. Theory and calculation of alternating current phenomena . of another sine wave, 32 ALTERNA TING-CURRENT PHENOMENA I (Fig. 23), their resultant sine wave, 7o, has the rectangularcomponents a^ = (a + a), and 60 = (& + h). To get from the rectangular components, a and 6, of a sinewave its intensity, i, and phase, d, we may combine a and h bythe parallelogram, and derive i = Va^ + 62. 6tan 6 ^ — a Hence we can analytically operate with sine waves, as with forces in mechanics, by resolving theminto their rectangular components. 28. To distinguish, however, thehorizontal and the vertical com-ponen
. Theory and calculation of alternating current phenomena . of another sine wave, 32 ALTERNA TING-CURRENT PHENOMENA I (Fig. 23), their resultant sine wave, 7o, has the rectangularcomponents a^ = (a + a), and 60 = (& + h). To get from the rectangular components, a and 6, of a sinewave its intensity, i, and phase, d, we may combine a and h bythe parallelogram, and derive i = Va^ + 62. 6tan 6 ^ — a Hence we can analytically operate with sine waves, as with forces in mechanics, by resolving theminto their rectangular components. 28. To distinguish, however, thehorizontal and the vertical com-ponents of sine waves, so as not to beconfused in lengthier calculation, wemay mark, for instance, the verticalcomponents by a distinguishing index,or the addition of an otherwise mean-ingless symbol, as the letter j, andthus represent the sine wave by the expression 7 = a + jh, which now has the meaning that a is the horizontal and 6 thevertical component of the sine wave I, and that both componentsare to be combined in the resultant wave of intensity,. and of phase. tan 6 i = Va2 + 62,h Similarly, a — jh means a sine wave with a as horizontal,and — 6 as vertical, components, etc. Obviously, the plus sign in the symbol, a -f jh, does notimply simple addition, since it connects heterogeneous quan-tities—horizontal and vertical components—but implies com-bination by the parallelogram law. For the present, j is nothing but a distinguishing index, andotherwise free for definition except that it is not an ordinarynumber. 29. A wave of equal intensity, and differing in phase from thewave, a -\- jh, by 180°, or one-half period, is represented in SYMBOLIC METHOD 33
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