Credit to: Geogebra - A B Cron for sharing publicly: Adding sine and cosine function graphically
3rd harmonic frequency injected to fundamental frequency
Created by Apolinario "Sam" Ortega using
Geogebra 7/9/2016
Credit to: Geogebra - Malin Christersson for sharing publicly: superposition of waves
Modified by Apolinario "Sam" Ortega for inductive circuit voltage, current, and power profile. Date created 7/9/2016
Shown above is the decomposition of distorted or noisy waveform. It is made up of fundamental waveform, (f1) 3rd harmonics (f2) and 5th harmonics (f3), and DC component (f4)
French mathematician Jean Baptiste Fourier demonstrated that any distorted periodic sine waveform can be decomposed into several fundamental sine wave.
To illustrate his discovery, look at the white color distorted periodic sine wave shown below. What Fourier discovered is he can recreate the distorted periodic sine wave by simply adding three perfect sine waveforms but at different cycle. For example if you add the 1 cycle sine wave + 3 cycle sine wave + 5 cycle sine wave the resulting wave form is the distorted sine wave form white color .
Why this knowledge is very important? This knowledge is very important because of the growing power electronics in solar farms and wind farms. The power electronics are known
to distort the perfect sine wave. With frequency spectrum analyzer we can see the dominant harmonic content that was added to the perfect sine wave. For example,
if we discovered from analyzing frequency spectrum that 3rd harmonics (meaning for US electric grid, 60 Hz is the electric frequency value,
3rd harmonics is 60 hz times 3 = 180 Hz) is causing the distortion of perfect sine voltage waveform or perfect current waveform then before we connect
to electric grid the inverter based power electronics we can install a filter that removes the 3rd harmonics generated by power electronics. Disclaimer:
Please note this example was used for illustration only.
Freq: 1 Phase: 0 Click the animation button to start
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Injecting 3rd harmonics to fundamental at 180 HzFreq: 2 Phase: 0
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Injecting 5th harmonics to fundamental at 300 HzFreq: 3 Phase: 0
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Addition of Waves 1, 2, and 3Creating complex waveforms can be made by adding three series of sine waves with different frequencies. Use the above button to Increase or decrease the frequencies and increment the phase of the component sine waves to see how they add together.
Fourier analysis is a mathematical tool you use to show how many series of sine waves frequencies you need to add to create a complex distored sine waveform. As you can see from above waveform the total of seven series of sine waves with different frequencies generated a new complex distorted sine waveform that looks like a square waveform. The process of adding together series of sine waves with different frequencies are called Fourier synthesis. . Transformation of sine wave in time domain to frequency domain can be done using fast Fourier transform or fft. Shown below is an illustration of fft using Matlab software.
As we progress in 4th industrial revolution, waveform analysis knowledge and understanding is very important. Shown below is an example of waveform analysis use in system protection relay and synchrophasor phasor measurement units for electrical system network situational awareness.
Review of Sampling Frequency
Fundamental (60Hz) + 3rd (1 peak) + 5th (1 peak) + 7th (1 peak) Harmonic Addition
Electro Magnetic Transient Program Simulation Waveform (EMTP) shared in LinkedIn by Bahram Khodabakhchian
Given the distorted waveform from EMTP study how can we do fast fourier transform (fft) to discover the dominant frequency of voltage or current?
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Fast Fourier Transform shared in LinkedIn by: Source Credit to:
Bingsen Wang, PhD
Global Manager, HV Safety Integration at Stellantis, SMIEEE
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