CHAPTER V. DISTRIBUTED SERIES CAPACITY. 43. The capacity of a transmission line, cable, or high-poten- tial transformer coil is shunted capacity, that is, capacity from conductor to ground, or from conductor to return conductor, or shunting across a section of the conductor, as from turn to turn or layer to layer of a transformer coil. In some circuits, in addition to this shunted capacity, dis- tributed series capacity also exists, that is, the circuit is broken at frequent and regular intervals by gaps filled with a dielectric or insulator, as air, and the two faces of the conductor ends thus constitute a condenser in series with the circuit. Where the elements of the circuit are short enough so as to be represented, approximately, as conductor differentials, the circuit constitutes a circuit with distributed series capacity. An illustration of such a circuit' is afforded by the so-called " multi-gap lightning arrester," as shown diagrammatically in Fig. 90, which consists of a large number of metal cylinders p, q . . . , with small spark gaps between the cylinders, connected between line L and ground G. This arrangement, Fig. 90, can be represented diagrammatically by Fig. 91. Each cylinder has a capacity (70 against ground, a capacity C against the adja- cent cylinder, a resistance r, — usually very small, — and an inductance L. If such a series of n equal spark gaps is connected across a & constant supply voltage e0, each gap has a voltage e = — . If, Tl however, the supply voltage is alternating, the voltage does not divide uniformly between the gaps, but the potential difference is the greater, that is, the potential gradient steeper the nearer the gap is to the line L, and this distribution of potential becomes the more non-uniform the higher the frequency; that is, the greater the charging current of the capacity of the cylinder against ground. The charging currents against ground, of all 848 DISTRIBUTED SERIES CAPACITY 349 the cylinders from q to the ground G, Figs. 90 and 91, must pass the gap between the adjacent cylinders p and g; that is, the charging current of the condenser represented by two adjacent -00000000000000-1 Fig. 90. Multi-gap lightning arrester. cylinders p and q is the sum of all the charging currents from qtoG', and as the potential difference between the two cylinders p and q is proportional to the charging current of the condenser 'filMI — i T Fig. 91. Equivalent circuit of a multi-gap lightning arrester. formed by these two cylinders, C, this potential difference increases towards L, being, at each point proportional to the vector sum of all the charging currents, against ground, of all the cylinders between this point and ground. The higher the frequency, the more non-uniform is the poten- tial gradient along the circuit and the lower is the total supply voltage required to bring the maximum potential gradient, near the line L, above the disruptive voltage, that is, to initiate the discharge. Thus such a multigap structure is discriminating regarding frequency; that is, the discharge voltage with increas- 350 TRANSIENT PHENOMENA ing frequency, does not remain constant, but decreases with increase of frequency, when the frequency becomes sufficiently high to give appreciable charging currents. Hence high fre- quency oscillations discharge over such a structure at lower voltage than machine frequencies. For a further discussion of the feature which makes such a multigap structure useful for lightning protection, see A. I. E. E. Transactions, 1906, pp. 431, 448, 1907, p. 425, etc. 44. Such circuits with distributed series capacity are of great interest in that it is probable that lightning flashes in the clouds are discharges in such circuits. From the distance traversed by lightning flashes in the clouds, their character, and the disruptive strength of air, it appears certain that no potential difference can exist in the clouds of such magnitude as to cause a disruptive discharge across a mile or more of space. It is probable that as the result of condensation of moisture, and the lack of uni- formity of such condensation, due to the gusty nature of air currents, a non-uniform distribution of potential is produced between the rain drops in the cloud; and when the potential gradient somewhere in space exceeds the disruptive value, an oscillatory discharge starts between the rain drops, and grad- ually, in a number of successive discharges, traverses the cloud and equalizes the potential gradient. A study of circuits containing distributed series capacity thus leads to an under- standing of the phenomena occurring in the thunder cloud during the lightning discharge.* Only a general outline can be given in the following. 45. In a circuit containing distributed resistance, conductance, inductance, shunt, and series capacity, as the multigap lightning arrester, Fig. 90, represented electrically as a circuit in Fig. 91, let r = the effective resistance per unit length of circuit, or per circuit element, that is, per arrester cylinder; g = the shunt conductance per unit length, representing leakage, brush dis- charge, electrical radiation, etc.; L = the inductance per unit length of circuit; C = the series capacity per unit length of cir- cuit, or circuit element, that is, capacity between adjacent arrester cylinders, and <70 = the shunt capacity per unit length of circuit, or circuit element, that is, capacity between arrester cylinder and * See paper, "Lightning and Lightning Protection," N.E.L.A., 1907. Reprinted and enlarged in " General Lectures on Electrical Engineering," by Author. DISTRIBUTED SERIES CAPACITY 351 ground. If then / = the frequency of impressed e.m.f., the series impedance per unit length of circuit is Z'=r-j(x-xc); (1) the shunt admittance per unit length of circuit is Y - g - jb, (2) where x = 2 nfL, 1 b - 2 xfCt; or the absolute values are (3) z = Vr2 + (x~xcy and (4) y = If the distance along the circuit from line L towards ground G is denoted by Z, the potential difference between point I and ground by E, and the current at point I by 7, the differential equations of the circuit are * f-Z7 (5) and . l-3^ <6> Differentiating (5) and substituting (6) therein gives (7) tM Equation (7) is integrated by where a = VYZ = a - //?, (9) « = ^\{yz + gr — b (x — xc)} and J- (10) /? = * Section III, Chapter II, paragraph 7. 352 TRANSIENT PHENOMENA Substituting (10) in (8) and eliminating the imaginary expo- nents by the substitution of trigonometric functions, E = A,£-al (cos pi + j sin pi) + A2e+al (cos pi - j sin pi). (11) 46. However, if n = the total length of circuit from line L to ground G, or total number of arrester cylinders between line and ground, for I = n, E = 0, (12) and for I = 0, E = e0 = the impressed e.m.f, (13) Substituting (12) and (13) into (11) gives 0 = Ajfi— * (cos pn + j sin pri) + A2e+an (cos pn- - j sin pri) and eo = «i ~t~ ^jJ hence, 1 "" 1 - £-2an (cos 2 ph f j sin 2 /?n) ' (14) A2 = - A^"2"*1 (cos 2pn + j sin 2 /?n), and the potential difference against ground is c'-a'(cos pl+j sin pi) - £-*(2*-f) [cos /? (2 n-Z) +/ sin p(2n-l)] 1 - £~2*n (cos 2pn + j sin 2 /?n) (15) From equation (5), substituting (15) and (9), we have (cos pl + j sin pl)+e~a(2n-l) [cos p(2n-l) +j sin ft (2 n-Z)] 1 _ £-2aw(cos 2pn + j sin (16) Reduced to absolute terms this gives the potential difference against ground as 4/j-s ll + e-2-C2»-o._2e : twcos2/?(n -Z) 6==eoV r- -^i^ — o.-2 , that is, for a very large number of lightning arrester cylinders, where e~2an is negligible, as in the case where the discharge passes from the line into the arrester without reaching the ground, equations (17), (18), (19) simplify to • -V"1, (20) (21) 1 £ and e> = e0xc \Jy-t-*l\ (22) that is, are simple exponential curves. Substituting (4) and (3) in (21) and (22) gives c 2 + (23) C2j[l - (2^/ and ; = 27r/Ce'; (24) or, approximately, if r and