CHAPTER VI. TRANSITION POINTS AND THE COMPLEX CIRCUIT. 40. The discussions of standing waves and free oscillations in Chapters III and V, and traveling waves in Chapter IV, apply directly only to simple circuits, that is, circuits comprising a con- ductor of uniformly distributed constants r, L, g, and C. Indus- trial electric circuits, however, never are simple circuits, but are always complex circuits comprising sections of different con- stants, — generator, transformer, transmission lines, and load, — and a simple circuit is realized only by a section of a circuit, as a transmission line or a high-potential transformer coil, which is cut off at both ends from the rest of the circuit, either by open- circuiting, i = 0, or by short-circuiting, e = 0. Approximately, the simple circuit is realized by a section of a complex circuit, connecting to other sections of 'very different constants, so that the ends of the circuit can, approximately, be considered as reflection points. For instance, an underground cable of low L and high (7, when connected to a large reactive coil of high L and low C, may, approximately, at its ends be considered as having reflection points i = 0. A high-potential transformer coil of high L and low C, when connected to a cable of low L and high (7, may at its ends be considered as having reflection points e = 0. In other words, in the first case the reactive coil may be considered as stopping the current, in the latter case the cable considered as short-circuiting the transformer. This approximation, however, while frequently relied upon in engi- neering practice, and often permissible for the circuit section in which the transient phenomenon originates, is not permissible in considering the effect of the phenomenon on the adjacent sections of the circuit. For instance, in the first case above mentioned, a transient phenomenon in an underground cable connected to a high reactance, the current and e.m.f. in the cable may approx- imately be represented by considering the reactive coil as a reflection point, that is, an open circuit, since only a small current 498 TRANSITION POINTS AND THE COMPLEX CIRCUIT 499 exists in the reactive coil. Such a small current in the reactive coil may, however, give a very high and destructive voltage in the reactive coil, due to its high L, and thus in the circuit beyond the reactive coil. In the investigation of the effect of a transient phenomenon originating in one section of a complex circuit, as an oscillating arc on an underground cable, on other sections of the circuit, as the generating station, even a very great change of circuit constants cannot be considered as a reflection point. Since this is the most important case met in industrial practice, as disturbances originating in one section of a complex circuit usually develop their destructive effects in other sections of the circuit, the investigation of the general problem of a com- plex circuit comprising sections of different constants thus becomes necessary. This requires the investigation of the changes occurring in an electric wave, and its equations, when passing over a transition point from one circuit or section of a circuit into another section of different constants. 41. The equations (50) to (57), while most general, are less convenient for studying the transition of a wave from one circuit to another circuit of different constants, and since in industrial high-voltage circuits, at least for waves originating in the circuits, q and k are very large compared with s and h, as discussed in paragraph 16, s and h may be neglected compared with q and k. This gives, as discussed in paragraph 9, h = o-s, k = + (7/2 C22 C32 (263) (264) f = tan «, TT = tan r, U3 C ' -+• = tan ^ (265) gives (266) TRANSITION POINTS AND THE COMPLEX CIRCUIT 501 and (267) 42. In these equations (266) and (267) ^ is the distance coordinate, using the velocity of propagation as unit distance, and at a transition point from one circuit to another, where the circuit constants change, the velocity of propagation also changes, and thus, for the same time constants s and q, h and k also change, and therewith kl, but transformed to the distance variable I, qX remains the same; that is, by introducing the distance variable X, the distance can be measured throughout the entire circuit, and across transition points, at which the circuit constants change, and the same equations (266) and (267) apply throughout the entire circuit. In this case, however, in any section of the circuit, (268) where Lt and Ct are the inductance and the capacity, respect- ively, of the section i of the circuit, per unit length, for instance, per mile, in a line, per turn in a transformer coil, etc. In a complex circuit the time variable t is the same throughout the entire circuit, or, in other words, the frequency of oscillation, as represented by q, and the rate of decay of the oscillation, as represented by the exponential function of time, must be the same throughout the entire circuit. Not so, however, with the distance variable Z; the wave length of the oscillation and its rate of building up or down along the circuit need not be the same, and usually are not, but in some sections of the circuit the wave length may be far shorter, as in coiled circuits as transformers, due to the higher L, or in cables, due to the higher C. To extend the same equations over the entire complex circuit, it therefore becomes necessary to substitute for the distance variable / another distance variable X of such character that the wave length has the same value in all sections of the complex circuit. As the wave length of the section i is — = > this is done by changing the unit distance by the factor cr = VLf!