CHAPTER II. LONG-DISTANCE TRANSMISSION LINE. 3. If an electric impulse is sent into a conductor, as a trans- mission line, this impulse travels along the line at the velocity of light (approximately), or 188,000 miles per second. If the line is open at the other end, the impulse there is reflected and returns at the same velocity. If now at the moment when the impulse arrives at the starting point a second impulse, of opposite direction, is sent into the line, the return of the first impulse adds itself, and so increases the second impulse; the return of this increased second impulse adds itself to the third impulse, and so on; that is, if alternating impulses succeed each other at intervals equal to the time required by an impulse to travel over the line and back, the effects of successive impulses add themselves, and large currents and high e.m.fs. may be produced by small impulses, that is, low impressed alternating e.m.fs., or inversely, when once started, even with zero impressed e.m.f., such alternating currents traverse the lines for some time, gradually decreasing in intensity by the energy consumption in the conductor, and so fading out. The condition of this phenomenon of electrical resonance thus is that alternating impulses occur at time intervals equal to the time required for the impulse to travel the length of the line and back; that is, the time of one half wave of impressed e.m.f. is the time required by light to travel twice the length of the line, or the time of one complete period is the time light requires to travel four times the length of the line; in other words, the number of periods, or frequency of the impressed alternating e.m.fs., in resonance condition, is the velocity of light divided by four times the length of the line; or, in free oscillation or resonance condition, the length of the line is one quarter wave length. 279 280 TRANSIENT PHENOMENA If then I = length of line, S = speed of light, the frequency of oscillations or natural period of the line is — '• "4? or, with I given in miles, hence S = 188,000 miles per second, it is , 47,000 /o = — j- cycles. (2) To get a resonance frequency as low as commercial frequencies, as 25 or 60 cycles, would require Z == 1880 miles for /0 = 25 cycles, and Z = 783 miles for./, - 60 cycles. It follows herefrom that many existing transmission lines are such small fractions of a quarter-wave length of the impressed frequency that the change of voltage and current along the line can be assumed as linear, or at least as parabolic; that is, the line capacity can be represented by a condenser in the middle of the line, or by condensers in the middle and at the two ends of the line, the former of four times the capacity of either of the two latter (the first approximation giving linear, the second a para- bolic distribution). For further investigation of these approximations see "Theory and Calculation of Alternating-Current Phenomena/' 4th edition, pages 225 to 233. If, however, the wave of impressed e.m.f. contains appreciable higher harmonics, some of the latter, may approach resonance frequency and thus cause trouble. For instance, with a line of 150 miles length, the resonance frequency is /0 = 313 cycles per second, or between the 5th harmonic and the 7th harmonic, 300 and 420 cycles of a 60-cycle system; fairly close to the 5th har- monic. The study of such a circuit of distributed capacity thus becomes of importance with reference to the investigation of the effects of higher harmonics of the generator wave. In long-distance telephony the important frequencies of speech probably range from 100 to 2000 cycles. For these fre- er quencies the wave length varies from — = 1880 miles down to L 94 miles, and a telephone line of 1000 miles length would thus LONG-DISTANCE TRANSMISSION LINE 281 contain from about one-half to 11 complete waves of the im- pressed frequency. For long-distance telephony the phenomena occurring in the line thus can be investigated only by consider- ing the complete equation of distributed capacity and inductance as so-called "wave transmission" and the phenomena thus essentially differ from those in a short energy transmission line. 4. Therefore in very long circuits, as in lines conveying alter- nating currents of high value at high potential over extremely long distances, by overhead conductors or underground cables, or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance, which consumes e.m.fs. in phase with the current, and of the line reactance, which consumes e.m.fs. in quadrature with the current, is not sufficient for the explanation of the phenomena taking place in the line, but several other factors have to be taken into account. In long lines, especially at high potentials, the electrostatic capacity of the line is sufficient to consume noticeable currents. The charging current of the line condenser is proportional to the difference of potential and is one-fourth period ahead of the e.m.f. Hence, it either increases or decreases the main current, according to the relative phase of the main current and the e.m.f. As a consequence the current changes in intensity, as well as in phase, in the line from point to point; and the e.m.fs. con- sumed by the resistance and inductance, therefore, also change in phase and intensity from point to point, being dependent upon the current. Since no insulator has an infinite resistance, and since at high potentials not only leakage but even direct escape of electricity into the air takes place by " brush discharge," we have to rec- ognize the existence of a current approximately proportional and in phase with the e.m.f. of the line. This current represents consumption of power, and is therefore analogous to the e.m.f. consumed by resistance, while the condenser current and the e.m.f. of inductance are wattless or reactive. Furthermore, the alternating current passing over the line pro- duces in all neighboring conductors secondary currents, which react upon the primary current and thereby introduce