LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
ALTERNATING CURRENT 
 
 COMMUTATOR MOTORS 
 
 BY A. S. M'ALLISTER 
 
ALTERNATING 
 CURRENT COMMUTATOR MOTORS 
 
 A thesis presented to the University Faculty of Cornell 
 University for the Degree of Doctor of Philosophy. 
 
 BY 
 A. S. M'ALLISTER 
 
 JUNE, 1905 
 
 OF THE 
 
 ( UNIVERSITY 
 
 V OF / 
 
 XC*L!FORNX 
 
 ^^^ i . .-^^-^~ 
 
CONTENTS. 
 ALTERNATING CURRENT COMMUTATOR MOTORS. 
 
 PAGE. 
 
 REPULSION MOTOR 18 
 
 REPULSION-SERIES MOTOR 33 
 
 PLAIN SERIES MOTOR 49 
 
 INDUCTIVELY COMPENSATED SERIES MOTOR 56 
 
 CONDUCTIVELY COMPENSATED SERIES MOTOR 57 
 
 INDUCTION SERIES MOTOR 61 
 
 COMPENSATED SERIES MOTOR WITH SHUNTED FIELD COIL 75 
 
 APPENDIX 79 
 
 173262 
 
Alternating Current Commutator Motors. 
 Repulsion Motor. 
 
 A. S. M 
 
 Reprinted from The Sibley Journal of Engineering, Oct., 1904. 
 
,TV 
 
 r OF / 
 
 ALTERNATING CURRENT COMMUTATOR MOTORS. 
 REPULSION MOTOR.* 
 
 A. s. M'AUJSTER. 
 
 In dealing with the phenomena connected with the operation 
 of alternating current motors of the commutator type, it must 
 be constantly borne in mind that the machine possesses simul- 
 taneously the electrical characteristics of both a direct current 
 motor and a stationary alternating current transformer. The 
 statement just made must not be confused with a somewhat sim- 
 ilar one which is applicable to polyphase induction motors, since 
 only with regard to its mechanical characteristics does an induc- 
 duction motor resemble a shunt-wound direct current machine, 
 its electrical characteristics being equivalent in all respects to 
 those of a stationary transformer. 
 
 Before discussing the performance of repulsion motors, it is 
 well to investigate a few of the properties common to all com- 
 mutator type, alternating current machines. It will be recalled 
 that when current flows through the armature of a direct cur- 
 rent machine, magnetism is produced by the ampere turns of 
 the armature current, such magnetism tending to distort the 
 flux from the field poles. In the familiar representation of the 
 magnetic circuit of machines, the two pole model, the arma- 
 ture magnetism is at right angles to the field magnetism, the 
 armature current producing magnetic poles in line with the 
 brushes. The amount of this magnetism depends directly on 
 the value of the armature current and the permeability of the 
 magnetic path. When alternating current is used, the change 
 of the magnetism with the periodic change in the current pro- 
 duces an alternating e.m.f. which being proportional to the rate 
 of change of the magnetism will be in time- quadrature to the 
 current. The armature winding thus acts in all respects simi- 
 larly to an induction coil. 
 
 It is not essential that the current to produce the alternating 
 flux flow through the armature coils in order that the alter- 
 nating e.m.f. be developed at the commutator. Under whatso- 
 ever conditions the armature conductors be subject to changing 
 flux a corresponding e.m.f. will be generated, in mechanical line 
 
 * Abstract of thesis for Ph.D. degree, Cornell University. 
 
Repulsion Motor. 19 
 
 with the flux and in time- quadrature to it. Referring to Fig. i 
 which represents a direct current armature situated in an alter- 
 nating field, having two pair of brushes, one in mechanical line 
 with the alternating flux and one in mechanical quadrature 
 thereto. When the armature is stationary an e.m.f. will be 
 generated at the brushes A and A due to the transformer action 
 of the flux, but no measurable e,m.f. will exist between B and 
 B. As seen above, this e.m.f. is in time- quadrature with the 
 field (transformer) flux and as will be seen later, its value is un- 
 altered by any motion of the armature. At any speed of the 
 armature, there will be generated at the brushes B and B an 
 e.m.f. proportional to the speed and to the field magnetism and 
 in time-phase with the magnetism. At a certain speed this 
 " dynamo " e.m.f. will be equal in effective value to the " trans- 
 former " e.m.f. at A and A, though it will be in time-quadra- 
 ture to it. This critical speed will hereafter be referred to as 
 the " synchronous " speed, and with the two-pole model shown 
 in Fig. i , it is characterized by the fact that in whatsoever posi- 
 
 F/G.1- ELECTROMOTWE FORCES PRODUCED . 
 M M* ALTER MATING FIELD. 
 
2O The Sibley Journal of Engineering. 
 
 tion on the armature a pair of brushes be placed across a diam- 
 eter, the e.m.f. between the two brushes will be the same and 
 will have a relative time-phase position corresponding to the 
 mechanical position of the brushes on the commutator. 
 
 A little consideration will show that the individual coils in 
 which the maximum e.m.f. is generated by transformer action 
 are situated upon the armature core under brushes B or B, al- 
 though the difference of potential between the brushes B and B 
 is at all times of zero value as concerns the transformer action. 
 A similar study leads to the conclusion that the e.m.f. generated 
 by dynamo speed action appears as a maximum for a single coil 
 when the coil is under brush A or A. Assuming as zero posi- 
 tion, the place under brush A and that at synchronous speed the 
 e.m.f. generated in a coil at this position is e. Then the e.m.f. 
 in a coil at b will equal e also. A coil a degrees from this posi- 
 tion will have generated in it a speed e.m.f. of e cos a and a 
 transformer e.m.f. of e cos (a 90) =^= q= e sin a. Since these 
 two component e.m.f.s are in time quadrature the resultant will 
 be V= N/ (>cosa) 2 + ( e sin a)' 2 = e an d is the same for all values 
 of a. The time-phase position of the resultant, however, will 
 vary directly with a or with the mechanical position of the coil. 
 From these facts it is seen that at synchronous speed the effective 
 value of the e.m.f. generated per coil at all positions is the same 
 and that there is no neutral e.m.f. position on the commutator. 
 
 In a repulsion motor as commercially constructed, the secondary 
 consists of a direct current armature upon the commutator of 
 which brushes are placed in positions 1 80 electrical degrees apart 
 and directly short circuted upon themselves, as shown in the 
 two-pole model of Fig. 2. The stationary primary member con- 
 sists of a ring core containing slots more or less uniformly 
 spaced around the air-gap. In these slots are placed coils so 
 connected that when current flows in them definite magnetic 
 poles will be produced upon the field core. The brushes on the 
 commutator are given a location some 15 degrees from the line 
 of polarization of the primary magnetism, or more properly ex- 
 pressed, the brushes are placed about 15 degrees from the true 
 transformer position. That component of the magnetism which is 
 in line with the brushes produces current in the secondary by trans- 
 former action, and this current gives a torque to the rotor due 
 to the presence of the other component of magnetism in me- 
 chanical quadrature to the secondary current. 
 
Repulsion Motor. 
 
 21 
 
 It is possible to make certain assumptions as to the relative 
 values of the magnetism in mechanical line with, and in me- 
 chanical quadrature to the brush line and thus to derive the 
 fundamental equations of the machine. It is believed, how- 
 ever, that the facts can be more clearly presented and the treat- 
 ment simplified without sacrifice of accuracy if the assumption 
 be made that the primary coil is wound in two parts, one in me- 
 chanical line and the other in mechanical quadrature with the 
 axial brush position as shown in Fig. 2. It will be noted that 
 the two fields produced by the sections of the primary coil if 
 
 F/G.&* - TWf)- POLE MODEL or IDEAL REPULSION MOTOR. 
 
 there were no disturbing influence present, would have a result- 
 ant position relative to the brush line depending upon the ratio 
 of the strengths of the two magnetisms. The angle which the 
 resultant field would assume can be represented by /8 having a 
 
 value such that cotan (3 = 4 where < t is the flux through trans- 
 
 9r 
 former coil and < f is flux through field coil. If n be the ratio of 
 
 turns on the transformer poles to those on the field poles, then for 
 any value of current in these coils (no secondary current) 
 
 -7 = n or n = cotan 
 
 (i) 
 
22 The Sibley Journal of Engineering. 
 
 It is understood that in Fig. 2, the core material is considered 
 to be continuous and that in the two-pole model represented 
 both field poles and both transformer poles are supposed to be 
 properly wound. 
 
 In Fig. 2, let it be assumed that the machine is stationary and 
 that a certain e.m.f. , , is impressed upon the primary circuits, 
 the secondary being on short circuit. The flux which the 
 primary current tends to produce in the transformer pole pro- 
 duces by its rate of change an e.m,f. in the secondary, and this 
 e.m.f. causes opposing current to flow in the closed secondary 
 circuit. If the transformer action is perfect and the transformer 
 coil and armature circuits are without resistance and local leak- 
 age reactance, then the magnetomotive force of the armature 
 current equals that of the current in the transformer coil, and 
 the resultant impedance effect of the two circuits is of zero 
 value, so that the full primary e.m.f., , is impressed upon the 
 field coil, that is to say, with armature stationary t = O, and 
 E,-E. 
 
 It remains now to investigate the effect of speed on the 
 electromotive forces of the transformer and field coils. Assume 
 a certain flux 3> f in the field coil. At speed ,5 the armature con- 
 ductors will cut this flux and at each instant there will be gen- 
 erated an e.m.f. therein proportional to S<f> v and therefore, in 
 time-phase with the flux. This e.m.f. would tend to cause cur- 
 rent to flow in the closed armature circuit, which current would 
 produce magnetism in line with the brushes, and, since the 
 armature circuit has zero impedance, (assumed) the flux so pro- 
 duced will be of a value such that its rate of change through the 
 armature coils just equals the e.m.f. generated therein by speed 
 action. At synchronous speed, the secondary being closed, the 
 flux in line with the brushes must equal that in line with the 
 field poles, since the e.m.f. generated by the rate of change of 
 the flux in the direction of the brushes must equal that gene- 
 rated at the brushes due to cutting the field magnetism, and at a 
 speed which has been termed synchronous these two fluxes are 
 equal, as previously discussed. At this speed the two fluxes 
 are equal but they are in time- quadrature one to the other. At 
 other speeds the two fluxes retain the quadrature time-phase 
 position, but the ratio of the effective values of the two fluxes 
 varies directly with the speed. 
 
 Giving to synchronous speed a value of unity, at any speed, 
 
Repulsion Motor. 23 
 
 S, the transformer flux may be expressed by the equation 
 
 * t = S4> t (2) 
 
 effective values being used throughout. Letting < be the max- 
 imum value of the field flux and reckoning time in electrical de- 
 grees from the instant when the field flux is maximum, at any 
 time 8, the instantaneous field flux is 
 
 < f =<cos8 (3) 
 
 and the transformer flux is 
 
 < t = S<f> sin (4) 
 
 These are the fundamental magnetic equations of the ideal 
 repulsion motor. 
 
 If at a certain speed S, the effective value of e.m.f. across the 
 field coil be F, requiring an effective flux of < f , then across the 
 transformer coil there will be an effective e.m.f. of 
 
 T=nSF (5) 
 
 due to the flux S<f> r Since the fluxes are in time-quadrature, 
 the e.mf.s are likewise in time quadrature, so that the impressed 
 e.m.f. E must have a value such that 
 
 E = Vp* + T 2 (6) 
 
 This is the fundamental electromotive force equation of the re- 
 pulsion motor. 
 
 The current which flows through the field coil is 
 
 7 =T (7) 
 
 where X is the inductive reactance of the field coil. Equation 
 (7) gives the value of the primary circuit current and is the 
 fundamental primary current equation. 
 
