17. IMPEDANCE AND ADMITTANCE 82. In direct-current circuits the most important law is Ohm's law, e -i or e r ir, or r = -.> where e is the e.m.f. impressed upon resistance r to produce current i therein. Since in alternating-current circuits a current i through a resistance r may produce additional e.m.fs. therein, when apply- a ing Ohm's law, i — - to alternating-current circuits, e is the IMPEDANCE AND ADMITTANCE ' 99 total e.m.f. resulting from the impressed e.m.f. and all e.m.fs. produced by the current i in the circuit. Such counter e.m.fs. may be due to inductance, as self-induc- tance, or mutual inductance, to capacity, chemical polarization, etc. The counter e.m.f. of self-induction, or e.m.f. generated by the magnetic field produced by the alternating current i, is repre- sented by a quantity of the same dimensions as resistance, and measured in ohms: reactance x. The e.m.f. consumed by reactance x is in quadrature with the current, that consumed by resistance r in phase with the current. Reactance and resistance combined give the impedance, + x2; or, in symbolic or vector representation, Z = r + jx. In general in an alternating-current circuit of current i, the e.m.f. e can be resolved in two components, a power component ei in phase with the current, and a wattless or reactive com- ponent e2 in quadrature with the current. The quantity e_i _ power e.m.f., or e.m.f. in phase with the current _ i current is called the effective resistance. The quantity 62 _ reactive e.m.f., or e.m.f. in quadrature with the current _ i current is called the effective reactance of the circuit. And the quantity 21 = Vr!2 + x2 or, in symbolic representation, Zi = ri + jxi is the impedance of the circuit. If power is consumed in the circuit only by the ohmic resist- ance r, and counter e.m.f. produced only by self-inductance, the effective resistance TI is the true or ohmic resistance r, and the effective reactance Xi is the true or inductive reactance x. 100 ELEMENTS OF ELECTRICAL ENGINEERING By means of the terms effective resistance, effective reactance, and impedance, Ohm's law can be expressed in alternating- current circuits in the form • = - e m y / 9 T ~ 9; ^ ' Zi vVi2 + Xi2 or, e = izi = i V^i2 + Zi2; (2) or, «! = Vri8 + a;ia = p (3) or, in symbolic or vector representation, or, E = IZl = /(n+jxi); (5) 7^7 or, Zi = ri + jzi = j- (6) In this latter form Ohm's law expresses not only the intensity but also the phase relation of the quantities; thus ei = iri = power component of e.m.f., ez = ix\ = reactive component of e.m.f. p 83. Instead of the term impedance z — - with its components, I? the resistance and reactance, its reciprocal can be introduced. e " z ' which is called the admittance. The components of the admittance are called the conduc- tance and the susceptance. Resolving the current i into a power component i\ in phase with the e.m.f. and a wattless component iz in quadrature with the e.m.f., the quantity i\_ _ power current, or current in phase with e.m.f. e e.m.f. . = 9 is called the conductance. The quantity _*2_ _ reactive current, or current in quadrature with e.m.f. e e.m.f. is called the susceptance of the circuit. The conductance represents the current in phase with the IMPEDANCE AND ADMITTANCE 101 e.m.f., or power current, the susceptance the current in quad- rature with the e.m.f., or reactive current. Conductance g and susceptance b combined give the admittance y = Vg2 + 62; (7) or, in symbolic or vector representation, Y = g - jb. (8) Thus Ohm's law can also be written in the form i = ey = e Vg2 + &2; (9) or, i or, y = Vg* + V = 7; (11) or, in symbolic or vector representation, I = EY = E(g-jb); (12) or, E = - or, Y = g - jb = |- (14) and i\ = eg = power component of current, ii = eb = reactive component of current. 84. According to circumstances, sometimes the use of the terms impedance, resistance, reactance, sometimes the use of the terms admittance; conductance, susceptance, is more convenient. Since, in a number of series-connected circuits, the total e.m.f., in symbolic representation, is the sum of the individual e.m.fs., it follows that in a number of series-connected circuits the total impedance, in symbolic expression, is the sum of the impedances of the individual circuits connected in series. Since, in a number of parallel-connected circuits, the total current, in symbolic representation, is the sum of the individual currents, it follows that in a number of parallel-connected cir- cuits the total admittance, in symbolic expression, is the sum of the admittances of the individual circuits connected in parallel. 