e.m.fs. of mutual inductance into the primary circuit. Mutual induc- tance is neither in phase nor in quadrature with the current, 282 TRANSIENT PHENOMENA and can therefore be resolved into a power component of mutual inductance in phase with the current, which acts as an increase of resistance, and into a reactive component in quadrature with the current, which decreases the self-inductance. This mutual inductance is not always negligible, as, for instance, its disturbing influence in telephone circuits shows. The alternating potential of the line induces, by electrostatic influence, electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence requires the expenditure of a current proportional to the e.m.f. and consisting of a power component in phase with the e.m.f. and a reactive com- ponent in quadrature thereto. The alternating electromagnetic field of force set up by the line current produces in some materials a loss of power by mag- netic hysteresis, or an expenditure of e.m.f. in phase with the cur- rent, which acts as an increase of resistance. This electro- magnetic hysteresis loss may take place in the conductor proper if iron wires are used, and may then be very serious at high fre- quencies such as those of telephone currents. The effect of eddy currents has already been referred to under " mutual inductance," of which if is a power component. The alternating electrostatic field of force expends power in dielectrics by what is called dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is com- paratively large, the dielectric hysteresis may at high potentials consume considerable amounts of power. The dielectric hystere- sis appears in the circuit as consumption of a current whose component in phase with the e.m.f. is the dielectric power current, which may be considered as the power component of the charging current. Besides this there is the apparent increase of ohmic resistance due to unequal distribution of current, which, however, is usually not large enough to be noticeable at low frequencies. Also, especially at very high frequency, energy is radiated into space, due to the finite velocity of the electric field, and can be represented by power components of current and of voltage respectively. 5. This gives, as the most general case and per unit length of line, LONG-DISTANCE TRANSMISSION LINE 283 E.m.fs. consumed in phase with the current, I, and = r/, repre- senting consumption of power, and due to resistance, and its apparent increase by unequal current distribution; to the power component of mutual inductance: to secondary currents; to the power component of self -inductance: to electromagnetic hysteresis; and to electromagnetic radiation. E.m.fs. consumed in quadrature with the current, I, and = xl, reactive, and due to self-inductance and mutual inductance. Currents consumed in phase with the e.m.f., E, and = gE, representing consumption of power, and due to leakage through the insulating material, including brush discharge; to the power component of electrostatic influence; to the power component of capacity, or dielectric hysteresis, and to electrostatic radiation. Currents consumed in quadrature with tJw e.m.f., E, and = bE, being reactive, and due to capacity and electrostatic influence. Hence we get four constants per unit length of line, namely: Effective resistance, r; effective reactance, x; effective conduc- tance, g, and effective susceptance, b = - bc (bc being the absolute value of susceptance). These constants represent the coefficients per unit length of line of the following: e.m.f. consumed in phase with the current; e.m.f. consumed in quadra- ture with the current; current consumed in phase with the e.m.f., and current consumed in quadrature with the e.m.f. 6. This line we may assume now as supplying energy to a receiver circuit of any description, and determine the current and e.m.f. at any point of the circuit. That is, an e.m.f. and current (differing in phase by any desired angle) may be given at the terminals of the receiving circuit. To be determined are the e.m.f. and current at any point of the line, for instance, at the generator terminals; or the impedance, Zt = rl - jxv or admittance, Yl = g1 + jblt of the receiver circuit, and e.m.f., E0, at generator terminals are given; the current and e.m.f. at any point of circuit to be deter- mined, etc. 7. Counting- now the distance, I, from a point 0 of the line which has the e.m.f. + je,f and the current 284 TRANSIENT PHENOMENA and counting I positive in the direction of rising power and negative in the direction of decreasing power, at any point I, in the line differential dl the leakage current is Egdl and the capacity current is - jEb dl; hence, the total current consumed by the line differential dl is dl = E (g - jb) dl = EY dl, %-YE. (1) In the line differential dl the e.m.f. consumed by resistance is Irdl, the e.m.f. consumed by inductance is - jlxdl; hence, the total e.m.f. consumed by the line differential dl is dE = I (r - jx) dl = IZ dl, f-Zl. (2) These fundamental differential equations (1) and (2) are sym- metrical with respect to / and E. Differentiating these equations (1) and (2) gives dl (3) d*E and LONG-DISTANCE TRANSMISSION LINE 285 and substituting (1) and (2) in (3) gives the differential equa- tions of E and I , thus : %-YZB (4) and ;r=FZ-- (5) These differential equations are identical, and consequently I and E are functions differing by their integration constants or by their limiting conditions only. These equations are of the form f - and are integrated by V - where e is the basis of the natural logarithms, = 2.718283. Choosing equation (5), which is integrated by 7 -- At™, (6) and differentiating (6) twice gives and substituting (6) in (5), the factor Asvl cancels, and we have V2 = ZY, or V -= VZY, (7) hence, the general integral, / = A,e + vl - A2£-vl. (8) By equation (1), E ld' Y~dl' and substituting herein equation (8) gives E -A,™ + A*-" , (9) 286 TRANSIENT PHENOMENA or, substituting (7), E =\/A1e+vl+A,e-vi . (10) The integration constants A1 and A2 in (8), (9), (10), in general, are complex quantities. The coefficient of the exponent, F, as square root of the product of two complex quantities, also is a complex quantity, therefore may be written V = a - jp, (11) and substituting for F, Z and Y gives (a - j/?)2 = (r - jx) (g - jb), or (a2 - /?2) - 2 jap = (rg - xb) - j (rb + gx), and this resolves into the two separate equations a2 — ft2 = rg — xb ) 2 aQ = rb + \ (12) since, when two complex quantities are equal, their real terms as well as their imaginary terms must be equal. Equations (12) sauared and added give (a2 + /?2)2 = (rg - xb)2 + (rb + xg)* = (r2 + z2) tf + 62) hence, «2 + P = *0, (13) and from (12) and (13), a = \^(zy + rg - xb) and (14) ft = V%(zy - rg + xb). I (15) Equations (8) and (10) now assume the form and E = LONG-DISTANCE TRANSMISSION LINE 287 Substituting for the exponential function with an imaginary exponent the trigonometric expression e±m - cos 02 ± j sin pi, (16) equations (15) assume the form / = 4l£+a'(cos0/ - /sin /#) - 42£~a'(cos0Z + / sin pi) /z( E =\ -\Af +al(cos pi - / sin 00 +42e ~ a'(cos pi +/ sin 00 where Al and A2 are the constants of integration. The distribution of current / and voltage E along the circuit, therefore, is represented by the sum of two products of expo- nential and trigonometric functions of the distance I. Of these terms, the one, with factor As+al, increases with increasing dis- tance /, that is, increases towards the generator, while the other, with factor A^-*1 , decreases towards the generator and thus increases with increasing distance from the generator. The phase angle of the former decreases, that of the latter increases towards the generator, and the first term thus can be called the main wave, the second term the reflected wave. At the point / = 0, by equations (17) we have 7 - A A * o "~ .1 ~~ *lv and the ratio ^ - = m (cos T + / sin r), Ai where r may be called the angle of reflection, and m the ratio of amplitudes of reflected and main wave at the reflection point. 8. The general integral equations of current and voltage dis- tribution (17) can be written in numerous different forms. Substituting — A2 instead of + A2, the sign between the terms reverses, and the current appears as the sum, the voltage as difference of main and reflected wave. 288 TRANSIENT PHENOMENA Rearranging (17) gives / = (Aie+t* - A2e~al) cos pi - j (Ale+al+ A2e~al) sin/?/ and E == Substituting (7) gives and substituting and or and (18) = --- Y V Y' (19) *- B " -v changes equations (17) to the forms, 7=7 \Ble+al(co8pl-j sin pi) -B^'"1 (cos pi +j sin.pl) I and or f +al(cos pl-j sin al (cos ^ +j sin pi) , (20) I = Y \ C^ +aZ (cos pi — j sin pi) - C2e al (cos pi +j sin pi) > +aZ(cos pl-j sin pi) +C2e~ I = and sn (21) LONG-DISTANCE TRANSMISSION LINE 289 Substituting in (17) 4, VY VY gives -D2£-aZ(cos/?Z+/sin 0Z) and (23) (22) Reversing the sign of Z, that is, counting the distance in the opposite direction, or positive for decreasing power, from the generator towards the receiving circuit, and not, as in equations (17) to (22), from the receiving circuit towards the generator, exchanges the position of the two terms; that is, the first term, or the main wave, decreases with increasing distance, and lags; the second term, or the reflected wave, increases with the dis- tance, and leads. Equations (47) thus assume the form / -= 4l£-a< (cos pi + j sin pi) - A2e+ai (cos pi - j sin pi) and fz ( E= y - J41e-rf(cos0Z + /sin0Z) + A2s+al(cospl-jsinpl) and correspondingly equations (18) to (22) modify. 9. The two integration constants contained in equations (17) to (23) require two conditions for their determination, such as current and voltage at one point of the circuit, as at the generator or at the receiving end; or current at one point, voltage at the other; or voltage at one point, as at the generator, and ratio of voltage and current at the other end, as the impedance of the receiving circuit. Let the current and voltage (in intensity as well as phase, that is, as complex quantities) be given at one point of the circuit, and counting the distance Z from this point, the terminal con- ditions are 2-0, r- '{• - s' + ?v> and E = E0 = e0 + jeQ'. (24) 290 TRANSIENT PHENOMENA Substituting (24) in (17) gives and hence, and = V/f (4 + 4,); and substituted in (17) gives _ 1 • ~2 - jsin/?/) + /sin/?/) and 0 - /0 V E- (cos /?Z sn (25) If then 70 and ^Oare the current and voltage respectively at the receiving end or load end of a circuit of length 1Q, equations (25) represent current and voltage at any point of the circuit, from the receiving end Z = 0 to the generator end I = 10. If 70 and E0 are the current and voltage at the generator terminals, since in equations (17) I is counted towards rising power, in the present case the receiving end of the line is repre- sented by I = — Z0; that is, the negative values of I represent the distance from the generator end, along the line. In this case it is more convenient to reverse the sign of /, that is, use equations (22) and the distribution of current and voltage at distance I from the generator terminals. 7 , E are then given by LONG-DISTANCE TRANSMISSION LINE 291 • = \ and v f )*~ o V |) * +^ o V f)£~a' (cos fti + j sin ~ Sn (26) 10. Assume that the character of the load, that is, the impe- 771 T dance, — -1 =Z.=r,—/jx., or admittance, 77 =F= — = &+$*, Ti • i i of the receiving circuit and the voltage E0 at the generator end of the circuit be given. Let /0 = length of circuit, and counting distance I from the generator end, for I = 0 we have E = E- this substituted in equation (23) gives However, for I (27) E 7 . J = Z» substituting (23) herein gives Z = y/^ 4 te - ^ (cos fflp + j sin (cos ^0 - j sin ffl0) § F AlS-* (cos/?/0+ j sin/?Z0)- 4/+afo (cos/?Z0- /si hence, substituting (19) and expanding, or A VZ*~Z:- — ^A« • —c VZ - Z' 42 - 4, vzl ';"° (cos 2 /?