CHAPTER 3. FILTER CIRCUITS47|Hhigh| =|Hlow|⇒1RCωc=1.(3.16)Thereforeωc =1RC(3.17)is the corner frequency of the ﬁlter. At the corner frequencyH(jωc)=11+ jωcRC=11+ j=1− j2,(3.18)|H(jωc)| =1√2.(3.19)We say that the output is down by 1/√2 at the corner frequency.3.3.2Approximate IntegraterThe low-pass ﬁlter acts as an approximate integrater at high frequencies. Assumevin(t)= vejωt(3.20)and integrate to obtainvout = vejωtdt =vjωejωt + vout(t =0).(3.21)The DC term is unimportant and may be dropped to obtainvout =1jωvin.(3.22)We deﬁneHintegrate≡voutvin=1jω.(3.23)For a low-pass ﬁlter at high frequencies ωωc andHhigh =1jωRC=1RCHintegrate.(3.24)Thus the low-pass ﬁlter integrates at high frequencies but also attenuates the signal by1/(RC).