Capacitors, Magnetic Circuits, and Transformers is a free introductory textbook on the physics of capacitors, coils, and transformers. See the editorial for more information....  # Harmonics

Under steady conditions of constant voltage and constant frequency, the current in a reactor, or the exciting current (the no-load current in a transformer), is a periodic function of time. A periodic function can be represented by a Fourier series as follows [5-20]

The constant a0 corresponds to a d-c component. Under steady-state conditions, if the applied voltage does not contain a d-c component, the constant a0 is equal to zero. With a-c 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 5-5 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 [5-21]

In Eq. 5-21, T is the period or the time for the wave to undergo a complete cycle. Hence, ωT = 2π radians, and Eq. 5-21 can be reduced to [5-22] Figure 5-5. Symmetrical wave

If the nth harmonic in the cosine series is to satisfy the conditions for symmetry [5-23]

but [5-24]

then, from Eqs. 5-23 and 5-24, symmetry requires that and [5-25]

Equation 5-24 can be satisfied only if n is odd; as for even values of n, cos nπ is 1. Similarly [5-26]

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 iron-core reactor or the exciting current in an iron-core transformer when the flux wave is symmetrical. Such a current can therefore be represented by omitting a0 and all the even harmonics in Eq. 5-20 as follows [5-27]

Equation 5-27 can also be put into the following form [5-28]

where [5-29] [5-30]

Hence, if I1 is the effective value of the fundamental component of current, and I3, I5, etc. are the effective values of the harmonic components, the exciting current can be expressed by [5-31]

A sinusoidal flux variation with respect to time produces a distorted current in an iron-core reactor or in iron-core transformers, which can be divided into a fundamental component and into odd harmonic components. The third harmonic is the predominating harmonic. Figure 5-6. Exciting current and its fundamental, 3rd, and 5th harmonic components

The amplitudes of the harmonics generally decrease as the order of the harmonics increases, harmonics of higher orders than the 9th usually having negligible amplitudes. Figure 5-6 shows a current wave with its fundamental, 3rd harmonic, and 5th harmonic components.

Last Update: 2011-01-14