EiBgineermgf library .' ..*. v -' DESIGN OF POLYPHASE GENERATORS AND MOTORS McGraw-Hill BookCompatiy Pujfi&s/iers offioo/br Electrical World The Engineering and Mining Journal Engineering Record Engineering News Railway Age G azette American Machinist Signal LnginGer American Engineer Electric Railway Journal Coal Age Metallurgical and Chemical Engineering Power DESIGN OF POLYPHASE GENERATORS AND MOTORS BY HENRY M. HOB ART, u Consulting Engineer, General Electric Company; Member American Institute Electrical Engineers; Institution Electrical Engineers; Institution Mechanical Engineers; Member Society for the Promotion of Engineering Education McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E. C. 1913 Engineering Library ;' COPYRIGHT, 1912, BY THE MCGRAW-HILL BOOK COMPANY THE. MAPLE . PKKS8- YOUR. PA PREFACE DURING several recent years the author has given courses of lectures at three technical schools in London, on the subject of the design of electric machinery. These three schools were: the Northampton Institute of Technology; Faraday House; and University College. Various methods of procedure were employed and these ultimately developed into a general plan which (so far as it related to the subjects of Polyphase Genera- tors and Polyphase Motors), has been followed in the present treatise. It was the author's experience that the students attend- ing his lectures took an earnest interest in calculating designs of their own, in parallel with the working out of the typical design selected by the author for the purpose of his lectures. At the outset of the course, each member of the class was assigned the task of working out a design for a stipulated rated output, speed and pressure. Collectively, the designs undertaken by the class, constituted a series of machines, and co-operation was encouraged with a view to obtaining, at the conclusion of the course of lectures, a set of consistent designs. If a student encountered difficulty or doubt concerning some feature of his design, he was encouraged to compare notes with the students engaged in design- ing machines of the next larger and smaller ratings or the next higher and lower speeds. Ultimately the results for the entire group of designs were incorporated in a set of tables of which each student obtained blue prints. At two of these three colleges, the " sandwich " system was in- operation, that is to say, terms of attendance at the college were " sandwiched " with terms during which the student was employed in an electrical engineering works. The result of the author's opportunities for making comparisons is to the effect that students who were being trained in accordance with the " sandwich " system were particularly eager in working out their v 257829 vi PREFACE designs. Their ambition to obtain knowledge of a practical character had been whetted by their early experiences of prac- tical work; they knew what they wanted and they were deter- mined to take full advantage of opportunities for obtaining what they wanted. One could discuss technical subjects with them quite as one would discuss them with brother engineers. There was no need to disguise difficulties in sugar-coated pills. It will be readily appreciated that under these circumstances there was no necessity to devote time to preliminary dissertations on fundamental principles. The author did not give these lectures in the capacity of one who is primarily a teacher but he gave them from the standpoint of an outside practitioner lectur- ing on subjects with which he had had occasion to be especially familiar and in which he took a deep interest. It is the function of the professional teacher to supply the student with essential preparatory information. In the author's opinion, however, a considerable knowledge of fundamental principles can advan- tageously be reviewed in ways which he has employed in the present treatise, namely; as occasion arises in the course of working out practical examples. Attention should be drawn to the fact that various important fundamental principles are there stated and expounded, though without the slightest regard for conventional methods. Of the very large number of college graduates who, from time to time, have worked under the author's direction in connec- tion with the design of dynamo-electric machinery, instances have been rare where the graduate has possessed any consider- able amount of useful knowledge of the subject. Such knowledge as he has possessed on leaving college, has been of a theoretical character. Furthermore, it has been exceedingly vague and poorly assimilated, and it has not been of the slightest use in practical designing work. One is forced to the conclusion that teachers are lecturing completely over the heads of their students far more often than is generally realized. The author is of the opinion that the most effective way to teach the design of electric machinery is to lead the student, step by step, through the actual calculations, simultaneously requiring him to work out designs by himself and insisting that he shall go over each design again and again, only arriving at the final design as the result of a study of many alternatives. It is only by making shoes that one learns PREFACE vii to be a shoemaker and it is only by designing that one learns to be a designer. While the author has not hesitated to incorporate in his treatise, aspects of the subject which are of an advanced character and of considerable difficulty, he wishes to disclaim explicitly any pro- fession to comprehensively covering the entire ground. The professional designer of polyphase generators gives exhaustive consideration to many difficult matters other than those dis- cussed in the present treatise. He must, for example, make careful provision for the enormous mechanical stresses occurring in the end connections on the occasions of short circuits on the system supplied from the generators, and when a generator is thrown on the circuit with insufficient attention to synchronizing. He will often have to take into account, in laying out the design, that the proportions shall be such as to ensure satisfactory opera- tion in parallel with other generators already in service, and it will, furthermore, be necessary to modify the design to suit it to the characteristics of the prime mover from which it will be driven. This will involve complicated questions relating to permissible angular variations from uniform speed of rotation. The designer of high-pressure polyphase generators will find it necessary to acquire a thorough knowledge of the properties of insulating materials; of the laws of the flow of heat through them; of their gradual deterioration under the influence of prolonged subjection to high temperatures and to corona influences. He must also study the effects on insulating materials of minute traces of acids and of the presence of moisture; and, in general, the ageing of insulation from all manner of causes. The designer of polyphase generators can employ to excellent advantage a thorough knowledge of the design of fans; in fact, a large part of his attention will require to be given to calculations relating to the flow of air through passages of various kinds and under various conditions. The prevention of noise in the operation of machines which are cooled by air is a matter requiring much study. There occur in all generators losses of a more or less obscure nature, appropriately termed " stray " losses, and a wide experience in design is essential to minimizing such losses and thereby obtaining minimum temperature rise and maximum efficiency. Furthermore, mention should be made of the import- ance of providing a design for appropriate wave shape and insur- viii PREFACE ing the absence of objectionable harmonics. To deal compre- hensively with these and other related matters a very extensive treatise would be necessary. While it will now be evident that the designer must extend his studies beyond the limits of the present treatise in acquaint- ing himself with the important subjects above mentioned, the familiarity with the subject of the design of polyphase generators and motors which" can be acquired by a study of the present treatise, will nevertheless be of a decidedly advanced character. It can be best amplified to the necessary extent when the problems arise in the course of the designer's professional work, by con- sulting papers and discussions published in the Proceedings of Electrical Engineering Societies. Just as the design of machinery for continuous electricity crystallizes around the design of the commutator as a nucleus, so in the design of polyphase generators, a discussion of the pre- determination of the field excitation under various conditions of load as regards amount and phase, serves as a basis for acquiring familiarity with the properties of machines of this class. The author has taken the opportunity of presenting a method of dealing with the subject of the predetermination of the required excitation for specified loads, which in his opinion conforms more closely with the actual occurrences than is the case with any other method with which he is acquainted. In dealing with the design of polyphase induction motors, the calculations crystallize out around the circle ratio, and this may be considered the nucleus for the design. The insight as regards the actual occurrences in an induction motor which may be acquired by accustoming one's self to construct mentally its circle diagram, should, in the author's opinion, justify a much wider use of the circle diagram than is at present the case in America. The author desires to acknowledge the courtesy of the Editor of the General Electric Review for permission to employ in Chap- ter VI, certain portions of articles which the author first published in the columns of that journal, and to. Mr. P. R. Fortin for assistance in the preparation of the illustrations. HENRY M. HOBART, M.Inst.C.E. November, 1912. CONTENTS PAGE PREFACE . . . . v CHAPTER I INTRODUCTION 1 CHAPTER II CALCULATIONS FOR A 2500-KVA. THREE-PHASE SALIENT POLE GENERATOR 3 CHAPTER III POLYPHASE GENERATORS WITH DISTRIBUTED FIELD WINDINGS 99 CHAPTER IV THE DESIGN OF A POLYPHASE INDUCTION MOTOR WITH A SQUIRREL- CAGE ROTOR 105 CHAPTER V SLIP-RING INDUCTION MOTORS 195 CHAPTER VI SYNCHRONOUS MOTORS VERSUS INDUCTION MOTORS 202 CHAPTER VII THE INDUCTION GENERATOR 213 CHAPTER VIII EXAMPLES FOR PRACTICE IN DESIGNING POLYPHASE GENERATORS AND MOTORS 225 APPENDIX 1 247 APPENDIX II 252 APPENDIX III 256 APPENDIX IV 257 INDEX 259 ix DESIGN OF POLYPHASE GENERATORS AND MOTORS CHAPTER I INTRODUCTORY POLYPHASE generators may be of either the synchronous or the induction type. Whereas tens of thousands of synchronous generators are in operation, only very few generators of the induc- tion type have ever been built. In the case of polyphase motors, however, while hundreds of thousands of the induction type are in operation, the synchronous type is still comparatively seldom employed. Thus it is in accord with the relative importance of the respective types that the greater part of this treatise is devoted to setting forth methods of designing synchronous generators and induction motors. Brief chapters are, however, devoted to the other two types, namely induction generators and synchronous motors. The design of an induction generator involves consider- ations closely similar to those relating to the design of an induc- tion motor. In fact, a machine designed for operation as an induc- tion motor will usually give an excellent performance when employed as an induction generator. Similarly, while certain modifications are required to obtain the best results, a machine designed for operation as a synchronous generator will usually be suitable for operation as a synchronous motor. In dealing with the two chief types: the author has adopted the plan of taking up immediately the calculations .for a simple design with a given rating as regards output, speed, periodicity and pressure. In the course of these calculations, a reasonable amount of familiarity will be acquired with the leading principles involved, and the reader will be prepared profitably to consult 2 POLYPHASE GENERATORS AND MOTORS advanced treatises dealing with the many refinements essential to success in designing. A sound knowledge of these refinements can only be acquired in the course of the practice of designing as a profession. The subject of the design of dynamo electric machin- ery is being continually discussed from various viewpoints in the papers contributed to the Proceedings of engineering societies. Attempts have been made to correlate in treatises the entire accumulation of present knowledge relating to the design of dyna- mo-electric machinery, but such treatises necessarily extend into several volumes and even then valuable fundamental outlines of the subject are apt to be obscured by the mass of details. No such attempt is made in the present instance; the discussion is restricted to the fundamental outlines. There are given in Appendices 1 and 2, bibliographies of papers contributed to the Journal of the Institution of Electrical Engineers and to the Transactions of the American Institute of Electrical Engineers. These papers deal with many important matters of which advanced designers must have a thorough knowledge. After acquiring proficiency in carrying through the fundamental calculations with which the present treatise deals, a study of these papers will be profitable. Indeed such extended study is essential to those engineers who propose to adopt designing as a profession. CHAPTER II CALCULATIONS FOR A 2500-KVA. THREE-PHASE SALIENT POLE GENERATOR LET us at once proceed with the calculation of a design for a three-phase generator. Let its rated output be 2500 kilo olt amperes at a power factor of 0.90. Let its speed be 375 revolutions per minute and let it be required to provide 25-cycle electricity. Let it be further required that the generator shall provide a ter- minal pressure of 12 000 volts. We shall equip the machine with a Y-connected stator winding. The phase pressure will be: 12000 12000 The Number of Poles. Since the machine is driven at a speed of (- =)6.25 revolutions per second, and since the required ,60 periodicity is 25 cycles per second, it follows that we must arrange for: 25 \ r- = ) 4.0 cycles per revolution, .^o / Consequently we must provide four pairs of poles, or (2X4 = )8 poles. Denoting by P the number of poles, by R the speed in revolu- tions per minute, and by ~ the periodicity in cycles per second, we have the formula: R In this treatise the power factor will be denoted by G. Since for our machine the rated load is 2500 kva. for (7 = 0.90, we may 3 4 POLYPHASE GENERATORS AND MOTORS also say that the design is for a rated output of (0.90X2500 = ) 2250 kw. at a power factor of 0.90. So many alternators have been built and analyzed that the design of a machine for any particular rating is no longer a matter which should be undertaken without any reference to accum- ulated experience. From experience with many machines, design- ers have arrived at data from which they can obtain in advance some rough idea of the proportions which will be most appro- priate. It is not to be concluded that the designing of a machine by reference to these data is a matter of mere routine copying. On the contrary, even by making use of all the data available, there is ample opportunity for the exercise of judgment and origin- ality in arriving at the particular design required. The Air-gap Diameter. Let us denote by D the internal diameter of the stator. Usually we shall express D in centimeters (cm.), but occasionally it will be more convenient to express it in millimeters (mm.), and also occasionally in decimeters (dm.), and in meters (m). Since the internal diameter of the stator is but slightly in excess of the external diameter of the rotor, it is often convenient briefly to describe D as the " air-gap diameter," but we must not forget that strictly speaking, it is the internal diameter of the stator and is consequently a little greater than the external diameter of the rotor. The Polar Pitch. Let us further denote the polar pitch, (in cm.), by T. By the polar pitch is meant the distance, measured at the inner circumference of the stator, from the center of one pole to the center of the next adjacent pole. The polar pitch T is the very first dimension for which we wish to derive a rough preliminary value. * The values of T given in Table 1 are indicated by experience to be good preliminary values for designs of the numbers of poles, the periodicity and the output shown against them in the Table. The table indicates that the polar pitch T should have a value of 70 cm. Since the machine has 8 poles, the internal periphery of the stator is : 8X70 = 560 cm. CALCULATIONS FOR 2500-KVA. GENERATOR TABLE 1. VALUES OF T AND . P=4. P=6. P=8. P = 12 R = R = R = R = Rated Output in Kva. / 750 for 25 ^ \1500for50 / 500 for 25- 1 1000 for 50 / 375 for 25 \ 750 for 50 - 1 250 for 25 \ 500 for 50 T r T T 1 r 250 56 1.12 48 1.15 44 1.20 38 1.22 500 68 1.25 58 1.30 50 1.35 43 1.40 o> 1000 78 1.38 65 1.45 57 1.55 50 1.65 "3 2000 92 1.45 77 1.55 67 1.75 60 1.90 10 4000 117 1.55 97 1.70 90 1.85 82 2.00 .s 2 | 4,000 | 3,000 2,000 1,000 *. ~~ ^. __ _ - m ' X ^ A (S A ' / / / / / / 1 / y / / / / 1 c^ iiiiiiiiiii 1 1 i I -rjT CD CO C> j ^ CM 51 CO i V w 3 3 ; $ { mmf per Field Spool in ats FIG. 23. No-load Saturation Curve of 2500-kva. Generator with an Air-gap of 18 mm. THE PRESSURE REGULATION The subject of this section is one to which much study and discussion have been devoted. Nevertheless we still find wide differences of opinion with reference to the problems involved in the estimation of the excitation required in any given case and for any given conditions of operation. The method which will be described has the merit of brevity combined with at least as much exactness, so far as regards the results obtained, as can be shown to be possessed by any other method. The chief defect in the method relates to the theoretical indefensibility from the quantitative standpoint of certain steps 40 POLYPHASE GENERATORS AND MOTORS in the calculations. Since, however, the occurrences assumed to take place are qualitatively in accordance with the facts, it is believed that the admitted defect is of minor importance. Before proceeding to explain the method in applying it to our 2500-kva. design, let us bring together the leading data which we have now worked out. This is done in the following specification: SUMMARY OF THE NORMAL RATING OF THE DESIGN Number of poles 8 Output at full load in kva 2500 Corresponding power-factor of external load . 90 Corresponding output in kw 2250 Speed in r.p.m 375 Periodicity in cycles per second 25 Terminal pressure in volts 12 000 Number of phases 3 Connection of phases Y Pressure per phase ( =- J . . . . . 6950 THE LEADING DATA OF THE DESIGN External diameter of stator core 2300 mm. Diameter at bottom of stator slots 1894 mm. Internal diameter of stator (D) 1780 mm. External diameter of rotor 1744 mm. Diameter at bottom of pole shoes 1580 mm. External' diameter of magnet yoke 1180 mm. Internal diameter of magnet yoke 928 mm. Polar pitch (T) 700 mm. Gross core length (Xgr) 1180 mm. Number of vertical ventilating ducts 20 Width each duct . 15 mm. Per cent insulation between laminations 10 per cent Net core length (1180-300) X0.90 (Xn) 792 mm. Length pole shoe parallel to shaft 1140 mm. Pole arc 420 mm. Area of pole face (114X42) 4790 sq.cm. Extreme length magnet core parallel to shaft. . . 860 mm. Extreme width magnet core 260 mm. Area of cross-section of magnet core 2080 sq.cm. CALCULATIONS FOR 2500-KVA. GENERATOR 41 Length of yoke parallel to shaft ............... 1200 mm. Radial depth of yoke ........................ 126 mm. Cross-section of yoke ................ . ....... 1510 sq.cm. Number of stator slots ...................... 120 Number stator slots per pole per phase Depth slot ................................. 57 mm. Width slot ................................ 24 mm. Width slot opening ......................... 12 mm. Sketches of the design have already been given in Figs. 20 and 21. These preliminary sketches show wide-open slots. Let us, however, employ slots with an opening of only 12 mm. in accordance with the above tabulated specification. The stator winding consists of 10 conductors per slot. The bare dimensions of each conductor are 12 mm. X 3.3 mm., and the 10 conductors are arranged one above the other as already indicated in Fig. 1 on page 11. The mean length of one armature turn is 656 cm. The winding is of the type which we have termed a " half-coiled " winding. That is to say, only half the field poles have opposite to them, armature coils belonging to any one phase. A diagram of one phase of a typical half-coiled winding for a 6-pole machine has been given in Fig. 15 on p. 20. L / Q\ and. is seen to havef ) = three coils per phase. Since our 2500 kva. machine has 8 poles, there are four coils in each phase. Each side of each coil comprises the contents of five adjacent slots. Since each slot contains 10 conductors, there are (5 X 10 = ) 50 turns per coil, and consequently (4X50 = ) 200 turns in series per phase. Denoting by T the number of turns in series per phase we have ^ = 200. In Fig. 24 is given a winding diagram for one of the three phases, and in Fig. 25, is given a winding diagram containing all three phases. /1 9 000\ For a pressure of ( - - ) = 6950 volts per phase on open \ v 3 / circuit, the armature flux per pole (denoted by M), is obtained as follows: 42 POLYPHASE GENERATORS AND MOTORS We may take the leakage factor as 1.15. Consequently the flux in the magnet core and yoke is: 1.15X31.3 = 36.0 megalines. FIG. 24. Winding Diagram for one of the Three Phases of the 2500-kva. three-phase Alternator. The estimation of the no-load saturation curve has previovsly been carried out in earlier sections. A summary of the component and resultant magnetomotive forces (mmf.) for 6000, 6950, and 7500 volts, is given in the following table: 6000 Volts. 6950 Volts. 7500 Volts. Air-gap 10 000 ats. 11 560 ats. 12 500 ats Teeth 90 " 330 " 720 " Magnet core Magnet yoke .... Stator core 770 " 200 " 140 " 2000 " 320 " 190 " 5300 " 420 " 230 " Total mmf 11 200 ats. 14 400 ats. 19 170 ats. CALCULATIONS FOR 2500-KVA. GENERATOR 43 A no-load saturation curve passing through these three points has been given in Fig. 23, on page 39. The Armature Interfering mmf. The current per phase at rated load of 2500 kva. is: 2500000 1 ~ 3X6950 " 12U ' FIG. 25. Complete Winding Diagram for all Three Phases of the 2500-kva. Generator. We have seen that there are 200 turns in series in each phase. Consequently there are: and 200 or , -Q- = 25 turns per pole per phase o 25X120 = 3000 (rms.) ats. per pole per phase. It has already been stated that many designers resort to theoretical reasoning in ascertaining from the mmf. of one phase, 44 POLYPHASE GENERATORS AND MOTORS the resultant mmf. exerted by the three phases. But, in practice, the distribution of the stator and rotor windings, the ratio of the pole arc to the pitch, and other details of the design, so complicate the matter as considerably to invalidate any theoretical deduc- tions. But working backward from a very large collection of experimental observations, the conclusion is reached that inde- pendently of the various relative dispositions of the winding and of other features of the design, results consistent with practice are obtained by taking the resultant mmf. of the three phases as equal to: 2.4 times the mmf. of each phase. Consequently for our design, we have, at full load, an armature mmf. of: 2.4X3000 = 7200 ats. It is only at zero power-factor that these armature ats. have the same axis as the field ampere-turns. If, when the power- factor of the external load is zero, the output is 120 amperes, then, if the current is lagging, the resultant mmf. acting to send flux round the magnetic circuit is obtained by subtracting 7200 ats. (the armature mmf.) from the excitation on each field pole. If, with the power-factor again equal to zero, the current is lead- ing, then the resultant mmf. acting to send flux round the magnetic circuit is obtained by adding 7200 ats. (the armature mmf.) to the excitation on each field pole. For this same current of 120 amperes, but at other than zero power-factor, the armature mmf. does not affect the resultant mmf. to so great an extent. Later we shall consider a method of determining the extent of the influence of the armature mmf. when the power-factor is other than zero. It follows as a consequence of the preceding explanations that the armature mmf. will exert a less disturbing influence on the terminal pressure the greater the mmf. provided on the field spools, and that for a given required closeness of pressure regulation the higher the armature mmf., the higher must also be the field mmf. The modern conception of preferable conditions is not based on such close inherent pressure regulation as was formerly con- sidered desirable. The alteration in conceptions in this respect CALCULATIONS FOR 2500-KVA. GENERATOR 45 does not, however, decrease the importance of having at our dis- posal means for accurately estimating the excitation required under various conditions of load as regards pressure, power-factor and amount. The Field Excitation Required with Various Loads. The required excitation is chiefly dependent upon three factors : 1. The no-load saturation curve of the machine; 2. The armature strength in ats. per pole; 3. The inductance of the armature winding. In our 2500-kva three-phase machine, the armature strength at rated load is equal to 7200 ats. The Position of the Axis of the Armature mmf . If the arma- ture winding had no inductance, then for an external load of unity power-factor, the axis of the armature magnetomotive force would be situated just midway between two adjacent poles; that is to say, there would be no direct demagnetization. At the other extreme, namely for the same current output at zero power-factor, the axis of armature demagnetization would cor- respond with the field axis. The two cases are illustrated dia- grammatically in Figs. 26 and 27. In our machine, when loaded with full-load current of 120 amperes at zero power-factor, the demagnetization would amount to 7200 ats. and this demagnet- ization could only be offset by providing 7200 ats. on each field pole. For power-factors between 1 and 0, the axis of armature demagnetization would be intermediate, as indicated diagram- matically in Fig. 28. But we are not concerned with imaginary alternators with zero-inductance armature windings, but with actual alternators. In actual alternators, the armature windings have considerable inductance. At this stage we wish to determine the inductance of the armature windings of our 2500-kva alternator. The Inductance of a i-turn Coil. Let us first consider a single turn of the armature winding before it is put into place in the stator slots. If we were to send one ampere of continuous electricity through this turn, how many magnetic lines would be occasioned? If the conductor were large enough to practically fill the entire slot, then with the dimensions employed in modern alternators, the general order of magnitude of the flux occasioned may be ascertained on the basis that some 0.3 to 0.9 of a line would be 46 POLYPHASE GENERATORS AND MOTORS Direction of dotation FIG. 26 Diagrammatic Representation of Relative Positions of Axes of Field mmf . and Armature mmf . for a Load of Unity Power-factor Neglecting Armature Inductance. Direction of Rotation FIG. 27. Diagrammatic Representation of Relative Positions of Axes of Field mmf. and Armature mmf. for a Load of Zero Power-factor. Direction of Rotation FIG. 28. Diagrammatic Representation of Relative Positions of Axes of Field mmf. and Armature mmf. for a Load of Intermediate Power-factor. CALCULATIONS FOR 2500-KVA. GENERATOR linked with every centimeter of length of the turn. Taking the mean value of 0.6 line per cm., then since in our design the length of a turn is 656 cm., the flux occasioned by 1 amp. of continuous electricity is 656X0.6 = 394 lines. The inductance (expressed in henrys) of a 1-turn coil is equal to 10~ 8 times the number of lines linked with the turn when 1 amp. of continuous electricity is flowing through the turn. Conse- quently the inductance is, in this case, equal to 10~ 8 X 394 = 0.00000394 henry. The Inductance of a Coil with More than One Turn. The inductance of any coil is equal to the product (when 1 amp. of continuous electricity is flowing through the coil), of the flux linked with the coil and the number of turns in the coil. This definition is framed on the assumption that the entire flux is linked with the entire number of turns. Where this is not the case, appropriate factors must be employed in order to arrive at the correct result. In a two-turn coil, the mmf . is, when a current of 1 amp. of continuous electricity is flowing through the coil, twice as great as in a one-turn coil of the same dimensions. Consequently for a magnetic circuit of air, the flux will also be twice as great, since in air the flux is directly proportional to the mmf. occa- sioning it. But since this doubled flux is linked with double the number of turns, the total linkage of flux and turns is four times as great. In other words, the inductance increases as the square of the number of turns. In Fig. 24, it has been shown that the winding of any one phase of our eight-pole machine is composed of four coils in series. Let us first consider one of those four coils. Each side com- prises the contents of five slots. Since there are ten conductors per slot, we see that we are dealing with a fifty-turn coil. On the sufficient assumption that the incomplete linkage of flux and turns is provided for by calculating from the basis of only 0.5 line per centimeter of length, instead of from the value of 0.6 line per centi- 48 POLYPHASE GENERATORS AND MOTORS meter of length which we employed when dealing with the one- turn coil, we obtain for the inductance the value: 502X^1x0.00000394 = 0.0082 henry. The Inductance and Reactance of One Phase. The winding of one phase comprises four such coils in series, and consequently we have: Inductance per phase = 4 X 0.0082 = 0.0328 henry The reactance is obtained from the formula: Reactance (in ohms) where the periodicity in cycles per second is denoted by ^ and the inductance in henry s by 1. We consequently have: Reactance per phase = 6.28X25X0.0328 = 5. 15 ohms. The Reactance Voltage per Phase. For our machine the full-load current per phase is 120 amperes. Consequently when carrying full-load current we have: Reactance voltage per phase = 120X5. 15 = 618 volts. The Inductance and Reactance of Slot-embedded Windings. But up to this point we have considered that throughout their length the windings are surrounded by air. In reality the wind- ings are embedded in slots for a certain portion of their length. For this embedded portion of their length, the flux, in lines per centimeter of length, set up in a one-turn coil when one ampere of continuous electricity flows through it, is considerably greater than for those portions of the coil which are surrounded by air. Suitable values may be obtained from Table- 9 : TABLE 9. DATA FOR ESTIMATING THE INDUCTANCE OF THE EMBEDDED LENGTH. No. of Lines per cm. Concentrated windings in wide-open, straight-sided slots .......... 3 to 6 Thoroughly distributed windings in wide-open, straight-sided slots 1.5 to 3 Concentrated windings in completely-closed slots .......... ....... 7 to 14 Thoroughly-distributed windings in completely-closed slots ........ 3 to 6 Partly distributed windings in semi-closed slots ................... about 5 CALCULATIONS FOR 2500-KVA. GENERATOR 49 In order to illustrate the sense centrated," " thoroughly-distrib- uted" and " partly-distributed" windings are employed in the above table, the three winding diagrams in Figs. 29, 30 and 31 have been prepared. Evidently for the windings of any one phase of our machine, the value of 5 lines per cm. of embedded length is sufficiently representative. The embedded portion of the length of a turn is equal to twice the net core length. For our machine we have : in which the terms " con- FIG. 29. Concentrated Winding. Embedded length = 2X79 = 158 cm.; Mean length of a turn = 656 cm. " Free " length (i.e., the portion in air) = 656 - 158 = 498 cm. ury~ui^n^njij"yru^^ FIG. 30, Thoroughly Distributed Winding. FIG. 31. Partly Distributed Winding. We have calculated the inductance which our coil would have if the entire 656 cm. of its length were surrounded by air (i.e., were " free " length). We can now readily obtain the value of 50 POLYPHASE GENERATORS AND MOTORS that part of the inductance which is associated with the actual " free " length. It amounts to: iX 0.0328 = 0.0248 henry. 