Theory and Calculation of Transient Electric Phenomena and Oscillations — part 6 of 20

*~phenomena in high potential circuits containing inductance and capacity are the electric oscillations produced by a change of circuit conditions, as starting, opening circuit, etc.

2. These phenomena are essentially independent of the fre- quency and the wave shape of the impressed e.m.f., but de- pend upon the conditions under which the circuit is changed, as the manner of change and the point of the impressed e.m.f. and current wave at which the change occurs.

3. The electric oscillations cccurring in connecting a trans- mission line to the generator are not of dangerous potential, but the oscillations produced by opening the transmission circuit under load may reach destructive voltages, and the oscillations caused by interrupting a short-circuit are liable to reach voltages far beyond the strength of any insulation. Thus special pre- cautions should be taken in opening ‘a high potential circuit under load. But the most dangerous phenomenon is a low resistance short-circuit in open space.

4. The voltages produced by the oscillations in open-circuiting a transmission line under load or under short-circuit are mod- erate if the opening of the circuit occurs at a certain point of the e.m.f. wave. This point approximately coincides with the moment of zero current. ;

. CHAPTER IX. | DIVIDED CIRCUIT. 72. A circuit consisting of two branches or multiple circuits 1 and 2 may be supplied, over a line or circuit 3, with an impressed | e.m.f., é,. . Let, in such a circuit, shown diagrammatically in Fig. 31, r,, L,, C, and r,, L,, C, = resistance, inductance, and capacity, respectively, of the two branch circuits 1 and 2; 1r,, L,, Cy= Ce | X% ‘fi | | om rh lL, Ly oO, | | c, Fig. 81. Divided circuit. resistance, inductance, and capacity of the undivided part of the _ circuit, 3. Furthermore let e = potential difference at terminals of branch circuits 1 and 2, 7, and 7, respectively = currents in branch circuits 1 and 2, and 7, = current in undivided part of circuit, 3. Then 1, =%4, +1, (1) and e.m.f. at the terminals of circuit 1 is . a, 1 ¢. ear t Lata fia (2) of circuit 2 is . a, 1 ¢. é art LG tg Sint (3) 121

122 TRANSIENT PHENOMENA and of circuit 3 is ; eet ryt Lette figae (4) 0 0o"3 0 at C, so Instead of the inductances, L, and capacities, C, it is usually preferable, even in direct-current circuits, to introduce the reactances, x = 22fL = inductive reactance, xz, = es = con- densive reactance, referred to a standard frequency, such as Sf = 60 cycles per second. Instead of the time ¢, then, an angle 6 = 2zxft (5) is introduced, and then we have da ox dh dé dy L— So Oo ee ET dt 2xf d6 dt dé and 1 i (6) GJ tat = arf, yan a J ae, dt since q = 2af.

Hereby resistance, inductance, and capacity are expressed in the same units, ohms.

Time is expressed by an angle @ so that 360 degrees correspond to as of a second, and the time effects thus are directly com- parable with the phenomena on a 60-cycle circuit.

A better conception of the size or magnitude of inductance and capacity is secured. Since inductance and capacity are mostly observed and of importance in alternating-current cir- cuits, a reactor having an inductive reactance of + ohms and 2 amperes conveys to the engineer a more definite meaning as regards size: it has a volt-ampere capacity of ?z, that is, the approximate size of a transformer of half this capacity, or of a 2

-watt transformer. A reactor having an inductance of L henrys and 7 amperes, however, conveys very little meaning to

DIVIDED CIRCUIT 123 the engineer who is mainly familiar with the effect of inductance in alternating-current circuits. Substituting therefore (5) and (6) in equations (2), (8), (4), gives the e.m.f. in circuit 1 as ean teat t x, fia; (7) | in circuit 2 as e=ry,t+ 2,93 + ty f ty; (8) in circuit 3 as . di . y= 6 + Taly + ty ot + ty f i, a8; (9) hence, the potential differences at the condenser terminals are €= Xe, fica =e Tt — 2, (10) a= 1, fis = 6 — 14, — 2 a (11) and = ty fi, dd = 6, — € — rly — 2S (12) 3 CO 3 0 o”s i) ao Differentiating equations (7), (8), and (9), to eliminate the integral, gives as differential equations of the divided circuit:

• Pi, di . de . x, - + "7 + 2,0, = iB’ (13) . Pi, dt . de , 25 p +r, 7 + tot, = 7 (14) Pi dy. . de, de and Taam t Toqy t Labs = 9 OH" (15) Subtracting (14) from (13) gives ; Ps $ 1S 1 @ ' ae

124 TRANSIENT PHENOMENA Multiplying (15) by 2, and adding thereto (13) and (14), gives, by substituting (1), 74, = 7, + 2,, a di . (22, + 2,) “at Qr tr) + Qty + 2), + ar. a . |. de (22, + 2,) at (27, + 1;) a + (24, + %)ip = 2 F2- (17) These two differential equations (16) and (17) are integrated by the functions i,=t/ + Ae ™ and (18) i, =t + Ag®, where 7,’ and 7,’ are the permanent values of current, and 1,” = Aye and i,” = A,™ are the transient current terms. Substituting (18) in (16) and (17) gives Fi! a,’ , Pi! di,’
(:, 77 + "wo +1,3/) — (2. + ap + Tete ) +A“ (@x,—ar, + 2,,.) —A,e% (@z,—ar,+2,,)=0 (19) and © Pi,’ « di,’ 7 (22x, + 2,) mr + (27, + "a + (2 L + X,,)0,’ + (2 24+ 2,) Fi! ai! . a + (27, +-7,) =a + (2.4,, + t,t, + Aye” {a?(2 x, +2,) —a(2r,+7,)+ (22, + 2,)} + Age” @ (22, + 2,) de — (27, +74) + (2x4 + 4,)} = 27 (20) 73. For 6 = «, the exponential terms eliminate, and there remain the differential equations of the permanent terms 2,’ and 2,’, thus | ra) op ra 7 ; (z, sale +r, “ + z, i, - (:, o +r, or + x, i,') 20 QI) | and Pi,’ di,’ - | (2.x) + 2,) = + (27, + 7,) 7 + (24, + 2.) t,/+(2%,+ 2,) ! i rr a! . de, at (2r, + 1,) “a t (2 £,,+ Xe.) ty -25° (22) |