/. + / sin 292 TRANSIENT PHENOMENA and denoting the complex factor by V7 - 7 C = l t -2 "'"(cos 2 pi. + j sin fa), (28) 4, = QAV which may be called the reflection constant, we have and by (27), 4j = — K<> ~IY (29) and hence, substituted in (22), / = and E = -° U~a/ (cos /?Z + y sin /?/) + Cs +aZ (cos /M - j sin /?Z) (30) 11. As an example, consider the problem of delivering, in a three-phase system, 200 amperes per phase, at 90 per cent power factor lag at 60,000 volts per phase (or between line and neutral) and 60 cycles, at the end of a transmission line 200 miles in length, consisting of two separate circuits in multiple, each consisting of number 00 B. and S. wire with 6 feet distance between the conductors. Number 00 B. and S. wire has a resistance of 0.42 ohms per mile, and at 6 feet distance from the return conductor an inductance of 2.4 mh. and capacity of 0.015 mf. per mile. The two circuits in multiple give, at 60 cycles, the following line constants per mile: r = 0.21 ohm, L = 1.2 X 10" 3 henry, and C = 0.03 X 10" 6 farad; hence, x = 2 TT/L = 0.45, z = 0.21 - 0.45 y, z = 0.50, and, neglecting the conductance (g = 0), b = 27T/C = IIXHT8, Y = -- 11 x 10- 6y, y = 11 X 10~6, and and LONG-DISTANCE TRANSMISSION LINE a = 0.524 X 10~ 3, P ** 2.285 X 10" 3, V = (0.524 - 2.285?) 10- 3, = ^ = (4.53 - 0.9 j) 10~3 Zi p = ^= (0.208 + 0.047 /) 10 + 3. 293 (31) Counting the distance I from the receiving end, and choosing the receiving voltage as zero vector, we have E = E0 = e0 = 60,000 volts, and the current of 200 amperes at 90 per cent power factor, 87 j, and substituting these values in equations (25) gives / = (226 + 14.4 j) e+al (cos pl-j sin pi) - (46-72.6 j) e~al (cos ftl + j sin /#), in amperes, and £"= (46.7 + 13.3 ?>+aZ (cos pi- /sin (13.3- 13.3 j) (32) (cos /?Z + j sin /?/), in kilo volts, where a and /? are given by above equations (31). From equations (32) the following results are obtained. Receiving end of line, 7 = 180 + 87 / E= 60 X 103 Middle of line, 7 = 177+ 18 / I =0 i = 200 amp. e = 60,000 volts I = 100 i = 178 amp. E= (66.2 - 6.9 /) 103 e = 66,400 volts Generator end of line, /= 165.7 - 56 / E= (69 - 15 /) 103 I = 200 i = 175 amp. e = 70,700 volts tan 0, = 0.483 6, = 26° 0,= 0 power factor, 0.90 lag tan 6l= + 0.102 6l = 6° tan 02 =-0.104 62 = - 6° d, - 02 =J~^ 12° power factor, cos 6 = 0.979 lag. tan0t =-0.338 0, = -19° tan 02 = - 0.218 02 = -12° Ol - 02 = 0 = - 7° power factor, cos 0 = 0.993 lead. 294 TRANSIENT PHENOMENA As seen, the current decreases from the receiving end to the middle of the line, but from there to the generator remains prac- tically constant. The voltage increases more in the receiving half of the line than in the generator half. The power-factor is practically unity from the middle of the line to the generator. 12. It is interesting to compare with above values the values derived by neglecting the distributed character of resistance, inductance, and capacity. From above constants per mile it follows, for the total line of 200 miles length, r0 = 42 ohms, x0 = 90 ohms, and 60 = 2.2 X 10~3 mho; hence, Z0 = 42 - 90 / and F0 = - 2.2 j 10- 3. (1) Neglecting the line capacity altogether, with 70 and EQ at the receiver terminals, at the generator terminals we have and hence, /! = 180 + 87; t\ = 200 amp. tan 01 = 0.483 6l =+ 26° E,= (75.4 - 12.6/) 103 e, = 76,400 volts tan 62 = - 0.167 62 = - 9° Ol - 62 = d =+ 34° power factor, cos 6 = 0.83 lag. These values are extremely inaccurate, voltage and current at generator too high and power factor too low. (2) Representing the line capacity by a condenser at the generator end, that is, adding the condenser current at the generator end, (» - f.-> y0?i and ^i = ^o + ^o(0; hence, 7, = 152 - 89 / t, = 176 amp. tan 0l =- 0.585 0t = -30° E,= (75 A - 12.6 /)103 el = 76,400 volts tan 02 = -0.167 02 = - 9° 0, - 02 = e =- 2iQ power factor, cos 0 = 0.93 lead. LONG-DISTANCE TRANSMISSION LINE 295 As seen, the current is approximately correct, but the voltage is far too high and the power factor is still low, but now leading. (3) Representing the line capacity by a condenser at the receiving end, that is, adding the condenser current at the load, /, = 7. + FA and El = Et + ZtIl; hence, 7, = 180 - 45 / t\ = 186 amp. tan 6,=- .250 0l = - 14° EI = (63.5 - 18.1 /) 103 el = 66,000 volts tan 02 = - .285 62 = -16° 6l-62 = 6 = + 2° power factor, cos 6 = 1.00 In this case the voltage el is altogether too low, the current somewhat high, but the power factor fairly correct. (4) Taking the average of the values of (2) and of (3) gives /, = 166 - 67 / i\ = 179 amp. tan 0l = - 0.403 0l = - 22° El = (69.4 - 15.3 /) 103 el = 71,100 volts tan 0a = - 0.220 02 = - 12° 01 - 62 = 0 = - 10° power-factor, cos 6 = 0.985 lead. As seen by comparing these average values with the exact result as derived above, these values are not very different, but constitute a fair approximation in the present case. Such a close coincidence of this approximation with the exact result can, however, not be counted upon in all instances. 13. In the equations (17) to (23) the length 1.-J- (33) 2x is a complete wave length, which means that in the distance — the phases of the components of current and of e.m.f. repeat, and that in half this distance they are just opposite. Hence, the remarkable condition exists that in a very long line at different points the currents are simultaneously in oppo- site directions and the e.m.fs. are opposite. 296 TRANSIENT PHENOMENA The difference of space phase T between current I and e.m.f. E at any point I of the line is determined by the equation Tjl m (cos r + j sin r) = j > (34) where w is a constant. Hence, r varies from point to point, oscillating around a medium position, r^ , which it approaches at infinity. This difference of phase, TW, towards which current and e.m.f. tend at infinity, is determined by the expression m (COST.+ /sin r J = - , M -h = oo or, substituting for E and 7 their values from equations (23), and since e~al = 0, and Ateal (cos jtt — j sin pi) cancels, >/~Z V - m (cos r^ + j sin rj == \ — = - = hence, tan r^ = — ~^L . (35) ag + po 14. This angle, r^ = 0; that is, current and e.m.f. come more and more in phase with each other when ab — pg — 0; that is, <* + P = 9 + &, or 2ap 2gb substituting (12) gives gr - bx ^ g* -V gx + br ~ 2gb hence, expanding, r -r- x = g -?