CHAPTER IX. 

INDUCTIVE DISCHARGES. 

64. The discharge of an inductance into a transmission line 
may be considered as an illustration of the phenomena in a 
complex circuit comprising sections of very different constants; 
that is, a combination of a circuit section of high inductance and 
small resistance and negligible capacity and conductance, as a 
generating station, with a circuit of distributed capacity and 
inductance, as a transmission line. The extreme case of such a 
discharge would occur if a short circuit at the busbars of a gen- 
erating station opens while the transmission line is connected 
to the generating station. 

Let r = the total resistance and L = the total inductance of 
the inductive section of the circuit; also let g = 0, C= 0, and 
L0 = inductance, <70 = capacity, r0 = resistance, g0 = conduc- 
tance of the total transmission line connected to the inductive 
circuit. 

In either of the two circuit sections the total length of the 
section is chosen as unit distance, and, translated to the velocity 
measure, the length of the transmission line is 


and the length of the inductive circuit is 

~T = 0; (374) 


that is, the inductive section of zero capacity has zero length 
when denoted by the velocity measure X, or is a "massed induc- 
tance." 

It follows herefrom that throughout the entire inductive 
section X = 0, and current i1 therefore is constant throughout this 
section. 

Choosing now the transition point between the inductance and 
the transmission line as zero of distance, A = 0, the inductance 

635 


536 


TRANSIENT PHENOMENA 


is massed at point ^ = 0, and the transmission line extends from 
X = 0 to X = V 

Denoting the constants of the inductive section by index 1, 
those of the transmission line by index 2, the equations of the two 
circuit sections, from (290), are 


Cj) cos qt — 


-(51 + 7)1)sin^},l 
- A-DJsing*}; j 


(376) 


i2 = £-Uot{s +sA [A2 cos q (X - t) + B2 sin g (^ - t)] 
— s~s* [C9 cos q (A + t) + Z), sin 0 (^ + £)"]}. 

L / Z\ / ^ 1\ x-!)^ 

e2 = ca e-rt {[£+^ [A2 cos g (^ - 0 + J52 sin q (I - t)] 

+ £~sA [Ca cos g (^ + 0 + D2 sin q (Jl + 0]}, 

and the constants of the second section are related on those of 
the first section by the equations (285) : 

A - n A 4- 7) n C1 

Aj - «!/!.! "I' Oil/ti ^2 

D „ D ^\ T\ T\ 

•D n == U/fJJ-i t/1iyi. -^'•5 

where 

«,=VC-' 

(286) 


+ 


(349) 


and 


(287) 


65. In the inductive section having the constants L and r, 
that is, at the point A = 0 of the circuit, current il and voltage 
et must be related by the equation of inductance, 


(377) 


Substituting (375) in (377), and expanding, gives 
ci { (Ai + CJ cos qt - (B^ - DJ sin qt] 

= (r + w0L) { (At - C,) cos qt - (B^ + D,) sin qt} 
+ ^L { (Al - CJ sin ^ + (B1 + DJ cos qt}, 


INDUCTIVE DISCHARGES 

and herefrom the identities 

Cl (A, + C,) = (r + ^) (^i ~ Ci) + ^ (*i + *>i), 
Cl (B,-D,) = (r + u0L) (5, + ig - gL (Ax- CJ. 

Writing 
and 


gives 

cx (AA + CJ = (r + w0^) M + 3LN ] 

and | 

c, (5, - Dx) = (r + w0L) A^ - gLM, J 

which substituted in (349) gives 

A2 = — { (c + r + w0£) ^ + ^^ } ^ - (M + p^V), 

J52 = 1 { (c + r + u0L) N - qLM} ~*:(N - pM), 
z c £ 


537 


(378) 


(379) 


(380) 


- 

Z C 


- (c - r - 


-f (c - r- m0L) 


- M), 


JV), 


(381) 


where in the second expression terms of secondary order have 
been dropped. 


P 


qL 


Then substituting in (375) gives the equations of massed 
inductance : 

it = e ~M { M cos qt - N sin qt } 


(382) 

If at t = 0, £j = 0, that is, if at the beginning of the transient 
discharge the voltage at the inductance is zero, as for instance 
the inductance had been short-circuited, then, substituting in 


538 


TRANSIENT PHENOMENA 


(382), and denoting by i0 the current at the moment t = 0, or at 
the moment of start, we have 


t = 0, i\= %,«! = 0; hence, 
M = i 


(383) 


and 


r T -}- u Li 

h = V~"u* 1 cos <# + -r2" sm ^ 

gL 

+ (r + V')2 , 


(384) 


In this case 

i 
A*=2' 


2 + (r + uQL)2 + c (r 

\ - \J s f / 

(385) 


+ (r + ^0L)2 - c (r + 


66. In the case that the transmission line is open at its end, 
at point A = ^0, 

>l = 1 


?•-*-?' 

i. 


hence, this substituted in (376), expanded and rearranged as 
function of cos qt and sin qt, gives the two identities 

£+sA° (A, cos g;o+ B2 sin gao) « fi-^ (C2 cos q*0 + D2 sin ^, 

and 

c+«fc// 


(386) 


Squared and added these two equations (386) give 
(A? + £2) . ,-2^o C2 + D 2> 


(387) 


INDUCTIVE DISCHARGES 


539 


Divided by each other and expanded equations (386) give 

(Af, - B2DJ sin 2 gA0 - (A2D2 + B2C2) cos 2 qXQ. (388) 

Substituting (381) into equations (387) and (388) gives 

2^{(qL)2 + (c + r + u0L)2}=e-*s**{(qL)2+(c-r-uQL)2} (389) 

{ (qL)2 + (r + u,L)2 - c2 \ sin 2 qXQ = 2 cgL cos 2 ^0. (390) 

Since 2 s^0 is a small quantity, in equation (389) we can sub- 


stitute 


hence, rearranging (389) and substituting 

s = u0 — u 
gives 

c (r + u0L) - (u - u0) ;0 { (qL)2+(r + u0L)2 + c2 } = 0. (391) 

Since (r + u0L) is a small quantity compared with c2 (qL)2, it 

can be neglected, and equations (390) and (391) assume the form 

{ (qL)2 - c2 } sin 2 gA0 = 2 cqL cos 2 q*Q (392) 

c (r + u,L) - (u - UQ) *0 { (qL)2 + c2} = 0, (393) 

and, transformed, equation (392) assumes the form 

2cqL 


tan 2 


or 


or 


(qL)2 - c2 ' 


q = - - tan ql0, 


(394) 


q = + -f cot g^0, 
L 

hence tan 2 q^ is positive if #L > c, as is usually the case. 
Expanded for u0, equation (393) assumes the form 


+cL 


or 


(L+r-) 
V wl 


u cL + ^0 { (qL)2 + c2 } 
s = - (u - u0). 


(395) 


540 


TRANSIENT PHENOMENA 


From equations (394) q is calculated by approximation, and 
then from (395) u0 and s. 

As seen, in all these expressions of q, uw s, etc., the integration 
constants M and N eliminate; that is, the frequency, time atten- 
uation constant, power transfer, etc., depend on the circuit con- 
stants only, but not on the distribution of current and voltage 
in the circuit. 

67. At any point X of the circuit, the voltage is given by equa- 
tion (376), which, transposed, gives 

e = c£-wo<{£+^ [(42 cos q% + %2 sin qX) cos qt 
+ (A2 sin qX — B2 cos qX) sin qt] 
+ e~sA [(C2 cos qX + D2 sin qX) cos qt 
— (C2 sin qX — D2 cos qX) sin qt] } , 

or approximately, 

e = cs~u<* {[(A2 + C2) cos qX + (B2 + D2) sin qX] cos qt 
+ [(A2 - C2) sin g^ - (B2 - D2) cos ^] sin qt}. 

Similarly to equation (381), 

A2 + C2 = pN', 
A2-C2 = M; 
B* + D2 = N; 


(396) 


where 
then 


T) — • 


cos ^A+c sin gd) (JV cos ^ + M sin ^), 


(cos ^ _ sin 


cos g« - N sin $); 


^1 (N cos qt + M sin $), 
(M cos qt-N sin g«). 


