CHAPTER VIII. REFLECTION AND REFRACTION AT TRANSITION POINT. 58. The general equation of the current and voltage in a sec tion of a complex circuit, from equations (290), is - £-sA [C cos q 0* + 0 + D sin q (A + 0]} e = C£-Uot {e+8* [A cos g (J - 0 + # sin g (A - 0] where A = l< 0 is section 1, A>0 is section 2, equations (285) assume the form A2 = B2 = C2 = D2 = blCv (349) From equations (349) and (286) it follows that c2 (A* - C22) = ct (A* - C,2) 1 and (350) c2 (B2 - D2) = c, (B2 - D2). J If now a wave in section 1, A B, travels towards transition point A = 0, at this point a part is reflected, giving rise to the reflected wave C D in section 1, while a part is transmitted and appears as main wave A B in section 2. The wave C D in sec- tion 2 thus would not exist, as it would be a wave coming towards A = 0 from section 2, so not a part of the wave coming from section 1. In other words, we can consider the circuit as com- prising two waves moving in opposite direction : (1) A main wave AJ$V giving a transmitted wave A 2B2 and reflected wave CJ)r (2) A main wave C2D2, giving a transmitted wave C/D/ and reflected wave A2B2. The waves moving towards the transition point are single main waves, AJ$i and C2D2, and the waves moving away from the transition point are combinations of waves reflected in the sec- tion and waves transmitted from the other section. 69. Considering first the main wave moving towards rising A: in this C2 = 0 = D2, hence, from (349), and (351) and herefrom and C2 ~C1A -,+c, (352) REFLECTION AND REFRACTION 527 which substituted in (349) gives and 2c, -, , 1 . a, a, Cl + c Then for the main wave in section 1, = £-v £+«,A ^ cog _+B sin (353) and sing (A- t)}. (354) When reaching a transition point A = 0, the wave resolves into the reflected wave, turned back on section 1, thus: and (355) The transmitted wave, which by passing over the transition point enters section 2, is given by and e = c (356) D The reflection angle, tan (t/) = ~ , is supplementary to the B impact angle, tan (ij = + -~, and transmission angle, tan (i'2) Reversing the sign of ^ in the equation (355) of the reflected wave, that is, counting the distance for the reflected wave also in the direction of its propagation, and so in opposite direction as 528 TRANSIENT PHENOMENA in the main wave and the transmitted wave, equations (355) become C2+C1 (357) and then or c, 2 ' "1 V1J (358) (1) In a single electric wave, current and e.m.f. are in phase with each other. Phase displacements between current and e.m.f. thus can occur only in resultant waves, that is, in the com- bination of the main and the reflected wave, and then are a function of the distance ^, as the two waves travel in opposite direction. (2) When reaching a transition point, a wave splits up into a reflected wave and a transmitted wave, the former returning in opposite direction over the same section, the latter entering the adjoining section of the circuit. (3) Reflection and transmission occur without change of the phase angle ; that is, the phase of the current and of the voltage in the reflected wave and in the transmitted wave, at the transi- tion point, is the same as the phase of the main wave or incoming wave. Reflection and transmission with a change of phase angle can occur only by the combination of two waves traveling in opposite direction over a circuit; that is, in a resultant wave, but not in a single wave. (4) The sum of the transmitted and the reflected current equals the main current, when considering these currents in their respective direction of propagation. REFLECTION AND REFRACTION 529 The sum of the voltage of the main wave and the reflected wave equals the voltage of the transmitted wave. The sum of the voltage of the reflected wave and the voltage of the transmitted wave reduced to the first section by the ratio g of voltage transformation — , equals the voltage of the main wave. c\ c (5) Therefore a voltage transformation by the factor — ci IT Q = V rT 7- occurs at the transition point; that is, the trans- T C2 Li mitted wave of voltage equals the difference between main wave £ and reflected wave multiplied by the transformation ratio — : c ci e2 = — (e^ — e"). As result thereof," in passing from one section C2 of a circuit to another section, the voltage of the wave may ^ . decrease or may increase. If — > 1, that is, when passing from ci a section of low inductance and high capacity into a section of high inductance and low capacity, as from a transmission line into a transformer or a reactive coil, the voltage of the wave is s* increased; if — < 1, that is, when passing from a section of high ci inductance and low capacity into a section of low inductance and high capacity, as from a transformer to a transmission line, the voltage of the wave is decreased. This explains the frequent increase to destructive voltages, when entering a station from the transmission line or cable, of an impulse or a wave which in the transmission line is of relatively harmless voltage. The ratio of the transmitted to the reflected wave is given by 2 VLjC, 2 and 2c2 L2C, (359) 530 TRANSIENT PHENOMENA 60. Example: Transmission line Lt = 1.95 X 1(T3 Ct = 0.0162 X 10-' ct = 346 ^7, = 0.56 i* And in the opposite direction Transformer 0.4 X 10-6 1580 .