- 6; (36) that is, J/ie ratio of resistance to inductance equals the ratio of leakage to capacity. LONG-DISTANCE TRANSMISSION LINE 297 This angle, r^ = 45°; that is, current and e.m.f. differ by one-eighth period if + ab — fig = ag + J3b, or which gives rg + xb = 0, (37) which means that two of the four line-constants, either g and x or g and 6, must be zero. The case where g = 0 = x, that is, a line having only resistance and distributed capacity but no self-inductance, is approxi- mately realized in concentric or multiple-conductor cables, and in these the space-phase angle tends towards 45 degrees lead for infinite length. 15. As an example are shown the characteristic curves of a transmission line of the relative constants, r : x : g : b = 8 : 32 : 1.25 X 1(T4 : 25 X 10~4 and e = 25,000, i = 200 at the receiving circuit, for the conditions (a) Non-inductive load in the receiving circuit, Fig. 80. (6) Wattless receiving circuit of 90 time-degrees lag, Fig. 81. (c) Wattless receiving circuit of 90 time-degrees lead, Fig. 82. These curves are determined graphically by constructing the topographic circuit characteristics in polar coordinates as explained in "The Theory and Calculation of Alternating- Current Phenomena," fourth edition, Chapter VII, paragraphs 42 to 44, and deriving corresponding values of current, potential difference, and phase angle therefrom. As seen from these diagrams, for wattless receiving circuit, current and e.m.f. oscillate in intensity inversely to each other, with an amplitude of oscillation gradually decreasing when passing from the receiving circuit towards the generator, while the space-phase angle between current and e.m.f. oscillates between lag and lead with decreasing amplitude. Approximately maxima and minima of current coincide with minima and maxima of e.m.f. and zero phase angles. For such graphical constructions, polar coordinate paper and two angles a and d are desirable, the angle a being the angle between current and change of e.m.f., tan a = - = 4, and the 298 TRANSIENT PHENOMENA Distam L 4.2Q 100 -80- -60, -40 Fig. 80. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Non-inductive load. 480 •140 -40 -80 ag f C \^ I & l/\ \ Dis anc 100 Fig. 81. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Inductive load. LONG-DISTANCE TRANSMISSION LINE 299 angle 3 the angle between e.m.f. and change of current, tan d = - = 20 in above instance. 9 With non-inductive load, Fig. 80, these oscillations of intensity have almost disappeared, and only traces of them are noticeable in the fluctuations of the space-phase angle and the relative values of current and e.m.f. along the line. Towards the generator end of the line, that is, towards rising power, the curves can be extended indefinitely, approaching more and more the conditions of non-inductive circuit. Towards decreasing power, however, all curves ultimately reach the conditions of a wattless receiving circuit, as Figs. 81 and 82, at the point where the total energy input into the line has been consumed therein, and at this point the two curves for lead and for lag join each other as shown in Fig. 83, the one being a prolongation of the other, and the power in the line reverses. Thus in Fig. 83 energy flows from both sides of the line towards the point of zero power marked by 0, where the current and e.m.f. are in quadrature with each other, the current being leading with regard to the power from the left and lagging with regard to the power from the right side of the diagram. 16. It is of interest to investigate some special cases of such circuits of distributed constants. (A) Open circuit at the end of the line. Assuming a constant alternating e.m.f. El impressed upon a circuit at one end while the other end of the circuit is open. Counting the distance I from the open end of the line, and denoting the length of the line by Z0, for I = 0, i = /o = o, and for I = L, hence, substituting in equations (17), 0 = A, - A2, 4i€+*(c°s#o-^ ; hence, A2 = Al = A 300 TRANSIENT PHENOMENA Fig. 82. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Anti-inductive load. En x*< IH t : now o Fig. 83. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. LONG-DISTANCE TRANSMISSION LINE 301 and e+tU° (cos/?/,, - j sin/?/0) + £-"<« (cos /H0 - j sin /?/0) hence, substituting in (17), T 771 and 77? 77T . ~ . 1 cos - £—') sin cospl0-j (£+a*° - e-^sin ®pl - j (e+al - e-al)smftt - J - ff-^o} si e— ") sin /?Z0 (38) At Z = 0, or the open end of the line, by equations (38), and 2E, -«-«•) -sin ft (39) The absolute values of / and E follow from equations (38) and (39) : p Jy_ v/ lV V sn T Tj1 \/ *L which expanded gives and and ; J + 2cos2/?/0 + 2 cos (40) 2 cos 2 /#0 (41) 302 TRANSIENT PHENOMENA As function of I, the e.m.f. E or the current 7 is a maximum or minimum for £-2al ± 2 COS 2 /?/ = Oj hence, £-2ttZ) = ± 2 ft sin 2 01. (42) For Z = 0, and since a is a small quantity, the left side of (42) also is small, and for values of sin 2 fll approximating zero, that is, in the neighborhood of I = --} or where fil is a multiple Z p of a quadrant, equation (42) becomes zero. At fil = 2 n-, or the even quadrants, E is a maximum, / a minimum, at 01 ~ (2 n — 1) - , or the odd quadrants, E is a minimum, 7 a £ maximum. The even quadrants, therefore, are nodes of current and wave crests of e.m.f., and the odd quadrants are nodes of e.m.f. and crests of current. A maximum voltage point, or wave crest, occurs at the open end of the line at / = 0, and is given by equation (41). As func- tion of the length 10 of the line this is a maximum for t~2al* + 2 cos 2 /?/0 - 0, or a (e+2al° - e-2«*>) = 2 /? sin 2 0Z0, or approximately at (43) that is, when the line is a quarter wave length or an odd multiple thereof. LONG-DISTANCE TRANSMISSION LINE 303 Substituting in (41), /?