° DIVIDED CIRCUIT 125 The solution of these equations (21) and (22) is the usual equation of electrical engineering, giving 2,’ and 7,’ as sine waves if the e.m.f., e,, is a sine wave; giving 7,’ and 1,’ as constant quantities if e, is constant and z,, and either z,, or z,, or both vanish, and giving 7,’ and 7,’ = O if either x, or both x, and 1, differ from zero.

Subtracting (21) and (22) from (19) and (20) leaves as dif-

ferential equations of the transient terms 2,” and 1,””,

e™ {A, (az, — ar, + 2.) — A, @z, — ar, + 4,,)} =0 (28)

and

e “(A [@(22,+2,)-a(2z,+7,) + 22, + 2,)] + A, (a (24, + x.) —a(2r, + 1r,) + (2 2,, +2,,)}} = 0. (24)

Introducing a new constant B, these equations give, from (23), \

A, = B (@x, — ar, + 4,,) and (25) A, = B(@zr, — ar, + 1.,); then substituting (25) in (24) gives (@r, — ar, + 4.) (@(2 4, + 2,)-a(2r, + 7,) + (24, + 2,,)]

• (x, — ar, + x,)[@(2 2, + 2.) —a(2r, + 7,)+(2 2,
• z,,)] = 0, (26) while B remains indeterminate as integration constant.

Quartic equation (26) gives four values of a, which may be all real, or two real and two conjugate imaginary, or two pairs of conjugate imaginary roots.

Rearranged, equation (26) gives G@ (ror, + Igt, + 2,2) -@ {7 (x, + x.) tr, (a + z;)

+7, (Zot xf +o {(rory + rer, + yr.) + Teg (LZ, + 25)

• Ze, (Lot Ly) + Le, (Lot 4,)}— a {2,(7,+ 7.) + 2%, (Tot 73)

• z., (r, + r)} + (ZX, + Leghcy + L.Teq) = 0. (27)

Let a,, @,, @,, a, be the four roots of this quartic equation (27);

Wo

. 126 TRANSIENT PHENOMENA then 1,=%, +B, (a@2r,—a,r, + 2,,) 7% + B, (a72,—a,7, + 2,,)

• B, (afz,—ay,+ 2.) + B, (afz,— ag,+ z.,) e~%" (28) and 1, =1, + B, (2, —a,r, + 2,,.) -% + B, Gx, —a,r, +,,) «7
• B, (a22,— ar, + z,,) «+ B, (afz,— ay, + 2,,) e~%* (29) | where the integration constants B,, B,, B, and B, are deter- mined by the terminal conditions: the currents and condenser | potentials at zero time, 0 = 0. The quartic equation (27) usually has to be solved by approxi- . mation. .

74. Special Cases: Continuous-current divided circuit, with resistance and inductance but no capacity, e,= constant. rs L, to Ly r L, Fig. 82. Divided continuous-current circuit witbout capacity. In such a circuit, shown diagrammatically in Fig. 32, equations ; (7), (8), and (9) are greatly simplified by the absence of the integral, and we have . di e=7rt,+ zsh (30) . di e=r,,4+ 2, 7 (31) . di, and e, =etrt, t+ mst (32) | (30) and (31) combined give . . di di. . rt, — Ty, + 2 -2,33 = 0. (33) |

DIVIDED CIRCUIT 127 Substituting (1), 7, = 7, + 7,, in (32), multiplying it by 2 and adding thereto (30) and (31), gives . . di 2e,= (2r,+ 7,)1,+ (27 + 7.) 13+ (22+ x,) a di, (34)

• (2 Iyt Z,) w . Equations (33) and (34) are integrated by i, =i + Aen™ | and ; (35) 1=t + Ay”. | Substituting (35) in (33) and (34) gives (r,t, _ r1,') + eA — az,) _ A,r, _- az,)§ =0 and 2e=(2retr,) 1 + (27, +7,) 1, +e“ {A, [2 r, + 1,) | —a(2z,+ 2,))+ A,[2r, + 7,)— a (22, + 24) }. J These equations resolve into the equations of permanent state, thus rt, — 7T,./ =0 and (2r,+7,)2f +(27r, + 17,) 2,’ = 2e,. Henee, if = e3 (36) and 1! = G5? where Parr, tr, try, (37) and the transient equations having the coefficients A, (r, — az,) — A, (7, — az,) = 0 and A,(2r,+7,) —@ (22, + 2,)] + A,[(27, + 7,) —a(2z, + 2,)) =0.

