3.6 Passive RCL Filters

Chapter 3.6 Passive RCL Filters

Physics Lecture Notes – Phys 395 Electronics Book
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Physics Lecture Notes – Phys 395 Electronics Book

  • CHAPTER 3. FILTER CIRCUITS54falls off−6 dB/octave at both the low- and high-frequency extremes.For small ω (ie. ωωH and ωωL) H≈ jω/ωH ;|H(jω)| = ω/ωH ;log10 H =log10 ω + C .Thusd(log10 |H|)d(log10 ω)=1⇒ 6db/octave.(3.49)For large ω (ie. ωωH and ωωL) H≈−jωL/ω;|H(jω)| = ωL/ω;|H(jω)| =− log10 ω + C .And thusd(log10 |H|)d(log10 ω)=−1 ⇒−6dB/octave.(3.50)3.6Passive RCL FiltersSequential RC filters always have real poles and hence smooth rounded corners. To improvethese filters we introduce an inductor (which is good for high frequencies).3.6.1Series RCL CircuitConsider the RCL circuits as shown in figure 3.9.Each has a low-frequency and high-frequency approximation. Considering the band-reject filter (figure 3.6d) we obtain for thetransfer functionH=1/(jωC)+ jωLR +1/(jωC)+ jωL(3.51)=1− ω2LC(1− ω2LC)+ jωRC.(3.52)The approximations are:ω→ 0;H→ HL = 1and(3.53)ω→∞; H→ HH =1.(3.54)We notice a zero in the transfer function at ω0 =1/√LC. In the low-medium frequencyrangeω< ω0;H→ HLM =1jωRC∝ ω−1,(3.55)for high-medium frequenciesω> ω0;H→ HHM =−ω2LCjωRC=jωLR∝ ω+1.(3.56)Solving for the corner frequencies we have