Theory and Calculation of Electric Circuits — part 10 of 15

. branch of the motor characteristic from the starting point, g, up to the maximum torque point, c, is unstable on a load requiring con- ; stant torque. ;

At load torque, D’ = 10, the motor can not start the load, can not carry it below b, S = 0.35; at speeds from b to a, S = 0.35 to 0.905, the motor speeds up; at speeds above a, S = 0.905, the motor slows down, and drops into stable condition at a.

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204 ELECTRIC CIRCUITS

With a load torque, D’o.= 5, the motor starts and runs up to speed ai, S = 0.96.

D’ = 7.2, point g, thus, is the maximum load torque which the

motor can start.

103. Suppose now, while running in stable condition, at point a, with the load torque, D’ = 10, the load torque is momentarily increased. If this increase leaves D’ lower than the maximum motor torque, Do = 14.3, the motor speed slows down, but re- mains above c, and thus when the increase of load is taken off, the motor again speeds up to a.

If, however, the temporary increase of load torque exceeds the maximum motor torque, Do = 14.3—for instance by starting a line of shafting or other mass of considerable momentum—then the motor speed continues to drop as long as the excess load exists, and whether the motor will recover when the excess load is taken off, or not, depends on the loss of speed of the motor during the period of overload: if, when the overload is relieved, the motor has dropped to point d, in Fig. 102, its speed thus is still above b, the motor recovers; if, however, its speed has dropped to dz, be- low the speed b, S = 0.35, at which the motor torque drops below the load torque, then the motor does not recover, but stops.

With a lighter load torque, D’o, which is less than the starting torque, g, obviously the motor will always recover in speed

The amount, by which the motor drops in speed at temporary overload, naturally depends on the duration of the overload, and on the momentum of the motor and its moving masses: the higher the momentum of the motor and of the masses driven by it at the moment of overload, the slower is the drop of speed

• of the motor, and the higher thus the speed retained by it at the moment when the overload is relieved.

Thus a motor of low starting torque, that is, high speed regula- tion, may be thrown out of step by picking up a load of high

momentum rapidly, while by adding a flywheel to the motor, it

would be enabled to pick up thisload. Or, it may be troublesome to pick up the first load of high momentum, while the second load of this character may give no trouble, as, due to the momentum ; of the load already picked up, the speed would drop less.

: Thus a motor carrying no load, may be thrown out of step by a load which the same motor, already partly loaded (with a load of . considerable momentum), would find no difficulty to pick up.

. The ability of an induction motor, to carry for a short time

INSTABILITY OF CIRCUITS 205 without dropping out of step a temporary excessive overload, naturally also depends on the excess of the maximum motor torque (at c in Fig. 102) over the normal load torque of the motor. A motor, in which the maximum torque is very much higher— several hundred per cent.—than the rated torque, thus could momentarily carry overloads which a motor could not carry, in which the maximum torque exceeds the rated torque only by 50 per cent., as was the case with the early motors. However, very high maximum torque means low internal reactance and thus high exciting current, that is, low power-factor at partial loads, . and of the two types of motors:

(a) High overload torque, but poor power-factor and efficiency at partial loads; (b) Moderate overload torque, but good power-factor and efficiency at partial loads; the type (b) gives far better average operating conditions, except in those rare cases of operation at constant full-load, and is there- fore preferable, though a greater care is necessary to avoid mo- mentary excessive overloads. Gradually the type (a) had more and more come into use, as the customers selected the motor, and the power supply company neglected to pay much attention to power-factor, and it is only in the last few years, that a realization of the harmful effects of low power-factors on the economy of operation of the systems is again directing attention to the need of good power-factors at partial loads, and the industry thus is returning to type (0), _ especially in view of the increasing tendency toward maximum output rating of apparatus. In distributing transformers, the corresponding situation had been realized by the central stations since the early days, and good partial load efficiencies and power-factors secured. ; 104. The induction motor speed-torque curve thus has on a : constant-torque load a stable branch, from the maximum torque ‘point, c, Fig. 102, to synchronism; and an unstable branch, from standstill to the maximum torque point. : However, it would be incorrect to ascribe the stability or in- stability to the induction motor-speed curve; but it is the char- acter of the load, the requirement of constant torque, which makes a part of the speed curve unstable, and on other kinds of load no instability may exist, or a different form of instability. Thus, considering a load requiring a torque proportional to

; 206 ELECTRIC CIRCUITS

the speed, such as would be given, approximately, by an electric generator at constant field excitation and constant resistance as load. .