(397) 
(398) 


INDUCTIVE DISCHARGES 


541 


If 


hence, 


e, = 0 for t = 0, 
N = 0; 

h = V ~^ cos $> 


'2 " 0 


, <1L • n 

(cos qX - - sin g/) cos 
c 


c sn 


Writing 


(M2 


the effective values of the quantities are 


I2 = 


E2 = I0e~u<* (qL cos q\ + c sin gA). . 

Herefrom it follows that 

/2 = 0 for A = ^0 by the equation 

cos aAn - q- sin gA0 = 0, 


or 

while 
gives 

that is, 


q - zco 


«- - 


^L cos ql0 + c sin gA0 = 0; 


(399) 


(400) 


(401) 


(402) 
(403) 


0 at 


542 


TRANSIENT PHENOMENA 


At the open end of the line ^ = 
tuting (402) into (401) is 


the voltage E2 by substi- 


cos 


If 


+ 


At the grounded end of the line X = 
stituting (403) into (401), is 

7 ~ — U(jt, 

V 


(404) 
the current 72, by sub- 


(405) 


An inductance discharging into the transmission line thus 
gives an oscillatory distribution of voltage and current along the 
line. 

68. As example may be considered the three-phase high- 
potential circuit, comprising a generating system of r = 2 ohms 
and L = 0.5 henry per phase and connected to a long-distance 
transmission line of r0= 0.4 ohm, L0 = 0.002 henry, gQ= 0.2 X 
10~6 mho, (70= 0.016 X 10~6 farad per mile of conductor or 
phase, and of 1Q = 80 miles length. 


c =\/- = 


c2 = 125,300; 


<TO = VL0C0 - 5.66 X 10-6; 
^o = Zo°"o = 0.453 X 10-3; 
l/rn 


L = 7°8' 


and herefrom, substituting in equations (394) and (395), 
q = - 708 tan (0.0259 q)° (zero voltage) 
= + 708 cot (0.0259 q)° (zero current), 


u 0.618 (f 10-6 + 1.28 


H = 

100.35° 

185.64° 

273.83° 

362.89° 

452.32° 

541.94° 

? = 

3875 

7168 

10,572 

14,010 

17,463 

20,920 

u 

0.0946 

0.0302 

0.0142 

0.00816 

0.0047 

0.0037 

W0 = 

95.8 . 

102.8 

104.5 

105.1 

105.5 

105.6 

jr- 

10.2 

3.2 

1.5 

0.87 

0.5 

0.4 

INDUCTIVE DISCHARGES 543 

By equation (401) the effective values of the first six har- 
monics are given as 

(1) Quarter-wave: 100.35°. 

q, = 3875; 
u0 = 95.8; 

7 = if-"* (cos qX - 5.48 sin qX)', 
E == 1939 v'^ (cos qX + 0.182 sin qX). 

(2) Half-wave: 185.64°. 

q2 = 7168; 
UQ = 102.8; 

7 = if-"* (cos qX - 10.14 sin qX); 
E = 3585 iQr^ (cos ^ + 0.098 sin qX). 

(3) Three-quarter wave: 273.83°. 

qz -- 10,572; 
u0 = 104.5; 

7 = v~^ (cos g^ - 14.90 sin qX); 
E = 5287 i0e-Uot (cos ^>i + 0.067 sin qX). 

(4) Full wave: 362.89°. 

?4 •• 14,010; 
w0 = 105.1; 

7 = v~^ (cos qX - 19.8 sin qX); 
E = 7005 v~^ (cos qX + 0.050 sin qX). 

(5) Five-quarter wave: 452.32°. 

<?5 » 17,463; 
w0 = 105.5; 

7 = v'7*0' (cos qX - 24.65 sin gfl; 
E = 8732 1>-^ (cos qX + 0.040 sin gj). 


544 TRANSIENT PHENOMENA 

(6) Three-half wave: 541.94°. 

& = 20,920; 
UQ = 105.6; 

7 = if-** (cos qX- 29.6 sin gd); 
jE/ = 10,460 v~Wo< (cos g>l + 0.033 sin qX). 


APPENDIX 


VELOCITY FUNCTIONS OF THE ELECTRIC FIELD 

IN the study of the propagation of the electric field through 
space (wireless telegraphy and telephony), a number of new 
functions appear (Section III, Chapter VIII). 

. By the following equations these functions are defined, and 
related to the " Sine-Integral" Si x, the " Cosine-Integral" Ci x, 
and the " Exponential Integral," Ei x, of which tables were 
calculated by J. W. L. Glaisher (Philosophical Transactions of 
the Royal Society of London, 1870, Vol. 160) : 


col x = — du 

u* u5 u7 du 


l_v? }_u? l 

3\3+55~7 


|2 [4 |6 
= cosxj--^+p-p+-...j 

fl |3 |S |7 
H-sinz -§-^ + ^-^5 

I 


71 /^ sin u 
2"Jo « 


du. 

545 


546 

sil x= I du 


APPENDIX 


rx cos u 

ll x= ~>1~( 

Jx 

C*\i-- --- 1-^ 


1 u2 1 u* 1 


IX4 


l |3 |5 |7 
— -^+i=-^ 
a;2 a^ x6 x8 


l 2 |4 |5 
--%4^-ty 

X X3 X5 X7 


fx 
^du; 


where 


r= 0.5772156 . . . 
/•<? t* /•« _„ 

Explx= -dt*= ( — 
Jx u J_x u 


u 


du 
u 


1 M2 1 M3 1 


/« 


where 


l 1 2 

J 


r-x^ 
J« ~v^du' 


x4 x5 


q is given by : expl q = 0 as : q = 0.37249680 . 


APPENDIX 


547 


Expl ( 


f» -u 
—du 
U 


du 


/. 1 u2 1 u3 1 u4 

= /logtt— u + 2 T^- Q 1^+4 17- +• • • 

/ ^ |^S O |O T: |T: 


1 x2 1 x3 1 14 


du. 


Tables of these four functions, reduced from the Glaisher 
tables, are given in the following for 6 decimals, and their 
first part plotted in Fig. 103. 

These functions have the following properties: 


col 0 = ^ 

sil 0 •-•= + oc 
expl 0 = + oo 
expl q = 0 


col 


= 0 


sil oo =0 

expl ( + oo ) = - oo 
expl (— co)=0 


col x has maxima at the even, sil x at the odd quadrants, 
and these maxima are alternately positive and negative; that is, 


col ^2s =max. 


sil o(2s— l)=max. 


= integer number. 


548 


APPENDIX 


.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 

BIG. 103. 


APPENDIX 549 

For large values of s, the numerical values of these maxima 
approach the values: 


for: s>40, the approximation is correct to the 6th decimal. 
For small values of x, the approximations hold : 

si\x = log- -0.5772156, 


For large values of x, it is : 

2 sin x 2 sin x 


2 cos x 2 cos 

- < col x < — 


where n\ and n2 are the two successive quadrants, between 
which x lies. For instance : 


sin x = sin 1.5^"= 0.707, 


cos x = cos 1.5^-= —0.707, 


550 APPENDIX 

and 0.01 725 <sil x< 0.01805, 

-0.01805 < col x< -0.01725. 
It is, approximately, for large values of x, if : 


col x = co\ —ucos v, 


sil x = sil pr-ucos v, 


where 


and u is the nearest even quadrant to x, for col x, and the nearest 

odd quadrant to x, for sil x, and col ^ u or sil ^ u are given by 
Table VII. 