656 The inductance of the " embedded " length is j^XJ^X 0.0328 = 0.0790 henry. The total inductance per phase is 0.0248+0.0790 = 0.104 henry. It is interesting to note that 0.0248. 0.104 X 100 = 23.8 per cent of the total inductance, is, in the case of this particular machine, associated with the end connections. Our estimate of the inductance has been so seriously interrupted by explanatory text that it is desirable to set it forth again in a more orderly form, and taking each step in logical order : Mean length of turn .................. 656 cm. " Free " length ...................... 498 cm. " Embedded " length .................. 158 cm. Flux per ampere-turn per j 0.5 line for " free " length, centimeter of length [ 5.0 lines for "embedded " length, J (249 lines for " free " length, Flux per ampere-turn j ?go ^ for M embedded length< Total flux per ampere-turn ( = 249+790=) 1040 lines. Number of turns in one phase per pair of poles (i.e., per coil) ................................. 50 CALCULATIONS FOR 2500-KVA. GENERATOR 51 Inductance of one coil (1040X50 2 X10~ 8 = ) ...... 0.0260 henry Number of coils (also pairs of poles) per phase ..... 4 Inductance of one phase (4X0.0260 = ) ........... 0.104 henry Reactance of one phase at 25 cycles (6.28X25X0.104 = ) 16.3 ohms Reactance voltage of one phase at 25 cycles and 120 amperes (120X16.3) = ..................... 1960 volts Physical Conditions Corresponding to this Value of the Reactance Voltage. This value of 1960 volts for the reactance voltage, is of the order of the value which we should obtain experimentally under the following conditions: Twenty-five-cycle current is sent into the stator windings from some external source, while the rotor (unexcited) , is, by means of a motor, driven at the slowest speed consistent with steady indica- tions of the current flowing into the three branches of the stator windings. Under these conditions, some 1960 volts per phase would be found to be necessary in order to send 120 amperes into each of the three windings. Theta (0) and Its Significance. The value of the reactance voltage thus determined, enables us to ascertain the angular distance from mid-pole-face position at which the current in the stator windings passes through its crest value. Let this angle be denoted by 0. For a load of unity power- factor, is the angle whose tangent is equal to the reactance voltage divided by the phase voltage. Thus we have: ! reactance voltage 6 = tan J r . phase voltage The conception of may possibly be made clearer by stat- ing that it represents the angular distance by which the center of a group of conductors belonging to one phase has traversed beyond mid-pole-face position when the current in these conduc- tors reaches its crest value. Theta at Unity Power-factor. For our example we have (for unity power-factor) : tan~ 1 0.282 = 15.9.* * In making calculations of the kind explained in this Chapter, the Table of sines, cosines and tangents in Appendix III. will be found useful. 52 POLYPHASE GENERATORS AND MOTORS The diagram is shown in Fig. 32. Strictly speaking, we ought to take into account in the diagram, the IR drop in the armature. FIG. 32. ^Diagram Relating to the Explanation of the Nature and Significance of the Angle Theta (6). The resistance per phase (at 60 Cent.), is 0.685 ohm. Conse- quently for the full-load current of 120 amperes, we have: IR drop = 120X0.685 = 82 volts. The corrected diagram (taking into account the IR drop), is shown in Fig. 33. In this diagram we have: 6-tBa-g^=tan->^taa-0^8*i5^ FIG. 33. More Exact Diagram for Obtaining 0. Relation between Theta and the Armature Interference. The armature demagnetization for any value of 6 is obtained by multiplying the armature strength by sin 0. For 120 amperes the armature strength is : 7200 ats. We also have: sin 6 =sin 15.8 = 0.270. CALCULATIONS FOR 2500-KVA. GENERATOR 53 Under these conditions (120 amperes output at unity power- factor of the external load), the armature mmf. is equal to: 0.270 X 7200 = 1940 ats. The Hypothenuse of the 6-Triangle Has no Physical Exist- ence. It is desirable to lay strong emphasis on the fact that the vector sum of 7032 volts and 1960 volts does not represent an actually-existing internal pressure corresponding to an actual flux of magnetic lines. The quantity which, earlier in this chapter, has been termed the " reactance voltage ", is made up of two parts, associated respectively with the " embedded " length and with the " free " length. While the portion associated with the " embedded " length manifests itself in distortion of the magnetic flux, the portion associated with the " free " length acts in the same manner as would an equal inductance located in an independent inductance coil connected in series with a non- inductive alternator. (More strictly, it is only that portion of the " free " length which is associated with the end connections which should be thus considered, and the portion associated with the ventilating ducts should be placed in a different cate- gory. But in practice the small margin provided by taking the entire " free " length, is desirable.) We have seen that the inductance of the " free " length of the windings of our 2500-kva. machine is 23.8 per cent of the total inductance ; 0.238X1960 = 466 volts. The True Internal Pressure and Its Components. The three components of the total internal pressure of our machine, when the external load is 120 amperes at unity power-factor, are, per phase : Phase pressure . . . 6950 volts IRdrop 82 " Reactance drop 466 ' ' 54 POLYPHASE GENERATORS AND MOTORS When these are correctly combined, as shown in Fig. 34, the internal pressure is ascertained to be: V?032 2 +466 2 = 7050 volts. The influence of the reactance voltage is thus (for these par- ticular conditions of load), practically negligible, so far as concerns occasioning an internal pressure appreciably exceeding the result- ant of the terminal pressure and the IR drop. FIG. 34. Pressure Diagram Corresponding to 6950 Terminal Volts and 120 Amperes at Unity Power-factor. Total mmf . Required at Full Load and Unity Power-factor. From the no-load saturation curve in Fig. 23 on p. 39, we see that we require: 15200 ats. to overcome the reluctance of the magnetic circuit when the internal pressure is: 7050 volts. We require further: 1940 ats. to offset the armature demagnetization for these conditions of load (120 amperes at unity power-factor). Consequently we require a total mmf. per field spool, of: 15 200+1940 = 17140 ats. That is to say, for full-load conditions (6950 volts per phase and 120 amperes at unity power-factor), we require an excitation of: 17 140 ats. The Inherent Regulation at Unity Power-factor. We can now ascertain from the saturation curve the value to which the CALCULATIONS FOR 2500-KVA. GENERATOR 55 pressure will rise, when, maintaining constant this excitation of 17 140 ats., we decrease the load to zero. We find the value of the pressure to be: 7350 volts. Thus the pressure rise occurring when the load is decreased to zero, is: /^-6950 xxtAft \ KQ ( 6950 X100 = J5.8 per cent. This is expressed by stating that at unity power-factor the inherent regulation is: 5.8 per cent. ESTIMATION OF SATURATION CURVE FOR UNITY POWER FACTOR AND 120 AMPERES Let us now proceed to calculate values from which we can plot a load saturation curve extending from a pressure of volts up to a pressure of 7500 volts for an external load of 120 amperes at unity power-factor. We already have one point; namely: 17 140 ats. for 6950 volts. For this unity power-factor, 120-ampere saturation curve, the terminal pressure will be varied from up to a phase pressure of say 7500 volts while the current is held constant at 120 amperes. The diagrams for obtaining fop 7500, 5000, 2500 and volts are shown in Fig. 35. For these four cases we have: - 1 0.258 = 14.5 sin 14.5 = 0.250 2. = tan~ 1 = tan- 1 0.386 = 21.1 sin 21.1 = 0.360 oUo-^ 3. 6 = tan- 1 ~? = tan- 1 0.758 = 37.1 sin 37.1 = 0.605 4. 6 = tan- 1 ^ =tan~ 1 23.9 = 87.5 sin 87.5 = 0.999 56 POLYPHASE GENERATORS AND MOTORS 7500 5000 2500 ,82 FIG. 35. Theta Diagrams for 120 Amperes at Unity Power-factor. Since the armature current is, in all four cases, 120 amperes, the armature strength remains 7200 ats. The armature demag- netization amounts, in the four cases, to : 1. 0.250X7200 = 1800 ats.; 2. 0.360X7200 = 2590 " 3. 0.605X7200 = 4350 " 4. 0.999X7200 = 7200 " CALCULATIONS FOR 2500-KVA. GENERATOR 57 2500 FIG. 36. Pressure Diagrams for 120 Amperes at Unity Power-factor. The Armature Reaction with Short-circuited Armature. It is interesting to note that in the last diagram in Fig. 35, i.e., in the diagram relating to zero terminal pressure (short- circuited armature) the angle is practically 90. Con- sequently the armature reaction with short-circuited armature, is practically identical with the armature strength expressed in ampere-turns per pole. The Required Field Excitation for Each Terminal Pressure. The field excitation at each pressure, comprises two components. The first of these components must be equal to the armature demagnetization (in order to neutralize it), and the second com- ponent must be of the right amount to drive the required flux through the magnetic circuit in opposition to its magnetic reluct- ance. This latter value may be obtained from the no-load saturation curve in Fig. 23 (on p. 39), and must correspond to the 58 POLYPHASE GENERATORS AND MOTORS four pressures obtained from the four diagrams in Fig. 36- These four pressures are: 1. V7582 2 +466 2 = 7590 volts. 2. V5082 2 +466 2 = 3. \/2582 2 +466 2 = 4. V82 2 +466 2 = The saturation ats. for these four pressures are found from Fig. 23 (on p. 39), to be as follows: 1. Pressure 7590 Sat. Ats 22000 2. 5100 9100 3. 2630 4700 4. 476 820 The derivation of the total required mmf . is arranged below in tabular form : Terminal Pressure. Saturation Ats. Ats.for Offsetting Armature Demagnetization. Total Required Ats. 1 7500 22000 1800 23800 2 5000 9100 2950 11690 3 2500 4700 4350 9050 [4 820 7200 8020 The unity-power-factor, 120-ampere, saturation curve, thus derived, is plotted in Fig. 37, where the no-load saturation curve is also reproduced from Fig. 23. CALCULATIONS FOR 2500-KVA. GENERATOR 59 7,000 6,000 | 5,000 ri 4,000 2,000 .1,000 -cfy N ^ 1 164. I r ^^^ 5000 5000 /164 ^ 2 3 > 1 164 2500 2500 J164 1 p> 1 /164. . / OT ^j 05 FIG. 38. Theta Diagrams for / = 240 FIG. 39. Pressure Diagrams for andG = 1.00. / = 240 and G = 1 .00. Consequently : 1 ft tin 1 """" + nvl 1 H X*] 1 <~>7 1 cnr ^71 H A^ft i. u tan lyfttsA tan u.oii 44.1 sin^/.i u.^oo 2. 6 = tan- 1 Irl^tan- 1 0.760 = 37.2 sin 37.2 = 0.605 olb4 oqorj 3. 6 = tan- 1 ~^ = tan- 1 1.47 =55.7 sin 55.7 = 0.826 oqor) 4. 6 = tan~ 1 ^^ = tan- 1 23.8 =87.5 sin 87.5 = 0.999 CALCULATIONS FOR 2500-KVA. GENERATOR 61 The armature strength is now; 2x7200 = 14400ats. Consequently in the four cases, we now have for the armature demagnetizing ats. : 1. 0.456X14400 = 6580 ats. 2. 0.605X14400 = 8700 " 3. 0.826X14400 = 11900 " 4. 0.999X14400 = 14400 " The internal inductance pressure is now: 2X466 = 932 volts. The four internal pressures and the corresponding saturation ats. are: Internal Pressures. 1 . V7664 2 +932 2 = 7720 2. V5164 2 +9322 = 5250 3. V2664 2 +932 2 4. V 1642+9322 = 945 Sat. Ats. 29 500 9 350 5150 1700 The total required ats. are shown in the last column of the following tabulated calculation: Term. Pres. Saturation Ats. Ats. for Offsetting Armature Demagnetization. Total Required Ats. 1 7500 29500 6580 36080 2 5000 9350 8700 18050 3 2500 5150 11900 17050 4 1700 14400 16 100 62 POLYPHASE GENERATORS AND MOTORS The values in the last column are the basis for the unity power-factor, 240-ampere saturation curve shown in Fig. 40. 8,000 c3 J N c c eo co > o P mmf per Field Spool, in ats , FIG. 40. Saturation Curves for Various Values of 7 and for (r = 1.00. The unit}' power-factor, 120-ampere curve is also reproduced from Fig. 37 and the 7=0 curve from Fig. 23. SATURATION CURVES FOR POWER-FACTORS OF LESS THAN UNITY Let us now return to a load of 120 amperes, but let the power- factor of the load on the generator be 0.90. Let us estimate the required mmf. under these conditions, for terminal pressures of 7500, 5000, 2500, and volts, and then, from these four results, let us plot a 0.90-power-f actor, 120-ampere, saturation curve. The angle 0, i.e., the angle by which the conductors have passed mid-pole-face position when carrying the crest current, is now obtained as follows: CALCULATIONS FOR 2500-KVA. GENERATOR 63 When the power-factor of the external load is 0.90, the current lags behind the terminal pressure by 26.0, since cos 26 = 0.90. The 0-diagram for 120 amperes and a terminal pressure of 6950 volts, is now as shown in Fig. 41. The entire object of this diagram is to obtain the angle 0, i.e., the angle by which the con- FIG. 41. Theta Diagram for 7 = 120 FIG. 42. Pressure Diagram for and G = 0.90. 7 = 120 and G=0.90. ductors have passed mid-pole-face position when the current is at its crest value. AB = BCsm2Q = 6950X0.438 = 3040 AC= 0.90X6950 = 6250 = AB+BE ~ AC+DE ^3040+1960 "6250+ 82 5000 Therefore 6332 = 0.790. 6 = 38.3 sin 38.3 = 0.620. Therefore : Armature demagnetizing ats. = 0.620X7200 = 4450 ats. 64 POLYPHASE GENERATORS AND MOTORS The diagram for obtaining the internal pressure is shown in Fig. 42. By scaling off from this diagram, we find that the internal pressure is 7250 volts. From the no-load saturation curve we find : Saturation mmf. for 7250 volts = 16 700 ats. Thus the total required mmf. for a phase pressure of 6950 volts with a load of 120 amperes at a power-factor of 0.90, is: 4450+16 700 = 21 150 ats. For loads of other than unity power-factor, the most expedi- tious method of arriving at the results is usually that by graphical constructions. In the chart of Fig. 43 which relates to the graphical derivation of the saturation curve for 120 amperes at 0.90 power-factor, the diagrams in the right-hand column relate to the determination of the internal pressure. The first, second, third and fourth horizontal rows relate respectively to the diagrams for phase pressures of 7200, 5000, 2500 and volts. The left-hand vertical row of diagrams relates to the construc- tions for the determination of 6 for these four terminal pressures. From the internal-pressure diagrams in Figs. 42 and 43 and from the no-load saturation curve in Fig. 23 we find: Phase Pressure. Internal Pressure. Saturation Ats. 472 1000 2500 2800 5000 5000 5290 9300 6950 7250 16700 7200 7500 19200 From the diagrams w r e obtain the following results : Phase Pressure. Tan e. Angle 6. Sin 0. Ampere-turns Required to Offset Armature Demagnetization. 24 87.5 0.999 7200 2500 1.35 53.5 0.804 5780 5000 0.918 42.5 0.675 4850 6950 0.790 38.3 0.620 4450 7200 0.785 38.1 0.617 4440 CALCULATIONS FOR 2500-KVA. GENERATOR 65 Theta Diagrams Pressure Diagrams FIG. 43. Theta and Pressure Diagrams for 7 = 120 and = 0.90. We are now in a position to obtain the total ats. for each value of the phase pressure. The steps are shown in the following table : Phase Pressure. Saturation Ats. Ats. Required to Offset Armature Demagnetization. Total Ats. for 120 Amp. and 0.90 Power-factor. 1 000 7200 8200 2500 5000 5780 10780 5000 9300 4850 14150 6950 16 700 4450 21 150 7200 19200 4440 23640 66 POLYPHASE GENERATORS AND MOTORS These values for 120 amperes at 0.90 power-factor and those previously obtained for 120 amperes at unity power-factor, give us the two load-saturation curves plotted in Fig. 44. We see that for a phase pressure of 6950 volts, when the current is 120 amperes and at 0.90 power-factor, the required excitation is 21 150 ats. From the no-load saturation curve we find that an 8,000 7,000 I I of ? > oo- $ 3 3 g ? g g - g o- mmf per Field Spool, in ats FIG. 44. Saturation Curves for 7 = 120 and for (7 = 1.00 and 0.90. excitation of 21 150 ats. occasions, at no load, a phase pressure of 7600 volts 7600-6950 6950 X 100 = 9.4. The inherent regulation at 0.90 power-factor is, for this machine, 9.4 per cent. In other words, if, for an output of 120 amperes at a power-factor of 0.90 we adjust the excitation to such a value as to give a phase pressure of 6950 volts, and if, keeping CALCULATIONS FOR 2500-KVA. GENERATOR 67 the excitation constant at this value, the load is decreased to zero, the pressure will rise 9.4 per cent. ESTIMATION OF SATURATION CURVE FOR 120 AMPERES AT A POWER FACTOR OF 0.80 Now let us carry through precisely similar calculations for 120 amperes at a still lower power-factor, namely, a power-factor of 0.80. We shall first estimate the required mmf. (at 120 amperes and 0.80 power-factor) for phase pressures of 7200, 5000, 2500 and volts, and from these four results we can plot the required satu- ration curve. We have the relation; cos" 1 0.80 = 37.0. The reactance voltage and the internal IR drop remain the same as in the diagrams of Fig. 43. From these data we readily arrive at the diagrams of Fig. 45. From the internal-pressure diagrams in Fig. 45 and from the no-load saturation curve in Fig. 23, we arrive at the following results : Phase Pressure. Internal Pressure. Saturation Ats. 472 1000 2500 2850 5100 5000 5330 9400 7200 7560 20000 From the 6 diagrams in Fig. 45 we obtain the following results ; Phase Pressure. 0. Sin 6. Ats Required to Offset Armature Demagnetization. 87.5 0.999 7200 2500 60.2 0.868 6250 5000 51.0 0.777 5600 7290 47.5 0.737 5300 68 POLYPHASE GENERATORS AND MOTORS -82 ,82 ?=60.2 Theta Diagram Pressure Diagram FIG. 45. Theta and Pressure Diagrams for 7 = 120 and G = 0.80. From the data in the two preceding tables, we can obtain the total ats. for each value of the phase pressure. This is worked through in the following table: CALCULATIONS FOR 2500-KVA. GENERATOR 69 Phase Pressure. Saturation Ats. Ats. Required to Offset Armature Demagnetization. Total Ats. for 120 Amperes and 0.80 Power-factor. 1000 7200 8200 2500 5 100 6250 11 350 5000 9400 5600 15000 7200 20000 5300 25300 CALCULATION OF SATURATION CURVE FOR ZERO POWER FACTOR AND 120 AMPERES For 120 amperes at zero power-factor, the calculations are so simple that no diagrams need be drawn. The internal pressures are: 472 volts for a phase pressure of . . volts (466+2500 = )2966 volts for a phase pressure of 2500 ' ' (466+5000 = )5466 volts for a phase pressure of 5000 ' ' (466 +7200 = ) 7666 volts for a phase pressure of 7200 " Theta is Equal to 90 for Loads of Zero Power-factor. The angle 0, by which the conductors have passed beyond mid-pole- face position when the instant arrives at which they are carrying the crest current, is substantially 90 for all four cases, and consequently the ats. required to offset armature demagnetiza- tion remain constant at 7200. The calculations are completed in the following table: Phase Pressure. Internal Pressure. Saturation Ats. Ats. Required'to Offset Armature Demagneti- zation. Total Ats. for 120 Amperes for Zero Power-factor. 472 1000 7200 8200 2500 2966 5100 7200 12300 5000 5466 9900 7200 17100 7200 7666 23000 7200 30200 In Fig. 46 are given saturation curves for loads of 120 amperes at 1.00, 0.90, 0.80 and power-factors. In Fig. 47 the same results are thrown into a set of curves, each relating to a particular phase-pressure, the ordinates repre- senting excitation and the abscissae representing power-factor. 70 POLYPHASE GENERATORS AND MOTORS , Phase Pressure in Volts ,^-* 1^=* -^ ^=5 ^z ~ , " 3 / x x Xx x^ ^ " ( / /; ? / / / 7 / ^/ ^ J o/ 2 ^/ 'ft 1 I/ v/ u ^ // / // / fl / /J fra 7 m r IN/ w W\ B 8 5 I 1 g f j gj ^ g go" o" gj 3J mmf per Field Spool, in ats FIG. 46. Saturation Curves for 7 = 120 at Various Power-factors. 32,000 30,000 28,000 26,000 ..24,000 22,000 ^ 20,000 g 18,000 1*000 S 14,000 1 12,000 1 10,000 W 8,000 6,000 4,000 2,000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Power Factor FIG. 47. Curves showing the Required Excitation for 7 = 120 at Various Phase Pressures and Power-factors. CALCULATIONS FOR 2500-KVA. GENERATOR 71 THE EXCITATION REGULATION The inherent regulation is not the only kind of regulation which it is necessary to take into consideration in connection with the performance of a generator. There is also the " excitation regulation." This, for a given power-factor may be defined as the percentage increase in excitation which is required in order to maintain constant pressure when the out- put is increased from no-load to any particular specified value of the current. For our design we have estimated that for a phase pressure of 6950 volts at no load, the required excitation is : 14 400 ats. For this same phase pressure but with an output of 120 amperes per phase, the required excitations are : 17 140 ats. for G = 1.00 21 150 ats. for G = 0.90 22 600 ats. for G = 0.80 25000 ats. for G = The corresponding values of the excitation regulation are: = 19 - per Cent f r G = IW oi i KH _ 14400 KH _ 14400 \ 14 400 X 100= ) 46 ' 8 P er cent for = 0.90 (25 000-14 Excitation Regulation Curves. Curves plotted for given values of G, and of the phase pressure, with excitation as ordinates and with current output per phase as abscissae, are termed excitation regulation curves. We have values for such curves so far as relates to 7 = and 7 = 120, but with respect to higher values, we have but one point, namely : 7 = 240 Phase pressure = 6950 G = 1.00 Excitation = 23 000. 72 POLYPHASE GENERATORS AND MOTORS Let us work out corresponding values for 7 = 240 and with the other power-factors, namely, G = 0.90, G = 0.80 and = 0. OS68 SUIBJSBJQ For these power-factors and also for G=1.00, the theta and pressure diagrams are drawn in Fig. 48. With the values obtained from these diagrams the estimates may be completed as follows : CALCULATIONS FOR 2500-KVA. GENERATOR 73 Ats. for G. e Sin 6- Offsetting Armature Internal Phase Saturation Ats Total Ats Demagneti- Pressure. zation. 1.00 28.9 0.483 7000 7130 15600 22730 0.90 49.2 0.758 10900 7490 18900 26390 0.80 54.9 0.817 11800 7650 22200 29850 90 1.00 14400 7880 35000 43 000 The values in the last column and corresponding values already obtained for 7 = 120 and for 7 = 0, are brought together in the following table: Excitation for 6950 Volts and: rt 1=0. 7 = 120. I =240. 1.00 14 400 ats. 17 140 ats. 22 730 ats. 0.90 14400 " 21 150 " 26390 " 0.80 14400 '' 22600 " 29850 " 14400 " 25000 " 43000 " These values are plotted in the excitation regulation curves of Fig. 49. 42,000 40,000 38,000 36,000 34,000 32,000 30,000 28,000 26,000 24,000 22,000 20,000 18,000 16,000 14,000' 12,000 10,000 5 / / / / / / / ^ / f * ^1 ^ '. / ? / & ^ ^ / / \S ^ ; / / rr ^ o^ ^ ' /, ^ ^ ^. ^ , ' ^ ^^ ~-~* < ^ r f *~~~ 20 40 60 80 100 120 140 160 180 200 220 240 Amperes per Phase FIG. 49. Excitation Regulation Curves for 6950 Volts. 74 POLYPHASE GENERATORS AND MOTORS Saturation Curves for 240 Amperes. In the course of the previous investigation we have had occasion to obtain the excita- tion required for a phase pressure of 6950 volts and with an output of 240 amperes. These values are: G Excitation. 1.00 22 730 ats. 0.90 26390 " 0.80 29850 " 43000 " It is not necessary to indicate the steps in working out cor- responding values for 5000 and 2500 volts, and it will suffice to state simply that the values are those set forth below : Excitation for Phase Pressures of: Q Volts. 2500 Volts. 5000 Volts. 6950 Volts. 1.00 16 300 ats. 17 000 ats. 18 200 ats. 22 730 ats. 0.90 16300 " 19000 " 21 700 ' ' 26390 " 0.80 16300 " 19500 " 22700 " 29850 " 16300 " 20600 " 25400 " 43000 " These 240-ampere saturation curves are plotted in Fig. 50. 7,000 6,000 5,000 (3 3,000 * 2,000 1,000 rf ^ >' cd ^ y 3 jo" of o gj ,J ^ oo" g g| ^ Excitation in ats per Field Spool FIG. 50. Saturation Curves for Various Values of the Power-factor and for 7=240. CALCULATIONS FOR 2500-KVA. GENERATOR 75 VOLT-AMPERE CURVES From the data in Figs. 23, 46 and 50, relating respectively to saturation curves for 7 = 0, 7 = 120 and 7 = 240, we can construct curves which may be designated " volt-ampere " 8000 7000 .2 5000 I c 1 g 4000 3000 1000 \ \ \ 20 40 60 80 100 120 140 160 180 200 220 240 260 Current^per Phase (in Amperes) FIG. 51. Volt-ampere Curves for Various Power-factors and for a mmf. of 17 140 ats. per Field Spool. curves, since they are plotted with the phase pressure in volts as ordinates and with the current per phase, in amperes, as abscissae. For any particular volt-ampere curve the excitation and the power-factor are constants. For the volt-ampere curves in Fig. 51, the excitation is maintained constant at 17 140 ats., the mmf. required at 6950 volts, 120 amperes and unity power-factor. 76 POLYPHASE GENERATORS AND MOTORS Comment is required on the matter of the value at which the volt-ampere curves cut the axis of abscissae. This is seen to be at 254 amperes. For this current, the mmf. required to over- come armature demagnetization is obviously: 254 X 7200 = 15 300 ats. 120 Since the excitation is maintained constant at 17 140 ats., there is a residue of : 17 140 -15 300 = 1840 ats., and this suffices to provide the flux corresponding chiefly to the reactance of the end connections. We have seen on pp. 51 and 53 that the reactance of the end connections amounts to : (16.3XP-238 = )3.88 ohms. Consequently for 254 amperes the reactance voltage is: 254X3.88 = 990 volts. The IR drop is: 254X0.685 = 174 volts. The impedence voltage on short-circuit with 250 amperes is consequently : \/990 2 +174 2 = 1000 volts. From the no-load saturation curve of Fig. 23, we see that 1800 ats. are required for a phase pressure of 1000 volts. THE SHORT-CIRCUIT CURVE When the stator windings are closed on themselves with no external resistance, then the field excitation required to occasion a given current in the armature windings must exceed the armature mmf. by an amount sufficient to supply a flux corre- sponding to the impedance drop. The impedance is made up of two parts : CALCULATIONS FOR 2500-KVA. GENERATOR 77 THE REACTANCE OF THE END CONNECTIONS AND THE RESISTANCE OF THE WINDINGS We have seen (on p. 51) that the reactance of one phase at 25 cycles, is 16.3 ohms. Furthermore we have seen (on p. 53) that the reactance of the end connections is 23.8 per cent of this value, or: 0.238X16.3 = 3.88 ohms. Also we have seen (on p. 22) that the resistance per phase, at 60 C., is: 0.665 ohm. Consequently, at 25 cycles, the impedance is : V3.88 2 +0.665 2 = 3.94 ohms. For any particular value of the current, the impedance drop is obtained by multiplying the current by 3.94 ohms. Thus for 100 amperes we have an impedance drop of: 100X3.94 = 394 volts. From the no-load saturation curve of Fig. 23, we find that for a pressure of 394 volts per phase, a mmf . of 700 ats. per field spool is required. There are 25 turns per pole per phase. Consequently for a current of 100 amperes per phase, the armature mmf. amounts to: 2.4X25X100 = 6000 ats. Thus to send 100 amperes per phase through the short-circuited stator windings, there is required a mmf. of: 700+6000 = 6700 ats. per field spool. 78 POLYPHASE GENERATORS AND MOTORS Making corresponding calculations for 200 and 300 amperes we arrive at the following results, which are plotted in Fig. 52 : Current in Armature. Impedance Drop. Saturation mmf. Armature mmf. Total Required AtS. amp. volts Oats. Oats. Oats. 100 " 394 " 700 " 6000 " 6700 " 200 " 788 " 1400 " 12000 " 13400 " 300 " 1182 " 2100 " 18000 " 20100 " 20 000 i 18,000 16,000 14,000 12,000 10,000 8,000 6 000 x x xl x x X* ' x x X ^ ^ 4,000 2.000 X x X ^x 20 40 100 120 140 160 180 200 220 240 260 280 300 Current per Phase, in Amperes FIG. 52. Short-circuit Curve for 2500-kva. Alternator with an 18-mm. Air-gap. Up to 300 amperes, the short-circuit characteristic is a straight line, but at some exceedingly-high current values it may, in machines designed with strong fields, bend upward quite a little owing to saturation. Let us now investigate the effect of employing a lower mmf. in our design. INFLUENCE OF MODIFICATIONS IN THE NO-LOAD SATURATION CURVE Let us make the single change of decreasing the radial depth of the air-gap to 6 mm. in place of the original value of 18 mm. The component and resultant mmf. at no load will now differ from CALCULATIONS FOR %500-KVA. GENERATOR 79 those given in the table on p. 42 to the extent indicated in the following table: 6000 Volts. 6950 Volts. 7500 Volts. Air-gap 3340 ats. 3850 ats. 4180 ats. Teeth 90 " 330 " 720 " Magnet core 770 " 2000 " 5300 " Yoke Stator core 200 " 140 " 320 " 190 " 420 " 230 " Total mmf 4540 ats. 6690 ats. 10 850 ats. These values are plotted in the no-load saturation curve of Fig. 53. 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 t rH 5000 .S 4000 3000 2000 1000 'Excitation Coutant at 14.400 ats, 9,140 ats. \ l\ 20 40 60 80 100 120 140 160 180 200 220 240 Current per Phase, in Amperes. FIG. 55. Volt-ampere Curves for 2500-kva. Alternator with 6-mm. Air-gap. generators whose volt-ampere characteristics turn down nearly so abruptly as do those in Fig. 55 which represent these two values. In Fig. 55, the right-hand portions of the curves have been drawn dotted, as it has not been deemed worth while to carry through the calculations necessary for their precise predetermina- tion. > Let us now revert to our original design with the 18-mm. air-gap which we have shown to possess the more appropriate attributes. 84 POLYPHASE GENERATORS AND MOTORS THE DESIGN OF THE FIELD SPOOLS Our generator's normal rating is 2500 kva. at a power-factor of 0.90 and a phase pressure of 6950 volts. The current per phase is then 120 amperes. For these conditions the required excitation is: 21 150 ats. per field spool. The field spools must be so designed as to provide this mmf . with an ultimate temperature rise of preferably not more than 45 Cent, above the temperature of the surrounding air. The question of the preferable pressure to employ for exciting the field, is one which can only be decided by a careful considera- tion of the conditions in each case. The pressure employed in the electricity supply station for lighting and other miscellaneous purposes, is usually appropriate, although it is by no means out of the question that it may be good policy in some cases to provide special generators to serve exclusively as exciters. These exciters should, however, be independently driven. In other words, their speed should be independent of the speed of the generator for which the excitation is provided. It is the worst conceivable arrangement to have the exciter driven from the shaft of the alter- nator, as any change in the speed will then be accompanied by a more than proportional change in the excitation. In general, the larger the generator or the more poles it has, the higher is the appropriate exciting pressure. But it is difficult to make any statement of this kind to which there will not be many exceptions. Let us plan to excite our 2500-kva. generator from a 500-volt circuit and let us so arrange that when the machine is at its ultimate temperature of (20+45 = ) 65 Cent., 450 volts at the slip rings shall correspond to an excitation of 21 150 ats. The remaining (500450 = ) 50 volts will be absorbed in the con- trolling rheostat. It would not be prudent to plan to use up the entire available pressure of 500 volts when obtaining the mmf. of 21 150 ats., for this would leave no margin for discrepan- cies between our estimates and the results which we should actually obtain on the completed machine. CALCULATIONS FOR 2500-KVA. GENERATOR 85 Thus we have 450 volts for the eight spools in series, or: 450 r- = 56.3 volts per spool. o In Fig. 56 is shown a section through the magnet core and the spool. 