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Square, Sine, and Triangular Wave
Fourier Synthesis Approximation
The equation below is telling us that if you add many perfect sine wave you will get distorted sine wave in the shape of square wave, triangular wave, and sawtooth wave.
Application in Digital Signal Processing
Oscilloscope signal wave generator can be simulated by manipulating mathematical model that represent sine wave, triangular wave, square wave, and saw tooth wave.
1. By changing the sine wave amplitude ( represented by variable " a " )
You are doing Amplitude Modulation (AM) of signal. You can assigned a message to the sine wave amplitude. AM Radio is an example of this application. Try creating your own sine wave. Then play by changing the value of a = 1, a=2, a=3.
2. By changing the sine wave phase angle ( represented by variable " c " )
Your are doing Phase Modulation (PM) of signal. You can assigned a message to the sine wave phase angle. Try creating your own sine wave. Then play by changing the value of c = 1 , c = 0
3. By changing the sine wave frequency ( represented by variable " domain " )
You are doing Frequency Modulation (FM) of signal. You can assigned a message to the sine wave frequency. FM Radio is an example of this application. Try creating your own sine wave. Then play by changing the value of domain = 1, 2, 3,4, and so on.
4. By changing the number of series of sine waves to be added
( represented by variable " n ") you are smoothing the waveform . Change the value of n = 38. You will create a smoother saw tooth signal.
5. By adjusting the value of " p " you can smooth the waveform to create a sine wave. Move the slider button to make the value of p = 5. Your circuit will generate a sine wave signal. Now you know how to generate a sine wave generator signal.
Note : domain is simply the number of cycles. For example domain = 4
means 4 cycles. Frequency is cycle per second, for example 4 cycle/sec or 4 Hz. Try counting the number of cycles of sawtooth wave shown below and you will notice domain number is equal to the number of waveform cycles. Since we assumed the signal cycle will continue to go on as long as its signal generator is present, it is acceptable to do mathematical model of one (1) cycle or four (4) cycles since the waveform pattern are all periodic, meaning the pattern repeats every period. Understanding how to design and interpret mathematical equations and waveform model that represent the periodic pattern of digital signal is very important knowledge to acquire. This knowledge will help engineer doing design review to see if all the specifications are satisfied. Maintenance engineer and technician can also benefit from acquiring this knowledge so they can communicate better with the circuit designer.
Shared by jeromeawhite from GeoGebra
Triangular Wave By: Sam Ortega Credit: Desmos Graphing Click edit graph on Desmos
Square Wave By: Sam Ortega Credit: Desmos Graphing Click edit graph on Desmos
Saw Tooth Wave By: Sam Ortega Credit: Desmos Graphing Click edit graph on Desmos
Copy the variables values of k, a, c, d, b, h, p to graph various wave shapes using Desmos Graphing
| Triangular wave | k = 20 | a = 3 | c = 1 | d = 1 | b =1 | h =1 | p =2 |
| Saw Tooth Wave | k = 20 | a = 2 | c = 1 | d =1 | b =1 | h =0 | p =1 |
| Sine Wave | k = 20 | a = 4 | c = 1 | d =1 | b =1 | h =0 | p =5 |
| Square Wave | k = 20 | a = 2 | c = 0 | d =0 | b =2 | h =1 | p =1 |
MATHEMATICAL MODEL IS THE DIGITAL TWIN
Practice creating your own signal wave generator. Remember we need mathematical model so we can control, design, and manipulate actual physical things that our mathematical model is representing. Today we call the mathematical model as the digital twin of the physical thing.
Credit: Desmos Graphing
Shown above is 1 sample every 2 cycles. This is important in understanding Phasor Measurement Unit (PMU).
Explain the derivation of 1 sample per 2 cycle in PMU conversation? The common sampling rate in PMU configuration setting is
30 samples / second multiply that by 1 second / 60 cycles is the derivation of this 1 sample every 2 cycles [ This conversion is valid if your power frequency is rated 60 Hz, for 50 Hz multiply it by 1 / 50 cycles ]. Aligning the time between protection relay with 16 samples per cycle versus synchrophasor PMU with 1 sample every 2 cycle is challenging but it is possible as you see from both graphs. Since most protection relay total tripping time is in the range of 2 cycles to 5 cycles therefore the synchrophasor data collection of fault magnitude and duration can be time aligned with protection relay using 2 cycles time interval.
32 Samples per cycle
24 Samples per cycle
16 Samples per cycle
12 Samples per cycle
8 Samples per cycle
6 Samples per cycle
4 Samples per cycle
Voltage Sag Plotted in SEMI-F47 Curve
15 steps square wave generator
To edit go to the bottom at right hand corner and click edit graph on Desmos
By Sam Ortega, Created with Desmos Graphing
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