Theory and Calculation of Transient Electric Phenomena and Oscillations — part 4 of 20

' ; 60 TRANSIENT PHENOMENA In the same manner, substituting (37) in (36), rearranging, and substituting (38), gives .-s-t (TB, + gB rB,— qB, . e,=e-e aL! Et 005 sa! + ay sing (40) : B, and B, are now the two integration constants, determined by the terminal conditions. That is, for¢ = 0, let « = 7, = cur- rent and e, = e, = potential difference at condenser terminals, and substituting these values in (39) and (40) gives t = B, . and 1 rB, + qB hence, B, =i, and 2 (e — &) — Ti, (41) B, = ———“"—". q Substituting (41) in (39) and (40) gives the general equations of condenser oscillation: the current is . can't. gd, , 2@-e@)—m%. . t=e | ice seg SEO sin 57, , (42) and the potential difference at condenser terminals is co r(e—e,) ~ +% 4, e,=e—e 22‘ | (e—e,) cos fy et sin 2 ¢f. ’ ° 2L q 2L (43) Herefrom follow the equations of condenser charge and dis- charge, as special case: For condenser charge,i, = 0; ¢, = 0, we have ° . Qe -s7!. 4g | t= 7 sin 5"! (44)

CONDENSER CHARGE AND DISCHARGE 61 and ~t, eSt—e 2 (eog £44 sin 2. )! | e, e}1 € (cos oft + 7 sin 35 t)§) (45) and for condenser discharge, 1, = 0, e = 0, we have . 2e, -aE!. t= are 2L" sin si! (46) - and — ee Seog © 14 Tsin el. €, = &¢ Joon f+ Eoin oF (47) 40. As an example is shown the oscillation of condenser charge in a circuit having the constants, e = 1000 volts; L = 100 mh., and C = 10 mf. ptt A 1200 b BSS, <//) eal valh L ERs ai. He HSE 4 PYERCEEEEEEEEEPEr ert gaa aan ane fe Fapelvola | ees obetEEEEN EH Heer te fe EEE EEE eff CCN Cee =» Degrees | =10°4)Sec.| | Ae ANE REE NEE? ono |_| 10} |_20 | | 0 EEE Fig. 18. Charging a condenser through a circuit having resistance and induc- : tance, Constant potential. Oscillating charge. | (a) In Fig. 13, r = 100 ohms, hence, g = 173 and the current is | t = 11.55 «- ' sin 866 ¢; | the condenser potential is | e, = 1000{1 — « ©* (cos 866 ¢ + 0.577 sin 866 £)}. ; (b) In Fig. 14, r = 40 ohms, hence, g = 196 and the current . is . t = 10.2 -— ™* sin 980 ¢; . the condenser potential is e, = 1000 {1 — «— ™! (cos 980 ¢ + 0.21 sin 980 #)}.

62 TRANSIENT PHENOMENA 41. Since the equations of current and potential difference (42) to (47) contain trigonometric functions, the phenomena are periodic or waves, similar to alternating currents. They r | differ from the latter by containing an exponential factor « 7”, which steadily decreases with increase of ¢. That is, the suc- 1600 VIN [| [elmo volts |_| LPI hy | a Heie HERE ee I= 4 | : jor pee H . 4-5-200 } SI Lett ES BD sem a Ol

+—}—+ ‘a | ae ct a ae) g? \0 |= |Degrees| 7 Va a al a ra | 320 Vaan $40} 29 —= : Pa bac at HN fo Het Ay pam sodeeso SESESEEEEEs! Fig. 14. Charging a condenser through a cirouit having resistance and induc- tance. Constant potential. Oscillating charge. cessive half waves of current and of condenser potential pro- gressively decrease in amplitude. Such alternating waves of progressively decreasing amplitude are called oscillating waves. Since equations (42) to (47) are periodic, the time ¢ can be represented by an angle 9, so that one complete period is denoted by 2 z or one complete revolution, q _ 4 2af= Sp hence, the frequency of oscillation is

aes fa (49) or, substituting ~ 4L 1-Vq-" gives the frequency of oscillation as , 1 / 1 ry =-_ “-—(—_). 50 f 2-2" LC (; 7) (60) |

CONDENSER CHARGE AND DISCHARGE 68 This frequency decreases with increasing resistance 7, and ry 1 . 4L wa becomes zero for (5) “To that is, 7? = @ or the critical case, where the phenomenon ceases to be oscillating. If the resistance is small, so that the second term in equa- tion (50) can be neglected, the frequency of oscillation is : 1 =———- 51 f on VEG (51) Substituting 4 for ¢ by equation (48) 1-744 q in equations (42) and (43) gives the general equations, _t (e—e,) 54 t=e 2 },008 0 + ———-— sin a}. (52) _t, ree) t%y, e,=e—e 7 | (@ — @,) cos 6+ —— -sin 6 | ,(53) ‘6 = 2xft (48) ~ and a tyl _ cy) f= gana ce7 (si) - (50) 42. If the resistance r can be neglected, that is, if r? is small 4b . . compared with ral the following equations are approximately exact: 2 vz 1=2VG (54) and f- 1 22nV IC : or, (55) 1 Daf = ig :

| 64 TRANSIENT PHENOMENA | Introducing now zx = 2 2fL = inductive reactance and 1 . ae v= fC = capacity reactance, and substituting (55), we have | = VE | —"@ and vi v=Va hence, y=2, , . that is, the frequency of oscillation of a circuit containing inductance and capacity, but negligible resistance, is that frequency f which makes the condensive reactance 2’ = rd | equal the inductive reactance x = 2 xfL: ii = = ——e | Then (54), . g=2r, : (57) and the general equations (52) and (53) are +, (e- ¢) 5%, t=e 2 i i ry sind | ; (58) e,=e- ¢ 2F° (e —e,) cos @ + ened 2 Te sin of ; (59) Cc and by (48) and (55): t @=—. VIC yO