The load-torque curves, then, would be straight lines going through the origin, as shown by D’,, D’:, D’s, ete., for increasingly larger values of load, in Fig. 103. The motor-torque curve, D, is the same as in Fig. 102. As seen, all the lines, D’, intersect D at points, a1, d2,G3 . . ., at which the speed is stable, since Pitt ttt tt TAT Tt Aol Ted Ta! PTT TTT ye eae ee er ptt tt AL | Al | Oh P| Pt tT tT Ett TAT oar EO eT | ae SERRE AZ CaR 4/4 aur Ht eA | AN ha SRRREP ARP Ane Cana t..a PT LEVY MA ean ert | IAL YI A et NAS I BREAN A02CZ24E2 4RReeDiee A pate tt tA aN S27 724¢ aR eee —— 627240 20 RRR

| VBE CEE EEC

DOZER ERE Aitreta tat sets telat st ht |

Fia. 103. dD’ _ dD dS 7 ds"

Thus, with this character of load, a torque required propor- tional to the speed, and the motor-torque curve, D, no instability exists, but conditions are stable from standstill to synchronism, just as in Fig. 101. That is, with increasing load, the speed de- creases and increases again with decreasing load.

If, however, the motor curve is as shown by Dy in Fig. 103, that is, low starting torque and a maximum torque point close to synchronism, as corresponds to an induction motor with low resistance secondary, then for a certain range of load, between

INSTABILITY OF CIRCUITS 207 | D’ and D’o, the load-torque line, D’:, intersects the motor curve, Do; in three points ba, dy, he. At be, S = 0.925, and at hs, S = 0.375, conditions are stable; at ds, S = 0.75, instability exists. . Thus with this load, D’s, the motor can run at two different speeds in stable conditions: a high speed, above co, and a low speed, be- low b; while there is a third, theoretical speed, dz, which is unstable. In the range below hs, the motor speeds up to hg; in the range between hg and de, the motor slows’ down to hs; in the range between dz and bs, the motor speeds up to bs, and in the range . above bs, the motor slows down to b;. There is thus a (fairly narrow) range of loads between D’ and — D'y, in which an unstable branch of the induction motor-torque _curve exists, at intermediate speeds; at low speed as well as at high speed conditions are stable. For loads less than D’, conditions are stable over the entire range of speed; for loads above D’y, the motor can run only at low speeds, hs, hy, but not at high speeds; but there is no load at which the motor would not start and run up to some speed. ' Obviously, at the lower speeds, the current consumed by the motor is so large, that the operation would be very inefficient. _ It is interesting to note, that with this kind of load, the “maxi- . mum torque point,’’ c, is no characteristic point of the motor- torque curve, but two points, cand b, exist, between which the op- eration of the motor is unstable, and the speed either drops down below 8, or rises above ¢. 105. With a load requiring a torque proportional to the square of the speed, such as a fan, or a ship propeller, conditions are al- | most always stable over the entire range of speed, from standstill to synchronism, and an unstable range of speed may occur only in | motors of very low secondary resistance, in which the drop of torque below the maximum torque point, c, of the motor character- | istic is very rapid, that is, the torque of the motor decreases more rapidly than with the square of the speed. This may occur with : very large motors, such as used on ship propellers, if the secondary resistance is made too low. : ‘ : More frequently instability with such fan or propeller load or other load of similar character may occur with single-phase motors, as in these the drop of the torque curve below maximum torque is much more rapid, and often a drop of torque with in- | creasing speed occurs, especially with the very simple and cheap |

208 ELECTRIC CIRCUITS starting devices economically required on very small motors, such as fan motors.

Instability and dropping out of step of induction motors also , may be the result of the voltage drop in the supply lines, and furthermore may result from the regulation of the generator vol- tage beingtooslow. Regarding hereto, however, see ‘Theory and Calculation of Electrical Apparatus, ’’ in the chapter on “Stability of Induction Machines.”

. D. Hunting of Synchronous Machines

106. In induction-motor circuits, instability almost always assumes the form of a steady change, with increasing rapidity, from the unstable condition to a stable condition or to stand- still, etc.

Oscillatory instability in induction-motor circuits, as the result of the relation of load fo speed and electric supply, is rare. It

has been observed, especially in single-phase motors, in cases of considerable oversaturation of the magnetic circuit.

Oscillatory instability, however, is typical of the synchronous machine, and the hunting of synchronous machines has probably been the first serious problem of cumulative oscillations in electric circuits, and for a long time has limited the industrial use’ of syn- chronous machines, in its different forms:

(a) Difficulty and failure of alternating-current generators to operate in parallel. ;

(6) Hunting of synchronous converters.