 The secondary armature current in general consists of two 
 components, that equal in magnetomotive force and opposite in 
 phase to the primary transformer current, and that necessary 
 to produce the flux in line with the brushes. With a ratio of 
 effective armature turns to field turns of a, the opposing trans- 
 former current is 
 
 and the current which produces the transformer poles is 
 
 /,=f 
 
24 The Sibley Journal of Engineering. 
 
 These component currents are in time- quadrature, so that the 
 resultant secondary current is 
 
 This is the fundamental equation for the secondary current. 
 Combining (8) (9) and (10) 
 
 It has been seen that the e.m.f. T is in time- quadrature to the 
 field circuit e.m.f., F. Now the current is in time- quadrature 
 with F, and hence, is in time-phase with T. Therefore, of the 
 total primary e.m.f. E, the part T is in phase with the current, 
 from which fact it is seen that the power factor is 
 
 COS0 = J (12) 
 
 Power, 
 
 Torque, 
 
 P IT ISnF 
 
 D = InF = InXI = PnX ( 14) 
 
 E 2 = F 2 + T 2 = F 2 (i+SV) (15) 
 
 
 when n = i, that is at /8 = 45 see (i) 
 
 
 
 I = and is constant at all speeds. 
 aX 
 
 when 6* = i, that is at synchronism for any value of n. 
 
 77 
 
 7 a = - which is seen to be equal to the primary 
 aX 
 
 current at starting, (when a = i ) 
 
 when S = i the secondary current 
 
Repulsion Motor. 
 
 and leads the primary current by angle cotan" 1 n = ft or angle 
 of brush shift. See equation (i). 
 
 f/C.3. - DIAGRAM Of IDEAL. REPULSION MOTOR 
 
 The above equations can be expressed graphically by a simple 
 diagram as shown in Fig. 3. The diagram is constructed as 
 follows : OE is the constant line e.m.f. OA at rt. angles to OE 
 is the line current at starting, OB A is a semicircle, OF in phase 
 opposition to OA is the secondary current at starting. ODFis 
 a semicircle. OG, in phase with OA, is the secondary current 
 at infinite speed. OHG is a semicircle. It will be noted that 
 the ratio OA to OG is na : i and ratio of OA to OF is a : n. 
 
 Distances measured from P in the direction of T represent 
 speed. 
 
 The characteristics of the machine may be found at once from 
 Fig. 3. Assuming any speed as PS, draw OS intersecting the 
 circle OB A at B. From point G draw line GK parallel to OS. 
 Join O and K. 
 
26 The Sibley Journal of Engineering. 
 
 OK is secondary current ; 
 OB is primary current ; 
 EOS is primary angle of lag ; 
 BC\$ power component of primary current ; 
 BC is power (to proper scale) ; 
 OC is torque (to proper scale) ; 
 DOK is angle of lead of secondary current. 
 At synchronous speed (S = i), cotan 6 = n, hence scale of 
 speed can readily be located. 
 OD = 7 t , see equation (7). 
 OH= I v see equation (8). 
 The proof of the construction of diagram of Fig. 3 is as follows : 
 
 cos0 = J Bq. (ii) 
 
 E* = r 2 + F* Bq. (6) 
 
 T= SnF Bq. (5) 
 
 ^=l+SV (20) 
 
 T Sn 
 
 '-ST+sw 
 
 Power component of primary current 
 
 (22) 
 
 Quadrature component of primary current 
 
 /q = AT(i -f 5V 
 
 cotan = Sn (25) 
 
 The cotangent of the angle of lag is directly proportional to 
 the speed, the proportionality constant being the ratio of trans- 
 former to field turns. 
 
Repulsion Motor. 27 
 
 (26) 
 
 Torque is proportional to quadrature component of the 
 primary (for given e.m.f.) the proportionality constant being the 
 ratio of transformer to field turns. 
 
 (27) 
 
 Torque varies as the square of the primary current and in this 
 respect is independent of the speed or the e.m.f. 
 
 A comparison of equations (26) and (27) reveals an interest- 
 ing property of a circle. In Fig. 3 assuming the diameter A O 
 to be unity, O C at all valnes of angle 6 equals the square of OB. 
 
 From equation (27) it is seen that the torque is at all times 
 positive, even when 6* is negative. Hence machine acts as 
 generator at negative speed. For the determination of the 
 generator characteristics it is necessary to construct the semi- 
 circle omitted in each case in Fig. 3. 
 
 It is interesting to observe that the construction of the diagram 
 of Fig. 3 can be completed at once when points F, O, G and A 
 and E are located. Thus the complete performance of the ideal 
 repulsion motor *;can be determined when E, X, n and a are 
 known. In the construction for ascertaining the value of the 
 secondary current, it will be seen that O K is equal to the vector 
 sum of O D and O H, giving the vector O K. From the proper- 
 ties of vector co-ordinates it will be noted that the point K 
 is located on the semicircle F K G whose center lies in the line 
 FOG. Therefore if G and F be located, the inner circles 
 F D O and O HG need not be drawn, since the point K can be 
 found as the intersection of the line drawn parallel to O B from 
 G with the circular arc FK ' G. 
 
 It is to be carefully noted that the above discussion refers to 
 ideal conditions which can never be realized. The circuits have 
 been considered free from resistance and leakage reactance while 
 all iron losses, friction, and brush short circuiting effects have 
 been neglected. The resistance and leakage reactance effects 
 can quite easily be taken into account, but the remaining dis- 
 turbing influences are subject to considerable error in approxi- 
 mating their values, due primarily to the difficulty in assigning 
 to iron any constant in connection with its magnetic phenomena. 
 It is to be regretted that the so-called complete equations for ex- 
 pressing the characteristics of this type of machinery with al- 
 
28 The Sibley Journal of Engineering. 
 
 most no exception neglect these disturbing influences, and yet 
 these same equations are given forth by the various writers as 
 though they represented the true conditions of operation. 
 In the ideal motor the apparent impedance is 
 
 +. 5V (28) 
 
 apparent resistance is 
 
 R = Zcos = XSn (29) 
 
 since 
 
 cos = _ ; T= SnF\ and E = 
 E 
 
 hence 
 
 ~ ^ V = v/i + 5V 
 
 apparent reactance is 
 
 since 
 
 i + 5V Vi + 5V 
 
 R t = resistance of field coil 
 R t = resistance of transformer coil 
 R & = resistance of armature coil 
 X & = reactance of armature coil 
 X t = reactance of transformer coil 
 X i = reactance of field coil 
 then copper loss of motor circuits will be 
 
 7 2 (R t + R^} 4- I*R. 
 
 hence 
 
 E Vn* + 5 2 
 y '-A-^/ I + 5V 
 
 /= ^7^ ( J 7) 
 
 /.-^ (32) 
 
 and copper loss will be 
 

 >m 
 
 \(R t 
 
 Repulsion Motor. 29 
 
 (33) 
 
 where R m is the effective equivalent value of the motor circuit 
 resistance, that is 
 
 R, = R, + JR t + -*. (34) 
 
 Similarly it may be shown that the effective equivalent value 
 of the leakage reactance of the motor circuits is 
 
 . (35) 
 
 If these valves be added to the apparent resistance and react- 
 ance of the ideal motor the corresponding effects will be repre- 
 sented in the resultant equations thus 
 
 ^^\ & ( 36) 
 
 a J 
 
 R = XSn + X t + X t 
 
 and 
 
 t (37) 
 
 a 
 + X* from (36) and (37) (38) 
 
 E , ^. 
 
 =z (39) 
 
 Input = .7 cos (40) 
 
 output = Ycos B PR m = P (41 ) 
 
 p 
 
 torque = = D, etc. (42) 
 
 *J 
 
 It will be noted that the short circuiting by the brush of a 
 coil in which an active e.m.f. is generated has thus far not been 
 considered. Referring to Fig. 2, it will be seen that at any 
 speed 5 there will be generated in the coil under the brush by 
 dynamo speed action an e.m.f. 
 
 E, = K^S (43) 
 
 where K is constant. This e.m.f. is in time-phase with the 
 flux < t . In this coil there will also be generated an e.m.f. by 
 the transformer action of the field flux, such that, 
 
 E t = A> f (44) 
 
 This e.m.f. is in time- quadrature to < f . Since < f and < t are in 
 time quadrature the component e.m.f.s acting in the coil under 
 
30 The Sibley Journal of Engineering. 
 
 the brush are in time-phase (opposition) so that the resultant 
 e.m.f. is 
 
 - -S^t) (45) 
 
 Eq. (2) (46) 
 
 Since for constant frequency of supply current, F is propor- 
 tional to < f we may write < f = C F, C being a constant depend- 
 ing on the number of field turns. 
 
 ^,= CE=^ - 2 =~ Eq. (16) (47) 
 
 hence 
 
 which becomes zero at d= ^S = i , that is at synchronism when 
 operated as either a motor or a generator. Above synchronism 
 /s b increases rapidly with increase of speed. 
 
 The friction loss can best be taken into account by considering 
 the friction torque as constant (= d) and subtracting this value 
 from the delivered electrical torque so that the active mechanical 
 torque becomes, 
 
 Torque = D d (49) 
 
 While the effect of the iron loss is relatively small as concerns 
 the electrical characteristics of the machine it is obviously in- 
 correct to neglect it when determining the efficiency. For pur- 
 pose of analysis it is convenient to divide the core material into 
 three parts, the armature the field and the transformer portions. 
 Since the frequency of the reversal of the flux in both the trans- 
 former and the field portions is constant the losses therein will 
 depend only upon the flux. Thus considering hysteresis only, 
 the transformer iron loss is 
 
 H, = L$l* (50) 
 
 where L is a constant depending upon the mass of the core 
 material similarly the field iron loss is 
 
 H t -Mtf* (51) 
 
 M being a constant 
 
 H,+ H t =tf"(M+SL} Eq. (2) (52) 
 
 Since both the field and the transformer fluxes pass through 
 the armature core and these two fluxes are of the same frequency 
 but displaced in quadrature both in mechanical position and in 
 
Repulsion Motor. 31 
 
 time-phase relation, the resultant is an elliptical field revolving 
 always at synchronous speed, having one axis in line with the 
 transformer and the other in line with the field, the values being 
 v/2 ^t an d \/2 < i > f respectively: The value of the two axes may 
 be writen thus 
 
 x/2 -S^t and -v/2 $f 
 
 At synchronous speed of the armature the two become equal and 
 since no portion of the iron is then subjected to reversal of mag- 
 netism the iron loss of the armature core is of zero value. At 
 other speeds, while the revolving elliptical field yet travels syn- 
 chronously, the armature does not travel at the same speed, so 
 that certain sections of the armature core are subjected to 
 fluctuations of magnetism while others are subjected to complete 
 reversals, the sections continually being interchanged. It is due 
 to this fact that no correct equation can be formed to represent 
 the core loss of the armature at all speeds, since the behavior of 
 iron when subjected to fluctuating magnetism cannot be reduced 
 to a mathematical expression. 
 
ALTERNATING CURRENT COMMUTATOR MOTORS. 
 II. REPULSION-SERIES MOTOR.* 
 
 A. s. 
 
 A type of motor closely related to the repulsion machine in 
 the performance of its magnetic circuits is the compensated 
 series motor shown in Fig. 4. Its electrical circuits seem to be 
 those of a series machine with the addition of a second set of 
 brushes, AA, placed in mechanical line with the field coil and 
 short-circuited upon themselves. The transformer action of this 
 closed circuit is such that the real power which the motor re- 
 ceives is transmitted to the armature through this set of brushes, 
 while the remaining set, BB, which in the plain series motor re- 
 ceives the full electrical power of the machine, here serves to 
 supply only the wattless component of the apparent power. 
 
 TWO POLl MODL Of IOAL 
 RCPVL&ON-SEWS MOTOR. 
 FIG.-+. 
 
 This complete change in the inherent characteristics of the 
 series machine by the mere addition of two brushes renders the 
 study of this type of motor especially interesting. 
 
 For purpose of analysis, assume an ideal motor without resist- 
 ance or local leakage reactance and consider first the conditions 
 
 * Abstract of thesis for Ph.D. degree, Cornell University. 
 
34 
 
 when the armature is at rest. When a certain e.m.f., E, is im- 
 pressed upon the motor terminals, the counter magnetizing 
 effect of the current in the brush circuits, AA, is such that the 
 e.m.f. across the transformer coil is of zero value, while that 
 across the armature is E, Thus when S=o, letting E t = trans- 
 former e.m.f. and E & armature e.m.f., 
 
 t = o, and a =E (53) 
 
 It is evident also that when S = O the flux through the arma- 
 ture in line with the brushes A A will be of zero value, so that 
 
 4>t = (54) 
 
 L,et <j> { be the flux through the armature in line with the 
 brushes BB. This flux, neglecting hysteretic effects, is in time- 
 phase with the line current and produces by its rate of change 
 through the armature turns a counter e.m.f. of value E^=E, 
 giving to the armature circuit a reactance when stationary, of X. 
 The relation which exists between the flux, the frequency, and 
 the number of armature turns can be expressed thus, 
 
 (55) 
 
 V 2 I0 
 
 where 
 
 f= frequency in cycles per second 
 
 N = effective number of armature turns 
 
 < m = maximum value of flux. 
 