^ The distance unit of 502 TRANSIENT PHENOMENA the new distance variable ^ then is the distance traversed by the wave in unit time, hence different in linear measure for the different sections of the circuit, but offers the advantage of carrying the distance measurements across the entire circuit and over transition points by the same distance variable A. This means that the length ^ of any section i of the complex circuit is expressed by the length ^ = a-^. The introduction of the distance variable A also has the advan- tage that in the determination of the constants r, L, g, C of the different sections of the circuit different linear distance measure- ments I may be used. For instance, in the transmission line, the constants may be given per mile, that is, the mile used as unit length, while in the high-potential coil of a transformer the turn, or the coil, or the total transformer may be used as unit of length I, so that the actual linear length of conductor may be unknown. For instance, choosing the total length of conductor in the high-potential transformer as unit length, then the length of the transformer winding in the velocity measure ^ is >10 = \/L0C0, where L0 — total inductance, C0 = total capacity of transformer. The introduction of the distance variable ^ thus permits the representation in the circuit of apparatus as reactive coils, etc., in which one of the constants is very small compared with the other and therefore is usually neglected and the apparatus considered as "massed inductance,'7 etc., and allows the investi- gation of the effect of the distributed capacity of reactive coils and similar matters, by representing the reactive coil as a finite (frequently quite long) section ^0 of the circuit. 43. Let y*0, Av >^2, ... kn be a number of transition points at which the circuit constants change and the quantities may be denoted by index 1 in the section from ^0 to Xv by index 2 in the section from Xl to X2, etc. At X = ^ it then must be i1 = i2, e1 = e2; thus substituting A = ^ into equations (266) and (267) gives cos q2(^+t-d2. (269) TRANSITION POINTS AND THE COMPLEX CIRCUIT 503 Herefrom it follows that «, = sin (274) 504 TRANSIENT PHENOMENA or — e~s^ [C cos q (A + 0 + D sin q (X +'01} and j~f e =\/-£-(u+s» {s+s*[Acosq()( - t)+Bsinq(l-t)] (275) where s may be positive or negative. From equation (269) it then follows that ul + sl = u2 + s2 = M3 + s3 = . . . = un + sn = w0, (276) where w17 w2, w3, etc., wn are the time constants of the individual sections of the complex circuit, ^ ( 7 + ^ )> an<^ uo may ^e callet^ 2 \L LI the resultant time decrement of the complex circuit. 45. Equation (269), by canceling equal terms on both sides, then assumes the form A1e+'>*' cos [q (^- t) - «J - 5^— '*« cos [g (^, + 0 - &] = A2^+S2^ cos b (A - 0— «21 - ^2^"S2Al cos [q (^ + t) - ft], and, resolved for cos <# and sin qt, this gives the identities coB (q^ - aj - 5^-'^ cos (^x - ft) • cos (g/l, - «2) - 52£-S2;i cos (q^ - ft), sin (q^ - at) + J?^-'1^ sin (g^ - /?x) = sin fe^1 - «a) + 52£~S2yl1 sin (^, - ft). (277) These identities resulted by equating it = i2 from equation (272). In the same manner, by equating el and e2 from equation (272) there result the two further identities [Af**** cos (q^ - a,) + BlS-^ cos (ql, - ft)} cos (g^ - «2) + J5,e-^ cos (q^ - ft) }, TRANSITION POINTS AND THE COMPLEX CIRCUIT 505 sn I, - a2) - Bf^ sin fort, - &) }. (278) Equations (277) and (278) determine the constants of any section of the circuit, A2, B2, a2, /?2, from the constants of the next section of the circuit, Av Bv av /?r Let cos sin - a) == A'; - a) = A"; - /?) = B' , -v& \ a Then 2c2A2' = (c, + c2) A/ + (c, - c2)fi/, 2c.fi/ - (cl + c,)Bl' + (c1 -cJAt', 2c2A2" = (cl + c,)A1" -(c, -c,)B,', and since by (279) : A'2 + etc., substituting herein (281), 4 c22A22£+2S!'1' = (c, + c2)2 At2e+2"^ + (c,- c2) (279) (280) (281) (282) (283) 506 and tan TRANSIENT PHENOMENA ct-ca B! 2 j sinCg^-ft) JL — i * / \ sm 1 i Cl~C2 £i -2^ COS (g^- ft) *"l r ~r~£ ^ ^ \ tan sn cos - tan (g^- ft). (284) In the same manner, equating, for ^ = ^, in equations (275) the current t\, corresponding to the section from >10 to Xv with the current iv corresponding to the section from ^ to \v and also the e.m.fs., e2 = ev gives the constants in equations (275) and (274), of one section, ^ to Xv expressed by those of the next adjoining section, X0 to Ait as where cos 2 sin 2 cos 2 sin 2 sin 2 g^) } cos 2 g/lj } sin 2 g/lj } cos 2 g/ij } (285) _ - (286) (287) TRANSITION POINTS AND THE COMPLEX CIRCUIT 507 46. The general equation of current and e.m.f. in a complex circuit thus also consists of two terms, the main wave A in equations (272), (273), and its reflected wave B. The factor e-(»+^ = e-^ jn equations (273) and (275) repre- sents the time decrement, or the decrease of the intensity of the wave with the time, and as such is the same throughout the entire circuit. In an isolated section, of time constant u, the time decrement, from Chapters III and V, is, however, e~ut; that is, with the decrement e~ut the wave dies out in the isolated sec- tion at the rate at which its stored energy is dissipated by the power lost in resistance and conductance. In a section of the circuit connected to other sections the time decrement e~U(* does not correspond to the power dissipation in the section; that is, the wave does not die out in each section at the rate as given by the power consumed in this section, or, in other words, power transfer occurs from section to section during the oscillation of a complex circuit. If s is negative, u0 is less than u, and the wave dies out in that particular section at a lesser rate than corresponds to the power consumed in the section, or, in other words, in this section of the complex circuit more power is consumed by r and g than is sup- plied by the decrease of the' stored energy, and this section, therefore, must receive energy from adjoining sections. Inversely, if s is positive, u0 > u, the wave dies out more rapidly in that section than its stored energy is consumed by r and g] that is, a part of the stored energy of this section is transferred to the adjoining sections, and only a part — occasionally a very small part — dissipated in the section, and this section acts as a store of energy for supplying the other sections of the system. The constant s of the circuit, therefore, may be called energy transfer constant, and positive s means transfer of energy from the section to the rest of the circuit, and negative s means reception of energy from other sections. This explains the vanishing of s in a standing wave of a uniform circuit, due to the absence of energy transfer, and the presence of s in the equations of the traveling wave, due to the transfer of energy along the circuit, and in the general equations of alternating-current circuits. It immediately follows herefrom that in a complex circuit some of the s of the different sections must always be positive, some negative. 508 TRANSIENT PHENOMENA In addition to the time decrement s~(u + s)i = £~uot the waves in equations (273) and (275) also contain the distance decrement £+sl for ^e mam wave, e~s* for the reflected wave. Negative s therefore means a decrease of the main wave for increasing X, or in the direction of propagation, and a decrease of the reflected wave for decreasing X, that is, also in the direction of propagation; while positive s means increase of main wave as well as reflected wave in the direction of propagation along the circuit. In other words, if s is negative and the section consumes more power than is given by its stored energy, and therefore receives power from the adjoining sections, the electric wave decreases in the direction of its propagation, or builds down, showing the gradual dissipation of the power received from adjoining sections. Inversely, if s is positive and the section thus supplies power to adjoining sections, the electric wave increases in this section in the direction of its propagation, or builds up. In other words, in a complex circuit, in sections of low power dissipation, the wave increases and transfers power to sections of high power dissipation, in which the wave decreases. This can still better be seen- from equations (272) and (274). Here the time decrement e~ut represents the dissipation of stored energy by the power consumed in the section by r and g. The time distance decrement, e+s^-'> for the main wave, £~s^+<) for the reflected wave, represents the decrement of the wave for con- stant U - t) or (A + t) respectively; that is, shows the change of wave intensity during its propagation. Thus for instance, following a wave crest, the wave decreases for negative s and increases for positive s, in addition to the uniform decrease by the time constant e~ut'} or, in other words, for positive s the wave gathers intensity during its progress, for negative s it loses intensity in addition to the loss of intensity by the time con- stant of this particular section of the circuit. 47. Introducing the resultant time decrement UQ of the com- plex circuit, the equations of any section, (273) and (275), can also be expressed by the resultant time decrement of the entire complex circuit, uw and the energy transfer constant of the individual section; thus s = u0 - u, (288) TRANSITION POINTS AND THE COMPLEX CIRCUIT 509 and e (289) or - £~sA [C cos q (J + 0 + D sin q (A + 0]} ,j (290) e = - The constants A, B, C, D are the integration constants, and are such as given by the terminal conditions of the problem, as by the distribution of current and e.m.f. in the circuit at the starting moment, for t = 0, or at one particular point, as A = 0. 