For instance : 


vi= -0.785 v2= +0.785 

col 26 £ = -0.0245 sil 25 £ = 0.0254 

_ l 

'cos (-0.785) =0.707 cos 0.785 = 0.707 

col 40 = -0.01730 sil 40 - 0.01795 

From the series expressions of these functions follows: 
col ( — x) =n — col x 
sil (— x) =sil x 
expl (jx) = sil x + y col x - / 


APPENDIX 551 

. 7T 

expl ( — jx)=su x—] col £ + 7 o 

£ 

col (±jx) = ±|iexpl x-expl ( — x)\ +/TT 

1 

sil (±jx) =^-{expl x + expl ( — x)} 


552 


APPENDIX 


TABLE I 

Col x and sil x from 0.00 to 1.00 


X 

cola; 

J 

sil x 

J 

.00 

+ 1 . 570 796 

in onn 

GO 

.01 
.02 
.03 
.04 

1.560 796 
1 . 550 797 
1.540 798 
1 . 530 800 

9 999 
9 999 
9 998 

Q QQ7 

+ 4.027 980 
3.334 907 
2.929 567 
2.642 060 

693 073 
405 340 

287 507 

999 QIC 

ftK 

1 520 803 

2 419 149 

zzz yio 

9QQC 

.06 
.07 
.08 
.09 

1.510 808 
1.500 815 
1.490825 
1.480837 

9 993 
9 990 

9 988 

9QO r 

2.237 095 
2.083 269 
1.950 113 
1.832 754 

loZ U4/ 
153 826 
133 156 
117 359 

.10 

1.470 852 

yoo 

1 . 727 868 

104 886 

.11 
.12 
.13 
.14 

1.460 870 
1.450 892 
1.440 918 
1.430 949 

9 982 
9 978 
9 974 
9 969 

9QA K 

1.633 083 
1 . 546 646 
1.467 227 
1.393 793 

94 785 
86 437 
79 419 
73 434 

.15 

1.420 984 

1.325 524 

68 269 

.16 
.17 
.18 
.19 

1.411 024 
1.401 069 
1.391 120 
1.381 177 

9 960 
9 955 
9 949 
9 943 

9QQ7 

1.261 759 
1.201 957 
1.145 672 
1.092 527 

63 765 
59 802 
56 285 
53 145 

.20 

1.371 240 

\)6i 

1.042 206 

50 321 

.21 
.22 
.23 
.24 

1.361 310 
1.351 387 
1.341 471 
1.331 563 

9 930 
9 923 
9 916 
9 908 ' 
9onn 

.994 437 
.948 988 
.905 656 
.864 266 

47 769 
45 449 
43 332 
41 390 

.25 

1.321 663 

yuu 

.824 663 

39 603 

.26 
.27 

.28 
.29 

1.311 771 
1.301 888 
1.292 013 

1.282 148 

9 892 
9 883 
9 875 
9 865 

.786 710 
.750 287 
.715 286 
.681 610 

37 953 
36 423 
35 001 
33 676 

.30 

1.272 292 

006 

.649 173 

32 437 

.31 
.32 
.33 
.34 

1.262 447 
1.252 611 
1.242 786 
1.232 972 

9 845 
9 836 
9 825 
9 814 

.617 896 
.587 710 
.558 549 
.530 355 

31 277 
30 186 
29 161 
28 194 

.35 

1.223 169 

803 

.503 076 

27 279 

.36 
.37 

.38 
.39 

1.213 378 
1.203 599 
1.193 832 
1 . 184 077 

9 791 
9 779 
9 767 
9 755 

9*7,49 

.476 661 
.451 067 
.426 252 
.402 178 

26 414 
25 594 
24 815 
24 074 

.40 

1.174 335 

/4Z 

.378 809 

23 369 

.41 
.42 
.43 
.44 

1 . 164 606 
1.154 891 
1.145 189 
1.135 501 

9 729 
9 715 
9 702 
9 688 

.356 114 
.334 062 
.312 625 
.291 776 

22 695 
22 052 
21 437 
20849 

.45 

1 . 125 828 

O/o 

.271 492 

20 284 

.46 
.47 

.48 
.49 

1.116 170 
1.106 526 
1.096 898 
1.087 285 

9 658 
9 644 
9 628 
9 613 

.251 749 
.232 526 
.213 803 
.195 562 

19 743 
19 223 
18 723 
18 241 

.50 

+ 1 . 077 689 

_ O.M) 

+ 0.177 784 

17 778 

APPENDIX 
TABLE I— Continued 


553 


X 

col x 

J 

sil x 

J 

50 

+ 1 077 689 

9 596 

+ 0 177 784 

17 778 

9C01 

1 7 oo-i 

.51 
.52 
.53 
.54 

1.068 108 
1.058 545 
1.048998 
1.039448 

9 563 
9 547 
9 530 

9r 1 9 

.160453 
.143 554 
.127 071 
.110 990 

11 OOl 

16 899 
16 483 
16 081 
i £ AQH 

55 

1 029 956 

Ol/ 

095 300 

lo oyu 

9JOK 

i ^ *?i d. 

.56 
.57 

.58 
.59 

1.020 461 
1.010 985 
1.001 527 
.992 088 

9476 
9 458 
9439 

94.91 

.079 986 
.065 037 
.050 441 
.036 190 

14 949 
14 596 
14 251 

1 Q Q1 Q 

60 

982 667 

022 271 

9AC\1 

.61 
.62 
.63 
.64 

.973 266 
.963 885 
.954 523 
.945 182 

4U1 

9 381 
9 362 
9 341 
9 001 

+ .008 675 
- .004 e06 
- .017 582 
- .030 260 

lo olo 
13 281 
12 976 
12 678 

1 9 QQfl 

.65 

.935 861 

641 

- .042 650 

iz oyu 

.66 
.67 
.68 
.69 

.926 561 
.917 282 
.908 024 

.898 788 

9 300 
9 279 

9 258 
9 236 

991 J. 

- .054 758 
- .066 591 
- .078 158 
- .089 463 

12 108 
11 833 
11 567 
11 305 

nnco 

.70 

.889 574 

- .100 515 

.71 
.72 
.73 

.74 

.880 382 
.871 213 
.862 066 
.852 942 

9 192 
9 169 
9 147 
9 124 
91 on 

- .111 318 
- .121 879 
- .132 203 
- .142 296 

10 803 
10 561 
10 324 
10 093 

9QAQ 

75 

843 842 

luu 

— 152 164 

oOo 

9fY77 

9RAK 

.76 

.77 
.78 
.79 

.834 765 
.825 713 
.816 684 
.807 680 

9 052 
9 029 
9 004 
8osn 

- .161 810 
- .171 240 
- .180 458 
- . 189 470 

O4O 

9 430 
9 218 
9 012 
8 s no 

.80 

.798 700 

you 

- .198279 

ouy 

.81 
.82 
.83 
.84 

.789 746 
.780 817 
.771 913 
.763 035 

8 954 

8929 
8904 

8 878 

80 r -I 

- .206 889 
- .215 305 
- .223 530 
- .231 568 

8 601 
8416 
8 225 
8 038 

85 

754 184 

OOl 

239 423 

OOO 

8 one 

7fi7 ^ 

.86 

.87 
.88 
.89 

.745 358 
.736 560 

.727 788 
.719 043 

8 798 
8 772 
8 745 

871 Q 

- .247 098 
- .254 597 
- .261 923 
- .269 079 

7 499 
7 326 
7 156 

6QQQ 

.90 

.710 325 

- .276 068 

.91 
.92 
.93 
.94 

.701 636 
.692 974 
.684 340 
.675 735 

8 689 
8 662 
8 634 
8 605 

8K77 

- .282 893 
- .289 558 
- .296 064 
.302 415 

6 825 
6 665 
6 506 
6 351 

61 QQ 

.95 

.667 158 

.30S 614 

iyy 

.96 

.97 
.98 
.99 

.658 610 
.650 092 
.641 602 
.633 143 

8 548 
8 518 
8 490 
8 459 

8 A Of\ 

- .314 662 
- .320 563 
- .326 319 
- .331 931 

6 048 
5 901 
5 756 
5612 

5170 

1.00 

+ 0.624 713 

4oU 

-0.337 404 

•1M 

554 


APPENDIX 


TABLE II 

Expl x and expl (-x) from 0.00 to 1.00 


X 

expl x 

J 

expl (-a;) 