 If C be the actual number of conductors on the armature, the 
 
 
 
 actual number of turns will be . These turns are evenly dis- 
 
 tributed over the surface of the armature, so that any flux which 
 passes through the armature core will generate in each individual 
 turn an e.m.f. proportional to the product of the cosine of the 
 angle of displacement from the position giving maximum e.m.f. 
 and the value of the maximum e.m.f. generated by transformer 
 action in the position perpendicular to the flux, or the average 
 
 e.m.f. per turn will be 2 times the maximum. The turns are 
 
 7T 2 
 
 connected in continuous series, the e.m.f. in each half adding in 
 parallel to that in the other half, so that the effective series turns 
 
 equal . Thus, finally 
 
 N= 2 ^=^ (56) 
 
 7T 4 2 7T 
 
35 
 
 and ^,= (57) 
 
 V 2 I0 8 
 
 The value of the reactance will depend inversely upon the re- 
 luctance of the paths through which the armature current must 
 force the flux. The major portion of the reluctance is found in 
 the air-gap, and with continuous core material and uniform air- 
 gap around the core, the reluctance will be practically constant 
 x in all directions and will be but slightly affected by the change 
 in specific reluctance of the core material, provided magnetic 
 saturation is not reached. In the following discussion it will be 
 assumed that the reluctance is constant in the direction of both 
 sets of brushes, and that the core material on both the stator and 
 rotor is continuous. 
 
 When dealing with shunt circuits it is convenient to analyze the 
 various components of the current at constant e.m.f., or assuming 
 an e.m.f. of unity, to analyze the admittance and its components. 
 When series circuits are being considered, however, the most 
 logical method is to deal with the e.m.f.'s for constant current, 
 or to assume unit value of current and analyze the impedance 
 and its various components. In accordance with the latter plan, 
 it will be assumed initially that one ampere flows through the 
 main motor circuits at all times and the various e.m.f.'s (im- 
 pedances) will thus be investigated. 
 
 An inspection of Fig. 4 will show that one ampere through 
 the armature circuit by way of the brushes BB will produce a 
 definite value of flux independent of any changes in speed of the 
 rotor, since there is no opposing magneto-motive force in any 
 inductively related circuit. From this fact it follows that on 
 the basis of unit line current < a has a constant effective value, 
 although varying from instant to instant according to an 
 assumed sine law. As will appear latter, while both the current 
 through the armature and the flux produced thereby have un- 
 varying, effective values and phase positions, the apparent 
 reactance of the armature is not constant, but follows a parabolic 
 curve of value with reference to change in speed. 
 
 When the armature travels at any certain speed the conductors 
 cut the flux which is in line with the brushes BB and there is 
 generated at the brushes A A an electro-motive force proportional 
 at each instant to the flux <f> f and hence in time-phase with < f , or 
 with the armature current through BB. 
 
36 
 
 m = maximum value of < f , then the maximum value of 
 the e.m.f. generated at A A due to dynamo speed action will be, 
 
 *.-%? 
 
 where V is revolutions per second. The virtual value of this 
 electro-motive force will be 
 
 - (59) 
 
 A comparison of (59) and (57) will show that at a speed V 
 revolutions per second such that Vf in cycles per second, 
 ^ v = ^ f for any value of < m . Consequently, the speed e.m.f. 
 due to any flux threading the armature turns, at synchronism 
 becomes equal to the transformer e.m.f. due to the same flux 
 through the same turns. E t is in time- quadrature and E^ in 
 time-phase with the flux at any speed, hence, E y is in time- 
 quadrature with t or in time-phase with the line current. 
 
 The brushes AA remain at all times connected directly to- 
 gether by conductor of negligible resistance so that the resultant 
 e.m.f. between the brushes must remain of zero value. On this 
 account when an e.m.f. E v is generated between the brushes by 
 dynamo speed action, a current flows through the local circuit 
 giving a magneto-motive force such that the flux produced 
 thereby generates in the armature conductors by its rate of 
 change, an e.m.f. equal and opposite to E y . This flux, < t , is 
 proportional to E y and being in time- quadrature thereto, is in 
 time-phase with E^ or in time-quadrature with < r 
 
 From the transformer relations it is seen that 
 
 ' ow 
 
 V 2 I0 8 
 
 where </ 2 $ t i g maximum value of flux due to current through 
 brushes A A . See (57). 
 
 V =G^ (61) 
 
 where G is a proportionality constant. 
 
 L,et 6* be the speed, with synchronism as unity, then 
 
 E V = SE< (62) 
 
 and 
 
 *t=S* r , (63) 
 
 effective values being used. This is the fundamental magnetic 
 equation of the repulsion-series motor. 
 
37 
 
 Flux < t passes through the transformer turns on the stator in 
 line with the brushes AA as shown in Fig. 4 and generates 
 therein by its rate of change an e.m.f. r such that 
 
 E, = n E v (64) 
 
 where n is the ratio of effective transformer to armature turns. 
 This e.m.f. is in phase with E v , in quadrature with E t and hence 
 is in phase opposition with the line current and produces the 
 effect of apparent resistance in the main motor circuits. 
 Combining (62) and (64) 
 
 E,= SnE, (65) 
 
 Since E t is the transformer e.m.f. in the armature circuits due 
 to constant effective value of flux from one ampere, we may 
 write 
 
 E t ~X (66) 
 
 where X is the stationary reactance of the armature circuit, so 
 that the apparent resistance of the transformer circuit is 
 
 R = SnX (67) 
 
 Under speed conditions the armature conductors cut the flux 
 in line with the brushes A A, and there is generated thereby an 
 e.m.f. which appears as a maximum at the brushes BB. This 
 e.m.f. is in phase with < t , in quadrature with < f and in phase 
 opposition to E r If E s be the value of this e.m.f. we may write, 
 
 E, = <V*. *SJ? (68) 
 
 N/2 I0 8 
 
 from dynamo speed relations. Comparing (60) and (68) and 
 remembering that /is unity in terms of speed, there is obtained 
 
 * = $, (69) 
 
 from (62) and (66) 
 
 s = S * t .= S 2 X (70) 
 
 Therefore the e.m.f. across the armature at B B will be 
 
 E m *. t -. = X(i-S') (71) 
 
 This e.m.f. is in quadrature with the line current and is in 
 effect an apparent reactance, so that the apparent reactance of 
 the motor circuits which is confined to the armature winding is 
 
 X=X(i-S*) (72) 
 
 The apparent impedance of the motor circuits at speed 6* is 
 
 X* + X*i-S** (73) 
 
38 
 
 This is the fundamental impedance equation of the ideal repul- 
 sion-series motor. 
 The power factor is 
 
 cos 6 = * = SnX 
 
 7 - / /" t~* -- 1S-\ 2 i V2 / _ O2\2 \/T-/ 
 
 The line current is 
 
 (75) 
 
 The power is, 
 
 /72 Q* V"-M 
 
 (76) 
 
 It will be noted that both the power and the power factor re- 
 verse when S is negative. Thus the machine becomes a gener- 
 ator when driven against its torque. 
 
 The wattless factor is, 
 
 Sin. = 4=- _*('">) (77) 
 
 Z ' 
 
 and becomes negative when 6* is greater than i, so that above 
 synchronism when operated as either a generator or motor the 
 machine draws leading wattless current from the supply system. 
 At ,S = i, Sin 0=o, which means that the power factor is unity 
 at synchronous speed, as may be seen also from eq. (74). 
 At S = o, 
 
 ' * 
 
 At 5 = i, /= - That is, at 
 
 n X 
 
 synchronism the line current is equal to the current at start di- 
 vided by the ratio of transformer to armature turns. If N = i, 
 the current at synchronism is of the same value as at start but 
 the power factor which at start was o has a value of i at syn- 
 chronism. This interesting feature will be touched upon later. 
 The torque is 
 
 * n X (78) 
 
 * S* ri> + X 2 ( i - S 2 )" 2 
 
 and is maximum at maximum current and retains its sign when 
 6* is reversed. 
 
 When S = o the secondary current, / 8 , is nl, and is in phase 
 
39 
 
 opposition with the transformer current /. See Fig. 4 When 
 
 / = 
 
 + 
 
 and at any speed S, 
 P = ! ^/n* + S 2 (79) 
 
 2 -f- 
 
 (8o) 
 
 
 
 
 
 
 
 
 \ 
 
 *~, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 *" 
 
 ^ 
 
 
 
 
 
 
 
 fi 
 
 
 
 
 
 
 I'** 
 
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 ^ 
 
 
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 Sjk -34 -Z -/-* -if -It -12. -* -4 O +.4 ** +I.Z +i + ZJ> +M 423 'K 
 
 CHARACTERISTICS Of IDEAL REPULSION - SERIES MOTOR. 
 FIG- cf. 
 
 Ill Fig. 5 are shown the results of calculations for a certain 
 ideal repulsion-series motor of which X= i and n = 2. It is 
 seen that with speed as abscissa, the curve representing the ap- 
 parent resistance of the motor circuits is a right line while that 
 for the apparent reactance is a parabola. At any chosen speed 
 the quadrature sum of these two components gives the apparent 
 impedance of the motor. Since the scale for representing the 
 speed is in all respects independent of that used for the apparent 
 resistance, it is possible always so to select values for the one 
 scale that a given distance from the origin may simultaneously 
 represent both the resistance and the speed. This method of 
 plotting the values leads to a very simple vector diagram for rep- 
 resenting both the value and phase position of the apparent im- 
 pedance at any speed, and for determining the power- factor from 
 inspection. Thus at any speed such as is shown at G the dis- 
 
40 
 
 tance OG is the apparent resistance, the distance GP is the ap- 
 parent reactance, OP is the apparent impedance while the angle 
 POG is the angle of lead of the primary current and its cosine is 
 the power- factor. 
 
 SPCD IN PfJC/VT. 
 
 CHARACTERISTICS OF IDEAL APVLS/<W-SWS MOTOR. 
 
 f/G.-e. 
 
 Fig. 6 gives the complete performance charactersitics of the 
 above ideal repulsion-series motor at various positive and nega- 
 tive speeds when operated at an impressed e.m.f. of 100 volts. 
 
 It will be noted that the armature e.m.f., which has a certain 
 value at standstill, decreases with increase of speed, becomes zero 
 at synchronism and then increases at higher speeds. The trans- 
 former e.m.f. is zero at starting, increases to a maximum at syn- 
 chronism and then continually decreases with increase of speed. 
 
 The inductive portion of the impedance is contained wholly by 
 the armature circuit, while the non-inductive is confined to the 
 transformer coil; thus the power-factor is zero at standstill, 
 reaches unity at synchronism and then decreases due to the lag- 
 ging component of the motor impedance (leading wattless cur- 
 rent). In comparison with the ordinary compensated series 
 motor whose armature e.m.f. is, for the most part, non-inductive 
 and continually increases with increase of speed, and whose in- 
 ductive field e.m.f, decreases continually with increase of speed 
 and whose power-factor never reaches unity, the repulsion-series 
 motor furnishes a most striking contrast. The machine resem- 
 
bles the repulsion motor in regard to its magnetic behavior, but 
 the performance of its electric circuits differs from that of the 
 repulsion motor due to the fact that the speed e.m.f. introduced 
 into the armature circuit BB (Fig. 4) which has been substituted 
 for the field coil of the repulsion motor (See Fig. 2) is in a di- 
 rection continually to decrease the apparent reactance of the 
 field circuit and thus to decrease the inductive component of the 
 impedance of the circuits and to improve the power factor and 
 the operating characteristics. It is an interesting fact that 
 under all conditions of operation the e.m.f. in the coils short cir- 
 cuited by the brushes BB is of zero value, so that no objectional 
 features are introduced by substituting the armature circuit for 
 the field coil of the repulsion motor, while the performance is 
 materially improved. Experiments show that even with currents 
 of many times normal value and at the highest commercial fre- 
 quency no indication of sparking is found at the brushes BB. 
 This feature will be treated in detail later. 
 
