IV. Self-inductance 12. The effect of self -inductance is ^ similar to that of armature reaction, FlQ 50._Diagram of e m fs> and depends upon the phase relation in loaded generator, in the same manner. If EI = real generated voltage, 0i = lag of current behind generated voltage EI, the magnetic flux produced by the arma- ture current I is in phase with the current, and thus the counter e.m.f. of self-inductance is in quadrature behind the current, and therefore the e.m.f. consumed by self-inductance is in quadrature ahead of the current. Thus in Fig. 50, denoting OEi = EI the generated e.m.f., the current is 01 = 7; lagging 61 behind OEi, the e.m.f. consumed by self -inductance OE "i, is 90 degrees ahead of the current, and the virtual generated e.m.f. E2, is the resultant of OEi and OE'\. As seen, the diagram of e.m.f s. of self -induc- tance is similar to the diagram of m.m.fs. of armature reaction. 134 ELEMENTS OF ELECTRICAL ENGINEERING 13. From this diagram we get the effect of load and phase re- lation npon the e.m.f. of an alternating-current generator. Let E — terminal voltage per machine circuit, 7 = current per machine circuit, and 0 = lag of the current behind the terminal voltage. Let r = resistance, x = reactance of the alternator armature. FIG. 51. — Diagram showing combined effect of armature reaction and arma- ture self-inductance. Then, in the vector diagram, Fig. 51, OE = E, the terminal voltage, assumed as zero vector. 01 = I, the current, lagging by the angle EOI = 0. _The e.m.f. consumed by resistance is OE \ = Ir in phase with 01. The e-m-i^ consumed by reactance is OEfz — Ix, 90 degrees ahead of 01. The real generated e.m.f. is found by combining OE and OE\ to SYNCHRONOUS MACHINES 135 The virtual generated e.m.f. is OEi and OE'Z combined to = E2. The m.m.f. required to produce -this e.m.f. Ez is OF = F, Fa I E, FIG. 52. — Diagram of generator e.m.fs. and m.m.fs. for non-reactive load. 90 deg. ahead of OE2. It is the resultant of the armature m.m.f. or armature reaction and of the impressed m.m.f. or field excita- tion. The armature m.m.f. is in phase with the cur- rent 7, and is nl in a single-phase machine, n \/2 / in a quarter-phase machine, 1.5 \/2 nl in a three- phase machine, if n = number of armature turns per pole and phase. The m.m.f. of armature reaction is represented in the diagram by OFa of Fa in phase with 01, and the impressed m.m.f. or field excitation OFo = FQ is the side of a parallelogram with OF as diag- onal and OFa as other side; or, the m.m.f. consumed by armature reaction is represented by OF'a = Fa in opposition to 01. Combining OF'a and OF gives OFQ = FQ, the field excitation. F, FIG. 53. — Diagram of generator, e.m.fs. and m.m.fs. for lagging reac- tive load. Power-factor 0 . 50. FIG. 54. — Diagram of generator e.m.fs. and m.m.fs. for leading reac- tive load. Power-factor 0.50. 136 ELEMENTS OF ELECTRICAL ENGINEERING In Figs. 52, 53, 54 are drawn the diagrams for 0 = zero or non-inductive load, 0 = 60 degrees, or 60 degrees lag (inductive load of power-factor 0.50), and 0 = — 60 deg., or 60 deg. lead (anti-inductive load of power-factor 0.50). Thus it is seen that with the same terminal voltage E a much higher field excitation, FQ, is required with inductive load than with non-inductive load, while with anti-inductive load a much lower field excitation is required. With open circuit the field excitation required to produce the terminal voltage W E would be ~r F = FQQ, or less than the field excitation J^o with JCJQ non-inductive load. Inversely, with constant field excitation, the voltage of an al- ternator drops with non-inductive load, drops much more with inductive load, and drops less, or even rises, with anti-inductive load.