(c) Hunting of synchronous motors.

While considerable theoretical work has been done, practically

; all theoretical study of the hunting of synchronous machines has been limited to the calculation of the frequency of the transi- ent oscillation of the synchronous machine, at a change of load, frequency or voltage, at synchronizing, etc. However, this transient oscillation is harmless, and becomes dangerous only if the oscillation ceases to be transient, but becomes permanent and cumulative, and the most important problem in the study of hunt- ing thus is the determination of the cause, which converts the transient oscillation into a cumulative one, that is, the determina- tion of the source of the energy, and the mechanism of its trans- fer to the oscillating system. To design synchronous machines, so as to have no or very little tendency to hunting, obviously re-

| , . INSTABILITY OF CIRCUITS 209 : quires a knowledge of those characteristics of design which are instrumental in the energy transfer to the oscillating system, and thereby cause hunting, so as to avoid them and produce the great- est possible inherent stability. If, in an induction motor running loaded, at constant speed, the load is suddenly decreased, the torque of the motor being in ex- cess of the reduced load causes an acceleration, and the speed in- creases. As in an induction motor the torque is a function of the speed, the increase of speed decreases the torque, and thereby de- creases the increase of speed until that speed is reached at which the motor torque has dropped to equality with the load, and thereby acceleration and further increase of speed ceases, and the motor continues operation at the constant higher speed, that is, the induction motor reacts on a decrease of load by an increase of speed, which is gradual and steady without any oscillation. Tf, in a synchronous motor running loaded, the load is suddenly decreased, the beginning of the phenomenon is the same as in the induction motor, the excess of motor torque causes an ac- celeration, that is, an increase of speed. However, in the ~ synchronous motor the torque is not a function of the speed, but in stationary condition the speed must always be the same, synchronism, and the torque is a function of the relative position of the rotor to the impressed frequency. The increase of speed, due to the excess torque resulting from the decreased load, causes the rotor to run ahead of its previous relative position, and thereby decreases the torque until, by the increased speed, the motor has run ahead from the relative position corresponding to the pre- vious load, to the relative position corresponding to the decreased load. Then the acceleration, and with it the increase of speed, stops. But the speed is higher than in the beginning, that is, is above synchronism, and the rotor continues to run ahead, the torque continues to decrease, is now below that required by the load, and the latter thus exerts a retarding force, decreases the speed and brings it back to synchronism. But when synchron- ous speed is reached again, the rotor is ahead of its proper position, thus can not carry its load, and begins to slow down, until it is brought back into its proper position. At this position, however, the speed is now below synchronism, the rotor thus continues to drop back, and the motor torque increases beyond the load, : thereby accelerates again to synchronous speed, etc., and in this manner conditions of synchronous speed, with the rotor position 14 _ |

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210 ; ELECTRIC CIRCUITS

behind or ahead of the position corresponding to the load, alter-

nate with conditions of proper relative position of the rotor, but

below or above synchronous speed, that is, an oscillation results

which usually dies down at a rate depending on the energy losses

resulting from the oscillation.

107. As seen, the characteristic of the synchronous machine is, that readjustment to a change of load requires a change of . relative position of the rotor with regard to the impressed fre- quency, without any change of speed, while a change of relative

; position can be accomplished only by a change of speed, and this results in an over-reaching in position and in speed, that is, in an oscillation. :

, Due to the energy losses caused by the oscillation, the success-

ive swings decrease in amplitude, and the oscillation dies down.

If, however, the cause which brings the rotor back from the posi-

tion ahead or behind its normal position corresponding to the

changed load (excess or deficiency of motor torque over the

; torque required by the load) is greater than the torque which

opposes the deviation of the rotor from its normal position, each

swing tends to exceed the preceding one in amplitude, and if the

energy losses are insufficient, the oscillation thus increases in amplitude and becomes cumulative, that is, hunting. .

— In Fig.'104 is shown diagrammatically as p, the change of the relative position of the rotor, from p; corresponding to the pre- vious load to pe the position further forward corresponding to the decreased load.

v then shows the oscillation of speed corresponding to the oscillation of position.

The dotted curve, w:, then shows the energy losses resulting from the oscillation of speed (hysteresis and eddies in the pole faces, currents in damper windings), that is, the damping power, assumed as proportional to the square of the speed.