 An inspection of Fig. 5 and of equation (73) will reveal the 
 fact that at synchronism the apparent impedance is n times its 
 value at stand still. If n be made unity, the apparent impedance 
 at synchronism will be equal to that at stand still, while between 
 these speeds it varies inappreciably. This means that from zero 
 speed to .synchronism the primary current varies but slightly, 
 and that the torque, which is proportional to the square of the 
 primary current is practically constant throughout this range of 
 speed. These facts show that a unity ratio repulsion series- 
 motor is a constant torque machine at speeds from negative to 
 positive synchronism, the relative phase position of the current 
 and the e.m.f. changing so as always to cause them to give by 
 their vector product the power represented by the torque at the 
 various speeds. Above synchronism the torque decreases con- 
 tinually, tending to disappear at infinite speed. 
 
 Any desired torque-speed characteristic within limits can be 
 obtained by giving to n a corresponding value, the torque at 
 synchronism being equal to the starting torque divided by the 
 square of the ratio of transformer to armature turns. 
 
 In connection with the discussion of the expression for deter- 
 mining the value of the torque it is well to mention the fact that 
 the commonly accepted explanations as to the physical phenom- 
 ena involved in the production of torque must be somewhat 
 modified if actual conditions of operation known to exist are to 
 
be represented. Referring to Fig. 4, it will be noted that when 
 the armature is stationary there exists no magnetism in line with 
 the brushes A A, so that the current which enters the armature 
 by way of the brushes BB could not be said to produce torque 
 by its product with magnetism in mechanical quadrature with it. 
 Similarly, the flux in line with the brushes BB could not be said 
 to be attracted or repelled by magnetism which does not exist. 
 That the current through AA produces torque by its product 
 with the magnetism due to current through BB would be con- 
 trary to accepted methods of reasoning, since both currents flow 
 in the same structure, yet, as concerns the torque, the effect is 
 quite the same as though the flux in line with the brushes BB 
 were due to current in a coil located on the field core. (As 
 shown in Fig. 2 for the ordinary repulsion motor.) 
 
 The calculated impedance characteristics shown in Fig. 5 are 
 based on arbitrarily assumed constants of a repulsion-series 
 motor under ideal conditions. It is obviously impossible to ob- 
 tain such characteristics from an actual motor, since all losses 
 
 -/_ 
 
 'SPEED IN IQO R.P.M. 
 
 Q +2, +4- 
 
 +8 
 
 -1-4 
 
 T< Y 
 
 *H 
 
 -z 
 
 -3 
 
 
 TE&T OF REPULSION SERIES MOTOR-ACTIVE FACTORS OF OPERATION 
 
 nc 7. 
 
 and minor disturbing influences have been neglected in deter- 
 mining the various values. As a check upon the theory given 
 above, the curves of Figs. 7 and 8, as obtained from tests of a 
 repulsion-series motor, are presented herewith. It will be ob- 
 
43 
 
 served that the apparent resistance of the transformer coil varies 
 directly with the speed and becomes negative at negative speed, 
 while the apparent reactance of the armature decreases with in- 
 crease of speed in either direction and, following approximately 
 a parabolic law, reverses and becomes negative at speeds slightly 
 in excess of synchronism. A comparison of the general shape 
 of the curves of Fig. 7 and Fig. 5 will show to what extent the 
 assumed ideal conditions can be realized in practice, and it would 
 indicate that, as concerns the active factors of operation, the 
 equations given represent the facts involved. The neglect of 
 the local resistance of the transformer circuit leads to the dis- 
 crepancy between the theoretical and observed curves as found 
 at zero speed, the latter curve indicating a certain apparent re- 
 sistance when the armature is stationary. Similarly at synchro- 
 nous speed the observed apparent reactance of the armature is 
 not of zero value due to the local leakage reactance of the circuit. 
 In the determination of the theoretical curves only active 
 factors have been considered, and it has been shown that the 
 apparent reactance of the motor circuits is confined to the arma- 
 ture, while the e.m.f. counter generated in the transformer coil 
 gives the effect of apparent resistance located exclusively within 
 this coil. The neglected disturbing factors, the apparent resist- 
 ance of the armature and the apparent reactance of the trans- 
 former, are of relatively small and practically constant value 
 SPEED IN 100 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 : f- 
 
 
 
 
 
 <a 
 
 V 
 
 .*> 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 /J 
 
 
 
 
 
 
 c 
 
 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 / 
 
 / 
 
 -f- 3 
 
 V 
 
 ^ 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 ^/ 
 
 / 
 
 
 + 2- 
 
 
 
 ^e 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ,/ 
 
 P 
 
 
 
 
 
 
 
 ^ 
 
 & 6 
 
 
 
 3^ 
 
 n^ 
 
 /Urn 
 
 i oft 
 
 
 
 
 f^ 
 
 
 
 
 
 \ 
 
 
 
 
 <r - 
 
 "of( 
 
 e c 
 
 c 
 
 fo 
 
 TC e 
 
 f 
 
 Tr 
 
 a n i 
 
 fo, 
 
 T7 
 
 T* 
 
 
 
 
 
 *->> 
 
 
 ^ 
 
 & 
 
 4 T| 
 
 
 
 
 
 
 s 
 
 
 
 $ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 \i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 *i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 TEST Of REPULSION iSR/E$ MOTOR-DISTURBING FACTORS OF OPERATION. 
 
 FIG. Q. 
 
44 
 
 throughout the operating range of speed from negative to posi- 
 tive synchronism, but they become of prime importance when 
 the speed exceeds this value in either direction, as shown by the 
 curves of Fig. 8 obtained from the test of a repulsion-series 
 motor giving the curves of Fig. 7. The predominating influence 
 of the disturbing factors above synchronism is attributable 
 largely to the effect of the short circuit by the brushes A A 
 (Fig. 4) of coils in which there is produced an active e.m.f. by 
 combined transformer and speed action. This short circuiting 
 effect will be treated in detail later. 
 
 The resistance and local leakage reactance of the coils may be 
 included in the theoretical equations as follows : 
 I v et R t = resistance of transformer coil 
 
 R & = resistance of armature circuit 
 
 R a = resistance of secondary circuit 
 
 X t = leakage reactance of transformer 
 
 X A = leakage reactance of armature 
 
 X n leakage reactance of secondary circuit 
 then copper loss of motor circuits will be 
 
 7'(* t + *.)+/.'*,, (81) 
 
 /.-^^/STf-J" 1 (82) 
 
 /' IX + ^a + <X + ^ 2 ) *J = / 2 * m (83) 
 
 where R m is the effective equivalent value of the motor circuit 
 resistance, that is, 
 
 *m = ^t + R* + O 2 + ^ 2 ) ^s (84) 
 
 Similarly it may be shown that the effective equivalent value 
 of the leakage reactance of the motor circuits is 
 
 X m = X t + X^ + ( + S') X. (85) 
 
 combining equations (84) and (85) with (73) the expression for 
 the apparent impedance of the motor circuits becomes 
 
 Z= V(R + R m Y + (X + XJ* = (86) 
 
 - 5 1 ) + X t + X & -f (** + 5 2 ) X a y 
 . __ _E _ 
 
 = ~* 22 
 
 ___ Sn_X_+Jt_ 
 = 
 
45 
 
 Input = E /cos (89) 
 
 Output = E /cos - P m = P (90) 
 
 E*(Sn 
 (SnX+R m y + 
 
 E 1 R 
 
 (SnX+ ^ m ) 2 + [* (i - S) -f A;,] 2 
 
 /7 2 C 7; "V" 
 
 ./^ - ;: J- kJ // -^L 
 
 torque = D = = TnX (93) 
 
 o 
 
 The above equations, though incomplete on account of neglect- 
 ing the brush shortening effect and the magnetic losses in the 
 cores, represent quite closely the electrical characteristics of the 
 repulsion-series motor when operated between negative and posi- 
 tive synchronism, throughout which range of speed the disturb- 
 ing factors are of secondary importance. 
 
 The e.m.f. in the coils short circuited by the brushes can be 
 treated by a method similar to that used with the repulsion 
 motor. Referring to Fig. 4, the coil under the brush A is sub- 
 jected to the transformer effect of the flux, <j> t , in line with the 
 brushes, BB, and the dynamo speed effect of the flux, < t , in line 
 with the brushes AA. 
 
 Effective values being used throughout, the transformer e.m.f. 
 will be, assuming C actual conductors on the armature, 
 
 4 
 
 in volts for one coil. See equation (57). This e.m.f. is in time- 
 quadrature with </>,.. 
 
 The dynamo speed e.m.f. in volts for one coil will be, 
 
 CTT y,~. 
 
 (95) 
 
 4 
 
 See equation (59). This e.m.f. is in time-phase with < t and 
 
 hence is time-quadrature with < r Thus the electro- motive force 
 in the coil under the brush A is 
 
 (96) 
 
4 6 
 
 But F = /5and < t = 5> f (97) 
 
 See equation (63), hence 
 
 F4> t = sy* p ( 9 8) 
 
 so that 
 
 ^ = ^' (i - 52) v! (99) 
 
 This resultant electromotive force has a value at standstill, 
 when 6* is zero, of 
 
 ' IirX 
 
 = - - (100) 
 
 C 2 C 
 
 .4 
 See equation (66) Thus 
 
 finally, E^ = /7r -^ ( i - 5") ( 101 ) 
 
 2 C 
 
 When the armature is stationary the electro- motive force in the 
 coil short circuited by the brush A has the value given by equa- 
 tion (100), which, with any practical motor, is of sufficient value 
 to cause considerable heating if the armature remains at rest, or 
 to produce a fair amount of sparking as the armature starts in 
 motion. At synchronous speed, however, this electro- motive 
 force disappears entirely, and the performance of the machine as 
 to commutation is perfect. As the speed exceeds this critical 
 value in either the positive or negative direction, the electro- 
 motive force in the short-circuited coil increases rapidly, resulting 
 in a return in an augmented form of the sparking found at lower 
 speeds and producing the disturbing factors shown by the curves 
 of Fig. 8. 
 
 Since the e.m.f. in the coil under the brush A reduces to zero 
 at both positive and negative synchronism and reverses with 
 reference to the time-phase position of the line current at speeds 
 exceeding synchronism in either direction, it possesses at high 
 speeds the same time-phase position when the machine is operated 
 as a generator as when it is used as a motor. The time-phase of 
 its reactive effect upon the current which flows in the armature 
 through the brushes BB is of the same sign at high positive and 
 negative speeds, but reversed from the phase position of the 
 effect at speeds below synchronism. A study of the test curves 
 of Fig. 8 will show the magnitude of these effects, and the 
 
47 
 
 reversal of their time-phase positions in accordance with the 
 theoretical considerations. 
 
 With reversal of direction of rotation, the time-phase position 
 of the flux threading the transformer coil (Fig. 4) reverses with 
 reference to the line current, and hence in its reactive effect 
 upon the transformer flux the current in the coil short circuited 
 by the brush A becomes negative at speeds above negative syn- 
 chronism, though positive above synchronism in the positive 
 direction. At speeds below synchronism, when the flux is large 
 the e.m.f. is small, and vice versa, so that the reactive effect is in 
 any case relatively small and of more or less constant value. See 
 Fig. 8. 
 
 It will be noted that in analyzing the disturbing factors no ac- 
 count has been taken of the short-circuiting effect at the brushes 
 BB, Fig. 4. This treatment is in accord with the statement 
 previously made that the component e.m.f.s generated in the 
 coils under these brushes are at all times of values such as to 
 render the resultant zero. The proof of this fact is as follows: 
 
 The transformer e.m.f. in the coil under B due to flux, </> t , in 
 line with brushes A A is 
 
 <-l f & - (I02) 
 
 See equation (94). This e.m.f. is in time- quadrature with < t . 
 The dynamo speed e.m.f. is 
 
 This e.m.f. is in time-phase with <,., in time quadrature with 
 t , and is in phase opposition to e r Thus the resultant e.m.f. is 
 
 E* = *, ~ e, - - ~ C/>, -V+d ( 104) 
 
 Since V =f S and < t = S <f> t from equations (97) and (63), 
 
 and 
 
 ^ b = (106) 
 
 This theoretical deduction is substantially corroborated by ex- 
 perimental evidence, as has been noted above. Even upon super- 
 ficial examination such a result is to be expected, since the 
 vector sum of all e.m.f.s in the armature in mechanical line with 
 the short-circuited brushes A A must be zero, while the e.m.f. 
 in the coil at brush B must equal its proper share of this e.m.f. or 
 
4 8 
 
 r O'TT 
 
 Jg; *_.<> (107) 
 
 A similar course of reasoning allows of the determination of 
 the electro-motive force under the brush A. See equation (71). 
 