CONDENSER CHARGE AND DISCHARGE 65 a, 43. Due to the factore 7” , successive half waves of oscilla- tion decrease the more in amplitude, the greater the resistance r. The ratio of the amplitude of successive half waves, or the decrement of the oscillation, is A = 6 aL" where ¢, = duration of one half wave or one half cycle, = xy 0 ” NESE ol KIT TTT TT Pe TTT TT IN Lapsed tt tt ot IN TTT TTT rT TTT ttt ty PTT IN EET TT TE TT TEE TE ott FT TCING EET TTT Te tT Pre TTP NTT et te tt att TTT AN ITT TTT TT tt Ett ttt te tH oLLLIT TTT tT fl TT Pee Ty | 0 ol 02 Os 04 0.5 0.6 07 «608 09 8610 Fig. 16. Logarithmic decrement of oscillation. Hence, from (50), : i= R ay Vie~ (ez) and oe ew Vvi£ Ase tae Ve (60) Denoting the critical resistance as 4L r? = Cc , (61) . we have A=e*=e vGy- or, (62) nz d=; vey- r

; 66 TRANSIENT PHENOMENA that is, the decrement of the oscillating wave, or the decay of the oscillation, is a function only of the ratio of the resistance of the circuit to its.critical resistance, that is, the minimum resistance which makes the phenomenon non-oscillatory. In Fig. 15 are shown the numerical values of the decrement A, for different ratios of actual to critical resistance —- . , 1 As seen, for r > 0.21 r,, or a resistance of the circuit of more than 21 per cent of its critical resistance, the decrement A is below 50 per cent, or the second half wave less than half the first one, etc.; that is, very little oscillation is left. Where resistance is inserted into a circuit to eliminate the danger from oscillations, one-fifth of the critical resistance, or r=0.4 vz. seems sufficient to practically dampen out the oscillation. : N

CHAPTER VI. OSCILLATING CURRENTS.

44. The charge and discharge of a condenser through an inductive circuit produces periodic currents of a frequency depending upon the circuit constants.

The range of frequencies which can be produced by electro- dynamic machinery is rather limited: synchronous machines or ordinary alternators can give economically and in units of larger size frequencies from 10 to 125 cycles. Frequencies below 10 cycles are available by commutating machines with low frequency excitation. Above 125 cycles the difficulties rapidly increase, due to the great number of poles, high periph- eral speed, high power required for field excitation, poor regu- lation due to the massing of the conductors, which is required . because of the small pitch per pole of the machine, etc., so that 1000 cycles probably is the limit of generation of constant potential alternating currents of appreciable power and at fair efficiency. For smaller powers, a few kilowatts, by using shunted capacity to assist the excitation, and not attempting to produce constant potential, single-phase alternators have been built and are in commercial service giving 10,000 and even 100,000 cycles, and 200,000-cycle alternators are being designed for wireless telegraphy and telephony.

Still, even going to the limits of peripheral speed, and sacri- ficing everything for high frequency, a limit is reached in the frequency available by electrodynamic generation.

It becomes of importance, therefore, to investigate whether by the use of the condenser discharge the range of frequencies can be extended.

Since the oscillating current approaches the effect of an alternating current only if the damping is small, that is, the resistance low, the condenser discharge can be used as high frequency generator only by making the circuit of as low resist- ance as possible.

67

| 68 TRANSIENT PHENOMENA

This, however, means limited power. When generating oscillat- ing currents by condenser discharge, the load put on the circuit, that is, the power consumed in the oscillating-current circuit, represents an effective resistance, which increases the rapidity of the decay of the oscillation, and thus limits the power, and, when approaching the critical value, also lowers the frequency. This is obvious, since the oscillating current is the dissipation of the energy stored electrostatically in the condenser, and the higher the resistance of the circuit, the more rapidly is this energy dissipated, that is, the faster the oscillation dies out.

With a resistance of the circuit sufficiently low to give a fairly well sustained oscillation, the frequency is, with sufficient approximation,

| p-—t

22 VLC 45. The constants, capacity, C, inductance, L, and resistance, r, have no relation to the size or bulk of the apparatus. For instance, a condenser of 1 mf., built to stand continuously a potential of 10,000 volts, is far larger than a 200-volt condenser of 100 mf. capacity. The energy which the former is able to : store ia = 50 joules, while the latter stores only 2 joules,

and therefore the former is 25 times as large.

