CHAPTER IV. INDUCTANCE AND RESISTANCE IN ALTERNATING- CURRENT CIRCUITS. • 26. In alternating-current circuits, the inductance L, or, as it is usually employed, the reactance x = 2 nfL, where / = fre- quency, enters the expression of the transient as well as the permanent term. At the moment 0 = 0, let the e.m.f. e = E cos (0 — 00) be impressed upon a circuit of resistance r and inductance L, thus inductive reactance x = 2 xfL; let the time 6 = 2 xft be counted from the moment of closing the circuit, and 00 be the phase of the impressed e.m.f. at this moment. In this case the e.m.f. consumed by the resistance = ir, where i = instantaneous value of current. The e.m.f. consumed by the inductance L is proportional to L and to the rate of change of the current, — , thus, is L — , at at or, by substituting 6 = 2 nft, x = 2 nfL, the e.m.f. consumed by inductance is x — • do Since e = E cos (0 — 00) = impressed e.m.f., di E cos (6 - 00) = ir + x — (1) is the differential equation of the problem. This equation is integrated by the function i = 7 cos (6 - d) + A£~a°, (2) where e = basis of natural logarithms = 2.7183. Substituting (2) in (1), E cos (6 - 00) = Ir cos (6 - i) + Ars~a0 - Ix sin (d-d)- Aaxs'"', or, rearranged: (E cos 00 - Ir cos § - Ix sin d) cos 0 + (E sin 00 - Ir sin 8 + /x cos d) sin 0 - ^e~a" (ax - r) = 0. 41 42 TRANSIENT PHENOMENA Since this equation must be fulfilled for any value of 6, if (2) is the integral of (1), the coefficients of cos 6, sin 0, £~a9 must vanish separately. That is, E cos 00 — Ir cos d — Ix sin d = 0, E sin 00 - Ir sin d + Ix cos d = 0, and Herefrom it follows that ax — r = 0. Substituting in (3), r a= - tan 0X = and where (3) (4) (5) and and herefrom and z = W2 + lag angle and z = impedance of circuit, we have E cos 00 - 70 cos (# -0^=0 E sin 00 - 7z sin (d - 0X) = 0, (6) Thus, by substituting (4) and (6) in (2), the integral equation becomes E --« i = - cos (0 - 00 - 0X) + As x , (7) where A is still indefinite, and is determined by the initial con- ditions of the circuit, as follows : for 0 = 0, i = 0; hence, substituting in (7), E 0 = -cos (00 + 0J + A, ALTERNATING-CURRENT CIRCUITS 43 or, A -_|cos & + *!), <(8) z and, substituted in (7), i = -z | cos (I? - 00- 0J- i~x° cos (00 + OJ j (9) is the general expression of the current in the circuit. If at the starting moment 0 = 0 the current is not zero but = iw we have, substituted in (7), A = ^--(508(0 i =-- cos (d - 60 - ^)-cos (00 + 0,)- e* . (10) 27. The equation of current (9) contains a permanent term E — cos (0 — 00 — dj, which usually is the only term considered, E -~e and a transient term — e x cos (00 + 0t). z The greater the resistance r and smaller the reactance x, the more rapidly the term :- e ;c cos (00 -f 0t) disappears. This transient term is a maximum if the circuit is closed at the moment 00 = — 6V that is, at the moment when the E permanent value of current, — cos (0 — 00 — 0t), should be a maximum, and is then The transient term disappears if the circuit is closed at the moment 00 = 90° — Ov or when the stationary term of current passes the zero value. 44 TRANSIENT PHENOMENA As example is shown, in Fig. 7, the starting of the current under the conditions of maximum transient term, or 6Q = — dlt X in a circuit of the following constants: — = 0.1, corresponding approximately to a lighting circuit, where the permanent value GO ^ \ 1 1 Fig. 8. Starting current of an inductive circuit. X Of the last case, — = 10, a series of successive waves are r plotted in Fig. 8, showing the very gradual approach to perma- nent condition. ALTERNATING-CURRENT CIRCUITS 45 Fig. 9 shows, for the circuit — = 1.5, the current when closing the circuit 0°, 30°, 60°, 90°, 120°, 150° respectively behind the zero value of permanent current. The permanent value of current is usually shown in these diagrams in dotted line. ^^\Ss m 1.5 120 180 240 300 Degrees 480 640 Fig. 9. Starting current of an inductive circuit. 28. Instead of considering, in Fig. 9, the current wave as consisting of the superposition of the permanent term / cos (6 — Q0) and the transient term — h c cos 00 the current wave can directly be represented by the permanent term 0 4 3 2 1 0 -1 -2 -3 -4 -ft ^ \ X / \ \ s. > ^ ""S^ "N / ^~*. --^ \ ^^. - — . / / \ 1 / \ 1 \ / \ / V */ Fig. 10. Current wave represented directly. I cos (0 - 00) by considering the zero line of the diagram as -r-e deflected exponentially to the curve h '' cos 00 in Fig. 10. That is, the instantaneous values of current are the vertical 46 TRANSIENT PHENOMENA distances of the sine wave 7 cos (6 — 60) from the exponential --fi curve /s cos Q0, starting at the initial value of perma- nent current. In polar coordinates, in this case 7 cos (0 — 00) is the circle, -r-e h x cos 00 the exponential or loxodromic spiral. As a rule, the transient term in alternating-current circuits containing resistance and inductance is of importance only in circuits containing iron, where hysteresis and magnetic saturation complicate the phenomenon, or in circuits where unidirectional or periodically recurring changes take place, as in rectifiers, and some such cases are considered in the following chapters.