 \~* 2 2 * (^ 
 
 for unit current. For / amperes this becomes. 
 
 .= (i-S') (109) 
 
 See equation (101). 
 
 From the facts just indicated it would seem that perfect com- 
 mutation dictates that the electro-motive force across a diameter 
 ninety electrical degrees from the brushes upon the armature be 
 at all times of zero value. Methods for approximating this 
 condition will be discussed in a later paper. 
 
 It has been stated that the magnetic circuits of the repulsion- 
 series motor are quite the same as those of the repulsion motor. 
 The fluxes in line with the two brush circuits under all condi- 
 tions are ir time-quadrature and have relative values varying 
 with the speed such that at all times 
 
 4> t = S4> t (no) 
 
 There exists, therefore, at all speeds a revolving magnetic field 
 elliptical in form as to space representation. At standstill the 
 ellipse becomes a straight line in the direction of the brushes BB 
 (Fisjp4), at infinite speed in either direction the ellipse would 
 again be a straight line in the direction of the brushes AA, while 
 at either positive or negative synchronism the ellipse is a true 
 circle, the instantaneous maximum value of the revolving mag- 
 netism traveling in the direction of motion of the armature. At 
 synchronous speed, therefore, the magnetic losses in the arma- 
 ture core disappear, while the losses in the stator core are evenly 
 distributed around its circumference. 
 
ALTERNATING CURRENT COMMUTATOR MOTORS. 
 III. COMPENSATED SERIES MOTORS.* 
 
 A. s. M'AUJSTKR. 
 
 The combined transformer and motor features of commutator 
 type of alternating current machinery are well exemplified in the 
 plain series motor as illustrated in Fig. 9. When the rotor is 
 
 SPEED E.MF 
 
 or Ccm*cwr 
 
 FIG. 9. Plain Series Motor. 
 
 stationary, the field and armature circuits of the motor form two 
 impedances in series. Assuming initially an ideal motor without 
 
 * Abstract of thesis for Ph.D. degree, Cornell University. 
 
50 Compensated Series Motor. 
 
 resistance and local leakage reactance, each impedance consists of 
 pure reactance, the current in the circuit having a value such 
 that its magneto-motive force when flowing through the armature 
 and field turns causes to flow through the reluctance of the mag- 
 netic path that value of flux the rate of change of which generates 
 in the windings an electro-motive force equal to the impressed. 
 
 If E be the impressed e.m.f., E f the counter transformer e.m.f. 
 across the field coil and E^ the counter transformer e.m.f. across 
 the armature coil, when the armature is stationary 
 
 = ,+ , (in) 
 
 From fundamental transformer relations there is obtained the 
 equation 
 
 EI = 2 ,-^ s > see e< i- (55) (112) 
 
 V 2 IO 
 
 where f = frequency in cycles per second 
 N t = effective number of field turns 
 <f> f = maximum value of field flux. 
 Similarly 
 
 *-H^ 
 
 where N & = effective number of armature turns 
 < a = maximum value of armature flux. 
 
 Since the field and armature circuits are electrically series con- 
 nected and are mechanically so placed as not to be inductively 
 related, with uniform reluctance around the air gap the fluxes in 
 mechanical line with the two circuits being due to the magneto- 
 motive force of the same current will be proportional to the 
 effective number of turns on the two circuits. 
 
 Therefore 
 
 If n be the ratio of effective field to armature turns 
 
 N t =nN^ (115) 
 
 and 
 
 <f> f =n<t> & (116) 
 
 L,et Cbe the actual number of conductors on the armature, then 
 
 ^a = (seeeq. 56) (117) 
 
 2 7T 
 
 Under speed conditions the armature conductors cut the field 
 magnetism and there is generated by dynamo action a counter 
 
Sibley Journal of Engineering. 
 
 e.m.f. proportional to the product of the field flux and the speed, 
 in time-phase with the flux, in leading time quadrature with the 
 field e.m.f., Z? f and the armature e.m.f. E A and in phase opposi- 
 tion with the current. 
 Thus 
 
 ^ = ^J (see eq - 59) < II8 > 
 
 V 2 IO 
 
 where Fis revolutions per second of bipolar model. 
 Combining (117) and (118) 
 
 If ,5* be the speed with synchronism as unity, then 
 
 V=Sf (120) 
 
 and 
 
 combining (113) (116) and (121) 
 
 combining (112) (115) and (121) 
 
 *-*&- 
 
 comparing (122) and (123) 
 
 E,-*<E. 
 
 Under speed conditions the impressed e.m.f. is balanced by 
 three components, E v in time phase opposition with the line cur- 
 rent and E t and E^ both in leading time quadrature with the 
 line current. 
 
 Thus 
 
 + E f E (125) 
 
 This is the fundamental electro-motive force equation of the plain 
 series motor having uniform reluctance around the air-gap. 
 
 On the basis of unit line current the electro-motive forces may 
 be treated as impedances, as was done with the repulsion-series 
 motor, so that the impedance equation becomes 
 
 ^^a^S^-f (i +rc 2 ) 2 (127) 
 
 where SnX & = R and (i + n 2 ) X^ = X 
 
52 Compensated Series Motor. 
 
 The power factor is 
 
 = 0080= Sn (128) 
 
 which reverses when 5* becomes negative and continually ap- 
 proaches unity with increase of 5" in either direction. 
 When 5" = i , or at synchronism 
 
 cos6 = - n (129) 
 
 Vn* + (i + n*y 
 
 which when n = i or for unity ratio of field to armature turns 
 becomes 
 
 (I30) 
 
 and decreases with either an increase or decrease of n. It is 
 apparent therefore that the power factor of such a machine is in- 
 herently very low and cannot be improved by a mere change in 
 the ratio of field to armature turns. 
 The line current is 
 
 /= ^= E _ , 
 
 = ~Z 22 
 
 The power is 
 
 which becomes negative when 5* reverses, or the machine operates 
 as a generator when driven against its natural tendency to rota- 
 tion. 
 
 The torque is 
 
 which is maximum at maximum current and retains its sign 
 when 5* is reversed. 
 
 At starting the torque is 
 
 ^-|-(iTW (I34) 
 
 At synchronous speed, the torque is 
 
 and 
 
 A- (I+** 1 ) 1 (I36) 
 
 D +!+ 
 
The Sibley Journal of Engineering. 53 
 
 which when n is negligibly small approaches a value of unity and 
 when n is infinitely large also tends to reach a value of unity. 
 When n = i equation (136) reduces to 
 
 - 8 
 
 ths interpretation of which is that the torque of the unity-ratio 
 single-phase, plain series motor with uniform reluctance around 
 the air-gap varies only 20 per cent, from standstill to synchro- 
 nism, and therefore, that such a machine is unsuited for traction. 
 This statement applies to the ideal single-phase motor without 
 internal losses and must be somewhat modified to include true 
 operating conditions. The method of treating the various losses 
 has previously been discussed and will further be enlarged upon 
 in connection with the compensated types of series machines. A 
 little consideration will show that such modifications as must be 
 introduced have a detrimental effect upon the characteristics of 
 the machine, and tend to lay greater stress upon the statement 
 just made. These facts are graphically represented in the per- 
 formance (impedance) diagram of Fig. 9. OA is the power and 
 AB the reactive component of the apparent field impedance at 
 starting while BC and CD are the corresponding power and re- 
 active components of the apparent armature impedance. The 
 power component of apparent armature impedance due to dynamo 
 speed action is shown as DE or DF giving the resultant imped- 
 ance under speed conditions of OE or O^and indicating an angle 
 of lag of the circuit current behind the impressed e.m.f. of EOA 
 or FOA. The variation in torque due to increase of speed from 
 synchronism to double synchronism with a unity ratio constant 
 reluctance machine, as represented in Fig. 9, would be as the 
 square of the ratio of OF to OE. 
 
 An inspection of equation (136) will reveal the fact that a 
 a change in the value of n does not improve the torque charac- 
 teristics of the machine unless such change be accompanied with 
 an increase in reluctance of the magnetic structure in line with 
 the brushes B v B 2 (Fig. 9). That is to say, if the mechanical 
 construction is such that equation (114) may be written 
 
 -" (I38) 
 
 . . 
 
 where m is a constant of a value many times unity, the oper- 
 ating characteristics of the machine become much improved. 
 Thus equation (116) becomes 
 
54 Compensated Series Motor. 
 
 o, === mn o i I^Q j 
 
 ~f r a V A O7/ 
 
 and equation (122) is changed to 
 
 I? O 
 
 (HO) 
 (HO 
 WT/ + OS; + f )-^(^y+(A + r y (I42) 
 
 S) 2 (I ) 
 
 (144) 
 
 m n' 
 
 + 
 mn 
 
 J(^y + ( I + j_v j5 + (^i^ 
 
 ^v7 V OTMV ~ ^ mn J 
 
 (145) 
 
 when ,S = i or at synchronism 
 
 i 
 
 mn 
 
 (I46) 
 
 With an excessively large reluctance of the magnetic structure 
 in line with the brushes B^B^ (Fig. 9), that is, with an enor- 
 mous value of m, the power factor at synchronous speed 
 approaches 
 
 Cos0=- L = ( I47 ) 
 
 Vl + w 2 
 
 the interpretation of which equation is that the operating power- 
 factor of such a machine is largely dependent upon the ratio of 
 field to armature turns. A little study will show that at any 
 chosen speed, whether synchronous or not, the cotangent of the 
 angle of lag is directly proportional to the ratio of armature to 
 field turns, and that the power- factor, the corresponding cosine, 
 can be given any desired value by a proper proportioning of the 
 windings. This feature will be treated more in detail when deal- 
 ing with compensated motors. 
 
 The current of the high brush-line-reluctance machine is 
 
The Sibley Journal of Engineering. 55 
 
 ,. E En i 
 
 z x t v/^T^^+Ty (148) 
 
 V m n J 
 The power is 
 
 2 rc 5 i 
 
 (i49) 
 
 mn J 
 
 The torque is 
 
 = 5 =: ~X^ ' 5 ,, ^^ 2 + iV = ~^T ( r 5o) 
 
 v mn J 
 At starting the torque is 
 
 _ ^ 2 ^ 
 
 ^rwrc 2 -fiy (150) 
 
 v #r y 
 
 At synchronous speed the torque is 
 
 (152) 
 
 m n 
 which ratio, with an enormous value of m, approaches 
 
 g-H^- ('53) 
 
 the significance of which is that the change of torque from stand- 
 still to synchronism can be altered at will by change in the ratio 
 of field to armature turn and that a relatively low value of n 
 would produce a machine suitable for traction. 
 
 By using projecting field poles thus leaving large air-gaps in 
 the axial brush line and thereby increasing the reluctance of the 
 structure inline with the magneto- motive force of the armature cur- 
 rent, the flux produced by the armature current may be materially 
 reduced, thus giving to m a relatively large value, and the power 
 factor will be thereby correspondingly increased with a resultant 
 improvement in the torque characteristics of the machine. Even 
 under the most favorable conditions, however, it is impossible to 
 reduce the reactance of the armature circuit to an inappreciable 
 
56 Compensated Series Motor. 
 
 value, that is, to give to m an enormous value, due to the in- 
 evitable presence of the magnetic material of the projecting poles. 
 The most satisfactory method of reducing the inductive effect 
 of the armature current is to surround the revolving armature 
 winding with properly disposed stationary conductors through 
 
 COf* 
 
 A 
 
 FiG. 10. Inductively Compensated Series Motor. 
 
 which current flows equal in magneto-motive force and opposite 
 in phase to the current in the armature. This compensating 
 current may be produced inductively by using the stationary 
 winding as the short circuited secondary of a transformer of 
 which the armature is the primary, as illustrated diagram- 
 matically in Fig. 10, or the main line current may be sent di- 
 
The Sibley Journal of Engineering. 
 
