfactory manner. There exist no general analytical solutions of the problem. The well known analytical solutions given by F. W. Carter1 3, 14 , relate to the particular cases; namely, (1) to the case when the reluctance of a pole may be considered as equal to zero and generally negligible, and (2) to the case of highly saturated poles or teeth, which case occurs in electrical machines of usual commercial type. The same remark applies also to a rece nt pa~er by , Coe an~4 But neither of those cases occurs in medium and high fre-Taylor quency inductor alternators; the reluctance of poles and rotor teeth does not play a negligible part in determining the coefficient of flux oscillation Q , nor do the magnetic induction densittes in them, in the working part of the saturation curve, reach its rectilinear saturated part. detail later. This point will be considered in Furthermore, the mentioned particular solutions lose their accuraqy as soon as one has to deal with toothed peripheries of both a rotor and stator, ·and that is precisely the case of inductor alternators. The gt'aphical methods of solution devised by Lehmannl5 are painstaking, require much time and labor and represent, in a geometrical form, the application of the method of successive approximations. loses That method becomes even more cumbersome and lts accuracy when the reluctance of the teeth and poles is to be taken into account as is the case with inductor alternators. 1 a,nd,f~1f11 5 Although Lehmann '/\ method has already been used by Messrs Roth, Metz~. St o renea ia: and Wieseman in application to inductor alternators, other methods more handy and convenient are wanting in determining the value of • The method which has been adopted in the present paper by the author embodies the working out of an empirical formula tor on a rational basis connected with a special and more definite view taken of the saturation curve of inductor alternators and of the causes that make it differentiate from the usual type of saturation curve. The numerical results obtained on the basis of the outlined method are checked against the experimental data observed on a number of inductor alternators of various capacities, frequencies, tooth arrangements, as well as of different make and application. The results of the verification are ~ satisfactory. Attention has been directed to the known peculiar property of the saturation curve of the ind~ctor alternator, namely: to the existence of a maximum EMF at no load for a definite value of the field current so that a further increase in the current is connected with the decrease in the EMF. A further step has been made in the present paper by relating the maximum EMF obtainable at no load to the values of flux density in the poles and stator teeth corresponding to the beginning of the knee in the magnetization curve of the material of which the said poles and rotor teeth are built. In other words, since the bending down of the saturation curve of the inductor alternator is rather sharp at the beginning and then tends to assume an assymptotic form, it is clear that this bending down and consequently the maximum EMF which belongs to that part of the saturation curve can be connected only with the knee in the magnetization curve of the corresponding material. The examination of magnetization curves has shown that in general the beginning of the knee corresponds to flux densities from 8000 to 9000 for steel castings and to 10000-11500 for laminations, steel forgings and solid iron. From the moment that those flux densities have been reached in the poles or stator teeth, the EMF must be close to the maximum voltage obtainable at no load. This is the basis of the method for the determination of the maximum EMF and the value of Bmax (in the air gap) corresponding to it. The outlined method has been applied by the author with ~ satisfactory results and will be developed further in detail. Since that part of the saturation curve which extends beyond the maximum EMF has but a little practical importance, if any, the problem is confined to the determination of O only in the region of the magnetization curve not extending beyond the · beginning of the knee in the poles and stator teeth. The successive steps by which the formula for f has been worked out and verified can be represented as follows: L The formula for was first established by the author on the basis of general considerations t~gether with experimental and constructional data relating to the following inductor alternators: No. 1. 1000 cycle inductor homopolar alternator investigated by the author at the laboratory of The Siemens-Halske Company of Petrograd. No. 2. 500 cycle inductor homopolar alternator designed by the author and built by The Siemens-Halske Company of Petrograd. Nd. 3. 