Theory and Calculation of Transient Electric Phenomena and Oscillations — part 7 of 20

An increase of ampere-turns from 5000 to 7000, corresponding to an increase of current in the shunt field winding by 2 amperes, increases the generated e.m.f. from 500 to 600 volts, and the magnetic flux from 10 to 12, or by 2 megalines per pole. In the induction range covered by the overcompounding from 500 to 600 volts, 1 ampere increase in the shunt field increases the flux by 1 megaline per pole, and so, with n, = 1000 turns, gives 10° magnetic interlinkages per pole, or 8 X 10° interlinkages with 8 poles, per ampere, hence 80 X 10° interlinkages per unit current or 10 amperes, that is, an inductance of 80 henrys. Reduced to the main circuit this gives an inductance of 1.2? x 10°° X 80 = 115.2 X 10-* henrys. This is the inductance due to the magnetic flux in the field poles, which interlinks with shunt and series coil, or the mutual inductance, M = 115.2 x 10-° henrys.

Assuming the total inductance Z, of the shunt field winding as 10 per cent higher than the mutual inductance M, that is, assuming 10 per cent stray flux, we have

L, = 1.1M = 126.7 X 10-° henrys. . | |

| MUTUAL INDUCTANCE 151 . In the main circuit, full load is 1000 amp. at 600 volts. This . gives the effective resistance of the main circuit as r = 0.6 ohm. The quantities referring to the main circuit may be denoted without index. The total inductance of the main circuit depends upon the character of the load. Assuming an average railway motor load, the inductance may be estimated as about L = 2000 x 10° henrys. In the present problem the impressed e.m.fs. are not constant but depend upon the currents, that is, the sum 7 + 7,, where 1, = shunt field current reduced to the main circuit by the ratio of turns. The impressed e.m.f., e, is approximately proportional to the magnetic flux ®, hence less than proportional to the current, in consequence of magnetic saturation. Thus we have e = 500 volts for 5000 ampere-turns, or? +%, = 2 4170 amp. and e = 600 volts for 7200 ampere-turns, ori += oo 6000 amp.; hence, 1830 amp. produce a rise of voltage of 100, or 1 amp. . 100 1 ; raises the voltage by 1830 7 183° ; At 6000 amp. the voltage ise = 328 volts higher than at _0 amp., that is, the voltage in the range of saturation between 500 and 600 volts, when assuming the saturation curve in this range as straight line, is given by the equation _ t+ ty e= 272+ 33 The impressed e.m.f. of the shunt field is the same, hence, reduced to the main circuit by the ratio of turns, a = 1.2 x 107°, is ~(o72 4 +4) 4 e, = (272 +24) 12 x 10>,

152 TRANSIENT PHENOMENA Assuming now as standard frequency, f = 60 cyeles per sec., the constants of the two mutually inductive circuits shown diagrammatically in Fig. 38 are: Main Circuit. | Shunt Field Circuit. : Current..........., vamp. , tjamp. Impressed e.m.f.. e= 272+ ae volts | e, = (21248) ax va Resistance........! r= .6ohms _ r, = 0.144X 10-3 ohms Inductance...... | L = 2000X 10—-* henrys L,= 126.7X 10—* henrys Reactance, 22/L..| x = 755X10-% ohms ! x, = 47.8X 10—* ohms Mutual inductance M = 115.2 x 10—-* henrys Mutual reactance. Im= 43.5 X 10—* ohms 2 . ee This gives the differential equations of the problem as t+% 2... -.,di di, - . 272 + 183 7 0.64 + 0.755 5 + 0.0435 = (26) and a+ di di 2(o72 + 224) = 014i, 4 478% 4 4352. 27 1.2 (272 + ae) 0.14 i, + ATST + 43.55 (27)

• edt . .

88. Eliminating 77, from equations (26) and (27) gives di, — ; a 0.695 ¢ — 0.0712 7, — 338. (28) Equation (28) substituted in (26) gives . di . i, = 13.07 3 + 9.951 — 4950. (29) ° Equation (29) substituted in (28) gives di, di ar | an 0.93 Fi 0.0157 +15. (30) | Equation (29) differentiated, and equated with (30), gives . ay di . - | 7 + 0.828 7) + 0.001157 — 1.15 = 0. (31)

MUTUAL INDUCTANCE 153 Equation (31) is integrated by . | i=i,+ Ae, | Substituting this in (31) gives Ae {a? — 0.828 a + 0.00115} + {0.00115 % — 1.15} =0, | hence, 7, = 1000, A is indefinite, as integration constant, and a — 0.828 a + 0.00115 = 0; | thus a = 0.414 + 0.4126, | | and the roots are | a, = 0.0014 and a, = 0.827. | Therefore ; | t = 1000 + Aye 000 + Aygo 7, (82) . Substituting (32) in (29) gives , | i, = 5000 + 9.932 Ae? 0.85 Aye "7", (38) | | Substituting in (32) and (33) the terminal conditions @ = 0. _ t=0, and i, = 4170, gives A, + A, = — 1000 and 9,932 A, — 0.85 A, = — 830, that is, A, = — 156 and A, = — 844. Therefore i = 1000 — 156 0"? — Big one (34) and ; i, = 5000 — 1550 °° 4 720°"; (85) or the shunt field current 7, reduced back to the number of turns of the shunt field by the factor a = 1.2 x 107° is i = 6 — 1.86 g7 0700140 + 0.86 coe (36)

