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Home Excitation Characteristics of IronCore Reactors Harmonics  
See also: Third Harmonics in 3Phase Transformer Operation  
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Harmonics
Under steady conditions of constant voltage and constant frequency, the current in a reactor, or the exciting current (the noload current in a transformer), is a periodic function of time. A periodic function can be represented by a Fourier series as follows
The constant a_{0} corresponds to a dc component. Under steadystate conditions, if the applied voltage does not contain a dc component, the constant a_{0} is equal to zero. With ac excitation only, the exciting current has symmetrical wave form, i.e., the negative half cycle has the same shape as the positive half cycle. Figure 55 shows a symmetrical wave, i.e., one in which f[t + (T/2)] = f(t). A symmetrical wave contains odd harmonics only because the presence of even harmonics leads to dissymmetry. Then, for a constant frequency, the angular velocity ω must be constant, and the wave is symmetrical when
In Eq. 521, T is the period or the time for the wave to undergo a complete cycle. Hence, ωT = 2π radians, and Eq. 521 can be reduced to
If the nth harmonic in the cosine series is to satisfy the conditions for symmetry
but
then, from Eqs. 523 and 524, symmetry requires that
and
Equation 524 can be satisfied only if n is odd; as for even values of n, cos nπ is 1. Similarly
only if n is odd. It follows, therefore, that only odd harmonics are present in a periodic function that is symmetrical, as is the case for an ironcore reactor or the exciting current in an ironcore transformer when the flux wave is symmetrical. Such a current can therefore be represented by omitting a_{0} and all the even harmonics in Eq. 520 as follows
Equation 527 can also be put into the following form
where
Hence, if I_{1} is the effective value of the fundamental component of current, and I_{3}, I_{5}, etc. are the effective values of the harmonic components, the exciting current can be expressed by
A sinusoidal flux variation with respect to time produces a distorted current in an ironcore reactor or in ironcore transformers, which can be divided into a fundamental component and into odd harmonic components. The third harmonic is the predominating harmonic.
The amplitudes of the harmonics generally decrease as the order of the harmonics increases, harmonics of higher orders than the 9^{th} usually having negligible amplitudes. Figure 56 shows a current wave with its fundamental, 3^{rd} harmonic, and 5^{th} harmonic components.


Home Excitation Characteristics of IronCore Reactors Harmonics 