/0 = - gives (44) -f £-2a (46) which is the maximum voltage that can occur at the open end of a line with voltage El impressed upon it at the other end. Since, approximately, P= Vxb by (44) we have (47) the frequency which at the length of line Z0 produces maximum voltage at the open end. For the constants in the example discussed in paragraph 11 we have 10 = 200 miles, r = 0.21 ohm, L = 1.2 X 10~3 henry, C = 0.03 X 1Q-6 farad, g = 0, / = 208 cycles per sec., x = 1.57 ohms, z =-- 1.58 ohms, b = 39 X 10"6 mho, a = 0.53 X 10-3, and #0 = 9.3 Er (B) Line grounded at the end. 304 TRANSIENT PHENOMENA 17. Let the circuit be grounded or connected to the return conductor at one end, I = 0, and supplied by a constant impressed e.m.f. E1 at the other end, I = 1Q. Then for I = 0, 771 __ 777 f~\ and for I = 10, hence, substituting in (17), 0 and hence, 4i = ~~ 4z = 4) and •*^- ~x~ i «? — 7_ \ and, substituting in (17), (e+al + £~al) cos/?Z - j (e+al-£-al) sin : * * ~a0 cos 0 - - £a° + £- cos ?Z - g+af + £-aZ sin (48) - £-0 cos 0 - At the grounded end, I = 0, j = = (49) (s a^ — s a^j cos plQ — j(^ ~t~£ ) sin pLQ Substituting (49) in (48) gives / = J / { (fi+°* + e-^) cos ^Z - j (e+ai - e~ai) sin pi} +tl = \ IQ - (e+«« -e-«0 cos /?Z - j (e LONG-DISTANCE TRANSMISSION LINE 305 In this case nodes of voltage and crests of current appear at I = 0 and at the even quadrants, pl = 2n - , and nodes of current 2 and crests of voltage appear at the odd quadrants, [ft = (2n — 1) - • & (C) Infinitely long conductor. 18. If an e.m.f. EQ is impressed upon an infinitely long con- ductor, that is, a conductor of such length that of the power input no appreciable part reaches the end, we have, for I = 0, E = E0 and for I = oo, E = 0 and / = 0; •hence, substituting in (23) gives and A2= 0 hence, fy I = E0 V 77£~al (cos pi + j sin pi) (51) and E = E^-*1 (cos pi + j sin pi). From (51) it follows that 7 Y' that is, an infinitely long conductor acts like an impedance, Z.-V/^-r,-/*,, and the current at every point of the conductor thus has the same space-phase angle to the voltage, tan «, = -1 - T< 306 TRANSIENT PHENOMENA The equivalent impedance of the infinite conductor is 7 a - //? z, YY~g-jb .flg-ab and the space-phase angle is If <7 = 0 and x = 0, we have 'and tan at = 1, or «t = 45°; that is, current and e.m.f. differ by one-eighth period. This is approximately the case in' cables, in which the dielectric losses and the inductance are small. An infinitely long conductor therefore shows the wave propa- gation in its simplest and most perspicuous form, since the reflected wave is absent. (D) Generator feeding into a closed circuit. 19. Let / = 0 be the center of the circuit; then El = - E_t and It = /_,; hence, E = 0 at I = 0, and the equations are the same as those of a line grounded at the end I = 0, which have been discussed under (B). (E) Line of quarter wave length. 20. Interesting is the case of a line of quarter wave length. Let the length 10 of the line be one quarter wave of the im- pressed e.m.f. fl. - \ • (54) LONG-DISTANCE TRANSMISSION LINE 307 To illustrate the general character of the phenomena, we may as first approximation neglect the energy losses in the circuit, that is, assume the resistance r and the conductance g as neg- ligible compared with x and 6, r = 0 = g. These values substituted in (14) give a = 0 and = Vxb. (55) Counting the distance I from the end of the line 10 we have for = 0, EQ = e0 + je0' and = and at the beginning of the line for I — 10, and and by (54) and (55), L = ~2Vxb Substituting (56), (57), and (54) in (17) gives (56) (57) (58) and or B. - VI (4, + 4.) = Vs (4, + and ^=-/V^(4-42)--yV-(41-42); hence, eliminating Ax and A2 gives the relations between the electric quantities at the generator end of the quarter-wave line, Ev I v and at the receiving end, Ew 70: 308 TRANSIENT PHENOMENA and and the absolute values are and (59) (60) which means that if the supply voltage E1 is constant, the output current 70 is constant and lags 90 space-degrees behind the input voltage; if the supply current II is constant, the output voltage E0 is constant, and lags 90 space-degrees, and inversely. A quarter- wave line of negligible losses thus converts from constant potential to constant current, or from constant cur- rent to constant voltage. (Constant-potential constant-current transformation.) Multiplying (60) gives or E -- hence, if 70 = 0, that is, the line is open at the end, E0 = oo , and with a finite voltage supply to a line of quarter-wave length, an infinite (extremely high) voltage is produced at the other end. Such a circuit thus may be used to produce very high voltages. Since x0 = I0x = total reactance and b0 = I0b = total sus- ceptance of the circuit, by (58) we have (61) or the condition of quarter-wave length. LONG-DISTANCE TRANSMISSION LINE 309 Substituting x0 = 2 7r/L0 and 60 = 2 7r/(70, we have 1 or f 16/2' 1 (62) (63) the condition of quarter-wave transmission. 21. If the resistance, r, and the conductance, g, of a quarter- wave circuit are not negligible, substituting (56), (54) and (57) in (17) we have, for I = 0, - and and for I = L and From (64) it follows that and and substituting in (65) and rearranging we have a*0 \ c~a*0 c+a^0 _ c-a T • T£ (64) (65) (66) and (67) 310 TRANSIENT PHENOMENA or, __ £ - 2al0 and (68) or, analogous to equation (59), • ° v z and (69) In these equations the second term is usually small, due to the factor (e+a*° — e~a*°), and the first term represents constant potential-constant current transformation. 