Low Pass Filter

  1. Hi,
    I am trying to understand the impact of cut off frequency on the gain and trying to relate it in time domain but unable to do so.
    For e.g., I have built a single pole RC filter wherein R and C values are chosen to match the cut off frequency of 1K (R = 10KΩ/C = 16nF). The input is a square wave 10V @ 1Khz. There should be loss of 3db at the output but I am unable to validate it.
    Please help.

    Thanks,
    - Hemant
     
  2. jcsd
  3. Do you observe an output voltage higher or lower than what you predicted?
     
  4. The output is lower than the input as expected.
    Input = 0-10V square wave @ 1KHz
    Output = Hi-9.5V; Lo-0.4 @ 1Khz not a square wave though.

    My expectation was to see the output amplitude being around 7 V but that's not the case.
    Am I looking at it in the right way?

    Thanks,
     
  5. The harmonics of your 10V square wave have amplitudes of 10*4/(nπ).
    Your 1KHz fundamental (n=1) has an amplitude of 12.7V. The filter will attenuate it to 9V.
     
  6. Are you familiar with Fourier series?

    http://en.wikipedia.org/wiki/Fourier_series

    Basically, your 1kHz square wave is formed by a combination of sine waves of frequency n*1000. 1kHz is your fundamental frequency, and the other wine waves that make the square wave are integer multiples of the fundamental frequency.

    What does that mean? In a square wave, the fundamental frequency has a higher amplitude than the resulting square wave (see the first picture in the wikipedia link I gave you). Since you have a lowpass filter with a 1k cutoff frequency, only the fundamental sine wave can pass through your filter (with a -3dB gain).

    Your output signal should look like a sine wave, since the next sine wave that forms your square wave has a frequency of 2kHz.

    Sorry for the crappy expanation, english isn't my primary language.
     
  7. I'd like to add that the only signals that can pass through a hardware* filter without being deformed are sine waves. Why? Every other signal is actually formed by a combination (sum) of sine waves of different frequencies. Since these individual sine waves all have a different gain when going through a filter, the output wave (given by the sum of the individual sine waves) will have a different enveloppe.

    *Obviously if you use a software to filter signals, it can be programmed to "fix" this issue.
     
  8. Thanks for the help!
     
  9. psparky

    psparky 864
    Gold Member

    I can answer your question mathematically......

    What's your transfer function?

    Let me guess.......1/(JωRC+1)

    That is found by taking your voltage divider across your cap

    1/jwc/(1/jwc + R)

    Multiply top and bottom by jwc...and you get...

    1/(JωRC+1)

    Your break frequency happens when ω= 1/RC.

    So when ω= 1/RC.....you are left with

    1/(j+1)

    What is the magnitude and value of this lovely function?

    Well....I'll simplify so you can see. 1/(1.41 at -45 degrees)

    Simply further and you get .7 at -45 degrees.

    Ok....now to your actual question.....where does the -3 dB come from?

    You know that you find dB from this equation ~ 20 log |gain|

    20 log .7 = ~ -3dB.
    I also just proved that your phase angle will be -45 degrees at the break as well.

    However....all that may go away if you are talking a non ideal situation...........
     
  10. No offense but have you read his question? He said he chose his R and C to get -3dB at 1kHz. Obviously he was able to find and solve the transfer function.



     
  11. vk6kro

    vk6kro 4,059
    Science Advisor

    That filter seems to be for 10 KHz.

    1/ F = 2 * π * R * C = 2 * π * 1000 * 16 * 10^-9

    F = 9947 Hz
     
  12. Did you drop a zero somewhere, good sir?
     
  13. vk6kro

    vk6kro 4,059
    Science Advisor

    Yes, I see the 10 K now. Thanks.

    I read the 1 K frequency as the resistance.
     
  14. psparky

    psparky 864
    Gold Member

    I write that out mathematically because most students are told it drops 3 dB at the break without knowing WHY. They see all the graphs of filters with the 3 db drop and just take it as that. For all of you who already know this, I apologize for insulting your intelligence with such a trivial post. However, if just one person learned something....then my post was a success.

    To this day I have not met one student or EE in person that knows why it drops 3 dB mathematically. Therefore, I assume it to be a weak point for most.
     
    Last edited: Jun 3, 2012
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