ALTERNATE CURRENT GENERATOR. 77 



Then the vectors to a,, a„ a,, &c., represent completely in ampli- 

 tude and phase the different harmonics of the armature current, the 

 subscribed numbers indicating the orders of the harmonics ; and those 

 to ttj, a^, a^, &c., represent completely, in the same way, the different 

 harmonics of the induced alternating field current. 



Again {see section 10) if we rotate the vector drawn to the middle 

 point of a^ a, backwards through a right angle we obtain the vector 

 OE, that represents Ijmw into the first harmonic of the total e.m.f. 

 E generated in the armature ; and if we rotate backwards through 

 7r/2 the vector to the middle point of a, a^ we obtain the vector OE3 

 that represents Ij^mio into the third harmonic of E ; and similarly for 

 the other harmonics of E. 



In the same way, by rotating backwards through 7r/2 the vector 

 to the middle point of a, a,, we obtain the vector OH, that represents 

 1/2 mw into the fundamental harmonic of the e.m.f. H induced in the 

 field circuit, and so on for the other harmonics of H. 



Again, the mean torque exerted on the generator is equal to m/2 

 into the sum of the areas of the triangles a^Oa,, a,Oa,, u.Oaj, a^Oa^, 

 <fcc., these triangles, in the case of any generator, being all taken as 

 positive. 



14. When, for any generator, the f, t, operators have been cal- 

 culated for a particular load (see section 5) a geometrical solution can 

 easily be obtained to a high degree of accuracy by aid of a ruler, scale, 

 slide rule, and protractor. 



Thus, if we neglect the harmonics a, a„ &c., then, drawing any 

 vector from the origin to represent a,, we can construct for a,„ as a„ = 

 — ^,a, (see section 6). Erom a^ we can construct for r^u^, and as 

 aj + T^a^ + a, = 0, the triangle of vectors gives us a,. Proceeding in 

 this' way, we obtain in succession a^, a,, a,, a,, a„, which represent the 

 harmonics of x and i correct as regards relative phase and relative 

 amplitude. But, as a^ is equal to twice the exciting current, we have 

 a scale for our diagram, and hence obtain a complete solution. 

 The fact that for a practical alternator the r operators are very 

 approximately pure numbers (see sections 8, 22) renders this method 

 of solution both easy and expeditious. 



15. If a source of constant e.m.f. be included in the armature 

 circuit as well as in the field circuit of the simple alternator indicated 

 in Fig. 1, equations 1, section 1 become — 



I'X -{- j7i ^"^ + ^«t cos u)f j = e 

 p^ + -T^ I Av + mx cos {3it\ ■=■ T] 



and both the armature and field currents will now contain harmonics 

 of all orders, odd and even. 