22. In a quarter-wave line, at constant impressed e.m.f. Ev the current output 1 0 is approximately constant and lagging 90 degrees behind El ; it falls off slightly, however, with increasing load, that is, increasing 7 v due to the second term in equation (68); the voltage at the end of the line, E0, at constant impressed voltage, is approximately proportional to the load, but does not reach infinity at open circuit, but a finite, though high, limiting value. Inversely, at constant current input the voltage output is approximately constant and the output current proportional to the load. The deviation from constancy, at constant Ev of 70, or at constant I v of EQ, therefore, is due to the second term, with factor (e+^- £-'a*°). Substituting (54), an hence, al0 is usually a very small quantity, and e~ ak = s ?2 thus can be represented by the first terms of the series : LONG-DISTANCE TRANSMISSION LINE 311 lM2 + ^LM°+±, a: TT hence, and and, by (69), and 2 £ — 2 If r and gi are small compared with x and 6, ft = VJ (ay - rg + xb) = \/2> and a = «/ + rg - xb) ; substituting, by the binomial theorem, fc2) = x6 j [l + Q'J (70) zy = l + gives bx/r and 312 TRANSIENT PHENOMENA The quantity a may be called the time constant of the circuit. The equations of quarter-wave transmission thus are (72) and and the maximum voltage Ea, at the open end of the circuit, at constant impressed e.m.f. Ev is and E0 = , (73) and the current input is where, approximately, \IZ -\ V 17 ~V 7 Y T c (75) 23. Consider as an example a high potential coil of a trans- former with one of its terminals connected to a source of high potential, for testing its insulation to ground, while the other terminal is open. Assume the following constants per unit length of circuit: r = 0.1 ohm, L = 0.02 mh., C = 0.01 X 10-6 farad, and g = 0; then, with a length of circuit 1Q = 100, the quarter-wave fre- quency is, by (47), / = -= = 177 cycles per sec., LONG-DISTANCE TRANSMISSION LINE 313 or very close to the third harmonic of a 60-cycle impressed voltage. If, therefore, the testing frequency is low, 59 cycles, the circuit is a quarter wave of the third harmonic. Assuming an impressed e.m.f. of 50,000 volts and 59 cycles, containing a third harmonic of 10 per cent, or El = 5000 volts at 177 cycles, for this harmonic, we have x = 22.2 ohms and 6 - 11.1 X 10- 6 ohm; hence, u = 0.00225 and therefore at El = 5000 volts, E? = 1,415,000 volts; that is, infinity, as far as insulation strength is concerned. Quarter- wave circuits thus may be used, and are used, to pro- duce extremely high voltages, and if a sufficiently high frequency is used — 100,000 cycles and more, as in wireless telegraphy, etc. - the length of the circuit is moderate. This method of producing high voltages has the disadvantage that it does not give constant potential, but the high voltage is due to the tendency of the circuit to regulate for constant current, which means infinite voltage at infinite resistance or open circuit, but as soon as current is taken off the high potential point the voltage falls. The great advantage of the quarter- wave method of producing high voltage is its simplicity and ease of insula- tion; as the voltage gradually builds up along the circuit, the high voltage point or end of circuit may be any distance away from the power supply, and thus can easily be made safe. 24. As a quarter-wave circuit converts from constant poten- tial to constant current, it is not possible, with constant voltage impressed upon a circuit of approximately a quarter-wave length, to get constant voltage at the other or receiving end of the circuit. Long before the circuit approaches quarter-wave length, and as soon as it becomes an appreciable part of a quarter wave, this tendency to constant current regulation makes itself felt by great variations of voltage with changes of load at the receiving end of the circuit, constant voltage being impressed upon the generator end ; that is, with increasing length of transmission lines the volt- age regulation at the receiving end becomes seriously impaired 314 TRANSIENT PHENOMENA hereby, even if the line resistance is moderate, and the operation of apparatus which require approximate constancy of voltage but do not operate on constant current — as most synchronous apparatus — becomes difficult. Hence, at the end of very long transmission lines the voltage regulation becomes poor, and synchronous machines tend to instability and have to be provided with powerful steadying devices, giving induction motor features, and with a line approaching quarter-wave length, voltage regulation at the receiving end ceases. In this case the constant potential-constant current trans- formation may be used to produce constant or approximately constant voltage at the load, by supplying constant current to the line; that is, the transmission line is made a quarter- wave length by modifying its constants, or choosing the proper fre- quency, the generators are designed to regulate for constant current and thus give a voltage varying with the load, and are connected in series (with constant current generators series con- nection is stable, parallel connection unstable) and feed constant current, at variable voltage, into the quarter-wave line. At the receiving end of the line, constant voltage then exists with varying load, or rather a voltage, which slightly falls off with the load, due to the power loss in the line. To maintain constant receiver voltage at all loads, then, would require a slight increase of generator current with increase of load, that is, increase of generator voltage, which can be produced by compounding regulated by the voltage. In such a quarter-wave transmission the voltage at the receiv- ing end then remains constant, while the current output from the line increases from nothing at no load. At the generator end the current remains approximately constant, increasing from no load to full load by the amount required to take care of the line loss, while the voltage at the generators increases from nearly nothing at no load, with increasing load, approximately proportional thereto. 