1149mm-- Section on A-B Looking in Direction of Arrows FIG. 56. Sketches of Magnet Pole and Field Spool for 2500-kva. Alternator. The inner periphery of the spool is 26X71+2X60 = 82+120 = 202 cm. The outer periphery is 46X^+2X60 = 144+120 = 264 cm. For the mean length of turn we have : 202+264 mlt. = = 233 cm. 86 POLYPHASE GENERATORS AND MOTORS Suppose we were to provide our normal excitation of 21 150 ats. by means of a single turn carrying 21 150 amperes. For this excitation the pressure per spool is 56.3 volts. Consequently the pressure at the terminals of our hypothetical turn carrying 21 150 amperes is also the 56.3 volts allocated to each of the eight spools. The resistance of the turn must consequently be: 56 ' 3 0.00266 ohm. 21150 We have mlt. =233 cm. Therefore since the specific resistance of copper per centimeter cube, at 65 Cent., is 0.00000204, we have: 233X0.00000204 Cross-section of the conductor = U.UlLJbo = 0.179 sq.cm. Now if we were to provide the entire excitation by a single turn per spool as above suggested, the loss in field excitation would be: 500X21150 l kw - This is over five times the output of our machine. Consequently its efficiency would be low say some 18 per cent. But also, we should be running our conductor at a density of : 21 150 -f 7_^- = 118 000 amperes per sq.cm. u. 1 1 y and it would fuse long before this density could be reached. It would, in fact, fuse with a current of the order of only some 1000 amperes. Also the loss of 10 600 kw. in the field spools would suffice, even if the heat could be uniformly distributed through the whole mass of the machine, to raise it to an exceed- ingly high temperature. Even a very spacious engine room would be unendurable with so great a dissipation of energy taking place within it. CALC ULA TIONS FOR 2500-K VA . GENERA TOR 87 So let us look into the merits of employing 10 turns per spool instead of only one turn per spool. Since we require an excitation of 21 150 ats. per spool, the 21 150 current will, in this case, be only = 2115 amperes. Since the mlt. is equal to 233 cm., the 10 turns will have a length of 10X233 = 2330 cm. The resistance of the 10 turns must now be: fff -0.0266 ohm. Then we have : Q ,. 2330X0.00000204 ni7n Section = - = 0.179 sq.cm. This is the same value as before. In fact, for, firstly, a given pressure at the terminals of a spool; secondly, a given excitation to be provided, and thirdly, a given mlt., the cross-section of the conductor is independent of the number of turns employed to provide that excitation, and it is convenient to determine upon the cross-section by first assuming that a single turn will be employed. Obviously the greater the number of turns per spool, the less will be the current, and since the terminal pres- sure is fixed, the less also will be the power. Thus with our second assumption of a 10-turn coil, and only 2115 amperes in the exciting circuit, the excitation loss is reduced to: 500X2115 - =1060kw - and the efficiency of our 2500-kva. machine would rise to over 65 per cent. With the endeavor to obtain a reasonably low excitation loss, it is obviously desirable to employ as many turns as we can arrange in the space at our disposal. We have already seen that this space provides a cross-section of 10X20 = 200 sq.cm. Our conductor has a cross-section of 17.9 sq. mm. and should thus 88 POLYPHASE GENERATORS AND MOTORS have a bare diameter of 4.77 mm. The curves in Fig. 57 give the thicknesses of the insulation on single, double and triple cptton- covered wires of various diameters. If, in this case, we employ a double cotton covering, the insu- lated diameter will be 4.77 +(2X0. 18) =5.13 mm. The term 0.3 I 0.2 0.1 4P w ^ a 2.0 4.0 6.0 8.0 Diameter of Bare Conductor, in mm 10.0 12.0 FIG. 57. Thicknesses of Insulation on Cotton-covered Wires. " space factor " as applied to field spools, is employed to denote the ratio of the total cross-section of copper in the spool, to the gross area of cross-section of the winding space. Attainable values for the " space factor " of spools wound with wires of various sizes and with various insulations, are given in the curves in Fig. 58. For the case we are considering, we ascertain from the curves that the space-factor may be equal to 0.55. That is to say: 55 CALCULATIONS FOR 2500-KVA. GENERATOR 89 per cent of the cross-section of the winding space will be copper and the remaining 45 per cent will be made up of the insulation and the waste space. Thus the aggregate cross-section of copper will be 200X0.55 = 110 sq.cm. Consequently the number of turns is: 110 0.179 = 615. 0.5 0.4 0.2 0.1 1234 5678 Bare Diameter of Wire in mm FIG. 58. Curves showing "Space Factors" of Field Spools Wound with Wires of Various Diameters. For the normal magnetomotive force of 21 150 ats. per spool, the exciting current is: 21 150 615 = 34.4 amperes. The total excitation loss is: 500X34.4 = 17200 watts. 90 POLYPHASE GENERATORS AND MOTORS But of this loss of 17 200 watts: crj ^rX 17 200 = 1720 watts ouU are dissipated in the field regulating rheostat, and only : X 17 200 = 15 500 watts are dissipated in the field spools. The loss per spool is thus: 15500 o 1940 watts. The next step is to ascertain whether this will consist with a suitably-low temperature rise. The peripheral speed of our rotating field is: 375 -^- = 35.0 meters per second. At this speed and with this very open general construction with salient poles, it will be practicable to restrict the temperature rise to some 1.3 rise per watt per sq.dm. of external cylindrical radiating surface of the field spool. We have: External periphery of the spool = 26.4 dm. Length of spool = 2.0 dm. External cylindrical radiating surface = 26.4 X 2.0 = 52.8 sq.dm. Watts per sq.dm. = = 36.8. OZ.o Ultimate temperature rise = 36.8X1. 3 = 48 Cent. This is a high value and would not be in accordance with the terms of usual specifications. But the modern tendency is to take advantage of the increasing knowledge of the properties of insulating materials and to permit higher temperatures in low-pressure windings, provided offsetting advantages are thereby CALCULATIONS FOR 2500-KVA. GENERATOR 91 obtained. This is the case in moderate-speed and high-speed polyphase generators. The entire design profits in great measure by compressing the rotor into the smallest reasonable compass. In designing on such lines, the space available for the field spools is necessarily restricted. In the case of the design under con- sideration, we could, by decreasing the air-gap, decrease the required excitation and consequently also the temperature rise of the field spools. But we have already seen that the char- acteristics of the machine would be impaired by doing this. For a power-factor of 1.00 the mmf. required per field spool is only 17 200 ats. The loss in the field spool decreases as the square of the mmf. Consequently were the machine required for an output of 2500 kva. at exclusively unity power-factor, the temperature rise would be only; 1720Q\ 2 ,21 1507 X 48 = 32 Cent. Thus for an putput of 2500 kilowatts at unity power-factor, the field spools are actually 33 per cent cooler than for an output of only 2250 kilowatts at a power-factor of 0.90 (i.e., for an out- put of 2500 kva. and a power-factor of 0.90). If on the contrary, the machine were required for an output of only 2000 kilowatts but at a power-factor of 0.80 (again 2500 kilovolt-amperes), the temperature rise of the field spools would be: It is impressive to note that although this output (2000 kilo- watts at 0.80 power-factor), is 20 per cent less than an output of 2500 kilowatts at unity power-factor, the temperature rise of the field spools is: K4 _ 00 ^f^X 100 = 69 per cent greater. The losses and temperature rise in the other parts of the machine will be the same for both these conditions, since the kilovolt-ampere output of the machine is 2500, in both cases. 92 POLYPHASE GENERATORS AND MOTORS It is the object of this treatise to explain methods of design, and consequently no particular advantage would be gained by carry- ing through modified calculations for the purpose of providing field spools which would permit of carrying the rated load at 0.90 power-factor with a temperature rise of only 45 Cent, instead of a temperature rise of 48 Cent. An inspection of Figs. 21 and 56 shows that there is room for more spool copper should its use appear desirable. It would also be practicable to decrease the internal diameter of the magnet yoke from the present 928 mm. down to say 878 mm. and increase the radial length of the magnet core (and consequently also the length of the winding space), by 25 mm, THE CORE LOSS In polyphase generators, the core losses may be roughly pre- determined from the data given in Table 10. TABLE 10. DATA FOR ESTIMATING THE CORE Loss IN POLYPHASE GENERATORS. Density in Stator Core in Lines per Square Centimeter. Core Loss in Stator Core, in kw. per (Metric) Ton for Various Periodicities. ^=15. ~=25. ^=50. 6000 1.1 2.2 5.0 8000 1.7 3.0 7.4 10000 2.2 4.0 10.0 12000 2.6 5.2 14000 3.2 6.2 In the case of the 2500-kva., 25-cycle generator which we have taken for our example, the density in the stator core is 10 000, lines per square centimeter. The core loss will thus be on the basis of some 4.0 kw. per (metric) ton. We must now estimate the weight of the stator core: Estimation of Weight of Stator Core. External diameter of stator core = 230 cm. Internal diameter of stator core = 178 cm. CALCULATIONS FOR 2500-KVA. GENERATOR 93 Area of the surface of an annular ring with the above external and internal diameters is equal to: |(230 2 - 178 2 ) = 16 700 sq.cm. Area of the 120 slots = 120X5.72X2.41 = 1650 sq.cm. Net area of surface of stator core plate = 16 700 1650 = 15 050 sq.cm. In = 79.2 cm. 79 2X15 050 Volume of sheet steel in stator core = ' =1.19 cu.m. 1 UUU UUU Weight of 1 cu.m. of sheet iron = 7.8 (metric) tons. Weight of stator core = 7.8Xl. 19 = 9.3 (metric) tons. On the basis of a loss of 4.0 kw. per ton, we have: Core loss = 4.0X9.3 = 37.2 kw. Friction Loss. No simple rules can be given for estimating the windage and bearing friction loss. The ability to form some rough idea of the former can, in a design of this type, only be acquired by long experience. It must suffice to state that, for the present design, a reasonable value is: Windage and bearing friction loss = 20 kw. Excitation Loss. The loss in the exciting circuit is made up (seep. 90) of: Loss in regulating rheostat = 1.7 kw. Loss in field spools = 15. 5 kw. Stator I 2 R Loss. The resistance of the stator winding (see p. 22) is: 0.665 ohm per phase. 94 POLYPHASE GENERATORS AND MOTORS and consequently we have: Stator PR loss at rated load = 3 - X 12 1 ^' 665 = 28.6 kw. 1UUU Total Loss. The total loss at full load is the sum of these various losses: I. Stator 7 2 # = 28 600 watts II. Field spool I 2 R = 15500 " III. Field rheostat PR = 1700 " IV. Core loss = 37200 " V. Friction loss = 20 OOP " Total loss at full load=" 103000 ' ' Output at full load =2250000 ' ' Input at full load = 2 353 000 ' ' 2250 Full -load efficiency = ^^X 100 = 95.6 per cent. CONSTANT AND VARIABLE LOSSES Of the five component losses, the last four remain fairty con- stant at all loads, whereas the first loss (the stator PR loss) varies as the square of the load. It is true that the sum of the second and third losses decreases slightly with decreasing load, but in the present machine the total decrease is only in the ratio of the mmf. at full load and no load. The twommf.are: 21 150 ats. at rated load. 14 400 ats. at no load. The corresponding values of the total excitation loss are : 17 200 watts at full load (and 6950 volts per phase) and 200 - 1 70 watts at no load < and 695 volts P er phase) CALCULATIONS FOR 2500-KVA. GENERATOR 95 Thus the sum of the last four losses decreases from: 17 200+37 200+20 000 = 74 400 watts at full load down to 11 700+37 200+20 000 = 68 900 watts at no load. This decrease only amounts to 74400-68900 74400 X 100 = 7.4 per cent. Thus, taken broadly, we may take the last four component losses as making up an aggregate which we may term the " con- stant loss," and in contradistinction we may term the first com- ponent the " variable loss." In our design we have: Variable loss =28 600 watts. Constant loss = 74 400 watts. If we ignore the 7 per cent decrease in the constant loss, we may readily obtain the efficiencies at various loads. The method will be clear from an inspection of the following estimates: EFFICIENCY AT ONE-FOURTH OF FULL LOAD Variable loss = 0.25 2 X 28 600 =1 800 watts Constant loss = 74400 " Total loss at one-fourth of full load= 76200 " Output = 0.25 X 2 250 000 = 564 000 " Input =640200 watts Kfi4. nnn rj(at one-fourth load) = ~ = 0.882. 96 POLYPHASE GENERATORS AND MOTORS EFFICIENCY AT HALF LOAD Variable loss = 0.50 2 X 28 600 = 71 50 watts Constant loss = 74400 " Total loss at half load = 81550 ' Output at half load = 1 125 000 " Input = 1 207 000 watts ij(at half load) EFFICIENCY AT 50 PEK CENT OVERLOAD Variable loss = 1 .50 2 X 28 600 = 64 500 watts Constant loss = 74400 " Total loss at 50 per cent overload = 138 900 " Output at 50 per cent overload =3 380 000 " Input = 3 519 000 watts ooorj rj(at 50% overload) = = 0.961 . LOAD CORRESPONDING TO MAXIMUM EFFICIENCY When the variable losses have increased until they equal the constant losses, the efficiency will be at its maximum. The corresponding load is : 400 X 2250 = 1.61X2250 = 3640kw. The efficiency is then: 3640+74.4+74.4 3789 0.962. CALCULATIONS FOR 2500-KVA. GENERATOR 97 From this point upward, the efficiency will decrease. This brief method of estimating the efficiencies at several loads, gives slightly too low results at low loads and slightly too high results at high loads. But the errors are too slight to be of practical importance; in fact the inevitable errors in deter- mining the component losses are of much greater magnitude. In Fig. 59 is plotted an efficiency curve for the above-cal- culated values which correspond to a power-factor of 0.90. 1000 2000 3000 4000 5000 Output in Kilowatts FIG. 59. Efficiency Curve of 2500-kva. Alternator for G=0 90. DEPENDENCE OF EFFICIENCY ON POWER-FACTOR OF LOAD Let us consider the load to be maintained at 2500 kva. but with different power-factors. The field excitation will be: For G= 1.00: For G = 0.90: For G = 0.80: 17200 21 150 22500 21 150 X 17 200 = 14 000 watts. 17 200 watts. X 17 200 = 18 300 watts. POLYPHASE GENERATORS AND MOTORS The total losses become: G 1.00 0.90 0.80 Total Losses 99 800 watts 103 000 " 104 100 " The outputs, inputs and efficiencies become: G. Output Corresponding to 2500 kva. Input. n 1.00 2500 kw. 2600 kw. 0.962 0.90 2250 " 2353 " 0.956 0.80 2000 " 2104 " 0.949 This brings out the importance of employing a considerable proportion of over-excited synchronous motors to offset the lag- ging load corresponding to induction motors. This question is further discussed in Chapter VI, entitled " Synchronous Motors versus Inductor Motors." CHAPTER III POLYPHASE GENERATORS WITH DISTRIBUTED FIELD WINDINGS THE type of polyphase generator with salient poles which has been described in the last chapter has served excellently as a basis for carrying through a set of typical calculations. Salient-pole generators are chiefly employed for slow- and medium- speed ratings. But for the high speeds associated with steam- turbine-driven sets, rotors with distributed field windings are practically universally employed in modern designs. It is not proposed to carry through the calculations for a design of this type. While there are a good many differences in detail, to which the professional designer gives careful attention, the underlying considerations are quite of the same nature as in the case of salient-pole designs. In Figs. 60 and 61 are shown photographs of a salient-pole rotor and a rotor with a distributed field winding. The former (Fig. 60), is for a medium speed (514 r.p.m.), water-wheel generator with a rated capacity for 1250 kva. The latter (Fig. 61), is for a 750- r.p.m., steam-turbine-driven set, with a rated capacity for 15000 kva. The former has 14 poles and the latter 4 poles. The former is for a periodicity of 60 cycles per second, and the latter for a periodicy of 25 cycles per second. An inherent characteristic of high-speed sets relates to the great percentage which the sum of the core loss, windage and excitation bears to the total loss. In our 2500-kva. salient-pole design for 375 r.p.m., the core loss amounted to some 37 000 watts, the windage and bearing friction to 20 000 watts, and the excitation to 17 000 watts, making an aggregate of 74 000 watts for the " constant " losses, out of a total loss at full-load of 103 000 watts. But in a design for 2500 kva. at 3600 r.p.m., (i.e., for very nearly 10 times as great a speed), the core loss, 99 100 POLYPHASE GENERATORS AND MOTORS bearing-friction, windage, and excitation would together amount to some 72 000 watts out of a total of some 80 000 watts. A FIG. 60. Salient Pole Rotor for a 14-pole, 1250-kva., 60-cycle, 514 r.p.m., 3-phase Alternator, built by the General Electric Co. of America. representative distribution of the losses for a 2500-kva., 0.90- power-f actor, 12 000- volt polyphase generator would be: Armature PR loss 8 000 watts Excitation PR loss 9 000 " Core loss 32000 Windage and bearing friction loss 31 000 Total loss at full load 80 000 " Output at full load 2 250 000 Input at full load 2 330 000 " Efficiency at full load 96.6 per cent WITH DISTRIBUTED FIELD WINDINGS 101 A result of the necessarily large percentage which the "constant" losses bear to the total losses, is that the efficiency falls off badly with decreasing load, In this instance we have: Variable losses = 8 kw. Constant losses = 72 kw. The efficiencies at various loads work out as follows: Load. Efficiency. 88.5 per cent i 93.9 " 1.00 . 96.6 " FIG. 61. Rotor with Distributed Field Winding for a 4-pole, 15 000-kva., 25-cycle, 750 r.p.m., 3-phase Alternator, built by the General Electric Co. of America. On pp. 94 to 96 the efficiencies of the 375-r.p.m. machine for this same rating were ascertained to be: Load. i 1.00. . Efficiency. 88.2 per cent 93.1 " . 95.6 " 102 POLYPHASE GENERATORS AND MOTORS For a 100-r.p.m., 2250-kw., 0.90-power-f actor, 25-cycle design, the efficiencies would have been of the following order: Load. Efficiency. J 88.0 per cent i 92.6 " 1.00 94.6 " The values may be brought together for comparison as follows : EFFICIENCIES Load. 100 r.p.m. 375 r.p.m. 3600 r.p.m. a 88.0 88.2 88.5 i 92.6 93.1 93.9 1.00 94.6 95.6 96.6 While it is within the designer's power to modify the inherent tendencies corresponding to the rated speeds, nevertheless the values above set forth are representative. Thus while a full- load efficiency of 96.6 per cent is obtained for the 3600-r.p.m. design, the full-load efficiency of the 100-r.p.m. design is only 94.6 per cent. At quarter load the efficiency is practically as high for the 100-r.p.m. design as for the 3600-r.p.m. design. In judging of improvements in efficiency, it is the decrease in the percentage of losses which should be considered. Thus when the efficiency is increased from 94.6 per cent to 96.6 per cent, the losses are decreased from 5.4 per cent of the input to 3.4 per cent of the input. The decrease in the losses is thus: per Cent< In a polyphase generator driven by a steam turbine, it is now almost universal practice totally to enclose the generator, except so far as relates to the provision of suitable inlets and outlets for the circulating air, which is usually driven through the machine by means of fans located on the rotor. Consequently, in this type of machine, a step in the calculation relates to esti- mating the supply of air required suitably to limit the temperature WITH DISTRIBUTED FIELD WINDINGS 103 rise, and so to proportion the passages as to transmit the air in the quantities thus ascertained to be necessary. In our 3600-r.p.m., 2250-kw. generator, the losses at full load amount to 80 kw. If the heat corresponding to this loss is to be carried away as fast as it is produced, then we must circulate sufficient air to abstract: 80 kw.-hr. per hr. A convenient starting point for our calculation is from the basis that: 1.16 w.-hr. raises 1 kg. of water 1 Cent. The specific heat of air is 0.24; that is to say, it requires only 0.24 times as much energy to raise 1 kg. of air by 1 degree Cent, as is required to raise 1 kg. of water by 1 degree Cent. Consequently, to raise by 1 degree Cent., the temperature of 1 kg. of air, requires the absorption of : f 1.16X0.24 = 0.278 w.-hr. One kilogram of air at atmospheric pressure and at 30 degrees Cent, occupies a volume of 0.85 cu.m. Therefore, to raise 1 cu.m. of air by 1 degree Cent, requires: If, for the outgoing air, we assume a temperature 25 degrees above that of the ingoing air, then every cu.m. of air circulated through the machine will carry away: 0.327X25 = 8.2 w.-hr. We must arrange for sufficient air to carry away: 80 000 w.-hr. per hour. 104 POLYPHASE GENERATORS AND MOTORS \ Consequently we must supply: 80000 8.2 or: = 9800 cu. m. per hour; 9800 1ft0 -- = 163. cu. m. per minute. In dealing with the circulation of air it appears necessary to make the concession of employing other than metric units. We have: 1 cu.m. = 35.4 cu.ft. Therefore in the case of our 2250-kw. generator, we must circulate : 163X35.4 = 5800 cu.ft. per min. CHAPTER IV THE DESIGN OF A POLYPHASE INDUCTION MOTOR WITH A SQUIRREL-CAGE ROTOR THE polyphase induction motor was brought to a commer- cial stage of development about twenty years ago. Many tens of thousands of such motors are now built every year. The design of polyphase induction motors has been the subject of many elaborate investigations and there has been placed at the disposal of engineers a large number of practical rules and data. The design of such a motor may proceed from any one of many starting points and each designer has his preferred method. The author proposes to indicate the method which he has found to be the most useful for his purposes. It must not be inferred that any set of rules can be framed which will lead with certainty to the best design for any particular case. The most which can be expected is that the rules shall lead to a rough preliminary design which shall serve to fix ideas of the general orders of dimen- sions. Before he decides upon the final design, the enterprising designer will carry through a number of alternative calculations in which he will deviate in various directions from the original design. A consideration of the several alternative results at which he will thus arrive, will gradually lead him to the most suitable design for the case which he has in hand. The method of design will be expounded in the course of working through an illustrative example. ILLUSTRATIVE EXAMPLE Let it be required that a three-phase squirrel-cage induction motor be designed. The normal rating is to be 200 hp. and the motor is to be operated from a 1000-volt, 25-cycle circuit. It is desired that its speed shall be in the close neighborhood of 500 r.p.m. 105 106 POLYPHASE GENERATORS AND MOTORS Determination of the Number of Poles. Denoting the speed in revolutions per minute by R, then the speed in revolutions -p per second is equal to ~~. If we denote the number of poles by P, and the periodicity in cycles per second by ^, then we have: In our case we have : ~ = 25 72 = 500. Therefore : 2X60X25 500 If we design the motor with 6 poles, the " synchronous " speed will be 500 r.p.m. At no-load, the motor runs at practi- cally its " synchronous " speed; that is to say, its " slip " is practically zero. The term " slip " is employed to denote the amount by which the actual speed of the motor is less than the " synchronous " speed. An appropriate value for the slip of our motor at its rated load, is some 2 per cent, or even less. Taking it for the moment as 2 per cent, we find that, at rated load, the speed will be 500-0.02X500 = 490 r.p.m. The variations in the speed between no load and full load are so slight that at many steps in the calculations the speed may be taken at the approximate value of 500 r.p.m., thus avoiding superfluous refinements which would merely complicate the calculations and serve no useful purpose. Rated Output Expressed in Watts. Since one horse-power is equal to 746 watts, the rated output of our*200-h.p. motor may also be expressed as 200X746 = 149 200 watts. Determination of T, the Polar Pitch. The distance (in cm.) measured at the inner circumference of the stator, from the center of one pole to the center of the next adjacent pole, is termed the " polar pitch " and is denoted by the letter T. Rough pre- POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 107 liminary values for T are given in Table 11 for designs for 25 and 50 cycles for a wide range of outputs and speeds. These values have been found by experience to be appropriate. TABLE 11. PRELIMINARY VALUES FOR T (THE POLAR PITCH) FOR USE IN DESIGNING THREE-PHASE INDUCTION MOTORS Rated Output. 25 Cycles. H.P. Watts. P=4. R =750. P=6. #=500. P=8. R =375. P=10. R =300. P = 12. R =250. P=14. R=218. 5 10 20 3720 7460 14900 17.0 19.0 22.0 14.0 16.0 19.0 12.0 14.0 17.0 13.0 15.5 14.0 40 60 80 29900 44700 59700 26.0 28.0 30.0 22.0 24.0 26.0 20.0 22.0 24.0 18.0 20.0 21.5 17.0 19.0 20.5 16.0 18.0 19.0 100 150 200 74600 112000 149000 31.5 35.0 38.0 27.0 30.0 32.5 24.5 27.0 29.5 23.0 25.0 27.5 21.5 23.5 26.0 20.0 22.0 24.0 300 400 500 224000 299000 373000 43.0 47.0 50.0 37.0 40.0 42.0 33.0 36.0 37.5 30.5 33.0 34.5 29.0 31.0 32.5 27.0 29.0 30.0 Rated Output 50 Cycles. ! HP. Watts P = 4 72=1500 P=6 72=1000 P = 8 72 = 750 P=10 R = 600 P=12 72=500 I P=14 72 = 429 P=16 72 = 375 P=20 72=300 P = 24 72 = 250 P = 28 72 = 214 5 10 20 3720 7460 14900 16.0 17.0 19.0 13.0 14.0 16.0 11.0 12.0 14.0 11.0 13.0 12.0 11.5 40 60 80 29900 44700 59700 22.0 24.0 25.5 19.0 20.5 22.0 17.0 18.5 19.5 15.5 17.0 18.0 14.5 16.0 17.0 13.5 15.0 16.0 13.0 14.5 15.5 13.5 14.5 13.5 100 150 200 74600 112000 149000 27.0 29.5 31.5 23.0 25.0 27.0 20.5 22.5 24.5 19.0 21.0 23.0 18.0 20.0 21.5 17.0 18.5 20.5 16.0 17.5 19.5 15.0 16.5 18.0 14.0 15.5 17.0 13.5 15.0 16.9 300 400 500 224000 299000 373000 35.0 38.0 40.0 30.0 32.5 34.5 27.0 29.5 31.5 25.0 27.0 29.0 24.0 25.5 27.0 22.5 24.0 25.5 21.5 23.0 24.0 20.0 21.5 22.5 18.5 20.0 21.5 17.5 19.0 20.5 For our 200-h.p., 6-pole, 25-cycle design, we find from Table 11, the value: cm. 108 POLYPHASE GENERATORS AND MOTORS THE OUTPUT COEFFICIENT The next step relates to the determination of a suitable value for ?, the " Output Coefficient," which is defined by the following formula : w in which W = Rated output in watts, (which is equal to 746 times the rated output in h.p.), D = Diameter at air-gap, in decimeters, i.e., the internal diameter of the stator, Xg = Gross core length, in decimeters, R = rated speed, in revolutions per minute. As in the earlier chapters of this treatise, the symbols D and Xgf will sometimes be employed for denoting respectively the air-gap diameter and the gross core length, expressed, as above, in decimeters, but more usually they will denote these quantities as expressed in centimeters. The student can soon accustom himself to distinguishing, from the magnitudes of these quan- tities, whether decimeters or centimeters are intended, and thus will not experience any difficulty of consequence, in this double use of the same symbols. Values of 5 suitable for preliminary assumptions are given in Table 12. In this table, is given as a function of P and T. For our design we have : p = 6, T = 32.5 cm. The corresponding value of ? in Table 12 is about 2.0. But let us for our design be satisfied with a less exacting value and take: = 1.80. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 109 Therefore : 1.80. 1492 149 200 TABLE 12. PRELIMINARY VALUES FOR (THE OUTPUT COEFFICIENT) FOR USE IN DESIGNING THREE-PHASE SQUIRREL-CAGE INDUCTION MOTORS. (The figures at the heads of the vertical columns give the numbers of poles.) r, the polar pitch (in cm.)! 4. 6. : 8. 10. 12. 14. 16. 18. 20. 22. 24. 11 0.80 1.07 1.12 1.15 1.14 1.12 1.11 1.10 1.08 1.05 12 0.95 1.14 1.19 1.21 1.20 1.20 1.18 1.16 1.14 1.12 14 1.13 1.28 1.30 1.31 1.30 1.28 1.26 1.26 1.25 1.25 16 0.85 1.28 1.40 1.41 1.41 1.41 1.40 1.40 1.40 1.40 1.40 18 1.11 1.40 1.50 1.50 1.51 1.51 1.51 1.51 1.51 1.51 1.51 20 1,27 1.52 1.59 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 22 1.42 .63 1.69 1.70 1.70 1.70 1.70 1.70 1.70 1.69 1.69 24 1.53 .72 1.78 1.79 1.80 1.80 1.80 1.80 1.80 1.80 1.80 26 1.63 .82 1.86 1.87 1.87 1.87 1.88 1.88 1.88 1.88 1.88 28 1.71 .90 1.93 1.93 1.94 1.94 1.95 1.95 1.95 1.95 1.95 30 1.79 .97 1.99 2.00 2.00 2.01 2.01 2.01 2.02 2.02 2.02 35 1.90 2.08 2.10 2.11 2.11 2.11 2.12 2.12 2 12 2.12 2.12 40 1.98 2.17 2.20 2.20 2.21 2.21 2.22 45 2.04 2.21 2.26 2.28 50 2.07 2.23 2.30 But D (in crn.), is equal to: Therefore : 6X32.5 110 POLYPHASE GENERATORS AND MOTORS Therefore :. 166 (in dm.) = r- dm. or 43.0 cm. D and \g and T are the three characteristic dimensions of the design with which we are dealing. D 2 \g (with D and \g expressed in decimeters), is also, in itself, a useful value to obtain at an early stage of the calculation of a design. We have, for our motor; PRELIMINARY ESTIMATE OF THE TOTAL NET WEIGHT A rough preliminary idea of the total net weight of an induc- tion motor may be obtained from a knowledge of its D 2 \g. The " Total Net Weight " may be taken as the weight exclusive of slide rails and pulley. In Table 13, are given rough repre- sentative values for the Total Net Weights of induction motors with various values of D 2 \g. TABLE 13. VALUES OF THE TOTAL NET WEIGHT OF INDUCTION MOTORS. 10 20 40 60 80 100 150 200 250 300 350 400 Total Net Weight in Metric Tons, (i.e., in Tons of 2204 Lbs.). 0.27 0.40 0.73 0.98 1.20 1.40 1.90 2.30 2.70 3.00 3.25 3.45 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 111 PRELIMINARY ESTIMATE OF THE TOTAL WORKS COST The Total Works Cost will necessarily also be a very indefinite quantity, susceptible to. large variations with variations in the proportions and arrangement of a design. It is even more greatly dependent upon the equipment and management of the Works at which the motor is manufactured. Neverthless, some rough indication appropriate for squirrel-cage induction motors is afforded by the data in Table 14. TABLE 14. TOTAL WORKS COST OF SQUIRREL-CAGE INDUCTION MOTORS. Total Net weight of Motor in Metric Tons. 0.20 0.40 0.60 0.80 1.00 1.50 2.00 2.50 3.00 3.50 4.00 Total Works Cost per Ton, in Dollars. 310 300 290 285 280 270 260 250 240 230 225 For our 200-h.p. motor, we have D 2 \g = 166. From Table 13, we ascertain that the Total Net Weight is some 2.00 tons. From Table 14, it is found that the Total Works Cost is of the order of $260. per ton. Consequently we have Total Works Cost -2.00X260 = $520. ALTERNATIVE METHOD OF ESTIMATING THE TOTAL WORKS COST An alternative method of estimating the T.W.C. of an induction motor is based on the following formula: TWC (in dollars) =KxDXfrg+0.7i), 112 POLYPHASE GENERATORS AND MOTORS where D, Xg, and T are given in centimeters. K is obtained from Table 15. TABLE 15 VALUES OF K IN FORMULA FOR T. W. C. Air-gap Diameter, D. in Centimeters. Values of K. 10 20 40 0.098 0.105 0.112 60 80 100 0.120 0.128 0.133 150 200 0.140 0.148 In the 200-h.p. motor which is serving us as an example, we have: T = 32.5, 0.7T = 22.7, D = 62.0, K (from Table 15) =0.122, TWC = 0.122X62X65.7 = $508. Thus the results by the two methods are $520 and $508 which are in good agreement with each other. Usually the agreement will be far less close and an average of the two values is preferable as a guide. It is interesting to note that the TWO per rated horse-power is, in the case of this motor, some: 514 -*2 57 200" The motor could not, of course, be bought at any such price, for the TWO merely covers all the costs incurred up to the delivery of the completed machine to the shipping department, to be packed. Selling and shipping expenses, as also profits, must be added to the TWC to arrive at the price at which the POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 113 machine should be sold and it is not unusual to find the selling price over double the Total Works Cost. This is one of the penalties of the competitive system of supplying the needs of mankind. The Peripheral Speed. It is well, before proceeding further with the design, to calculate the peripheral speed. Let us denote by S, the peripheral speed, expressed in meters per second, R ' 100 A 60' For our design : xX62X500 ' 100X60 = meters P er second. In this instance the peripheral speed is very low, and does not constitute a limiting consideration from the point of view of mechanical strength. For other speeds, ratings, and peri- odicities, the preliminary data as derived from the rules which have been set forth, might lead to an undesirably-high peripheral speed. Consequently it is well to ascertain the peripheral speed at an early stage of the calculations and arrange to reduce D and T in cases where the electrical design ought to be sacrificed in some measure in the interests of improving the mechanical design. PERIPHERAL LOADING We shall next deal with the determination of the number of conductors to be employed. The product of the number of conductors and the current per conductor, (i.e., the ampere- conductors), constitutes a quantity to which the term " peripheral loading " may be applied. Designers find from experience that it is desirable to employ certain definite ranges of values for the peripheral loading per centimeter of periphery, measured at the air-gap. In Table 16, are given values which will serve as pre- liminary assumptions for a trial design. 