{"id":1067,"date":"2013-09-18T21:57:48","date_gmt":"2013-09-18T19:57:48","guid":{"rendered":"http:\/\/www.sciencetronics.com\/greenphotons\/?p=1067"},"modified":"2025-06-16T18:38:14","modified_gmt":"2025-06-16T16:38:14","slug":"about-filters-and-cut-off-frequencies","status":"publish","type":"post","link":"https:\/\/www.sciencetronics.com\/greenphotons\/?p=1067","title":{"rendered":"About filters and cut-off frequencies"},"content":{"rendered":"

A good question was brought up by my students in the lab exercise in the course Electronics I yesterday: What actually makes the cut-off frequency of a filter circuit (1st order) so special? Why do we look out for a point which happens to be located \"SI{3}{dB}\" or a factor of \"sfrac{1}{sqrt{2}}\" under the amplitude of the pass band? Why not \"sfrac{1}{3}\", \"sfrac{1}{sqrt{5}}\", \"SI{5}{dB}\"?<\/p>\n

The first answer, which has some historical relevance, would be to look at a resistor \"R\" and a signal with an amplitude \"V_0\" and another signal with an amplitude \"V_1=V_0cdotsfrac{1}{sqrt{2}}\" while calculating the electrical power:
\n[ P_0 = frac{V_0^2}{R}qquad P_1 = frac{V_1^2}{R} = frac{left(V_0,sfrac{1}{sqrt{2}}right)^2}{R} = frac{1}{2},P_0 ]<\/p>\n

The power has been reduced to half the original power, it could be the electric power which is converted to sound waves in a speaker, or the power converted to electromagnetic waves in an antenna.<\/p>\n

But already here we can make one important observation: \"20log_{10}left(sfrac{1}{sqrt{2}}right)=SI[output-decimal-marker={.}]{-3.0103}{dB}approxSI{-3}{dB}\"<\/p>\n

Is this everything then? It still sounds random to choose exactly this point. If we for example look at a simple RC filter then we can show another reason which makes this point’s frequency so special. In a low-pass filter we have a constant amplitude \"sfrac{v_{ut}}{v_{in}}=1\" for low frequencies while we observed mathematically and experimentally that the amplitude drops with \"SI{-20}{dB}\" per decade of frequency for high frequencies. If we draw out these two asymptotic lines in a Bode diagram, then we will find exactly one point where the lines intersect – you guessed it: it’s at the cut-off frequency (the same holds of course true for a high-pass filter). We can see this if we zoom into the area around the cut-off frequency in a Bode diagram:
\n

\"Bode<\/a>
Bodediagram of a low-pass filter – zoomed into the area around the cut-off frequency.<\/figcaption><\/figure><\/p>\n

Then was this everything now? Is there more about the cut-off frequency?<\/p>\n

As we also saw we have a phase angle of \"SI{+45}{degree}\" (high-pass) or \"SI{-45}{degree}\" (low-pass) between the input- and the output signal at the cut-off frequency. In the complex notation of the \"j,omega\"-method this correlates to the fact that the real and the imaginary part of the transfer function \"A_v=sfrac{v_{ut}}{v_{in}}\" have the same magnitude at this point. The relation between real part, imaginary part, amplitude and phase angle I have plotted below for both the low-pass and the high-pass filter.<\/p>\n

It is interesting to observe that the imaginary part actually reaches a minimum at this point (\"=num{-0.5}\" for the low-pass filter) or a maximum (\"=num{+0.5}\" for the high-pass filter), respectively.<\/p>\n

You can see that the cut-off frequency actually is a remarkable point for a filter circuit.<\/strong><\/p>\n

Low-pass filter<\/strong><\/p>\n

\"Low-pass<\/a>
Low-pass filter: RC and RL.<\/figcaption><\/figure>\n

\"impedance_lowpass\"<\/a><\/p>\n

High-pass filter<\/strong><\/p>\n

\"High-pass<\/a>
High-pass filter: RC and RL.<\/figcaption><\/figure>\n

\"impedance_highpass\"<\/a><\/p>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

A good question was brought up by my students in the lab exercise in the course Electronics I yesterday: What actually makes the cut-off frequency of a filter circuit (1st order) so special? Why do we look out for a point which happens to be located or a factor of under the amplitude of the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"enabled":false},"version":2}},"categories":[11,8,27],"tags":[],"class_list":["post-1067","post","type-post","status-publish","format-standard","hentry","category-electronics_en","category-everything_en","category-science_en","entry"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p48grL-hd","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/posts\/1067","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1067"}],"version-history":[{"count":24,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/posts\/1067\/revisions"}],"predecessor-version":[{"id":2465,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=\/wp\/v2\/posts\/1067\/revisions\/2465"}],"wp:attachment":[{"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1067"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1067"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sciencetronics.com\/greenphotons\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1067"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}