10000 cycle inductor homopolar alternator described by B. G. Lamme16 . II. The formula obtained has been verified by the experimental results observed on the following inductor alternators: No. 4. 12.Q_cycle No. 5. 8000 cycle inductor homopolar alternator described by Karl Schmidt 18 . No. 6. 1 6500 cycle inductor homopolar alternator } Both described 9 by Laffoon in the It tt tt Electric Journal. cycle 30000 No. 7. No. 8. inductor homopolar alternator described by Karl Metzler17 . - ' 20000 cycle inductor homopolar alternator described by V. p. Vologdin20 . No. 9. 500 cycle inductor homopolar alternator described by K. Schmidt 21. ~ No. 10. 500 cycle inductor homopolar alternator described by K. Schmidt 22 . This list of alternators will be referred to in the next section. III. Besides the experimental information contained in the above papers and articles, some supplementary data relating to constructioaal features and observed magnitudes as the pole and tooth form, number of parallel circuits in the armature winding, maximum EMF observed at no load etc., in the listed alternators have been kindly communicated to the author, in reply to his letters, by Messrs. Bethenod of Paris, Calvert of Westinghouse Electric and Manufacturing Company, Schmidt of C. Lorenz A. G., Berlin, Alexanderson and Wieseman of General Electric Company, Vologdin of Leningrad. IV. In some particular limiting cases it has proved possible to - - -- -- - - - - -- get from the empirical formula for the same results as those obtainable by means of Carter's analytical method as well as by tUu{_ t-, 1"hJ Lehman ' graphical solution of the problem in the form given by R. Wieseman 23. It has been possible to determine the optimum ratiJ {tth from the empirical formula worked out for in the. present paper and by the graphical method according to Wieseman, and in both cases the results have proved to be sufficiently close to one another. This is a l j ,o regarded as a substantiation of the empirical formula for • Furthermore, the optimum ratio ar and the optimum pole Pr width (opt. ar) determined from that formul a . are in good agreement with the respective magnitudes in the alternators referred to. Likewise some examples of the optimum ratio ar given by Karl Schmidt 22 are in satisfactory agreement wlthpthe results obtained on the basis of the said formula for The following ls a set of expressions for general and particular form. • given in They satisfy the requirements imposed upon them by the foregoing paragraphs I, II, III and IV. The general formula can be presented as follows: + (See Notation and Figs. 10 and 11); ar, Pr, as, ps' are measured in millimeters when introduced in (46); C and q may, for a certain type of generator, be considered as constant and taken, for a cylinder rotor type of the alternator which is most widely used (see Figs. 1, la, 2, 2a), C :::=. 10-7 q - 2 ··················•• - •·'47) These values hold for the part of saturation curve, not exceeding the maximum voltage at no load. Furthermore, C must have appro- priate dimensions to satisfy the requirement of dimensional homogeneity. The coefficients A and y are slightly dependent upon the d , but within certain domains for . L may be regarded Pr Pr as constant; Bznax represents the maximum flux density at no load ratio in the air gap along the central line of a pole at the instant when the said line coincides with the central line of a stator . tooth. ., With reference to the preceding paragraph I, of this section, the general expression (46) is found to assume the following particular forms. Inductor alternator class a. high frequency, 10000 cycles and up: d > 0.0794 Pr = -Si"',¼_ + -J -'z,.. / OtJ --------- ./5: 11,£ 10 -7 This expression is obtained from the general one (46) by putting A -= 10 and y- 3. Inductor alternator class b, medium frequency. 8000 cycles and down: - /+ , A :::: 12 for - J1 -2. 0.01 J' Pr A = 11 for 0.01 A = lOfor L Pr - - ~ 0.025 Pr / 0.025 The particular expression (49) is obtained from the general one (46) by putting y - 1.5 Both the expressions (48) and .(49) have been worked out on the basis of certain general considerations and the experimental results obtained on the alternators Nos. 1, 2, and 3, and later they have been verified using the structural data and experimental tests of the alternators Nos. 4, 5~ 6, 7, 8, 9, 10, (see paragraph II of this Sect1on)with the supplementary information contained in the correspondence referred to in the paragraph III of this Section. Furthermore, it is found that it is advisable, although not cf L necessary, to subdivide the class b ( Pr frequency) into two sub-classes: for 0.01 b1 0.49, medium ~ L- 0.49 and ~~~ T is recommended to be used,L~ Inductor alternator sub-class b 2 , LPr b 2A· The following expression for L - o. 01 . which expression is obtained from the general one (46) by putting y: 1 and A= 12. The expression (49) holds for sub-class bf For all the alternators