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• What is a filter• Passive filters• Some common filters
Lecture 23. Filters I
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What 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)
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Bode Plots of Common Filters
FrequencyFrequency
Low Pass
Frequency
Band Pass
Frequency
Band Reject
Gai
nG
ain
Gai
nG
ain
High Pass
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Passive vs. Active filters
– Passive filters: RLC components only, but gain < 1
– Active filters: op-amps with RC elements, and gain > 1
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Passive Filters
• Passive filters use R, L, C elements to achieve the desired filter
Some 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 frequency
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First-Order Filter Circuits
L+–
VSC
R
Low Pass
High Pass
HR = R / (R + sL)
HL = sL / (R + sL)
+–
VSR
High Pass
Low Pass
GR = R / (R + 1/sC)
GC = (1/sC) / (R + 1/sC)
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Second-Order Filter Circuits
C+–
VS
R
Band Pass
Low Pass
LHigh Pass
Band Reject
Z = R + 1/sC + sL
HBP = R / Z
HLP = (1/sC) / Z
HHP = sL / Z
HBR = HLP + HHP
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Higher 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 filter
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Frequency & Time Domain Connections
• First order circuit break frequency: break = 1/• Second order circuit characteristic equation
s2 + 20 s + 02 [ = 1/(2Q) ]
(j)2 + 2(j) + 1 [ = 1/0 ]
s2 + BW s + 02
s2 + R/L s + 1/(LC) [series RLC]Q value also determines damping and pole types
Q < ½ ( > 1) overdamped, real & unequal rootsQ = ½ ( = 1) critically damped, real & equal roots
Q > ½ ( < 1) underdamped, complex conjugate pair
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Time 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?
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Other 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 complicated
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Butterworth 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.
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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 frequency
As n approaches infinity, it becomes a rectangle function
The poles of this expression occur on a circle of radius ωc at equally spaced points
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Class Examples
• Example 10-1 and 10-2• Drill Problem 10-1