128 TRANSIENT PHENOMENA Herefrom it follows that A, = B(r, — ax,) and (38) A, = B(r, — az,), and @ (xr, + rot, +2,2,) — alr, (rz, + 2,) + 7, (x, + Z,)

• 7, (to + 2,)) + (ror, + Tor, + 7,72) = 9, (39) B = indefinite. (40) Substituting the abbreviations, Lol, + Lol, + UX, = VL, rr, tr, tyr, =P, and (41) ry (rt, + 2,) + 7, (Ly + 2,) + 7, (Lo + 2,) = 2, (7, + 17,)
• 2, (r, + r,) + q, (ro + r,) = 8’, gives (39) ar —ar +r =0, (42) hence two roots, ea ¢ a, = ve and (43) s+¢ "= OB! where fg = Vs — 4 Pr. (44) The two roots of equation (42), a, and a,, are always real, since in ¢ si>4rr, as seen by substituting (41) therein. The final integral equations thus are . r. - 2-F, - etary ty =C95 + ("2 4,2) Be #* +(r,-a,2,)Bye ?* and (45) oF _#=@, _#ta, | 1, =) 5 + (7,—4,2,) Be 22 + (r,—@,2,) By ae |

DIVIDED CIRCUIT 129

B, and B, are determined by the terminal conditions, as the currents 7, and 7, at the start, 0 = 0.

Let, at zero time, or @ = 0,

i= 1,° and (46) 1, = 1,°; then, substituting in (45), we have . r.

• if =e, 3 + (r, — 4,2,) B, + (7, — @,2,) B, |. and (47) . rT 1 = 7 + (r, ~ a,Z,) B, + (r, ~ a,2,) B;; and herefrom calculate B, and B,,.

76. For instance, in a continuous-current circuit, let the impressed e.m.f.,e, = 120 volts; the resistance of the undivided part of the circuit, r, = 20 ohms; the reactance, r, = 20 ohms; the resistance of one of the branches, r, = 20 ohms; the reactance, z, = 40 ohms, and the resistance of the other branch, r, = 5 ohms, the reactance, z, = 200 ohms.

Thus one of the branches is of low resistance and high react- ance, the other of high resistance and moderate reactance.

The permanent values of the currents, (7? = 600), are

2,’ = 1 amp. and _ 1,/ = 4 amp.

(a) Assuming now the resistance r, suddenly decreased from r, = 20 ohms to r, = 15 ohms, we have the permanent values ~ of current as

a’ = 1.265 amp. and 1,/ = 5.06 amp.

The previous values of currents, and thus the values of currents at the moment of start, @ = 0, are

2,° = Lamp. and . i,° = 4 amp.

130 TRANSIENT PHENOMENA therefrom follow the equations of currents, by substitution in the preceding, , 4, = 1.265 + 0.455 27%" — 0.720 «om? and ~ i, = 5.06 — 1.038 «— ™* — 0,022 «754,

• [ wee (b) Assuming now the resistance r, suddenly raised again | from r, = 15 ohms to r, = 20 ohms, leaving everything else } the same, we have ;

‘ 2,° = 1.265 amp. and } 2,0 = 5.06 amp.; and then 7, = 1 — 0.528 e~°79 + 0.793 «0? and } i, = 4 + 1.018 °°"? + 0.042 0? (c) Assuming now the resistance r, suddenly raised from r, = 20 ohms to r, = 25 ohms, gives 1, = 0.828 — 0.374 «°°? + 0.546 -°™? and } 1, = 3.312 + 0.649 «°°? + 0.039 °™?. (d) Assuming now the resistance 7, lowered again from r, = 25 ohms to r, = 20 ohms, gives a, = 1 + 0.342 e 0807 _ 0.514 e040 and i, =4— 0.660 eg 000076 0.028 g7 OHO 76. In Fig. 33 are shown the variations of currents 2, and 2,, resultant from a sudden variation of the resistance r, from 20 to 15, back to 20, to 25, and back again to 20 ohms. As seen, the readjustment of current 7,, that is, the current in the induc- tive branch of the circuit, to its permanent condition, is very ‘slow and gradual. Current 7,, however, not only changes very rapidly with a change of 7, but overreaches greatly; that is, a decrease of r, causes 7, to increase rapidly to a temporary value | far in excess of the permanent increase, and then gradually i,

DIVIDED CIRCUIT 131 falls back to its normal, and inversely with an increase of r,. Hence, any change of the main current is greatly exaggerated in the temporary component of current 7,; a permanent change . of about 20 per cent in the total current results in a practically instantaneous change of the branch current 7,, by about 50 per cent in the present instance.

Thus, where any effect should be produced by a change of current, or of voltage, as a control of the circuit effected thereby, the action is made far more sensitive and quicker by shunting © the operating circuit 7,, of as low inductance as possible, across

Pee eee Leiria eee ee - Inductive Branch: [re{—|5 ohms! | | | { | | { | | eae 6 EERE EEE EE EEE Pu RREEEEEEEEEEEEEE EE “i PT ty eer ee al SCL pete pelea 7°) 2d Oi FP EE A Ca 6=0 © @ 0 © © 0 20 4 #O 2 4 : Fig. 88, Current in divided continuous-current circuit resulting from sudden variations in resistance. a high inductance of as low resistance as possible. The sudden and temporary excess of the change of current 7, takes care of the increased friction of rest in setting the operating mechanism in motion, and gives a quicker reaction than a mechanism operated directly by the main current.

This arrangement has been proposed for the operation of arc lamps of high arc voltage from constant potential circuits. The operating magnet, being in the circuit 7,, more or less anticipates the change of arc resistance by temporarily over- reaching.

77. The temporary increase of the voltage, e, across the branch circuit, 7,, corresponding to the temporary excess current’ of this circuit, may, however, result in harmful effects, as de- struction of measuring instruments by the temporary excess voltage.