154 TRANSIENT PHENOMENA and the terminal voltage of the machine is a+ = 272 +. . e 2 + 18.3 ? or, e = 600 — 93.2 «7 °°"? — 6.8 2-9" . (37) As seen, of the two exponential terms one disappears very quickly, the other very slowly. Introducing now instead of the angle 6 = 2 xft the time, ¢, gives the main current as 7 = 1000 — 156 «7°! — 844 7", the shunt field current as i’ = 6 — 1.86 °! + 0.8627", 5 ‘ (38) and the terminal voltage as e = 600 — 93.2 e-°! — 6.8 6-!. 89. Fig. 39 shows these three quantities, with the time, ¢, as abscissas. Seconds o t=0.01_ 0.02 (0.03 _ 0.04 0.05 0.06 0.07 __—0.08 50 , eee eee on abe oer Ty let 900 4.6 t ae +—|—| EL ee es ee ise eS Be @ ot NU -— 2 i Poe a | ; we ery TT ape 300 5.6}4- | | = yt 2a eee 560 200 5.4 a SoH +4 e 540 ww 5.2 4 | BI | | J | se —- | | 520 100. to mw | | rac | r= 016 ohms a A 800. 46 1 nie tL = 2mh.| | me “/-eeaemaioise cai 4 a2¥ 1 | | eet —}+ Il 96amh—|—_} | is | | i, Shunt field current | | | | . t—1 2 8 tecona® 6 7 8 Fig. 39. Building-up of over-compounded direct-current generator from 600 volts no load to 600 volts load. The upper part of Fig. 39 shows the first part of the curve with 100 times the scale of abscissas as the lower part. As seen, the transient phenomenon consists of two distinctly different

MUTUAL INDUCTANCE 156 periods: first a very rapid change covering a part of the range of current or e.m.f., and then a very gradual adjustment to the final condition.

So the main current rises from zero to 800 amp. in 0.01 sec., but requires for the next 100 amp., or to rise to a total of 900 amp., about a second, reaching 95 per cent of full value in 2.25 sec. During this time the shunt field current first falls very rapidly, from 5 amp. at start to +.2 amp. in 0.01 sec., and then, after a minimum of 4.16 amp., at ¢ = 0.015, gradually and very slowly rises, reaching 5 amp., or its starting point, again after somewhat more than a second. After 2.5 sec. the shunt field . current has completed half of its change, and after 5.5 sec. 90 per cent of its change.

The terminal voltage first rises quickly by a few volts, and then rises slowly, completing 50 per cent of its change in 1.2 . sec., 90 per cent in 4.5 sec., and 95 per cent in 5.5 sec.

Physically, this means that the terminal voltage of the machine rises very slowly, requiring several seconds to approach station- ary conditions. First, the main current rises very rapidly, at a rate depending upon the inductance of the external circuit, to the value corresponding to the resistance of the external circuit | and the initial or no load terminal voltage, and during this period of about 0.01 sec. the magnetizing action of the main current is neutralized by a rapid drop of the shunt field current. Then gradually the terminal voltage of the machine builds up, and the shunt field current recovers to its initial value in 1.15 sec., and then rises, together with the main current, in corre- spondence with the rising terminal voltage of the machine.

It is interesting to note, however, that a very appreciable time elapses before approximately constant conditions are reached.

90. In the preceding example, as well as in the discussion of the building up of shunt or series generators in Chapter II, the e.m.fs. and thus currents produced in the iron of the magnetic field by the change of the fiell magnetization have not been considered. The results therefore directly apply to a machine with laminated field, but only approximately to one with solid iron poles.

In machines with solid iron in the magnetic circuit, currents produced in the iron act as a second electric circuit in inductive

156 TRANSIENT PHENOMENA relation to the field exciting circuit, and the transition period thus is slower.

As example may be considered the excitation of a series booster with solid and with laminated poles; that is, a machine with series field winding, inserted in the main circuit of a feeder, for the purpose of introducing into the circuit a voltage propor- tional to the load, and thus to compensate for the increasing drop of voltage with increase of load.

Due to the production of eddy currents in the solid iron of the field magnetic circuit, the magnetic flux density is not uniform throughout the whole field section during a change of the mag- netic field, since the outer shell of the field iron is magnetized by the field coil only, while the central part of the iron is acted upon by the impressed m.m.f. of the field coil and the m.m.f. of the

, eddy currents in the outer part of the iron, and the change of magnetic flux density in the interior thus lags behind that of the outside of the iron. As result hereof the eddy currents in the different layers of the structure differ in intensity and in phase.

A complete investigation of the distribution of magnetism in this case leads to a transient phenom- enon in space, and is discussed in

Section III. For the present purpose, where the total m.m.f. of the eddy currents is small compared with that TIN of the main field, we can approxi- . mate the effect of eddy currents in the iron by a closed circuit second- ary conductor, that is, can assume uniform intensity and phase of _ - secondary currents in an outer layer Fi8- 40. _ Section of a mag- of the iron, that is, consider the outer netic elreult- layer of the iron, up to a certain depth, as a closed circuit secondary.