whose data and tests have been used for ~orking out or verifying the expressions ( see paragraph II) we have either {--> O. 0794 ; and y = 1.5 respectively. ( 48), ( 49) with y : 3 ~ Therefore it is only natural to assume for the intermediate class of alternators with 0.049 ·. , < 0.0794 that y = 2"\1 ·5 = 2.25, the average of the above values of y for the two classes of alternators. Thus the third class of alter- nators, c, arises whose characteristic feature is that 0.049 d Pr ( 0.0794 and y = 2.25. it is obtained for 0.049 Sr ( 0.0794 Inductor alternator, class c; f == I + Therefore putting in (46) y.:::2.25, 0.049 0.0794 It is seen that in the general expression for (46) y is a function of fr such that it·, increases a.s the ratio ' increases; but it is found that within wide limits y can be considered as a constant so that three values of y are sufficient to cover practically all cases, while the range of frequencies extends from 450 to 30000 cycles inclusive and that of the charac teristic ratio _sf_ varies from 0.01 to 0.113. Although it is Pr quite possible to come out with three values of y for the classes of alternators designated a, b, c respectively in order to practically meet all requirements, it is advisable to assign a special L- 0.01. but t his is not Pr necessary as the experimental verification of the formulas for value of y- 1 for very small ratios will sh ow The expressions for ing the optimum ratio pitch. ;~ af~ord the possibility of determinof the width of a pole to the pole Taking the expression for in its general form (46) and differentiating it with respect to the ratio~, it is found Pr to be a ..... . .. ( 5 2) Optimum_!:_ a y¾l Pr It Therefore for the class a ~ = 0.333 Pr 11 . . .... (53) b " " " " ~ = o.4165 Pr tt tt It ar = 0.361 C " " Pr ar- , reaches its maximum and For these values o f Pr therefore EMF will have its highest value. The values obtained for optimum _!:r_ from the expressions Pr for are in satisfactory agreement with the experimental results, on the one hand, and those obtained by Wieseman 23 by means of graphical methods, on the other; this is regarded as a strong evidence in favor of the expressions for Q this article. presented in . ' Furthermore, it is possible to predetermine from the same formulas the changes in EMF at no load with the variations of the ratio~- In one particular case it has been found possible Pr to compare the results obtained by the methods presented in this paper with those arrived at by means of a graphical method used by Wieseman in the quoted article, and both the results have proved to be close to one another. This is also regarded as one of the substations of the formulas and expressions for l • Normal form of the inductor alternators If _ar = 0.5 and~ is very close to l(see Figs. 10 and Pr Pe 11), the inductor alternator is, in the present article, called and referred to as normal. In this particular case the general 2a a express ion for 46) in which the ratios ~ and Ps are to be Pr s put equal to 1 will assume the form as follows T( T 12 for _J_ f: 0.01 Pr L 0.025 A = 11 for 0.01 . ( L Pr _ 7 0.025 A - 10 for _£_ Pr If the case is taken in which Bmax is very small so that where as before A w - it can be co~sidered negligible, i.e. Bmax:::::: o, then the foregoing expression for Q (54) is reduced to the following one .., - I with the same va+ue of A as in (54) or in (49). This is the case where can be determined using Carter's analytical method as presented in his article 13. a a When __r_: 0.5, or, in other words, ar =brand __e_ = 1 Pr Pa and the val.ue of the flux density B is not taken into consideration, which amounts to the foregoing condition that in the expression for T (54) Bznax is taken equal to zero, then it follows from Carter's analytical theory of the distribution of the magnetic flux in the air-gap and slot that I /-o 2 .L tan -l___,£r _ 1. 466 _g_log ( 1 + 41 Pr ) ..... ( 57) 90 fJ' Pr 111'-In Notation of the present article(Figs. 10 and 11), the angle be- where C" - o- ing expressed in degrees and the logarithm being a common one (Carter's paper, p.885). It is obvious that in the expression for O (56) the meaning of the symbol is different from that in the Table I and in the set III of expressions {23), {24), (25). Although the application of Carter's analytical method to inductor alternators is very restricted being confined to the considered case (as=Ps;ar:br- ~r, Bmax 0), and the case itself has but a little practical importance, its methodological significance lies in the possibility of chec.king the expression for limiting case of Bmax ,;::::: O. The result of this verification is very satisfactory, since the values of formula for T in the T computed from the author's 1 (55) are very close to those obtained on the basis of Carter analytical theory from the expressions (56) an1 ( 57), for different values of