If there is no lag of the synchronizing force behind the position displacement, the synchronizing force, that is, the force which tends to bring the rotor back from a position behind or ahead of the position corresponding to the load, would be—or may ap- proximately be assumed as—proportional to the position dis- placement, p, but with reverse sign, positive for acceleration when

; p is negative or behind the normal position, negative or retarding when p is ahead. The synchronizing power, that is, the power exerted by the machine to return to the normal position, then is

INSTABILITY OF CIRCUITS 211 derived by multiplying —p with », and is shown dotted as w2 in Fig. 104. Asseen, it has a double-frequency alternation with zero as average.

’ The total resultant power or the resulting damping effect which restores stability, then, is the sum of the synchronizing power ws: and the damping power 1;, and is shown by the dotted CTE CLES pi y it | | eee EE er at Pte PE aS eS BIT NETS ptt Pt Tt TT yy FERREEEERREEEEP PCP NEVET REPT Pi ttt tT ttt tee ee ed Nt RK RST PC IANZSSPEEECEE | SWARM PLATE EET TE TEE Tt NGA eS eS Se Perry Pi tt te ttt et TTT TT EES SS PPT ZENSZASEr ET PT RNAT IAP | Tt ty tt Pt ieee ttt ee tT TT . Fie. 104, curve w. As seen, under the assumption or Fig. 104, in this case a rapid damping occurs. If the damping winding, which consumes a part of all the power, w1, is inductive—and to a slight extent it always is—the current in the damping winding lags behind the e.m-f. induced in it by the oscillation, that is, lags behind the speed, v. The power, 1,

212 ELECTRIC CIRCUITS or that part of it which is current times voltage, then ceases to be continuously negative or damping, but contains @ positive period, and its average is greatly reduced, as shown by the drawn curve, 1, in Fig. 104, that is, inductivity of the damper winding is very harmful, and it is essential to design the damper winding as non- inductive as possible to give efficient damping. . With the change of position, p, the current, and thus the ar- mature reaction, and with it the magnetic flux of the machine,

• changes. A flux change can not be brought about instantly, as it represents energy stored, and as a result the magnetic flux of the machine does not exactly correspond with the position, p, but lags behind it, and with it the synchronizing force, F, as shown in Fig. 104, lags more or less, depending on the design of the machine. The synchronizing power of the machine, F», in the case of a lag- ging synchronizing force, F, is shown by the drawn curve,w:. As seen, the positive ranges of the oscillation are greater than the negative ones, that is, the average of the oscillating synchronizing , power is positive or supplying energy to the oscillating system, which energy tends to increase the amplitude of the oscillation—in other words, tends to produce cumulative hunting.

The total resulting power, w = w, + wz, under these condi- tions is shown by the drawn curve, w, in Fig. 104. As seen, its average is still negative or energy-consuming, that is, the oscilla- tion still dies out, and stability is finally reached, but the average value of w in this case is so much less than in the case above dis-_ - cussed, that the dying out of the oscillation is much slower.

If now, the damping power, w1, were still smaller, or the aver- age synchronzing power, ws, greater, the average w would become positive or supplying energy to the oscillating system.

In other words, the oscillation would increase and hunting result. ;

That is:

If the average synchronizing power resulting from the lag of the synchronizing force behind the position exceeds the average damping power, hunting results. The condition of stability of the synchronous machine is, that the average damping power ex- ceeds the average synchronizing power, and the more this is the case, the more stable is the machine, that is, the more rapidly the transient oscillation of readjustment to changed circuit con-

. . ditions dies out.

INSTABILITY OF CIRCUITS 213 Or, if a = attenuation constant of the oscillating system, ; a<0 gives cumulative oscillation or hunting. a>0 gives stability. : 108. Counting the time, ¢, from the moment of maximum back- ward position of the rotor, that is, the moment at which the load on the machine is decreased, and assuming sinusoidal variation, and denoting @ = 2aft = ut (1) where f = frequency of the oscillation (2) the relative position of the rotor then may be represented by p = —pre™* cos ¢, where Po = pz — Pi = position difference of rotor resulting from change of load, (3) a = attenuation constant of oscillation. (4) The velocity difference from that of uniform rotation then is p= P= oP = wpoe (sing + 0.0084) (5) Let a=tana; 1+ a? = A? (6) hence, . a 1 sina = 7; coea=F (7) it is v = wpoAe~* sin (¢ + a). (8) Let : y = lag of damping currents behind e.m.f. induced in damper windings (9) the damping power is 7 Wi = — cw, = —Cw* po? Are gin (p-+a)sin(¢+a—y) (10) where | c= S = damping power per unit velocity and vy is 2, lagged by angle y. (11) : i