Let Fig. 40 represent a section of the magnetic circuit of the machine, and assume uniform flux density. If ® = the total magnetic flux, /,= the radius of the field section, then at a distance | from the center, the magnetic flux enclosed by a

2 circle with radius [ is (;) ®, and the e.m.f. generated in the

MUTUAL INDUCTANCE 157 . ‘ l 3 zone at distance / from the center is proportional to (7) ®, that 2 ise =@ (7) ®. The current density of the eddy currents in this zone, which has the length 2 zl, is therefore proportional to 73 orist = n°. This current density acts as a m.m.f. upon | " 2 the space enclosed by it, that is, upon (7) of the total field section, and the magnetic reaction of the secondary current at | 2 distance / from the center therefore is proportional to i(?) , or is ¥ -F ®, and therefore the total magnetic reaction of the eddy currents is |

1. c \ = Fdl=-®., : On | J dl Z At the outer periphery of the field iron, the generated e.m.f. is e, = a®, the current density therefore 7, = 74, and the | magnetic reaction F, -+ ®, and therefore . | y= 4 F 5 that is, the magnetic reaction of the eddy currents, assuming uniform flux density in the field poles, is the same as that of the currents produced in a closed circuit of a thickness £, or one- fourth the depth of the pole iron, of the material of the field pole and surrounding the field pole, that is, fully induced and fully

• magnetizing. The eddy currents in the solid material of the field poles thus can be represented by a closed secondary circuit of depth 4 surrounding the field poles. : The magnitude of the depth of the field copper on the spools

158 TRANSIENT PHENOMENA is probably about one-fourth the depth of the field poles. Assum- ing then the width of the band of iron which represents the eddy current circuit as about twice the width of the field coils

• —since eddy currents are produced also in the yoke of the machine, etc. — and the conductivity of the iron as about 0.1 that of the field copper, the effective resistance of the eddy current circuit, reduced to the field circuit, approximates five times that of the field circuit.

Hence, if r, = resistance of main field winding, r, = 5r, = resistance of the secondary short circuit which represents the eddy currents.

Since the eddy currents extend beyond the space covered by the field poles, and considerably down into the iron, the self- inductance of the eddy current circuit is considerably greater than its mutual inductance with the main field circuit, and thus may be assumed as twice the latter.

91. As example, consider a 200-kw. series booster covering the range of voltage from 0 to 200, that is, giving a full load value of 1000 amperes at 200 volts. Making the assumptions set forth in the preceding paragraph, the following constants are taken: the armature resistance = 0.008 ohms and the series field winding resistance = 0.004 ohm; hence, the short circuit — or eddy current resistance —r, = 0.02ohm. Further- more let M = 900 X 107° henry = mutual inductance between main field and short-circuited secondary; hence, 2, = 0.34 ohm = mutual reactance, and therefore, assuming a leakage flux of the secondary equal to the main flux, L, = 1800 x 10-° henry and x, = 0.68 ohm.

The booster is inserted into a constant potential circuit of 550 volts, so as to raise the voltage from 550 volts no load to 750 volts at 1000 amperes.

The total resistance of the circuit at full load, including main circuit and booster, therefore is r = 0.75 ohm.

The inductance of the external circuit may be assumed as L = 4500 X 107° henrys; hence, the reactance at f = 60 cycles per sec. is z = 1.7 ohms. The impressed e.m.f. of the circuit is e = 550 + e, & being the e.m.f. generated in the booster. Since at no load, for 7 = 0, e? = 0, and at full load, for 1 = 1000, e’ = 200, assuming a straight line magnetic characteristic or saturation curve, that is, assuming the effect of magnetic satura-

MUTUAL INDUCTANCE 159 | tion as negligible within the working range of the booster, we have e = 550 + 0.2 (2 + 2,). This gives the following constants: ‘ ‘ Main Circuit. Eddy Current Circuit. a _Current... ......,...] t amp. t, amp. | Impressed e.m.f......| e=550+0.2 (t+7,) volts.| 0 volts. Resistance...........] r==0.75 ohm. r,=0.02 ohm. ' Inductance. .........| L=4500< 10~ henrys. L =1800x 10-* henrys. | ; Reactance...........] 2=1.7 ohms, 2,=0.68 ohm. | : Mutual inductance. .. M-= 900 X 10 henrys. ' Mutual reactance... Im = 0.34 ohm. J This gives the differential equations of the problem as di di 4-7) ory ; _ at 550 — 0.557 + 0.21, — 1.7 7} 0.34 70 0 (39) | and . dy di, | 0.02 7, + 0.34 7) + 0.68 a 0. (40) | . . Adding 2 times (39) to (40) gives 1100 — 1.17 + 0.422, — 3.06 2 = 0, (41) or t, = 7.28 “ + 2.627 — 2620, (42) herefrom: 0.027, = 0.1456 “ + 0.0524 7 — 52.4, (43) di, Pt di and 0.685 = 4,95 w + 1.78 ’ (44) substituting the last two equations into (40), a di . 2 7 + 0.458 7) + 0.0106 * — 10.6 = 0. (45) If t=t, + Ae~™, (46) then Ae~™(a? — 0.458 a + 0.0106) + 0.0106 *, — 10.6 = 0.