A reactive coil of 0.1 henry inductance, designed to carry

,2 continuously 100 amperes, stores = 500 joules; a reactive coil of 1000 times the inductance, 100 henrys, but of a current- carrying capacity of 1 ampere, stores 5 joules only, therefore is only about one-hundredth the size of the former.

A resistor of 1 ohm, carrying continuously 1000 amperes, is a ponderous mass, dissipating 1000 kw.; a resistor having a resistance a million times as large, of one megohm, may be a lead pencil scratch on a piece of porcelain.

, Therefore the size or bulk of condensers and reactors depends not only on C and L but also on the voltage and current which can be applied continuously, that is, it is approximately pro-

,2 portional to the energy stored, ¢ and a , or since in electrical

OSCILLATING CURRENTS 69 engineering energy is a quantity less frequently used than power, condensers and reactors are usually characterized by ; the power or rather apparent power which can be impressed upon them continuously by referring to a standard frequency, for which 60 cycles is generally used. . That means that reactors, condensers, and resistors are rated” in kilowatts or kilovolt-amperes, just as other electrical appa- ‘ ratus, and this rating characterizes their size within the limits of design, while a statement like “a condenser of 10 mf.” or “a reactor of 100 mh.” no more characterizes the size than a statement like “an alternator of 100 amperes capacity” or “a transformer of 1000 volts.” A bulk of 1 cu. ft. in condenser can give about 5 to 10 kv-amp. at 60 cycles. Hence, 100 kv-amp. constitutes a very large size of condenser. In the oscillating condenser discharge, the frequency of oscil- lation is such that the inductive reactance equals the condensive reactance. The same current is in both’at the same terminal voltage. That means that the volt-amperes consumed by the inductance equal the volt-amperes consumed by the capacity. The kilovolt-amperes of a condenser as well as of a reactor are proportional to the frequency. With incréasing frequency, at constant voltage impressed upon the condenser, the current varies proportionally with the frequency; at constant alter- nating current through the reactor, the voltage varies propor- tionally with the frequency. If then at the frequency of oscillation, reactor and con- denser have the same kv-amp., they also have the same at 60 cycles. A 100-kv-amp. condenser requires a 100-kv-amp. reactive coil for generating oscillating currents. A 100-kv-amp. react- ive coil has approximately the same size as a 50-kw. trans- former and can indeed be made from such a transformer, of ratio 1 : 1, by connecting the two coils in series and inserting into the magnetic circuit an air gap of such length as to give the rated magnetic density at the rated current. ‘ A very large oscillating-current generator, therefore, would ; consist of 100-kv-amp. condenser and 100-kv-amp. reactor. 48. Assuming the condenser to be designed for 10,000 volts alternating impressed e.m.f. at 60 cycles, the 100-kv-amp. con- .

70 TRANSIENT PHENOMENA “6 denser consumes 10 amperes: its condensive reactance is ; E . 1 Z=7= 1000 ohms, and the capacity C= afte 2.65 mf.

Designing the reactor for different currents, and therewith different voltages, gives different values of inductance L, and therefore of frequency of oscillation /.

From the equations of the instantaneous values of the con- denser discharge, (46) and (47), follow their effective values, or mean square,

& .-s7! e,=—“Lé 2L " V3 and (63) ; 2% v2 Fi! _ to y/E 20! q V2" L , and thus the power, Me e aud 1 ve ye -5 P= et => L‘ L , (64) since for small values of r Ve ot? q= 2 C we N ow ~ Herefrom would follow that the energy of each discharge is w =f pat = VOL 65 _ ° pat = or ° ( ) Therefore, for 10,000 volts effective at 60 cycles at the con- denser terminals, the e.m.f. is e, = 10,000 V2, and the condenser voltage is te e, = 10,000« 7”.

Designing now the 100-kv-amp. reactive coil for different

voltages and currents gives for an oscillation of 10,000 volts: | |

OSCILLATING CURRENTS 71 e- ‘ . { | Frequency H Reactive Coil. | React- | Inductance. soutien, Oscillating [Foe en a ee | 7 . 1! 100,000 | 105 | 265 6 1 10 ; 10] 10,000 ; 10° 2.65 60 10 100 / 100] 1,000 | 10 2.65 x 10-? 600 100 1,000 ' 1,000 100 10-! | 2.65x10-* | 6,000 | 1,000 | 10,000 | 10,000 10 | 10-3 | 2.65x10-* | 60,000 | 10,000 | 100,000 ' 100,000 1 | 10-§ | 2.65% 10-# | 600,000 | 100,000 |1,000,000 r r ; xe 2! xe 2! As seen, with the same kilovolt-ampere capacity of con- denser and of reactive coil, practically any frequency of oscil- lation can be produced, from low commercial frequencies up to . hundred thousands of cycles. At frequencies between 500 and 2000 cycles, the use of iron in the reactive coil has to be restricted to an inner core, and at frequencies above this iron cannot be used, since hysteresis and eddy currents would cause excessive damping of the oscil- lation. The reactive coil then becomes larger in size. ' 47. Assuming 96 per cent efficiency of the reactive coil and 99 per cent of the condenser, r = 0.05 2, : gives L r= oo, since z=2cfL, ; 1 It =O? 227VLC and the energy of the discharge, by (65), is e 2 W =~* VIG = 10e? C volt-ampere-seconds; 2r ° thus the power factor is cos 6, = 0.05.