A 

.00 

+ GO 

00 

.01 
.02 
.03 
.04 

+ 4.017 929 
3.314 707 
2.899 116 
2.601 257 

703 223 
415 591 

297 859 

+ 4.037 930 
3.354 708 
2.959 119 
2.681 264 

684 222 
395 589 
277 855 

05 

2 367 885 

Zoo otZ 

2 467 898 

213 366 

.06 
.07 
.08 
.09 

2.175 283 
2.010 800 
1.886 884 
1.738 664 

JL\)A DU^ 

164 483 
143 916 
128 220 

2.295 307 
2.150 838 
2.026 941 
1.918 745 

i/z oyl 
144 469 
123 897 
108 196 

10 

1 622 813 

llo ool 

1 822 924 

95 821 

1 f\ C 0 CA 

.11 

.12 
.13 
.14 

1.516 959 
1.419 350 
1.328655 
1.243 841 

97 609 
90695 

84 814 

1.737 107 
1.659 542 
1.588 899 
1.524 146 

oO ol/ 

77 565 
70643 
64 753 

15 

1 164 086 

/y i o4 

1 464 462 

59 684 

7 e o re 

.16 
.17 
.18 
.19 

1.088 731 
1.017 234 
.949 148 
.884 096 

to oOo 
71 497 

68 087 
65 052 

1.409 187 
1.357 781 
1.309 796 
1.264 858 

oo ^/o 
51 406 

47 985 
44 938 

.20 

.821 761 

o^ ooo 

1.222 651 

42 207 

.21 
.22 
.23 
.24 

.761 872 
.704 195 
.648 529 
.594 697 

59 889 
57 676 
55 666 
53 832' 

1.182 902 
1.145 380 
1.109 883 
1.076 235 

39 749 
37 522 
35 497 
33 648 

.25 

.542 543 

QL loo 

1.044 283 

31 952 

.26 
.27 
.28 
.29 

.491 932 
.442 741 
.394 863 
.348 202 

50 611 
49 191 

47 878 
46 662 

1.013 889 
.984 933 
.957 308 
.930 918 

30 394 
28 956 
27 625 
26 390 

.30 

.302 669 

4o Ooo 

.905 677 

25 241 

.31 
.32 
.33 
.34 

.258 186 
.214 683 
.172 095 
. 130 363 

44 482 
43 503 
42 588 
41 732 

.881 506 
.858 335 
.836 101 
.814 746 

.24 171 
23 171 
22 234 
21 355 

.35 

.089 434 

4u y^y 

.794 215 

20 531 

.36 
.37 
.38 
.39 

.049 258 
+ .009 790 
- .029 Oil 
- .067 185 

40 176 

39 468 
38 801 
38 173 

.774 462 
.755 441 
.737 112 
.719437 

19 753 
19 021 
18 329 
17 675 

.40 

- .104 765 

61 Ool 

.702 380 

17 057 

.41 
.42 
.43 
.44 
45 

- .141 786 
- .178278 
- .214 270 
- .249 787 
284 855 

37 021 
36492 
35 991 
35 517 
35 068 

.685 910 
.669 997 
.654 613 
.639 733 

fl'JK '-JQ1 

16 470 
15 913 
15 384 
14 880 
14 402 

.46 
.47 
.48 
.49 

.319 497 
- .353 735 

- .387 589 
- .421 078 

34 642 
34 238 
33 854 
33 489 

.611 387 
. 597 877 
.584 784 
.572 089 

13 944 
13 510 
13 093 
12 695 

.50 

-0.454 220 

33 142 

+ 0.559 774 

12 315 

APPENDIX 
TABLE II— Continued 


555 


X 

expl x 

J 

expl ( — x) 

A 

.50 

-0.454 220 

33 142 

+ 0.559 774 

12 315 

.51 

.52 
.53 
.54 

- .487 032 
- .519 531 
- .551 730 
- .583 646 

32 812 

32 498 
32 200 
31 915 

o-i nA r 

.547 822 
.536 220 
.524 952 
.514 004 

11 952 
11 602 
11 268 
10 948 

.55 

- .615 291 

.503 364 

10 640 

.56 
.57 

.58 
.59 

- .646 677 
- .677 819 
- .708 726 
- .739 410 

31 387 
31 141 
30 907 
30 684 

.493 020 
.482 960 
.473 173 
.463 650 

10 344 
10 060 

9 787 
9 523 

.60 

- .769 881 

6(J QiZ 

.454 380 

9 270 

.61 
.62 
.63 
.64 

- .800 150 
- .830 226 
- .860 119 
- .889 836 

30 269 
30 076 
29 892 
29 717 
90 £^n 

.445 353 
.436 562 
.427 997 
.419 652 

9 027 
8 791 
8 565 
8 345 

.65 

- .919 386 

zy oou 

.411 517 

135 

.66 
.67 
.68 
.69 

- .948 778 
- .978 019 
-1.007 116 
-1.036 077 

29 392 
29 241 
29 079 
28 960 

.403 586 
.395 853 
.388 309 
.380 950 

7 931 
7 733 
7 544 
7 359 

.70 

-1.064 907 

28 80 1 

.373 769 

7 181 

.71 
.72 
.73 
.74 

-1.093 615 
-1.122 205 
-1.150 684 
-1.179 058 

28 707 
28 590 
28 479 
28 374 

.366 760 
.359 918 
.353 237 
.346 713 

7 009 
6 842 
6 681 
6 524 

.75 

-1.207 333 

Zo Z. O 

.340 341 

6 3,2 

.76 

.77 
.78 
.79 

-1.235 513 
-1.263 605 
-1.291 613 
-1.319 542 

28 181 
28 092 
28 008 
27 929 

97 Q££ 

.334 115 
.328 032 
.322 088 
.316 277 

6 226 
6 083 
5 944 
5 811 

.80 

1.347 397 

Z< OOO 

.310 597 

5 680 

.81 

.82 
.83 

.84 

-1.375 182 
-1.402 902 
-1.430 561 
-1.458 164 

27 785 
27 720 
27 659 
27 603 

.305 043 
.299 611 
.294 299 
.289 103 

5 554 
5 432 
5 312 
5 196 

.85 

-1.485 714 

27 ooU 

.284 019 

5 084 

.86 

.87 
.88 
.89 

-1.513 216 
- 1 . 540 673 
- 1 . 568 089 
- 1 . 595 467 

27 502 
27 457 
27 416 
27 379 

97 QA £ 

.279 045 
.274 177 
.269 413 
.264 749 

4 974 
4 868 
4 764 
4 664 

.90 

-1.622 812 

£i O4O 

.260 184 

565 

.91 
.92 
.93 
.94 

-1.650 126 
-1.677 413 
-1.704 677 
-1.731 920 

27 314 

27 287 
27 264 
27 243 

97 99R 

.255 714 
.251 336 
.247 050 
.242 851 

4 470 
4 378 

4 286 
4 199 

95 

— 1 759 146 

238 738 

llo 

97 91 1 

.96 
.97 
.98 
.99 

-1.786 357 
-1.813 557 
-1.840 749 
-1.867 935 

Z/ ZLL 

27 200 

27 192 
27 186 

97 1 QQ 

.234 708 
.230 760 
.226 891 
.223 100 

UoU 

3 948 
3 869 
3 791 

1.00 

-1.895 118 

Zt loo 

+ 0.219 384 

/ lo 

556 


APPENDIX 


TABLE III 

Col x and sil x from 0.0 to 5.0 


X 

col x 

J 

sil x 

A 

.0 

+ 1.570 796 

oo 

.1 
.2 
.3 
.4 

1.470 852 
1.371 240 
1.272 292 
1.174 335 

+ 1.727 868 
1.042 206 
.649 173 
.378 809 

1 H77 ASQ 

mnoA 

.6 

.7 
.8 
.9 

.982 667 
.889 574 
.798 700 
.710 325 

+ .022 271 
- .100 515 
- .198 279 
- .276 068 

10 

R94. 71 "^ 

007 A(\A 

_i_ 09 cno 

1.1 
1.2 
1.3 
1.4 

.542 111 
.462 749 

.387 838 
.314 570 

79 362 
75 911 

72 269 

f»o A cn 

- .384 873 
- .420 459 
- .445 739 
- .462 007 

+ 4/ 4by 
35 586 
25 279 
16 -268 

1.5 

.246 113 

Do 4O/ 

- .470 356 

o4y 

1.6 
1.7' 

1.8 
1.9 

.181 616 
.121 204 
.064 979 
+ .013 021 

64 497 
60412 
56 225 
51 958 

A 7 ftQQ 

- .471 733 
- .466 968 
- .456 811 
- .441 940 

+ 1 377 
- 4 765 
-10 157 

-14 871 

2.0 

- .034 617 

*±l DOo 

- .422 981 

— 18 959 

2.1 
2.2 
2.3 
2.4 

- .077 902 
- .116 839 
- .151 411 

- .18 689 

43 285 
38 927 
34 582 
30 278 ' 