160 TRANSIENT PHENOMENA As transient and permanent terms must each equal zero, 7% = 1000 and a’ — 0.458 a + 0.0106 = 0, wherefrom a = 0.229 + 0.205; the roots are a, = 0.024 and a, = 0.434; then we have t = 1000 + Aye"? + Ay ***? (47) and i, = 245 Aye 9 — 0,55 Aye Oe, (48) With terminal conditions 6 = 0, 7 = 0, and 2, = 0, A, = — 183 and A, = — 817. ' If 0 = 22ft = 377.51, we have t = 1000 — 183 6 © 97! — 817 eo 14 t i, = — 450 fereert — grime} (49) and e = 750 —127 #97! 737, = | | | L_| | | {Laminated Poles | ee CS ae Nha | 1 Seed FeeEEEE EERE EEE 4 .U2 5 LO4 Beconde 1.07 03 0,09 a10 Fig. 41. Building up of feeder voltage by series booster. . In the absence of a secondary circuit, or with laminated field poles, equation (39) would assume the form 7, = 0, or . -- di 550 + 0.2% = 0.751 4 7S; (50) di . hence, — = 0.323 (1000 — 2) dé and i = 1000 (1 — 7 °™4); or t = 1000 (1 — «7 ™4) ) and e = 750 — 200 e—'!; (51 . | |

, MUTUAL INDUCTANCE 161 that is, the e.m.f., e, approaches final conditions at a more rapid rate.

Fig. 41 shows the curves of the e.m.f., e, for the two conditions, namely, solid field poles, (49), and laminated field poles, (51). (B) Mutual inductance in circuits containing self-inductance and capacity. 92. The general equations of such a pair of circuits, (3) and (4), differentiated to eliminate the integral give de . ad ig ay 77] =2,,+7. 5 + ae + tag (52) and de . di Cia) at Gime + Ge + SP + tm (53) and the potential differences at the condensers, from (3) and (4), are ., . di di e,/ = za f i,d0 =e, = ri, — 2, = te (54) and - . . di di ef = ty find? =e iy GE tm Gt (65) If now the impressed e.m.fs., e, and e,, contain no transient term, that is, if the transient values of currents 7, and 7, exert no appreciable reaction on the source of e.m.f., and if 2,’ and 7,’ are the permanent terms of current, then, substituting 7,’ and i,’ in equations (52) and (53), and subtracting the result of this substitution from (52) and (53), gives the equations of the transient terms of the currents 7, and 7,, thus: . di Cig} Pi. O=2,i,+7, = +2, mn yh tase (56) and . di dr. Pi . 0 = tat, + 1a + tage + tm ae (57) | . de, de, If the impressed e.m.fs., e, and e,, are constant, 77 and 7

162 TRANSIENT PHENOMENA equal zero, and equations (52) and (53) assume the form (56) and (57); that is, equations (56) and (57) are the differential equations of the transient terms, for the general case of any e.m.fs., e, and e,, which have no transient terms, and are the general differential equations of the case of constant impressed e.m.fs., e, and e,. From (56) it follows that Fi . at Cia Im Tan =~ Fey 7M 77 ae ° (58) Differentiating equation (57) twice, and substituting therein (58), gives dy at et Xl om + (7,2, + 7,2,) ie + (%,,0_ + Lq_l, + 17, — Im) oR dt .

• (%p,7, + Lex) 3 + 2,1 = 0. (59) This is a differential equation of fourth order, symmetrical in r,2,t,, and r,z,2,,, which therefore applies to both currents, v,and7,. ° The expressions of the two currents 7, and 7, therefore differ only by their integration constants, as determined by the ter- minal conditions. Equation (59) is integrated by i= Ac™ (60) and substituting (60) in (59) gives for the determination of the exponent a the quartic equation z,2t,a° — (r,t, + 742,) a + (2,2, + 4,2, + 7,7, — Im”) @ — (4,7, + Lar) 4 + 2%, = 9, or a — (4 72) a + (4 fo Oe te) @ Z, 2, QZ, ot, 2&2, Lt, — (2 + eg + le = 0, (61) T,2, TT, TL, The solution of this quartic equation gives four values of a, and thus gives t= Ayn + Agen + Age + Agen ™, (62)

MUTUAL INDUCTANCE 163

The roots, a, may be real, or two real and two imaginary, or ; all imaginary, and the solution of the equation by approxima- tion therefore is difficult.

In the most important case, where the resistance, 7, is small compared with the reactances z and x, — and which is the only case where the transient terms are prominent in intensity and duration, and therefore of interest — as in the transformer and the induction coil or Ruhmkorff coil, the equation (61) can be solved by a simple approximation.

In this case, the roots, a, are two pairs of conjugate imaginary numbers, and the phenomenon oscillatory.

The real components of the roots, a, must be positive, since the exponential «~~ must decrease with increasing 0.

The four roots thus can be written:

. a, =a, — 98, a,=a,+ 78, 2 1 : 1 (63) a, =a, — jf, a,=a,+ 98, where a and f are positive numbers,

In the equation (61), the coefficients of a* and a are small, since they contain the resistances as factor, and this equation thus can be approximated by

2 at + (+ lat + 8 =; (64) Z, Lt, U4L L,2, hence, e=- s(2+% -Z)+ V (24 fy _ tat Lata), : 2(\z, 2, 2,2. Xt, Lye zt, )’ (65) that is, a? is negative, having two roots, b,=—8 and 6, = — 8. This gives the four imaginary roots of @ as first approximation: a= + | | . 66

• 78; (66)

164 TRANSIENT PHENOMENA If a,, a,,.@,, a, are the four roots of equation (61), this equation can be written f(a) = (a— a,) (a — a,) (a — a,) @ — a,) = 0; or, substituting (63), F(a) = {(@— a,) + BY} {(@— a, + B"} = 0, (67) and comparing (67) with (61) gives as coefficients of a’ and of a, T r 2 =t42 (1 + 2) qT, + Ty and (68) 2 2) — Tele t Tol 2 (a@,8,7 + a,8,”) am and since £,? and 8,? are given by (65) and (66) as roots of equa- tion (64), @,, a, 8,, 8,, and hereby the four roots a,, a,, a,, a, of equation (61) are approximated by (64), (65), (66), (68).

