Slide 1What are filters?Bode Plots of Common FiltersPassive vs. Active filtersPassive FiltersFirst-Order Filter CircuitsSecond-Order Filter CircuitsHigher Order FiltersFrequency & Time Domain ConnectionsTime Domain Filter ResponseOther types of filtersButterworth filtersSlide 13Class Examples• What is a filter• Passive filters• Some common filtersLecture 23. Filters I12What are filters?•Filters are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signal components and/or enhance wanted ones.•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)3Bode Plots of Common FiltersFrequencyFrequencyLow PassFrequencyBand PassFrequencyBand RejectGainGainGainGainHigh Pass4Passive vs. Active filters–Passive filters: RLC components only, but gain < 1–Active filters: op-amps with RC elements, and gain > 15Passive Filters•Passive filters use R, L, C elements to achieve the desired filterSome Technical Terms:•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 frequency6First-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)7Second-Order Filter CircuitsC+–VSRBand PassLow PassLHigh PassBand RejectZ = R + 1/sC + sLHBP = R / ZHLP = (1/sC) / ZHHP = sL / ZHBR = HLP + HHP8Higher 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 filter9Frequency & 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 pair10Time 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?11Other types 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 complicated12Butterworth filters•Butterworth – The Butterworth filter is designed to have a frequency response which is as flat as mathematically possible in the passband. Another name for them is 'maximally flat magnitude' filters.Example: A 3rd order Butterworth low pass filter.C2 = 4/3 farad, R4 = 1ohm, L1 = 3/2 and L3=1/2 H.Butterworth filtersnth order Butterworth filter.where n = order of filter ωc = cutoff frequency (approximately the -3dB frequency) G0 is the DC gain (gain at zero frequencyAs n approaches infinity, it becomes a rectangle function The poles of this expression occur on a circle of radius ωc at equally spaced points14Class Examples•Example 10-1 and 10-2•Drill Problem
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