the cha:rP.cteristic ratio LPr The numerical results will be given in t~e next section where the verification of the author's method will be presented. It might be stated that Carter's analytical methods have been devised for generators of commercial type, and whatever conclusions may be reached concerning their extension to inductor alternators, they will not lessen the importance of Carter's theory for commercial generators and motors. There are other limiting cases in which the results obtained from the general expression for l (46) coincide with those ob- tained theoretically with indisputable precision. It is clear that for great values of either J' or Bmax, T must assume the value equal to 1, since for great values of the air-gap cf there must be f1J max = ,9' min' and to great values of Bmax in the air-gap cf between a pole and armature core correspond hie;h flux densities in the poles and in the teeth of a stator, which causes very high reluctance in the corresponding parts of the magnetic circuit of an inductor alternator, thus obliterating dlfi'erencee between <J max and ¢ min and ma,C).ng O equal to 1. The same conclusions are drawn from the general ex~ression for f ( 46) 1 putting either cf' or &nax or both equal to very _great values. Furthermore, in justifyine the introduction of the [<y -t 1) -y 2ar__ Pr 2 ar into the right-l1and side of the Pr ,a expression for (46) it is saen that for __r_ - 0.5 the rightPr hand side assumes the form of the expression for (54) as it fact or 1 must be for the normal type of the inductor alternator according to the foregoing co~siderations. 2ar Pr= 0, i.e. for ar Moreover O is reduced to 1 for = 0, as it must also happen, since this would mean that the rotor periphery is smooth and J' is great, the shrinkin5 of the poles making the rotor periphery smooth and the air gap J great. The reader is referred to Figs. 10 and 11. Besides satisfying the limiting conditions, the above factor accounts for the maximum value of of the ratio - ar Pr O being confined to the region = . . (See (52) and (53) ). _ ar- L O 5 such that O ,/ Pr That this is actually the case is clear from the considerations as follows: ar =- 0.5pr Suppose that starting from the width of a pole is decreased; then, although. both )2>max and f min are decreased in their absolute values, the rate of the decrease is different, being the greater for min on account of the greater increase in ~he slot reluctanc ~e~ J;)1~4J to (.IL ~ in the neighborhood of ar = O.Spr . Sine , ror s.z. = O, and is less than for ar in the region ~~V/; equals 1 = O.Spr, there exists a maximum for a o L.. :X L o.5p . Pr r It is only natural to assume a simple form for the factor accounting for the above changes in with the variations of the ratio~ and further to adapt it to experiPr mental results arising from the tests of alternators of the type under consideration. It has been found by the author to be possible to solve this problem by means of the factor 2s.z. 2ar ........... ( 58) Y+ l - y - Pr Pr (46) in the general expression for T The same or analogous considerations apply to the second factor in the expression for y O , namely to 1- ~ ~ ........... '59) y Pa Pa 2a a where the ratio .:::a plays the same role as the ratio Pr in Pa Th/4 i1J ZetAJ r the first factor (58). When as ahl'i~o t.o no:tR.--¼ftg so that in the limit as - o, it amounts to say1ns tn:e:t there is no iron core within the coils of the armature winding, and consequently the air•gap J assumes a great value inconsistent with the varia• tions of the magnetic flux, with the final result that f = 1. The same conclusion is obtained from the general expression for (46 ) , putting as ::::::: O, on the basis of the factor (59). O On the other hand, if as -== p 8 the case will be that of the normal type of the inductor alt~rnator, and at the same time, " in agreement with this result the factor (59) is reduced to 1. It is found unnecessary to make the general a expression for Q ( 46) fit the cases where P~-7 o. 5, since in practice such cases could hardly occur for the simple reason that the value of steadily decreases to 1, as ar is increased Pr In this case min will increase, while max 1J will remain practically invariable, with the result that O . . from 0.5 to 1. which is equal to ¢ m~x ~ will decrease toward 1. Since this -ymTn a would be an obvious disadvantage, the arrangement with -2:___7 0.5 is not commendable. Pr In discussing these questions the reader is referred to Figs. 10 and 11. Now it remains to derive some expressions for e. the <\termination of the flux densities Br and Bs in poles and stator teeth respectively. It has already been pointed out that the problem of determining the maximum EMF at no load has bee n connect ed with f inding those · flux densities in the poles and stator teeth wfiich correspond to the beginning of the knee in the