48. The constants u0 and q depend upon the circuit conditions. If the circuit is closed upon itself — as usually is the case with an electrical transmission or distribution circuit — and A is the total length of the closed circuit, the equations must give for A = A the same values as for ^ = 0, and therefore q must be a complete cycle or a multiple thereof, 2 nn; that is, ?=2-- (291) and the least value of q} or the fundamental frequency of oscilla- tion, is 4. - v (292) A and q = nq0. (293) If the complex circuit is open at both ends, or grounded at both ends, and thus performs a half-wave oscillation, and At = total length of the circuit, q, = and q = nq0, (294) 510 TRANSIENT PHENOMENA and if the circuit is open at one end, grounded at the other end, thus performing a quarter-wave oscillation, and A2 = total length of circuit, it is 3 = (2ra"1)5»' (295) while, if the length of the complex circuit is very great compared with the frequency of the oscillation, q0 may have any value; that is, if the wave length of the oscillation is very short com- pared with the length of the circuit, any wave length, and there- fore any frequency, may occur. With uniform circuits, as trans- mission lines, this latter case, that is, the response of the line to any frequency, can occur only in the range of very high fre- quencies. Even in a transmission line of several hundred miles7 length the lowest frequency of free oscillation is fairly high, and frequencies which are so high compared with the fundamental frequency of the. circuit that, considered, as higher harmonics thereof, they overlap (as discussed in the above), must be extremely high — of the magnitude of million cycles. In a com- plex circuit, however, the fundamental frequency may be very much lower, and below machine frequencies, as the velocity of propagation - - may be quite low in some sections of the cir- VLC curt, as in the high-potential coils of large transformers, and the presence of iron increases the inconstancy of L for high frequen- cies, so that in such a complex circuit, even at fairly moderate frequencies, of the magnitude of 10,000 cycles, the circuit may respond to any frequency. 49. The constant UQ is also determined by the circuit constants. Upon u0 depends the energy transfer constant of the circuit sec- tion, and therewith the rate of building up in a section of low power consumption, or building down in a section of high power consumption. In a closed circuit, however, passing around the entire circuit, the same values of e and i must again be reached, and the rates of building up and building down of the wave in the different sections must therefore be such as to neutralize each other when carried through the entire circuit; that is, 'the total building up through the entire complex circuit must be zero. This gives an equation from which un is determined. TRANSITION POINTS AND THE COMPLEX CIRCUIT 511 In a complex circuit having n sections of different constants and therefore n transition points, at the distances A (296) where An+i = ^i + A, and A = the total length of the circuit, the equations of i and e of any section i are given by equations (290) containing the constants A& Bi} Ci} Dt-. The constants A, B, C, D of any section are determined by the constants of the preceding section by equations (285) to (287). The constants of the second section thus are determined by those of the first section, the constants of the third section by those of the second section, and thereby, by substituting for the latter, by the constants of the first section, and in this manner, by successive substitutions, the constants of any section i can be expressed by the constants of the first section as linear func- tions thereof. Ultimately thereby the constants of section (n + 1) are expressed as linear functions of the constants of the first section : An+l - afAl + af'Bt + a'"^ + Bn+l = VA, + b"B, + b'"Ct + Cn+l « cfA, + cf'B, + c"'Cl + Dn+1 = (297) where a', a", of" ', a"" ', b', b", etc., are functions of st and ^. The (n + l)st section, however, is again the first section, and it is thereby, by equations (290) and (296), Bn+l = (298) and substituting (298) into (297) gives four symmetrical linear equations in 'Alt Bly Clt Dlt from which these four constants can be eliminated, as n symmetrical linear equations with 512 TRANSIENT PHENOMENA n variables are dependent equations, containing an identity, thus: >t + (&// _ e-<^) B, + amC, + ft""/^ r 0; c^i + c^i + (cr// - £+'SlA) Ct + c""/), = 0; d'Ai + d"Bi + d'"Ci + (d"" - e+s^) D, = 0, and herefrom (299) d'" (d"" - £+s'A) = 0. (300) Substituting in this determinant equation for st- the values from (276) Si = u0- Ui (301) gives an exponential equation in uw thus : F (uQyuif.^Ci) = 0, (302) from which the value UQ, or the resultant time decrement of the circuit, is determined. In general, this equation (302) can be solved only by approxi- mation, except in special cases.