132 TRANSIENT PHENOMENA

Let, for instance, in a circuit of impressed continuous e.m.f., €, = 600 volts, as an electric railway circuit, the resistance of the circuit equal 25 ohms, the inductive reactance 44 ohms. This gives a permanent current of 7’ = 24 amperes.

Let now a small part of the circuit, of resistance r, = 1 ohm, but including most of the reactance z, = 40 ohms — as a motor series field winding — be shunted by a voltmeter, and r, = 1000 ohms = resistance, z, = 40 ohms = reactance of the volt- meter circuit.

In permanent condition the voltmeter reads y X 600 = 24 volts, but any change of circuit condition, as a sudden decrease or increase of supply voltage e,, results in the appearance of a temporary term which may greatly increase the voltage impressed upon the voltmeter.

In this divided circuit, the constants are: undivided part of the circuit, r, = 24 ohms; x, = 4 ohms; first branch, voltmeter (practically non-inductive), r, = 1000 ohms, z, = 40 ohms; second branch, motor field, highly inductive, r, = 1 ohm, z, = 40 ohms.

(a) Assuming now the impressed e.m.f., e,, suddenly dropped from e, = 600 volts to e, = 540 volts, that is, by 10 per cent, gives the equations

1, = 0.0216 — 0.0806 «°** + 0.0830 e7*"° and %, = 21.6 + 2.407 «9? — 0.007 ~7*.

(6) Assuming now the voltage, e,, suddenly raised again from

e, = 540 volts to e, = 600 volts, gives the equations 4, = 0.024 + 0.0806 <°** — 0.0830 e"19

and | . 1, = 24 — 2.407 9 + 0.007 1".

The voltage, e, across the voltmeter, or on circuit 1, is e=ri,t+ 2 = 10001,’ F 77.9 e994 6.2 8, . r where i/= es .

DIVIDED CIRCUIT 1338 | ' Hence, in case (a), drop of impressed voltage, e,, by 10 per cent, — @ = 21.6 — 77.9 9 + 6.2 B18, and in (b), rise of impressed voltage, e = 24.0 + 77.9 6-9? — 6.2 6-819, : This voltage, e, in the two cases, is plotted in Fig. 34. As seen, during the transition of the voltmeter reading from 21.6 _ to 24.0 volts, the voltage momentarily rises to’ 95.7 volts, or $0 tee ttt ttt } 44 i |_| mth SECC @ EEE EEE CE . Tas en ore SINT Saumiau a Berit tice Lele Phe i eo ese == os Saeee ! | | Volts a= from 640|t0 600 —

) mS, a Pa a | | | | |_| PRAT tata =n Farah os a , +} }—jindustire fpperatna: Medobmy pio im oa | Voltmeter: 7;=1000 ohmis.| “,=40\ohms. Ce) cc eae st 8 4 SCO OE 8 8 Fig. 34. Voltage across inductive apparatus in series with circuit of high resistance. | four times its permanent value, and during the decrease of "permanent voltage from 24.0 to 21.6 volts the voltmeter momen- tarily reverses, going to 50.1 volts in reverse direction.

In a high voltage direct-current circuit, a voltmeter shunted . across a low resistance, if this resistance is highly inductive, is in danger of destruction by any sudden change of voltage or current in the circuit, even if the permanent value of the voltage is well within the safe range of the voltmeter.

CAPACITY SHUNTING A PART OF A CONTINUOUS-CURRENT . CIRCUIT.

78. A circuit of resistance r, and inductive reactance z, is shunted by the condensive reactance z,, and supplied over the resistance r, and the inductive reactance x, by a continuous impressed e.m.f., é, as shown diagrammatically in Fig. 35.

184 TRANSIENT PHENOMENA . In the undivided circuit, . . di, . dt eet re tis +i) +2,(Git G3): (48) In the inductive branch, . e=ri,+ 2, ot : (49) In the condenser branch, e=2, f i, a0. (50) ©, 2, %% Lo o ny Ly% Fig. 85. Suppression of pulsations in direct-current circuits by series induc- tance and shunted capacity. | Eliminating e gives, from (48) and (49), . di . di €y = (To + 7,) t, + (% + 2) + rot, + 0 FR (51) | and from (49) and (50), 2, fi, = rit oS. (62) Differentiating (52), to eliminate the integral, . di a iy = et GR (53)

DIVIDED CIRCUIT 135

Substituting (53) in (51), and rearranging,

. 1 di e= (+ 7)4 + z, (ro, + XL + x,%,) 7 dt Fi

• (7,2, + 1,2) PP + rene ; (54) a differential equation of third order. This resolves into the permanent term . — = (ry + 7,) YY, yr So hence, a n+ 7 (55) and a transient term 4” = AeW®; (56) that is, . - - e - tat + Ace = ht Ae al (57) Equation (57) substituted in (54) gives as equation of a, Z (rot) — Or, + Tt) + 2L,) +a (rer, + 17,2) — a zr, = 0, or o-o(2 4%) 4o(Mi4 eS) 2GtN) (58) oO ot, yy Lol, while A remains indefinite as integration constant.

Equation (58) has three roots, a,, a,, and a,, which either are all three real, when the phenomenon is logarithmic, or, one real and two imaginary, when the phenomenon is oscillating.

The integral equation for the current in branch 1 is

a & —a6 —a6 —a,6.

ts +r, + Aye + Ag + Aygo*; (59)

the current in branch 2 is by (53) . Alfa Cig i, = =(r D + =F) = = { — 4, (7, — 4,2,) A,e~*? — 4, (7, - a,2,) Ayo”? — a, (7, — ayr,) Ag}, (60) |

136 TRANSIENT PHENOMENA and the potential difference at the condenser is . . di e- 2, [id =ri, + 2453 . = He + (7, — az, Ago" + (7, — ar, Ag rot,

• (r, — ayz,) Ag ™*. (61)

In the case of an oscillatory change, equations (59), (60), and (61) appear in complex imaginary form, and therefore have to be reduced to trigonometric functions.