magnetization curve of the material of which the corresponding part of an alternator has been made. Now consider, with reference to Figs. 8 and 9 and equation(l4) , , th e expression ~ max + y)min ::. Bmax~ Bminprlr •• (60) Since the radia l length of a pole is, in general, such that only an insignificant part of the slot flux f min reaches the bottom of a rotor sl ot, practically the entire flux 9>min enters a pole at its lateral sides, and the expression (60) represents a . total flux within a pole in the neighborhood of its root, max being a flux at the top of a pole. Then, denoting the width of a pole near its root by a'r, and putting the 2a' ratio _ _r ::= t, it is found to be Pr 1 9) max+ f 'm1n Since 1 - = Bmax 2 Bmin Prl = ~ax _2_B_m___a___x _2a_,~_r_1 ..... (61) a' rl 0 min _ Bmax 1 + Bmin ) t Bmax = k 1 ... _L ) and Br -- (/?max Bm,1n on account of {20) and Notation, it follf ws Br _ Bmax 2 - k 1 ... i_) where the nume~ical value of k ............... {6 2) 1s to be taken from Table I or computed by means of the expression (21). The expression (62) holds for solid poles; in the case of laminated rotor teeth it is necessary to introduce in the formula {62) the iron space factor r ; then for laminated poles B _ r - Bmax [ 2 - k ( 1 .. _L \ yt \ l J ................ ( 63) Expressions {62) and (63) give the possibility of determining B under various conditions connected with the maximum flux r density Bmax in the air-gap. For finding the expression for Bs the following derivation is available with respect to the trapezoid wave form of the field. (17 bis} From Fig. 9 it is found to be on account of equation B ) Jiap + a'p min 2 O (/} ( 1 - _L ) = B (/) i : o<.1 r max 1 max= Bmax max 1r p -r a' p) (1' - o(.) 2 1 a P + a' P) 2 Now denoting wit! a' 8 th~ f a stator tooth -a t the level of a stator sl--e-t. so that a' ~ theAwidth of a stator slot and p 8 p ... b, where b is s s the stator slot pitch (See Figs. 10 and 11), the expression for Bs becomes ¢ max _ Bmax B 6 - r a 1 8 1 - -r a. p + a' n l • ol 1 - 1. 2a - S •........... ( 64) r; Furthermore, using the equationT (19) where 1 - ).. = 1 --1.... as in Fig. 9, it is seen tllat l or a and remembering tnat av = trapezoidal wave B _ k Pa Bmax ................... (65) s -ra ·s k'::: -=---....,..,,.---::-""T""""---,er-----.,,---~ f .... ( 66) ap 0.5 ) 1 + 1 ... ap + a P where f) ' It is seen that the coefficient k' depends upon the wave shape of the trapezoidal field flux, namely upon the ratio a.P ap + a 'p (Fig. 9) For the rectangular wave form of the field k' == 1, since ap = a' p' and the expression ( 65) becomes simply (rectangular we.ve)' Ba -- 8max Pa -::-, a s 'f .................. (67) For sine flux oscillation on account of (14 bis) there is (/> max (/ max </J max </ min - _g_ 8ma.x :,r ( 1 ... .-L)a -2.ff Bmax 2 1 !ff 1 _ -L.) Bmax p s 1 T Pr l 8min) ~ ( See Fig. 8) Pr 1 1 - ~ ~ where Pr _ p 8 2 - and using the equation (17) where A-=- _L Bs - ------------k-:-,Bmax 2 ·, P s y ff a 6 ( Sine wave field flux oscillat1on)B 6 = k' r where k' _ - 2 n Pa B a' 6 max it obtains }J •••.•...........• (68 ) k, and the numerical values for k are to be taken from Table I or computed from the formula (16 bis). It is clear that the expressions (65), (66), (67), (68) can be applied to the determination of the flux density at the upper part of a pole as distinguished from the flux density at its root, with the only difference that the value -of the width of a pole at its top or upper part must be substituted for a's• namely:~ should be used instead of a' 8 • The predetermination of the saturation curve by means of the equations (35), or (38), or (39) on the one hand, and the expressions for O ( 46), or ( 48), or ( 49), or (50), or (51), on the other, must be accompanied by the computation of the flux densities Br and Bs in both rotor and stator teeth, for which purpose the method presented in this section provides a set of expressions and formulas. When Br or Bs assumes the critical value corresponding to the beginning of the knee in the magnetization curve of the material which a corresponding tooth is made of, it means that E 0 is close to its maximum value obtainable at no load. Usually the critical values of Br and Bs occur almost simultaneously hand in hand; but it is not always so due to some particular arrangement of a toothed periphery on either the rotor or stator or on both. .' For instance, in the case of a tooth arrangement shown in Fig. la, it is clear that the magnetization phenomena occurring in the poles will be of much more importa nce than those in the stator teeth, since the latter become decidedly narrowed only in one crosswsectional plane, and their radial length is considerably less than that of the poles, so that with the circular slot or semi-circular slot arrangement on one of the two elements of an inductor alternator the critical effect