Summation of harmonics with random phase angles

The magnitude of the sum of a number of harmonics, of the same order but with random phase angle, is derived, subject to certain assumptions regarding the phase distribution. Computation of the net magnitude is carried out for different types of individual harmonic amplitudes. When the number of individual harmonics is greater than about ten, it is shown that simplified expressions for the sum may be used. These lead to a simple approach, based on r.m.s. values, to the problem of determining an assessed level of harmonic content which would have a specific probability of being exceeded. Consideration is also given to the situation when just one or two harmonics are dominant, and a consequent simplification of an assessed level. The main results apply for instantaneous values of the net magnitude, but some subsidiary results given an indication of the likely behaviour over a continuous period of time.