Transcript Passive Filters - Arizona State University

Passive Filters
Dr. Holbert
April 21, 2008
Lect22
EEE 202
1
Introduction
• 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 > 1
Lect22
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2
Filter 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
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3
Low Pass
Gain
Gain
Bode Plots of Common Filters
Frequency
High Pass
Frequency
Gain
Gain
Band Pass
Frequency
Lect22
Band Reject
Frequency
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Passive 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 unity
Lect22
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First-Order Filter Circuits
High Pass
+
–
VS
R
C
Low Pass
Low
Pass
VS
+
–
R
L
GR = R / (R + 1/sC)
HR = R / (R + sL)
GC = (1/sC) / (R + 1/sC)
HL = sL / (R + sL)
Lect22
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High
Pass
6
Second-Order Filter Circuits
Band Pass
Z = R + 1/sC + sL
R
VS
+
–
Low
Pass
High
Pass
Lect22
HBP = R / Z
C
Band
Reject
L
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 + 20 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 roots
Q = 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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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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Class 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 P10-1
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