The integration constants A,, A,, As, A, now follow from the terminal conditions.

93. As an example may be considered the operation of an inductorium, or Ruhmkorff coil, by make and break of a direct- current battery circuit, with a condenser shunting the break, in

the usual manner.

Let e, = 10 volts = impressed emf.; 7, =04 ohm = resistance of primary circuit, giving a current, at closed circuit and in stationary condition, of 1,= 25 amp.; 7,= 0.2 ohm = resistance of secondary circuit, reduced to the primary by the square of the ratio of primary + secondary turns; z,= 10 ohms = primary inductive reactance; z, = 10 ohms = secondary inductive reactance, reduced to primary; z, = 8 ohms = mutual inductive reactance; z,, = 4000 ohms = primary condensive reactance of the condenser shunting the break of the interrupter in the battery circuit, and 2, = 6000 ohms = secondary condensive reactance, due to the capacity of the terminals and the high tension winding.

Substituting these values, we have _

. e, = 10 volts 1, = 25 amp. r,=04ohm 2z,=10ohms~ 1, = 4000 ohms (69) r,=0.2ohm 2,=10ohms_ 2, = 6000 ohms Im = 8 ohms.

MUTUAL INDUCTANCE 165 | These values in equation (61) give f(@) = at — 0.06 a® + 999.36 a? — 32a + 240,000 = 0, (70) and in equation (64) they give F, (a) = at + 999.36 a? + 240,000 = 0 . and @ = — (499.68 + 98.39) = — 598.07, or = — 401.29; hence, B, = 24.5 and B, = 20.0. From (68) it follows that a, + a, = 0.03 Oo and 598.07 a, + 401.29 a, = 16; hence, a, = 0.02 and a, = 0.01. Introducing for the exponentials with imaginary exponents the trigonometric functions gives | a7, = °° 1A, cos 24.50 + A, sin 24.5 0}

• e~°"* {B, cos 206+ B, sin 20 6} and (71) i, = 67° {C, cos 24.54 + C, sin 24.5 0} |
• «~*"? {D, cos 200+ D, sin 20 6}, . | where the constants C and D depend upon A and B by equations (56), (57), or (58), thus: . Substituting (71) into (58), gee y 40003, + 0.4 + 10.7 = 0 (58) | gives an identity, from which, by equating the coefficients of |

• cog b# and e~™ sin b6 to zero, result four equations: | A, = — 2.4C, — 0.004C,, | A, = — 24C, + 0.004 C,, B, = — 800D, + 0.8 D,, B, = + 800 D, + 0.8 D,, | .

166 TRANSIENT PHENOMENA

or with sufficient approximation, A, =-— 24C,, A, = —2A4C,, 7 B, = — 800 D, B, = + 800D,:

hence,

. a, = — 2.467%" 9 {C, cos 24.56 + C, sin 24.5 6}

— 800 <~°"? {D, cos 20 — D, sin 206}, (73) and substituting (71) and (73) in the equations of the condenser potential, (54) and (55), gives

e,/ = 10 + 392e~°"° {C, cos 24.56 — C, sin 24.5 6} — 160,000 «°'"* { D, cos 204 + D, sin 20 6} and (74) ey, = 225e—°* {C, cos 24.56 — C, sin 24.56} — 128,000 e °°"? { D, cos 204 + D, sin 20 6}. 94. Substituting now the terminal conditions of the circuit: | At the moment where the interrupter opens the primary circuit the current in this circuit is 7, = <2 = 25 amp. The condenser in the primary circuit, which is shunted across the break, was short-circuited before the break, hence of zero poten- tial difference. The secondary circuit was dead. This then gives the conditions @ = 0; 1, = 25, i, = 0, e,/ = 0, and e,/ = 0. ~ Substituting these values in equations (71), (73), (74) gives 25 = — 2.4C, — 800 D,, 0 = Ci, + D,, 0 = 10 + 392C, — 160,000 D,, 0 = 225 C,, — 128,000 D,; hence, C, = + 0.158 x 10° = 0, ' C, = — 0.09, D, = — 0.158 x 107° =0, D, = — 0.0312, |

| MUTUAL INDUCTANCE 167 and 1, = 0.216 e~ °? sin 24.50 + 256—%" cos 200, , 1, = — 0.09 e~°® sin 24.5 6 — 0.0312 e~°® sin 20 0, e, =10— 35.367 °* cos 24.50 + e—°® (25.3 cos 20 8 (75)

• 5000 sin 20 4), €,/ = 20.2 °° cos 24.50 — e~°™* (20.2 cos 20 0 — 4000 sin 206). Approximately therefore we have i, = 252" cos 20 8, i, = — {0.00 «°? sin 24.5 0 + 0.0312 e™ ® sin 20 6}, e/ = 5000 e~" sin 208, e,! = 4000 ¢~ °%¢ sin 20 6. The two frequencies of oscillation are 1470 and 1200 cycles per sec., hence rather low. | The secondary terminal voltage has a maximum of nearly 4000, reduced to the primary, or 400 times as large as corre- _ ; sponds to the ratio of turns. In this particular instance, the frequency 1470 is nearly suppressed, and the main oscillation is of the frequency 1200.