- .400 512 
- .375 075 
- .347 176 
- .317 292 

-22 469 
- 25 437 

-27 899 
-29 884 

2.5 

- .207 724 

Zo Ooo 

- .285 871 

— 31 421 

2.6 
2.7 

2.8 
2.9 

- .229 598 
- .247 416 
- .261 300 
- .271 394 

21 874 
17 818 
13 884 
10 094 

- .253 337 
- .220 085 
- . 186 488 
- .152 895 

-32 534 
-33 252 
-33 597 
-33 593 

3.0 

- .277 856 

4oZ 

- .119 630 

— 33 265 

3.1 
3.2 
3.3 
3.4 

- .280 863 
- .280 605 
- .277 284 
- .271 118 

+ 3 007 
258 
- 3 319 
- 6 166 

- .086 992 
- .055 257 
- .024 678 
+ .004 518 

-32 638 
-31 735 
-30 579 
-29 196 

3.5 

- .262 329 

to\J 

.032 129 

— 27 610 

3.6 
3.7 

3.8 
3.9 

- .251 152 
- .237 825 
- .222 594 
- .205 705 

— 11 177 
- 13 323 
-15 231 

- 16 889 

.057 974 
.081 901 
.103 778 
.123 499 

-25 846 
-23 927 
-21 877 
-19 721 

4.0 

- .187 407 

— lo Z\)o 

. 140 982 

— 17 483 

4.1 

4.2 
4.3 
4.4 

- .167 947 
- .147 572 
- .126 524 
- .105 038 

— 19 460 
-20 375 
-21 048 
-21 486 

.156 165 
.169 013 
.179 510 
.187 660 

-15 183 
- 12 848 
- 10 497 
- 8 150 

4.5 

- .083 344 

— ZL by4 

.193 491 

- 5 831 

4.6 
4.7 

4.8 
4.9 

- .061 664 
- .040 209 
- .019 179 
+ .001 237 

-21 680 
-21 455 
-21 030 
-20 416 

.197 047 
.198 391 
.197 604 
.194 780 

- 3 556 
- 1 344 

+ 787 
2 824 

5.0 

+ 0.020 865 

— iy o^o 

+ 0.190 030 

+ 4 750 

APPENDIX 


557 


TABLE IV 
expl x and expl ( — x) from 0.0 to 5.0 


X 

expl x 

J 

expl (-x) 

A 

.0 

+ 00 

00 

.1 
.2 
.3 
.4 

+ 1.622 813 

+ .821 761 - 
+ .302 669 
. 104 765 

+ 1.822 924 
1.222 651 
.905 677 
.702 380 

AX.A 99fl 

CCQ 77/1 

.6 

.7 
.8 
.9 

- .769 881 
- 1.064 907 
- 1.347 397 
- 1.622 812 

.454 380 
.373 769 
.310 597 
.260 184 

i f\ 

i on c 1 1 Q 

91 Q ^84. 

1.1 
1.2 
1.3 
1.4 

- 2.167 378 
- 2.442 092 
- 2.721 399 
- 3.007 207 

272 260 
274 714 
279 306 

285 809 

9Q/L O7S 

. 185 991 
.158 408 
.135 451 
.116 219 

oo oyo 
27 583 
22 957 
19 232 

1 ft 1 QQ 

1.5 

- 3.301 285 

zy<± u/o 

. 100 020 

ID iyy 

1.6 
1.7 

1.8 
1.9 

- 3.605 320 
- 3.920 963 
- 4.249 868 
- 4.593 714 

304 034 
315 643 

328 904 
343 846 

.086 308 
.074 655 
.064 713 
.056 204 

13 712 
11 653 
9 942 
8 509 

2 0 

4 954 234 

ooU OZL 

048 901 

olio 

070 nni 

69S7 

2.1 

2 2 
2~3 
2.4 

- 5.333 235 
- 5.732 615 
- 6.154 381 
- 6.600 670 

399 379 
421 766 
446 289 

A7Q flQA 

.042 614 
.037 191 
.032 502 
.028 440 

Zo< 
5423 
4 689 
4 062 

3FQr 

2.5 

- 7.073 766 

4/O U»O 

.024 915 

oZo 

2.6 

2.7 
2.8 
2.9 

- 7.576 115 
- 8.110 347 
- 8:679 298 
- 9.286 024 

502 349 
534 233 
568 950 
606 726 

.021 850 
.019 182 
.016 855 
.014824 

3 065 
2 668 
2 327 
2 031 

3 0 

— 9 933 833 

647 808 

013 048 

776 

3.1 
3.2 
3.3 
3.4 

-10.626 300 
-11.367 303 
-12 161 041 
-13.012 075 

oyz 4oo 
741 002 
793 739 
851 034 

.011 494 
.010 133 
.008 939 
.007 891 

OO4 

1 361 
1 194 
1 048 

Q91 

3.5 

13 925 354 

yio z/y 

.006 970 

yji 

3.6 
3.7 

3.8 
3.9 

-14.906 254 
-15.960 619 
-17.094 802 
-18.315 714 

980 900 
1.054 365 
1.134 183 
1.220 912 

.006 160 
.005 448 
.004 820 
.004 267 

810 
712 
628 
553 

4 0 

— 19 630 874 

. olo loU 

003 779 

48» 

1X17 CQO 

jorv 

4.1 
4.2 
4.3 
4.4 

-21.048 467 
-22.577 401 
-24.227 380 
-26.008 973 

.'I I/ O\)Z 

1.528 934 
1.649 979 
1.781 593 

1Q94. 79*} 

.003 349 
.002 969 
.002 633 
.002 336 

4oU 

380 
336 
297 

4 5 

— 27 933 697 

002 073 

Juo 

4.6 

4.7 
4.8 
4.9 

-30.014 099 
-32.263 860 
-34.697 890 
-37.332 451 

. UoU 4Uo 

2.249 760 
2.434 030 
2.634 561 

20C9 QQ r 

.001 841 
.001 635 
.001 453 
.001 291 

Z6Z 
206 
182 
162 

5.0 

-40.185 275 

.00^ o^o 

+ 0.001 148 

14o 

.558 


APPENDIX 


TABLE V 

Col x, sil x, expl x, and expl ( — x) from 0 to 15 


X 

col x 

sil x 

expl x 

expl (-x} 

x 

x° 

0 

+ 1 . 570796 

GO 

00 

00 

0 

0 

1 
2 
3 
4 

+ .624713 
- .034617 
- .277856 
- .187407 

-.337404 
- .422981 
-.119630 
+ .140982 

1.895118 
4.954234 
9.933833 
19.630874 

+ .219384 
.048901 
.013048 
.003779 

1 

2 
3 
4 

57.2958 
114.5916 
171.8874 
229 . 1832 

5 

+ .020865 

+ .190093 

40.185275 

.001148 

5 

286.479 

6 

7 
8 
9 

+ .146109 
+ .116200 
- .003391 
- .094244 

+ .068057 
-.076695 
- . 122434 
-.055348 

85.990 
191 . 505 
440 . 380 
- 1037.878 

.000360082 
.000115482 
.000 37666 
.000012447 

6 

7 
8 
9 

343.775 
401.071 
458.366 
515.662 

10 

- .087551 

+ .045456 

2492.229 

.000004157 

10 

572.958 

11 

12 
13 
14 

- .007511 
+ .065825 
+ .071435 
+ .014585 

+ .089563 
+ .049780 
-.026764 
-.069396 

- 6071.406 
- 14959 
- 37198 
- 93193 

.000001400 
.000000475 
.000000162 
.000000056 

11 
12 
13 
14 

630.254 
687.550 
744 . 846 
802 . 142 

15 

- .047398 

-.046279 

-234956 

+ .000000019 

15 

859.438 

APPENDIX 


559 


TABLE VI 

col x and sil x 


X 

co' x 

sil x 

x 

col x 

sil x 

0 

+ 1.570 796 

GO 

150 

+ .004 6.0 

+ .004 796 

5 
10 
15 
20 
25 
30 
35 
40 
45 

+ .020 865 
- .087 551 
- .047 398 
+ .022 555 
+ .039 314 
+ .004 040 
- .026 126 
- .016 189 
+ .012 081 

+ . 190 093 
+ .045 456 
-.046 279 
-.044 420 
+ .006 849 
+ .033 032 
+ .011 480 
-.019 020 
-.018 632 

