**2. Definition of harmonics**

The term *harmonic* comes from acoustics. It refers to the vibration of a column of air at a fre‐ quency which is a multiple of the basic frequency of repetition.

In electric signals, a harmonic is defined as the signal content at a specific frequency, which is a multiple integral of the current frequency system or main frequency produced by the gen‐ erators. With an oscilloscope, it is possible to observe a complex signal in the domain of time. At any moment in the given time, the amplitude of the waveform is displayed. If the same signal is applied to a high‐fidelity amplifier, the result in sounds is a mix of frequencies. The phase relationship does not affect the audible effects, which is acceptable in acoustics. But this is not the case with electric signals. The position of harmonics and the phase relationship in the harmonic from a different source can considerably alter the effects in electric signals. To define harmonic, it is important to first define the quality of the voltage wave, which must have a constant amplitude and frequency, as well as the sinusoidal form. **Figure 1** depicts the waveform without any content of harmonics, with a constant frequency of 60 Hz and a constant amplitude of 1 pu.

When a periodic wave does not have a sinusoidal form, it is said to have a harmonic content. This can alter its peak value and/or its RMS value causing alterations in the normal function‐ ing of any equipment that undergoes this voltage. The frequency of the periodic wave is known as the fundamental frequency and the harmonics are the signals whose frequency is an integer multiple of this frequency. **Figure 2** shows a voltage wave with a content of 30% of the fifth harmonic.

**Figure 1.** Wave without harmonic content.

electrical power system applications, the harmonics and their effects on power quality are a topic of concern. Currently in the United States, only 15–20% of the utility distribution load‐

Nowadays, the recommendation from IEEE Std. 519 imposed by utilities is becoming stricter due to the increase in proportion of nonlinear load. The problems of harmonic in the electrical power systems are low, but their analysis can help to increase plant power system reliability. The harmonics are a problem when their magnitude produces an electrical power system

The analysis and modeling of the harmonics are supported for the Fourier analysis. In the eighteenth and nineteenth century, J. B. Joseph Fourier (1768–1830) and other mathematicians performed basic calculations of harmonics. In the 1920s and 1930s, the distortion in voltage waveforms caused by power converters was noticed and studied. In the 1950s and 1960s, the study of harmonics in power converters extended to the transmission of voltage in the electri‐ cal power system. Currently, the electrical power systems have a large number of nonlinear elements that generate other waves at different frequencies. They generate these waves from sinusoidal waveforms to network frequency. This causes a phenomenon known as harmon‐ ics. Harmonics are phenomena that cause problems for both the users and the electricity sup‐

pliers. They have various harmful effects on the equipment in the electrical network.

quency which is a multiple of the basic frequency of repetition.

The term *harmonic* comes from acoustics. It refers to the vibration of a column of air at a fre‐

In electric signals, a harmonic is defined as the signal content at a specific frequency, which is a multiple integral of the current frequency system or main frequency produced by the gen‐ erators. With an oscilloscope, it is possible to observe a complex signal in the domain of time. At any moment in the given time, the amplitude of the waveform is displayed. If the same signal is applied to a high‐fidelity amplifier, the result in sounds is a mix of frequencies. The phase relationship does not affect the audible effects, which is acceptable in acoustics. But this is not the case with electric signals. The position of harmonics and the phase relationship in the harmonic from a different source can considerably alter the effects in electric signals. To define harmonic, it is important to first define the quality of the voltage wave, which must have a constant amplitude and frequency, as well as the sinusoidal form. **Figure 1** depicts the waveform without any content of harmonics, with a constant frequency of 60 Hz and a

When a periodic wave does not have a sinusoidal form, it is said to have a harmonic content. This can alter its peak value and/or its RMS value causing alterations in the normal function‐ ing of any equipment that undergoes this voltage. The frequency of the periodic wave is known as the fundamental frequency and the harmonics are the signals whose frequency is an integer multiple of this frequency. **Figure 2** shows a voltage wave with a content of 30% of

ing consists of nonlinear loads.

44 Fourier Transforms - High-tech Application and Current Trends

**2. Definition of harmonics**

constant amplitude of 1 pu.

the fifth harmonic.

resonance.

**Figure 2.** Voltage waveform with harmonic content.

## **3. Fourier analysis**

The analysis of harmonics is the process of calculating the magnitudes and phases of the fun‐ damental and high order harmonics of the periodic waveforms. The resulting series is known as Fourier series. It establishes a relation between a function in the domain of time and a func‐ tion in the domain of frequency.

The Fourier's theorem states that every nonsinusoidal periodic wave can be decomposed as the sum of sine waves through the application of the Fourier series, given the following conditions:


*Coefficients and Fourier series.* The Fourier series of a periodic function *x*(*t*) is expressed as:

Luegrêners and Fourier series. In he Fourier series or a permutation  $\mathbf{x}(t)$  is expressed as:

$$\mathbf{x}(t) = a\_0 + \sum\_{n=1}^{\bullet} \left( a\_n \cos\left(\frac{2\pi nt}{T}\right) + b\_n \sin\left(\frac{2\pi nt}{T}\right) \right) \tag{1}$$

This constitutes a representation of periodic function in the domain of the frequency.