114 POLYPHASE GENERATORS AND MOTORS TABLE 16. PRELIMINARY ASSUMPTIONS FOR THE PERIPHERAL LOADING OF AN INDUCTION MOTOR. 5 (in Centimeters). Stator Ampere Conductors per Centi- meter of Gap Periphery. 25 Cycles. 50 Cycles. 15 20 25 140 180 220 180 220 270 30 40 60 270 320 370 310 350 400 80 100 120 380 390 400 420 430 440 For motors wound for very low pressures, (say 250 volts), somewhat higher values may be employed, especially in the larger sizes. On the contrary, for motors wound for high pres- sures say 2500 volts or more it is necessary to employ for the ampere conductors per centimeter of periphery, lower values than those indicated in the table, especially in motors of very small diameter. Indeed the loss of space in providing for slot insulation renders it very undesirable to wind small motors for very high pressures. It is better, in such cases, to interpose step-down transformers between the supply system and the motor or motors. Experienced designers will find occasions where the periphery may, and should, be loaded with ampere conductors much more highly than corresponds with the data in Table 16. On the other hand, there are often difficult ratings, (as regards overload capacity and other features which will later come in for con- sideration), where the peripheral loading should (and must) be much lower than the values indicated in Table 16. As a matter of fact, the rated speed and output, the periodicity and pressure, and also the stipulated instantaneous overload capacity, all require to be taken into consideration. But at this early stage in the design, it is useful to take a value from Table 16. For our case, D is equal to 62, and the corresponding value for the peripheral loading is seen to be 372 ampere conductors per centimeter. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 115 Consequently the total peripheral loading is: 62XxX372 = 72 500 ampere-conductors; or 72 500 ~- = 24 200 ampere-conductors per phase. From this product we wish next to segregate the ampeers and the conductors. We may do this by dividing the ampere- conductors by the full-load current per phase. We cannot estimate the precise value of the full-load current per phase until the design has been completed, as its precise value depends upon the efficiency and power-factor at full load. But we cannot complete the design without determining upon the suitable number of conductors to employ. Hence it becomes necessary to have recourse to tables of rough approximate values for the full-load efficiency and the power-factor of designs of various ratings. Such values are given in Tables 17 and 18. EFFICIENCY AND POWER-FACTOR For the case of squirrel-cage induction motors for moderate pressures, and in the absence of any specially exacting require- ments as regards capacity for carrying large instantaneous over- loads, we may proceed from the basis of the rough indications in Tables 17 and 18. From these tables we obtain: Full-load efficiency = 91 per cent. Full-load power-factor = 0.91. The required estimation of the full-load current may be carried out as follows: Horse-power output at rated load = 200 Watts output at rated load = 200 X 746 = 149 200 Efficiency at rated load =0.91 1 4Q 200 Watts input at rated load - = 164 000 u.yi 116 POLYPHASE GENERATORS AND MOTORS Watts input per phase at rated load Power-factor at rated load 164 OOP 3 0.91 = 54700 54 700 Volt-amp, input per phase at rated load = -777^- = 60 200 u.yi Pressure between terminals (in volts) = 1000 Phase pressure Current per phase at rated load 60200 577 = 104 TABLE 17. PRELIMINARY VALUES OF FULL LOAD EFFICIENCY, IN PER CENT, FOR POLYPHASE SQUIRREL-CAGF INDUCTION MOTORS. THE VALUES GIVEN CORRESPOND TO THOSE OF NORMAL MOTORS. Rated Output in Horse- power. Efficiencies for the Following Periodicities and Synchronous Speeds. Periodicity =12.5 Cycles. Periodicity =25 Cycles. Periodicity =50 Cycles. 375 188 94 750 375 188 1500 750 375 r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. 10 76.7 72.8 65.0 82.5 79.1 74.6 84.1 80.6 75.7 20 79.0 74.7 69.4 84.0 80.6 76.6 85.5 82.6 78.1 30 80.7 77.3 72.3 85.0 82.0 78.1 87.3 84.1 80.4 40 81.4 78.0 73.5 86.5 83.3 79.8 88.8 86.0 82.1 50 83.0 80.0 75.6 87.5 84.5 81.1 90.0 87.3 84.0 60 84.0 81.1 77.1 88.0 85.5 82.5 90.8 88.3 85.3 80 86.0" 83.3 80.1 89.8 87.3 84.8 92.5 90.3 .87.8 100 87.0 85.0 82.1 91,3 88.8 86.5 93.8 91.8 89.3 300 89.0 87.0 84.8 92.5 90.8 88.8 95.0 93.5 90.8 500 89.8 88.0 86.0 93.2 91.5 89.8 95.5 94.0 91.8 700 90.5 88.8 87.0 93.8 92.3 90.5 95.7 94.3 92.3 1000 91.0 89.8 88.5 94.0 92.8 91.3 95.8 95.5 92.8 Conductors per phase. For a preliminary estimate we may proceed as follows: Conductors per phase Conductors per pole per phase 24200 104 232 = 232. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 111 We must round off this value to some suitable whole number, say 39 or 40, taking whichever leads to the best arrangement of the winding. An inspection indicates that we should consider the following alternatives: 40 conductors arranged 8 per slot in (~o~= ) 5 slots. " ^2_\ ' U / P \o (u r 40 39 40 10 13 20 TABLE 18. PRELIMINARY VALUES FOR FULL-LOAD POWER FACTOR OF POLY- PHASE SQUIRREL-CAGE INDUCTION MOTORS OF NORMAL DESIGN. Power-factor for the Following Periodicities and Synchronous Speeds. Rated Output Periodicity = 12.5 Cycles. Periodicity =25 Cycles. Periodicity =50 Cycles. in Horse- power. 375 188 94 750 375 188 1500 750 375 r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m. r.p.m 10 0.896 0.820 0.770 0.882 0.820 0.760 0.898 0.830 0.750 20 0.906 0.829 0.780 0.892 0.830 0.770 0.904 0.840 0.760 30 0.916 0.838 0.790 0.900 0.837 0.780 0.908- 0.847 0.770 40 0.921 0.847 0.800 0.906 0.844 0.790 0.914 0.852 0.780 50 0.928 0.855 0.810 0.911 0.850 0.800 0.917 0.857 0.790 60 0.931 0.864 0.820 0.916 0.856 0.810 0.921 0.862 0.800 80 0.941 0.872 0.830 0.924 0.862 0.820 0.926 0.867 0.810 100 0.947 0.880 0.840 0.930 0.871 0.830 0.928 0.872 0.820 300 0.955 0.887 0.850 0.938 0.880 0.840 0.931 0.876 0.830 500 0.960 0.896 0.860 0.942 0.886 0.850 0.932 0.880 0.840 700 0.962 0.904 0.870 0.945 0.892 0.860 0.932 0.884 0.850 1000 0.963 0.912 0.880 0.946 0.900 0.870 0.932 0.888 0.860 Number of Slots per Pole per Phase. There cannoj; be given any absolute rule as regards the number of slots per pole per phase which should be employed. In a general way it may be 118 POLYPHASE GENERATORS AND MOTORS stated that the quality of the performance of the motor is higher, the greater the number of slots per pole per phase. But the overall dimensions and the weight and the Total Works Cost increase with increasing subdivision of the winding amongst many slots, and consequently the designer should endeavor to arrive at a reasonable compromise between quality and cost. The Slot Pitch. We may designate as the slot pitch the dis- tance (measured at the air-gap) from the center line of one slot to the center line of the next adjacent slot. Since this quantity is usually small, it is generally convenient to express it in mm. Good representative values for the stator slot pitch are given in Table 19. The values in the table may be taken as applying to designs for moderate pressures. The higher the pressure, the more must one depart from the tabulated values in the direc- tion of employing fewer slots, TABLE 19. VALUES OF STATOR SLOT PITCH FOR INDUCTION MOTORS. T, the Polar Pitch (in cm.). Stator Slot Pitch, (in mm.) 12 15.0 14 16.4 16 17.7 18 18.9 20 20.0 25 22.3 30 24.4 35 26.0 40 27.0 From Table 19 we select as appropriate for our motor, a trial slot pitch of about 25 mm. Since T is equal to 325 mm., we should have: 325 25 = 13 slots per pole. But the number finally chosen should be a multiple of 3, the number of phases. Consequently let us employ 12 slots per pole, POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 119 (12 \ = ) 4 slots per pole per phase (pppp). Thus we must o / revise the stator slot pitch to: = 27.1 mm. \& The Total Number of Stator Slots. Since our machine has 6 poles, we shall have: 12X6 = 72 stator slots. The appropriate slot layout will be based on 10 conductors per slot and : 4X10 = 40 conductors pppp. and 6X40 10 _, ^ = 120 turns in series per phase. & Let us denote the turns in series per phase by T. Then: ^=120. THE PRESSURE FORMULA In our discussion of the design of generators of alternating electricity we have become acquainted with the pressure formula : V=KXTX~XM. In this formula we have: y = the phase pressure in volts; il = a coefficient; T = turns in series per phase; ~ = periodicity in cycles per second; M = flux per pole in megalines. For a motor, the phase pressure in the above formula must, for full load, be taken smaller than the terminal pressure, to the extent of the IR drop in the stator windings. But at no load 120 POLYPHASE GENERATORS AND MOTORS the phase pressure is equal to the terminal pressure divided by V3. Therefore, 1000 Phase pressure = :r-, = 577 volts. The coefficient K depends upon the spread of the winding and the manner of distribution of the flux. For the conditions pertaining to a three-phase induction motor with a full-pitch winding we have: # = 0.042. For other winding pitches, the appropriate value of K may be derived by following the rule previously set forth on pp. 16 to 18 of Chapter II, where the voltage formula for generators of alternating electricity is discussed. Since our motor is for operation on a 25-cycle circuit, we have: Thus at no load we have: 577 = 0.042X120X25XM; M = 4.57 megalines. THE MAGNETIC CIRCUIT OF THE INDUCTION MOTOR In Fig. 62 are indicated the paths followed by the magnetic lines in induction motors with 2, 4, and 8 poles. One object of the three diagrams has been to draw attention to the dependence of the length of the iron part of the path, on the number of poles. Thus while in the 2-pole machine, some of the lines extend over nearly a semi-circumference, in the stator core and in the rotor core; their extent is very small in the 8-pole design. As a consequence, the sum of the magnetic reluctances of the air-gap and teeth constitutes a greater percentage of the total magnetic reluctance, the greater the number of poles. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 121 Fig. 63 has been drawn to distinguish that portion of the total flux which corresponds to one pole. It is drawn to correspond to a 6-pole machine. 2 Pole 4 Pole 8 Pole FIG. 62. Diagrammatic Sketches of 2-, 4-, and 8-pole Induction-motor Cores, showing the Difference in the Mean Length of the Magnetic Path. At this stage of our calculations, we wish to ascertain the cross- section which must be allowed for the stator teeth. A crest density of 15 500 lines per sq. cm. is appropriate for the stator FIG. 63. Diagrammatic Representation of that Portion of the Total Path which Corresponds to One Pole of an Induction Motor. [The heavy dotted lines indicate the mean length of the magnetic path for one pole.] teeth in such a design as that which we are considering. A skilled designer will, on occasions, resort to tooth densities as high as 19 000, but it requires experience to distinguish appropriate cases 122 POLYPHASE GENERATORS AND MOTORS for such high densities and the student will be well advised to employ lower densities until by dint of practice in designing, he is competent to exercise judgment in the matter. In general the designer will employ a lower tooth density the greater the number of poles. This is for two reasons: firstly, as already mentioned in connection with Fig. 62, the magnetic reluctance of the air- gap and teeth constitutes a greater percentage of the total reluc- tance the greater the number of poles; and secondly, (for reasons which will be better understood at a later stage), a high tooth density acts to impair the power-factor of a machine with many poles, to a greater extent than in the case of a machine with few poles. In Figs. 64 and 65 are shown two diagrams. These represent the distribution of the flux around the periphery of our 6-pole motor at two instants one-twelfth of a cycle apart. Since the periodicity is 25 cycles per second, one-twelfth of a cycle occupies ( ^= Wfoth f a second. After another g^th of a second the flux again assumes the shape indicated in Fig. 64, but dis- placed further along the circumference, as indicated in Fig. 66. In other words, as the flux travels around the stator core, its distribution is continually altering in shape from the typical form shown in Fig. 64, to that shown in Fig. 65, and back to that shown in Fig. 66 (which is identical with Fig. 64, except that it has advanced further in its travel around the stator) . Successive positions of the flux, each ^th second later than its predecessor, are drawn in Figs. 67 to 70. Comparing Fig. 70 with Fig. 64, we see that they are identical except that in Fig. 64, a south flux occupies those portions of the stator, which, in Fig. 64, were occupied by a north flux. In other words, a half cycle has occurred in the course of the (dhr^iroth of a second which has elapsed while the flux has traveled from the position shown in Fig. 64, to that shown in Fig. 70. A whole cycle will have occurred in u^th of a second (the periodicity is 25 cycles per second) ; and the flux will then have been displaced to the extent of the space occupied by one pair of poles. At the end of the time occupied by 3 cycles (*\ths of a second) the flux will have completed one / (\ \ revolution around the stator core, since the machine has ( 9 = ) 3 pairs of poles. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 123 nates are Arbitrary Figures Proportional to the Flux Density around the Air Gap s l.S l.G 1.1 1.2 1.0 0.8 O.G 0.4 0.2 ' l 5 / \ / \ 1 \ 1 \ / \ / \ \ 1 \ / \ 1 i f \ i _? L \ 7 5 A c 1 _^ d \ 1 ^ \ 1 j / \ / \ 1 \ / 1 \ 1 \ \ j 1 \ S 1.8 1.6 1.4 1.2 1.0 0.8 O.G 0.4 0.2 ^ ^ C C f ^ f ^ n / \ \ \ / \ / s \ f / ^ /I / \ / \ / \ / \ / \ I \ A r I < \ A J \ / " \^ / I I \ t v / I ^ I \ / \ 1 \ 1 \ 1 \ / \ 1 \ / \ / \ 1 \ 1 \ 1 \ I S o S 1.8 l.G 1.4 1.2 1.0 0.8 O.G 0.4 0.2 / \ 1 \ / 5 / \ 1 \ / \ / \ \ \ \ I \ \ \ 1 I / i P [ i } V 1 / \ / \ / \ / \ 1 \ I \ I \ / \ / \ / \ 1 \ \ 5 o S 1.8 l.G 1.4 1.2 1.0 0.8 0.6 0.4 0.2 ( C c c f 1 (_ \ / \ \ 1 \ / \ 1 \ \ \ 1 I / \ 1 \ 1 \ 1 L 1 \ 1 \ 1 \ \ \ \ \ 1 V S *> 1 i / ! / A 1 !. i 1 V 1 \ 1 \ I \ / \ 1 \ 1 \ \ \ \ \ j \ \ 1 S g 1.8 1.6 1.4 1.2 1.0 0.8 O.G 0.4 0.2 \ \ I \ / ^ / \ / \ 1 \ / i r \ \~ ~? r \\ -c \ \ \ \ ^ 1 V !5 ] 1 \ 1 1 \ 1 1 / \ 1 \ 1 \ 1 \ j \ 1 \ 1 \ \ \ \ \ 2 i 1.8 l.C 1.4 1.2 1.0 0.8 O.G 0.4 0.2 j f f ^ / \ / \ / \ / \ / \ / \ j \ The Ordi / \ / / V 1 \ / \ / / \ / / \ \ / \ / 3 i \J S / J 5 1 \ 1 \ / \ \ 1 \ \ 1 V / \ / \ 1 1 \ g o S 1.8 1.6 1.4 1.2 1.0 0.8 O.G 0.4 0.2 ^ \ \ / \ i \ / \ / \ / \ / \ 1 \ I ~l \ 1 \ / \ t - i J i *( ! 1 / \ / / 1 \ / \ / \ / \ / \ / / \ / \ / \ \ \ 1 * 0.5 1 1.5 2 2.5 3 Cycles per Sec. i I 1 i i i i U 50 25 50 25 10 25 Time in Seconds FlGS. 64 to 70. Diagrams Flux as it Indicating the Variations in the Shafce of the Around the Stator Core. 124 POLYPHASE GENERATORS AND MOTORS In estimating the magnetomotive force (mmf.) which must be provided for overcoming the reluctance of the magnetic circuit, we must base our calculations on the crest flux density. This corresponds to the flux distributions represented in Figs. 65, 67 and 69. It can be shown * that the crest density indicated in these figures is 1.7 times the average density. In other FIG. 71. Diagram Illustrating that the Crest Density in the Air-gap and Teeth of an Induction Motor is 1.7 Times the Average Density. words, the crest density with the flux distribution correspond- ing to the peaked curve in Fig. 71, is 1.7 times the average * This is demonstrated, step by step, on pp. 380 to 390 of the 2d edition of the author's "Electric Motors" (Whittaker & Co., London and New York, 1910;, POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 125 density indicated by the rectangle in the same figure. Each tooth is, in turn, located at the center of the rotating flux and is in turn subjected to this crest density. Thus for our assumption of a crest density of 15 500 lines per square centimeter we shall have : 15 500 Average density in stator teeth = ^ = 9100 lines per sq.cm. Since we have a flux of 4.57 megalines, we must provide, per pole, a tooth cross-section of: 4570000 _ nn = 500 sq.cm. There are 12 stator teeth per pole. The cross-section of each tooth must thus be: - = 41.6 sq.cm. Before we can attain our present object of determining upon the width of the tooth, we shall have to digress and take up the matter of the proportioning of the ventilating ducts. VENTILATING DUCTS The employment of a large number of ventilating ducts in the cores of induction motors, renders permissible, from the temperature standpoint, the adoption of much higher flux den- sities and current densities than could otherwise be employed, and thus leads to a light and economical design. In Table 20 are given rough values for the number of ducts, each 15 mm. wide, which may be taken as suitable, under various circumstances of peripheral speed and values of \g. In our case, where the peripheral speed is 16.2 meters per second and Xg is 43, the table indicates 1.8 ducts per dm., or a total of 126 POLYPHASE GENERATORS AND MOTORS 1.8X4.3 = 7.7 ducts, to be a suitable value. Eight ducts will be employed and they will require .8X1.5 = 12.0 cm. TABLE 20. VENTILATING DUCTS FOR INDUCTION MOTORS. Peripheral Speed in Meters per Second. Number of Ventilating Ducts (Each 15 mm. Wide), which Can Appropriately be Used, per Decimeter of \g. \g=W. X0=30. X0=50. 10 2.2 2.3 2.4 15 1.7 1.9 2.1 20 1.5 1.7 1.9 25 1.3 1.5 1.7 30 1.1 1.3 1.5 35 1.0 1.2 1.3 40 0.9 1.1 1.2 The varnish by means of which adjacent core plates are insulated from one another, will occupy some 10 per cent of the total depth occupied by the insulated core plates. Thus for Xw, the net core length, we arrive at the value: Xn=(43-12)X0.9 = 3lX0.9 = 27.9 cm. WIDTH OF STATOR TOOTH We may now complete the calculation of the width of the stator tooth at its narrowest part. The tooth will be of the form indicated in Fig. 72, and the narrowest part (or neck) will be at a diameter not appreciably greater than the air-gap diameter. It is not import- FIG. 72. Stator Tooth r^ . n , . ,, of 200 H.P. Induction ant tO be s P eciall y exact m the matter > Motor, so let us make the rough preliminary assumption that the diameter at this narrowest part is 1 cm. greater than D, the air-gap diameter. We have: D = 62.0 cm. I <-14.9mro-> h 22m POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 127 Diameter to neck of tooth =62.0+1.0 = 63.0 cm. Tooth pitch at neck = ~72~ = ^7.5 mm. Required cross-section of tooth at neck = 4 1.6 sq.cm. Width of tooth at neck = -= = -7^77- = 14.9 mm. An 279 Width of Slot. The slot will have parallel sides, and its width will be: 27.5 -14.9 = 12.6 mm. This will be its width when punched. Owing to inevitable slight inaccuracies in building up the stator core from the individual punchings, the assembled width of the slot will be some 0.3 mm. less, or 12.6-0.3 = 12.3 mm. This allowance of 0.3 mm. is termed the " slot tolerance." Having now determined the width of the stator slot, it would appear in order to proceed at once to determine its depth. But this depends upon the copper contents for which space must be provided. Consequently we must now turn our attention to the determination of: The Dimensions of the Stator Conductor. The current density in the stator conductor is determined upon as a compro- mise amongst a number of considerations, one of the chief of which is the permissible value of the watts per square decimeter (sq.dm.) of peripheral radiating surface at the air-gap. This value is itself influenced by such factors as the peripheral speed and the ventilating facilities provided, hence the current density will also be influenced by these considerations. The value of the watts per square decimeter of peripheral radiating surface at the air-gap, cannot, unfortunately, be ascer- tained until a later stage when we shall have determined not only the copper losses, but also the core loss. If, at that later stage, the value obtained for the watts per square decimeter of peripheral radiating surface at the air-gap, shall be found to be unsuitable, it will be necessary to readjust the design. 128 POLYPHASE GENERATORS AND MOTORS Table 21 has been compiled to give preliminary representa- tive values for the stator current density for various outputs and peripheral speeds, for designs of normal proportions. TABLE 21. PRELIMINARY VALUES FOR THE STATOR CURRENT DENSITY. Rated Output in h.p. Current Density for Various Peripheral Speeds in Meters per Second (mps). 10 mps. 20 mps. 30 mps. 40 mps. 5 400 10 380 400 50 350 370 390 100 320 340 360 380 500 290 320 340 350 1000 280 300 310 The Slot Insulation. The fact that the slot has an assembled width of 12.3 mm. is not to be taken as indicating that this width is available for the conductors. The iron core must be separated from the conductors by a considerable thickness of insulation. This insulation consists preferably in specially-manufactured tubes of high-grade insulating material. Suitable thicknesses are indicated in Table 22. TABLE 22. VALUES FOR THE THICKNESS OF THE STATOR SLOT LINING. Normal Pressure (in Volts) for Which the Induction Motor is Wound. Thickness of Slot Lining in mm. 500 0.9 1000 1.4 2000 2.3 3000 2.9 4000 3.3 6000 4.0 8000 4.7 10000 5.2 12000 5.6 In Fig. 73 a sketch is given of the slot with the insulating tube in place, and before winding. The precise depth has not POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 129 yet been determined, but the width of the available space is seen to be 12.3 2X1.4 = 9.5 mm. Thus we see that a space 9.5 mm. wide is available for the insulated conductors. From Table 21 we ascertain that the stator conductor should be proportioned for about 300 amperes per square centimeter. Since we have, I 12.3 mm > for the full-load current, 7= 104 amperes, the cross-section should be : 104 = 0.347 sq.cm. or 34.7 sq.mm. oUU In the interests of securing a suit- able amount of flexibility in the process of winding, let us divide this aggregate cross-section into two conductors which shall be in parallel and each of which shall have a cross-section of some: - = 17.4 sq.mm. The diameter of a wire with a cross- section of 17.4 sq.mm. is: D = 4.70 mm. -9.5 -1.4mm FIG. 73. Stator Slot of 200 H.P. Induction Motor, showing Insulating Tube in Place. The bare diameter of each wire would, on this basis, be 4.70 mm. In Table 23 are given the thicknesses of insulation on suitable grades of cotton-covered wires employed in work of this nature. We see that our wire of 4.70 mm. diameter would, if double cotton covered, have a thickness of insulation of about 0.18mm. Consequently its insulated diameter would be: 4.70+2X0.18 = 4.70+0.36 = 5.06 mm. But the width of the winding space is seen from Fig. 73 to be only 9.5 mm. The natural arrangement in this case, would be 130 POLYPHASE GENERATORS AND MOTORS to place the two components side by side. Consequently the insulated diameter must not exceed: . -^- = 4.75 mm. TASLE 23. VALUES OF THE THICKNESS OF COTTON COVERING. Diameter of Bare Conductor (in mm.). Thickness of Insulation (in mm.). Single Cotton Covered. Double Cotton Covered. Triple Cotton Covered. 1 0.060 . 0.100 2 0.080 0.127 0.180 3 0.098 0.150 0.207 4 0.112 0.167 0.227 5 0.123 0.183 0.244 6 0.133 0.196 0.258 8 0.147 0.214 0.279 10 0.220 0.288 12 0.290 So let us reduce the bare diameter to : 4.75-0.36 = 4.39 mm. The readjusted conductor has a cross-section of only: 7X4.39 2 = 15.1 sq.mm. A The two component conductors make up a cross-section of: 2X0.151 = 0.302 sq.cm. The revised current density is: 104 ^-57^ = 345 amperes per sq.cm. This density is considerably higher than the value in Table 21, but we have made a rather liberal provision for ventilating ducts POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 131 and there is no reason, at this stage, to anticipate that we have exceeded permissible values. However, this must be put to the test at a later stage when we shall have sufficient data to estimate the temperature rise on the basis of the watts lost per square decimeter of peripheral radiating surface at the air-gap. Since we must provide for 10 conductors (20 component wires) per slot, the height of the winding space must be at least 10X4.75 = 47.5 mm. But it will be impracti- cable to thread the wires into place with complete avoidance of any lost space. So let us add 5 per cent to the height of the winding space, making it: L4mm 10 Ins. Conds. 47.5 mm 1.4mm 2.2mm 2.5mm 54.0mm 1.05X47.5 = 50 mm. The slot with the wires in place, is drawn in Fig. 74. It is seen from this figure that the total depth is 54 mm. The slot open- ing is 6 mm. wide. The Slot Space Factor. The total area of cross- section of copper in the slot amounts to 10X0.302 = 3.02 sq.cm. The product of depth and punched width of slot is equal to 1.26X5.40 = 6.80 sq.cm. 3 02 Space factor of stator slot = r 1 ^ = 0.445. b.oU It is to be distinctly noted that this slot design is merely a preliminary layout. Should it at a later stage not be found to fulfil the requirements as regards sufficiently-low temperature- FIG. 74. Stator Slot of 200 H.P. Motor with Winding in Place. 132 POLYPHASE GENERATORS AND MOTORS rise at rated load, it will be necessary to consider ways and means of so modifying the design as to fulfil the requirements. Preliminary Proportions for the Rotor Slot. For reasons which will appear later, there will be a number of rotor slots not differing greatly from the number of stator slots and these rotor slots will be of about the same order of depth as the stator slots, but considerably narrower. The result will be that the rotor tooth density will be fully as low or. even lower than the stator tooth density. Let us for the present, con- sider that the rotor slots are 54 mm. deep and that the crest density in the rotor teeth is, at no load, 15 500 lines per square cen- timeter. Let us further assume for the present that the rotor slots are nearly wide open, the shape of a rotor slot being somewhat as FIG. 75. Rotor Slot indicated in Fig. 75. In the final design, of 200 H.P.Squirrel- the width of the rotor slot opening may cage Motor. readily be so adjusted as to constitute about 20 per cent of the rotor tooth pitch at the surface of the rotor, the tooth surface thus constituting some 80 per cent of the tooth pitch. Determination of Cross-section of Air-gap. The stator tooth pitch at the air-gap is: 620 Xic 72 = 27.1 mm. Stator slot opening = 6.0 mm. The stator tooth surface thus constitutes: 27 l 6 ' 271 X 100 = 78.0 per cent of the stator slot pitch. Considering the average value of this percentage on both sides of the air-gap, we find it to be : 78.0+80.0 = 79.0 per cent. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 133 If there were no slot openings, the cross-sections of the surfaces from which the lines of each pole emerge and into which they enter, could be taken as Xn X T. But the slot openings bring this cross-section down to 0.79XXnX-r. In crossing the air-gap, however, the lines spread out and may be considered to occupy a greater cross-section when half way across. They then gradually converge as they approach the surfaces of the teeth at the other side of the gap. To allow for this spreading, we may increase the cross-section by 15 per cent, bringing it up to 1.15X0.79XXnXT. For our 200-h.p. induction motor we have: \n = 27.9 cm., T = 32.5 cm. Cross-section of air-gap = 1.15X0.79X27.9X32.5 = 825 sq.cm. 4 570 000 Average air-gap density (at no load) = 5^ = 5550 lines per sq. cm. 825 Crest density = 1 .7 X 5550 = 9450. Radial Depth of Air-gap. Let us denote the radial depth of the air-gap in mm. by A. Appropriate preliminary values for A are given in Table 24. TABLE 24. APPROPRIATE VALUES FOR A THE RADIAL DEPTH OF THE AIR- GAP FOR INDUCTION MOTORS. D, the Air-gap Diameter, A, the Radial Depth of the Air-gap (in mm.), for Various Values of the Peripheral Speed in Meters per Second (mps.). (in cm.). 10 mps. 20 mps. 30 mps. 40 mps. 20 0.65 0.75 0.87 1.00 40 0.87 1.05 1.25 1.45 60 1.10 1.35 1.70 1.90 80 1.30 1.7 2.0 2.3 100 1.57 2.0 2.8 2.8 120 1.77 2.3 2.8 3.3 134 POLYPHASE GENERATORS AND MOTORS i For our 200-h.p. motor, the air-gap diameter is 62 cm. and the peripheral speed is 16.2 mps. Consequently the radial depth of the air-gap is A = 1.3 mm. Preliminary Magnetic Data for Teeth and Air-gap. We have now obtained (or assumed) the densities in the teeth and in the air-gap and we have the lengths of these portions of the mag- netic circuit. These data are: Length (in cm.) of Portions of Magnet Circuit. Crest Density at no Load (in Lines per sq.cm.). Stator teeth Rotor teeth 5.4 5 4 15500 15500 Air-gar) 0.13 9450 The magnetomotive force (mmf.) required to overcome the reluctance of the above-tabulated portions of the magnetic circuit, will, in most designs, constitute a predominatingly-large percentage of the total mmf. As the designer gains experience, he will often be able, after calculating this portion of it, safely to use his judgment in assigning (without detailed calculations), a suitable and relatively-small amount, to provide for the mmf. required for overcoming the reluctance of the stator and rotor cores. In the great majority of cases, this further amount con- stitutes so small a percentage of the total mmf. per pole, that a very considerable divergence from the value which would be obtained by detailed calculations, would not seriously influence the total. That the length of the magnetic circuit in the stator and rotor cores is much less the greater the number of poles has already been pointed out on page 120 and has been illustrated by the sketches of the 2-, 4-, and 8-pole magnetic circuits in Fig. 62. Until, however, considerable experience has been gained in calculating the magnetic circuits of induction motors, it is not desirable to trust to obtaining sufficient accuracy by the process of multiplying by a suitable factor the mmf. required for the teeth and the air-gap. It is, however, not necessary to consume time in accurately estimating the mean length of the magnetic path in the stator and rotor cores. On the contrary, it suffices to adopt the assumption that the portions corresponding to one pole may be POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 135 taken as equal to the circumferences corresponding to the mean core diameter of the stator and rotor respectively, divided by 2P, where P is the number of poles. But as a step toward obtaining these values, it is necessary to obtain the external diameter of the stator core discs and the internal diameter of the rotor core discs. These data, in turn, are dependent upon the densities which should be employed behind the slots in the stator and rotor respectively. Densities in Stator and Rotor Cores. The densities to be employed in the stator and rotor cores are quantities which may be varied between wide limits. In general, however, the core densities should be lower, the greater the value of T the polar pitch, and ~ the periodicity. For preliminary assumptions, the values given in Table 25 will be found suitable. TABLE 25. DENSITIES IN STATOR AND ROTOR CORES. Periodicity in Cycles per Second. Values of Polar Pitch (in cm.) . Density in Magnetic Lines per sq.cm. Stator. Rotor. i, j 15 20 30 and larger 11500 10 500 10000 14000 13500 13000 , { 15 20 30 and larger 10000 9000 8500 13000 12500 12000 In the present instance we find from the above table that for a periodicity of 25 cycles per second and a polar pitch of 32.5 cm., the stator density should be 10 000 lines per square centimeter, and the rotor density 13 000 lines per square centimeter. Since the total flux (at no load) is 4.57 megalines per pole, the cross-sections required in the stator and rotor cores are respectively : Cross-section of stator core = 4 570 OOP 2X10000 229 sq.cm. ^ .. , 4570000 1I7 . Cross-section of rotor core = Artn = 176 sq.cm. ^ X J-O UUU ~kn is equal to 27.9 cm. 136 POLYPHASE GENERATORS AND MOTORS Consequently : Radial depth of stator punchings (exclusive of slot depth) 229 Radial depth of rotor punchings (exclusive of slot depth) 176 27.9 = 6.3 cm. External diameter of stator punchings = 62.0+2X5.4+2X8.2 = 62.0+10.8+16.4 = 89.2 cm. Internal diameter of rotor punchings = 62.0-2X0.13-2X5.4-2X6.3 = 62.0-0.26-10.8-12.6 = 38.3 cm. Diameter at bottom of stator slots = 62.0+2X5.4 = 72.8 cm. Diameter at bottom of rotor slots = 62.0 -2X0.13 -2X5.4 = 50.9 cm. 89.2+72.8 Mean diameter of stator core ~ =81.0 cm. Mean diameter of rotor core '^ L =44.6 cm. Length of sta. mag. circ. per pole = ' -=21.2 cm. z X o Length of rotor mag. circ. per pole= '- = 11.7 cm. ^ X Compilation of Diameters. It is of interest at this stage to draw up an orderly list of the leading diameters: External diameter of stator core 892 mm. Diameter at bottom of stator slots 728 mm. Internal diameter of stator (D) 620 mm. External diameter of rotor (D 2A) 617.4 mm. Diameter at bottom of rotor slots 509 mm. Internal diameter of rotor core. . . 