160 
170 
180 
190 
200 
300 
400 
500 
600 
700 

-.006 089 
+ .005 529 
-.003 349 
+ .000 377 
+ .002 414 
- .000 085 
- .001 319 
- .001 770 
-.001 665 
001 1 Q8 

-.001 409 
-.002 006 
+ .004 432 
-.005 249 
+ .004 378 
+ .003 332 
+ .002 124 
+ .000 932 
- .000 076 

OOO 77Q 

50 

+ .019 179 

+ .005 628 

800 
QOO 

-.000 559 
_i_ 000 07^ 

-.001 118 

001 1OQ 

cc 

_i_ 000 079 

4- 01 8 173 

60 

cc 

- .015 949 
oofi fi? ^ 

+ .004 813 

012 847 

i ooo 

+ .000 563 

-.000 826 

70 
75 
80 
85 
90 
95 

+ .009 201 
+ .012 217 
- .001 535 
- .011 602 
- .004 867 
+ .007 760 

-.010 922 
+ .005 332 
+ .012 402 
+ .001 935 
-.009 986 
-.007 110 

2 000 
3 000 
4 000 
5 000 
6 000 
7 000 
c 000 

-.000 183 
-.000 325 
-.000 182 
+ .000 031 
+ .000 151 
+ .000 123 
4- 000 008 

- .000 465 
-.000 073 
+ .000 171 
+ .000 198 
+ .000 071 
-.000 072 

000 19^ 

100 

+ .008 571 

+ .005 149 

9 000 

-.000 088 

-.000 068 

110 

1 Of) 

- .009 084 

+ OO7 894. 

+ .000 320 

004- 781 

10 000 

-.000 095 

+ .000 031 

130 
140 

- .002 880 
- .001 363 

+ .007 132 
-.007 Oil 

11 000 
100 000 
1 000 000 

-.000 026 
-.000 010 
+ 000 001 

+ .000 087 
-.000 000 
4- 000 000 

150 

+ .004 630 

+ .004 796 

560 


APPENDIX 
TABLE VII 

7T' 7t 

MAXIMA AND MINIMA OF col —x and sil —x 


X 

col-^c 

2 . Tt 

col— re 

7CX 2 

x 

i*; 

2 .. 7T 

• sil —x 

TtX 2 

0 
2 

+ 1.570 796 
- .281 141 

1 
3 

- .472 001 
+ .198 408 

4 

+ .152 645 

5 

- .123 772 

6 

8 

— .103 966 

+ .078 635 

7 
9 

+ .089 564 
- .070 065 

x + l 

10 

- .063 168 

494 

11 

+ .057 501 

374 

12 

+ .052 762 

290 

13 

- .048 742 

229 

14 

- .045 289 

184 

15 

+ .042 292 

149 

16 

18 

+ .039 665 
- .035 281 

123 

94 

17 

19 

- .037 345 
+ .033 432 

103 
74 

20 

+ .031 767 

64 

21 

- .030 260 

55 

22 

- .028 889 

48 

23 

+ .027 637 

42 

24 

+ .026 489 

37 

25 

- .025 432 

33 

26 

28 

— .024 456 
+ .022 713 

29 
23 

27 
29 

+ .023 552 
- .021 931 

26 
21 

30 

- .021 202 

19 

31 

+ .020 519 

17 

32 

+ .019 879 

15 

33 

- .019 277 

14 

34 

- .018 711 

13 

35 

+ .018 177 

12 

36 

38 

.Ull D/o 

- .016 744 

: 

10 

37 

39 

- .017 196 
+ .016 315 

10 
9 

40 

+ .015 907 

9 

41 

- .015 520 

8 

42 

- .015 151 

8 

43 

+ .014 799 

7 

44 

+ .014 462 

7 

45 

-- .014 141 

6 

46 

48 

— .013 834 
+ .013 258 

6 
5 

47 
49 

+ .013 540 
- .012 988 

5 

4 

50 

- .012 728 

4 

51 

+ .012 480 

3 

52 

+ .012 239 

3 

53 

- .012 008 

3 

54 

- .011 786 

3 

55 

+ .011 572 

3 

56 

58 

.UII ODD 

- .010 974 

2 

57 
59 

- .011 166 
+ .010 788 

3 

2 

60 

+ .010 608 

2 

61 

- .010 434 

2 

62 

- .010 266 

2 

63 

+ .010 103 

2 

64 

+ .009 945 

2 

65 

- .009 792 

2 

66 

68 

— .009 644 
+ .009 360 

2 

67 
69 

+ .009 500 
- 009 225 

2 
2 

70 

- .009 093 

2 

71 

+ .008 965 

1 

72 

+ .008 841 

1 

73 

- .008 719 

1 

74 

7fi 

- .008 602 

+ nno 07 c 

1 

75 

+ .008 487 

1 

<D 

78 

. UUo o/O 

- .008 161 

. 
1 

77 
79 

- .008 267 
+ .008 057 

1 
1 

80 

+ .007 957 

1 

81 

- .007 858 

0 

x> 

80 

j»fe 

79 

X 

sil|s=(-l) 

+ 1 
2~2 

INDEX 

PAGE 

Acceleration constant of traveling wave 466 

Air blast, action in oscillating-current generator 75 

pressure required in oscillating-current generator 75 

Alternating-current circuit and transient term of fundamental frequency 473 

distribution in conductor 369 

transformer operating oscillating-current generator 87 

transmission, equations of traveling wave 477 

wave as traveling wave without attenuation 472 

Alternator control by periodic transient term of field excitation 223 

Aluminum cell rectifier 222 

effective penetration of alternating current 378 

Amplitude of traveling wave 465 

of wave 438 

Arc, and spark 249 

continuity at cathode 249 

lamp, control by inductive shunt to operating mechanism 131 

machine 230 

as rectifier 221 

current control 220 

properties 249 

rectification 249 

rectifiers 222 

resistivities 9 

starting 249 

Arcing ground on lines and cables, as periodic transient phenomenon . . 23 

Armature reactance, reaction and short-circuit current of alternator 199 

Attenuation of alternating magnetic flux in iron 361 

constants 434 

of traveling wave, and loading 462 

Booster, response to change of load 158 

Brush arc machine 221, 230, 242, 248 

Building up of direct-current generator 32 

of overcompounded direct-current machine 49 

Cable, high-potential underground, standing waves 452 

opening under load 112, 118 

short-circuit oscillation 113, 118 

starting 111,117 

transient terms and oscillations 98, 102 

561 


562 INDEX 

PAGE 

Capacity, also see Condenser. 

and inductance, equations 48 

and velocity of propagation 400, 401 

distributed series 348 

energy of complex circuit 517 

in mutual inductive circuit 161 

of electric circuit 112 

range in electric circuit 13 

representing electrostatic component of electric field 5 

shunting direct-current circuit 133 

specific, numerical values 11 

suppressing pulsations in direct-current circuit 134 

Cast iron, effective penetration of alternating current 378 

Cathode of arcs 249 

Charge of condenser 51 

of magnetic field 27 

Circuit, complex, see Complex circuit. 

control by periodic transient phenomena 220, 223 

electric, speed of propagation in 422 

Closed circuit transmission line 306 

Col al 392, 394 

Commutation and rectification 222 

as transient phenomenon 40 

Commutator, rectifying 229 

Complex circuit, of waves 498 

power and energy 513 

resultant time decrement 504 

traveling wave 468 

Compound wave at transition point 532 

Condenser, also see Capacity. 

charge, inductive 18 

noninductive 18 

circuit of negligible inductance 55 

equations 48 

oscillation, effective value of voltage, current and power. ... 70 

efficiency, decrement and output 72 

frequency 62 

general equations 60 

size and rating 69 

starting on alternating voltage 94 

voltage in inductive circuit 49 

Conductance, shunted, effective 12 

Conductors at high frequency 403 

Constant-current mercury arc rectifier 250 

rectification 221, 230 

potential-constant-current transformation by quarter-wave line 308 

mercury arc rectifier 251 

rectification 221, 230 


INDEX 563 

PAGE 

Control of circuits by periodic transient phenomena 220 

Conversion by quarter-wave circuits 313 

Copper, effective penetration of alternating currents 378 

ribbon, effective high frequency impedance 408 

wire, effective high frequency impedance 408 

Cosine wave, traveling 434 

Critical case of condenser charge and discharge 53 

resistance of condenser oscillation 66 

start of condenser on alternating voltage 95 

Current density, in alternating-current conductor 372 

effective, of oscillating-current generator 81 

transformation at transition point of wave 529 

Damping of condenser oscillation 66, 72 

Decay of continuous current in inductive circuit 17 

of wave of condenser oscillation 72 

Decrement of condenser oscillation 65, 72 

resultant time, of complex circuit 504 

Destructive voltages in cables and transmission lines 120 

Dielectric constant, numerical values 11 

strength, numerical values 11 

Dielectric also see Electrostatic. 