25. There is, however, a serious limitation imposed upon quarter-wave transmission by considerations of voltage; to use the transmission line economically the voltage throughout it should not differ much, since the insulation of the line depends on the maximum, the efficiency of transmission, however, on the LONG-DISTANCE TRANSMISSION LINE 315 average voltage, and a line in which the voltage at the two ends is very different is uneconomical. To use line copper and line insulation economically, in a quarter-wave transmission, the voltages at the two ends should be approximately equal at maximum load. These voltages are related to each other and to the current by the line constants, by equations (72). By these equations (neglecting the term with u), reduced to absolute values, we have approximately -V?, and and if el = e0, hence, the power is = V or e* = Po V - ; (76) y hence, the voltage e0 required to transmit the power pQ without great potential differences in the line depends on the power p0 and the line constants, and inversely. 26. As an example of a quarter- wave transmission may be considered the transmission of 60,000 kilowatts over a distance of 700 miles, for the supply of a general three-phase distribution system, of 95 per cent power factor, lag. The design of the transmission line is based on a compromise between different and conflicting requirements: economy in first cost requires the highest possible voltage and smallest con- ductor section, or high power loss in the line ; economy of opera- tion requires high voltage and large conductor section, or low power loss; reliability of operation of the line requires lowest 316 TRANSIENT PHENOMENA permissible voltage and therefore large conductor section or high power loss; reliability of operation of the receiving system requires good voltage regulation and thus low line resistance, etc., etc. Assume that the maximum effective voltage between the line conductors is limited to 120,000, and that there are two sepa- rate pole • lines, each carrying three wires of 500,000 circular mils cross section, placed 6 feet between wires, and provided with a grounded neutral. If there were no energy losses in the line and no increase of capacity due to insulators, etc., the speed of propagation would be the velocity of light, S = 188,000 miles per second, and the quarter-wave frequency of a line of 10 = 700 miles would be S / = — =67 cycles per sec. ; 4 LQ hence, fairly close to the standard frequency of 60 cycles. The loss of power in the line, and thus the increase of induc- tance by the magnetic field inside of the conductor (which would not exist in a conductor of perfect conductivity or zero resistance loss), the increase of capacity by insulators, poles, etc., lowers the frequency below that corresponding to the velocity of light and brings it nearer to 60 cycles. In a line as above assumed the constants per mile of double conductor are: r = 0.055 ohm; L = 0.001 henry, and C = 0.032 X 10~6 farad, and, neglecting the conductance, g = 0, the quarter-wave frequency is / = — = = 63 cycles per sec. Either then the frequency of 63 cycles per second, or slightly above standard, may be chosen, or the line inductance or line capacity increased, to bring the frequency down to 60 cycles. Assuming the inductance increased to L = 0.0011 henry gives / = = 60 cycles per second, and the line constants then are £0 = 700 miles; / = 60 cycles per second; r = 0.055 ohm; L = 0.0011 henry; C = 0.032 X 10~6 farad, and g = 0; hence, x = 0.415 ohm; z = 0.42 ohm; Z = 0.055 - 0.415 j ohm; LONG-DISTANCE TRANSMISSION LINE 317 6 = 12.1 X 10"6 mho; y = 12.1 X 10~6 mho, and Y = - j 12.1 X 10~6 mho, and v/ - = 186, y 0.066, p = 2.247 X 10-3 a = up = 0.148 X 10-3. At 60,000 kilowatts total input, or 20,000 kilowatts per line, 120 000 and 120,000 volts between lines, or - '-1— = 69,000 volts per line, and about 95 per cent power factor, the current input at full load is 306 amp. per line (of two conductors in multiple). To get at full load p = 20 X 106 watts, approximately the same voltage at both ends of the line, by equation (67), we must have y or e = 61,000 volts. Assuming therefore at the receiving end the voltage of 110,000 between the lines, or, 63,500 volts per line, and choosing the output current as zero vector, and counting the distance from the receiving end towards the generator, we have for / = 0, . ~ . 0 0> and the voltage, at 95 per cent power factor, or Vl — 0.952 = 0.312 inductance factor, is E = E0 = e0 (0.95 - 0.312 j) = 60,300 - 19,800 j. Substituting these values in equations (72) gives . \JE1 - 0.104 (60,300 - 19,800 j)} *° = 186 + 12 j 60,300 - 1 9,800 / = (186 + 12 j) {j I, - 0.104 i0}; 318 TRANSIENT PHENOMENA hence, JE1 = (186 + 12 j) iQ + (6250 - 2060 j) and .(^ 60,300 -19,800f+ Q ^ ^ = 317 - 128 j + 0.104 i0, and the absolute values are and e, = V(186 i0 + 6250)2 + (12 t0 - 2060)' \ = V(317 + 0.104 i, 1282 70 L^ Cos. 0-1— 80 32 70 28 500 50 20 400 40 16 300 30 12 100 10 4 6 10 14 18 Power Output per Phase Po .Megawatts 26 Fig. 84. Long-distance quarter-wave transmission. herefrom the power output and input, efficiency, power factor, etc., can be obtained. In Fig. 84, with the power output per phase as abscissas, are shown the following quantities : voltage input e1 and output e0, LONG-DISTANCE TRANSMISSION LINE 319 in drawn lines; amperes input t\ and output t0, in dotted lines; power input pl and output p0, in dash-dotted lines, and efficiency and power factor in dashed lines. As seen, the power factor at the generator is above 93 per cent leading, and the efficiency reaches nearly 85 per cent. At full load input of 20,000 kilowatts per phase, and 95 per cent power factor, lagging, of the output, the generator voltage is 58,500, or still 8 per cent below the output voltage of 63,500. The generator voltage equals the output voltage at 10 per cent overload, and exceeds it by 14 per cent at 25 per cent overload. To maintain constant voltage at the output side of the line, the generator current has to be increased from 342 amperes at no load to 370 amperes at full load, or by 8.2 per cent, and inversely, at constant-current input, the output voltage would drop off, from no load to full load, by about 8 per cent. This, with a line of 15 per cent resistance drop, is a far closer voltage regulation than can be produced by constant potential supply, except by the use of synchronous machines for phase control.