 57 
 
 rectly through the compensating coil as shown in Fig. n. In 
 the former case the transformer action is such that the com- 
 pensation is practically complete, giving minimum combined 
 reactance of the two circuits while in the latter ease, the propor- 
 tion of compensation can be varied at will. It is found that in 
 any case the best general effects are produced when the com- 
 pensation is complete, and experiments seem to indicate that 
 under such conditions the two methods of compensation differ 
 
 r/ELO CO/L 
 
 FIELD CO/L 
 
 FIG. ii. Conductively Compensated Series Motor. 
 
 inappreciably for strictly alternating current work, but that 
 for direct current operation where the forced compensation can 
 
58 Compensated Series Motor. 
 
 be used to prevent field distortion and improves the commutation, 
 the latter method is preferable. 
 
 Referring to Figs. 10 and n, assume an ideal series motor 
 with complete compensation, letting n be the ratio of effective 
 field to armature turns, at any speed 6 1 with synchronism as unity, 
 the apparent impedance of the motor circuits will be 
 
 z=x, Jj+i (154) 
 
 of which 
 
 X-X, (155) 
 
 represents the reactance of the motor circuits which is confined 
 to the field coil, and of which 
 
 represents the apparent resistance effect of the dyamo speed 
 e.m.f. counter generated at the brushes B V B. L due to the cutting 
 of the field flux by the armature conductors, (See eq. 123). 
 
 The power factor is 
 
 R 
 ~Z 
 
 ~=r= (157) 
 
 ^r + I 
 
 which continually approaches positive or negative unity with in- 
 crease of speed in the corresponding direction. 
 At synchronism when 5 = i the power factor is 
 
 cos = ss===s- (see eq. 147) (158) 
 
 v i -|- n 2 
 
 The line current is 
 
 , E E i_ 
 
 ^Z^X^-f^ ('59) 
 
 NJF" 
 
 The power is 
 
 5 
 />= EIco$0= - - _ (160) 
 
 The torque is 
 
The Sibley Journal of Engineering. 59 
 
 r> 772 / 2 xt^ 
 
 D= s = o = ~~ (see eq - I5o) 
 
 The ratio of the torque at synchronous speed to that at stand- 
 still is 
 
 D i n* 
 
 L = ~ - = T (seeeq. 153) (162) 
 
 which in a practical machine can be made as much smaller than 
 unity as desired by a proper proportioning of the field and arma- 
 ture windings. It is evident, therefore, that such a machine can 
 be made suitable for traction when a proper value of n is chosen. 
 The above equations refer to ideal motors without resistance 
 and local leakage reactance and devoid of all minor disturbing 
 influences. A close approximation for the effect of the resistance 
 and leakage reactance may be obtained as follows : 
 Let r { = resistance of field coil 
 
 r c = resistance of compensating coil (reduced to a i to i, 
 
 armature ratio) 
 r a = resistance of armature 
 x t = local reactance of field coil 
 x^ = combined leakage reactance effect of armature and 
 
 compensating coils. 
 Then the apparent impedance is 
 
 Z= J( '+ r r + r e + O' + (*, + ^ +*J ( I 6 3 ) 
 
 ^ v. n 
 Power factor is 
 
 SXj + r l + r c + r, 
 Cos e =z-= 175^ = (I64) 
 
 j (165) 
 
 n 
 Power input is 
 
 The copper loss and equivalent effective resistance loss will be, 
 
 _ *(r t +r c + rj 
 
 ( l66 ) 
 
 ,SX 
 
 (n 
 
 / v ? 
 
 { UNIVERSITY j 
 
 ti&r 
 
60 Compensated Series Motor. 
 
 Electrical output is 
 
 The torque is 
 
 D /2 D 7"2 
 
 / 
 
 (167) 
 
 (see eg. 161) (168) 
 
 The equations here given are represented graphically in the 
 diagrams of Figs. 10 and n, which show the impedance (e.m.f. 
 for unit current) characteristics of the machines. 
 
 OA=r { 
 
 BC = r a + r c 
 
 AB = X t + * f 
 
 at speed 5* 
 n 
 
 OE = Z at speed 5 1 
 
 cos EOA = cos = power factor at speed ,5" 
 These characteristics together with the brush short circuiting 
 effect and other minor modifying influences will be discussed in 
 detail in a later paper. It is sufficient here to state that the effect 
 of the short circuit by the brush of a coil in which an active e.m.f. 
 is generated, both by transformer and speed action, tending to 
 increase the apparent impedance effects at high speeds is to some 
 extent balanced by the fact that the flux which causes the gener- 
 ation of a counter e.m.f. by dynamo speed action is out of phase 
 and lagging with respect to the line current and that tne counter 
 e.m.f. therefore, tends to lag behind the current or to cause the 
 current to become leading with respect to the counter e.m.f., so 
 that the neglected disturbing influences tend to render the final 
 effect quite small, the result being that the incomplete equations 
 and corresponding graphical diagrams as given above, represent 
 quite closely the observed performance characteristics of the com- 
 pensated series motors. 
 
ALTERNATING CURRENT COMMUTATOR MOTORS. 
 IV. INDUCTION-SERIES MOTOR.* 
 
 BY A. s. M'ALLISTKR. 
 
 Excellent performance of the conpensated alternating- current 
 motor may be obtained by using the field coil as the load circuit 
 from the compensating coil employed as the secondary of a trans- 
 former, the armature being used as the primary, as diagrammat- 
 ically represented in Fig. 12. The current which enters the 
 
 SPEED EMF 
 
 FIG. 12. Induction-Series Motor. 
 
 * Thesis for Ph.D. degree, Cornell University. 
 
62 The Sibley Journal of Engineering. 
 
 armature winding through the brushes B l 2 causes the formation 
 on the armature core of magnetic poles having the mechanical 
 direction of the axial line joining the brushes, and the rate of 
 change of the magnetism generates an electromotive force in the 
 compensating coil. Due to this electromotive force, current 
 flows through the locally-closed circuits around the compensat- 
 ing and field coils, and produces magnetic poles in the stationary 
 field- cores. 
 
 Consider now the load-circuit surrounding the quadrature 
 field-cores. Since to this winding there is no opposing secondary 
 circuit, the magnetism in the core will be practically in time- 
 phase with the current producing it. This current is the second- 
 ary load- current of the transformer. As is true in any trans- 
 former, there will flow in the primary coil a current in phase op- 
 position to the secondary current in addition to and superposed 
 upon the primary no-load exciting-current. It is thus seen, that 
 the load- current in the primary (or armature) coil will be in 
 time-phase opposition with the magnetism in the quadrature core. 
 And, since this current and the magnetism reverse signs together, 
 the torque, due to their product and relative mechanical position, 
 will remain always of the same sign though fluctuating in value. 
 Hence the machine operates similarly to a direct- current series 
 motor. 
 
 When the armature revolves at a certain speed, the motion 
 of its conductors through the quadrature magnetic field, generates 
 in the armature winding an electromotive force which appears at 
 the brushes B l B 2 as a counter e.m.f. This weakens the effec- 
 tive electromotive force and therewith the armature- current, the 
 armature-core magnetism, the field-current and the field-core 
 magnetism. Thus there results from increased speed of the arm- 
 ature a reduced torque, just as occurs in direct-current series 
 motors. By increasing the applied electromotive force, an in- 
 crease of torque can be obtained even at excessively high speeds, 
 and the motor tends to increase indefinitely the speed of its arm- 
 ature as the applied electromotive force is increased, or as the 
 counter torque is decreased. There is no tendency to attain a 
 definite limiting speed as is found to be true with revolving 
 field induction-motors and repulsion motors. 
 
 Let E & be the counter transformer e.m.f. across the armature 
 coil, the armature being stationary. 
 
Induction- Series Motor. 63 
 
 Then 
 
 ~ 2 7T / ' N < f \ 
 
 E * = ,- a 8 a > see eq. (55) (169) 
 
 V 2 IO 
 
 where /" = frequency in cycles per second 
 
 N & = effective number of armature turns 
 < a = maximum value of armature flux cutting the 
 compensating coil 
 
 ^ / a = - seeeq. (56) (170) 
 
 2 7T 
 
 where C is the actual number of conductors on the armature, a 
 bipolar model being assumed. 
 
 t70 
 
 Let N Q = effective number of turns on the compensating coil, 
 
 4- 
 
 where /s c = transformer e.m.f. of the compensating coil. 
 Let N t = effective number of turns on the field coil 
 
 then t =-'- (173) 
 
 V 2 IO 
 
 where E t = impre,ssed e.m.f. of the field coil 
 
 < f = maximum value of field flux 
 
 E t = E , hence N <t> & = N, ^ ( 1 74) 
 
 and 
 
 ^ = C'75) 
 
 ^>a ^f 
 
 Let ^" v be the e.m.f. counter generated at the brushes B^ B^ 
 (Fig. 12) by speed action due to the cutting of the flux < f by the 
 armature conductors C at speed V revolutions per second, then 
 
 '- seeeq. (59) (176) 
 
 V- Sf (177) 
 
 where 5* is the speed with synchronism as unity. 
 Combining (171), (176) and (177) 
 
 * t = ^=^ ( I7 8) 
 
 Let n be the ratio of effective field to compensating coil turns. 
 
 o r? 
 
 v = -- - a seeeq. (123) (179) 
 
 n 
 
64 The Sibley Journal of Engineering. 
 
 This electromotive force is in time-phase with the field flux 
 <j> f , is in phase opposition with the live current and hence is in 
 time quadrature (leading) with respect to the e.m.f. E & . The 
 impressed electromotive force E is balanced by the two com- 
 ponents, E v and E & so that 
 
 + E* (180) 
 
 -E W 
 
 *\^> 
 
 + i (181) 
 
 On the basis of unit line current, the electro-motive forces may 
 be treated as impedances, as was done with the repulsion-series 
 and compensated-series motors. 
 
 where = R and X t = x 
 
 n 
 
 X t being the combined reactance effect of the field, compensating 
 coil and armature circuits. 
 The power factor is, 
 
 5 
 
 R ___ *L= S ( I8 3) 
 
 cos0=Z~ ~ 
 
 which when 6* = i or at synchronism, reduces to 
 
 the interpretation of which is that the power factor at synchro- 
 nism can be caused to approach unity quite closely by the use of 
 a small value of n, that is, by employing a small ratio of field to 
 compensating coil turns. With increase of speed the power fac- 
 tor continually increases for any value of n. 
 The line current is 
 
 The power is 
 
 5 
 
 n n 
 
 _^_^__^ (I86) 
 
Induction-Series Motor. 65 
 
 which becomes negative when S reverses, or the machine oper- 
 ates as a generator when driven against its natural tendancy to 
 rotation. 
 
 The torque is 
 
 D= 5~x t (s* + n>) = ^ir (l8y) 
 
 which is maximum at maximum current and retains its sign 
 when >S is reversed. 
 
 At starting, the torque is 
 
 77* <yt 
 
 D = ^-'- (188) 
 
 X t n 
 
 at synchronous speed, the torque is 
 
 ^ ".'(!+') (I89) 
 
 and 
 
 t = rr^ (I90) 
 
 which when n = i reduces to 
 
 ^ = -4 =-5 (i9O 
 
 and can be given any desired value by a proper selection of n, 
 see eq. (153). A relatively low value of n would produce a 
 machine having the torque characteristics of the direct current 
 series motor and hence one suitable for traction. See eq. (162). 
 
 It remains to investigate the relation of the currents in the 
 compensating coil and in the armature circuit (the secondary and 
 primary of the assumed transformer. ) 
 
 Let i & be the current which would flow in the armature when 
 the field coil circuit is open. Then z' a is the exciting current of the 
 assumed transformer and it has a value such that its product with 
 the effective number of armature turns, forces the flux, < a , 
 demanded by the impressed e.m.f., through the reluctance of 
 their paths in the magnetic structure, in line with the brushes 
 B l B^ (Fig. 12). When the field circuit is closed there flows 
 through the field and compensating coil a current z f , of a value 
 such that its magnetomotive force when flowing through the field 
 turns N v produces the flux < f demanded by the e.m.f. E^ or E G . 
 The current i t is in time- phase with the flux <f> t and hence is in 
 time quadrature with the e.m.f. E G , The current z a is in phase 
 with the flux < a and in time quadrature with E & or E c . When 
 the field circuit is closed a current equal in magnetomotive force 
 
66 The Sibley Journal of Engineering. 
 
 and opposite in phase to i t is superposed upon z a in the primary 
 (armature) circuit. These two currents are directly in phase so 
 that the resultant current becomes 
 
 /= 4+M (192) 
 
 where p is a proportionality constant the value of which will be 
 discussed later. 
 