The three integration constants, A,, A,, and A,, are deter- mined by the three terminal conditions, at # = 0, i, = 2,°, 1, = 1,9,€ = &. ;

79. As numerical example may be considered a circuit having the constants, e, = 110 volts; r, = 1 ohm; z, = 10 ohms; 7, = 10 ohms; x, = 100 ohms, and x, = 10 ohms.

In other words, a continuous e.m.f. of 110 volts supplies,

, over a line of r, = 1 ohm resistance, a circuit of r, = 10 ohms resistance. An inductive reactance z, = 10 ohms is inserted into the line, and an inductive reactance +, = 100 ohms in the load circuit, and the latter shunted by a condensive reactance of z, = 10 ohms.

Then, substituting in equation (58),

@ — 0.2a? + Lila — 0.11 =0. ___ This cubic equation gives by approximation one root, a, = 0.1, and, divided by (a4 — 0.1), leaves the quadratic equation a@—Olat+11=0, which gives the complex imaginary roots a, = 0.05 — 1.047 j and a, = 0.05 + 1.047 7; then from the equation of current, by substituting trigonometric functions for the exponential functions with imaginary exponent, we get the equation for the load current as

a, = tf + Ayenwot® + °°? (B, cos 1.047 6 + B, sin 1.047 6), the condenser potential as

e= 107%, + «-°* {(5 B, + 104.7 B,) cos 1.047 6 — (104.7 B,

— 5B,) sin 1.047 6},

| | | DIVIDED CIRCUIT 137 . and the condenser current as 1, = 10.9 e-? { B, cos 1.047 0 + B, sin 1.047 6}. . At é = 110 volts impressed, the permanent current is 7,’ = 10 amp., the permanent condenser potential is e’ = 100 volts, and the permanent condenser current is 7,’ = 0. Assuming now the voltage, e,, suddenly dropped by 10 per cent, from e, = 110 volts to e, = 99 volts, gives the permanent current as 7,/=9 amp. At the moment of drop of voltage, 6=0, we have, however, 71, =7,° = 10 amp.; e =e = 100 volts, and 7, = 0; hence, substituting these numerical values into the above equations of 7,, e, ¢,, gives the three integration constants : A, =1; B, = 0, and B, = 0.0955; therefore the load current is t, =9+ °°? + 0.0955 -~° sin 1.047 0, | _ the condenser current is | t, = 1.05 e~°® sin 1.047 6, _ and the condenser, or load, voltage is e = 90 + «7% (10 cos 1.047 0 + 0.48 sin 1.047 6). . Without the condenser, the equation of current would be a=9+ 6°98, In this combination of circuits with shunted condensive reactance z,, at the moment of the voltage drop, or 6 = 0, the rate of change of the load current is, approximately, - di ‘7 = [— 0.167%? + 0.0955 X 1.047e7%? cos 1.047 6], = 0, while without the condenser it would be = (— 016-9}, = — O.1. 80. By shunting the circuit with capacity, the current in the circuit does not instantly begin to change with a change or fluctuation of impressed e.m.f. YY

| 138 “TRANSIENT PHENOMENA |

In Fig. 36 is plotted, with @ as abscissas, the change of the current, 2,, in per cent, resulting from an instantaneous change of impressed e.m.f., e,, of 10 per cent, with condenser in shunt to the load circuit, and without condenser. |

As seen, at 0 = 172°, both currents, 7, with the condepsér | and 7 without condenser, have dropped by the samé prhount, |

nl prepa ttt | “TT erase LL ry | 20 Pe nepatatae | LL |

• ae cionms| 1 | MI 7 ! 16 " pan x ohins EDA | Vt i | tut! || eer fA | | | | MCCAY ! BLL et ZT | ost | | Ast TT eA TT ot tit tt lea tt atta ttt Pett tT oa “Mt ty] tt tt tt | LCE EC | Om04 O8 12 16 20 24 28 | Fig. 86. Suppression of pulsations in direct-current circuits by series induc- | tance and shunted capacity. Effect of 10 per cent drop of voltage. | | 2.6 per cent. But at 6 = 57.3°, 7, has dropped only 4 per cent. | and 7 nearly 1 per cent, and at @ = 24°, 7, has not yet dropped at all, while 7 has dropped by 0.38 per cent.

That is, without condenser, all pulsations of the impressed

e.m.f., e,, appear in the load circuit as pulsations of the current, 7, of a magnitude reduced the more the shorter the duration of the pulsation. After @ = 60°, or ¢ = 0.00275 seconds, the pulsation of the current has reached 10 per cent of the pulsation of impressed e.m.f. .

With a condenser in shunt to the load circuit, the pulsation of current in the load circuit is still zero after @ = 24°, or after 0.001 seconds, and reaches 1.25 per cent of the pulsation of

impressed e.m.f., e,, after 0 = 60°, or ¢ = 0.00275 seconds.