of the flux density is to occur first in the teeth of the other one. In this case it would be quite justifiable to confine consideration to the alternator element (rotor or stator) other than that in which slots are circular or semi-circular. Sometimes the edges of a pole at the periphery, i.e. at the air-gap, are rounded as shown in Fig. 12; in this case the width of a pole ar at the top is considered in the cross-sectional plane where the narrowing of a pale due to the rounding of the edges begins, the latter being taken care of by assuming a special wave form of the field flux (sine - like or close to it). Section IV Verification of the theory and method presented in the foregoing . sections. ., The following criteria are used for t he verification of the theory and met hod presented in this article. 1. E0 saturation curve. 2. Maximum voltage obtainable at no load. 3. 4. Optimum ratio Pr ~ and optimum pole width ( opt. ar). Particular limiting cases including that of the normal Pr inductor alternator ( ar = , as~ Ps ) for small values 2 of Bmax· (Bmax-:::::::- o) • I. E Saturation curve Table IV Alternators numbered as in Section III I Voltage Observed Computed- - j t 116.5 81.3 Field Current i or Field ampere-turns ni 82 114.5 180 1 = 0.17 ampere i = O. 25 ampere i = 0.44 ampere 207.5 170 207 i = 0.36 amilere i - 0.48 No. 3; f = 10000 ,- 160.5 401 167 400 i = 0.64 i = l.8 No. 4; f = 450 88 ni = 1191 ni = 2477 n1 = 3326 ni =4000 No : 1; f - = 1000 185.5 No. 2; f = 500 174 83.3 162 191 198 l 165.5 190 195 "tt It is seen that the computed results are in, good agreement with the experimental ones. -61- 2. Maximum voltage ebtainable at no load and corresponding critical values of Bs and~, the flux densities in stator and rotor teeth reapectivelL Table V Computed Observed lo- Alternators numbared as in Sec.III Maximum I Bs volta~e formula ( 6r") I formu1,.13f &:t) No. 4; f = 450 191 198 10070 10530 398.5 Circular stator slots No. 5; f = 8000 Maximum Voltaf2e 8380 8980 125 10250 Rotor consider..., ed as steel forging 400 ·- 8950 fil Rotor consider-cd as steel casting I No. 6; f :: 6500 m I 10000 No. 7; rating f = 30000; operated at f - 13200 11 I 11000 • 7570 192.2 No . 8 ; f = 20000 182 6700 I Mo. 9; f = 500 2900 ! 11350 I 6300 250 7300 ~ 10000 Rotor considered as steel forging About 190 8700 Rotor considered steel I casting Semicircular rotor slots 2Q75 The available data relating to the alternators Nos. 5 and 8 do not specify the material of which their rotors are made. Therefore two possibilities are considered, and the results in both cases are substantially in good agreement with experiment. - -02- In accordance with the method adopted the critical values of B8 and Br determ1ning ~those of voltage at no load iri the neighborhood of its maximum lie within the region 10000 w 11350 for laminations and steel forgings, with a suitable reduction for steel 01. #JU~ castings; and besides that, in special cases of circularAslota 1n one of the alternator elements , rotor or stator, only flux densities in another are considered. In making the forePoing computations, average magnetic properties of materials have been considered. It is clear that the knowledge of individual magnetization curves will serve better the purpose of the method presented in this article. The reader is, for further discussion, referred to Appendix which contains detailed computations. 3. ar Optimum ralio of the width of a pole to the polepitch -. Pr a. According to Schmidt's experiment on an inductor alternator No. 10 with d= 1 mm, Pr -== 26 mm and characteristic ratio 6 -= 0.0385, the optimum w.idth of a pole ar = 10 mm. According to the presentelmethod (opt. ar) =- 10.8 mm, in the .£. -::::. ~ same alternator . . b. According to Schmidt, for J = 2 mm and Pr = 26 mm the optimum width of a pole is expected to bear ==- 8.5 mm According to the author's method (opt. ar) = 8.87 mm It 1s seen that t~e optimum width of a pole 1n both cases is in satisfactory agreement with the results ob- tained on the basis of the present theory. c. It is found to be in alternator No. 3 (opt. ar) = ~.33 mm ., computed; It ti No. 5 (opt. ar) = 5.05 mm 0 t1 " No. 6 (opt. ar) -::::: 10. 78 mm " ti ti No. 7 (opt. ar} :::: 2.37 mm It It No. 9 (opt. ar) = 32.6 mm " " ar-----3 .175 mm actual ar==4 • 7 mm. It . ar=l0.15 mm " . ar - 2.8 mm " ) . ar 30 mm " ) This coincidence shows that the alternators have been designed to r~alize the· optimum pole width and is regarded as an evidence in favor of the present method. d. Optimum ratio of the width of a pole to the pole pitch ar 1'r according to the present method, on the one hand, and Wieseman re• sults obtained by graphical method, on the other, are put together in Table Vl. Table VI results Author's results > 0.0794 (opt. ar } = .Q..!.222 Pr £. L=- o.049 (opt. ar} == .Q..,_4165 Pr (opt. ~)=: 0.38 - 0.49 Pr average =.9.