In this expression, *a*<sup>0</sup> is the average value of the function *x*(*t*), where *<sup>a</sup> n* and *b n* are the coefficients of the series besides being the rectangular components of the *n* harmonic. For the correspond‐ ing *n* harmonic its vector is:

$$A\_u \mathfrak{D}\_u = |au + jbm\rangle\tag{2}$$

With a magnitude and an angle of phase:

$$A\_u = \sqrt[n]{(a^2 n + b^2 n)}, \bigotimes\_n = \tan^{-1} b\_n \tag{3}$$

## **4. Harmonic sources**

Harmonics are the result of nonlinear loads which give a nonsinusoidal response to a sinusoi‐ dal signal. The main sources of harmonics are:


The AC electrical power system harmonic issues are mainly due to the substantial increase of nonlinear loads due to technological advances, such as the use of power electronics circuits and devices, in AC/DC transmission links, or loads in the control of power systems using power electronic or microprocessor controllers. Such equipment creates load‐generated har‐ monics throughout the electrical power system.

**Figure 3.** Current and voltage of a typical six‐pulse rectifier input.

In the case of a motor drive, the AC current at the input to the rectifier looks more like a square wave than a sine wave (see **Figure 3**).

The rectifier can be thought of as a harmonic current source and produces roughly the same amount of harmonic current over a wide range of electrical power system impedances. The characteristic current harmonics that are produced by a rectifier are determined by the pulse number. The following equation allows determination of the characteristic harmonics for a given pulse number:

$$h = \, kq \pm 1\tag{4}$$

where:

In this expression, *a*<sup>0</sup>

ing *n* harmonic its vector is:

**4. Harmonic sources**

excite the iron.

With a magnitude and an angle of phase:

46 Fourier Transforms - High-tech Application and Current Trends

dal signal. The main sources of harmonics are:

• Controlled sources for electronic equipment.

monics throughout the electrical power system.

• Static reactive power compensators.

• AC to DC converters (inverters).

• DC high voltage transmission stations.

cal values with regard to the fundamental harmonic:

*An* = √

is the average value of the function *x*(*t*), where *<sup>a</sup>*

\_\_\_\_\_\_\_\_\_

of the series besides being the rectangular components of the *n* harmonic. For the correspond‐

*An* ∅*<sup>n</sup>* = *an* + *jbm* (2)

Harmonics are the result of nonlinear loads which give a nonsinusoidal response to a sinusoi‐

• Arc furnaces and other elements of arc discharge, such as fluorescent lamps. Arc furnaces are considered as voltage harmonic generators more than current generators. Typically all harmonics (2nd, 3rd, 4th, 5th,...) appear but the odd harmonics are predominant with typi‐

• Magnetic cores in transformers and rotating machines require third harmonic current to

– The third harmonic represents 20%, and the fifth harmonic represents 10%. – The seventh harmonic represents 6%, and the ninth harmonic represents 3%.

• The inrush current of transformers produces second and fourth harmonics. • Adjustable speed controllers used in fans, pumps, and process controllers.

• Solid‐state switches which modulate control currents, light intensity, heat, etc.

• Rectifiers based on diodes and thyristors for welding equipment, battery chargers, etc.

The AC electrical power system harmonic issues are mainly due to the substantial increase of nonlinear loads due to technological advances, such as the use of power electronics circuits and devices, in AC/DC transmission links, or loads in the control of power systems using power electronic or microprocessor controllers. Such equipment creates load‐generated har‐

*n* and *b n*

(*a* <sup>2</sup> *n* + *b*<sup>2</sup> *n*), ∅*<sup>n</sup>* = tan<sup>−</sup><sup>1</sup> *bn* (3)

are the coefficients

*h* is the harmonic number (integer multiple of the fundamental),

*k* is any positive integer, and

*q* is the pulse number of the converter.

The harmonics 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th, etc., are the harmonics that a 6‐pulse rectifier will exhibit and which are multiples of the fundamental. The quotient of the fun‐ damental current and the harmonic number will result in the magnitudes of the harmonic currents (e.g., the magnitude of the 5th harmonic would be about 1/5th of the fundamental current). When it comes to a 12‐pulse systems, a small amount of the 5th, 7th, 17th, and 19th harmonics will be present (the magnitudes will be approximately a 10 percent of those for a 6‐pulse drive). The induction machines are quite affected by the harmonic currents pro‐ duced by inverters. Most of these harmonics produced are integer multiples of the inverter frequency and their magnitude will depend on the algorithm switching power semiconduc‐ tors of the inverter. It is common that there are "interharmonics" currents at the input or the output of the inverter but they do not necessarily occur at integer multiples of the power supply or inverter fundamental frequency. For this reason, it needs a good design of DC link to minimize the presence of interharmonics.

Some authors [1] agree to classify the sources of harmonic distortion in three groups: small and predictable (harmonics generated by residential consumers), large and transient (volt‐ age fluctuations produced by arc furnaces), and large and predictable (SVC and HVDC transmission causing characteristic and uncharacteristic harmonics). Now, if at a point com‐ mon coupling harmonic currents are not within the permissible limits, it is necessary to take appropriate measures to comply with regulations. For example the IEEE 519‐1981, "*IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters*," originally established levels of voltage distortion acceptable to the distribution system for individual nonlinear loads. This distortion is a steady‐state deviation from a sine wave of power fre‐ quency called waveform distortion [2]. Fourier series is generally used to analyze this nonsi‐ nusoidal waveform.