CHAPTER XI. GENERAL SYSTEM OF CIRCUITS,

(A) Circuits containing resistance and inductance only.

95. Let, upon a general system or network of circuits con- nected with each other directly or inductively, and containing resistance and inductance, but no capacity, a system of e.m.fs., e, be impressed. These e.m.fs. may be of any frequency or wave shape, or may be continuous or anything else, but are supposed to be given by their equations. They may be free of

transient terms, or may contain transient terms depending upon the currents in the system. In the latter case, the dependency ‘of the e.m.f. upon the currents must obviously be given. . Then, in each branch circuit, . adi dis e-ri- Lo —- Dems = 0, (1) where e = total impressed e.m.f.; r = resistance; Z = induc- tance, of the circuit or branch of circuit traversed by current 7, and M, = mutual inductance of this circuit with any circuit in inductive relation thereto and traversed by current 7. : The currents in the different branch circuits of the system depend upon each other by Kirchhoff’s law, . yi=0 (2) at every branching point of the system.

By equation (2) many of the currents can be eliminated by expressing them in terms of the other currents, but a certain number of independent currents are left.

Let n = the number of independent currents, denoting these currents by 7,, where « = 1, 2,...n. (3)

Usually, from physical considerations, the number of inde-

pendent currents of the system, n, can immediately be given. 168

GENERAL SYSTEM OF CIRCUITS 169 : For these n currents 7,, n independent differential equations | of form (1) can be written down, between the impressed e.m.fs. | e, or their combinations, and currents which are expressed by | the n independent currents 7,. They are given by applying equation (1) to a closed circuit or ring in the system. | These equations are of the form ;

• . - di, eq — “ bf4,— « cé— = 0, . I ps py dt (4) . where q=1,2,...n, where the n? coefficients b,? are of the dimension of resistance (5) and the n? coefficients c,? of the dimension of inductance. These n simultaneous differential equations of n variables 1, are integrated by the equations 1,= 1! + > Afe*!, (6) 1 where 7,’ is the stationary value of current 7,, reached fort = 0. | Substituting (6) in (4) gives n n -y n m n | ca Ye bt id — Se ok — Se bY Ast De ct | 1 1 dt 1 1 1 di a Ase = 0. (7) 1 Fort = o, this equation becomes e _>. boil — D+ ote 20. (8) q . ne « 7 g dt These n equations (8) determine the stationary components of the n currents, 7,’. Subtracting (8) from (7) gives, for the transient components of currents 7,, i = Di Agen, (9) : 1 | the n equations

: bf > Afe~@ 7 >: c! >: a,A,*e~ % = 0. (10) 1 1 1 1

a 170 TRANSIENT PHENOMENA Reversing the order of summation in (10) gives . Di eo! De AS OS — ae,%) = 0. (11) 1 1 "Then equations (11) must be identities, that is, the coefficients of «~** must individually disappear. Each equation (11) thus . gives m equations between the constants u, A, b, c, fort = 1, 2,...m, and since n equations (11) exist, we get altogether mn equations of the form \ >: A; (b,! ~ a¢,’) = 0,

1 (12)

where .

q=1,2,3,...n and 7 =1,2,3,...m.

In addition hereto, the n terminal conditions, or values of current 7,” for t = 0: 2,°, give by substitution in (9) n further equations,

if = Ay, (13) 1

There thus exist (mn + n) equations for the determination of the mn constants A;“ and the m constants a,, or altogether (mn + m) constants. That is,

m=n (14) and i, = 4 + i Aken %, (15) 1 | n | where >: A; (6,2 — ac’) = 0; (16) 1 Di Af = 4; (17) 1 : q=1,2,...n, «= 1,2,...2, (18) and t1=1,2,...0n.

|

• GENERAL SYSTEM OF CIRCUITS 171 Each of the n sets of n linear homogeneous equations in A; (16) which contains the same index 7 gives by elimination of A,“ the same determinant: ; b,'—ae,', b?—agc,’, bS—ae*...b,"—ac," b,'-ag,', b2—ac?, b,—ac,...b,"—ae," . [b—a¢.4| =|b—ae,', b2—ae?, bs—ae? .. . b,"—a,c,"|=0- (19) iDy Cn’; b,’—a¢,’, b, ac,” one b,"—ac,” | Thus the n values of a; are the n roots of the equation of nth degree (19), and determined by solving this equation. Substituting these n values of a; in the equations (16) gives n’ linear homogeneous equations in A,“, of which n (n — 1) are independent equations, and these n (n — 1) independent equa- tions together with the n equations (17) give the n? linear equations required for the determination of the n? con- stants A;*. The problem of determining the equations of the phenomena in starting, or in any other way changing the circuit conditions, ' in a general system containing only resistance and inductance, | with n independent currents and such impressed e.m.fs., ¢,, _ that the equations of stationary condition, | if =f. 0, ean be solved, still depends upon the solution of an equation of nth degree, in the exponents a; of the exponential functions which represent the transient term.