-? " 2'56 •J- -0.56. The ratio -^becomes a maximum, = GO, for -1 =77, but in ei ci ^2 this case e/' = 0; that is, no reflection occurs, and the reflected wave equals zero, the transmitted wave equals the incoming wave. hence, becomes a maximum for c2 = 0, or c^ = oo and then = 2; in which case e2 = 0. 2c, c2 + hence, becomes a maximum for ct (360) 0, or c2 = oo and then = 2; in which case i2 = 0. From the above it is seen that the maxi- mum value to which the voltage can build up at a single transi- tion point is twice the voltage of the incoming wave, and this occurs at the open end of the circuit, or, approximately, at a point where the ratio of inductance to capacity very greatly increases. hence, becomes a maximum, and equal to 1, for cl = 0, or c2 = oo. < = C2 ~ Cj «i c2 + ct has the same value as the current-ratio. (361) REFLECTION AND REFRACTION 531 61. Consider now a wave traversing the circuit in opposite direction; that is, C2D2 is the main wave, A ^B2 the reflected wave, C1D1 the transmitted wave, and Al= 0 = B^ In equa- tion (349) this gives C2 = and hence, ft = -ft and + l "I B,— Ac,--^!),.. J ~ & - I _ ^ D. (362) that is, the same relations as expressed by equations (352) and (353) for the wave traveling in opposite direction. The equations of the components of the wave then are : Main wave: 2 cos q {C2 cos q sn q D2 sin 0 } 0 } ; (363) Transmitted wave: sn sn ^(364) Reflected wave: i2 = if -«*«+*«* {(72cosg (>l — /)— D2sin^ (^— 0} - (365) 532 TRANSIENT PHENOMENA or, in the direction of propagation, that is, reversing the sign of ^ : 1 If -**$-'»* {C. cos0 (/ + 0+^2 sing (/ + 0 } I (.* i \ r ,£ J. \ / ) — C. cos sn q (366) 62. The compound wave, that is, the resultant of waves pass- ing the transition point in both directions, then is (367) 4 & & & s j In the neighborhood of the transition point, that is, for values ^ which are sufficiently small, so that e+8* and e~s* can be dropped as being approximately equal to 1, by substituting equations (354) to (356) and (363) to (366) into (367) we have = ~Uot cos - 0 + Bl sin q(l-t)}' '* {Al cos q (A + 0 - #1 sin q(l + t)} '1 + C2 '-^ {C2cosq (J + 0+ Da sin g(^ + 0}]; -Wo' [{A^osg (j _ Q + J5tsing (^ - 0 } '2 + cx 'l "» C2 t t p ~ \ C2 cos q (A — t) — D2 sin q (A — t) } '1 ~l~ C2 2,c, - { A l cos ^ (^ — t) + -Bj sin q (^ — 0 } ] 5 (368) e° = REFLECTION AND REFRACTION 533 lot [{C2 cos q (X + t) + D2 sin q (* + £)} cos -t - sn In these equations the first term is the main wave, the second term its reflected wave, and the third term the wave transmitted from the adjoining section over the transition point. Expanding and rearranging equations (368) gives 2 e ~ w°' *!° = -f1— [{(c^! - caCJ cos^ + C2 (#1- £2) sin^} cosqt - { (clBl + c2D2) cos qX - c2(Al+ C2) sin qt} sin qt]; 2c< 2 s~Uot (369) ci (Bi - #2) sin ql } cos qt 2 - { (clBl + c2D2) cos ql-cl(Al + C2) sin g;} sin qt]; cos e2" -- - £ ^Ltci^i + ki;cos2x+^c1tf1 + e2LJ2;sin2>ijcc Cl + C2 - { cl (Bl - D2) cos qX - (c±A l - c2C2) sin qX \ si] 63. This gives the distance phase angle of the waves: tan i o = C2 {£i ~ ^2) c°s $ + (Al + C2) sin qt} ^ (c^Ai - c2C2) cos qt — (clBl + c2D2) sin qt ' (c1Al- c2C2) cos qt - (clBl + c2D2) siuqt ' no (370) tan i2 hence, tan^° c2 T L/2'ul (c.B. + c,D2) cos qt + (c.^ - i fon p 0 V 1 "* V 1 1 tan e, — (/^/^\ /T-» T- c2{(Al + C2) cos^ - c2(?2) sin ^ o _ (cJS^ + c2D2) cos $ -f (CjAj — c2(72) I tl ' I t/O " if ^ ~A ^ ^ , -^v ^ ^ cos sin 9< - D2) sin §( } (371) (372) 534 TRANSIENT PHENOMENA hence, tan62°_C /LA. that is, at a transition point the distance phase angle of the wave changes so that the ratio of the tangent functions of the phase angle is constant, and the ratio of the tangent functions of the phase angle of the voltages is proportional, of the currents inversely proportional to the circuit constants c = y — • In other words, the transition of an electric wave or impulse from one section of a circuit to another takes place at a constant ratio of the tangent functions of the phase angle, which ratio is a constant of the circuit sections between which the transition occurs. This law is analogous to the law of refraction in optics, except that in the electric wave it is the ratio of the tangent functions, while in optics it is the ratio of the sine functions, which is con- stant and a characteristic of the media between which the tran- sition occurs. Therefore this law may be called the law of refraction of a wave at the boundary between two circuits, or at a transition point. The law of refraction of an electric wave at the boundary between two media, that is, at a transition point between two circuit sections, is given by the constancy of the ratio of the tangent functions of the incoming and refracted wave.