&2 for o.049 ~ ( 0.0794 r (opt. ar) Pr (opt. ~) =0.355 - 0.38 for f_ Pr (opt. ar} Pr for d Pr for - Pr L = - 0.01 ~61 (opt. ~)=0.34 • 0.355 Pr average = 0. m-5 p averagf; = 0,3675 - ~ a (opt . ...J:}=0 .49 - 0.5 Pr average = O. 495 Furthermore, •from Wieseman' s findinP:s ( by graphical method) it results that for cf Pr E _2_ -= 1.083, E1 = 0.05 while by the aut hor' a met hod, for the same value of the ratio __L= 17..bSPr E _2_ :::::. 1. 09 , El ·, ~E - 0.965 E1 - where E 1 is E M F corresponding to ar -=0.5, E2 to ar -= o.4 and Pr Pr ~ to ~ = 0.20 Pr This shows ~uito a satisfactory agreement between the results of the two methods. 4. Limiting cases. Normal type of inductor alternator(~= Pr, a 8 ~ 2, p8 ) and Bmax -:::::::: 0 Table VII Numerical values of for different values of Pr J Pr from Computed aj the formula ( 56 ) obtained on the basis of Carter's theory. from Computed t he formula 55 )obtained by the author's method e{ • ---- 0.125 1. 785 1.8 0.083 2.22 2.2 o. o625 2.595 2.6 0.05 2.98 3.0 0.04165 3.33 3.4 0.03125 4.035 4.2 0.025 4.65 4.635 0.01 9.32 9.32 ---- --- This table shows that in the limiting case considered the author's formula gives practically the same ' results as Carter's analytical method, but the latter, as it has already been pointed out, is not a pplicable to inductor alternators in actual cases. The other limitw lng cases have been considered in Section III and found to be in favor of the formula ( 46 . )presented by the author. Appendix In order to illustrate in detail the a pplication of the .' method presented in this paper the following examples are given. Inductor alternator No. 4 ~requency} The structural and experimental data relating to this inductor alternator are all the more interesting as they make it ' possible to use several criteria for the verification of the method presented in this paper. Predetermination of the saturation durve. Structural and experimental data are as follows : f = 450; J'= 2.25 mm; Pr = 62.8 mm; B.r = br 31.4 mm; as - 27.4 mm; Ps-=- 31.4 mm; 7 cm; u - 2825 cm p s; 5 = 2 x 20; z = 5; a' 23 mm; tJ s Pr 0.0359; ar as - -== 0.5; - - 0.873. This is the "normal" type in ·v1ew of the values Pr Ps o f ~ and aps· As a field flux wave form the trapezoid I is to be Pr s a assumed since the value of the ratio 3 in the alternator considered Ps a' .,.. is somewhat in excess of the value of ap for the trapezoid I ( er j) a a' P rJ ( p; ==0.873, ~ =- 0.833 ). The r efore in view of the values of O 1 - found later it follows fr om Ta.bles I and III th.at k = 1.055 and ~ ==-1. In view of the characteristic ratio ffr which is equal to 0.0359 the expression for ( ~ 9 ), with A = 10, must be applied to the alternator considered. Now using the expressions ( 6f" ) and ( 6 ) for B8 and Br, it is also possible, while predetermining the saturation curve, to find out the above flux densities and compare them with those determined by K. Metzler who applied a graphical method. -6 6 - The values of ~ax• the maximum flux densities in the air gap, have been computed by the . same writer from the saturation curve and data of the magnetic circuit and will be used in the following computation to predetermine the saturation curve, as well as B8 and Br according to the theory presented in this article. Point 1. Bmax- 2860-J == l + 62.8 x (2.5 '."' 1.5 x 0.873) x 0.873 = 3.8 22 s • -l- I 286 o2 107 1 = 1 ; consequently from the expression( 1 - _l_ =- 0. 737 ; Eo :: o. 737 x 2860 x 2825 x 5 x 2 x 20 x 7 x 10 against 88 ...3 = 35 ) 83.3 volts observed, according to the table published in p.70 and the saturation curve represented in p.58 of the article by K. Metzler. The use of the expression ( 56 ) based on Carter's theory gives E0 -::::::: 93.5 volts. Point 2. Bma,x==-5720 ; T ==- 3.535 1 ... r -==- 0.717 Eo - 0. 717 x 5720 x 2825 x 5 x 2 x 20 x 7 x 10·8 = against 165.5 observed. 162 volts The use of the expression ( 56 ) gives E0 = 186.5 volts. (On the basis of Carter ' s theory). Point 3. E 0 Bmax=-6850 > T = 3.4 ; 1 - f - 0.7o6 0.7o6 X 6850 X 2825 X 5 X 2 X 20 X 7 X 10 ...S = against 190 observed. E0 - 222.5 volts. 191 volts TUe use of the expression ( 56 ) gives (On the basis of Carter ' s theory ). -67- Point 4. Bmax = a=- 3.37 7150 • E 0 1 - T 1 = 0.703 =- 198 ~o l ts against 195 observed. The use of the expression ( b ) gives E 0 = 332 volts. It is seen that the saturation curve determined by the method of the present paper is very close to that actually observed. This is not, however, the case when the expression ( ~6 ) based on Carter's analytical theory is used. The next step will show that the flux densities B8 com- t.qJW~ c.-P!