, | 214 ELECTRIC CIRCUITS Let 8 = lag of synchronizing force behind position displace- ment p ‘ (12) F and B = why (18) where t = time lag of synchronizing force. (14) The synchronizing force then is F = bpoe** cos (@ — B) (15) . where b= = ratio of synchronizing force to po- sition displacement, or specific synchronizing force. (16) The synchronizing power then is ws = Fv = bupoAc ain (¢ + a) cos (¢ — 8). (17) The oscillating mechanical power is , : dte | dg = muBprA%e—? % sin (6 + a) {cos (¢ + a) —asin(@+a)} (18) where m = moving mass reduced to the radius, on which p is measured. (19) It is, however, w+uwe—-w=0 (20) hence, substituting (10), (17), (18) into (20) and canceling, b cos (¢ — B) — cwA sin (6 + @ — y) — | mwAcos (¢ + a) + mwtAasin(?+a) =0. (21) This gives, as the coefficients of cos ¢ and sin ¢ the equations. bcos 8B — cwA sin (a — y) — ma?A cosa + mwAasin a = 0 (22) bsin B — cwA cos (a — y) + mwA sina + mwA cos a = 0 Substituting (6) and (7) and approximating from (13), for 8 as a small quantity, , , cosB = 1; sin B = wb (23) gives , b — cw (acos y — sin y) — mw? (1 — a?) = 0 (24) bto — c (cos y + asin y) + 2mwa = 0

' INSTABILITY OF CIRCUITS 215 This gives the values, neglecting smaller quantities . c cosy — bto = 25)

• V4 mb — c? cos? y + to? (25) - o= a Wi mb —ccos27+ b%? +csiny} (26) = These equations (25) and (26) apply only for small values of a, but become inaccurate for larger values of a, that is, very rapid damping. However, the latter case is of lesser importance. a=0 a gives . blo = c cos y, . hence, , ¢> bt cos 7. or, (28) CCOsy | | bo < b are the conditions of stability of the synchronous machine. , If b = 0 . 7=0 it is a = ——*___, V4mb — c? _ Vim — o= 2m and, if also, : c= 0: it is @=4/— m | !

i CHAPTER XII ; REACTANCE OF INDUCTION APPARATUS

109. An electric current passing through a conductor is ac- companied by a magnetic field surrounding this conductor, and this magnetic field is as integral a part of the phenomenon, as is the energy dissipation by the resistance of the conductor. It is represented by the inductance, L, of the conductor, or the number of magnetic interlinkages with unit current in the conductor. Every circuit thus has a resistance, and an inductance, however

small the latter thay be in the so-called “‘non-inductive” circuit. .

With continuous current in stationary conditions, the inductance, L, has no effect on the energy flow; with alternating current of frequency, f, the inductance, L, consumes a voltage 2 xfLi, and is, therefore, represented by the reactance, x = 22fL, which is measured in ohms, and differs from the ohmic resistance, r, merely by being wattless or reactive, that is, representing not dissipation of energy, but surging of energy.

Every alternating-current circuit thus has a resistance and a reactance, the latter representing the effect of the magnetic field of the current in the conductor.

When dealing with alternating-current apparatus, especially those having several circuits, it must be realized, however, that the magnetic field of the circuit may have no independent exist- ence, but may merge into and combine with other magnetic fields, so that it may become difficult what part of the magnetic field is to be assigned to each electric circuit, and circuits may exist which apparently have no reactance. In short, in such cases, the magnetic fields of the reactance of the electric circuit may be merely a more or less fictitious component of the resultant mag- netic field.

The industrial importance hereof is that many phenomena, such as the loss of power by magnetic hysteresis, the m.m.f. required for field excitation, etc., are related to the resultant magnetic - field, thus not equal to the sum of the corresponding effects of the components.