Direct-current generator, self-excitation 32 

railway, transient effective resistance 379 

Disappearance of transient term in alternating-current circuit 43 

Discharge of condenser , . 51 

Geissler tube 9 

inductive, as wave 535 

into transmission line '. 542 

of motor field 29 

Disruptive strength, numerical values 11 

voltage in opening direct-current circuit 26 

Distance attenuation constant 434 

in velocity measure . 435 

Distortionless circuit 441, 447 

Distributed series capacity 348 

Distribution of alternating-current density in conductor 369 

of alternating magnetic flux in iron 355 

Divided circuit, general equations 122 

continuous-current circuit without capacity 126 

Dynamostatic machine 220 

Effective current of condenser discharge 70 

voltage and power of oscillating-current generator. . . 81 

layer of alternating-current conductor 377 

penetration of alternating current in conductor 376, 378 

power of complex circuit 514 

of cpndenser oscillation '70 

reactance of armature reaction 200 


561 INDEX 

PAGE 

Effective resistance of alternating-current distribution in conductor, 370, 376 

voltage of condenser oscillation 70 

Efficiency of condenser oscillation 72 

Electric circuit, general equations 428 

field, velocity of propagation 387 

Electrolytes, resistivities 8 

Electrolytic rectifiers 222 

Electromagnetic, also see Magnetic. 

axis of electric field 4 

Electrostatic, also see Dielectric. 

axis of electric field 4 

energy of complex circuit 517 

field, energy of 7 

Elimination of pulsations in direct-current circuit by capacity 134 

Energy of complex circuit 513 

condenser discharge 70 

electric field 4, 7 

transfer in complex circuit 507, 521 

constant of complex circuit 507 

Equations, general, of electric circuit 428 

Even harmonics of half-wave oscillations 483 

Excitation of motor field 27 

Exponential curve of starting current 45 

Field current at armature short-circuit . t 202 

electric, of conductor 3 

energy of 4 

velocity of propagation 387 

excitation, transient term 27 

exciting current, rise and decay 17 

regulation of generator by periodic transient terms 223 

resultant polyphase , 192 

Flat conductor, unequal current distribution 371 

Floating system of control 220 

Fluctuations of current in divided circuit 129 

voltage of direct-current generator with load 149 

Free oscillations 432, 478 

and standing waves 482 

Frequency, absence of effect on circuit oscillation 10 

and starting current of transformer 182 

constant of wave 433 

limit of condenser oscillation 73 

of condenser oscillation 62, 68 

field current at armature short-circuit 203 

oscillation of condenser, transmission line, cable. ... 99, 338 

recurrence of discharge in oscillating-current generator. . 81 

wave 433 

range of condenser oscillation 71 

electromagnetic induction 67 


INDEX 565 

PAGE 

Full-wave oscillation of complex circuit 508 

transmission line 330 

Fundamental frequency of oscillation, cables and transmission lines 103, 105 

Gas pipe, effective high frequency impedance 408 

General circuits with inductance and capacity 174 

without capacity 168 

equations of electric circuit 428 

Generator, direct-current over-compounded, building up 149 

self-excitation 32 

oscillating current 74 

German silver, effective penetration of alternating current 378 

Gradual approach to permanent value 21 

or logarithmic condenser charge and discharge 53 

term, also see Logarithmic. 

Graphite, effective penetration of alternating current 378 

Grounded transmission line 303 


Half-wave oscillation 483, 490 

of complex circuit 509 

transmission line 333 

rectification 221 

Harmonics, even, of half-wave oscillation 483 

Hertzian oscillators 388 

High frequency alternators, momentary short-circuit current 201 

conductor 370, 403 

discharge 388 

oscillating currents by periodic transient terms 220 

oscillations of cables and transmission lines 103, 105 

power surge of low-frequency 105 

stray field and starting current of transformer 189 

Impact angle at transition point of wave 527 

Impedance of conductor at high frequency 407 

effective high frequency 408, 413 

of radiation 396 

traveling wave 460 

Inductance and shunted capacity suppressing pulsations in direct-cur- 
rent circuit 134 

effective, of radiation 394 

energy of complex circuit 515 

in telephone lines 455, 462 

massed, in circuit 537 

of conductor without return 390 

electric circuit 12 

range in electric circuit 13 

representing magnetic component of electric field 5 


566 INDEX 

PAGE 

Induction, magnetic, and starting current of transformer 180 

motor circuit, starting 44 

Inductive discharges into transmission line 542 

shunt to non-inductive circuit 129 

Inductorium, equations 164 

Infinitely long conductor 305 

Input, see Power. 

Instantaneous power in complex circuit 514 

Insulators, resistivities 9 

Iron arc operated by oscillating-current generator 82 

effective penetration of alternating current 348 

wire, current distribution 370 

and pipe, effective high frequency impedance 408 

Laminated iron, alternating magnetic flux 355 

pole series booster, response to voltage change 158 

Layer, effective, of alternating-current conductor 377 

Leakage in telephone lines 455, 463 

Length of wave 433 

Lighting circuit, starting 27, 44 

Lightning arrester, multigap 348 

conductors 370 

discharges 388 

in thunder cloud 350 

Limit condition of condenser equations 50 

of frequency of condenser oscillations 73 

Loading of telephone lines 455, 462 

Local oscillations of cables and lines 103, 105 

Logarithmic decrement of con^jnser oscillation 65 

or gradual condenser charge and discharge 53 

start of condenser on alternating voltage 95 

Low frequency surge in cables and lines 103, 105 

stray field and starting current of transformer 189 

Loxodromic spiral of starting current 46 

Magnetic, also see Electromagnetic. 

density and starting current of transformer 180 

energy of complex circuit 515 

field, energy of 6 

flux, alternating, in iron 355, 361, 363, 365, 366, 367 

materials 10 

saturation, numerical values 10 

Magnet poles, solid, as mutual inductive circuit 155 

Main axes of electric field 46 

wave at transition point 531 

Massed inductance and electric wave 537 

Mechanical rectification 221, 229 

Mercury arc rectifier 250 


INDEX 567 

PAGE 

Metallic conductors, resistivities 8 

magnetic induction 10 

Minimum wave length of oscillating currents 74 

Motor circuit, alternating, starting 44 

field, excitation 27 

Multigap lightning arrester '. 348 

Mutual impedance and velocity of propagation 399 

inductance, equations 143 

and velocity of propagation 397 

inductive circuit with capacity 161 

without capacity 144 

of solid magnet poles 155 

reactance : 143 

Nominal generated e.m.f. and short-circuit current 200 

Noninductive condenser circuit 54 

shunt to inductive circuit 129 

Nonoscillatory, see Gradual or Logarithmic. 