96. As an example of the application of this method may be considered the following case, sketched diagrammatically in Fig. 42: An alternator of e.m.f. E cos (9 — 6,) feeds over resistance r, the primary of a transformer of mutual reactance z,,. The secondary of this transformer feeds over resistances 7, and r, the primary of a second transformer of mutual reactance Zm,, and the secondary of this second transformer is closed by resist- ance r,. Across the circuit between the two transformers and the two resistances r, and rs, is connected a continuous-current

172 TRANSIENT PHENOMENA e.m.f., €,, as a battery, in series with an inductive reactance z. The transformers obviously must be such as not to be saturated magnetically by the component of continuous current which _ traverses them, must for instance be open core transformers. . i, i, P is is VE con (8-6) cC> ——? c> (©) —, rs — % Cc _ CS O Zn Tmo : ry "s ", . , Fig. 42. Alternating-current circuit containing mutual and self-inductive reactance, resistance and continuous e.m.f. Let 1,, 15, U, ts, % = currents in the different circuits; then, at the dividing point P, by equation (2) we have ; 1, + 1, — 1, = 0; hence, 1 = 1, — 15, (20) leaving four independent currents 7,, 7,, 1,, ¢,. This gives four equations (4): . di E cos (6 — 8.) — ri, — tmae = 0, . di di, di. = 6) — Pip — tm G+ 2 (GE 3) = 0, , di di, di (21) & — Ti, — Ame Gt — (FP -2)- 0, and . a —_ Tare _- Xm Gf = 0. If now 1,’, 7,’, 7’,, 7,’ are the permanent terms of current, by substituting these into (21) and subtraction, the equations of the transient terms rearranged are:

GENERAL SYSTEM OF CIRCUITS 173 q: «= 1 2 3 4 . di } 1 rity tinge =0, di ., dt di, 2 magtnitegh mig =O . . . ) di ., dt di 3 —2z ptsttae + Imam = 0, diy, ; “4 . im +P yl, =0. These equations integrated by 4 . i ee (23) 1 | give for the determination of the exponents a, the determinant (19): rT, — aly, 0 0 | — Qn T,—at ar 0 _o. | 0 ax fy — ax —ar,,| — 0; (24) 0 0 = Alin, TY | . | or, resolved, ; f = Gn? Lmg + BL (Lm ye + Ling ,) — @ (Lm Pry + Let {Ts) | — QIr,Ty(T, + 73) + 77 ry = 0. (25) Assuming now the numerical values, | r= 1 Im = 10 r,= 1 Im, = 10 on r= 1 r=100[ | (26) | r, = 10 ; . equation (25) gives | f=a'+ lla —0.11@ — 0.24 + 0.001 = 0. (27) | The sixteen coefficients, A‘, 1 =1,2,3,4, kk =1,2,3,4, : are now determined by the 16 independent linear equations (12) | and (13). | | | i a

174 TRANSIENT PHENOMENA (B) Circuits containing resistance, self-inductance, mutual in- ductance and capacity. ; 97. The general method of dealing with such a system is the same as in (A). | Kirchhoff’s equation (1) is of the form | . di a a . e-ri-LG-Yi Mag fia =o. (28) | Eliminating now all the currents which can be expressed _ in terms of other currents, by means of equation (2), leaves n independent currents: iy © =1,2,...0. Substituting these currents 7, in equations (28) gives n inde- . pendent equations of the form . n n a n | — De belie — De 8 SD gs fide =0. eT BAM Se et a & | Resolving these equations for f 1, dt gives | ,.1 f¢. . di ef =a fied = Yor Yoi+ Bes (30) as the equations of the potential differences at the condensers. Differentiating (29) gives de, 2 . 2 a, Pi, a ee where q=1,2,...n. By the same reasoning as before, the solution of these equa- tions (31) can be split into two components, a permanent term, =f, (32) and a transient term, which disappears for t = 0, and is given by the n simultaneous differential equations of second order, thus : eG. di, ae " Ja re + 64 + 625 = 0. (33) x ? dt dt?

| GENERAL SYSTEM OF CIRCUITS 175

These equations are integrated by | ie = Dy Aster. (34)

1 e Substituting (34) in (33) gives . nom . « —at _

| a a Jat = abt + arc.t{, (85)

where q=1,2,...n,

«e=1,2,...0, (36)

and 4=1,2,...m.

Reversing in these n equations the order of summation,

f mm n

t eat > A} ‘a8 ~ ab! + ac,4 = 0, (37) 1 1 and this gives, as identity, the mn equations for the determina- tion of the constants: . : A; \ ae _ ab! + are | = 0, . 1 (38) where g=1,2,...n and 7=1,2,...m.

In addition to these mn equations (38), two sets of terminal conditions exist, depending respectively on the instantaneous _ current and the instantaneous condenser potential at the moment

of start.