, 72 TRANSIENT PHENOMENA Since the energy stored in the capacity is ‘ W, = ae joules, ] the critical resistance is cat /L. i= 2 vi , hence, 7 = 0.025, r, and the logarithmic decrement of the oscillation is A = 0.92, that is, the decay of the wave is very slow at no load.

Assuming, however, as load an external effective resistance equal to three times the internal resistance, that is, an elec- trical efficiency of 75 per cent, gives the total resistance as

r+r=0.22; hence, rt+r = 0.1, r and the decrement is A = 0.73; hence a fairly rapid decay of the wave. At high frequencies, electrostatic, inductive, and radiation losses greatly increase the resistance, thus giving lower effi- ciency and more rapid decay of the wave.

48. The frequency of oscillation does not directly depend

: upon the size of apparatus, that is, the kilovolt-ampere capacity of condenser and reactor. Assuming, for instance, the size, in kilovolt-amperes, reduced to *, then, designed for the same voltage, condenser and reactor, each takes a the current, that

is, the condensive reactance is m times as great, and therefore _ the capacity of the condenser, C’, reduced to , the inductance, L,

|

. OSCILLATING CURRENTS 73 is increased n-fold, so that the product CL, and thereby the frequency, remains the same; the power output, however, of the oscillating ¢urrents is reduced to.

The limit of frequency is given by the mechanical dimensions.

With a bulk of condenser of 10 to 20 cu. ft., the minimum length of the discharge circuit cannot well be less than 10 ft.; 10 ft. of conductor of large size have an inductance of at least ; 0.002 mh. = 2 x 10~°, and the frequency of oscillation would therefore be limited to about 60,000 cycles per second, even . without any reactive coil, in a straight discharge path.

The highest frequency which can be reached may be estimated about as follows:

The minimum length of discharge circuit is the gap between the condenser plates.

The minimum condenser capacity is given by two spheres, since small plates give a larger capacity, due to the edges.

The minimum diameter of the spheres is 1.5 times their . distance, since a smaller sphere diameter does not give a clean spark discharge, but a brush discharge precedes the spark. -

With e, = 10,000 V2, the spark gap length between spheres is e= 0.3 in., and the diameter of the spheres therefore 0.45 in. The oscillating circuit then consists of two spheres of 0.45 in., separated by a gap of 0.3 in.

This gives an approximate length of oscillating circuit of

-3 0.5 in., or an inductance L = ea = (0.125 x 10-* henry. ?

The capacity of the spheres against each other may be estimated as C = 10-* mf.; this gives the frequency of oscilla- tion as 3° 1

= ——— = 4. 0 f wis 4.5 X 10°, or, 4.5 billion cycles. At e, = 10,000 V2 volts, -sot ; e, = 10,000¢ 7” volts, i= 2.836 24" amp., and P, = 2836 " kv-amp.

/

74 TRANSIENT PHENOMENA

Reducing the size and spacing of the spheres proportionally, and proportionally lowering the voltage, gives still higher frequencies. ; .

As seen, however, the power of the oscillation decreases with increasing frequency, due to the decrease of size and therewith of storage ability, of capacity, and of inductance.

With a frequency of billions of cycles per second, the effective resistance must be very large, and therefore the damping rapid.

Such an oscillating system of two spheres separated by a gap would have to be charged by induction, or the spheres charged separately and then brought near each other, or the spheres may be made a part of a series of spheres separated by gaps and

. connected across a high potential circuit, as in some forms of lightning arresters.

, Herefrom it appears that the highest frequency of oscillation of appreciable power which can be produced by a condenser discharge reaches billions of cycles per second, thus is enormously higher than the highest frequencies which can be produced by electrodynamic machinery.

At five billion cycles per second, the wave length is about 6 cm., that is, the frequency only a few octaves lower than the lowest frequencies observed as heat radiation or ultra red light. ;

The average wave length of visible light, 55 x 10-° cm., corresponding to a frequency of 5.5 x 10" cycles, would require

. ‘spheres 10-° cm. in diameter, that is, approaching molecular ! dimensions. OSCILLATING-CURRENT GENERATOR.

“ 49. A system of constant impressed e.m.f., e, charging a con- denser C through a circuit of inductance L and resistance r, with : a discharge circuit of the condenser, C, comprising an air gap

in series with a reactor of: inductance L, and a resistor of resist- ance r,, is a generator of oscillating current if the air gap is set for such a voltage e, that it discharges before the voltage of the condenser C has reached the maximum, and if the resistance r, is such as to make the condenser discharge oscillatory, that is,

OSCILLATING CURRENTS 75

In such a system, as shown diagrammatically in Fig. 16, as soon, during the charge of the condenser, as the terminal voltage at C and thereby at the spark gap has reached the value e,, the condenser C' discharges over this spark gap, its potential dif- ference falls to zero, then it charges again up to potential differ- ence é€,, discharges, etc. Thus a series of oscillating discharges

L r to

| Lo

? c

"

i]

Fig. 16. Oscillating-current generator.

oceur in the circuit, L,, 7,, at intervals equal to the time required to charge condenser C' over reactor L and resistor r, up to the potential difference e,, with an impressed e.m.f. e.