216

REACTANCE OF INDUCTION APPARATUS 217 As the transformer is the simplest alternating-current appara- tus, the relations are best shown thereon. | Leakage Flux of Alternating-current Transformer 110. The alternating-current transformer consists of a mag- netic circuit, interlinked with two electric circuits, the primary circuit, which receives power from its impressed voltage, and the secondary circuit, which supplies power to its external circuit. ’ For convenience, we may assune the secondary circuit as re- duced to the primary circuit by the ratio of turns, that is, assume ratio of turns 1 + 1. ; Let Yo = g — jb = primary exciting admittance; Zo = ro + jxo = primary self-inductive impedance; Z, = r1 + jz: = secondary self-inductive impedance (reduced to the primary). . ; The transformer thus comprises three magnetic fluxes: the mutual magnetic flux, , which, being interlinked with primary and secondary, transforms the power from primary to secondary, ; and is due to the resultant m.m.f of primary and secondary cir- cuit; the primary leakage flux, 9, due to the m.m.f. of the primary circuit, Fo, and interlinked with the primary circuit only, which is represented by the self-inductive or leakage reactance, 29; and the secondary leakage flux, ’,, due to the m.m-f. of the secondary - circuit, F;, and interlinked with the secondary circuit only . which is represented by the secondary reactance, 2. : As seen in Fig. 1050, the mutual flux, 6—usually—has a closed . iron circuit of low reluctance, p, thus low m.m.f.,¥F, and high intens- ity; the self-inductive flux or leakage reactance flux, ©’) and #',, ! close through the air circuit between the primary and secondary _ electric circuits, thus meet with a high reluctance, po, respectively | pi, usually many hundred times higher than p. Their m.m.fs., Fo | and F,, however, are usually many times greater than F; the lat- ! ter is the m.m-f. of the exciting current, the former that of full primary or secondary current. For instance, if the exciting current is 5 per cent. of full-load current, the reactance of the transformer 4 per cent., or 2 per cent. primary and 2 per cent. secondary, then the m.m_f. of the leakage flux is 20 times that of the mutual flux, and the mutual flux 50 times the leakage flux, hence the reluctance of leakage flux 50 _X 20 = 1000 times that of the mutual or main flux: p: = 1000p.

1

REACTANCE OF INDUCTION APPARATUS 219 111. Usually, ‘as stated, the leakage fluxes are not considered as such, but represented by their reactances, in the transformer diagram. Thus, at non-inductive load, it is, Fig. 106, O@ = mutual, or main magnetic flux, chosen as negative ver- tical. OF = m.m.f. required to produce flux, 06, and leading it by the angle of hysteretic advance of phase, FO®. OE’, = e.m.f. induced in the secondary circuit by the mutual flux, and 90° behind it. ; TRANSFORMER DIAGRAM NON-INDUCTIVE LOAD . SHOWING MAGNETIC FLUXES | - : gi tet | HENCE F=2; @'=15: Oi=6 . , Fon 7.6; Do=-2.9; Go—7.6 Eo Ea, Fee ———— at / / = ; / / ™< cz fe} / E, + < % ee Eo , | oT ee bed FL TS | . 7s | | AF | . /|\I | a i %, ~~ i . @ ~s, | 0 | Fig. 106. I,z,; = secondary reactance voltage, 90° behind the secondary "current, and combining with OE’; to OE, = true secondary induced voltage. From this subtracts : the secondary resistance voltage, I,r:, leaving the sec- ondary terminal voltage, and, in phase with it at non- induetive load, the secondary current and secondary m.m.f., OF:. . From component, OF, and resultant, OF’, follows the other com- ponent, | 4

220 ELECTRIC CIRCUITS , ; OF) = primary m.m-f. and in phase with it the primary current. OE’, = primary voltage consumed by mutual flux, equal and opposite to OE’;. Ioxo = primary reactance voltage, 90° ahead of the primary current OF o.

From Iz 28 component and E’y as resultant follows the other component, OKo, and adding thereto the primary resistance vol- tage, Ioro, gives primary supply voltage.

In this diagram, Fig. 106, the primary leakage flux is represented by O@’o, in phase with the primary current, OF ;, and the secondary leakage flux is represented by 0®’;, in phase with the secondary

: current, OF.

As shown in Fig. 1050, the primary leakage flux, ’o, passes through the iron core inside of the primary coil, together with the resultant flux, &, and thesecondary leakage flux, ©’, passes through the secondary core, together with the mutual flux, ®. However, ~ at the moment shown in Fig. 1050, S’; and @ in the secondary core are opposite in direction. This obviously is not possible, and the flux in. the secondary core in this moment is ® — #’,, that is, the magnetic disposition shown in Fig. 1050 is merely nominal, but the actual magnetic distribution is as shown in — Fig. 105a; the flux in the primary core, @) = @ + ’o, the flux in the secondary core, $; = & — ©’),

As seen, at the moment shown in Fig. 1050 and 105a, all the leakage flux comes from and interlinks with the primary winding, none with the secondary winding, and it thus would appear, that — all the self-inductive reactance is in the primary circuit, none in the secondary circuit, or, in other words, that the secondary circuit of the transformer has no reactance.

