Active Filters

Simple lowpass and highpass filters are discussed in Sects. 29.3.1 and 29.3.2, the circuit of the simplest lowpass filter being shown again in Fig. 13.1. The ratio of the output voltage to the input voltage can be expressed using (29.3.1) as % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGafmyqaeKba0bacqGGOaakiqGacaWFQbGaeqyYdCNa % eiykaKIaeyypa0ZaaSaaaeaacuWGwbGvgaqhamaaBaaaleaaieGacq % GFWaamaeqaaaGcbaGafmOvayLba0badaWgaaWcbaGaa8xAaaqabaaa % aOGaeyypa0ZaaSaaaeaacqaIXaqmaeaacqaIXaqmcqGHRaWkcaWFQb % GaeqyYdCNaa8Nuaiaa-neaaaaaaa!4EA6! $$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} (j\omega ) = \frac{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V} _0 }} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V} _i }} = \frac{1} {{1 + j\omega RC}} $$ and is called the frequency response of the circuit. Replacing jω by jω+σ = s gives the transfer function: % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemyqaeKaeiikaGcceiGaa83CaiabcMcaPiabg2da % 9maalaaabaGaemitaWKaei4EaSNaemOvay1aaSbaaSqaaGqaciab+b % daWaqabaGccqGGOaakcaWF0bGaeiykaKIaeiyFa0habaGaemitaWKa % ei4EaSNaemOvay1aaSbaaSqaaiaa-LgaaeqaaOGaeiikaGIaa8hDai % abcMcaPiabc2ha9baacqGH9aqpdaWcaaqaaiabigdaXaqaaiabigda % XiabgUcaRiaa-nhacaWFGaGaa8Nuaiaa-neaaaaaaa!58E3! $$ A(s) = \frac{{L\{ V_0 (t)\} }} {{L\{ V_i (t)\} }} = \frac{1} {{1 + s RC}} $$ This is the ratio of the Laplace-transformed output and input voltages for signals of any time dependence. On the other hand, the transition from the transfer function A(s) to the frequency response A(jω) for sinusoidal input signals is made by setting σ to zero.