15. LOAD CHARACTERISTIC OF TRANSMISSION LINE 70. The load characteristic of a transmission line is the curve of volts and watts at the receiving end of the line as function of the amperes, and at constant e.m.f . impressed upon the generator end of the line. Let r = resistance, x = reactance of the line. Its impedance z = -y/r2 + x2 can be denoted symbolically by Z = r + jx. Let EQ = e.m.f. impressed upon the line. Choosing the e.m.f. at the end of the line as horizontal com- ponent in the vector diagram, it can be denoted by E = e. 86 ELEMENTS OF ELECTRICAL ENGINEERING At non-inductive load the line current is in phase with the e.m.f. e, thus denoted by 7 = i. The e.m.f. consumed by the line impedance Z — r + jx is E! = ZI = (r + jx) i = ri+jxi. (1) Thus the impressed voltage, ' Eo = E + Ei = e + ri + ja». (2) or, reduced, #o = V(e + n)2 + z2*2, (3) and _ 6 = ^o2 - z2*2 - n, the e.m.f. (4) p = d = i V-Eo2 - x2i2 - ri2, (5) the power received at end of the line. The curve of e.m.f. e is an arc of an ellipse. With open circuit i = 0, e = E0 and P = 0, as is to be expected. At short circuit, e = 0, 0 = \/#o2 — xzi2 — ri, and ° ; (6) X' that is, the maximum line current which can be established with a non-inductive receiver circuit and negligible line capacity. 71. The condition of maximum 'power delivered over the line '• i| f-* on that is, substituting (3): '! V#o2 - x*i* = e + ri, and expanding, gives e* = (r2 + x2) i2 (8) = z2i2; hence, e — zi, and - = z. (9) -T- = 7*1 is the resistance or effective resistance of the receiving circuit; that is, the maximum power is delivered into a non- LOAD CHARACTERISTIC OF TRANSMISSION LINE 87 inductive receiving circuit over an inductive line upon which is impressed a constant e.m.f., if the resistance of the receiving circuit equals the impedance of the line, TI = z. In this case the total impedance of the system is Z0 = Z + n = r + z + jx, (10) or, zo = V(r + z)2 + z2. (11) Thus the current is *o V(r + z)2 + x2 and the power transmitted is Eo2z (r that is, the maximum power which can be transmitted over a line of resistance r and reactance x is the square of the impressed e.m.f. divided by twice the sum of resistance and impedance of the line. At x = 0, this gives the common formula, Inductive Load 72. With an inductive receiving circuit of lag angle 6, or power-factor p = cos 8, and inductance factor q = sin 6, at e.m.f. E = e at receiving circuit, the current is denoted by I = I(p-jq); (15) thus the e.m.f. consumed by the line impedance Z = r -f jx is E! = ZI = I (p -jq)(r+jx) = I [(rp + xq) - j (rq - xp)], and the generator voltage is Eo = E + #1 = [e + / (rp + sg)]. - jl (rq - xp); (16) 88 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, #o = V+ 7 (rp + xq)}2 + P (rq - xp)*, (17) and e = \/EQ*- P(rq-xp)2 - I (rp + xq). (18) The power received is the e.m.f. times the power component of the current; thus P = elp __ = Ip \/Eo*- P(rq-xp)* - Pp (rp + xq). (19) The curve of e.m.f., e, as function of the current I is again an arc of an ellipse. At short circuit e = 0; thus, substituted, /-£• (20) the same value as with non-inductive load, as is obvious. 73. The condition of maximum output delivered over the line is that is, differentiated, V#o2 -I2(rq-xp)2 = e + I (rp + xq); (22) substituting and expanding, e = Iz] or y = z. (23) Zi = -j is the impedance of the receiving circuit; that is, the power received in an inductive circuit over an inductive line is a maximum if the impedance of the receiving circuit, z\y equals the impedance of the line, z. In this case the impedance of the receiving circuit is Zi = z(p +jq), (24) and the total impendance of the system is ZQ = Z -{- Zi = r + jx + z (p + jq) LOAD CHARACTERISTIC OF TRANSMISSION LINE 89 Thus, the current is /i = and the power is V(r-f 2 (z + rp + xq) EXAMPLES (25) (26) 74. (1) 12,000 volts are impressed upon a transmission line of impedance Z = r + jx = 20 + 50 j. How do the voltage \ \ \ V son mo VOLTS 11000 9000 7000 4000 20 40 60 80 100 120 140 160 .180 200 220 FIG. 39. — Non-reactive load characteristic^ of a transmission line. Con- stant impressed e.m.f. and the output in the receiving circuit vary with the current with non-inductive load? Let e = voltage at the receiving end of the line, i = current: thus = ei — power received. The voltage impressed upon the line is then Eo = e + Zi = e +ri + jxi; 90 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, Eo = V( Since EQ = 12,000, 12,000 = V(e + n)2 + xH* = V(e + 20 i)* + 2500 i\ e = V12,0002 - x2i2 - ri = Vl2,0002 - 2500^ - 20 The maximum current for e = 0 is thus, 0 = V12,0002 - 2500 i2 - 20 i\ i = 223. Substituting for i gives the values plotted in Fig. 39. i e p = ei 0 12,000 0 20 11,500 230 X 103 40 11,000 440 X 103 60 10,400 624 X 103 80 9,700 776 X 103 100 8,900 890 X 103 120 8,000 960 X 103 140 6,940 971 X 103 160 5,750 920 X 103 180 4,340 784 X 103 200 2,630 526 X 103 220 400 88 X 103 223 0 0