A Treatise On the Theory of Alternating Currents volume 1

Alexander Russell

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field we must have either OP or OQ equal to zero. Hence either Op = Op' or Oq= Oq' '; in either case the amplitudes of the two alternating fields must be equal in magnitude. In the first case (Fig. 110) Op and Op' must be in the same straight line but Fig. 110. Op and Op' are in the same line, when the alternating vectors produce a pure rotating field.
XIV] PURE ROTATING FIELD 289 pointing in opposite ways, so that the angle between Op and Op' is 7T.
Hence cf> + a = TT, where is... the angle between the directions of the two alternating vectors and a is their phase difference. For if (Fig. 110) Op be in the same line as Op', then > + = rOr' - rOp + r'Op' = r'6p + r'Op' = -jr. We can prove this important theorem analytically as follows. Let the strengths of the fields in the directions sr and s'r' (Fig. Ill) be given by H l cos cot and H. 2 cos (cot a) respectively, and let A B Fig. 111. When the angle rOr is the supplement of the phase difference between the fields alternating in the directions sr and s'r' we get a pure rotating field.

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