 Since both i & and i t reach their maximum values simultaneously 
 with < f , one is led to the highly interesting conclusion that even 
 the exciting current i & is effective in producing torque by its 
 direct product with the field magnetism, and, that under speed 
 conditions both z" a and p i t are equally effective (per ampere) in 
 producing power. 
 
 The relative values of / a and i t and of p may be approximated 
 as follows : 
 
 Assuming similar conditions for the three coils, the field, the 
 compensating and the armature circuits, equal reluctance 
 
 / N LN, 
 
 _J^_S = J_f ( I93 ) 
 
 <P a 9f 
 
 N,= nN G (194) 
 
 N *$* = N t$t seeeq. (174) (195) 
 
 * f = a d96) 
 
 From transformer relations there is obtained the equation 
 
 j=p seeeq. (192) (198) 
 
 Combining (197) and (198) 
 
 *r = T^i ('99) 
 
 Combining (199) and (192) 
 
 /= ^a( : + ^r) ( 20 ) 
 
 Comparing (199) and (200), 
 
 *; = __^ = _ (201) 
 
 T _1_ * 
 
 * I n 
 
 The relations above expressed depend upon certain assump- 
 tions as to the reluctance in line with the armature circuit and 
 
Induction- Series Motor. 67 
 
 the field coil, and will be modified if the assumptions made are 
 not applicable to the motor as constructed. As a method of re- 
 viewing the problem, in a general way, however, the assumption 
 made and the conclusions drawn therefrom are sufficiently exact. 
 In the determination of the equations used above, an ideal motor 
 has been considered, the resistance and local leakage reactance 
 effects being neglected. Actual operating conditions may be 
 more closely represented as follows : 
 
 Let 
 
 r t = resistance of field coil. 
 
 r c = resistance of compensating coil. 
 
 r a = resistance of armature. 
 
 x t = local leakage reactance of field coil. 
 
 x c = local leakage reactance of compensating coil. 
 
 x t = local leakage reactance of armature circuit. 
 Then the copper loss of the motor circuits will be 
 
 P * m = /V. + if (r, + O = P [r. + t] ( 2O2 ) 
 
 where R m is the effective equivalent value of the motor-circuit 
 resistance, that is, 
 
 (203) 
 
 Similarly it may be shown that the equivalent effective value 
 of the local leakage reactance of the motor- circuit is 
 
 Combining equations (182), (203), and (204), the apparent 
 impedance of the motor- circuits becomes 
 
 (205) 
 
 The power factor is 
 
 R= _ 
 
 C r ' +r ' 
 
 Z +r 
 
 r r ^ n 
 
 (206) 
 The current is = 7. (207) 
 
68 The Sibley Journal of Engineering. 
 
 The input = El cos 6. ( 208 ) 
 
 The output is P= El cos 6 PR m . (209) 
 
 />= 
 
 . 
 
 The torque is 
 
 see eq . (187) andeq. (168) (212) 
 
 >j n 
 
 The graphical diagram of Fig. 12. represents the above im- 
 pedance equations, (e.m.f. for unit current), where 
 
 (2I4) 
 
 (215) 
 
 O F= Zat speed 5 (216) 
 
 cos F O A = cos = power factor at speed 5 (217) 
 
 AlthouglTneglecting certain modifying eifects, the graphical 
 diagram represents quite closely the observed performance char- 
 acteristics of the induction-series motor. An inspection of equa- 
 tion (205) will show that certain values there given may be 
 represented by others of much simplified nature since various 
 terms there contained are constant in any chosen motor. 
 
 Let, therefore 
 
 P = -< (220) 
 
 n 
 
Induction- Series Motor. 69 
 
 then the apparent impedance becomes, 
 
 Z=* \/(X + PS)* + X* (221) 
 
 the power factor is, 
 
 which continually approaches unity with increase of speed. 
 
 Let rotation of the armature in the direction produced by the 
 electrical (its own) torque be considered positive. Then may 
 rotation in the contrary direction (against its own torque) be 
 considered negative. Since the power component of the motor 
 impedance has a certain value at zero speed, and increases with 
 increase of speed, it should follow that by driving the rotor in a 
 negative direction the apparent power component will reduce to 
 zero and disappear. The power factor then reduces to zero and 
 the current supplied to the motor will represent no energy flow- 
 ing either to or from the motor. 
 
 This will be apparent from the relations above set forth, as 
 well as by the relations algebraically expressed by the equation 
 
 - power - 
 
 the negative sign being due to the direction of rotation and the 
 expression reducing to zero for zero value of the apparent power 
 component J^ PS. A further increase of speed in the nega- 
 tive direction will cause the expression for the power- factor and 
 for the power, to become negative, the interpretation of which is 
 that the machine is now being operated as a generator and 
 hence is supplying energy to the line, that is, energy is flowing 
 from the machine. Fig. 13, which gives the observed perform- 
 ance characteristics of a certain induction-series motor, will 
 serve to show to what extent these theoretical deductions may 
 be realized in an actual machine. If, then, during operation as 
 a motor at a certain speed, the quadrature field flux be relatively 
 reversed with reference to the brush axial-line field flux, so as to 
 tend to drive the armature in the opposite direction, not only 
 will a braking effect be produced by such change but energy 
 will be transmitted from the machine to the line. 
 
 The effect of the short circuit by the brush of a coil in which 
 an active e.m.f. is generated, which has been omitted in the 
 above equation's, though completely included in the test curves, 
 
The Sibley Journal of Engineering. 
 
 may be treated as follows. Referring to Fig. 12 it will be seen 
 that at any speed 6" there will be generated in the coil under the 
 brush by dynamo speed action an e.m.f. 
 
 see eq. (43) (224) 
 
 FiG. 13. Test Characteristics of Induction-Series Motor. 
 
 where A" is constant. This e.m.f. is in time-phase with the flux 
 < a . In this coil there will also be generated an e.m.f., <? f , by 
 the transformer action of the field flux, such that 
 
 e f =K<t> t seeeq. (44) (225) 
 
 This e.m.f. is in time quadrature to < r Since < { and < a are in 
 time phase, the component e.m.f.'s acting in the coil under the 
 brush are in time quadrature, so that the resultant e.m.f. is 
 
 $+tf (226) 
 
 see eq. (175) (227) 
 
 *.- 
 
 S' + A 
 
 combining equations (169) and (181) 
 .g-g'/AT.*. 
 
 \/2 I0 8 
 
 (228) 
 
 (229) 
 
Induction- Series Motor. 71 
 
 combining (230) and (228) 
 
 p=^ (231) 
 
 'ji (232) 
 
 where A is a constant as found above. 
 
 When n= i, b is constant, independent of the speed, while 
 when ?z is very small E^ is large at zero speed and continually 
 decreases with increase of speed. When 6"= i or at synchronous 
 speed 
 
 quite independent of the value of n. 
 
 The relative impedance effect of E^ can be determined by com- 
 bining equations (232) and (185) thus 
 
 +1 (235) 
 
 B being a constant. The interpretation of equation (235) is 
 that the apparent impedance effect of the short circuit by the 
 brush, consists of two components in quadrature, one component 
 being of constant value and the other varying directly with the 
 speed. Experimental observations fully confirm these theoreti- 
 cal conclusions, and show that the increase in apparent reactive 
 effect with increase of speed for motor operation is approxi- 
 mately counterbalanced by the lagging counter e.m.f. (leading, 
 current) effect of the time-phase displacement between exciting 
 current and field magnetism as has been mentioned previously 
 and as will be dwelt upon subsequently. During generator 
 operation, that is, with negative value of S, the apparent re- 
 active effect of the short circuit at the brush adds directly to 
 the lagging field flux, counter e.m.f. effect and therefore, the 
 
72 The Sibley Journal of Engineering. 
 
 apparent reactance of the motor circuits increases rapidly with 
 increase of speed in the negative direction, though remaining 
 practically constant for all values of positive speed. These facts 
 will be appreciated from a study of the test characteristics of the 
 induction series machine throughout both its generator and 
 motor operating range as shown in Fig. 13. 
 
 Mention has frequently been made of the fact that in the 
 development of the equations for expressing the performance of 
 the various types of series motors the effect of the hysteretic 
 angle of time-phase displacement, between the magnetizing force 
 and the magnetism produced thereby has been neglected. In a 
 closed magnet path operated at a density below saturation the 
 tangent of the angle of time- phase displacement will be approxi- 
 mately unity depending for its exact value upon the quality of 
 the magnetic material. Consider the magnetic and electric 
 circuits of the machine treated as a stationary transformer. 
 The hysteresis loss will be, in watts, 
 
 where A = cross sectional area of magnetic path 
 
 / = length of magnetic path (in centimeters) 
 B m maximum magnetic density (c.g.s) 
 
 The electromotive force counter generated in the transformer 
 coil having N turns will be, in effective volts, 
 
 E= : (237) 
 
 The current to supply the hysteresis loss will be 
 
 With a permeability of /A the magnetizing .component of the 
 no-load current will be 
 
 /,--^L^- U.5J (239) 
 
 10 
 
 For a certain value of permeability, depending upon the mag- 
 netic density, the hysteresis current and the magnetizing current 
 become equal in value. Thus when the two components of the 
 no-load exciting current become equal //* = 7 h , 
 
 .002 1 /^' 6 ^-/-IO ( ^ 
 
 (24 ' 
 
Induction- Series Motor. 73 
 
 from which is obtained, 
 
 /x= 119 B^ (250) 
 
 The meaning of equation (250) is that with a permeability of 
 the value there designated, the hysteresis current and the no-load 
 exciting current are equal in value and that the resultant current 
 \X/ h 2 _j_ 7^2 is displaced from the flux by a time-phase angle whose 
 tangent (equal at all times to the ratio of 7/x, to 7 h ) is unity, as 
 stated previously. For commercial laminated steel operated at 
 densities below saturation, the permeability differs but slightly 
 from the value given by the equation (250), though with increase 
 of magnetic density above 7,000 lines per square centimeter the 
 permeability falls off rapidly and the tangent of the angle of dis- 
 placement between flux and current becomes correspondingly 
 increased. 
 
 In an open magnetic circuit the permeability of a portion of 
 the path reduces from the value approximately represented by 
 the equation (250) to a value of unity, producing a very marked 
 effect upon the hysteretic angle of displacement between flux 
 and current. 
 
 Let / = length of path in magnetic material of permeability /*, 
 
 d= length of path in air, 
 
 then, assuming that permeability is as represented by equation 
 (250), the tangent of the angle of time-phase displacement be- 
 tween flux and magnetizing force is such that 
 
 (251) 
 
 the significance of which equation is that the flux lags behind 
 the current producing it, by an angle which depends for its value 
 largely upon the ratio of the air-gap to the length of the mag- 
 netic path. Assigning values to /A, / and d, it will be seen that 
 in any practical case the angle 8 must be quite small, seldom 
 more than 2 degrees. 
 
 It should be carefully noted that a slight error is introduced 
 on account of the fact that the permeability of commercial mag- 
 netic material undergoes a cyclic change with each alternation of 
 the current, and that, independent of the angle of time-phase 
 displacement between flux and current, the shape of the waves 
 representing the time- values of the two can not both be sinu- 
 soidal, and that in assigning a value to the angle of time-phase 
 
74 The Sibley Journal of Engineering. 
 
 displacement between the flux and current, the lack of similarity 
 of the two waves has been neglected. 
 
 Under speed conditions the e.m.f. counter generated by the 
 cutting of the armature conductors across the field magnetism, 
 varies in value with the magnetism, and hence it must have a 
 wave shape of time- value similar in all respects to that of the 
 field flux, and must have a time-phase position with reference to 
 the field current quite the same as that of the magnetism. The 
 counter generated speed e.m.f. must, therefore, lag behind the 
 current by an angle whose tangent is as given by equation (251 ). 
 Now since the counter e.m.f. lags behind the current, the cur- 
 rent must lead the counter e.m.f. by the same angle a fact 
 which has been mentioned previously. 
 
