Passive FiltersIntroductionFilter NetworksBode Plots of Common FiltersSlide 5First-Order Filter CircuitsSecond-Order Filter CircuitsHigher Order FiltersFrequency & Time Domain ConnectionsTime Domain Filter ResponseTypes of FiltersClass ExamplesLect22 EEE 202 1Passive FiltersDr. HolbertApril 21, 2008Lect22 EEE 202 2Introduction•We shall explore networks used to filter signals, for example, in audio applications–Today: passive filters: RLC components only, but gain < 1–Next time: active filters: op-amps with RC elements, and gain > 1Lect22 EEE 202 3Filter Networks•Filters pass, reject, and attenuate signals at various frequencies•Common types of filters:–Low-pass: deliver low frequencies and eliminate high frequencies–High-pass: send on high frequencies and reject low frequencies–Band-pass: pass some particular range of frequencies, discard other frequencies outside that band–Band-rejection: stop a range of frequencies and pass all other frequencies (e.g., a special case is a notch filter)Lect22 EEE 202 4Bode Plots of Common FiltersFrequencyHigh PassFrequencyLow PassFrequencyBand PassFrequencyBand RejectGainGainGainGainLect22 EEE 202 5Passive Filters•Passive filters use R, L, C elements to achieve the desired filter•The half-power frequency is the same as the break frequency (or corner frequency) and is located at the frequency where the magnitude is 1/2 of its maximum value•The resonance frequency, 0, is also referred to as the center frequency•We will need active filters to achieve a gain greater than unityLect22 EEE 202 6First-Order Filter CircuitsL+–VSCRLow PassHigh PassHR = R / (R + sL)HL = sL / (R + sL)+–VSRHigh PassLow PassGR = R / (R + 1/sC)GC = (1/sC) / (R + 1/sC)Lect22 EEE 202 7Second-Order Filter CircuitsC+–VSRBand PassLow PassLHigh PassBand RejectZ = R + 1/sC + sLHBP = R / ZHLP = (1/sC) / ZHHP = sL / ZHBR = HLP + HHPLect22 EEE 202 8Higher Order Filters•We can use our knowledge of circuits, transfer functions and Bode plots to determine how to create higher order filters•For example, let’s outline the design of a third-order low-pass filterLect22 EEE 202 9Frequency & Time Domain Connections•First order circuit break frequency: break = 1/•Second order circuit characteristic equations2 + 20 s + 02[ = 1/(2Q) ](j)2 + 2(j) + 1 [ = 1/0 ]s2 + BW s + 02s2 + R/L s + 1/(LC) [series RLC]Q value also determines damping and pole typesQ < ½ ( > 1) overdamped, real & unequal rootsQ = ½ ( = 1) critically damped, real & equal rootsQ > ½ ( < 1) underdamped, complex conjugate pairLect22 EEE 202 10Time Domain Filter Response•It is straightforward to note the frequency domain behavior of the filter networks, but what is the response of these circuits in the time domain?•For example, how does a second-order band-pass filter respond to a step input?Lect22 EEE 202 11Types of Filters•Butterworth – flat response in the passband and acceptable roll-off •Chebyshev – steeper roll-off but exhibits passband ripple (making it unsuitable for audio systems)•Bessel – yields a constant propagation delay•Elliptical – much more complicatedLect22 EEE 202 12Class Examples•Compare the frequency responses of fourth-order Butterworth and Chebyshev low-pass filters [use Excel to compute and produce Bode magnitude plots]–Butterworth:(s² + 1.8478 s + 1)(s² + 0.7654 s + 1)–Chebyshev:(2.488 s² + 1.127 s + 1)(1.08 s² + 0.187 s + 1)•Drill Problem
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