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Cut-off Frequency of an RC Element

A RC element's frequency response depends on the ohmic resistance R and the capacitance C. However, the basic amplitude and phase characteristics (curve shapes) are always the same. A change in resistance merely results in a corresponding phase shift of the curves on the frequency axis. An RC element's characteristic parameter is its cut-off frequency fG, given by the following equation:

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Substituting this parameter in the amplitude response gives the following result:

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At the cut-off frequency, the amplitude or rms value of the output variable has dropped to 0.707 of the peak value. The phase response is given by:

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The phase shift between the input and output voltages at the cut-off frequency is precisely -45°. The following diagrams illustrate these relationships.

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The cut-off frequency depends less on the individual values of R and C and rather on their product R·C - termed the RC element's time constant T. The larger this time constant, the lower the RC element's cut-off frequency, i.e. the greater its attenuation of the low-frequency range. For instance, increasing an RC element's ohmic resistance by a factor of ten while simultaneously decreasing its capacitance by the same factor does not change the element's frequency response.

When displaying amplitude response in communications engineering, not only the frequency axes but also amplitude axes often have a logarithmic scale. The RC element's resulting characteristic falls linearly in the high frequency range. In the diagram below, the same amplitude response depicted earlier is plotted along a logarithmic amplitude axis.

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In this type of graph, the amplitude ratio is often specified in decibels (dB). An amplitude ratio of 1 corresponds to 0 dB, 0.1 (i.e. 10-1) to -20 dB, 0.01 (i.e. 10-2) to -40 dB and so on. The conversion formula is:

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An amplitude ratio of 1/Ö2 = 0.707 at the cut-off frequency thus corresponds to -3 dB.

The following interactive animation shows an RC element's amplitude response at different values of R and C , as well as the position of the cut-off frequency (red vertical line). On the right-hand edge of the display, you can select a variety of component parameters to observe their effect on the amplitude response.

 

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