The resistance, r, obviously should be as low as possible, to get good efficiency of transformation; the inductance, L, must be so large that the time required to charge condenser C' to potential e, is sufficient for the discharge over L,, r, to die out and also the spark gap e, to open, that is, the conducting products of the spark in the gap e, to dissipate. This latter takes a con- _ siderable time, and an air blast directed against the spark gap e,, by carrying away the products of the discharge, permits a more rapid recurrence of the discharge. The velocity of the air blast (and therefore the pressure of the air) must be such as to carry the ionized air or the metal vapors which the discharge forms in the gap e, out of the discharge path faster than the con- denser recharges.

Assuming, for instance, the spark gap, e,, set for 20,000 volts, or about 0.75 in., the motion of the air blast during successive discharges then should be large compared with 0.75 in., hence at least 3 to6 in. With 1000 discharges per second, this would | require an air velocity of r = 250 to 500 feet per second, with 5000 discharges per second an air velocity of v = 1250 to 2500 feet per second, corresponding to an air pressure of approximately p = 14.7 {(1 +2¥10-7)8* — 1} lb. per sq. in., or 0.66 to 2.75 Ib. in the first, 23 to 230 lb. in the second case.

| 76 TRANSIENT PHENOMENA

While the condenser charge may be oscillatory or logarithmic, efficiency requires a low value of r, that is, an oscillatory charge.

With a frequency of discharge in L,, r, very high compared with the frequency of charge, the duration of the discharge is short compared with the duration of the charge, that is, the

' oscillating currents consist of a series of oscillations separated by relatively long periods of rest. Thus the current in L does not appreciably change during the time of the discharge, and at the end of the condenser charge the current in the reactor, L,

. is the same as the current in L, with which the next condenser charge starts. The charging current of the condenser, C, in L thus changes from 7, at the beginning of the charge, or con- denser e.m.f.,e¢, = 0, to the same value 7, at the end of the charge, or condenser e.m.f., e, = @,.

50. Counting, therefore, the time, t, from the moment when the condenser charge begins, we have the terminal conditions:

t = 0, 7'=1,, e, = 0 at the beginning of the condenser charge. t=t, t=1%, e, =e, at the end of the condenser charge.

In the condenser discharge, through circuit L,, r,, counting the time ¢ from the moment when the condenser discharge begins, that is, i’ = ¢ — t,, we have

v=0, 7=0, e, =e, the terminal condition.

€,, thus, is that value of the voltage e, at which discharge takes place across the spark gap, and ¢, is the time elapsing between e, = 0 and e, = e,, or the time required to build up the voltage e, sufficiently to break down the spark gap.

Under the assumption that the period of oscillation of the condenser charge through L, 1, is large compared with the period of oscillation of the condenser discharge through L,, r,, the equations are:

(A) Condenser discharge:

} = 2% 5 aie" sin 0 ‘ Yo 2L, ’ (66)

• 0 rT, . ; €, = Oe 21," cos yt + resin ee, (67) where IL % = \ Ce a ne (68)

OSCILLATING CURRENTS 17 (B) Condenser charge:

• • 37'S. gq 2e—T, . gq a=e Yipes st + 7 sin tt, (69) v¢ q nan “st to q | =e-—e 3L i gin é =e-—e € cosy t+ 7 sin 55! , (70) where —_—__ . 4L q= vit -?7. (71) Substituting in (69) and (70) the above discussed terminal conditions, t=h, t=h, e, = &y gives . .
• a’, 4. “ta "inst tt lp =e figcos 5% ty +S sing 7 (72) and:
• ; q rei q =e—e 2b at =p . G&@=Ee-—€ € cos 57 fy + 7 sing7 4 (73) Denoting, for convenience, . r _

i . A (74) and r —_ = a, q and resolving (72) for 7,, gives . . 2e esing : eT Tm etc + asin g’ @) and substituting (75) in (73) and rearranging, _. 1—2ecosf +e7?? fo °T cos f + ae sing (76) | | |

78 TRANSIENT PHENOMENA The two equations (75), (76) permit the calculation of two of | the three quantities 7,, ¢,, t: the time, ¢,, of condenser charge appears in the exponential function, in s, and in the trigonometric function, in ¢. Since in an oscillating-current generator of fair efficiency, that is, when r is as small as possible, s is a small quantity, e— can be resolved into the series etelietoipen. (77) . Substituting (77) in (75), and dropping all terms higher than s*, gives s\ . ae (1-54 5)sing a 1—cos¢ + $ cos $ — 5 cos ¢ + asin d — assing Multiplying numerator and denominator by (1 + ) and rearranging, gives . 2e sin uC ets 1-7 * _ 9s ¢ + asing 2-8 7s) ~2@ sing — g 2s 2@ . | 2, t 2sin’'g tasing Substituting (77) in (76), dropping terms higher than s? and as, multiplying numerator and denominator by (1 + 5) and realranging, gives , ; 2 sin? : + . e&= 2e We : (79) 28° 3? . at 28in 5 +asind

; OSCILLATING CURRENTS 79

Substituting ¢, in (78) and (79) gives

in 2. . 2e 1 OL fo eT bh oat, atad, | O™ Po 4 O gin? 4 ~ gin —2 4L— t,t sin apt gays and 2 2 sin? t+ rity 4L 8L? aQe= 2e rs rr cr er (81) Thy 2 7 reg AL — ri,* 2sin wa + qn 21" as approximate equations giving 7, and e, as functions of ¢,, or the time of condenser charge.