A pulsation of the impressed e.m.f., e, of a frequency higher than 250 cycles, practically cannot penetrate to the load circuit, that is, does not appear at all in the load current 7’ regardless of how much a pulsation of the impressed e.m.f., e,, it is, and a

| DIVIDED CIRCUIT 139 pulsation of impressed e.m.f., €,, of a frequency of 120 cycles re- appears in the load current 7,, reduced to 1 per cent of its value. In cases where from a source of e.m.f., e,, which contains a ; slight high frequency pulsation — as the pulsation corresponding to the commutator segments of a commutating machine— a current is desired showing no pulsation whatever, as for instance for the operation of a telephone exchange, a very high inductive reactance in series with the circuit, and a condensive reactance in shunt therewith, entirely eliminates all high frequency pulsa- tions from the current, passing only harmless low frequency pulsations at a greatly reduced amplitude. 81. As a further example is shown in Fig. 37 the pulsation of a non-inductive circuit, z, = 0, of the resistance r, = 4 ohms, shunted by a condensive reactance z, = 10 ohms, and supplied over a line of resistance r, = 1 ohm and inductive reactance z, = 10 ohms, by an impressed e.m.f., e, = 110 volts. Due to z, = 0 equation (58) reduces to a —a(&+ 2) 4 (1 +7) -9; | r, n\ st; i or, substituting numerical values, | a — 264+ 1.25 =0 | and a, = 0.637, a, = 1.963; mo | that is, both roots are real, or the phenomenon is logarithmic. . We now have uy = uy + Ae o87? + Ay 8? t, = — 0.255 A,2~°""? — 0.785 Aye 19, : and e =r, = 4 if + Agee? + Aen 9), The load current is u,! = 22 amp. A reduction of the impressed e.m.f., e,, by 10 per cent, or from 110 to 99 volts, gives the integration constants A, = 3.26 and
A, = — 1.06; hence, t, = 19.8 + 3.26 67°97? — 1.06 9, t, = — 0.83 (27 0878 [rene F and e=4%,. :

140 TRANSIENT PHENOMENA Without a condenser, the equation of current would be i = 19.8 + 2.2 8,

In Fig. 37 is shown, with @ as abscissas, the drop of current 7, and 7, in per cent.

Although here the change is logarithmic, while in the former paragraph it was trigonometric, the result is the same—a very great reduction, by the condenser, of the drop of current imme- diately after the change of e.m.f. However, in the present case

S| Tsupply [eo = 10 folts | el it loleeed COE ORE ie CC bee *[ eee] rs [a gnats | Te T s a 4 eee mates i | T | : am si a as a Esl | | Cr Ty 7 LAG! ie || _ tal] fea Ko Sf s— 4 — ‘i || 2 ct bal as = Ka" | Kor | | | | i au daeae H is TA a FEE att dal got ee CE gy @=-02 04 O06 O08 10 12 14 #16 18 Fig. 37. Suppression of pulsations in non-inductive direct-current circuits by series inductance and shunted capacity. Effect of 10 per cent drop of voltage. the change of the circuit is far more rapid than in the preceding case, due to the far lower inductive reactance of the present case. For instance, after 6 = 0.1, the drop of current, with condenser, is 0.045 per cent, without condenser, 0.5 per cent. At 6 = 0.2, the drop of current is 0.23 and 0.95 per cent respectively. For longer times or larger values of 0, the difference produced by the condenser becomes less and less. This effect of a condenser across a direct-current circuit, of suppressing high frequency pulsations from reaching the circuit, requires a very large capacity.

| CHAPTER X. MUTUAL INDUCTANCE.

82. In the preceding chapters, circuits have been considered containing resistance, self-inductance, and capacity, but no mutual inductance; that is, the phenomena which take place in the circuit have been assumed as depending upon the impressed e.m.f. and the constants of the circuit, but not upon the phenomena taking place in any other circuit.

Of the magnetic flux produced by the current in a circuit and interlinked with this circuit, a part may be interlinked with a second circuit also, and so by its change generate an e.m.f. in the second circuit, and part of the magnetic flux produced by

| n Ly | .

H M :

t

: u |

| L %, H

Fig. 38. Mutual inductance between circuits.

the current in a second circuit and interlinked with the second circuit may be interlinked also with the first circuit, and a change of current in the second circuit, that is, a change of magnetic flux produced by the current in the second circuit, then generates an e.m.f. in the first circuit.

Diagrammatically the mutual inductance between two circuits ean be sketched as shown by M in Fig. 38, by two coazial coils, while the self-inductance is shown by a single coil L, and the resistance by a zigzag line.

141

142 TRANSIENT PHENOMENA The presence of mutual inductance, with a second circuit, | introduces into the equation of the circuit a term depending upon the current in the second circuit. | If 7, = the current in the circuit and r, = the resistance of the circuit, then 7,7, = the e.m.f. consumed by the resistance of the circuit. If L, = the inductance of the circuit, that is, total number of interlinkages between the circuit and the number of lines of magnetic force produced by unit current in the circuit, we have di ; Lt = e.m.f. consumed by the inductance, where, ¢ = time. If instead of time £ an angle 0 = 2 2ft is introduced, where f is some standard frequency, as 60 cycles, di . zq, 7 = e.m.f. consumed by the inductance, where x, = 2xfL, = inductive reactance. ! If now M = mutual inductance between the circuit and another circuit, that is, number of interlinkages of the circuit | “with the magnetic flux produced by unit current in the second | circuit, and 7, = the current in the second circuit, then adi . . . M 7 =e.m.f. consumed by mutual inductance in the first circuit, | M a = e.m.f. consumed by mutual inductance in the second | | circuit. | Introducing zm = 27fM = mutual reactance between the | two circuits, we have | . ai . . Zn a= e.m.f. consumed by mutual inductance in the first circuit, , di . . In a= e.m.f. consumed by mutual inductance in the second | circuit. | | | a |

