Revolving Vectors With Special Application to Alternating Current Phenomena

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As an example of the former, suppose a circuit, of resistance R and reactance Leo, includes two electromotive forces in series, the second lagging behind the first by a phase difference 6. It is assumed that the frequency is the same for both. The combined e. M. F. Expressed in a uniform circular formula is E = E! + E 2 = fi *% + *- E 2 = e'**[Ei + e ~ ' %] = ' w [#i + (cos 0-/sin 0)E 2 ], = e'^Ei +E 2 cos 0-jE 2 sin 0]. Let us assume that tan 6 f = 2 z, and derive corre- HARMONIC ELECTROMOTIVE... FORCES IN SERIES 31 spending values for sin Q' and cos 6' '. We then shall have 0' /sin 6' cos and cos, cos . The impedance of the circuit is R +jLa), and it produces a lag 6" in the current behind the combined e. M. F. The value of 6" is tan" 1 -^. We have also the relation H R + jLu) = J e "VR 2 + L 2 aj 2 . Therefore the current is 7= E Example. Assume ^i = 1000 volts, ^2 = 1200 volts, 6> = 60, R=12 ohms, Lw = 5 ohms, to find 6', 6", I, and E. Answers. 6' = i&ir l 0. 6495, 0" = tan" 1 -ft, 7= 146.

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