61. The time, ¢,, during which the condenser charges, increases with increasing é,, that is, increasing length of the spark gap in the discharge circuit, at first almost proportionally, then, as e, approaches 2 e, more slowly.

As long as e, is appreciably below 2 e, that is, about e, < 1.75 e, t, is relatively short, and the charging current 7, which increases from 7, to @ maximum, and then decreases again to 7,, does not vary much, but is approximately constant, with an average value very little above 7,, so that the power supplied by the impressed e.m.f., e, to the charging circuit can approximately be assumed as

Po = &q- (82)

The condenser discharge is intermittent, consisting of a series of oscillations, with a period of rest between the oscillations, which is long compared with the duration of the oscillation, and during which the condenser charges again.

The discharge current of the condenser is, (66),

. 2e -snt . %,. t= —e ~“™ sin = ¢, in amp., Go 2L, P and since such an oscillation recurs at intervals of ¢, seconds, the effective value, or square root of mean square of the dis- charge current, is . 1 to ia V/L fre (83) i=\ ido ? dt.

80 TRANSIENT PHENOMENA Long before ¢ = ¢,, 2 is practically zero, and as upper limit of the integral can therefore be chosen © instead of t,. Substituting (66) in (83), and taking the constant terms out of the square root, gives the effective value of discharge cur- rent as . 2, 1 J “ —-2, gq 4,=—V= Jc © ginr_to 1 % to r sin 5) L, t dt 24 1 { fez fea Yo, ue. =a Van tee ad —J, cos 7 tdty ; (84) however, fee dt =— bf] — bs, ° ur ", and by fractional integration, 2 [ent of 088 L, t dt | Ly =-—fo__ [ “ia! sin 204 to cos 21} T 1+ (=) ° L, % . Lo % | _ be Te + G0 hence, substituting in (84), . . 2D ! =e / Asn 85) | 5 OV trete + ae’) “ : Since | 41. ar we have, substituting in (85), . Cc 1 =e —_, 86 1=8%V Jip, (86) and, denoting by 1 f= i,’

OSCILLATING CURRENTS 81 the frequency of condenser charge, or the number of complete trains of discharge oscillations per second,

; Ch,

4 = ve ’ (87) that is, the effective value of the discharge current is propor- tional to the condenser potential, e,, proportional to the square root of the capacity, C, and the frequency of charge, f,, and inversely proportional to the square root of the resistance, r,, of the discharge circuit; but it does not depend upon the induc- tance L, of the discharge circuit, and therefore does not depend on the frequency of the discharge oscillation.

The power of the discharge is

. . e2C DP, = tr, = Ss; o ° (88) . _ esl. . .

Since “9 8 the energy stored in the condenser of capacity C at potential e,, and f, the frequency or number of discharges of this energy per second, equation (88) is obvious.

Inversely therefore, from equation (88), that is, the total energy stored in the condenser and discharging per second, the effective value of discharge current can be directly calcu- lated as

aig (A.

= v2 - %o V5 T,

The ratio of effective discharge current, 7,, to mean charging current, 2,, is

te {a

t 1° 2 r, (89) and substituting (80) and (81) in (89),

in? 1p 4 The |

i, ar ter

== VY (90)

“0 2% sin -2

2L°

82 TRANSIENT PHENOMENA The magnitude of this quantity can be approximated by

; ; ae 4b | neglecting r compared with % that is, substituting g = Ve and replacing the sine-function by the arcs. This gives

t, 1

OE | (91)

% V2 7,C. T, 1 that is, the ratio of currents is inversely proportional to the square root of the resistance of the discharge circuit, of the capacity, and of the frequency of charge.

( - 62. Example: Assume an oscillating-current generator, feed- ing a Tesla transformer for operating X-ray tubes, or directly supplying an iron arc (that is, a condenser discharge between iron electrodes) for the production of ultraviolet light.

The constants of the charging circuit are: the impressed e.m.f., e = 15,000 volts; the resistance, r = 10,000 ohms; the inductance, ZL = 250 henrys, and the capacity, C= 2 x 10-° farads = 0.02 mf.

The constants of the discharge circuit are: (a) operating Tesla transformer, the estimated resistance, r, = 20 ohms (effective) and the estimated inductance, L, = 60 x 10° henry = 0.06 mh.; (6) operating ultraviolet arc, the esti-

; mated resistance, r, = 5 ohms (effective) and the estimated inductance, L, = 4 X 10-° henry = 0.004 mh.