102 ELEMENTS OF ELECTRICAL ENGINEERING Thus in series connection the use of the term impedance, in parallel connection the use of the term admittance, is generally more convenient. Since in symbolic representation Y = ^ (15) or, ZY = 1; (16) that is, (r+jx)(g - jb) = 1; (17) it follows that (rg + xb) - j (rb - xg) = 1; that is rg -f zb = 1, rb - xg = 0. r = -f±- = ±, (18) 6 = JTqirji = •#> (21) or, in absolute values, y = ' (22) (23) (r2 + x2)(^2 + 62) = 1. (24) Thereby the admittance with its components, the conduc- tance and susceptance, can be calculated from the impedance and its components, the resistance and reactance, and inversely. If x = 0, z = r and g = — , that is, g is the reciprocal of the resistance in a non-inductive circuit; not so, however, in an inductive circuit. EXAMPLES 85. (1) In a quarter-phase induction motor having an im- pressed e.m.f. e = 110 volts per phase, the current is /0 = ii — jiz = 100 — 100 j at standstill, the torque = D0. The two phases are connected in series in a single-phase cir- cuit of e.m.f. e = 220, and one phase shunted by a condenser of 1 ohm capacity reactance. What is the starting torque D of the motor under these con- ditions, compared with Z>0, the torque on a quarter-phase cir- IMPEDANCE AND ADMITTANCE 103 cuit, and what the relative torque per volt-ampere input, if the torque is proportional to the product of the e.m.fs. impressed upon the two circuits and the sine of the angle of phase dis- placement between them? In the quarter-phase motor the torque is D0 = ae2 = 12,100 a, where a is a constant. The volt-ampere input is Qo = 2 e Vii2 + i22 = 31,200; hence, the "apparent torque efficiency," or torque per volt- ampere input, rjQ = D* = 0.388 a. The admittance per motor circuit is the impedance is Y = = 0.91 - 0.91 j, e _ 110 (100 + 100 j) _055+055/ I ~ 100- 100 j ~ (100.-100j)(100+100j)~ the admittance of the condenser is Yo = j; thus, the joint admittance of the circuit shunted by the con- denser is Yi= Y + 7o = 0.91 - 0.91 j + j = 0.91 +0.09 j; its impedance is 7 J_ _ L_ 0.91- 0.09 j Zl ~ F, ~ 0.91 + 0.09 j ~ 0.9P + 0.092 = X 3> and the total impedance of the two circuits in series is Z2 = Z + Zl = 0.55 + 0.55 j + 1.09 - 0.11 j = 1. 64 + 0.44 j. Hence, the current, at impressed e.m.f. e = 220. r . .. e 220 220 (1.64- 0.44 j) !ti ^2 ~ Z2~ 1.64 + 0.44 j~ 1.642 + = 125 - 33.5 j; 104 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, / = V1252 + 33.52 = 129.4 amp. Thus, the volt-ampere input, Q = el = 220 X 129.4 = 28,470. The e.m.fs. acting upon the two motor circuits respectively are Ei = /Zi = (125 - 33.5 j) (1.09 - 0.11 j) = 132.8 - 50.4 j and E' = IZ = (125 - 33.5 j) (0.55 + 0.55 j) = 87.2 + 50.4 j. Thus, the tangents of their phase angles are 50 4 tan 0i = + - = + 0.30; hence, Bl = + 21°; 50 4 tan tf = - = - 0.579; hence, 6' = - 30°; and the phase difference, 0 = 0i The absolute values of these e.m.fs. are ' = 51°. ei = x/132.8 + 50.42 = 141.5 and e' = V87.22 - 50.42 = 100.7; thus, the torque is D = ae\e' sin 6 = 11, 100 a; and the apparent torque efficiency is _D 11,100 a " ~ Q WTO" Hence, comparing this with the quarter-phase motor, the relative torque is D_ = 11,100 a Do 12,100 a and the relative torque per volt-ampere, or relative apparent torque efficiency, is it 0.39 a 0.388 a = 1.005. IMPEDANCE AND ADMITTANCE 105 86. (2) At constant field excitation, corresponding to a nominal generated e.m.f. JSQ — 12,000, a generator of synchro- nous impedance ZQ = r0 + J^o = 0.6 + 60 j feeds over a trans- mission line of impedance Z\ = ri + jx\ = 12 '+ 18 j, and of capacity susceptance 0.003, a non-inductive receiving circuit. How will the voltage at the receiving end, e, and the voltage at the generator terminals, e\, vary with the load if the line capacity is represented by a condenser shunted across the middle of the line? Let I = i = current in receiving circuit, in phase with the e.m.f., E = e. The voltage in the middle of the line is = e + 6 i + 9 ij. The capacity susceptance of the line is, in symbolic expression, Y = 0.003 j; thus the charging current is 72 = E2Y = 0.003 j (e + 6 i + 9 ij) = 0.027 i + j (0.003 e + 0.018 i), and the total current is /! = I + 72 = 0.973 i + j (0.003 e + 0.018 i). Thus, the voltage at the generator end of the line is = e + 6 i + 9 ij + (6 + 9 j)[0.973 i + j (0.003 e + 0.018 i)] = (0.973 * + 11.68 i) + j (17.87 i + 0.018 e), and the nominal generated e.m.f. of the generator is E0 = E! + Zo|i = (0.973 e + 11.68 i) + j (17.87 i + 0.018 e) + (0.6 + 60 j) [0.973 t + j (0.003 e + 0.018 t)] = (0.793 e + 11.18 i) + j (76.26 i + 0.02 e); or, reduced, and e0 = 12,000 substituted, e02 = 144 x 106 = (0.793 e + 11.18 i)2 + (76.26 i + 0.02 e)2; thus, e2 + 33 ei + 9450 i2 = 229 X 106, e = - 16.5 i + V229 X 106 - 9178 1*, 106 ELEMENTS OF ELECTRICAL ENGINEERING and ei = V(0.973e + 11.68 *)2 -f (17.87 i + O.OlSe)2; at i = 0, e = 15,133, a = 14,700; at e = 0, i = 155.6, d = 3327. P )WER CURRENT REC'D AMP V *OLT8 5000 3000 5000 3000 2000 10 20 JO 40 50 DO 70 80 90 100 110 120 130 140 150 FIG. 40. — Reactive load characteristics of a transmission line fed by synchronous generator with constant field excitation. Substituting different values for i gives i ' ei i e ei 0 15,133 14,700 100 10,050 11,100 25 14,488 14,400 125 7,188 8,800 50 13,525 13,800 150 2,325 4,840 75 12,063 12,730 155.6 0 3,327 which values are plotted in Fig. 40.