The current is | ig = + Di Asem (39) ' 1

and the condenser potential of the circuit q is n n n a e, = >: a fi. dt = ey — >D «bit, ~ Dre! oe (40) 1 . 1 1 dt hence, for ¢ = 0, toatl + Dias, (41) 1

176 TRANSIENT PHENOMENA | where «e=1,2,...n,

• . ° di? . and . Cy = & — 2 b,.1,° — 2" cf 1’ (42) ’ where, g=1,2...n; or, substituting (39) in (40), and then putting ¢ = 0,
• . di,’ ) a =[¢ — Sasi — o¢ Se | Plan Be a $44 — > Di ASO — ae). (43) 1 1 As seen, in (41) and (43), the first term is the instantaneous value of the permanent current 7’, and condenser potential e,’. These two sets of n equations each, given by the terminal conditions of the current, ?’, = i,° (42), and condenser potential, e,/ = €,° (43), together with the mn equations (38), give a total of (mn + 2n) equations for the determination of the mn con- stants A,* and the m constants a,, that is, a total of (mn + m) constants. . From mm+2n=mn+m it follows that m=2n. (44) We have, then, 2 » constants, a,, giving the coefficients in the exponents of the 2n exponential transient terms, and 2 n? coefficients, A,*, and for their determination 2? equations,

As (ge! ~ a,b! + a,7c,") = 0, (45) e 1 . n equations, di At = 4, (46) 1 and n equations, n 2n Dr Di At - ae,2) = ke, (47) | 1 1 f

| GENERAL SYSTEM OF CIRCUITS 177 | where be=[e —- deg fica] ; 48 @ a p2 9 e t=o ¢ ) or the difference between the condenser potential required by the permanent term and the actual condenser potential at time t = 0, where yg = 1,2,3,...2, « = 1,2,3,...n, (49) _ and t= 1,2,3,...2n. Eliminating A,‘ from the equations (45) gives for each of the 2n sets of n equations which have the same a, the determinant: 1G? 7 ab! + a?c,"|| = ‘gab, +aze), 9 —ab,+a7c/,...9,"—ab,"+a7c," ‘G2! —a,b,! +aZc,', 9s —ab?+azc,’, see 92" —ab,"+a7c," , ; gs' —a,b,' +a7c,', gs” — a,b,’ + 4,7c,’, . . . gs" — abs" +.4,7c5"_| =0.(50) In" —a,b,' +a,7c,.', Jn —a,” +a7Ze,’, ce Jn” _ ap,” vee The 2 n values of a, thus are the roots of an equation of 2 nth order. . Substituting these values of a, in equations (45), (46), (47), leaves 2 (n — 1) independent equations (45) and 27 inde- ' pendent equations (46) and (47), or a total of 2 n? linear equa- tions, for the determination of the 2 n? constants A*, which now can easily be solved. The roots of equation (50) may either be real or may be com- plex imaginary, and in the latter case each pair of conjugate roots gives by elimination of the imaginary form an electric oscillation. That is, the solution of the problem of n independent circuits leads to n transient terms, each of which may be either an oscillation or a pair of exponential functions. 98. The preceding discussion gives the general method of the determination of the transient phenomena occurring in any system or net work of circuits containing resistances, self-induc-

178 TRANSIENT PHENQMENA tances and mutual inductances and capacities, and impressed and counter e.m.fs. of any frequency or wave shape, alternating or con- tinuous.

It presupposes, however,

(1) That the solution of the system for the permanent terms of currents and e.m.fs. is given.

(2) That, if the impressed e.m.fs: contain transient terms depending upon the currents in the system, these transient

terms of impressed or counter e.m.fs. are given as linear functions of the currents or of their differential coefficients, that is, the rate of change of the currents.

(3) That resistance, inductance, and capacity are constant quantities, and for instance magnetic saturation does not appear.

The determination of the transient terms requires the solution of an equation of 2 nth degree, which is lowered by one degree for every independent circuit which contains no capacity.

Thus, for instance, a divided circuit having capacity in either branch leads to a quartic equation. A transmission line loaded with inductive or non-inductive load, when representing the capacity of the line by a condenser shunted across its middle, leads to a cubic equation.

i CHAPTER XII. MAGNETIC SATURATION AND HYSTERESIS IN ALTERNAT- ING-CURRENT CIRCUITS. . 99. If an alternating e.m.f. is impressed upon a circuit con- taining resistance and inductance, the current and thereby the magnetic flux produced by the current immediately assume their final or permanent values only in case the circuit is closed | at that point of the e.m.f. wave at which the permanent current is zero. Closing the circuit at any other point of the e.m.f. wave produces a transient term of current and of magnetic flux. So for instance, if the circuit is closed when the current 7 should have its negative maximum value — /,, and therefore the magnetic flux and the magnetic flux density also be at their negative maximum value — ®, and — @,—that is, in an inductive circuit, near the zero value of the decreasing e.m_f. wave — during the first half wave of e.m.f. the magnetic flux, which generates the counter e.m.f., should vary from — ®, to

• ®,, or by 2 ®,; hence, starting with 0, to generate the same counter e.m.f., it must rise to + 2 ®,, that is, twice its permanent value, and so the current 7 also rises, at constant inductance L, from zero to twice its maximum permanent value, 2/,. Since the e.m.f. consumed by the current during the variation from 0 to 2/, is greater than during the normal variation from — /, to + I,, less e.m.f. is to be generatéd by the change of magnetic flux, that is, the magnetic flux does not quite rise to 2 ®,, but remains below this value the more, the higher the resistance of the circuit. During the next half wave the e.m.f. has reversed, but the current is still mostly in the previous direction, and the generated e.m.f. thus must give the resistance drop, that is, the total variation of magnetic flux must be greater than 2 ®,, the more, the higher the resistance. That is, starting at a value . somewhat below 2,, it decreases below zero, and reaches a negative value. During the third half wave the magnetic flux, . starting not at zero as in the first half wave, but at a negative 179 :

180 TRANSIENT PHENOMENA