Therefore in the charging circuit,

q = 223,400 ohms, ; = 0.0448, 4 J. aL 4416.8, aL 20, D _ 0.025; Tr then . jpn ogg —————8HOB 8 in? 22% j 0.025 — + 2 sin’? 223.4 t,+ 0.0448 sin 446.8 ¢, and a (92) 223.4 t 200 t,? €, = 30,000 , 2 sin? 223.4 t, + 200 t, ; 0.035 — 1, + 2sin? 223.4 t,+ 0.0448 sin 446.8 ¢, |

OSCILLATING CURRENTS 83 Fig. 17 shows 1, and e, as ordinates, with the time of charge t, as abscissas. ee CE SCE mae COC 32 H+ 0=40.02-inf [| jl fo — Eeiuasiaaaetttt x BE errr eed bite | petciane aR EEC — 0.3 \—+ —+-—+ —+— + —F — +44 4 4 $+ +—j ie HA i HH at EEE Eee eer oF aEEUOEEEEEESRETESEESTEE erry 04 0.6 os 10 12 14 16 18 “2.0K 10 Sec, Fig. 17. Oscillating-current generator charge. The frequency of the charging oscillation is . -/ 2. . f Ink 71.2 cycles per sec.; for 1, = 0.365 amp., substituting in equations (69) and (70) we have i=e—! {0.365 cos 446.8 t+-0.118 sin 446.8 ¢}, in amp., ! and (93) , e, = 15,000 { 1—e~? [cos 446.8 t— 2.67 sin 446.8 ¢]}, in volts, the equations of condenser charge. From these equations the values of 7 and e, are plotted in Fig. 18, with the time ¢ as abscissas. | As seen, the value 7 = 7, = 0.365 amp., is reached again at the time ¢, = 0.0012, that is, after 30.6 time-degrees or about vs of a period. At this moment the condenser e.m.f. is e, = €, = 22,300 volts; that is, by setting the spark gap for 22,300 volts the duration of the condenser charge is 0.0012 second, or in other words, every 0.0012 second, or 833 times per second, discharge oscillations are produced. With this spark gap, the charging current at the beginning and at the end of the condenser charge is 0.365 amp., and the

| 84 TRANSIENT PHENOMENA average charging current is 0.3735 amp. at 15,000 volts, con- suming 5.6 kw.

Assume that the e.m.f. at the condenser terminals at the end of the charge is e, = 22,300 volts; then consider two cases, namely: (a) the condenser discharges into a Tesla transformer, and (6) the condenser discharges into an iron arc.

Fete HHA “eae CTT

wf eevee’ TTT TAT

PT ttt ttt Vt

wet tt tt Tt SEER EE

[|

eee a fett+t ttt iA tt tt aeetLt tT tI Yt {| tT ; wo tt tt tA TT tt

PTT ET Ye tt tt CCE... att 7 tT tt tt tt

PTT yt tT ttt tt AtltT7 TTT Ty tt HA++ tt Ce

{|

Vitti titi ttt tt

t=—02 04 06 08 10 1.2X10°8fe0c. ,

Fig. 18. Oscillating-current generator condenser charge.

(a) The Tesla transformer, that is, an oscillating-current transformer, has no iron, but a primary coil of very few turns (20) and a secondary coil of a larger number of turns (360), both immersed in oil.

While the actual ohmic resistance of the discharge circuit is only 0.1 ohm, the load on the secondary of the Tesla trans- former, the dissipation of energy into space by brush discharge, etc., and the increase of resistance by unequal current distribu- tion in the conductor, increase the effective resistance to many times the ohmic resistance. We can, therefore, assign the

OSCILLATING CURRENTS 85 following estimated values: r, = 20 ohms; L,= 60 X 10-° henry, and C = 2 x 10-® farad. Then , q. = 108 ohms, “e — 0.186, % =4¢ = 0.808 x 10°, = * = 0.1667 x 10° 2L, , 2L, , which give : ; E=A415 c—0-1687%10! sin 0.898 x 10° ¢, amp. and €, = 22,300 e~ 010710" { cog 0.898 x 10° ¢+0.186 sin 0.898 x 10° ¢}, volts. (94) The frequency of oscillation is 6 f= oe = 143,000 cycles per sec. (95) Fig. 19 shows the current 7 and the condenser potential e, during the discharge, with the time ¢ as abscissas. As seen, the discharge frequency is very high compared with the fre- ol J Pies TTT OP, 4=,0.06 mh. mit = SHANE eT] a* ol SECA rr ene re ct Pee : ag °* ONT 5S Sat “8 ot CELL HH 0 1.5 3 4.5 6 7.5 9 10.5 12 13.5 ' x10°% Seo. ; Fig. 19. Oscillating-current generator condenser discharge. quency of charge, the duration of discharge very short, and the damping very great; a decrement of 0.55, so that the oscil- lation dies out very rapidly. . The oscillating current, however, is enormous compared with the charging current; with a mean charging current of 0.3735 amp., and a maximum charging current of 0.378 amp. the maximum discharge current is 315 amp.,or 813 times as large as the charging current.

