Home Computer Science Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

# Harmonics in the Time Domain and Frequency Domain

In electrical engineering, “time domain” is a term used to describe the analysis of electrical signals with respect to time. In the graph below, we have an example of an ideal undistorted alternating voltage or current signal. The values of the signal fluctuate between positive and negative amplitudes over time. As time progresses, the graphical representation clearly displays the variance in amplitude. One method of describing the nonsinusoidal waveform is called its Fourier Series. Jean Fourier was a French mathematician of the early 19th century who discovered a special characteristic of periodic waveforms.

## A. Time Domain Model

In this model, it is assumed that the waveform under consideration consists of a fundamental frequency component and harmonic components with order of integral multiples of the fundamental frequency. It is also assumed that the frequency is known and constant during the estimation period. Consider a non-sinusoidal voltage given by a Fourier-type equation:

where v(t) is the instantaneous voltage at time t, n is the order of the harmonic, Vn is the voltage amplitude of harmonic n, and n is the phase angle of harmonic n, 0 is the fundamental frequency, and N is the total number of harmonics.

The equation (2-1) can be rewritten as a summation of trigonometric function:

where

and

With the DC component case: then equation (2-5) can be written as:

Figure 2.4 shows sinusoidal waveform (a) and non-sinusoidal waveform (b) in the time domain. The distorted waveform includes a number of harmonics components. Periodic waveforms are those waveforms comprised of identical values that repeat in the same time interval, like those shown above. Fourier discovered that periodic waveforms can be represented by a series of sinusoids summed together. The frequency of these sinusoids is an integer multiple of the frequency represented by the fundamental periodic waveform. The waveform on the above, for example, is described entirely by one sinusoid - the fundamental - since it contains no harmonic distortion.

Figure 2.4 Harmonics in time domain presentation

## B. Frequency Domain Model

All periodic waves can be generated with sine waves of various frequencies. The Fourier theorem breaks down a periodic wave into its component frequencies. Therefore, the frequency domain (FD) presentation allows us to see the amplitudes of the frequencies that make up a signal. Frequency domain presentation can be created by using Fourier series transforms, as shown in Figure 2.5.

The waveform can be defined as a summation of trigonometric function, and the complex Fourier series with к harmonics can be expressed as follows:

Trigonometry form:

where:

Exponential form: where:

Figure 2.5 Frequency domain graphs - frequency spectrums

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