383 mm. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 137 Sketch of Magnetic Portions of Design. We are now in a position to make a preliminary outline drawing of the magnetic parts of our machine. This has been done in Fig. 76 which shows the leading dimensions of the magnetic portions of the design and indicates the locations of the end connections of the stator windings. Magnetic Reluctance of Sheet Steel. In the design of the induction motor, our magnetic material is exclusively sheet steel. For this material the mmf. data in the first two columns of the table previously given on page 33 will give conservative results. Tabulated Data of Magnetic Circuit. We now have the lengths of the magnetic paths, the densities, and also data for ascertaining the mmf. required at all parts of the magnetic circuit. Thus for example: Density in stator core = 10 000 lines per square centimeter; Corresponding mmf. from column 2 of table on page 33 = 4.6 ats. per centimeter Length of magnetic circuit in stator core (per pole) =21.2 cm. Mmf. required for stator core = 4.6X21. 2 = 98 ats. As a further illustration we may give the calculation of the mmf. required for the air-gap: Crest density in air-gap = 9450 lines per sq.cm. Corresponding mmf. =X 9450 = 0.8X9450 = 7550 ats. per cm. Length of magnetic circuit in air-gap =0.13 cr . Mmf. required for air-gap = 7550X0. 13 =980 ats. These illustrations will suffice to render clear the arrangement of the calculations in the following tabulated form: TABLE 26. ARRANGEMENT OF MMF. CALCULATIONS. Part. U) Length of Mag. Circ. in cm. Density in Lines per sq.cm. ,<*) mmf. per cm. (AXB) Total mmf. Stator teeth. . Rotor teeth . . Air-gap 5.4 5.4 13 15500 15500 9450 22 22 7550 119 119 980 Stator core . . Rotor core. . . 21.2 11.7 10000 13000 4.6 9.5 98 111 Total mmf . per Dole 1427 ats. 138 POLYPHASE GENERATORS AND MOTORS POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 139 (980 \ since Tjoy X 100 = 68.5 j, the air-gap mmf. is 68.5 per cent of the total required mmf. The mmf. required for air-gap and teeth is: 980+119+119^., 1218 inA 1427 - X 100 = ^^X100 = 85.5 per cent of the total required mmf. Attention is drawn to these percent- ages to bear out the correctness of the assertion on page 134 that considerable inaccuracy in the estimation of the mean length of the magnetic circuit in the stator and rotor cores will not seriously affect the accuracy of the result obtained for the total mmf. per pole. Consequently the use of the rough but time- saving rule to divide by twice the number of poles, the mean periphery of these cores, is shown to be justified. Resultant mmf. of the Three Phases Equals Twice the mmf. of One Phase. It is a property of the three-phase windings of induction motors that the resultant mmf. of the three phases is twice that exerted by one phase alone. Consequently in our 1427 design, each phase must contribute a mmf. of 75 = 714 ats. 2i In our design, T, the number of turns in series per phase, is 120 equal to 120. The design has 6 poles. Thus we have -^- = 20 turns per pole per phase. Magnetizing Current. The magnetizing current per phase which will suffice to provide the required 714 ats. must obviously amount to : 714 = 35.7 crest amperes or 35 7 = = 25.2 effective amperes. Since the full-load current is 104 amperes, the magnetizing cur- 25 2 rent is -rX 100 = 24. 2 per cent of the full-load current. 140 POLYPHASE GENERATORS AND MOTORS No-load Current. The no-load current is made up of two components, the magnetizing current and the current correspond- ing to the friction and windage loss and the core loss, i.e., to the energy current at no load. It may be stated in advance that the energy current at no load is almost always very small in compar- ison with the magnetizing current. Since, furthermore, the mag- netizing current and the energy current differ from one another in phase by 90 degrees, it follows that their resultant, the no- load current, will not differ in magnitude appreciably from the magnetizing current. The calculation of the energy current is thus a matter of detail which can well be deferred to another stage. But to emphasize the relations of the quantities involved, let us assume that the friction, windage and core loss of this motor will later be ascertained to be a matter of some 4500 watts. This (1000 \ j=- J577 volts. Consequently the energy component of the current con- 1500 sumed by the motor at no load is -^== = 2.6 amperes. The no-load current thus amounts to V25.2 2 +2.6 2 = 25.3 amperes- In other words the no-load current and the magnetizing current differ from one another in magnitude by less than one-half of one per cent, in this instance. Although of but slight practical importance, it may be interesting to show that they differ quite appreciably in phase. For we have for the angle of phase differ- ence between the no-load current and the magnetizing current: 9 fin tan- 1 =" tan- 1 0.103 = 5.9. Thus the current in this motor when it is running unloaded, lags (90 5.9 = ) 84.1 behind the pressure; in other words, its power-factor is equal to (cos 84.1 = ) 0.104. y, the Ratio of the No-load Current to the Full-load Current. It is convenient to designate by y, the ratio of the no-load current to the full-load current. For our motor we have: T-Sg-0.242. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 141 Practical use will be made of this ratio at a later stage in the design of this motor. H, the Average Number of Slots per Pole. It is also con- venient to adopt a symbol for the average of the number of slots per pole on the stator and rotor. We have definitely determined upon the use of 12 slots per pole on the stator. We have reserved to a later stage of the calculations the determination of the precise number of rotor slots. However it may here be stated that it is preferable to employ a number of rotor slots not widely differ- ing from the number of stator slots. Thus as a preliminary assumption we may take 12 as the average of the numbers of slots per pole on statoi and rotor. Designating this quantity by H we have ; THE CIRCLE RATIO We have now all the necessary data for determining a quantity for which we shall employ the symbol a and which, for reasons which we shall come to understand as we proceed, we shall term the " circle ratio." This quantity is of great utility to the practical designer. Although a cannot be pre- determined with any approach to accuracy, it so greatly assists one's mental conceptions from the qualitative stand- point as to make ample amends for its quantitative uncertainty. We have seen that at no load, the current consumed by an induction motor lags nearly 90 degrees behind the pressure. Let us picture to ourselves a motor with no friction or core loss and with windings of no resistance. In such a motor the no- load current would be exclusively magnetizing and would lag 90 degrees behind the pressure. Let us assume a case where, at no load, the current is 10 amperes. The entire magnetic flux emanating from the stator wind- ings will cross the zone occupied by the secondary conductors (i.e., the conductors on the rotor) and pass down into the rotor core. If the circumstance of the presence of load on the motor were not to disturb the course followed by the magnetic lines, then the magnetizing component of the current flowing into the motor 142 POLYPHASE GENERATORS AND MOTORS would remain the same with load as it is at no load. If the motor were for 100 volts per phase, then, in this imaginary case where the flux remains undisturbed as the load comes on, we could cal- culate in a very simple way the current flowing into the motor for any given load. To illustrate; let us assume that a load of 3000 watts is carried 'by this hypothetical motor. A load of 3000 watts corresponds to an output of 1000 watts per phase. Assuming a motor with no internal losses, the input will also amount to 1000 watts per phase. Since the pressure per phase is 100 volts, the energy component of the current input per phase is (1000 \ -r^r = j 10 amperes. Since the magnetizing component is 10 amperes the resultant current per phase is Vl0 2 +10 2 = 14.1 amperes. The vector diagram corresponding to these conditions is given in Fig. 77. The resultant current lags behind the terminal pressure by tan" 1 jptan- 1 1.0 = 45. The power-factor is : cos 45 = 0.707. Let us double the load. The energy component of the cur rent increases to 2X10 = 20 amperes, as shown in Fig. 78. The total current increases to Vl0 2 4-20 2 = 22.4 amperes. The angle of lag becomes : tan- 1 ijptan- 1 0.5 = 26.6. The power-factor increases to: cos 26.6 = 0.894. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 143 Let us again double the load, thereby increasing the energy component of the current to 40 amperes, and the angle of lag to 14.0 as shown in Fig. 79. The total current is now 41.3 amperes and the power-factor is: 740 .0^ Ul.3 0.97. 10 FIG. 77 FIG. 78 FIG. 79 FIGS. 77 to 79. Vector Diagrams Relating to a Hypothetical Polyphase Induction Motor without Magnetic Leakage. At this rate we should quickly approach unity power-factor. The curve of increase of power-factor with load, would be that drawn in Fig. 80. It is to be especially noted that in the diagrams in Figs. 77, 78 and 79, the vertical ordinates indicate the energy 144 POLYPHASE GENERATORS AND MOTORS components of the total current and the horizontal ordinates indicate the wattless (or magnetizing) components of the current. 10 12 14 16 18 20 22 24 Output in Kilowatts 26 FIG. 80. Curve of Power-factor of Hypothetical Polpyhase Induction Motor without Magnetic Leakage. It would be very nice if we could obtain the conditions indicated in the diagrams of Figs. 77, 78 and 79. That is to say, it would be very nice if the magnetizing component of the current remained FIG. 81. Diagrammatic Representation of the Distribution of the Magneto- motive Forces in the Stator and Rotor Windings of a Three-phase Induction Motor. constant with increasing load. But this is not the case. As the load increases, the current in the rotor conductors (which was POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 145 negligible at no load), increases. The combined effect of the cur- rent in the stator and rotor conductors is to divert a portion of the flux out of the path which it followed at no load. In Fig. 81 are indicated diagramatically a few slots of the stator and rotor windings. Considering the 12 left-hand conductors, the current is indicated as flowing (at the moment) down into the plane of the paper in the stator conductors and up out of the plane of the paper in the rotor conductors. This has the same effect (as regards the resultant mmf. of the stator and rotor conductors), as would be occasioned by the arrangement indicated in Fig. 82, in which the stator and rotor conductors constitute a single spiral. Obviously the mmf. of this spiral would drive the flux along the air-gap between the stator and rotor surfaces. The reluctance of the circuit traversed by the magnetic flux thus increases grad- ually (with increasing current input), from the relatively low reluctance of the main magnetic circuit traversed by the entire flux at no load, up to the far higher reluctance of the circuit traversed by practically the entire magnetic flux with the rotor at stand- still, the pressure at the terminals of the stator windings being maintained constant throughout this entire range of conditions. Consequently the magnetizing component of the current consumed by the motor increases as the load increases. Thus instead of the diagrams in Figs. 77, 78 and 79, we should have the three diagrams shown at the right hand in Fig. 83. The corresponding diagrams at the left hand in Fig. 83 are simply those of Figs. 77, 78 and 79 introduced into Fig. 83 for comparison. In both cases, the no-load current is 10 amperes. But in the practical case with magnetic leakage, the loads calling respectively for energy components of 10, 20 and 40 amperes (loads of 3000, 6000 and FIG. 82. Diagram Indicating a Solenoidal Source of mmf. Occasioning a Flux Along the Air-gap, Equivalent to the Leakage Flux in an Induction Motor. 12000 watts) involve magnetizing components of 10.6, 12.1 and 18.2 amperes. 146 POLYPHASE GENERATORS AND MOTORS 10.0 10.6 10.0 12,1 10.0 18.2 No Magnetic Leakage Magnetic Leakage FIG. 83. Vector Diagrams for Hypothetical Motor without Magnetic Leakage (at Left), and for Actual Motor with Magnetic Leakage (at Right). The total current inputs are increased as follows : Load (in Watts). Current Input per Phase. On Assumption of no Magnetic Leakage. On Assumption of Magnetic Leakage. 3000 6000 12000 10.0 14.1 22.4 41.3 10.0 14.6 23.4 44.0 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 147 The power-factors for the two cases are: Load in Watts. Power-factor. No Magnetic Leakage. Magnetic Leakage. 3000 0.707 - .(-) 0.685 6000 0.894 /20.0 \ \23.4~/' 855 12000 0.970 nr^- 91 The curve of power-factor for the case with magnetic leakage is given in Fig. 84, and there has been reproduced dotted in this 2 4 10 12 14 16 18 20 22 24 Output in Kilowatts FIG. 84. Curves of Power-factor of Hypothetical Induction Motor without Magnetic Leakage (Dotted Line) and of Actual Motor 'with Magnetic Leakage (Full Line). figure the power-factor curve for the case with no magnetic leakage which has already been given in Fig. 80. 148 POLYPHASE GENERATORS AND MOTORS In Fig. 85, the hypothenuses of the right-hand diagrams of Fig. 83 have been superposed, and their right-hand extremities are seen to lie upon the circumference of a semi-circle with a diameter of 200 amperes. In Fig. 85 the magnetizing current of 10 amperes is denoted by A B. The diameter of the semi- circle is BD. We have: L _ 10 20 30 40 50 CO 70 80 90 100 120 130 140 150 160 170 180 190 200 210 Wattless Components of the Current FIG. 85. Circle Diagram for a Polyphase Induction Motor with a No-load Current of 10 Amperes and a Circle-ratio of 0.050. The quantity which we termed the " circle ratio " and which we designated by the symbol c, is the ratio of AB to BD. For this case we have: AJJ - 10 -0050- B5-266- - 050 ' a = 0.050. The semi-circle in Fig. 85 is the locus of the extremities of the vectors representing the current flowing into the stator winding. If, for any value of the current input, we wish to ascertain its phase relations, we draw an arc with A as a center and with the value of the current as a radius. The intersection of this arc with the semi-circle, constitutes one extremity of the vector POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 149 representing the current, and A constitutes its other extremity. The horizontal projection of this vector is its wattless component and the vertical projection is its energy component. Consequently we also have the convenient relation that the vertical component is, for constant pressure, a direct measure of the power absorbed by the motor. We shall give further attention to these important relations at a later stage. The circle ratio is a function of \g, T, A and H. Knowing Xgr, a, A and H we can obtain a rough value for a for any motor. In other words, having selected these four quantities for a motor which we are designing, we can obtain a. Having calculated A B, the magnetizing current, by the methods already set forth on page 139, we may divide it by a and thus obtain BD. I For we have: We are now in a position to construct, for the 200 h.p. motor which we are designing, a diagram of the kind represented in Fig. 85. We must first determine a. We have: T = 32.5 cm.; A = 1.3 mm.; H = 12. Knowing these four quantities, the " circle factor," a, may be obtained from Table 27. For our motor we find from the table, a = 0.041. The values of a in Table 27 apply to designs with intermediate proportions as regards slot openings. Should both stator and rotor slots be very nearly closed (say 1 mm. openings), the value of a would be increased by say 20 per cent or more. On the other hand, were both stator and rotor slots wide open, a would be decreased by say some 20 per cent below the values set forth in the table. It cannot be too strongly emphasized that we can- not predetermine a at all closely. We can, however, take O.C41 as a probable value for a in the case of our design. If, on test, the observed value were found to be within 10 per cent of 0.041, 150 POLYPHASE GENERATORS AND MOTORS ooooo rH 00 (M rH t^ OOOOO ooooo ooooo 00 Oi O O O O r- ( rH ooooo 88S23 ooooo O rH rH IQrH CO CD t^. 00 o o o o ooooo ooooo ooooo ooooo t^- O ooooo O rH rH O O 00 1O *f iO CO CO l> ooooo oo ooooo CO l^ t^ ooooo 00 rt< U3 O CO o o o o ooooo o o o o o co oq 01 o TP 10 co t^ oooo ooooo O rH rH - CO rH (N rH t^oooiO OO O rH ^ GO T^l -HH ooooo ooooo ooooo 00000 i i CO rjn (M O O5 O i i (M O i ' rH T 1 OOOOO O -*f . CO ooooo ooooo rf (N T-I ( O i ( (N CO O rH rH rH rH OOOOO ^ T^ O O rH - 1^- oo O5 o ills ooooo ooooo 00 CO C^ CO ooooo ooooo >O CO "* 00 T^I 00 OS CO rH C^l O O rH rH rH OOOOO O O O rH rH o d d o d CO rH 00 O 00 IQ CO CO OO 00 ooooo (XXfft SOO^ O iH rH W W O rH rH >'l ffl 00 (N OO^ O rH rH ei ffl a- O rH rH W S-1 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 151 tO O Tfri OS CO CO ^t 1 ^ ^t 1 lO O O O O O ddddd CO "* * to to doddd lO OS "* CM CM CO CO -* ddddd 00000 oo ^ os r* co CO ^ ^ tO CO ddddd CO TJH Tt< tO CO ddddo ^ O CO ** i I 8SSp8 I d d odd O CO CO t i GO ^ T^ iO CO CO CO CO CO CO CO ddddd t^ co i 1 1^ co CO Tf iO tQ CO o p o p o d d d d d 00000 00000 CO 00 rf i I OS CO CO Tt< lO lO p p p p p d d d d d 00 ffl SO O rt< 00 1 I ooooo ooooo GO CO C^l OS t^* CO T^ tO O CO dddoo ooooo rt< CM CO f~- do odd I ooooo oo ffi so o ^ O rH r-i (N CM O iH iH - I s - ooooo ooooo ooooo odd 1>- GO OS co co co oo'od CO C^ i i t-- GO ( >O O' ooooo CO CO CO T~H OS tO CO t^ GO GO ddddd tO GO Os CO -^ CO Tf< lO CO !> ooooo 00 CM SO 00 rH T- CM CM 00 CM SO O ^ d i-i TH / ^o 1f> r / \ / \ Sty aa \ / \ \ / X / \_ Wattless Component of Current FIG. 87. Revised Circle Diagram of 200 H.P. Motor. The mean length of turn (mlt.) may be roughly estimated from the formula: K is a function of the pressure for which the stator is wound, and may be taken from Table 29. TABLE 29. VALUES OF K IN FORMULA FOR MLT. Terminal Pressure for which Stator is Wound. K. 500 (or less) 2.5 1000 3.0 2000 3.5 4000 4.0 6000 4.5 8000 5.0 10000 5.5 12000 6.0 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 157 For our 1000-volt 200 h.p. design, we find the value: # = 3.0. Therefore : mlt. = 2X43.0+3,OX32.5 = 183.5 cm. For the number of stator turns in series per phase we have: T=120 Consequently the length of conductor per phase equals: 120X183.5 = 22000 cm. The cross-section of the conductor has already been fixed at: 0.302 sq.cm. The PR loss at 60 Cent, is desired. At this temperature, the specific resistance of commercial copper wire may be taken at 0.00000200 ohm per cm. cube. Thus at 60 Cent, the resistance of each phase of the stator winding is: 22000X0.0000020 For the stator I 2 R loss at full load we have: 3 X102 2 X 0.146 = 4540 watts. Stator IR Drop at Full Load. Instead of a flux corresponding /1000 \ to ( =- = 577 volts per phase, we shall, at full load, have a lesser flux. It will, in fact, correspond to : 577-102X0.146 = 577-15 = 562 volts. 158 POLYPHASE GENERATORS AND MOTORS This is an internal drop of: ~X 100 = 2.6 per cent. Oil Strictly speaking, we ought, therefore, to. take into account the decreased magnetic densities with increasing load. Cases arise where it would be of importance to do this, but in the present instance such a refinement would be devoid of practical inter- est and will not be undertaken. THE DETERMINATION OF THE CORE LOSS. For the stator core, the best low-loss sheet-steel should be employed notwithstanding that its cost is still rather high. The advantage in improved performance will much outweigh the very slight increase which its use occasions in the Total Works Cost. In the rotor, the reversals of magnetism are, during normal running, at so low a rate that the rotor core loss is of but slight moment. It is consequently legitimate to employ a cheaper grade of material in the construction of the rotor cores. But in practice it is usually more economical to use the same grade as for the stator cores notwithstanding the absence of need for the better quality. By the time the outlay for waste and the outlay for wages and for general expenses are added, there will be but trifling difference in the cost of the two qualities. The data given in Table 30 are well on the safe side. Individ- ual designers will ascertain by experience in how far they can rely upon obtaining better material. TABLE 30. DATA FOR ESTIMATING THE CORE Loss IN INDUCTION MOTORS. Density in Stator Core in Lines per Square Centimeter. Core Loss in Stator Core in Watts per kg. for Various Periodicities. ~=15 ~=25 ~=50 6000 8000 10000 1.1 1.7 2.2 2.2 3.0 4.0 5.0 7.4 10.0 12000 14000 2.6 3.2 5.2 6.2 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 159 For our 25-cycle motor, we have employed in the stator core, a density of 10 000 lines per sq.cm. and we shall consequently estimate the core loss on the basis of 4.0 watts per kg. of total weight of stator core. ESTIMATION OF WEIGHT OF STATOK CORE External diameter of stator core = 89. 2 cm. Internal diameter of stator core =62.0 cm. The gross area of a stator core plate, i.e., the area before deduct- ing the area of the slots, is equal to : |(89.2 2 -62.0 2 )=3260 sq.cm. Area of 72 stator slots = 72X5.4X1. 26 = 490 sq.cm. Net area of stator core plate =(3260 -490 = ) 2770 sq.cm . The volume of the stator core is obtained by multiplying this area by \n, i.e., by 27.9. Volume of sheet steel in stator core = 2770X27.9 = 77 400 cu.cm. Weight of 1 cu.cm. of sheet steel = 7.8 grams. Therefore : Weight of sheet steel in stator core = - = 603 kg. J.UUU The accuracy with which core losses can be estimated is not such as to justify dealing separately with the teeth and the main body of the stator core. It suffices simply to multiply the net weight in kg. by the loss in watts per kg. corresponding to the density in the main body of the stator core. Therefore: Stator core loss = 603X4.0 = 2410 watts. Core Loss in Rotor. The periodicity of reversal of magneti- zation in the rotor core is so low that there should not be much core loss in the rotor. But to allow for minor phenomena and to keep on the safe side, it is a good rule to assess the rotor core 160 POLYPHASE GENERATORS AND MOTORS loss at 10 per cent of the stator core loss. For our machine we have : . Rotor core loss = 2410 X 0.10 = 240 watts. Input to Motor and to Rotor at Rated Load. We cannot yet check our preliminary assumption of an efficiency of 91 per cent at rated load. On the basis of this efficiency, the input to the motor at its rated load is: 200X746 =164 000 watts. L/.y -L The losses in the stator amount to a total of : 4540+2410 = 6950 watts. Deducting the stator losses at full load from the input to the motor at full load we ascertain that: 164 000 - 6950 = 157 050 watts are transmitted to the rotor. A Motor is a Transformer of Energy. A motor receives energy in the form known as " electricity." An account can be rendered of all the energy received. In the case we are consider- ing, the full-load input is at the rate of 164 000 watts. Energy flows into the motor at the rate of 164 kw. hr. per hour.* In cer- tain instances it is more convenient to make some equivalent state- ment with other units of power and time. Thus we may say that energy flows into the motor at the rate of 164 000 watt seconds per second. The amount of energy corresponding to the expendi- ture of one watt for one second is termed by the physicist, one joule. 1 joule = 1 watt second. It requires 4190 joules to raise the temperature of I kg. of water by 1 cent. Thus we have: 1 kg. calorie (kg.cal.) =4190 joules. * The proposal to designate as 1 kelvin the amount of energy correspond- ing to 1 kw. hr., is gradually gaining favor amongst European engineers. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 161 The energy expended in lifting a weight of 1 kg. through a height of 1 meter is equivalent to 9.81 joules. In other words 1 kg.m. = 9.81 joules. The energy need not necessarily be expended in lifting a weight. If a force of 1 kg. is exerted through a distance of 1 meter, an amount of energy equal to 9.81 joules is expended in the process. The amount of energy corresponding to the expenditure of 1 h.p. for 1 sec. is equal to 746 joules; or 1 h.p. sec. = 746 joules. It is convenient to bring together these equivalents: 1 watt second = 1 joule; 1 h.p. second =746 joules; 1 kg.m. =9.81 joules; 1 kg.cal. =4190 joules. 1 kelvin = 3 600 000 joules. The rated output of our motor may be expressed as: 1. 200 h.p.; la. 200 h.p. sec. per sec.; 2. 149 200 watts; 2a. 149 200 joules per second. 3. = 15 2 kg.m. per second. /149 200 \ . \ 4190 = / kg.cal. per second. Let us concentrate our attention on Designation 3 for the rated output. In accordance with this designation, the motor's output at its rated load is 15 200 kg.m. per sec. If the shaft of our motpr is supplied with a gear wheel of 1 meter radius through which it transmits the energy to another engaging gear wheel, and thence to the driven machinery, then 162 POLYPHASE GENERATORS AND MOTORS for every revolution of the armature, the distance travelled by a point on the periphery of the gear wheel is 2Xx = 6.28 meters. At the motor's synchronous speed of 500 r.p.m., the peripheral speed of the gear wheel is 500 6.28 X-gQ- = 52.3 meters per sec. (Although such a high speed would not, in practice, be employed, it has been preferable, for the purpose of the present discussion, to consider a gear wheel of 1 meter radius.) Since at full load the motor's output is : 15 200 kg.m. per sec. and since the peripheral speed of the gear wheel is 52.3 meters per sec., it follows that the pressure at the point of contact between the driving and the driven gear teeth is This is the force exerted at a radius of 1 meter. We say that at full load the motor exerts a " torque " of 290X1 = 290 kg. at 1 meter leverage. Had the radius of the gear been only 0.5 m. instead of 1.0 meter, then the force would have been 2X290 = 580 kg., but the " torque " would still have been equivalent to 0.5X580 = 290 kg. at 1 meter leverage. In dealing with torque it is usually convenient to reduce it to terms of the force in kg. at 1 meter leverage, irrespective of the actual leverage of the point of application of the force. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 163 At rated load our motor will not run at the synchronous speed of 500 r.p.m., but at some slightly lower speed. If the " slip " is 1 per cent, then the full-load speed will be 0.99X500 = 495 r.p.m. If the " slip " is 2 per cent, then the full-load speed will be 0.98X500 = 490 r.p.m. Since the rated output will nevertheless be 200 h.p. in these two cases, the torque will be; not 290 kg. at 1 m. leverage, but and / 290 ( rTnn = ) 293 k S- * ' r tne case * * P er u.yy (290 = 1 296 kg. for the case of 2 per cent slip. \J i/O The torque (i.e., the force at 1 m. leverage) is only one com- ponent of the energy delivered. The other component is the distance traversed by a hypothetical point at 1 meter radius revolving at the angular speed of the rotor. For 2 per cent slip, the distance traversed in 1 sec. by such a hypothetical point, is 0.98X52.4 = 51. 3 meters. The energy delivered from the motor in each second is thus 296X51.3 = 15200kg.m. The power (or rate of deliverance of energy) is 15 200 kg.m. per sec.; or, 9.81 X 15 200 = 149 200 watts; or, 149 200 746 = 200 h.p. 164 POLYPHASE GENERATORS AND MOTORS The torque exerted by the rotor conductors is greater than that finally available at the gear teeth. The discrepancy cor- responds to the amount of the PR loss in the rotor conductors and to the amount of the rotor core loss and the windage and bearing friction. The rotor core loss and the friction come in just the same category as an equal amount of external load. Thus if 3 h.p. is required to supply the rotor core loss plus friction, then the output from the rotor conductors is 203 h.p. as against the ultimate output of 200 h.p. from the motor. But the PR loss in the rotor conductors comes in an altogether different category. The loss can only come about as the result of a cutting of the flux across the rotor conductors. In other words, the rotor conductors must not travel quite as fast as the revolving magnetic field. Consequently the rotor will run at a speed slightly less than synchronous; there will be a " slip " between the revolving magnetic field and the revolving rotor. It is only in virtue of such slip that the rotor conductors can be the seat of any force. Thus the torque is inseparably associated with the " slip " and the " slip " will be greater the greater the load. As the " slip " and torque increase, the rotor PR loss also increases and the speed of the rotor decreases. If the PR loss in the rotor conductors amounts to 1 per cent of the input to the rotor, then the " slip " will be 1 per cent. If the PR loss is increased to 2 per cent, then the " slip " increases to 2 per cent. If, finally, the PR loss amounts to 100 per cent of the input to the rotor, then the " slip " will be 100 per cent, i.e., the motor will be at rest, but it may nevertheless be exerting torque. For such a condition it is desirable to regard matters from the following standpoint. If the rotor is suitably secured so that it cannot rotate; then if electricity is sent into the motor a certain portion will be transmitted by induction to the rotor circuit, just as if it constituted the secondary circuit of a trans- former. The input to the rotor will under these conditions consist of the PR loss in its conductors and the core loss. There is no other outlet for the energy sent into the rotor and it all becomes transformed into energy in the form of heat in the rotor conductors and in the rotor core. Since under such conditions the rotor core loss is negligible in comparison with the rotor PR loss, we may regard the input to the rotor as practically POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 165 identical with the rotor PR loss. The rotor PR loss is 100 per cent of the input to the rotor and the slip is 100 per cent. The Locus of the Rotor Current in the Circle Diagram. Let us consider the hypothetical case of a rotor with a number of conductors equal to the number of stator conductors. , We can describe this as an arrangement with a 1 : 1 ratio of transformation. Any actual induction motor can be considered to have its equivalent with a 1 : 1 ratio, and it is convenient and usual to investigate certain properties of induction motors by considering the equivalent rotor wind- ing with a 1 : 1 ratio. The vector diagram of the stator and rotor currents is shown in Fig. 88 for our 200-h.p. motor with a l:l-ratio rotor. The diagram is drawn for full load and consequently the stator current, AE, is equal to 102 amperes. The rotor current, AF must have such direction and magnitude that the resultant, AB, shall be equal to the no-load magnetizing current, which we have already found to be 25.2 amperes. AF is consequently equal, as regards phase and magnitude, to EB and is found graphically to amount to 96 amperes. In practice, it is more convenient to represent the rotor current by lines drawn from B as the origin and connecting B to the points where the corresponding primary vectors intersect the circumference of the semi-circle. In Fig. 89 are drawn the stator and rotor vectors for two values of the stator cur- rent, namely 80 amperes and 300 amperes. The corresponding values of the rotor current are 72 and 286 amperes. The Rotor I 2 R Loss of the 200-H.P. Motor at Its Rated Load. It is still desirable to postpone to a later stage the design of the rotor conductors. But let us assume that the full-load slip will be 2.0 per cent. Then the rotor PR loss at full load will be 2.0 per cent of the input to the rotor. The input to the rotor FIG. 88. Vector Diagram Indicating the Primary and Secondary Currents in the 200-H.P. Induc- tion Motor at its Rated Load. 166 POLYPHASE GENERATORS AND MOTORS has been ascertained (on page 160) to be 157 050 watts. Thus the rotor PR loss at full load is 0.02X157 050 = 3140 watts. We have now determined (or assumed) all the full-load losses except windage and bearing friction. / // 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 Wattless Component of Current FIG. 89. Diagram Indicating Stator and Rotor Current Vectors. Friction Losses. It is very hard to generalize as regards friction losses. A reasonable estimate may, however, be made from Table 31. TABLE 31. DATA FOR ESTIMATING THE FRICTION Loss IN BEARINGS AND WINDAGE IN INDUCTION MOTORS. Z> 2 X0 (D and \g in dm.) Total Friction Loss in Watts, for Various Rated Speeds. 