86 TRANSIENT PHENOMENA The effective value of the discharge current, from equation (87), is 1, = 14.4 amp., or nearly 40 times the charging current. 63. (6) When discharging the condenser difectly, through an ultraviolet or iron arc, in a straight path, and estimating 7, = 5 ohms and L, = 4 X 10~* henry, we have dy = 27.84 ohms, “9 _ 0.1795, % -49. = 34810, 2 = 0.625 x 10°; 2L, — , 2L, , . then, 1= 1600 e~ °5*! sin 3.48 x 10° 2, in amp., and €, = 22,300 e- 5x10" ( eos 3.48 x 10° t+-0.1795 sin 3.48 x 10° t}, in volts, (96) and the frequency of oscillation is J, = 562,000 cycles per sec. ; (97) that is, the frequency is still higher, over half a million cycles; the maximum discharge current over 1000 amperes; however, the duration of the discharge is still shorter, the oscillations dying out more rapidly. The effective value of the discharge current, from (87), is 1, = 28.88 amp., or 77 times the charging current. A hot _ wire ammeter in the discharge circuit in this case showed 29 amp. | As seen, with a very small current supply, of 0.3735 amp., at e = 15,000 volts, in the discharge circuit a maximum voltage . of 22,300, or nearly 50 per cent higher than the impressed voltage, is found, and a very large current, of an effective value very many times larger than the supply current.

As a rule, instead of a constant impressed e.m.f., e, a low frequency alternating e.m.f. is used, since it is more conven- iently generated by a step-up transformer. In this case the

condenser discharges occur not at constant intervals of t, sec-

onds, but only during that part of each half wave when the

e.m.f. is sufficient to jump the gap e,, and at intervals which

are shorter at the maximum of the e.m.f. wave than at its , beginning and end.

| OSCILLATING CURRENTS 87 ' For instance, using a step-up transformer giving 17,400 volts effective (by the ratio of turns 1: 150, with 118 volts im- : pressed at 60 cycles), or a maximum of 24,700 volts, then during each half wave the first discharge occurs as soon as the voltage has reached 22,300, sufficient to jump the spark gap, and then a series of discharges occurs, at intervals decreasing with the increase of the impressed e.m.f., up to its maximum, and then with increasing intervals, until on the decreasing wave the e.m.f. has fallen below that which, during the charg- ing oscillation, can jump the gap e,, that is, about 13,000 volts. Then the oscillating discharges stop, and start again during the next half wave.

Hence the phenomenon is of the same character as investi- gated above for constant impressed e.m.f., except that it is intermittent, with gaps during the zero period of impressed voltage and unequal time intervals t, between the successive discharges.

64. An underground cable system can act as an oscillating- current generator, with the capacity of the cables as condenser, the internal inductance of the generators as reactor, and a short- circuiting arc as discharge circuit.

In a cable system where this phenomenon was observed the constants were approximately as follows: capacity of the cable system, C = 102 mf.; inductance of 30,000-kw. in gen- erators, L =6.4 mh.; resistance of generators and circuit up to the short-circuiting arc, r = 0.1 ohm and r = 1.0 ohm respec- tively; impressed e.m.f., 11,000 volts effective, and the fre- quency 25 cycles per second.

The frequency of charging oscillation in this case is

f= rea = 197 cycles per sec. since g- Vite = 15.8 ohms. Cc

Substituting these values in the preceding equations, and estimating the constants of the discharge circuit, gives enor- mous values of discharge current and e.m.f.

CHAPTER VII. RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES IN ALTERNATING-CURRENT CIRCUIT. ° 55. Let, at time ¢ = 0 or 6 = 0, thee.m/f., e = E cos (0 — 4), (1) be impressed upon a circuit containing in series the resistance, r, the inductance, L, and the capacity, C. The inductive reactance is x = 2 2fL i, . 1 (2) and the condensive reactance is x, = 2afC , where f = frequency and 60 = 2 aft. @) - Then the e.m.f. consumed by resistance is rv; the e.m.f. consumed by inductance is dy dy L a = aa i and the e.m.f. consumed by capacity is 1 ¢. . eG fiaas, fia, 4) where 7 = instantaneous value of the current. Hence, exritc Sts, fide, (5) di | or, Ecos (0 = 6,) = ri + 25+ me fias, 6) | and hence, the difference of potential at the condenser terminals | : . . di | ¢, = a, fidd = Boos (0 — 0) — ri - 25. (7) 88

RESISTANCE, INDUCTANCE, AND CAPACITY 89 Equation (6) differentiated gives . ig ai . Hsin (@—4)+taa+r at ta = 0. (8) The integral of this equation (8) is of the general form i = Ae™ + B cos (6 — a). (9) Substituting (9) in (8), and rearranging, gives Ae | a?x—ar+z,} +sin 6 {E cos 6,—rB coso—B (x—z,) sinc} — cos 6{E sin 6, — rBsine + B (zr — z,) cos «} = 0, and, since this must be an identity, _— ar—ar+2, =0, E cos 0, — rB cos o — B (x — z,) sine = 0, (10) E sin 6, — rBsine + B (x — z,) cosa = 0. Substituting s =Vr—4232,, ° 2= Vr + (x _ 2,)*, (11) r in equations (10) gives ; ant t3 ay a E B= 2, (12) o=O0,+4 and A = indefinite, and the equation of current, (9), thus is ; roe rts t= F 0s (6-0-7) + Ag 2? + Ay 22°, (13) °