( t6J puted from tlieA( 65 \ are very close to those obtained by K. Metzler by means of a graphical method. Predetermination of the flux densities Bs and.J2r in the stator and rotor teeth respectively for the foregoi!lB..points of the saturation curv~ Since in the case considered the field flux wave form, as has already been shown, is close to the trapezoid I, it is easy to 2ap . - 24 see that ap + ap - 22 (Fig. 9) and k :::::; l.055 for the foregoing values of T (from 3.37 to 3.8); then the computation gives the results as follows: Point 1. Br = 2860 (2 - 0.737 x 1.055) = 3550; Y, = O. 9 for medium frequency alternators Bs-- 31.4 - -123 0.9 2860 . 1 + o.s x o.737 x(24 _ 1) 22 again st 4300 as found by Metzler. Point 2. _ 4200 Br = 7120 ·> Bs = 8400 against 8600 as determined by Metzler. ~oint ). Br = 8}80; B6 = 10070 against 10300 according to Metzler. -68- Point 4, Br -= 8980; B - 10530. a ~ There is a close agreement between the results obtained by either method. Predetermination of the maximum voltage obtained at no load. According to the view expressed in Section III the maximum volta ge at no load corresponds to flux densities of 8000 - 9000 in steel castings and 10000 - 11500 in laminations, solid iron teeth and steel forgings. The examination of the values of B8 and Br shows that for Bmax6 850 - 7150(points 3 and 4) EMF which equals 191-198 volts must be close to the maximum. The maximum voltage observed at no load is equal to 195 volts (pp. 58 and 62 of Metzler's paper) which shows a good agreement between two results. It would be in place to point out how simple and short all those computations are when compared with the painstaking and lengthy graphical solution exemplified in Metzler's paper, while the accuracy of final results is practically the same. If the trapezoid II is assumed as the field flux wave the results remain substantially the same. Inductor Alternator No. 3 ( High frequency) Its characteristic features are as follows: Pr = 10 mm; ar = 3 . 175 mm ; p 6 = 5 mm ; a 8 = 4. 2o6 mm; cf-= O. 794mn; u = 10000 cm p s. The edges of the poles at the air- gap are rounded; L ~ 0.0794; a Pr 2 ax. = 0. 635; S == 400,· 1 = 2 7 cm· z = l· __§ = 0.8412·, ar = 0.3175. • ' 'Pa Pr Pr In view of the rounded edges of the poles as well as of the values of the ratios ~ and ar the field flu x oscillation is obviously Pr represented by a sine wave. Ps For i = l.8 amp. (field current) saturation curve, and Bmax E0 = 400 volts from the experimental = 6900. as determined from the magnetic circuit and the number of ampere-turns. JPr_ = 0.0794 the In view of the value of the . ratio expression for ( l.j J ; ) is to be applied with 0 :f ollows: A =- 10, hence it l -t" 10 x (4 -3 x o.635)x o.635 x(4 - 3 x o.8412)x o.8412 = 2 . 3 2 7. 94 + §2oo 107 1 1 - T = o.565 Accordingly for sine wave and 1-::::: 0.955 From the formula ( 3J- ) r=- 2.3 from Table III i 0 is found to be E == 0.955 x 0.565 x 10000 x 6900 x 2 x 200 x 2.7 x 10-8 = 401 volts 0 against 400 observed. It is found from the structural data and experimental r,,,n,{ saturation curve to be ) for 1 The computation gives = o.6375 amp., Bmax= 2400A E 0 = 167 volts. = 2.94, 1 - T=0.66. Since ~ -- 0.935 from Table III. Eo -= 0.935 x 0,66 x 10000 x 2400 x 400 x 2.7 x 10-8 =- 160 volts against 167 volts observed. Section alon.f C;T)lf.,l,LJ(Of<2.t. i C, ! s I EiG./ I CY-t.lNORICA.L RoroR /NovcroR N.M 060V/(~l'F .c i TYPe or rHe A, re RNA roR 1v s R 119a....~~ -A Fie, la !NovcToR N.M. 0/IQw<Ho,r,r AL7eRNATOR 'I r t L ~,~:.... Fio. J& CIRCl/LAR.STATlJ~ SLOT N 1'10t1ouXMtYF - - ---- El(;_i's I _rit;Za OPeCIAL rDRM or NM OaovxMo.rFI/ N0UCT0R ALTJ:"RNATOR SrATOR St.or f=oRM N.M.Oaou,r 110~~ /,.$ ~ ~ ~-,t Wil ~ "Flc 3 Dl.$K ROTOR lNOUCTOR NM. Ot1ou1<H•F,. r '•' I • /'i I' TYPe- A:~ERNATOR , ~ fl LATOI.IR's /NDLICTOR AI..URNATOR LArovR's INoucroR ALreRNAToR NMOatwtrHOl'I' 1111'1Oaov,r,wrl' rTimrn I I L-, r-' ~~----~, .~-------: I ~Ll II ~----4?-,' j vv,)'19 --l~l II . . Flc6 • BerHeNoo'.5 /N()(/CTORALTeRNA7DR /ILH. O1101/KH 01'I' R c7 CHARAcr1:R1snc ReLAT1 v 1= A:JJ1T101Vs OF RoroR ANO STATOR Fi~Lo f=Lvx 01AGRAM. N MOt10UlfHOl'I' 4NO fit;8 rieLD f"i.vx 01-'TRIBf.lTION /"DR TMREE CHARACTERl~TIC CA3E., - RECTANGLJ; TRAPEZOID, A/VO S1NELIKE DJ.JTRIBllTION. /'t/l(IJ-,rHO,-, I U - llZ IY-- X rtG. 9 TYPICAL WAve- fvRHJ or ricLD FLUX 4/YO AT /VO LOAD. N.HOaov,rNoFF COMBINED DIAGRAM. c MF I II. 5 ~-----1V'(,·-------- Elc;JO ricLo n. uJ< D1srR1Bur10N ""'' Trfll!!!: CHARACTeRl.5TIC RivlT/ON.lI IV MOaouKh'OFF I. _Fero~ FiG JC .:JPECIAI. FbLe FoHM ( ROTOR TOOT/1 rOR M/ NMtJ11ou1<HOFF n c. 11 .Fin.a F/.u)( D1.,TRIBVTION CMA;;c~Risr,c Po.,,riotv I N.M.°'1oUHMOFF I _, REFERENCES 1. Dr. K. Schmidt. Ein neuer Hochfrequenzsender fur drahtlose 'l''e legraphie, Elektrotechnische Zeitschr.ift(E T Z) No. 40, 1923. Dr. Dornig. ET Z Feb., 1925, p. 223. Dr. Pungs. The Present Statue of Radio Engineering in Germany. The Magazine of Wireless Telegraphy and Telephony No. 30, 1925. Nijny Novgorod, Russia. I J. Bethenod. Les Alternateurs ' a haute frequence. by A. Blondel, p. IV. 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