400 r.p.m. 800 r.p.m. 1600 r.p.m. 10 20 30 150 280 370 340 620 800 1200 2000 2500 40 50 100 460 550 800 1000 1250 1750 3100 3700 5600 200 1100 2400 8000 POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 167 For our motor we have : (i 40\ = ^M=2.06h.p. = 166. From Table 31 we ascertain by interpolation that the friction loss may be taken as 1300 watts. Output from Rotor Conductors. The output from the rotor conductors is made up of the 200-h.p. output from the motor, the friction, and the rotor core loss. The two latter amount to Thus we may take the output from the rotor conductors as 200+2.06 = 202 h.p. Referring back to page 163 we find that for 2 per cent slip and 200 h.p., we require 296 kg. at 1 meter leverage. Consequently the torque required to be exerted by the rotor conductors is : 202 -X296 = 299 kg. at 1 meter leverage. In other words, the full-load torque exerted by the rotor conductors is 299 kg. The Torque Factor. At full load, the input to the rotor is 157 050 watts. Thus the input to the rotor in watts per kg. of torque developed, which we may term the " torque factor," is: 157050__ . ~299~ This factor will be useful to us in studying the starting torque. The " Equivalent " Resistance of the Rotor. We have on p. 157 made an estimation of the stator resistance and have ascertained it to be 0.146 ohm per phase at 60 cent. The stator PR loss at full load is 3 X102 2 X 0.146 = 4540 watts. The rotor I 2 R loss at full load is 3140 watts. 168 POLYPHASE GENERATORS AND MOTORS We shall employ a squirrel-cage rotor (for which we shall soon design the conductors) and we may consider its " equivalent " resistance to be:. X 0.146 = 0.101 ohm. Without introducing any serious inaccuracy we may (when the motor is running in the neighborhood of synchronous speed), ascertain the rotor PR loss for any value of the stator current (except for very small loads) by multiplying this " equivalent " resistance by three times the square of the stator current. Rotor at Rest. But when the rotor is at rest, the currents circulating in its windings are of the line periodicity and the conductors have an apparent resistance materially greater than their true resistance. Attention was called to a related phe- nomenon in a paper presented by A. B. Field, in June, 1905, before the American Institute of Electrical Engineers and entitled " Eddy Currents in Large Slot- Wound Conductors."* Recently the application of the principle has been incorporated in the design of squirrel-cage induction motors to endow them with desired values of starting torque. The multiplier by which the apparent resistance may be obtained from the true resistance may be found approximately from the formula: Multiplier = 0. 15 X (depth of rotor bar in cm.) X Vperiodicity. In our case we have: Multiplier = 0. 15 X 5.4 X V25 = 4.05. This multiplier only relates to the embedded portions of the conductors. The portions of the length where the conductors cross the ventilating ducts are not affected, nor are the end rings subject to this phenomenon. A rough allowance for this can be made by reducing the multiplier to 0.8X4.05 = 3.24. Thus at standstill the " apparent " resistance of our rotor will be 3.24X0.101=0.327 ohm. "Vol. 24, p. 761. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 169 The " Equivalent " Resistance of the Motor. At standstill, the motor, regarded as a whole, and without distinction between primary and secondary may be considered as having a resistance of (0.146+0.327 = ) 0.473 ohm per phase. On page 154 we have seen that the reactance of the motor is: S= 0.902 ohm per phase. For the impedance of the motor at standstill we have: Vo.4732+0.902^1.03 ohm per phase. THE STARTING TORQUE In starting this motor we should not put it at once across the full pressure of 1000 volts but should apply only one-half or one-third of this pressure. Let us examine the conditions at standstill when half pressure is applied. We shall have ^ = 288 volts per phase. The stator current will be: OQQ = 280 amperes per phase. For the rotor PR loss we have 3X280 2 X0.327 = 77 000 watts. The torque developed is obtained by dividing this value by the torque factor, i.e., by 525. 77 000 Torque = _ = 146 kg. at 1 meter leverage. The full-load torque is 296 kg. Consequently with half the normal pressure at the motor, we shall obtain 146 49.5 per cent of full-load torque. 170 POLYPHASE GENERATORS AND MOTORS We obtain the half pressure at the motor by tapping off from the middle point of a starting compensator. The connections (for a quarter-phase motor), are as shown in Fig. 90. With this arrangement, the current drawn from the line will be only half of the current taken by the motor. Since the motor takes 280 amperes, the current from the line is only /280 \ , I - s-= ) 140 amperes. (140 \ TQO = ) 1-37 times full-load current from the line, we can start the motor with 50 per cent of full -load torque. FIG. 90. Connections for Starting Up an Induction Motor by Means of a Compensator (sometimes called an auto-transformer). This excellent result is achieved by employing deep rotor con- ductors and does not involve the necessity of resorting to high slip during normal running. Circle Diagram for These Starting Conditions. For half pressure, the magnetizing current will, strictly speaking, be a little less than half its former value of 25.2 amperes, since the magnetic parts are worked at lower saturation. But for simplicity (25 2 \ ~ = } 12.6 amperes. The circle ratio, 5 t^ 3 . ^ "_ \ o 80 ^ ? i G I 3 J70 / s . 3 < 1 ^ : ri \ 60 / x < 3 i S M ^ / x* ! j 9 ^ - 1 \ CD 40 3 ^x- ^ ^ i \ X" s 2 j x ^ ^. ^ "n \_ 1 1020304050607080 100 120 140 160 180 200 220 240\ 260 280 300 320 Wattless Component of Current FIG. 91. Starting Torque Diagram for 200-H.P. Motor when Connected for Starting from Half the Normal Line Pressure. A vertical line EF is drawn from E to the base line. This is the vertical projection of 280 amperes and represents the energy component of the input to the motor under these conditions of standstill. From the diagram we ascertain graphically that: #^=131 amperes. Consequently the input to the motor is: = 113000 watts. The stator PR loss = 3X280 2 X0.146 = 34 300 watts. Subtracting this from the input, we ought to obtain the PR loss in the rotor. Rotor PR loss = 113 000-34 300 = 78 700 watts. 172 POLYPHASE GENERATORS AND MOTORS This is in good agreement with the value of 77 000 watts which we obtained for the rotor PR loss by applying the analytical method. EF is, for constant pressure, a measure of the input to the motor and is to the scale of 113 000 = 865 watts per ampere. lol [It can also be seen that this would be the case from the cir- cumstance that at a pressure of 288 volts per phase, the input is equal to : (3X288X7) = (865 7) watts]. The rotor PR loss of 78 700 watts may then be represented by the height FG corresponding to ( = ) 91.0 amperes. \ oDO / and the stator PR loss may be represented by the remainder, GE, corresponding to =. amperes. It also necessarily follows that FG (and corresponding vertical heights for other conditions similarly worked out), is a measure of the torque. When used in this way, the scale is 146 j-r- r = 1.60 kg. (at 1 meter leverage) per amp. Some General Observations Regarding the Circle Diagram. It is this adaptability to the forming of mental pictures of the occurrences, which renders the circle diagram of great importance in the design of induction motors. All the various calculations involved in induction-motor design may be carried through by analytical methods but it is believed that these exclusively analyt- ical methods are inferior in that they disclose no simple picture of the occurrences. It is well known that in practice the locus of the extremity of the stator-current vectors is rarely more than POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 173 a very crude approximation to the arc of a circle and that the designer can only consider as rough approximations the results he deduces on the basis of the circle assumption. But when employed with judgment the plan is of great assistance and suffices to yield reasonable results. THE SQUIRREL CAGE Let us now design our rotor's squirrel cage. As yet we have merely prescribed that the full-load I 2 R loss in the squirrel cage shall be 3140 watts, that the slots shall be 54 mm. deep and that we shall employ a number of slots not differing very materially from 72, the number of stator slots. The Number of Rotor Slots. Were we to employ 72 rotor slots, the motor would have a strong " cogging " tendency. That is to say, if, with the rotor at rest, pressure were applied to the stator terminals, there would be a strong tendency for the rotor to " lock," at a position in which there would be a rotor slot directly opposite to each stator slot. This tendency would very markedly interfere with the development of the starting torque calculated in a preceding section. The choice of 71 or 73 slots would eliminate this defect, but might lead to an unbal- anced pull, slightly decreasing the radial depth of the air-gap at one point of the periphery and correspondingly increasing its depth at the diametrically opposite point. This excentricity once established, the gap at one side would offer less magnetic reluctance than the gap diametrically opposite, thus increasing the dead-point tendency. But by selecting 70 or 74 slots, this defect is also eliminated. Let us determine upon 70 rotor slots for our design. It may in general be stated that the tendency to dead points at starting will be less. 1. The smaller the greatest common divisor of the numbers of stator and rotor slots. 2. The greater the average number of stator and rotor slots per pole. 3. The less the width of the slot openings. 4. The greater the resistance of the squirrel-cage. 5. The deeper the rotor slots. 174 POLYPHASE GENERATORS AND MOTORS The influence of the last two factors will be better understood if it is pointed out that they determine the rotor PR loss at start- ing, and we have already seen that the starting torque is pro- portional to the PR loss in the rotor. The fluctuations in the starting torque arising from variations in the relative positions of the stator and rotor slots, will obviously be a smaller percentage of the average starting torque, the greater the absolute value of the average starting torque. Thus if the average starting torque is very low, a small fluctuation might periodically reduce it to zero; i.e., the< motor would have dead points. If, on the other hand, the average starting torque is high, these same fluctuations, superposed on this high average starting torque, would still leave a high value for the minimum torque, and there would be no dead points. The Pitch of the Rotor Slot. The external diameter of the rotor is (620-2X1.3 = ) 617.4 mm. The diameter at the bottom of the slots is : (617.4 -2X54 = ) 509.4 mm. Consequently the rotor slot pitch at the bottom of the slot is : 509.4 Xx 70 = 22.8 mm. The depth of the rotor conductor can be practically identical with the depth of the slot. Therefore depth of rotor conductor = 54 mm. Ratio of Transformation. We have 72 stator slots and 10 con- ductors per slot; hence a total of 720 stator conductors; as against only 70 rotor conductors. The ratio of transformation is thus: 720 : 70 = 10.3 : 1. We have already estimated that for a 1 : 1 ratio, the rotor current would, at full load, amount to 96 amperes. We are now able to state that with the actual ratio of 10.3 : 1, the current in the rotor face conductors will be 10.3X96 = 990 amperes. At each end of the rotor core, the rotor conductors will termin- ate in end rings. It will be desirable, for structural reasons, to POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 175 space these end rings (in this particular motor), 2.0 cm. away from the ends of the laminated core. Consequently the length of each conductor between end rings will amount to 43+2X2 = 47 cm. The PR loss in the 70 rotor conductors will be equal to that in a single conductor of the same (as yet undetermined) cross- section, but with a length of 70X47 = 3290 cm. We have seen that we wish, at full load, to have a loss of 3140 watts in the squirrel cage. Were the loss in the end rings negli- gible (the entire 3140 watts being dissipated in the 70 face con- ductors, then their section would be so chosen that they should have an aggregate resistance of: = 0.00320 ohm. Since at 60 cent., the specific resistance of commercial copper is 0.00000200, the required cross-section would be: 3290X0.00000200 0.00320 = 2.06sq.cm. Since the depth of a rotor conductor is 54 mm., its width would thus require to be: = 3.82 mm. -W, But we cannot afford to provide so much material in the end rings as to render them of practically negligible resistance. Let us plan to allow a loss of 628 watts (20 per cent) in the end rings, the remaining (3140 628 = ) 2512 watts occurring in the slot con- ductors. This will require an increase in the width of the slot conductors, to: = 4.78 mm. or 4.8mm. 176 POLYPHASE GENERATORS AND MOTORS Thus the slot conductors will have a cross-section of 54 mm.X4.8 mm. The rotor slot will also be 54 mm. deep and 4.8 mm. wide, and there will be no insulation on the rotor con- ductors. The End Rings. The object in minimizing the loss in the end rings will have been divined. Since we want to have the apparent resistance at starting, as great as possible for a given " slip " at normal load, we want to concentrate the largest prac- ticable portion of the loss in the slot conductors since it is these conductors which manifest the phenomenon of having, for high- periodicity currents, a loss greatly in excess of that occurring when they are traversed by currents of the low periodicity corresponding to the " slip." The Current in the End Rings. It can be shown* that the current in each end ring is equal to: Number of rotor conductors , ^ ; - X current per slot conductor. iuX number of poles For our motor we have: Full-load current in each end ring = -^X 990 = 3660 amperes. Since we have allowed 628 watts for the loss in the end rings ; we have a loss of 314 watts per end ring, and we have: Resistance for one end ring = - ^ 2 = 0.0000234 ohm. Each end ring will have a mean diameter of: 61.7+50.9 ^ = 56.3 cm. and a mean circumference of 56.3x = 177 cm. Consequently the cross-section of each end ring is : 177X0.00000200 Q 0.0000234 * The proof is given in Chapter XXIII (pp. 490 to 492) of the 2d Edition of the author's " Electric Motors " (Whittaker & Co., 1910). POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 177 Let us make the cross-section up with a height of 54 mm. and a width of 28 mm. A conductor of this cross-section will, at With 11.7 teeth <-4.8 mm 22 mm Section on A.B Looking in Direction of Arco.ws , FIG. 92. Outline of Squirrel-cage Rotor for 200-H.P. Polyphase Induction Motor. 25 cycles, be subject to a slight increase in resistance, due to ordinary skin effect, but this will not be of sufficient amount to add much to the starting torque. The increase in re- >j k-2.4mm. sistance will in fact amount to about 10 per cent. A sketch of the rotor is given in Fig. 92, and a section of a slot and two teeth, in Fig. 93. Since the slot pitch at the bottom of the slot is 22.8 mm. and since the slot is 4.8 mm. wide, we have a tooth width of 22.8-4.8 = 18.0 mm. (?=) per pole, the cross-section of the magnetic circuit at the bottom of the rotor teeth is FIG. 93. Rotor Slot and Teeth for 200- H.P. Squirrel-cage Motor. 11.7X1.8X27.9 = 588 sq.cm. 178 POLYPHASE GENERATORS AND MOTORS The crest density at the bottom of the rotor teeth is: 4 570 000 X 1.7 = 13 200 lines per sq.cm. 588 THE EFFICIENCY We have now estimated all the losses in our 200-h.p. motor. Let us bring them together in an orderly table : At full load we have : Stator PR loss 4 540 watts Stator core loss 2410 " Rotor PR loss 3140 " Rotor core loss 240 " Friction and windage loss 1 300 ' ' Total of all losses 11 630 watts Output at full load 149 200 ' ' Input at full load 160 830 watts 14.0 onrj Full-load efficiency = - X 100 = 93.0 per cent, lou ooU Our original assumption for the full-load efficiency was 91.0 per cent. Consequently, strictly speaking, we ought to revise several quantities, such as current input, stator PR loss, rotor PR loss, and ultimately obtain a still closer approximation to the efficiency. But the object of working through this example has been to convey information with respect to the methods of carrying out the calculations involved in the design of an. induc- tion motor ; and there is no special reason to undertake the above mentioned revision. THE HALF-LOAD EFFICIENCY At half load, the energy component of the full-load current input, will, sufficiently exactly for our purpose, be halved. This energy component is: GXl = 0.93X102 = 95 amperes. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 179 At half load, the wattless component will be slightly in excess of 25.2 amperes, the no-load current. Let us take it as 29 amperes. The total current input at half load will be: 55.5 amperes. The stator PR loss at half load will be: X 4540 = 1340 watts. /55.5\ 2 U027 +* 34 I 32 ^30 * 28 4J g 26 I 24 I 22 & 2 <3 18 W 16 14 12 10 7 Dia. 308-A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 Wattless Component of Current FIG. 94. Diagram for Obtaining Rotor Current at Half-load. The readiest way to obtain the corresponding value of the secondary current is to make the circle-diagram construction indicated in Fig. 94. From this construction we see that for a 1 : 1 ratio, the secondary current would be 47.5 amperes. We 180 POLYPHASE GENERATORS AND MOTORS have seen on page 165 that at full load, when the 1 : 1 secondary current was 96 amperes, the rotor PR loss was 3140 watts. Consequently at half load, the rotor PR loss will be: X 3140 = 770 watts. Thus at half load we have : Stator PR loss ' 1 340 watts Stator core loss 2 410 " Rotor PR loss 770 " Rotor core loss 240 ' ' Friction and windage loss 1 300 ' ' Total of all losses 6 060 watts Output at half load. 74 600 " Input at half load . 80 660 watts 74- fiOO Half-load efficiency = ^ ^ X 100 = 92.5 per cent. oU DOu Making similar calculations for other loads we obtain the follow ing inclusive table of the losses at various loads : TABULATION OF LOSSES AND EFFICIENCIES AT 60 CENT. Percentage of rated out- put 25 50 75 100 125 Amperes input per phase 25.2 35.0 55.5 77.0 102 127 Stator I 2 R loss .... 280 550 1340 2580 4540 7100 Stator core loss Rotor 7 2 B loss Rotor core loss. 2410 20 240 2410 300 240 2410 770 240 2410 1750 240 2410 3140 240 2410 4900 240 Friction and windage loss 1300 1300 1300 1300 1300 1300 Total of all losses Output in watts 4250 4800 37400 6060 74600 8280 112000 11630 149200 15950 186600 Output in horse-power . . 50 100 150 200 250 Input 4250 42200 80660 120280 160830 202550 0.885 0.925 930 930 921 These efficiencies are plotted in the curve of Fig. 95. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 181 Estimation of the Temperature Rise. In making a rough estimate of the heating of the motor the first step consists in cal- culating the " watts per square decimeter (sq.dm.) of radiating surface at the air-gap." The " radiating surface at the air-gap " is taken as the sur- face over the ends of the stator windings. While this surface will vary considerably according to the type of winding employed, a representative basis may be obtained from the formula: " Equiv. radia. surface at air-gap " 100 1 90 2 P4 80 d I 70 .2 I G 50 20 40 60 80 100 120 140 160 180 200 220 240 260 280 Output in Horse-power FIG. 95. Efficiency Curve for 200-H.P. Polyphase Induction Motor. The k in this formula is a factor which is a function both of the normal pressure for which the motor is built, and of the polar pitch, T. Suitable values will be found in Table 32. TABLE 32. DATA FOR ESTIMATING THE RADIATING SURFACE. Rated Pressure of the Motor. Values of k in the Formula: Equiv. rad. sur. =irXDX(\g +kr). T=60 cm. r =40 cm. r=20 cm. 1000 volts (or less). . . 2000 0.8 0.9 1.0 1.1 1.2 1.4 1.7 0.9 .0 .1 .3 .5 .8 2.0 1.1 1.2 1.4 1.7 2.0 2.3 2.8 4000 6000 8000 10000 12000 ' ' 182 POLYPHASE GENERATORS AND MOTORS For our design we have: D = 62.0; X0-43.0; r = 32.5; .'. " Equivalent radiating surface at air-gap = xX62.0X (43.0+29.3) = xX62.0X72.3 = 14000 sq.cm. = 140 sq.dm. The loss to be considered is, at full load: 11 630 watts. Thus we have: Watts per sq.dm. = T- =83.4. The data in Table 33, gives a rough notion of the temperature rise corresponding to various conditions. TABLE 33. DATA FOR ESTIMATING THE TEMPERATURE RISE IN INDUCTION MOTORS. Peripheral Speed in Meters per Second. Thermometrically determined ultimate temperature rise of open-protected types of induction motors, per watt of total loss per sq.dm. of equiva- lent radiating surface at the air-gap. Badly Ventilated. Fairly Ventilated. Very Well Ventilated. 10 20 30 40 50 60 0.59 0.49 0.47 0.45 0.43 0.41 0.41 0.40 0.38 0.36 0.34 0.32 0.36 0.34 0.32 0.29 0.26 0.24 Our motor is decidedly well supplied with ventilating ducts, and with reasonable care in the arrangement of parts, it will come in the class of " very well ventilated " motors. The per- POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 183 ipheral speed has already (on page 113) been ascertained to be 16.2 meters per second. From Table 33 we find that the value for the thermometric- ally determined temperature rise is: 0.35 per watt per sq.dm. This gives us a total (thermometrically determined) rise of 83.4X0.35 = 29 Cent. Although in the predetermination of temperature rises, no close accuracy is practicable, still, the margin indicated by the above result is so great that the designer should rearrange the final design in such a manner as to save a little material at the cost of a slight increase in the losses. Since a rise of fully 40 cent, is usually considered quite con- servative (45 often being adopted), an estimated rise of 35 would leave a sufficient margin of safety. THE WATTS PER TON Another useful criterion to apply in judging whether a motor is rated at as high an output as could reasonably be expected, is the " watts per ton." The weight taken, is exclusive of slide rails and pulley. In the case of a design which has not been built, the method of procedure is as follows : First estimate the weights of effective material. Weight Stator Copper. mlt. = 183.5 cm. Total number of turns = 3 T = 3 X 120 = 360; Cross-section stator cond =0.302 sq.cm.; Volume of stator copper = 183.5X360X0.302 = 19900 cu.cm.; Weight 1 cu.cm. of copper =8.9 grams; w . , . 19900X8.9 Weight stator copper = J-UUu = 177 kg. 184 POLYPHASE GENERATORS AND MOTORS Weight Rotor Copper. Total length rotor face 1 _ QOon 1 / -n-rr\ ' O^yU CHI.; conds (see page 175) J Cross-section =5.4X0.48 = 2.59 sq.cm.; Volume of rotor face conds = 3290X2.59 = 8550 cu.cm.; 8550X8.9 Weight rotor face conds = = 76 kg.; Mean circum. end ring =177 cm.; Cross-section =15.1 sq.cm.; 2X177X15.1X8.9 Weight two end rings = 1UUU = 47.5 kg.; Total weight rotor copper =76+48 = 124 kg.; Total weight copper I_ 177 , l9d , . i x f -L I I |~ JL^iTt (stator plus rotor) j = 301 kg. Weight Stator Core. This has already been estimated (on page 159) to be 603 kg. Weight Rotor Core. External diameter rotor core =61.7 cm.; Internal diameter rotor core =38.3 cm.; Gross area rotor core plate = |(61.7 2 -38.3 2 ) = 1840 sq.cm.; Area 70 slots = 70 X 5.4 X 0.48 = 180 sq.cm. ; Net area rotor core plates = 1660 sq.cm.; A w = 27.9 cm.; Volume of sheet steel in rotor core = 1660X27.9 = 46 300 cu.cm.; Weight of 1 cu.cm. of sheet steel =7.8 grams; 46300X7.8 Weight of rotor core = 360 kg.; Total weight sheet steel (stator , 603 +360 = 963 kg. plus rotor) POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 185 Thus we have: Weight copper Weight sheet steel = 301 kg.; = 963kg. Total net weight effec. material = 1264 kg. From experience in the design of motors of this type it has been determined that an adequate mechanical design is consistent with obtaining for the ratios of " Total Weight of Motor " to " Weight of Effective Material," the values given in Table 34. TABLE 34. DATA FOR ESTIMATING THE TOTAL WEIGHT OF AN INDUCTION MOTOR. D Diameter at Air-gap (in cm.). Ratio of Total Weight (exclusive of slide rails and pulley) to Weight of Net Effective Material. D/X, = 1.5. Z)/X, = 2.5. 20 1.80 2.20 40 1.60 1.82 80 1.54 1.78 For our motor we have: Total weight (exclusive of slide rails and pulley) = 1.56X1264 -1970kg. = 1.97 tons. It will be remembered (p. Ill) that our original rough estimate for the weight, was 2.0 tons. Even this last method, which depends upon obtaining a factor by which to multiply the cal- culated weight of effective material, can only be regarded as rough. We can, however, regard the value of 1.97 tons as fairly correct for the total weight of our motor. 186 POLYPHASE GENERATORS AND MOTORS Consequently we have : Watts per ton =ry|^ = 5900. The designer must gain his own experience as to the attainable values for the " watts per ton." It may, however, here be stated that 8000 watts per ton and even considerably higher values are FIG 96. An 8-pole, 50-H.P., 900-r.p.m., Squirrel-cage Induction Motor of the Open-protected Type. [Built by the General Electric Co. of America.! quite consistent with moderate temperature rise for a motor of this size and speed, if constructed in the manner generally designated the " open-protected type." In Fig. 96 is given a photograph of an open-protected type of squirrel-cage induction motor. The Breakdown Factor. It has been stated that so far as temperature-rise is concerned, the motor is rather liberally designed and that in a revised design it should be possible to save POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 187 a little in material. We must, however, take into account the limitation imposed by the heaviest load which the motor is capable of carrying before it pulls up and comes to rest. The ratio of this load to the rated load is termed the breakdown factor (bdf.) and the present custom is to require that a motor shall have a breakdown factor of about 2. In the case of our 200-h.p. motor, it ought to be capable of carrying an instantaneous load of 2X200 = 400 h.p. without pulling up and coming to rest. It has previously been explained that the vertical projection of the vector representing the stator current in the circle diagram, is, for constant terminal pressure, proportional to the watts input to the motor. Thus the current corresponding to the maximum input is the current AM in the diagram in Fig. 97. The power input corresponding to this current may be represented by MN, the vertical radius of the circle. We have: = VAN 2 +MN 2 = V333 2 +308 2 = V206000 = 454 amperes. Since the phase pressure is 577 volts, we have: Power input = 3 X 577 X 308 = 534000 watts. Consequently when used as a measure of the power input, the vertical ordinates are to the scale of: 3X577 = 1730 watts per ampere. But it is not the input which we wish, but the output. The PR losses at the full-load input of 102 amperes, amount to: 4540+3140 = 7680 watts. 188 POLYPHASE GENERATORS AND MOTORS Consequently when the input is 454 amperes, the I 2 R losses are: ~ X 7680 = 152 000 watts. The core losses and friction aggregate: 2410+240+1300 = 3950 watts. 240 220 .180 160 ft 140 100 W 80 60 40 20 A 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 Wattless Component of Current FIG. 97. Diagram for Preliminary Consideration of the Breakdown Factor. The total internal losses amount to: 152 000+4000 = 156 000 watts. Thus the output corresponds to that portion of the vertical radius MN, which is equal to : 534 000 - 156 000 = 378 000 watts. This is the vertical height, PN, laid off equal to : 378 000 -- = 218 amperes. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 189 This corresponds to an output of: 378 000 746 = 505h.p. This vertical radius does not, however, correspond to the point of maximum output. Let us draw, as in Fig. 98, the vectors corresponding to: 430, 400 and 370 amperes. .320 300 ^240 2220 |200 "olSO 160 0,140 g 120 60 20 I 20 40 60 SO 100 120 140 160 ISO 200 220 240 260 280 300 -SO 340 360 380 400-420440 Wattless Component of Current FIG. 98. Revised Diagram for Determining the Breakdown Factor. The corresponding vertical ordinates are: 307, 300 and 290 amperes. The inputs are: 1730X307 = 531 000 watts, 1730X300 = 520 000 watts, 1730X290 = 502 000 watts. 190 POLYPHASE GENERATORS AND MOTORS The / 2 # losses are: 2 X7680 = 137 00 watts > (4on\ 2 Y~j X 7680 = 118 000 watts, = 101 000 watts. Adding the remaining losses of 4000 watts, we obtain for the total losses in the three cases: 141 000, 122 000, and 105 000 watts. Deducting these losses from the respective inputs, we obtain as the three values for the output: 531 000-141 000 = 390 000 watts, 520 000-122 000 = 398 000 watts, 502 000 - 105 000 = 397 000 watts. Obviously the output is a maximum at an input of 400 amperes and then amounts to: 398 000 Thus for our motor the bd.f. is: S A rough, empirical formula for obtaining the breakdown factor is: * -f. For our motor we have : y = 0.248; .'. bdf= 0.041. 0.4X0.24? 0.041 = 2.42. POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 191 The use of this empirical formula is, in general, quite sufficient, as it is rarely of consequence to be able to estimate the bdf. at all closely. THE POWER-FACTOR It is a very simple matter to obtain a curve of power-factors for various values of the current input. In Fig. 99 are drawn 320 240 220 200 180 160 140 120 100 80 60 40 20 Wattless Component of the Current FIG. 99. Diagram for Determining the Power-factors. vectors representing stator currents ranging from the no-load current of 25.2 amperes, up to the break-down current of 400 amperes. In each case, the power-factor is the ratio of the vertical projection of the stator current, to the stator current. The values of the currents and of the vertical projections are recorded in Fig. 99, and also in the first two columns of the following table in which the estimation of the power-factor is carried out, and also the estimation of the efficiencies. 