5. SELF-INDUCTANCE AND MUTUAL INDUCTANCE 26. The number of inter-linkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in the circuit is called the inductance of the circuit. The number of interlinkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in a second electric circuit is called the mutual inductance of the second upon the first circuit. It is equal to the mutual induc- tance of the first upon the second circuit, as will be seen, and thus is called the mutual inductance between the two circuits. The number of interlinkages of an electric circuit with the lines of magnetic flux produced by unit current in this circuit and not interlinked with a second circuit is called the self- inductance of the circuit. If i = current in a circuit of n turns, = flux produced thereby and interlinked with the circuit, n$ is the total number 9? ^^ of interlinkages, and L = — r- the inductance of the circuit. If $ is proportional to the current i and the number of turns n, ni n2 . , $ = — , and L = — the inductance. 01 (K (ft is called the reluctance and ni the m.m.f. of the magnetic circuit. In magnetic circuits the reluctance (R has a position similar to that of resistance r in electric circuits. The reluctance (R, and therefore the inductance, is not con- stant in circuits containing magnetic materials, such as iron, etc. If (Ri is the reluctance of a magnetic circuit interlinked with two electric circuits of n\ and n% turns respectively, the flux produced by unit current in the first circuit and interlinked with the second circuit is -- and the mutual inductance of the first (HI upon the second circuit is M = , that is, equal to the Oil 22 ELEMENTS OF ELECTRICAL ENGINEERING mutual inductance of the second circuit upon the first circuit, as stated above. If no flux leaks between the two circuits, that is, if all flux is interlinked with both circuits, and LI = inductance of the first, L2 = inductance of the second circuit, and M = mutual induc- tance, then M2 = ZaL2. If flux leaks between the two circuits, then M 2 < LiL2. In this case the total flux produced by the first circuit con- sists of a part interlinked with the second circuit also, the mu- tual inductance, and a part passing between the two circuits, that is, interlinked with the first circuit only, its self-inductance. 27. Thus, if LI and L2 are the inductances of the two circuits, — and — is the total flux produced by unit current in the first n\ HZ and second circuit respectively. T Sf Of the flux --a part — is interlinked with the first circuit ni HI only, Si being its self-inductance or leakage inductance, and a part — interlinked with the second circuit also, M being the mutual inductance and — 1 = — + — HI HI n2 Thus, if LI and L2 = inductance, Si and Sz = self-inductance, M = mutual inductance of two circuits of n and n2 turns respectively, we have h = Sl + *L Lz = Sz M HI HI nz nz ~ HZ n\ or Li = Si + — M L2 = Sz + - M, HZ HI or M2 = (Li - Si)(Lz - Sz). The practical unit of inductance is 109 times the absolute unit or 108 times the number of interlinkages per ampere (since 1 amp. = 0.1 unit current), and is called the henry (h); 0.001 of it is called the milhenry (mh.). The number of interlinkages of i amperes in a circuit of SELF-INDUCTANCE AND MUTUAL INDUCTANCE 23 L henry inductance is iL 108 lines of force turns, and thus the e.m.f. generated by a change of current di in time dt is e = — -j-. L 108 absolute units r dt = — -T.L volts. at A change of current of 1 amp. per second in the circuit of 1 h. inductance generates 1 volt. EXAMPLES 28. (1) What is the inductance of the field of a 20-pole alternator, if the 20 field spools are connected in series, each spool contains 616 turns, and 6.95 amp. produces 6.4 mega- lines per pole? The total number of turns of all 20 spools is 20 X 616 = 12,320 Each is interlinked with 6.4 X 106 lines, thus the total number of interlinkages at 6.95 amp. is 12,320 X 6.4 X 106 = 78 X 109. 6.95 amp. = 0.695 absolute units, hence the number of in- terlinkages per unit current, or the inductance, is - 112 X I*- 112 h. 29. (2) What is the mutual inductance between an alter- nating transmission line and a telephone wire carried for 10 miles below and 1.20 m. distant from the one, 1.50 m. distant from the other conductor of the alternating line; and what is the e.m.f. generated in the telephone wire, if the alternating cir- cuit carries 100 amp. at 60 cycles? The mutual inductance between the telephone wire and the electric circuit is the magnetic flux produced by unit current in the telephone wire and interlinked with the alternating circuit, that is, that part of the magnetic flux produced by unit current in the telephone wire, which passes between the dis- tances of 1.20 and 1.50 m. At the distance lx from the telephone wire the length of mag- netic circuit is 2irlz. The magnetizing force / = —- if 7 = 24 ELEMENTS OF ELECTRICAL ENGINEERING current in telephone wire in amperes, and the field intensity d the 0.27 H = 0.4 TT/ = — — , and the flux in the zone dlx is j dlx. lx I = 10 miles = 1610 X 103 cm. thus, f1500.2// = I — i — dlx Jl20 *» = 322 X 10371ogei||° = 72 7 103; or, 72 7 103 interlinkages, hence, for 7 = 10, or one absolute unit, thus, M = 72 X 104 absolute units = 72 X 10~5 h. = 0.72 mh. 100 amp. effective or 141.4 amp. maximum or 14.14 abso- lute units of current in the transmission line produces a maximum flux interlinked with the telephone line of 14.14 X 0.72 X 10~3 X 109 = 10.2 megalines. Thus the e.m.f. generated at 60 cycles is E = 4.44 X 0.6 X 10.2 = 27.3 volts effective.