Open-circuit rectification 230 

Opening of cable or transmission line under load 112, 118 

of continuous-current circuit 26 

Open transmission line 299 

Oscillating-current generator 69, 74 

and charging current 85 

high-frequency currents by periodic transient terms 220 

Oscillation, also see Condenser discharge. 

free, of circuit 432, 478 

of rotating field in starting 197 

transmission line 322, 333, 336, 338 

Oscillatory approach to permanent value 21 

case of alternating circuit 93 

or trigonometric condenser charge and discharge 53 

start of condenser on alternating voltage 95 

Oscillograms of mercury arc rectifier 264 

of transformer starting current 190 

Output, also see Power. 

effective, of oscillating-current generator 81 

Overcompounded direct-current generator, building up 149 

Overlap of rectifying arcs 251 

Overreaching of condenser charge 19 

in noninductive branch of inductive circuit 131 

Oversaturated transformer flux of starting current 181 

Penetration, effective depth of, 405 

of alternating current in conductor 376, 378 

magnetic flux in iron 361, 363, 365 

Period of recurrence 218 

wave . . 433 


568 INDEX 

PAGE 

Periodic transient terms 22, 218 

Permanent term of alternating-current circuit 91 

values of electric quantities 16 

Permeability 9 

apparent,, of iron for alternating currents 355, 367 

Phase difference in transmission line 296 

of wave and transient term 45, 91 

Physical meaning of transient term 103 

Polyphase alternator short circuit 202, 204 

m.m.f ., resultant 192 

rectification 230 

Potential regulation by periodic transient terms 223 

Power of complex circuit 513 

component of high frequency impedance 412 

gradient of electric circuit 3 

electric field 48 

output, effective, of oscillating-current generator 81 

radiated by conductor 394, 397 

transfer in complex circuit 521 

constant of complex circuit 507 

Propagation constant of wave 441 

speed of, of wave 422 

field 387 

Pulsation of rotating field in starting 197 

Pyroelectrolytes, resistivities 9 

Quarter-phase rectification 230 

Quarter-wave circuit 313 

oscillation 483, 489 

of complex circuit 509 

transmission line 322 

transformer 312 

transmission line 306, 315 

Quartic equation of divided circuit 126 

Radiation, effective resistance 393 

power of conductor 393, 397 

resistance of conductor 403 

Rail, effective penetration of alternating current 378 

return of single-phase system 370 

transient effective resistance 379 

Railway, direct-current, transient rail resistance 379 

motor, self-excitation as generator 38 

single-phase, rail return 370 

Rating of capacity and inductance 122 

resistors, reactors, condensers 69 

Ratio of currents,1 oscillating-current generator 82 

Reactance, effective, of external field . . 406 


INDEX 569 

PAGE 

Reactance, effective (continued), 

of internal field 406 

of armature reaction 200 

mutual inductive 143 

Reaction, armature, and short-circuit current 199 

Reactor, size and rating 69 

Rectification, and commutation 222 

arc 249 

by periodic transient terms 22, 221 

constant-current 230, 242 

potential 230, 236 

mechanical 229 

open-circuit 230 

polyphase 230 

quarter-phase 230, 242 

reversal or change of circuit connections 221 

short-circuit 230 

single-phase 229, 231, 236 

Rectifier, mercury arc 250 

oscillograms 264 

Rectifying commutator 222, 229 

Recurrent transient terms 218 

Reflected waves 431 

at transition point 527, 531 

Reflection angle at transition point 527 

of wave 525 

Refraction law of wave 534 

of wave 525 

ratio at transition point 534 

Regulation of potential by periodic transient terms . 223 

Remanent magnetism in starting transformer 181 

Resistance, and starting current of transformer 185 

effective, of alternating-current conductor 370, 376 

of electric circuit 12 

radiation, of conductor 393, 403 

range in electric circuits 13 

representing power gradient of electric field 6 

specific, see Resistivity. 

thermal, of conductor 403 

Resistivity, numerical values 8 

Resistor, size and rating 69 

Resonators, Hertzian 388 

Resultant polyphase m.m.f 192 

time decrement of complex circuit 504 

Rise of continuous current in inductive circuit 17 

voltage by transformation at transition point 529 

Rotating field, polyphase 192 

Ruhmkorff coil, equations 164 


570 INDEX 

PAGE 

Salt solution, effective penetration of alternating current 378 

Saturation, magnetic, numerical values 10 

and transformer starting current 180 

Screening effect of alternating currents 378 

Self-excitation of direct-current generator 32 

railway motor as generator 38 

series generator 38 

shunt generator 37 

Self-inductance in direct-current circuits 26 

and short-circuit current of alternator 199 

Series capacity, distributed 348 

generator, self-excitation 38 

motor, self-excitation as generator 38 

Short-circuit current of alternator 199, 201 

oscillation of cables and lines 113, 118 

rectification 229 

Shunt generator, self-excitation 32 

motor field excitation, change 27 

Sil al 392, 394 

Silicon, effective penetration of alternating current 378 

Sine wave traveling 434 

Single-phase alternator short-circuit 202 

railway rail return 370 

rectification 229 

Solid magnet poles as mutual inductive circuit 155, 158 

Space induction, magnetic 10 

Spark discharge in cables and lines as periodic transient phenomenon. . 23 

of condenser as periodic transient phenomenon 22 

Speed, effect on discharge of motor field 29 

propagation in electric circuit 422 

of alternating magnetic flux in iron 366 

Standing waves 439, 442 

and free oscillations 482 

Starting current of transformer, calculation 188 

and frequency 182 

magnetic saturation 180 

remanent magnetism 181 

resistance 185 

stray field 184 

oscillograms 190 

of alternating current 43 

continuous current 27 

polyphase or rotating field 192, 197 

oscillation of cables and lines Ill, 117 

Static, see Electrostatic. 

phenomena 13, 105 

Stationary waves 439, 442 

Steel, effective penetration of alternating current 378 


INDEX 571 

PAGE 

Stored energy of complex circuit 515 

Stranded conductor, effective resistance of current distribution 370 

Stray field and starting current of transformer 184 

Suppression of pulsations of direct current by capacity and inductance. 134 

Synchronous reactance and short-circuit current 200 

rectifier 221 

Telegraph, wireless 388 

cable, submarine, standing waves 454 

Telephone 281 

circuit, long distance, standing waves 454 

Terminal conditions of condenser equations 50 

Tesla transformer and oscillating-current generator 82 

Thermal resistance of conductor 403 

Third harmonic of short-circuit current 213 

Thomson arc machine 221, 230 

Thunder cloud, lightning discharge in 350 

Time attenuation constant 434 

constant, resultant, of complex circuit 504 

decrement, resultant, of complex circuit 504 

local, of traveling wave 460 

Tirrill regulator 223 

Transfer constant of energy, of complex circuit 507 

of energy in oscillation of complex circuit 507, 521 

Transformation ratio at transition point of wave 529 

of voltage and current at transition point 529 

Transformer, alternating, operating oscillating-current generator 87 

distributed capacity 342 

quarter-wave oscillation 312 

starting 44 

and magnetic saturation 180 

Transient rail resistance with direct current 386 

terms, conditions of their appearance 21, 23 

of alternating-current circuit 91 

capacity and inductance, physical meaning 103 

fundamental frequency in alternating-current circuit 473 

periodic 22 

unequal current distribution 379 

Transition period at change of circuit condition 16 

points of wave 498 

Transmission angle at transition point 527 

line, characteristic curves 297 

closed 306 

constants 282 

conversion by 308, 313 

equations 284, 287 

approximate 294 

free oscillation . . 322 


572 INDEX 

PAGE 

Transmission line (continued), 

frequency 100, 280, 320, 338 

full-wave oscillation 336 

general equations of standing waves 449, 452 

grounded 303 

half-wave oscillation 333 

inductive discharges 542 

infinitely long 305 

natural period 280, 320 

open 299 

opening under load 112, 118 

phase difference 296 

quarter-wave 306, 313, 315 

oscillation 322 

radiation 283 

resonance frequency 279 

with higher harmonics 280 

short-circuit oscillation 113, 118 

starting 111,117 

transient terms and oscillations 98, 102 

Transmitted wave at transition point 527, 531 

Traveling sine and cosine waves 434 

waves, general equations 458 

without attenuation, as alternating waves 472 

Trigonometric or oscillatory condenser charge and discharge 53 

term, see Oscillatory. 

Turbo alternators, short-circuit current 201 

Unequal alternating-current distribution in conductor 370 

transient-current distribution in conductor , 379 

Velocity, also see Speed. 

measure o.f distance 435 

of propagation of electric field 387 

Voltage control by transient terms 223 

drop, high frequency conductor 413 

transformation at transition point of wave 529 

variation of direct-current generator, with load 149 

Voltmeter across inductive circuit, pulsation 132 

Water, effective penetration of alternating current 378 

Wave of alternating magnetism in iron 359 

direct or main, and reflected 431 

length of alternating magnetic flux in iron 361, 365 

constant 434 

minimum, of oscillating current 74 

transmission 281 

Wireless telegraphy 388 

X-ray apparatus, equations 82 


'1 883 


6 


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