192 POLYPHASE GENERATORS AND MOTORS I. II. III. IV. V. VI. VII. VIII. Stator Current Phlse. Vertical Com- ponent. Power Factor, (II -i-1). Input in Watts (3X577 XII). Losses. Output in Watts, (IV -V). Output in H.P. Efficiency, /VIX100\ (rv ) 25.2 2.3 0.09 4000 4000 70 61 0.87 105 000 7600 97400 130 0.93 120 110 0.92 190000 14600 175400 235 0.92 170 155 0.91 268000 25300 243000 326 0.91 220 197 0.90 342000 39700 302000 405 0.88 270 235 0.87 406000 57800 348 000 465 0.86 320 264 0.83 457 000 79300 378 000 506 0.83 370 290 0.78 502 000 105000 397 000 531 0.79 400 300 0.75 520000 122000 398000 534 0.77 In Figs. 100 and 101, these values for the power-factor and efficiency are plotted as ordinates and with amperes per phase as abscissae. In Figs. 102 and 103 they are again plotted as ordinates but with the output in horse-power as abscissae. i.UU 0.90 0.80 0.70 |0.60 ^0.50 0.40 0.30 0.20 0.10 2 ~~ i~. ^ ^ \ 50 100 150 200 250 300 350 400 Amperes Input FIG. 100. Efficiency Curve with Current Input as Abscissae. FIG. 101. Power-factor Curve with Current Input as Abscissae. THE SLIP Were it not that the apparent resistance of the slot-embedded portions of the conductors of the squirrel cage, varies with the periodicity of the rotor currents, the determination of the slip at any load would be a very simple matter. The effect of this POLYPHASE MOTOR WITH SQUIRREL-CAGE ROTOR 193 variation is too small to be of consequences at ordinary loads and may for most purposes be neglected. For constant apparent resistance, percentage slip is the percentage which the rotor PR loss constitutes of the input to the rotor. The Determination of the Rotor I 2 R Loss at Various Loads. By constructions similar to that illustrated in Fig. 94 on p. 179, we may obtain the values of the rotor current for various values of the stator current. A number of such values is tabulated below: Stator Current. Rotor Current for a 1:1 Ratio. Actual Rotor Current (Ratio of Trans, is 10.3 : 1). 25.2 amp. 70 " amp. 62 " amp. 640 " 120 " 113 " 1160 " 170 " 162 M 1670 " 220 " 210 " 2160 " 270 " 256 " 2640 " 320 " 305 " 3140 " 370 " 354 " 3640 " 400 " 384 " 3960 " 100 Output in Horsepower FIG. 102. Efficiency Curve with Horse-power Output as Abscissae. ! r-i w cq eo eo -^ TJI 10 Output in Horsepower FIG. 103. Power-factor Curve with Horse-power Output as Abscissae. With the full-load stator current of 102 amperes we have seen (on p 166.) that the squirrel-cage loss is 3140 watts. From the above table we see that for a stator current of 102 amperes, the rotor current is l^:XHQQ = j 985 amperes. Thus, referred 194 POLYPHASE GENERATORS AND MOTORS to the current in the face conductors, the squirrel cage may be considered as having an " equivalent " resistance of: 3140 985 2 = 0.00324 ohm. In the following table, the rotor I 2 R losses, the input to the rotor, the slip and the speed have been worked out: 6 Q I u . | 0-2 .? S of 05 <-< '~ ^ 03 0, e S x liT |* "o o S3 ts 02 o 6 1 S3 || 02 a flls gOOO o w |l tf S IH I- . V [Al [B] [C] [D] [E] [F] to; 500 25.2 280 2410 4000 1310 70 2140 2410 105 000 100 450 640 1330 1.32 130 493 120 6300 2410 190 000 181 300 1160 4350 2.40 235 488 170 12600 2410 268 000 253 000 1670 9000 3.55 326 482 220 21 100 2410 342 000 318 500 2160 15100 4.75 405 476 270 32000 2410 406 000 371 600 2640 22500 6.05 465 469 320 44800 2410 457000 409 800 3140 32000 7.81 506 461 370 60000 2410 502 000 439 600 3640 43000 9.80 531 451 400 70000 2410 520 000 447 600 3960 51 000 11.4 534 443 M 22 20 18 16 1 14 I 12 I 10 * 8 6 4 2 1 I / / } 1 / I / 1 / / / x x x ^^* Y^ ^-^ ^ -^^ ,^-~ DUU 500 400 300 200 100 -JT- - = ~ r"V ^^, ' \ \ 100 200 300 400 500 Output in Horsepower FIG. 104. Curves of Slip. 100 200 300 400 500 Output in Horsepower FIG. 105. Curves of Speed. These values for " slip " and speed are plotted in the full- line curves in Figs. 104 and 105; the dotted line curves being added to indicate qualitatively the general way in which the results would be affected by the influence of the frequency of the rotor currents in increasing the apparent resistance of the slot-embedded portions of the squirrel-cage. CHAPTER V SLIP-RING INDUCTION MOTORS PROBABLY considerably over 90 per cent of the total number of induction motors manufactured per annum are nowadays of the squirrel-cage type. The strong preference for this type is on account of the exceeding simplicity of its construction and the absence of moving contacts. Nevertheless occasions arise when it is necessary to supply definite windings on the rotor and to connect these up to slip rings. Sometimes this is done in order to control the starting torque and to obtain a higher starting torque for a given stator current than could be obtained with a squirrel-cage motor and sometimes the use of slip rings is for the purpose of providing speed control by regulating external resistances connected in series with the rotor windings. This is a very inefficient method of providing speed control, but cases occasionally arise when it is the economically correct method to employ. Since the controlling resistance is located external to the motor, there is no longer occasion to study to provide a deep rotor con- ductor in order to obtain a desired amount of starting torque. With freedom from this restriction, the density at the root of the rotor tooth would (for a design of the rating we have discussed in Chapter IV) be taken higher than the 15 500 lines per sq. cm. adopted for the squirrel-cage design. We shall do well to employ a somewhat shallower and wider slot, for it must be observed that it now becomes necessary to provide space for insulating the rotor conductors. Let us employ 70 slots, as before, but let the slots have a depth of only 30 mm. and a (punched) width of 8 mm. The diameter at the bottom of the slots will now be: 617.4-2X30 = 557.4 mm. 195 196 POLYPHASE GENERATORS AND MOTORS The rotor slot pitch is now: 557.4 XTC OK n -njQ =25.0 mm. Width tooth at root = 25.0 -8.0 = 17.0 mm. For the squirrel-cage design, the slot pitch (see p. 888) was 22.8 mm. and the width of tooth at the root, was: 22.8 -4.8 = 18.0 mm. Thus the tooth density will be increased in the ratio of 17 to 18, but since the length of the tooth has been decreased in the ratio of 54 to 30, the mmf . required for the rotor teeth will be no greater than for the squirrel-cage design. The thickness of the slot insulation will be 0.8 mm. Allowing 0.3 mm. tolerance in assembling the punchings, the assembled width of the slot will be: 8.0-0.3 = 7.7 mm. After deducting the portions of the width occupied by the slot insulation on each side, we arrive at the value of : 7.7-2X0.8 = 6.1 mm. for the width of the conductors, of which there will be 2 per slot, arranged one above the other. Of the depth of 30 mm., the insulation and the wedge at the top of the slot will require a total allowance of 6 mm. Consequently for the depth of each conduc- tor we have: 30-6 \ , = ) 12 mm. Thus each conductor has a height of 12 mm., a width of 6.1 mm. and a cross-section of : 12X6.1=73 sq.mm. SLIP-RING INDUCTION MOTORS 197 A drawing of the slot is given in Fig. 106. A two-layer, full-pitch, lap winding will be employed. There are 70 slots and (2X70 = ) 140 conductors. The winding will be of the 2-circuit type. It is impracti- cable in this treatise to deal with the extensive subject of armature windings. The reader will find a discussion of the laws of 2-circuit windings on page 3 of the Author's " Elementary Principles of Continuous-Current Dynamo Design " (Whittaker und" I lot of totor. Consequently conductor number 1 is connected (as indicated in Fig. 107), to conductor number (1+23 = ) 24; conductor number 24 to conductor number (24+23 = ) 47, etc., until the entire 140 conductors are interconnected to constitute a singly-re- entrant circuit. After the diagram has been carried out in this way in pencil, the next step consists in dividing up the (-=) 70 turns into six separate circuits. The six circuits 198 POLYPHASE GENERATORS AND MOTORS cannot contain equal numbers of turns. The nearest to this consists in subdividing the winding up into two circuits each comprising 22 conductors and four circuits each comprising 24 conductors. We have 22+22+24+24+24+24 = 140. FIG. 107. Winding Diagram for the Rotor of a Slip-ring Induction Motor. Let us start in with conductor number 1 and designate this beginning by A\, as shown in Fig. 107. After following through the 22 conductors: (1, 24, 47, 70, 93, 116, 139, 22, 45, 68, 91, 114, 137, 20, 43, 66, 89, 112, 135, 18, 41, 64), SLIP-RING INDUCTION MOTORS 199 we interrupt the winding and bring out a lead at the point Am. We then start in again, indicating the point as B\ and proceed next through conductor number (64+23 = ) 87. After passing through 22 more conductors we come out again at a point which we designate as Bm. 'the next circuit starts at C\ and ends at Cm. The remaining three circuits are A^A n , B^B n and C ^ -" ? & 1^* 850 K.V.A..32 Poles, 94 R.P.M. 2880 Volts per Phase, 25 Cycles Reactance Voltage per Phase at this load(unity P.F.J915 V Resultant Armature Strength 3900 Maximum rms. Current per Phase at Full Load and Unity Power Factor = 98.5_Ampa. Turns per Phase = 448 / / / / / 1 1 / / / / 2000 4000 6000 8000 10000 12000 14000 16000 Ampere Turns per Field Spool FIG. 118. No-load Saturation Curve for the 850-kva. Three-phase Generator Described in Paper No. IV. Draw a group of saturation curves (as indicated roughly in Fig. 119, for 98.5 amperes and for power-factors of 1.0, 0.8 andO. Plot excitation regulation curves of this machine for normal pressure of 2880 volts per phase, and for power-factors of 1.0, 0.8 and 0. PAPER NUMBER V Answer only one of the two following questions: (Do not necessarily attempt to do more than arrive at the approximate general outline for the machine. If, afterwards, you have time, work out and tabulate its leading properties. 230 POLYPHASE GENERATORS AND MOTORS But the chief consideration is that you should demonstrate your ability to make a rough estimate of the most probably correct design.) 1. Design a 4-pole, Y-connected, 30-cycle, three-phase, squir- rel-cage induction motor for a primary terminal pressure of 1000 volts (577 volts per phase) and for a rated output of 100 h.p. FIG. 119. Rough Indication of the Saturation Curves Called for in Paper No. IV. 2. Design a 50-cycle, Y-connectea poiypiiase generator for a rated output of 1500 kva. at a speed of 375 r.p.m. and for a ter- minal pressure of 5000 volts (2880 volts per phase). PAPER NUMBER VI For the induction motor shown in Fig. 120: 1. Make a rough estimate of a reasonable normal output to assign to the motor. 2. Estimate the no-load current. 3. Estimate the circle ratio. 4. Estimate the breakdown factor at the output you have assigned to the motor. 5. Estimate the losses and efficiency at the output you have assigned to the motor. 6. Estimate the temperature rise at the output you have assigned to the motor. 7. Estimate the power-factor at various loads. EXAMPLES FOR PRACTICE IN DESIGNING 231 (NOTE. If, rightly or wrongly, you consider that some essential data have not been included in Fig. 120, do not lose time over the matter, but make some rational assumption stating on your paper that you have done so and then proceed with the work.) Rotor Slot Cond. 6.5 x 2.5mm THREE-PHASE SQUIRREL-CAGE INDUCTION MOTOR. Number of poles Terminal pressure ( .'. Pressure per phase . . Synchronous speed in r.p.m. Stator connections 1000 V* 12 1000 volts = 578 volts 500 Y Stator. Number of slots 180 Dimensions of slot (depth Xwidth) 25 X 10 . 5 mm. Slot opening 3 mm. Conductors per slot 6 Dimensions of bare conductor. . . . 2.5 X6.5 mm. Number of end rings Rotor. 216 21.5X8.0 mm. 3 mm. 4.5X16.0 mm. 2 Section of end ring 20 X20 mm. Dimensions in centimeters and millimeters. FIG. 120. Sketches and Data of Induction Motor of Paper No. VI. PAPER NUMBER VII Design the following three-phase, squirrel-cage induction motor: Rated output = 30 h.p.; Synchronous speed = 1000 r.p.m.; Periodicity = 50 cycles per second; Pressure between terminals = 500 volts; Y-connected stator winding. I .'. Pressure per phase = ^= = 288 volts. ) 232 POLYPHASE GENERATORS AND MOTORS Proportion the squirrel-cage rotor for 4 per cent slip at rated load. Carry the design as far as time permits, but devote the last half hour to preparing an orderly table of your results. NOTE. If, rightly or wrongly, you conclude that some essential data have not been included in the above, do not lose time over the matter, but make some rational assumption stating on your paper that you have done so and then proceed with the work. PAPER NUMBER VIII For the induction motor design of which data is given in Fig. 121, estimate the losses at rated load. Draw its circle diagram. Stator Slot |< 2.16 | 3,Ducts each 1.3 wide Dimensions in cm Dimensions in cm. Output in H.P Number of poles Connection of stator windings Periodicity in cycles per second Volts between terminals Stator Winding. Total number of stator conductors . . . Number of stator conductors per slot . Dimensions of stator conductor (bare) . . Mean length of stator turn Rot-ir Windings. Total number of rotor conductors .... Number of rotor conductors per slot. . Number of phases in rotor winding. . . Dimensions of rotor conductor (bare) . Mean length of rotor turn Rotor Slot Rotor Slot. . 220 12 . Y 50 .5000 .3564 . 33, each consisting of two components . 2 in parallel, each 2.34 mm. diameter. , 159 cm. . 288 2 3 . 16X9 mm. . 150 cm. FIG. 121. Sketches and Data of the 220-H.P. Induction Motor of Paper No. VIII. Plot its efficiency, power-factor and output, using amperes input as abscissae, from no load up to the breakdown load. Estimate EXAMPLES FOR PRACTICE IN DESIGNING 233 the starting torque and the current input to the motor at starting, when half the normal voltage is applied to the terminals of the motor. NOTE. If you are of opinion that sufficient data have not been given you to enable all the questions to be answered, do not hesitate to make some reasonable assumption for such missing data. PAPER NUMBER IX Design a 300-h.p., three-phase, 40-cycle, 240-r.p.m., squirrel- cage induction motor for a terminal pressure of 2000 volts. Let the stator be Y-connected. Obtain y, the ratio of the no-load to the full-load current, a, the circle ratio, and bdf., the 'break- down factor; and further data if time permits. Employ the last half hour in criticising your own design, and in stating the changes you would make with a view to improving it, if you had .time. PAPER NUMBER X During the entire six hours at your disposal, design a three- phase squirrel-cage induction motor, to the following specification: Normal output in h.p 150 Periodicity in cycles per second 50 Synchronous speed (i.e. speed at no load), in r.p.m 250 Terminal pressure in volts 400 Connection of phases Y Pressure per phase in volts . 231 The last two hours, i.e., from 3 to 5 P.M., must be devoted to tabulating the results at which you have arrived, and to the preparation of outline sketches with principal dimensions. As to the electrical design, the following particulars will be expected to be worked out or estimated: 1. Ratio of no-load to full-load current, (y). 2. Circle ratio, (a). 3. Circle diagram to scale. 4. Breakdown factor (bdf.). 5. Per cent slip at rated load. 6. Losses in stator winding, at rated load. 234 POLYPHASE GENERATORS AND MOTORS 7. Losses in rotor winding (i.e., squirrel-cage losses at rated load). 8. Core losses. 9. Friction. 10. Efficiency at rated load. 11. Power-factor at rated load. 12. Estimate of thermometrically determined ultimate tem- perature rise at rated load. (NOTE. The students are permitted to bring in any books and notes and drawing instruments they wish. They are also permitted to fill out and to hand in as a portion of their papers, specification forms which they have prepared in advance of coming to the examination.) PAPER NUMBER XI Any notes or note-books and other books may be used, but students are put on their honor not to discuss any part of their work in the lunch hour. A three-phase, squirrel-cage induction motor complies with the following general specifications: Rated load in h.p = 90 Periodicity in cycles per second =40 Speed in r.p.m. at synchronism =800 Terminal pressure = 750 Connection of phases = Y (1) Given the following data, estimate y, the ratio of no- load to full-load current: Average air-gap density =3610 lines per sq.cm. Air-gap depth =0.91 mm. Air-gap mmf . -f- total mmf = . 80 Total number of conductor on stator .... = 576 . (2) Given the following data, estimate a, the circle ratio: Internal diameter of stator laminations =483 mm. Slot pitch of stator at air-gap =21.1 mm. Slot pitch of rotor at air-gap =24.8 mm. Gross core length; \g =430 mm. EXAMPLES FOR PRACTICE IN DESIGNING 235 (3) Draw the circle diagram. Determine the primary current : (a) At point of maximum power-factor. (6) breakdown load. (4) Plot curves between: (a) H.p. and YJ, the efficiency. (6) H.p. and G, the power-factor. (c) H.p. and slip. (d) H.p. and amperes input. Given the following data: Section of stator conductor =0.167 sq.cm. Dimensions of rotor conductor =1.27 cm.X0.76 cm. (One rotor conductor per slot) Length of each rotor conductor =49.5 cm. Diameter of end rings (external) =45 em. Number of end rings at each end = 2, each 2.54 cm. X 0.63 cm. Total constant losses =2380 watts. (5) Calculate the starting current and torque when com- pensators supplying 33, 40 and 60 per cent of the terminal pres- sure are used. (6) Calculate new end rings so that with a 2 : 1 compensator the starting torque shall be equal to one-half torque at rated load. (7) Designating the first motor as A and the second one (i.e., the modification obtained from Question 6) as B, tabulate the component losses at full load in two parallel (vertical columns) . Estimate the watts total loss per ton weight of motor (exclusive of side-rails and pul'ey). (8) If the squirrel-cage of the first motor A is replaced by a three-phase winding having the same equivalent losses at full load, calculate the resistance per phase which would be required, external to the slip rings, in order to limit the starting current to 70 amperes. What would be the starting torque, expressed as percentage of full-load torque, with this external resistance inserted? 236 POLYPHASE GENERATORS AND MOTORS PAPER NUMBER XII For the three-phase induction motor of which data is given below, make calculations enabling you to plot curves with amperes input per phase as abscissae and power-factor, efficiency and output in h.p. as ordinates. Rated load =60 h.p. Periodicity in cycles per second . = 50 Speed at synchronism = 600 r.p.m. Terminal pressure = 550 volts Connection of phases = Y Average air-gap density =3900 lines per sq.cm. Air-gap depth =0. 9 mm. Total mmf. -f- air-gap mmf = 1. 2 Total no. of conductors on stator = 720 Circle ratio =0.061 I 2 R losses at rated load =2840 Constant losses = 2050 PAPER NUMBER XIII The data given below are the leading dimensions of the stator and rotor of a 24-pole three-phase induction motor with a Y-con- nected winding suitable for a 25-cycle circuit. Ascertain approx- imately by calculation the suitable terminal voltage for this induction motor and give your opinion of the suitable rated out- put. Proceed, as far as time permits, with the calculation of the circle ratio and of the no-load current, and construct the circle diagram. Abbreviate the calculations as much as possible by reasonable assumptions for the less important steps, thus obtaining more time for the steps where assumptions can les safely be made. DIMENSIONS IN MM. Stator. External diameter 2800 Internal diameter 2440 Gross core length 385 Net core length 299 Winding 3-phase, Y-connected Number of slots 216 Depth X width of slot 51X21 Conductors per slot 12 Section of conductors, sq.cm 0.29 EXAMPLES FOR PRACTICE IN DESIGNING 237 Rotor. External diameter 2434.5 Internal diameter 2144 Winding 3-phase, Y-connected Number of slots 504 Depth X width of slot 35X9.5 Conductors, per slot 2 Section of conductor, sq.cm 0.811 [This paper to be brought in for the afternoon examination (see Paper No. XIV), as certain data in it will be required for the afternoon examina- tion.] PAPER NUMBER XIV The 24-pole stator which you employed this morning (see Paper No. XIII), for an induction motor, will, if supplied with a suitable internal revolving field with 24 poles, make an excellent three-phase, 25-cycle, Y-connected alternator. What would be an appropriate value for the rated output of this alternator? Without taking the time to calculate it, draw a reasonable no- load saturation curve for this machine. From this curve and from the data of the machine and your assumption as to the appro- priate rating, calculate and plot a saturation curve for the rated current when the power-factor of the external circuit is 0.80. PAPER NUMBER XV 1. Of two 50-cycle, 100-h.p., 500-volt, three-phase induction motors, one has 4 poles and the other has 12 poles. (a) Which will have the higher power-factor? (6) " " " efficiency? (c) " li " current at no load? (d) " " " breakdown factor? Of two 750 r.p.m., 100-h.p., 500-volt, three-phase induction motors, one is designed for 50 cycles, and the other for 25 cycles, (e) Which will have the higher power-factor? (/) " " " current at no load? (0) li " " breakdown factor? 2. Describe how to estimate the temperature rise of an induc- tion motor. 3. Describe how to estimate the magnitude and phase of the starting current of a squirrel-cage induction motor. 238 POLYPHASE GENERATORS AND MOTORS PAPER NUMBER XVI Answer one of the following two questions. Question I. For the three-phase squirrel-cage, induction motor of which data are given in Fig. 122: Rotor Slot h-i3,H mm Terminal pressure 750 Method of connection Y Pressure per phase 432 Speed in r.p.m SOO Full load primary current input per phase. . . '. 59 Number of primary conductors per slot 8 Periodicity in cycles per second 40 Length in cm. Density in Lines per sq.cm. Ampere-turns per cm. of Length. Total Ampere- turns. Stator teeth Stator core .... Rotor teeth Rotor core Air-gap Total mmf. per pole FIG. 122. Sketches and Data of the Three-phase Induction Motor of Question I in Paper No. XVI. (a) Work out the magnetic circuit calculations required to fill in the table indicated, and enter up the results in a similar table. (b) Obtain y and a for the motor. EXAMPLES FOR PRACTICE IN DESIGNING 239 Question II. For a certain three-phase, slip ring, induction motor the following data apply : Periodicity in cycles per sec 50 Speed at no load in r.p.m 500 Y (ratio of no-load to full-load current) . 36 a (circle ratio) 0.0742 Rated output in h.p 300 Terminal pressure 700 Pressure per phase 405 Connection of phases Y Stator resistance per phase (ohm) . 030 Rotor resistance per phase (ohm) . 022 Total core loss (watt) 4080 Friction and windage loss (watt) 2000 Ratio number of stator to number of rotor conductors ... 1 . 28 (a) Draw the circle diagram. (6) Plot curves of efficiency, power-factor, output in horse- power and slip, all as a function of the current input per phase. PAPER NUMBER XVII 1. For an armature having an air-gap diameter, Z) = 65 cms. and a gross core length \g = 35 cms. What would be a suitable rating for a 500-volt, 500-r.p.m. machine of these data: 1st. As a 25-cycle induction motor. 2d. As a 25-cycle alternator. Select one of these cases and work out the general lines of the design as far as time permits. 2. In Fig. 123 are given data of the design of an alternator for the following rating: 2500 kva., 3-phase, 25-cycle, 75 r.p.m., 6500-volts, Y-con- nected. The field excitations for normal voltage of 2200 volts and for 1.2 times normal voltage (2640 volts), are given. From these values the no-load saturation curve may be drawn. Estimate (showing all the necessary steps in the calculations). (a) The field ampere-turns required for full terminal voltage at full-load kva. at power-factors 1.0 and 0.8; and the pressure regulation for both these cases. 240 POLYPHASE GENERATORS AND MOTORS (b) The short circuit current for normal speed and with no-load field excitation for normal voltage. (c) The losses and efficiency at rated full load and J load at 0.8 power-factor. Also the armature heating. Scale 1-20 Mh 25 - 65 V<>lt Y Connected A.C. Generator Data No of Conductors per Slot 9 True Cross Section of 1 Conductor 1.29 sq.cm. Field Spool Winding Turns per Spool 42.5 Mean Length of 1 Turn 1620mm Cross Section of Conductor 1.78 sq. cms Saturation- Ampere Turns at 6500 Volts 8000 , .. .. 7800 ... 13000 Air Gap Ampere Turns at 6500 Volts goOO FIG. 123. Data of the Design of the 2500-kva. Alternator of Question 2 of Paper No. XVII. 3. In Fig. 121 are given data of the design of an induction motor for the following rating : 40-h.p., 600-r.p.m., 50-cycles, 500-volt A-connected, three- phase induction motor. Scale 1-10 Stator Slot opening 4mm Rotor Slot opening l.Smn 40 H.P. 600 R.P.M. 50f\J500 Volt, 3 Phase Induction Motor Stator Winding Rotor Winding Connection Winding A Squirrel Cage Type No.of Conductors pet- Slofr 18 Total No. of Bars 39 Cross Section of 1 Conductor 0.0685 sq.cms * Friction and Windage Losses 300 All Dimensions Total No.of Bars 39 in Millimeters Cross Section of Bar 1.0 sq.cm Cross Section of End Ring 3.0 sq.cm FIG. 124. Data of the Design of the 40-H.P. Motor of Question 3 of Paper No. XVII. Estimate (showing all the necessary steps in the calculations). (a) Circle ratio ( a) ; (b) No-load current in per cent of full-load current; (c) Breakdown factor ; (d) Maximum power-factor. EXAMPLES FOR PRACTICE IN DESIGNING 241 (e) Losses and efficiencies and heating at \ load and at rated full load. (/) Slip at full load. PAPER NUMBER XVIII (You may use notes, curves or any other aids.) 1. The leading particulars of a certain induction motor are given in Fig. 125. 80. H.P., 600 r.p.m., 50 CYCLES, 500 VOLTS, A-CONNECTED SQUIRREL-CAGE INDUCTION MOTOR Data: Number of stator slots 90 f Depth 36.0 Slot dimensions < Width 11.0 [ Opening 6.0 Number of rotor slots Ill f Depth 21.5 Slot dimensions { Width 6.5 [ Opening 1.5 WINDINGS: Stator Conductors per slot 12 Cross-section of conductor 0. 138 sq.cm. Rotor: Bars per slot 1 Cross-section of bar 1.0 sq.cm. Cross-section of end ring 3.6 sq.cm. All dimensions in millimeters. FIG. 125. Sketches and Data of the 80-H.P. Induction Motor of Question 1 of Paper No. XVIII. (a) Estimate y ti\e ratio of the no-load current to the current at normal rating. (6) Estimate a the circle ratio, and also estimate the maximum power-factor. (c) Estimate bdf., the breakdown factor. 242 POLYPHASE GENERATORS AND MOTORS (d) Estimate the copper losses at normal rating. (e) Estimate the core losses at normal rating. (/) Estimate the watts per square decimeter of equivalent gap surface. (g) Estimate the efficiency at normal rating. (h) Estimate the power-factor at normal rating. (i) Estimate the T.W.C. (the total works cost). 2. (a) Deduce the leading proportions for a three-phase alternator for a normal rating of 1500 kw., 1000 r.p.m., 50 cycles, 6 poles, 11 000 volts. Do not go into detail, but go far enough to give an opinion as to the suitable values for D, \g, number of slots, conductors per slot, and field excitation at no load. State briefly your reasons for choosing each of these values. (6) What, in a general way, should be the changes in the general order of magnitude of these quantities for a design for the same rated output and voltage at (1) 50 cycles, 250 r.p.m., 24 poles, and the changes necessary in this second design in order to obtain a design for (2) 25 cycles, 250 r.p.m., 12 poles. PAPER NUMBER XIX Work up a rough outline for a design for a 25-cycle, three- phase generator to supply at a pressure of 10 000 volts (Y-con- nected with 5770 volts per phase). The generator is to have a rated capacity of 3000 kva. at a power-factor of 0.90, and is to be run at a speed of 125 r.p.m. Design the machine to give good regulation of the pressure, and work out the inherent regulation at various power-factors. Work out any other data for which you find time, and devote the last half hour to an orderly tabu- lation of your results. i PAPER NUMBER XX Work up a rough outline for a design fc/r a 25-cycle, 8-pole, three-phase, squirrel-cage induction motor for a rated output of 80 h.p. The terminal pressure is 600 volte, i.e., 346 volts per phase, EXAMPLES FOR PRACTICE IN DESIGNING 243 and the s tat or windings should be Y-connected. The load will not be thrown on the motor until it is up to speed. During the last half hour prepare an orderly tabulation of the leading results which you have found time to work out. PAPER NUMBER XXI Commence the design of a three-phase, 100-h.p., squirrel- cage induction motor with a Y-connected stator for a synchro- nous speed of 375 r.p.m. when operated from a 25-cycle circuit with a line pressure of 500 volts. (The pressure per stator wind- 500 ing is consequently ^==288 volts.) Try and carry the design as far as determining upon the gap diameter and the gross core length, the number of stator slots, the number of stator conductors per slot, the flux per pole and the external diameter of the stator laminations and the internal diameter of the rotor laminations. Then tabulate these data in an orderly manner before proceeding further. Then, if time permits, make the magnetic circuit calculations and estimate the magnetizing current, PAPER NUMBER XXII (Answer one of the following two questions.) 1. Which would be the least desirable, as regards interfering with the pressure on a 50-cycle net work, low-speed or high- speed induction motors? Why? Of two 1000-volt, 100-h.p., 750-r.p.m., three-phase induction motors, which would have the highest capacity for temporarily carrying heavy overloads, a 25-cycle or a 50-cycle design? Wliich would have the highest power-factor? Which the lowest current when running unloaded? Of two 1000-volt, 100-h.p., three-phase induction motors for 25 cycles, one is for a synchronous speed of 750 r.p.in. and the other is for a synchronous speed of 150 r.p.m. Assuming rational design in both cases, which has the higher power-factor? Which the lower current when running light? Which the higher breakdown factor? Which the higher efficiency? Three 100- 244 POLYPHASE GENERATORS AND MOTORS h.p., 1000- volt designs have been discussed above. Their speeds and periodicities may be tabulated as follows: DpdirnatJrm Synchronous speed Periodicity in cycles inr.p.m. per sec. A 750 25 B 750 50 C 150 25 Assume that these are provided with low-resistance squirrel- cage windings. Make rough estimations of the watts total loss per ton of total weight of motor for each case. 2. A three-phase, Y-connected, squirrel-cage induction motor has the following constants: External diameter stator core 1150 mm. Air-gap diameter (D) 752 mm. Internal diameter rotor core 324 mm. Diameter at bottom of stator slots ... 841 mm. Diameter at bottom of rotor slots 710 mm. No. of stator slots 72 No. of rotor slots 89 Conductors per stator slot 5 Conductors per rotor slot 1 Width of stator slot 19 mm. Width of rotor slot 12 mm. Space factor stator slot 0.51 Space factor rotor slot 0.80 Gross core length (Xgr) 190 mm. Net core length (Xn) 148 mm. Polar pitch (T) 394 mm. Depth of air-gap (A) 1.5 mm. Peripheral speed 38.6 mps. Output coefficient () 2.37 The pressure per phase is 1155 volts, the pressure between terminals being 2000 volts. What is the rated output of the motor in h.p.? What is the speed in r.p.m.? What is the periodicity in cycles per second? What is the flux per pole in megalines? What is the stator PR loss? EXAMPLES FOR PRACTICE IN DESIGNING 245 What is the stator core loss? What cross-section must be given to the copper end rings in order that the slip at rated load shall be 2 per cent? Estimate the efficiency at J, J and full load. Estimate a the circle ratio. Estimate Y the ratio of the magnetizing current to the current at rated load. What is the power-factor at }-, J and full load? What is the breakdown factor? Estimate the probable temperature rise after continuous running at rated load. Remember that some of these questions are most readily solved by constructing a circle diagram and combining the dimensions scaled off from it, with slide-rule calculations. APPENDIX I A BIBLIOGRAPHY OF I. E. E. AND A. I. E. E. PAPERS ON THE SUBJECT OF POLYPHASE GENERATORS. 1891. M. I. PUPIN. On Polyphase Generators. (Trans. Am. Inst. Elec. Engrs., Vol. 8, p. 562). 1893. GEORGE FORBES. The Electrical Transmission of Power from Niagara Falls. (Jour. Inst. Elec. Engrs., Vol. 22, p. 484.) 1898. A. F. McKissiCK. Some Tests with an Induction Generator. (Trans. Am. Inst. Elec. Engrs., Vol. 15, p. 409.) 1899. M. R. GARDNER and R. P. HOWGRAVE-GRAHAM. The Synchronizing of Alternators. (Jour. Inst. Elec. Engrs., Vol. 28, p. 658.) 1900. B. A. BEHREND. On the Mechanical Forces in Dj^namos Caused by Magnetic Attraction. (Trans. Am. Inst. Elec. Engrs., Vol. 17, p. 617.) 