MUTUAL INDUCTANCE 148 If now e, = the e.m.f. impressed upon the first circuit and e, = the e.m.f. impressed upon the second circuit, the equations of the circuits are : . di. di. . qari tacit im ot +2, fi, d8 (1) and . dt. di . €,= Th, + eT + Im ra + &,, finde, (2) where r, = the resistance, z, = 22fL, = the inductive re- 1 . actance, and z, = Daft, = the condensive reactance of the 1 first circuit; 7, = the resistance, z, = 22fL, = the inductive 1 = __—_- = h i reactance, Z,, IafC, the condensive reactance of the second circuit, and zm = 27fM = mutual inductive reactance between the two circuits. | 83. In these equations, x, and z, are the total inductive reactance, L, and L, the total inductance of the circuit, that is, | the number of magnetic interlinkages of the circuit with the total flux produced by unit current in the circuit, the self- inductive flux as well as the mutual inductive flux, and not merely the self-inductive reactance and inductance respectively. In induction apparatus, such as transformers and induction . machines, it is usually preferable to separate the total reactance z, into the self-inductive reactance z,, referring to the magnetic flux interlinked with the inducing circuit only, but with no other circuit, and the mutual inductive reactance, rm, usually represented as a susceptance, which refers to the mutual induc- tive component of the total inductance: in which case Z=2,+ Im. This is not done in the present case. Furthermore it is assumed that the circuits are inductively related to each other symmetrically, or reduced thereto; that is, where the mutual inductance is due to coils enclosed in the first circuit, interlinked magnetically with coils enclosed in the second circuit, as the primary and the secondary coils of a transformer, or a shunt and a series field winding of a generator,

° 144 TRANSIENT PHENOMENA the two coils are assumed as of the same number of turns, or . reduced thereto. \ r) Ifa=-™@= No._turns second circuit second circu the currents in the n, No. turns first circuit second circuit are multiplied, the e.m.fs. divided by a, the resis- tances and reactances divided by a’, to reduce the second circuit to the first circuit, in the manner customary in dealing with transformers and especially induction machines.* If the ratio of the number of turns is introduced in the equa- . tions, that is, in the first equation “tm substituted for zm, in the 1 second equation “4 rm for 2m, and the equations then are 2 . di, nn, de . 6 = 7, + 7] + 7 do + x, fi do (3) and . di, on, dt, . — Tht tam at *, fi dp. (4) Since the solution and further investigation of these equations (3), (4) are the same as in the case of equations (1) and (2), except that n, and n, appear as factors, it is preferable to eliminate n, and n, by reducing one circuit to the other by the ratio of turns a= ot, and then use the simpler equations (1), (2). 1 84. (A) Circuits containing resistance, inductance, and mutual inductance but no capacity. In such a circuit, shown diagrammatically in Fig. 42, we have . di di . é, =T, + 2, ib + Im iD (5) . di di and €, = Tit, + Ty io + Im 7] ° (6) Differentiating (6) gives de di ia) Pi dy a9 + Tag + tm ape a

• See the chapters on induction machines, etc., in ‘‘ Theory and Calcula- tion of Alternating Current Phenomena.”

MUTUAL INDUCTANCE 14d from (5) follows ¢,-Tt,-2 a, | di, 1 m1 1 do me ® and, differentiated, di, 1 (de, a, a, iP salah ap ae ®) Substituting (8) and (9) in (7) gives de de . di re, +2, 7 — In = rr, + (7,2, + 7,2,) I a

• (2,2, ~ Im’) aR . (10) and analogously, de de . } re, + 2, FI — Im 0 = TP, + (7,2, + 7,2,) a i
• (2,2, — 2m?) 7 : (11) Equations (10) and (11) are the two differential equations of second order, of currents 7, and 7,.

If e,’, 2,’ and e,’, 1,’ are the permanent values of impressed e.m.fs. and of currents in the two circuits, and e,”, 7,” and e,”, t,/” are their transient terms, we have,

a= e,’ + e,”, ty = i’ + n”% ; e, =e, + e,”, ty = 15! + 1,1. Since the permanent terms must fulfill the differential equat‘ons (10) and (11), de,’ de,’ . ia re, + 139 _ ine =Tyrt! + (rt, + rp) . a!

• (2,2, — Im’) “ai? (12) and de,’ de,’ . di,’ 7 ,C,/ + 2, 7A _ ima =ryry/ + (7,2, + ra) &,/
• (1,2, — Im’) “ae (13)

146 TRANSIENT PHENOMENA subtracting equations (12) and (13) from (10) and (11) gives the differential equations of the transient terms, de,” de,/’ . a,” re’ + 2 ~ om = ryt,” + (7,2, + 7,2,) i ai ut

• (4,2, — In’) a (14) and ” “ur ,u re,” + ae - im Se = rr,” + (r,t, + rp) Se a.” .
• (@,2, — Im’) Fee (15) These differential equations of the transient terms are the same as the general differential equations (10) and (11) and the differential equations of the permanent terms (12) and (13).

85. If, as is usually the case, the impressed e.m.fs. contain no transient term, that is, the transient terms of current do not react upon the sources of supply of the impressed e.m.fs. and affect them, we have e,’=0 and e,” =0; hence, the differential equations of the transient terms are . . di ig 0 =r t+ (7,2, + 7,2,) at (z,2, — Tm) (16)

• and are the same for both currents 7,” and 7,”, that is, the transient terms of currents differ only by their integration — constants, or the terminal conditions.