However, at a later moment of the cycle, shown in Fig. 105c, all the leakage flux comes from and interlinks with the secondary, and this figure thus would give the impression, thatll the leakage . reactance of the transformer is in the secondary, none in the primary winding.

In other words, the leakage fluxes of the transformer and the mutual or main flux are not independent fluxes, but partly tra- verse the same magnetic circuit, so that each of them during a part

. of the cycle is a part of any other of the fluxes. Thus, the react- ance voltage and the mutual inductive voltage of the transformer

REACTANCE OF INDUCTION APPARATUS 221 . are not separate e.m.fs., but merely mathematical fictions, com- ponents of the resultant induced voltage, OF; and OFo, induced by the resultant fluxes, 0%, in the primary, and 04, in the sec- ondary core. :

112. In Fig. 107 are plotted, in rectangular coérdinates, the . magnetic fluxes: The mutual or main magnetic flux, $; The primary leakage flux, ®’o; The resultant primary flux, By = © + 4’; The secondary leakage flux, ©’; The resultant secondary flux, ®; = & — ®’;; a ) s be \f. o | | f @, \ | if e% ; Ae eI YW yj MAGNETIC FLUXES OF WJ TRANSFORMER O=-62 } 91-15 $1 = 1.06; O,=6 Oo=1,9 Go = 00: Oo=7.8, Fia. 107. and the magnetic distribution in the transformer, during the moments marked as a, b, ¢, d, e, f, g, in Fig. 107, is shown in Fig. 105.

• In Fig. 105a, the primary flux is larger than the secondary, and all leakage fluxes (17 and z:) come from the primary flux, that is, there is no secondary leakage flux.

In Fig. 105b, primary and secondary flux equal, and primary and secondary leakage flux equal and opposite, though small. In Fig. 105c, the secondary flux is larger, all leakage flux (zo and 21) comes from the secondary flux, that is, there is no primary leakage flux.

222 . ELECTRI C CIRCUITS

In Fig. 105d, there is no primary flux, and all the secondary flux is leakage flux. :

In Fig. 105e, there is no mutual flux, all primary flux is primary leakage flux, and all secondary flux is secondary leakage flux.

In Fig. 105f, there is no secondary flux, and all primary flux is leakage flux.

In Fig. 105g, the primary flux is larger than the secondary, and all leakage flux comes from the primary, the same as in 105a.

Figs. 105a to 105f, thus show the complete cycle, corresponding to diagrams, Figs. 106 and 107.

These figures are drawn with the proportions,

p+p +m =14125 + 12.5 F+Fo +F, =1+ 38 + 3 ++ 6, = 14+ 0.317 + 0.25. thus are greatly exaggerated, to show the effect more plainly. Actually, the relations are usually of the magnitude, p + po + px: = 1 + 1000 + 1000 Fe + Fo + Fy = 1+ 20.6 + 20 : @ + $n + H, = 1 + 0.02 + 0.02 113. In symbolic representation, denoting, = mutual magnetic flux. E = mutual induced voltage. $)= resultant primary flux. ’) = primary leakage flux. Eo = primary terminal voltage. Zo = primary current. Zo = ro + jto = primary self-inductive imped- ance. #, = resultant secondary flux. . $', = secondary leakage flux. E, = secondary terminal voltage. 71 = secondary current. Z, = 7, + jz: = secondary self-inductive im- pedance. and c=2nfn where n = number of turns. |

{ REACTANCE OF INDUCTANCE APPARATUS 2238 . It then is | of, = jxofo cP, = jai . | ce = EF = Ey — Zolo = Fit 2id1 Cho = Ey — nolo = E + jrolo : f= Btn = F—- jali Fo = h —- > | = «1 = +4, thus, the total leakage flux G = Po + $1 = Po — Fi.

114. One of the important conclusions from the study of the

actual flux distribution of the transformer is that the distinction

; between primary and secondary leakage flux, $’y and ©’,, is really . an arbitrary one. There is no distinct primary and secondary leakage flux, but merely one leakage flux, ®’, which is the flux passing between primary and secondary circuit, and which during

a part of the cycle interlinks with the primary, during another

. part of the cycle interlinks with the secondary circuit Thus the corresponding electrical quantities, the reactances, 2 and 2), are not independent quantities, that is, it can not be stated that there is a definite primary reactance, 2, and a definite secondary react- ance, 21, but merely that the transformer has a definite reactance, x, which is more or less arbitrarily divided into two parts; 7 = 2

• 21, and theone assigned to the primary, the other to the second- ary circuit.

