Fundamental frequencies of square wave and sine wave

In summary, the fundamental frequencies of a 50 kHz square waveform of 50% duty cycle and a 25 kHz sinusoidal waveform, respectively, are 50 kHz and 25 kHz, respectively.
  • #1
galaxy_twirl
137
1

Homework Statement



What are the fundamental frequencies for a 50 kHz square waveform of 50% duty cycle and a 25 kHz sinusoidal waveform, respectively? (The duty cycle of a square waveform is the ratio between the pulse duration and the pulse period.)

Homework Equations



2qv7di9.jpg

My teacher then gave an example to illustrate v(t) and cn. Hence, I have a feeling that I should use these formula to find the answer.
2lbdvk8.jpg

The Attempt at a Solution



I am at a loss of which equation to use because there are too many representations of v(t) and cn in my lecture notes. Nevertheless, I shall attempt o solve the above question.

For the square wave, with its frequency at 50kHz, which is f0, its period, 1/f0 is 2X10-5 seconds. I don't understand what does T stand for in the equation, but I assume it refres to the amount of time the signal was turned on. Since the signal is ON for half the time, T will be 1X10-5.

Hence, I will have
1zqav7d.png

Am I correct?

May I have some hints as to how to start the question for the part on sine wave?

Thank you everyone. :)
 
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  • #2
The question asks for the fundamental frequencies of the given waveforms, one a square wave and the other a sine wave. You should be able to get to the answers without any math at all :)

The Fourier series represents a sum of frequency components, all of which are sinusoidal. In effect it breaks down a given periodic waveform into a series of sinusoidal frequency components.

A pure sine wave has only one component frequency, namely that of the sine wave itself. So you should be able to dispose of the 25 KHz sinewave easily enough.

The square wave can also be dealt with by inspection if you picture the largest sinewave that will fit its outline.
 
  • #3
Hi gneill. Thank you for your help! :)

Ah I see. I determined the f0 in the lab by feeding in sinusoidal signals, square pulses and triangular waves into the FFT (Fast Fourier Transform) Oscilloscope and they all turned out to match the frequencies supplied by the signal generator regardless of the duty cycle. :)

gneill said:
The square wave can also be dealt with by inspection if you picture the largest sinewave that will fit its outline.

I know from my lecture notes that a sine wave can look like a square wave with squiggles when n becomes a larger number in sin(nx), as seen below:

3169gyb.jpg


Do you mean the above in your reply to my question on square wave? :) Just wondering, how do you know how large should the sine wave get into order to achieve a square waveform?

Thanks again! :D
 
  • #4
galaxy_twirl said:
Do you mean the above in your reply to my question on square wave? :) Just wondering, how do you know how large should the sine wave get into order to achieve a square waveform?

I meant that by looking at a given square wave you should be able to see the period (or frequency) of the fundamental sine wave that fits its shape:
Fig1.gif

Determining the magnitude of that fundamental frequency for the Fourier series of the square wave is another matter :)
 
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Likes galaxy_twirl
  • #5
Ah I see.

Thank you! :D
 

FAQ: Fundamental frequencies of square wave and sine wave

What is the difference between a square wave and a sine wave?

A square wave is a periodic waveform that alternates between two levels, high and low, with equal durations. It has abrupt changes between the two levels, resulting in a sharp, square-like shape. A sine wave, on the other hand, is a smooth, continuous waveform that varies in amplitude and frequency over time. It has a gentle, curved shape and is often used to represent pure tones in sound or vibrations in electrical signals.

What is the fundamental frequency of a square wave?

The fundamental frequency of a square wave is equal to the inverse of the period, or the time it takes for one complete cycle. This means that the fundamental frequency is determined by the width of the square wave's high and low levels. For example, a square wave with a high level of 1 volt and a low level of 0 volts would have a fundamental frequency of 1 Hz (1 cycle per second).

How is the fundamental frequency of a sine wave calculated?

The fundamental frequency of a sine wave is determined by its wavelength, which is the distance between two consecutive peaks or troughs. The formula for calculating the fundamental frequency is f = c/λ, where f is the frequency, c is the speed of the wave, and λ is the wavelength. In the case of sound waves, the speed of sound is approximately 343 meters per second, so the fundamental frequency of a sine wave with a wavelength of 1 meter would be 343 Hz.

What is the relationship between the fundamental frequency and harmonics?

The fundamental frequency is the lowest frequency component of a complex wave. Harmonics are higher frequency components that are integer multiples of the fundamental frequency. For example, if the fundamental frequency is 100 Hz, the first harmonic would be 200 Hz, the second harmonic would be 300 Hz, and so on. The presence and strength of harmonics in a wave can affect its overall shape and sound.

Why are square waves and sine waves important in science and technology?

Square waves and sine waves are two of the most fundamental waveforms used in science and technology. They are used in a variety of applications, such as in electronics, acoustics, and signal processing. Square waves are often used to represent digital signals in computers and communication systems, while sine waves are used to represent analog signals in music and audio equipment. Understanding the properties and behaviors of these waveforms is crucial in many fields of science and engineering.

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