 With motors having air-gaps of sizes demanded by mechanical 
 clearance, the inherent angle of lead is quite small, and its effect 
 upon the power- factor is neutralized by the effect of the short 
 circuit by the brush of a coil in which is generated an e.m.f. by 
 both transformer and speed action when the machine is operated 
 as a motor. When the machine is operated as a generator, how- 
 ever, the hysteretic angle and the angle due to the short circuit- 
 ing effect are in a direction such as to be additive to the station- 
 ary reactive effect of the motor circuits and, therefore, during 
 generator operation the power factor is lower than during motor 
 operation as shown in Fig. 13. 
 
 While the angle of lead due to the hysteretic effect, even when 
 the machine is running as a motor, is in any case quite small and 
 its good effects cannot be availed of, it is possible by means of 
 certain auxiliary circuits to give to the angle of time-phase dis- 
 placement between the line current and the flux any value de- 
 sired, and thus to cause the operating power factor to become 
 unity or to decrease with leading wattless current, as is shown 
 below. 
 
 Fig. 14 represents diagrammatically the circuits of a conduc- 
 tively compensated-series motor in parallel with the field coil of 
 which is placed a non-inductive resistance. Consider first, ideal 
 conditions in which the armature and compensating coils are 
 without resistance and the compensation is complete so that these 
 two circuits, treated as one, are without inductance. The field 
 coil is without resistance but constitutes the reactive portion of 
 the motor circuits. 
 
 When the armature is stationary the circuit through the resis- 
 
Induction- Series Motor. 
 
 75 
 
 tance being open, the current taken by the machine has a value 
 determined by the ratio of the impressed e.m.f. and the reactance 
 of the field coil. This current lags 90 time degrees behind the 
 e.m.f. across the field coils. When a resistance is placed in shunt 
 to the field coil, current flows therethrough, quite independently 
 of the field current. The current taken by the resistance is in 
 time-phase with the e.m.f. impressed upon the field coil. 
 
 COMPENSATING COIL 
 
 a SPEED CM/: 
 
 FIG. 14. Compensated Series Motor with Shunted Field Coil. 
 
76 The Sibley Journal of Engineering. 
 
 In Fig. 14 let O 7= 7 f represent the field current, assumed 
 always of unit value. O D = E t is the e. m. f . impressed across 
 the field coil and the shunted resistance. 7 r is the current taken 
 by the resistance. <9C=7, the current which flows through 
 the armature and compensating coil or the resultant current taken 
 by the motor has a value represented by the equation 
 
 7=v/7 f 2 + 7 r 2 (252) 
 
 and has a phase displacement /3 with reference to the field current 
 such that 
 
 tan = y (253) 
 
 With unit value of field current, under speed conditions, the 
 e.m.f., E s , (DjFoi Fig. 14) counter generated at the brushes, 
 due to the presence of the field flux, will be proportional directly 
 to the speed and in time-phase with the field current. Thus this 
 component of the counter e.m.f. of the motor is in no wise 
 affected by the presence of the current through the shunted 
 resistance. At a certain speed, the counter generated armature 
 e.m.f. will have a value represented by the line D F Fig. 14 the 
 resultant e.m.f. E OF being the vector (quadrature) sum of 
 the speed e.m.f. and the stationary e.m.f. E a that is 
 
 E = VEl + E* (254) 
 
 and has a time-phase a position with reference to the speed e.m.f. 
 E s such that 
 
 tana = ^ f (255) 
 
 An inspection of Fig. 14 will show that under operating condi- 
 tions, the angle of time-phase displacement between the current 
 and the electromotive force, 0, has a value represented by the 
 equation 
 
 0=P a (2 5 6) 
 
 or the current leads the e.m.f. by the angle 0. At a certain 
 critical speed for each value of shunted resistance, or at a certain 
 value of resistance for any given speed, the angle reduces ta 
 zero, and the power factor of the motor becomes unity. 
 
 It is interesting to observe the effect of removing the resistance 
 from in shunt with the field circuit. Since the current taken by 
 the resistance is 90 time-degrees from the field flux, the resultant 
 torque due to the product of this component of the current and 
 the flux is of zero value, the instantaneous torque alternat- 
 
Induction- Series Motor. 77 
 
 ing at double the circuit frequency. The current through the 
 resistance, therefore, contributes in no way to the power of the 
 machine or to the counter- generated, armature-speed e.m.f., and 
 when the circuit through the resistance is opened no effect what- 
 soever is produced upon the value of the current taken by the 
 field coil, the counter e.m.f. or the torque of the machine. It 
 is apparent, therefore, that the use of the shunted resistance in- 
 creases the circuit current in a certain definite proportion, the 
 added component being a leading { ' wattless ' ' current under 
 speed conditions. If a reactance be placed in parallel with the 
 field coil, the current which flows therethrough will be in time- 
 phase with the field flux, and the torque produced thereby will 
 add to the torque due to the field current and it will affect 
 directly the whole performance of the machine. The current 
 taken by a condensance in shunt with the field coil will be in 
 time-phase opposition to the field current and will tend to 
 decrease directly both the circuit current and the armature 
 torque. An excess of condensance will cause the torque to 
 reverse and the machine to act as a generator even when the 
 speed is in a positive direction. When the condensance and the 
 field reactance are just equal, the circuit current reduces to zero 
 and the torque disappears. Under the conditions here assumed, 
 the counter generated e.m.f. at the armature remains propor- 
 tional to the product of the field flux and the speed, and there 
 appears the remarkable combination of zero current being trans- 
 mitted over a certain counter e.m.f. (that is, through infinite 
 impedance) to divide into definite active currents at the end of 
 the transmission circuits. 
 
 From what has been demonstrated above, it is seen that 
 shunted condensance acts to take current in phase opposition and 
 to decrease the torque ; reactance takes current directly in 
 phase, and increases the torque, while resistance takes current 
 in leading quadratures with the field current and has no effect 
 upon the torque. It is evident that the improvement in power 
 factor due to the use of the resistance is advantageous provided 
 the losses caused by the resistance are not excessive. Referring 
 to Fig. 14, when the resistance is not used the power taken by 
 the machine under speed conditions is 
 
 />= <9/.0/^cos J ro/=/ f ^cosa = / f ; (257) 
 
 When the machine is stationary, the power absorbed by the 
 resistance is 
 
 I r E t (258) 
 
7 8 The Sibley Journal of Engineering. 
 
 When the motor is running with shunted field coil, the power 
 delivered to the machine is 
 
 /> = OC- OF- cos COF= IE cos 6 (259) 
 
 = ft-a (250) 
 
 cos = cos ft cos a + sin ft sin a (261 ) 
 
 P t = /cos ft'E cos a -f- /sin ft>E sin a (262) 
 
 The significance of equation (263) is that the energy absorbed 
 is that incident to the use of the resistance, and that for a given 
 current it is unaffected by the speed e.m.f. Thus the current 
 taken by the resistance multiplies into the stationary transformer 
 e.m.f. to give the actual watts absorbed while the same current 
 multiplies into the speed e.m.f. to give apparent leading wattless 
 power. 
 
 CURRENT 234-5 
 
 FlG. 15. Observed E.M.F. Current Characteristics of Plain Series Motor 
 with Shunted Field. 
 
 In the derivation of the above equations ideal conditions have 
 been assumed, which cannot be obtained in a practical motor, 
 motor. Fig. 15 represents the observed e.m.f. current charac- 
 teristics of a certain plain, uniform reluctance motor (see Fig. 9) 
 with shunted field coils, and serves to show that even such an 
 unfavorable machine may be caused to operate at unity power 
 factor at any speed greater than about one-half synchronism. 
 
APPENDIX. 
 
 In compliance with the request of the committee having in charge the 
 work of the candidate, there is given below a list of articles dealing 
 with alternating- current phenomena as published by him during his candi- 
 dacy for the degree of Doctor of Philosophy, at Cornell University. 
 
 Frequency Converters. Elec. W. and Eng. May n, 1901. 
 
 The Regulation of Alternating Current Generators. Amer. Elec. Aug., 1901. 
 
 Parallel Operation of Alternators. Amer. Elec. Sept., 1901. 
 
 Transformers. Amer. Elec. Oct., 1901. 
 
 Measurement of the Angle of Lag of Three-Phase Circuits with One Watt- 
 meter. Elec. W. and Eng. Nov. 23, 1901. 
 
 Rotary Converters. Amer. Elec. Dec., 1901. 
 
 Complete Commercial test of Polyphase Induction Motors Using One Watt- 
 meter and One Voltmeter. Elec. W. and Eng. Jan. n, 1902. 
 
 Measuring Three-Phase Circuits. Elec. W. and Eng. March 8, 1902. 
 
 Characteristic Performance of the Induction Motor. Amer. Elec. April, 1902. 
 
 The Constant-Current Transformer. Amer. Elec. May, 1902. 
 
 The Single-Phase Induction Motor. Amer. Elec. June, 1902. 
 
 Characteristic Performance of Alternators. Trans. Cornell Elec. Society. 
 1901-02. 
 
 Polyphase Induction Motors Operating on Single- Phase Circuits. Amer. 
 Elec. Aug., 1902. 
 
 Synchronous Commutating Machines. Amer. Elec. Oct. and Nov., 1902. 
 
 An Asynchronous Motor with Unity Power- Factor. Sibley Journal. Nov. , 
 1902. 
 
 Six-Phase Transformation. Amer. Elec. Dec., 1902. 
 
 Three-Phase Measurements. Elec. W. and Eng. Dec. 13, 1902. 
 
 Excitation of Asynchronous Generators by Means of Static Condensance. 
 Elec. W. and Eng. Jan. 17, 1903. 
 
 Action of a Shunt- Wound Motor when Driven by a Series- Wound Dynamo. 
 Amer. Elec. Feb., 1903. 
 
 Some Engineering Features of the Bedell System of Composite Transmis- 
 sion. Elec. W. and Eng. Feb. 28, 1903. 
 
 The Joint Transmission of Different Currents Bedell System. Elec. Age. 
 March, 1903. 
 
 The Bedell System of Composite Transmission. Elec. Review. March 14, 
 1903. 
 
 Circuits for the Transmission and Distribution of Electrical Energy. Amer. 
 Elec. April, 1903. 
 
 System for the Joint Transmission of Differing Currents. Mill Owners. 
 April, 1903. 
 
 Graphic Representation of Induction Motor Phenomena. Trans. Cornell 
 Elec. Soc.. 1903. 
 
 The Heyland Asynchronous Motor. Elec. Age. June, 1903. 
 
 The Heyland Induction Motor. Amer. Elec. July, 1903. 
 
 Asynchronous Generators. Amer. Elec. Nov., 1903. 
 
2 Appendix. 
 
 The Winter- Eichberg Single-Phase Railway System. Elec. W. and Eng. 
 Dec. 19, 1903. 
 
 Single-Phase Railway Motors. Elec. W. and Eng. Feb. 13, 1904. 
 
 Alternating-Current Railway Motors. Trans. Amer. Inst. Elec. Eng. Feb., 
 1904. 
 
 A Convenient and Economical Electrical Method for Determining Mechani- 
 cal Torque. Elec. W. and Eng. May 7, 1904. 
 
 Single-Phase Induction Motors. Trans. Amer. I. E. E., June, 1904. 
 
 Self Exciting Asynchronous Generators. Trans. Cornell Elec. Soc. 1903-04. 
 
 Efficiency Curves of Rotary Converters. Elec. W. and Eng. June 4, 1904. 
 
 The Repulsion Motor. Amer. Elec. Sept., 1904. 
 
 Single-Phase Railway Motors Elec. W. and Eng. Nov. 12.1904. 
 
 The Graphic Treatment of the Phenomena of Static Transformers and In- 
 duction Motors. Sibley Journal, Nov., 1904. 
 
 Electromagnetic Torque. Elec. W. and Eng. Dec. 3, 1904. 
 
 Alternating Current Commutator Motors. Trans. Cornell Elec. Society. 
 1904-05- 
 
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