1901. W. L. R. EMMET. Parallel Operation of Engine-Driven Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 18, p. 745.) ERNST J. BERG. Parallel Running of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 18, p. 753.) 247 248 POLYPHASE GENERATORS AND MOTORS P. 0. KEILHOLTZ. Angular Variation in Steam Engines. (Trans. Am. Inst. Elec. Engrs., Vol. 18, p. 703.) CHAS. P. STEINMETZ. Speed Regulation of Prime Movers and Parallel Operation of Alternators. (Trans. Am. Inst. Elec. Engrs. Vol. 18, p. 741.) WALTER I. SLIGHTER. Angular Velocity in Steam Engines in Rela- tion to Paralleling of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 18, p. 759.) 1902. C. 0. MAILLOUX. An Experiment with Single-Phase Alternators on Polyphase Circuits. (Trans. Am. Inst. Elec. Engrs., Vol. 19, p. 851.) Louis A. HERDT. The Determination of Alternator Characteristics. (Trans. Am. Inst. Elec. Engrs., Vol. 19, p. 1093.) C. E. SKINNER. Energy Loss in Commercial Insulating Materials when Subjected to High-Potential Stress. (Trans. Am. Inst. Elec. Engrs., Vol. 19, p. 1047.) 1903. C. A. ADAMS. A Study of the Heyland Machine as Motor and Gen- erator. (Trans. Am. Inst. Elec. Engrs., Vol. 21, p. 519.) W. L. WATERS. Commercial Alternator Design. (Trans. Am. Inst. Elec. Engrs., Vol. 22, p. 39.) A. S. GARFIELD. The Compounding of Self-Excited Alternating- Current Generators for Variation in Load and Power Factor. (Trans. Am. Inst. Elec. Engrs., Vol. 21, p. 569.) B. A. BEHREND. The Experimental Basis for the Theory of the Regulation of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 21, p. 497.) 1904. A. F. T. ATCHISON. Some Properties of Alternators Under Various Conditions of Load. (Jour. Inst. Elec. Engrs., Vol. 33, p. 1062.) H. W. TAYLOR. Armature Reaction in Alternators. (Jour. Inst. Elec. Engrs., Vol. 33, p. 1144.) MILES WALKER. Compensated Alternate-Current Generators. (Jour. Inst. Elec. Engrs., Vol. 34, p. 402.) J. B. HENDERSON and J. S. NICHOLSON. Armature Reaction in Alternators. (Jour. Inst. Elec. Engrs., Vol. 34, p. 465.) DAVID B. RUSHMORE. The Mechanical Construction of Revolving Field Alternators. (Trans. Am. Inst. Elec. Engrs., Vol 23, p. 253.) B. G. LAMME. Data and Tests on a 10 000 Cycle-per-Second Alter- nator. (Trans. Am. Inst. Elec. Engrs., Vol. 23, p. 417.) APPENDIX 249 H. H. BARNES, Jr. Notes on Fly-Wheels. (Trans. Am. Inst. Elec. Engrs., Vol. 23, p. 353.) H. M. HOBART and FRANKLIN PUNGA. A Contribution to the Theory of the Regulation of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 23, p. 291.) 1905. WILLIAM STANLEY and G. FACCIOLI. Alternate-Current Machinery, with Especial Reference to Induction Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 24, p.^51.) A. B. FIELD. Eddy Currents in Large, Slot-wound Conductors. (Trans. Am. Inst. Elec. Engrs., Vol. 24, p. 761.) W. J. A. LONDON. Mechanical Construction of Steam-Turbines and Turbo-Generators. (Jour. Inst. Elec. Engrs., Vol. 35, p. 163.) 1906. J. EPSTEIN. Testing Electrical Machinery and Materials. (Jour. Inst. Elec. Engrs., Vol. 38, p. 28.) A. G. ELLIS. Steam Turbine Generators. (Jour. Inst. Elec. Engrs., Vol. 37, p. 305.) SEBASTIAN SENSTIUS. Heat Tests on Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 25, p. 311.) MORGAN BROOKS and M. K. AKERS. The Self-Synchronizing of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 25, p. 453.) E. F. ALEXANDERSON. A Self-Exciting Alternator. (Trans. Am. Inst. Elec. Engrs., Vol. 25, p. 61.) 1907. B. A. BEHREND. Introduction to Discussion on the Practicability of Large Generators Wound for 22000 Volts. (Trans. Am. Inst. Elec. Engrs., Vol. 26, p. 351.) ROBERT POHL. Development of Turbo-Generators. (Jour. Inst. Elec. Engrs., Vol. 40, p. 239.) G. W. WORRALL. Magnetic Oscillations in Alternators. (Jour. Inst. Elec. Engrs., Vol. 39, p. 208.) [This paper is supplemented by another paper contributed by Mr. Worrall in 1908.] 1908. M. KLOSS. Selection of Turbo- Alternators. (Jour. Inst. Elec. Engrs., Vol. 42, p. 156.) S. P. SMITH. Testing of Alternators. (Jour. Inst. Elec. Engrs., Vol. 42, p. 190.) 250 POLYPHASE GENERATORS AND MOTORS G. STONE Y and A. H. LAW. High-Speed Electrical Machinery. (Jour. Inst. Elec. Engrs., Vol. 41, p. 286.) R. K. MORCOM and D. K. MORRIS. Testing Electrical Generators. (Jour. Inst. Elec. Engrs., Vol. 41, p. 137.) G. W. WORRALL. Magnetic Oscillations in Alternators. (Jour. Inst. Elec. Engrs., Vol. 40, p. 413.) [This paper is a continuation of Mr. Worrall's 1907 paper. | JENS BACHE-WIIG. Application of Fractional Pitch Windings to Alternating-Current Generators. (Trans. Am. Inst. Elec. Engrs., Vol. 27, p. 1077.) CARL J. FECHHEIMER. The Relative Proportions of Copper and Iron in Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 27, p. 1429.) 1909. S. P. SMITH. The Testing of Alternators. (Jour. Inst. Elec. Engrs., Vol. 42, p. 190.) J. D. COALES. Testing Alternators. (Jour. Inst. Elec. Engrs., Vol. 42, p. 412.) E. ROSENBERG. Parallel Operation of Alternators. (Jour. Inst. Elec. Engrs., Vol. 42, p. 524.) E. F. W. ALEXANDERSON. Alternator for One Hundred Thousand Cycles. (Trans. Am. Inst. Elec. Erigrs., Vol. 28, p. 399.) CARL J. FECHHEIMER. Comparative Costs of 25-Cycle and 60-Cycle Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 28, p. 975.) C. A. ADAMS. Electromotive Force Wave-Shape in Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 28, p. 1053.) 1910. MILES WALKER. Short-Circuiting of Large Electric Generators. (Jour. Inst. Elec. Engrs., Vol. 45, p. 295.) MILES WALKER. Design of Turbo Field Magnets for Alternate- Current Generators. (Jour. Inst. Elec. Engrs., Vol. 45, p. 319.) GEO. I. RHODES. Parallel Operation of Three-Phase Generators with their Neutrals Interconnected. (Trans. Am. Inst. Elec. Engrs., Vol. 29, p. 765.) H. G. STOTT and R. J. S. PIGOTT Tests of a 15 000-kw. Steam- Engine-Turbine Unit. (Trans. Am. Inst. Elec. Engrs., Vol. 29, p. 183.) E. D. DICKINSON and L. T. ROBINSON. Testing Steam Turbines and Steam Turbo-Generators. (Trans. Am. Inst. Elec. Engrs., Vol. 29, p. 1679.) APPENDIX 251 1911. J. R. BARR. Parallel Working of Alternators. (Jour. Inst. Elec. Engrs., Vol. 47, p. 276.) A. P. M. FLEMING and R. JOHNSON. Chemical Action in the Wind- ings of High-Voltage Machines. (Jour. Inst. Elec. Engrs., Vol. 47, p. 530.) S.' P. SMITH. Non-Salient-Pole Turbo-Alternators. (Jour. Inst. Elec. Engrs., Vol. 47, p. 562.) W. W. FIRTH. Measurement of Relative Angular Displacement in Synchronous Machines. (Jour. Inst. Elec. Engrs., Vol. 46, p. 728.) R. F. SCHUCHARDT and E. 0. SCHWEITZER. The Use of Power- Limiting Reactances with Large Turbo-Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 30.) 1912. H. D. SYMONS and MILES WALKER. The Heat Paths in Electrical Machinery. (Jour. Inst. Elec. Engrs., Vol. 48, p. 674.) W. A. DURGIN and R. H. WHITEHEAD. The Transient Reactions of Alternators. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) A. B. FIELD. Operating Characteristics of Large Turbo-Generators. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) H. M. HOBART and E. KNOWLTON. The Squirrel-Cage Induction Generator. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) E. M. OLIN. Determination of Power Efficiency of Rotating Elec- tric Machines. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) D. W. MEAD. The Runaway Speed of Water-Wheels and its Effect on Connected Rotary Machinery. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) D. B. RUSHMORE. Excitation of Alternating-Current Generators. (Trans. Am. Inst, Elec, Engrs., Vol. 31.) APPENDIX II A BIBLIOGRAPHY OF I. E. E. AND A. I. E. E. PAPERS ON THE SUBJECT OF POLYPHASE MOTORS 1888. NIKOLA TESLA. A New System of Alternate-Current Motors and Transformers. (Trans. Am. Inst. Elec. Engrs., Vol. 5, p. 308.) 1893. ALBION T. SNELL. The Distribution of Power by Alternate-Current Motors. (Jour. Inst. Elec. Engrs., Vol. 22, p. 280.) 1894. Louis BELL. Practical Properties of Polyphase Apparatus. (Trans. Am. Inst. Elec. Engrs., Vol. 11, p. 3.) Louis BELL. Some Facts about Polyphase Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 11, p. 559.) Louis DUNCAN, J. H. BROWN, W. P. ANDERSON, and S. Q. HAYES. Experiments on Two-Phase Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 11, p. 617.) SAMUEL REBER. Theory of Two- and Three-Phase Motors. (Trans. Am. Inst. Elec. Engrs., Vol 11, p. 731.) . CHAS. P. STEINMETZ. Theory of the Synchronous Motor. (Trans. Am. Inst. Elec. Engrs., Vol. 11, p. 763.) LUDWIG GUTMANN. On the Production of Rotary Magnetic Fields by a Single Alternating Current. (Trans. Am. Inst. Elec. Engrs., Vol. 11, p. 832.) 1897. CHAS. P. STEINMETZ. The Alternating-Current Induction Motor. (Trans. Am. Inst. Elec. Engrs., Vol. 14, p. 185.) 252 APPENDIX 253 1899. ERNEST WILSON. The Induction Motor. (Jour. Inst. Elec. Engrs., Vol. 28, p. 321.) 1900. A. C. EBORALL. Alternating Current Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 29, p. 799.) 1901. CHAS. F. SCOTT. The Induction Motor and the Rotary Converter and Their Relation to the Transmission System. (Trans. Am. Inst. Elec. Engrs., Vol. 18, p. 371.) 1902. ERNST DANIELSON. A Novel Combination of Polyphase Motors for Traction Purposes. (Trans. Am. Inst. Elec. Engrs., Vol. 19, p. 527.) CHAS. P. STEINMETZ. Notes on the Theory of the Synchronous Motor. (Trans. Am. Inst. Elec. Engrs., Vol. 19, p. 781.) 1903. C. A. ADAMS. A Study of the Heyland Machine as Motor and Generator. (Trans. Am. Inst. Elec. Engrs., Vol. 21, p. 519.) H. BEHN-ESCHENBURG. Magnetic Dispersion in Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 33, p. 239.) 1904. B. G. LAMME. Synchronous Motors for Regulation of Power Factor and Line Pressure. (Trans. Am. Inst. Elec. Engrs., Vol. 23, p. 481.) H. M. HOBART. The Rated Speed of Electric Motors as Affecting the Type to be Employed. (Jour. Inst. Elec. Engrs., Vol. 33, p. 472.) 1905. R. GOLDSCHMIDT. Temperature Curves and the Rating of Electrical Machinery. (Jour. Inst. Elec. Engrs., Vol. 34, p. 660.) D. K. MORRIS and G. A. LISTER. Eddy-Current Brake for Testing Motors. (Jour. Inst. Elec. Engrs., Vol. 35, p. 445.) P. D. IONIDES. Alternating-Current Motors in Industrial Service. (Jour. Inst. Elec. Engrs., Vol. 35, p. 475.) 254 POLYPHASE GENERATORS AND MOTORS C. A. ADAMS. The Design of Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 24, p. 649.) CHAS. A. PERKINS. Notes on a Simple Device for Finding the Slip of an Induction Motor. (Trans. Am. Inst. Elec. Engrs., Vol. 24, p. 879.) A. S. LANGSDORF. Air-Gap Flux in Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 24, p. 919.) 1906. \ J. B. TAYLOR. Some Features Affecting the Parallel Operation of Synchronous Motor-Generator Sets. (Trans. Am. Inst. Elec. Engrs. , Vol. 25, p. 113.) BRADLEY McCoRMiCK. Comparison of Two- and Three-Phase Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 25, p. 295.) A. BAKER and J. T. IRWIN. Magnetic Leakage in Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 38, p. 190.) 1907. L. J. HUNT. A New Type of Induction Motor. (Jour. Inst. Elec. Engrs., Vol. 39, p. 648.) R. RANKIN. Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 39, p. 714.) C. A. ADAMS, W. K. CABOT, and C. A. IRVING, Jr. Fractional-Pitch Windings for Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 26, p. 1485.) R. E. HELLMUND. Zigzag Leakage of Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 26, p. 1505.) 1908. R. GOLDSCHMIDT. Standard Performances of Electrical Machinery. (Jour. Inst. Elec. Engrs., Vol. 40, p. 455.) G. STEVENSON. Polyphase Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 41, p. 676.) H. C. SPECHT. Induction Motors for Multi-Speed Service with Particular Reference to Cascade Operation. (Trans. Am. Inst. Elec. Engrs., Vol. 27, p. 1177.) 1909. J. C. MACFARLANE and H. BURGE. Output and Economy Limits of Dynamo-Electric Machinery. (Jour. Inst. Elec. Engrs., Vol. 42, p. 232.) S. B. CHARTERS, Jr., and W. A. HILLEBRANDT. Reduction in APPENDIX 255 Capacity of Polyphase Motors Due to Unbalancing in Voltage. (Trans. Am. Inst. Elec. Engrs., Vol. 28, p. 559.) H. G. REIST and H. MAXWELL. Multi-Speed Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 28, p. 601.) A. MILLER GRAY. Heating of Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 28, p. 527.) 1910. R. E. HELLMUND. Graphical Treatment of the Zigzag and Slot Leakage in Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 45, p. 239.) C. F. SMITH. Irregularities in the Rotating Field of the Polyphase Induction Motor. (Jour. Inst Elec. Engrs., Vol. 46, p. 132.) WALTER B. N YE. The Requirements for an Induction Motor from the User's Point of View. (Trans. Am. Inst. Elec. Engrs., Vol. 29, p. 147.) 1911. T. F. WALL. The Development of the Circle Diagram for the Three-Phase Induction Machine. (Jour. Inst. Elec. Engrs., Vol. 48, p. 499.) N. PENSABENE-PEREZ. An Automatic Starting Device for Asyn- chronous Motors. (Jour. Inst. Elec. Engrs., Vol. 48, p. 484.) C. F. SMITH and E. M. JOHNSON The Losses in Induction Motors Arising from Eccentricity of the Rotor. (Jour. Inst. Elec. Engrs., Vol. 48, p. 546.) H. J. S. HEATHER. Driving of Winding Engines by Induction Motors. (Jour. Inst. Elec. Engrs., Vol. 47, p. 609.) THEODORE HOOCK. Choice of Rotor Diameter and Performance of Polyphase Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 30.) Gus A. MAIER. Methods of Varying the Speed of Alternating Current Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 30.) 1912. J. K. CATTERSON-SMITH. Induction Motor Design. (Jour. Inst. Elec. Engrs., Vol. 49, p. 635.) CARL J. FECHHEIMER. Self-Starting Synchronous Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) H. C. SPECHT. Electric Braking of Induction Motors. (Trans. Am. Inst. Elec. Engrs., Vol. 31.) P. M. LINCOLN. Motor Starting Currents as Affecting Large Trans- mission Systems, (Trans. Am, Inst, Elec, Engrs., Vol. 31.) APPENDIX III TABLE OF SINES, COSINES, AND TANGENTS FOR USE IN THE CALCULATIONS IN CHAPTER II. Angle. Sin. Cos. Tan. Angle. Sin. Cos. Tan. 1 .000 1 0.01745 0.999 0.01745 46 0.719 0.695 1.035 2 0.03490 0.999 0.03490 47 0.731 0.682 1.072 3 0.0523 0,998 0.0524 48 0.743 0.669 1.111 4 0.0697 0.998 0.0619 49 0.755 0.656 1.150 5 0.0871 0.996 0.0874 50 0.766 0.643 1.192 6 0.1045 0.994 0.1051 51 0.777 0.629 1.235 7 0.1218 0.992 0.1227 52 0.788 0.616 1.280 8 0.1391 0.990 0.1405 53 0.799 0.602 1.327 9 0.1564 0.988 0.1583 54 0.809 0.588 1.376 10 0.1736 0.985 0.1763 55 0.819 0.574 1.428 11 0.1908 0.982 0.1943 56 0.829 0.559 1.482 12 0.2079 0.978 0.2125 57 0.839 0.545 1.540 13 0.2249 0.974 0.2308 58 0.848 0.530 1.600 14 0.2419 0.970 0.2493 59 0.857 0.515 1.664 15 0.2588 0.966 0.2679 60 0.866 0.500 1.732 16 0.2756 0.961 0.2867 61 0.875 0.4848 1.804 17 0.2923 0.956 0.3057 62 0.883 0.4694 1.880 18 0.3090 0.951 0.3249 63 '0.891 0.4539 1.963 19 0.3255 0.945 0.3443 64 0.899 0.4383 2.050 20 0.3420 0.940 0.3639 65 0.906 0.4226 2.144 21 0.3583 0.934 0.3838 66 0.914 0.4067 2.246 22 0.3746 0.927 0.4040 67 0.920 0.3907 2.356 23 0.3907 0.920 0.4244 68 0.927 0.3746 2.475 24 0.4067 0.914 0.4452 69 0.934 0.3583 2.605 25 0.4226 0.906 0.4663 70 0.940 0.3420 2.747 26 0.4383 0.899 0.4877 71 0.945 0.3255 2.904 27 0.4539 0.891 0.509 72 0.951 0.3090 3.077 28 0.4694 0.883 0.532 73 0.956 0.2923 3.271 29 0.4848 0.875 0.554 74 0.961 0.2756 3.487 30 0.500 0.866 0.577 75 0.966 0.2588 3.732 31 0.515 0.857 0.601 76 0.970 0.2419 4.011 32 0.530 0.848 0.625 77 0.974 0.2249 4.331 33 0.545 0.839 0.649 78 0.978 0.2079 4.705 34 0.559 0.829 0.674 79 0.982 0.1908 5.14 35 0.574 0.819 0.700 80 0.985 0.1736 5.67 36 0.588 0.809 0.726 81 0.988 0.1564 6.31 37 0.602 0.799 0.754 82 0.990 0.1391 7.11 38 0.616 0.788 0.781 83 C.992 0.1218 8.14 39 0.629 0.777 0.810 84 0.994 0.1045 9.51 40 0.643 0.766 0.839 85 0.996 0.0871 11.4 41 0.656 0.755 0.869 86 0.998 0.0697 14.3 42 0.669 0.743 0.900 87 0.998 0.0523 19.1 43 0.682 0.731 0.932 88 0.999 0.03490 28.6 44 0.695 0.719 0.966 89 0.999 0.01745 57.3 45 0.707 0.707 1.000 90 1.000 Infinite 256 t> I-H P fc o, gg a o O001>tOTH 000 (M b- OS lO (M CO i 03 T i TP r~ CN Ososcoosoo TH !> ' IO T-H T-H I O T-H T-H T-H (M (M CO to GO to CO T-H CO CO 1> GO to i 1 T i !> CO 23i ^ O O O T-H CO O o o o o o OO T^ !> OS S 3: 00000 00000 C T-H OS OS O C^COT-Ht^TH (MtOt^OSfM COT-HI>THT-( (MT-HT-H GO TH CO (M CM T-H T I O O 00 to -^H OS iO to O CO O Os O to T-H T i to oo T-H oo oq ddddd doT-i,-H"co' O O Oco O O OT-H T-H CO oco oo oo 10 GO t^- CO (M tO CO OO T-H O T-H OS T-H >ot^o ooo rn' TH' go ... . IO T-H OS i-H CO (M CO CO IO >O CO 1> C- Os T i 03 to CO O (M COGOCO (N T-H GC CO O T-H T-H T-H (M C^ CO T 1 CO TH t>- CO ^ 1C GO (M O T-H T-H O TjH 10 o t^- oo co CO O OQ CO O o coos to CO l^ CO T i T- 1 1 0000 OS(M b- COO CO CO to dooo (NrHOOO t>- CO to TH (M ddddd (M T-H O OS 00 CO iO TH CO (N TH odd ddodod INDEX Air Circulation, see Ventilation Air-gap Density, Method of estimating for Synchronous Generators, 36 Diameter, Explanation of term, 4 (of) Induction Generators, 217, 219, 222-224 (of) Induction Motors, 132-134 Radiating Surface at the, Data for estimating, 181 (of) Synchronous Generator, 36, 37 A.I.E.E., Bibliography of papers on Polyphase Generators, 246- 251 on Polyphase Motors, 252- 255 Apparent Resistance of Squirrel-cage Rotor, 168 Armature Inference, 34 et seq; 43 et seq; 57 Relation between Theta and, 52 Magnetomotive Force, Axis of, 45, 46 Reaction with Short-circuited Arm- ature, 57 Resistance, Estimation for Syn- chronous Generator, 21 Strength, see Armature Interfer- ence Auto-transformer, Connections for starting up Induction Motor by means of, 170 Axis of Armature Demagnetization, 45,46 Bibliography of Papers on Circle Ratio, 153 Polyphase Generators 246-251, Polyphase Motors, 252-255 Breakdown Factor in Induction Motors, Determination of, 186- 191 Calculations for Slip Ring Induction Motor, 195- 201 Squirrel-cage Induction Motor, 105 et seq Design of Squirrel- cage for, 173-178 Synchronous Generator, 3 et seq Specification for, 40, 41 Deriva- tion of Design for Induction Generator from, 216-222 Circle Diagram of Squirrel-cage Motor, 154, 170 General Observations regarding, 172 Locus of Rotor Current in, 165 Circle Ratio, Bibliography of, Papers on, 153 (in) Induction Generator, 222 Kierstead's Formula, 152, 153 (in) Slip-ring Induction .Motor, 200 (in) Squirrel-cage Induction Mo- tors, Estimation of, 141 et seq Formula for estimating, 152, 153 Values of, Table, 150, 151 Circulation of Air, see Ventilation Compensator Connections for starting up an Induction Motor by means of, 170 Step-up, for Induction Generator, 223 Concentrated Windings, Inductance Calculations in, 48, 49 Conductors Copper, Table of Properties of, Appendix, 257 Rotor in Induction Generators, 214, 215, 217-219 in Induction Motors, 167 259 260 INDEX Conductors Continued (in) Slip-ring Induction Motors, 196-199 (in) Squirrel-cage Induction Motors, 113-117, 127 (in) Synchronous Generators, 9, 10 Constant Losses in Synchronous Generators, 94-96 at high speeds, 99-101 Copper Conductors, Table of Pro- perties of, Appendix, 257 Core Densities in Induction Motors, 135 Length in Induction Generators, 218 Induction Motors, 108 Synchronous Generators, 6, 26 Loss in Induction Motors, Data for esti- mating, 158 et seq Synchronous Generators, 92, 99, 100 Cosines, Table of, Appendix, 256 Cost, see Total Works Cost of Induc- tion Motors Crest Flux Density in Induction Motors, 121 Critical Speed of Vibration, 224 Current Density suitable for Induction Motors, 127, 128 Synchronous Generators, 9, 10 (in) End Rings of Squirrel- cage, 176 Ideal Short-circuit, 154, 200 Magnetizing of Induction Motor, 139, 140 (at) Rated Load Squirrel-cage Induction Motor, Values for 115, 116 Synchronous Generator, Estima- tion, of, 8 Curves Efficiency, 97, 181, 192, 193 Excitation Regulation, 71-73 Power-factor, 192, 193 Saturation, 74 No-load Satura- tion, 37-39 (of) Slip, 194 Short-circuit for Synchronous Gen- erators, 76-78 Speed, 194 Volt-ampere, 75, 83 Diameters of Squirrel-cage Induction Motors, 136 Synchronous Generators, Tabula- tion of , 28, 31, 37 Distributed Field Windings, Poly- phase Generators with, 99 et seq Dynamic Induction, Discussion of, 12, 13 Eddy Current Losses in Rotor Con- ductors as affecting the Torque, 168, 169, 177, 192, 194 Efficiency (of) Curves, 97, 181, 192, 193 Dependence of on Power-factor of Load, 97, 101, 102 Induction Generator, 219 Squirrel-cage Induction Motors, Methods of Calculating, 178 et seq.; 192, 193. Values for, 115, 116 Synchronous Generators, Methods of Calculating, 95-98; 101, 102 End Rings in Induction Generators, 218, 219 Squirrel-cage Rotor, 174-176 Use of Magnetic Material for, 206, 207 Energy, Motor Transformer of, 160 Equivalent Radiating Surface at Air- gap, Data for estimating, 181 Equivalent Resistanceof Squirrel-cage Induction Motor, 167, 168, 194 Examples for Practice in Designing Polyphase Generators and Mo- tors, 225-245 Excitation (of) Induction Generators supplied from Synchronous Apparatus on System, 213 Loss in Synchronous Generators, 93-96; 99, 100 Pressure for Synchronous Gener- ators, 84 et seq. Regulation Curves, 71-73 Field, A. B. on Eddy Current Losses in Copper Conductors, 168, 210 "Field Effect" for improving Start- ing Torque in Squirrel-cage In- duction Motors, 168; 210-212 INDEX 261 Field Excitation, 45 et seq. Calcula- tions for Synchronous Generator, 57,58 Field Spools, Design for Synchronous Generator, 84-92 Flux per Pole, Estimation of in Synchronous Generator, 18-22 Formula Circle Ratio, Kierstead's Formula for estimating, 152, 153 (for) Current in End Rings of Squirrel cage, 176 (for) Equivalent Radiating Surface at Air-gap, 181 Field Effect, 168 (for) Mean Length of Turn, 21, 156, 199 Output Coefficient, 6, 7, 108 Peripheral Speed, 113 Pressure Discussion leading up to Deriva- tion of, 13, 14 (for) Squirrel-cage Induction Motors, 119 Winding Pitch Factor in, 16, 20 Reactance, 48 Theta, 51 Total Works Cost, 111 Two-circuit Armature Winding, 197 Fractional Pitch Windings, 16-18 Friction Losses in Induction Motors, Data for esti- mating, 166 Synchronous Generators, 93-96; at high speeds, 99, 100 Full Load Power-factor, Estimation of, in Squirrel-cage Induction Motors, 155 Full Pitch Windings, 14-18; 120, 197 Generators Induction, see Induction Genera- tors Synchronous, see Synchronous Generators Synchronous versus Induction, 209, 210, 213 Goldschmidt, Dr. Rudolf, on Power- factors of Induction Motors, 202 Gross Core Length in Induction Generators, 218 Squirrel-cage Induction Motors, 108 Synchronous Generators, 6 Half-coiled Windings, 18-20; 41-43 I.E.E., see Institution of Electrical Engineers Ideal Short-circuit Current, 154; 200 Impedance, 76, 77 Inductance of Armature Windings of Synchronous Generators, 45-50; 53 Induction Generators, Design of, 1; 213-224 Derivation of Design from Design of Synchronous Generators, 216 222 Speeds, Appropriateness for ex- ceedingly high, 213 Synchronous Generators versus, 209, 210, 213 Ventilating, Methods of, 220. 221 Induction Motors, 1 Slip Ring, 195-201 Squirrel-cage, Design of, 105 et seq. Magnetic End Rings, Use of, 206, 207 Open Protected Type, 186 Slip Ring, Discussion of the rela- tive merits of Squirrel-cage and, 195-201 Squirrel-cage Design, 173-178 Synchronous Motors versus, 202-212 Inherent Regulation, 44; 54; 55; 66; 71; 80-82 Institution of Electrical Engineers, Bibliography of papers on Poly- phase Generators, 246-251 on Polyphase Motors, 252-255 Insulation (of) Field Spools, 88, 89 Lamination of Induction Motors, 126 Slot (in) Slip-ring Induction Motors, 196; 200 Squirrel-cage Induction Motors, 114; 128-130 Synchronous Generators, 10, 11 Kierstead's Formula for Circle Ratio, 152; 153 Lap Winding, 18; 19; 197 Leakage Factor, 22; 23; 42. See also Circle Ratio 262 INDEX Losses (in) Squirrel-cage Induction Motors, 155 et seq.; 166; 178 et seq. Synchronous Generator, 93-98 Effect of High Speed on, 99-101 Magnet Core, Material and Shape suitable for 2500 kva. Synchro- nous Generator, 23-25 Magnet Yoke, Calculations for Syn- chronous Generator, 29 Magnetic Circuit Squirrel-cage Induction Motors Design, 120 et seq. Magnetomotive Force, Esti- mation of, 124; 134 Mean Length of, 121 Sketch, 137; 138 Synchronous Generators Design, 22 et seq. Magnetomotive Force, Esti- mation of, 32 et seq. Mean Length of, 31 Data for Teeth and Air-gap in In- duction Motors, 134 End Rings, Use of in Squirrel-cage Motors, 206; 207 Flux, 12 et seq. Distribution in Induction Mo- tors, 122-125 Estimation of Flux per Pole, for Synchronous Generators, 18- 22 Materials, Saturation Data of va- rious, 32; 33 Reluctance of Sheet Steel, 137 Magnetizing Current of Induction Motor, 139; 140 Magnetomotive Force Axes of Field and Armature, 45; 46 Induction Generator Calculations, 222 Induction Motor Calculations, 134 per Cm. for various Materials, Table, 33 Synchronous Generator Calcula- tions, 32 et seq, 54 Tabulated data of, 32; 34; 38; 42; 58; 61; 64; 65; 67; 69; 73; 74; 137 Mean Length of Magnetic Circuit in Induction Motors, 121 in Synchronous Generators, 31 Turn of Winding, 21 in Slip-ring Induction Motors, 199 in Squirrel-cage Induction Mo- tors, 155-157 Metric Wire Table, Appendix, 257 Motors, Induction, see Induction Motors Motor is Transformer of Energy, 160 Net Core Length in Synchronous Generators, 26 No-load Current of Induction Motor, 140 Saturation Curves, 37-39 In- fluence of Modifications in, 78- 83 Open Protected Type of Squirrel-cage Induction Motor, 186 Output Coefficient Formula, Discussion of Significance of, 6; 7 Squirrel-cage Induction Motors, Values for, 108; 109 Synchronous Generator, Table of Values for, 5; 6 Output from Rotor Conductors in Induction Motors, 167 Partly Distributed Windings, In- ductance Calculations, 48, 49 Peripheral Loading, Appropriate Val- ues f.or Squirrel-cage Induction Motors, 113-115 for Synchro- nous Generators, 8 Peripheral Speed of Squirrel-cage Induction Motors, 113 Synchronous Generators, 25 Pitch Polar, see Polar Pitch Rotor Slot in Slip-ring Induction Motors, 196 Squirrel-cage Induction Motors, 174; 196 Slot, see Tooth Pitch Tooth, see Tooth Pitch (of) Windings, 16-18; 120; 197 INDEX 263 Polar Pitch, Suitable Values for Squirrel-cage Induction Motors, 106; 107 Synchronous Generators, 4; 5 Poles, Data for number of in Squirrel-cage Induction Motors, 106 Synchronous Generators, 3 Power-factor Curves, 192; 193 Efficiency, Dependence of on P.F. of Load, 97; 101; 102 Induction Motors versus Synchron- ous Motors, 202-212 Saturation Curves, Estimation of for various, 55 et seq Squirrel-cage Induction Motors, Estimation of, 115; 116; 155; 191-193 Pressure Formula Discussion leading up to deri- vation of, 13; 14 (for) Squirrel-cage Induction Motors, 119 Winding Pitch Factor in, 16; 120 Regulation, Method of Estimating, 39 et seq Total Internal of Synchronous Generator, 53 Radial Depth of Air-gap (for) Induction Motors, 133 Synchronous Generators, 36; 37 Radiating Surface at Air-gap, Data for Estimating, 181 Ratio of Transformation in Induction Generators, 217; 223 Slip-ring Induction Motors, 199 Squirrel-cage Induction Motors, 174 Reactance of Windings of Synchro- nous Generators, 48 Reactance Voltage, 53 Determina- tion of Value for Synchronous Generators, 48-51 Regulation, Excitation for Synchro- nous Generators, 71-73 Resistance of Squirrel-cage Rotor, 167; 168; 194 Robinson, L. T., Skin Effect Investi- gation on Machine-steel Bars, 208; 209 Rotor (of) Induction Generators, 223; 224 C&nductors in, 214 et seq Slots in, 217 Slip-ring Induction Motor, Slot Pitch, 196 Slots, 195-197 Windings for, 195; 197-199 Squirrel-cage Induction Motor Conductors, Eddy Current Losses as affect- ing the Torque, 168; 169; 177; 192; 194 Output from, 167 Core Densities in, 135 Loss in, 158; 159; 164-166; 193; 194 Material for, Choice of; 158 Resistance, 167; 168 Slots Design of, 132 Number, 173 Pitch, 174; 196 Squirrel-cage, Design of, 173- 178 Salient Pole Generator, Calculations for 2500 kva., 3 et seq Derivation of Design for Induction Generator from; 216-222 Specification of, 40 see also Synchronous Generator Saturation Curves for Synchronous Generator, 55 et seq; 74 No-load Curves, 37-39 Influ- ence of Modifications of, 78-83 Data of various Magnetic Mate- rials, 32; 33 Sheet Steel, Magnetic Reluctance of 137 Short-circuit Curve for Synchronous Generators, 76-78 Sines, Table of, Appendix, 256 Single-layer Windings, 18 Skin Effect to improve Starting Torque of Synchronous Motors, 207-209 Slip, 106; 163; 164; 192; 194; 215 Slip-ring Induction Motor, Discus- sion of relative merits of Squirrel- cage and, 195-201 264 INDEX Slot-embedded Windings, Inductance and Reactance of, 48-50 Slot Insulation in Induction Motors, 114; 128- 130; 196; 200 Synchronous Generator, 10; 11 Pitch, see Tooth Pitch Space Factor, 11; 131 Tolerance, 127; 196 Slots Rotor, for Induction Generators, 217 Slip-ring Induction Motors, 195- 197 Squirrel-cage Induction Motors, 132; 141; 173; 174; 196 Stator, for Induction Generators, 217 Induction Motors, 117-119; 127; 131; 141 Synchronous Generators, 9; 11 Space Factor (of) Field Spools of Synchronous Gen- erators, 88; 89 Slot, 11; 131 Specification of 2500 kva. Synchro- nous Generator, 40; 41 Speed Control, Methods of providing, 195 Curves, 194 High-speed Sets, Characteristics of, 99 Synchronous versus Induction Mo- tors for Low and High, 202-212 Spiral Windings, 18-20 Spread of Winding, 14-16; 120 Spreading Coefficients, 36 Squirrel-cage Induction. Motor, see Induction Motor Starting Torque of Squirrel-cage Motor, 169-172 Stator Conductors, Determination of Di- mensions for Induction Motors, 127 Core of Induction Motors Density, 135 Loss, 158; 159 Material preferable for, 158 Weight of, Estimation, 159 Stator Continued Core of Continued Synchronous Generators, 27; 28 Weight, 92 Current Density suitable for Induc- tion Motors, 127; 128 PR Loss in Induction Motors, 155-157 Synchronous Generators, 93-96 Slot Pitch, Values for Induction Motors, 118 Slots in Induction Generators, 217 Induction Motors, 119; 127; 131 Synchronous Generators, 11 Teeth, Data for Induction Motors, 121-127 Steam-turbine Driven Sets, Rotors with Distributed Field Windings for, 99 Circulating Air Calcu- lations, 102-104 Step-down Transformers for small Motors, 114 Step-up Transformer for Induction Generator, 223 Synchronous Generators, 1 Distributed Field Winding Type, 99 et seq Efficiency, Dependence of on Power-factor of Load, 97; 101; 102 Induction Generator versus, 209; 210; 213 Salient Pole Type Calculations for 2500 kva., 3 et seq Derivation of Design for Induc- tion Generator from, 216- 222 Specification, 40; 41 Synchronous Motors, 1 Induction Motors versus, 202-212 Tabulated Data of Magnetomotive Force Calculations, 32; 34; 38; 42; 58; 61; 64; 65; 67; 69; 73; 74; 137 Tabulation of Losses and Efficiencies in Squirrel- cage Induction Motors, 180 Squirrel-cage Induction Motor Di- ameters, 136 INDEX 265 Tabulation of Continued Synchronous Generator Diameters, 28, 31: 37 Tangents, Table of, Appendix, 256 Teeth, Stator, in Induction Motors, 121-127 Temperature Rise, Data for estima- ting, 90; 181-183 Theta and its Significance, 51 et seq Thoroughly Distributed Windings, Inductance Calculations, 48; 49 Tooth Densities in Induction Motors, 121-125 Pitch 26; 118; 196 Torque, 162; 195 Eddy Current Losses in Rotor Conductors as affecting, 168; 169; 177; 192; 194 Starting (of) Squirrel-cage Induction Motor, 169-172 Synchronous Motors appropri- ate for high, 204 et seq Torque Factor, 167 Total Net Weight of Induction Motors, 110; 111 Data for estimating, 185 Synchronous Generators, 7 Total Works Cost of Induction Motors, 158 Methods of esti- mating, 111-113; 118 Transformers Auto, for starting up Induction Motors, 170 Step-down, for small Motors, 114 Step-up, for Induction Generators, 223 Two-layer Winding, 18; 197 Variable Losses in Synchronous Gen- erators, 94-96 Ventilating Ducts for Induction Motors, 125; 126 Synchronous Generators, 25; 26 Ventilation of Induction Generators, Methods suitable for, 220; 221 Synchronous Generators with Dis- tributed Field Windings, 102-104 Vibration, Critical Speed of, 224 Volt-ampere Curves, 75, 83 Voltage Formula, see Pressure Formula Regulation, Method of Estimating, 39 et seq Watts per Ton for Squirrel-cage Induction Motor, 183-186 Weight of Induction Motors, 110; 111; 183- 185 Stator Core of Induction Motors, 159 Synchronous Generators, 7 Whole-coiled Windings, 18-20 Winding Pitch, 16-18; 120; 197 Pitch Factor, 16; 120 Spreads, 14-16; 120 Inductance and Reactance of, 45- 49 Types of, 16-20; 41-43 Distributed Field, Polyphase Generators with, 99 et seq (for) Slip-ring Induction Motors, 197-199 Wire Table, Metric, Appendix, 257 THIS BOOK IS DUE ON THE LAST BATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. - JUL 2 195T-P SEP 5 J.I) 21-20m-6,'32 YC 33501 257829