As the result hereof, “mutual magnetic flux” #, and the mutual induced voltage, EZ, are not actual quantities, but rather mathe- matical fictions, and not definite but dependent upon the distri- bution of the total reactance between the primary and the sec- ondary circuit.

This explains why all methods of determining the transformer reactance give the total reactance 29 + 71.

However, the subdivision of the total transformer reactance into a primary and a secondary reactance is not entirely arbitrary. Assuming we assign all the reactance to the primary, and consider the secondary as having no reactance. Then the mutual mag- netic flux and mutual induced voltage would be ;

ch = EF = Eo — [ro + j (to + 21)] Jo and the hysteresis loss in the transformer would correspond hereto, by the usual assumption in transformer calculations.

224 ELECTRIC CIRCUITS Assigning, however, all the reactance to the secondary circuit, and assuming the primary as non-inductive, the mutual flux and mutual induced voltage would be ch = EF = Ey — rofo, hence larger, and the hysteresis loss calculated therefrom larger than under the previous assumption. The first assumption would give too low, and the last too high a calculated hysteresis loss, in most cases. By the usual transformer theory, the hysteresis loss under load is calculated as that corresponding to the mutual induced voltage, E. The proper subdivision of the total transformer reactance, 2, into primary reactance, 2%, and secondary reactance, 21, would then be that, which gives for a uniform magnetic flux, , corresponding to the mutual induced voltage, E, the same hysteresis loss, as | exists with the actual magnetic distribution of By) = & + ’ in | the primary, and 4, = @ — &’; in the secondary core. Thus, if Vo is the volume of iron carrying the primary flux, &o, at flux den- | sity, Bo, Vi the volume of iron carrying the secondary flux, 4, at | flux density, B,, the flux density of the theoretical mutual mag- | netic flux would be given by BM VoBol® + ViB,!8 | ; Vot+ Vi | from B then follows %, FE, and thus x and 2. . This does not include consideration of eddy-current losses. For these, an approximate allowance may be made by using 1.7 | as exponent, instead of 1.6. Where the magnetic stray field under load causes additional losses by eddy currents, these are not included in the loss assigned to the mutual magnetic flux, but appear as an energy component | of the leakage reactances, that is, as an increase of the ohmic re- sistances of the electric circuits, by an effective resistance. | 115. Usually, the subdivision of x into 2 and 2, by this as- sumption of assigning the entire core loss to the mutual flux, | is sufficiently close to equality, to permit this assumption. ‘That is, the total transformer reactance is equally divided between | primary and secondary circuit. | This, however, is not always justified, and in some cases, the one circuit may have a higher reactance than the other. Such, . for instance, is the case in some very high voltage transformers, and usually is the case in induction motors and similar apparatus. It is more commonly the case, where true self-inductive fluxes

. —— REACTANCE OF INDUCTION APPARATUS 225 exist, that is, magnetic fluxes produced by the current in one circuit, and interlinked with this circuit, closing upon themselves in a path which is entirely distinct from that of thé mutual mag- netic flux, that is, has no part in common with it. Such, for in- stance, frequently is the self-inductive flux of the end connections of coils in motors, transformers, etc. To illustrate: in the high- , voltage shell-type transformer, shown diagrammatically in Fig. 108, with primary coil 1, closely adjacent to the core, and high-voltage secondary coil 2 at considerable distance:

The primary leakage flux consists of the flux in spaces, a, between the yokes of the transformer, closing through the iron core, C, and the flux through the spaces, b, outside of the trans- former, which enters the faces, F, of the yokes and closes through ; the central core, C.

The secondary leakage flux contains the same two components:

. the flux through the spaces, a, between the yokes closing, however, . | through the outside shells, S, and the flux through the spaces, b, outside of the transformer, and entering the faces, F, but in this case closing through the shells, S. In addition to these two com- | ponents, the secondary leakage flux contains a third component, | passing through the spaces, b, between the coils, but closing, through outside space, c, in a complete air circuit. This flux has no corresponding component in the primary, and ‘the total secondary leakage reactance in this case thus is larger than the total primary reactance. | Similar conditions apply to magnetic structures as in the in- . duction motor, alternator, etc. .

In such a case as represented by Fig. 108, the total reactance of the transformer, with (2) as primary and (1) as secondary, would be greater than with (1) as primary and (2) as secondary.

In this case, when subdividing the total reactance into primary ; ; reactance and secondary reactance, it would appear legitimate to divide it in proportion of the total reactances with (1) and (2) | as primary, respectively. That is,