Equation (16) is integrated by the function i= Aen, (17)

Substituting (17) in (16) gives | Ae“ f{rr, — a (rt, + 1,0,) + @ (2,2, — Lm?) } = 0;

hence, A = indefinite, as integration constant, and oe 1,2, + 7,2, a+ TT, -0 (18° L,Lq — Lm? oe ee

MUTUAL INDUCTANCE 147 The exponent a is given by a quadratic equation (18). This quadratic equation (18) always has two real roots, and in this respect differs from the quadratic equation appearing in a circuit containing capacity, which latter may have two imaginary roots and so give rise to an oscillation. Mutual induction in the absence of capacity thus always gives a logarithmic transient term; thus, a- (rz, + 742,) + Vz, — 7,2,)” + ATT om’ (19) 2 (2,2, — Lm’) . . As seen, the term under the radical in (19) is always positive, that is, the two roots a, and a, always real and always positive, since the square root is smaller than the term outside of it. Herefrom then follows the integral equation of one of the currents, for instance 7,, as 4, = 1 + Aye + Aye ™, (20) and eliminating from the two equations (5) and (6) the term os gives . 1 . di i, = nt Jit + (@,2, — Im’) wo + Ime, — ze,}, (21) leaving the two integration constants A, and A, to be deter- . mined by the terminal conditions, as @ = 0, 4,=122 and 1, =1,. 86. If the impressed e.m.fs. e, and e, are constant, we have | de, de, . wo =0 and a =0 ; hence, the equations of the permanent terms (J2) and (13) give ., @ . e a, _ and 1,/ 7 (22) thus: . us i, = = + Aye“ %° 4+ Aye and (23) t, = <! + Ajve™! + Aye, a e where, A,’ and A,’ follow from A, and A, by equation (21).

148 TRANSIENT PHENOMENA

If the mutual inductance between the two circuits is perfect, that is,

Im’ = TT (24) . soe a. way Bly — Te? equation (18) becomes, by multiplication with —_~——— , r,t, + rT, Tt, + Tt, , that is, only one transient term exists.

As example may be considered a circuit having the following constants: e, = 100 volts; e, = 0; r, = 5 ohms; r, = 5 ohms; z, = 100 ohms; x, = 100 ohms, and z,, = 80 ohms. This gives

a,’ = 20 amp. and 1,’ = 0, and ; a’ — 0.278 a + 0.00695 = 0; the roots are a, = 0.0278 and a, = 0.251 and 1, = 20 + Aye re + Ayo oe, By equation (21), _ . , oii t, =— 25 + 1.252, + 9) hence, i, = Aen 74 — Aye o*9, ‘ For 6 = 0 let 7° = 18 amp., or the current 10 per cent below | the normal, and 7,° = 0; then substituted, gives: 18 = 20+ A,.+A, and 0=4A,-A,, hence, A, =A,=—-1; and we have 4.=2- (e9.07780 + g7 02810) 1 ’ and ty =-_ (e 0-278 _ e7 O81 8)

MUTUAL INDUCTANCE 149 , 87. An interesting application of the preceding is the inves- tigation of the building up of an overcompounded direct-current generator, with sudden changes of load, or the building up, or down, of a compound wound direct-current booster. While it would be desirable that a generator or booster, under . sudden changes of load, should instantly adjust its voltage to the change so as to avoid a temporary fluctuation of voltage, actually an appreciable time must elapse. A 600-kw. 8-pole direct-current generator overcompounds from 500 volts at no load to 600 volts at terminals at full load of 1000 amperes. The circuit constants are: resistance of armature winding, r, = 0.01 ohm; resistance of series field winding, 7,/ = 0.003 ohm; number of turns per pole in shunt field winding, n,= 1000, and magnetic flux per pole at 500 volts, ® = 10 megalines. At 600 volts full load terminal voltage (or voltage from brush to brush) the generated e.m.f. is e + ir = 610 volts. From the saturation curve or magnetic characteristics of the machine, we have: At no load and 500 volts: 5000 ampere-turns, 10 megalines and 5 amp. in shunt field circuit. . At no load and 600 volts: 7000 ampere-turns and 12 megalines. At no load and 610 volts: ; 7200 ampere-turns and 12.2 megalines. At full load and 600 volts: 8500 ampere-turns, 12.2 megalines and 6 amp. in shunt | field. . Hence the demagnetizing force of the armature, due to the | shift of brushes, is 1300 ampere-turns per pole. | At 600 volts and full load the shunt field winding takes 6 amperes, and gives 6000 ampere-turns, so that the series field winding has to supply 2500 ampere-turns per pole, of which 1300 are consumed by the armature reaction and 1200 magnetize. At 1000 amp. full load the series field winding thus has 2.5 turns per pole, of which 1.3 neutralize the armature reaction and n, = 1.2 turns are effective magnetizing turns.

150 TRANSIENT PHENOMENA

The ratio of effective turns in series field winding and in shunt

field winding is a = is = 1.2 x 10-*. This then is the reduc- 1 . tion factor of the shunt circuit to the series circuit.

It is convenient to reduce the phenomena taking place in the shunt field winding to the same number of turns as the series field winding, by the factors a and a? respectively.

If then e = terminal voltage of the armature, or voltage impressed upon the main circuit consisting of series field winding and external circuit, the same voltage is impressed upon the shunt field winding and reduced to the main circuit by factor a, gives e, = ae = 1.2 X 10%.

Since at 500 volts impressed the shunt field current is 5 amperes, the field rheostat must be set so as to give to the shunt field circuit the total resistance of r,’ = a = 100 ohms.

Reduced to the main circuit by the square of the ratio of , turns, this gives the resistance,

r, = ar = 144 X 10 ohms.