UK EE 221 - Lecture 09: Filter Networks

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Lecture 09: Filter NetworksVijay Singh∗February 10, 2003AbstractDiscussion of Low-pass filters, High-pass filters, Band-pass filters, Band-rejectionfilters.11/sqrt(2)0w0wGain|Gv(jw)|TypicalIdeal characteristicFigure 1: Low-pass filter.11/sqrt(2)0w0wGain|Gv(jw)|Figure 2: High-pass filter.11/sqrt(2)0w0wGain|Gv(jw)|wLOwHIFigure 3:Band-pass filter.11/sqrt(2)0w0wGain|Gv(jw)|wLOwHIFigure 4:Band-rejection filter.1 Low-pass filter networkA simple low-pass filter:Gv(jω) = gain =VOV1=1jωC1jωC+ R∗Professor and Chairman, Department of Electrical & Computer Engineering, University of Kentucky,Lexington, KY, USA. E-mail: [email protected] Lecture presented on February 10, 2003. Typesetin LATEX1+ -V1RCVO+_Figure 5: Low-pass filter.GV(jω) =11 + jωRC(1)=11 + jωτ(2)where,τ = RCAmplitude: M(ω) =1p1 + (ωτ )2(3)Phase: Φ(ω) = −arctan(ωτ ) (4)At break frequency (also called half-power frequency), ωbτ = 1, ωb=1τ(M)ω=ωb=1√2(5)-20 dB/decade0-20 wwb = 1/ tau20log10(M)(dB)Phase0w-45o-90ow = 0Gv = 1Figure 6: Low-pass filter characteristics.22 High-pass filter network+ -V1CRVO+_Figure 7: High-pass filter.Gv(jω) =RR +1jωC=jωRC1 + jωRC=jωτ1 + jωτ; τ = RCM(ω) =ωτp1 + (ωτ )2Φ(ω) =π2− arctan(ωτ )Half-power frequency =ωb=1τ+20 dB/decade0-20 wwb = 1/ tau20log10(M)(dB)Phase0w45o90oFigure 8: High-pass filter characteristics.33 Band-pass filterGv(jω) =RR + j(ωL −1ωC)(1)M(ω) =ωRCp(ωRC)2+ (ω2LC − 1)2(2)At low frequency (ω small):M(ω) ≈ωRC1≈ 0 (3)At high ω:M(ω) ≈ωR Cω2LC=RωL≈ 0In the mid-frequency range:(ωRC)2À (ω2LC − 1)2andM(ω) ≈ 1At the center frquency (also called resonance frequency):ω0=r1LC+ -V1CRVO+_LFigure 9: Band-pass filter.At ωLO(lower cut-off frequency)ω2LC − 1 = −ωRCω2+ωRL− ω20= 0Solving, we get:ωLO=−RL+q(RL)2+ 4ω2024At ωHI(Upper cut-off frequency)ω2LC − 1 = +ωRCω2−ωRL− ω20= 0Solving, we get:ωHI=RL+q(RL)2+ 4ω202Bandwidth = BW = ωHI− ωLO=RL11/sqrt(2)0w0wGain|Gv(jw)|wLOwHIFigure 10: Band-pass filter


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UK EE 221 - Lecture 09: Filter Networks

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