EE 205 Dr. A. Zidouri
Electric Circuits II
Frequency Selective Circuits (Filters)
Bandstop Filters
Lecture #39
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EE 205 Dr. A. Zidouri
The material to be covered in this lecture is as follows:
o
o
o
o
Introduction to Bandstop filters
Center Frequency, Bandwidth and Quality factor
Bandstop Series RLC Circuit Analysis
Relationship between different parameters
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EE 205 Dr. A. Zidouri
After finishing this lecture you should be able to:
¾
¾
¾
¾
¾
Understand the behavior of Bandstop filters
Determine the different Bandstop filter parameters
Relate each parameter in terms of other parameters
Analyze Bandstop filters
Design a simple Bandstop filter
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EE 205 Dr. A. Zidouri
Introduction to Bandstop filters
Bandstop filters are those that pass voltages outside the band between the two cutoff frequencies
to the output while attenuating voltages at frequencies between the two cutoff frequencies.
Bandpass filters and Bandstop filters thus perform complementary functions in the frequency
domain.
Bandstop filters are also characterized by same parameters
o Two cutoff frequencies C1 , C 2
o
o
o
ω ω
The center frequency ω0
The bandwidth
β
The quality factor
Q
Only two of these five parameters can be specified independently
These parameters are defined in the same way for both types of filters. So we examine Bandstop
filters in the same way as we did with the Bandpass filters.
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EE 205 Dr. A. Zidouri
Bandstop Filter: Center Frequency, Bandwidth, and Quality Factor
The center frequency
ω0
is defined as the frequency for which a circuit’s TF is purely real. It is
also referred to as the resonant frequency or corner frequency
The center frequency is the geometric center of the stopband, that is,
The magnitude of the TF is minimum at the center frequency
The bandwidth
β
The quality factor
is the width of the stopband
β = ωC 2 − ωC1
ω0 = ωC1ωC 2
( )
H max = H jω0
(39-1)
(39-2)
(39-3)
Q is the ratio of the center frequency to the bandwidth Q =
ω0
β
(39-4)
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EE 205 Dr. A. Zidouri
Analysis of the Bandstop Series RLC Circuit
Consider the Bandstop Circuit of Fig. 39-1. Using the same arguments used for Low-pass and Highpass filters, so as before changing the frequency of the source results in changes to the impedance of
the capacitor and the inductor.
¾ At ω = 0 the impedance of the capacitor is 1 = ∞ so the
jωC
capacitor behaves like an open circuit, and the impedance of the
inductor is jω L = 0 so the inductor behaves like a short circuit.
Thus Vo = Vi as depicted in Fig. 39-1b
ω =0
ω =∞
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EE 205 Dr. A. Zidouri
ω =∞
1 = 0 so the capacitor behaves like a short circuit,
jωC
and the impedance of the inductor is jω L = ∞ so the inductor behaves like an open circuit. Thus
¾ At
the impedance of the capacitor is
Vo = Vi also, as depicted in Fig. 39-1c
¾ Between these two passbands, both the inductor and capacitor have finite impedances of
opposite signs.
¾ As the frequency is increased from zero, jω L increases and 1 decreases. The phase shift
jωC
between the input and the output approaches -90o as ω L approaches 1 . As soon as ω L
ωC
exceeds 1 , the phase jumps to +90o and then approaches zero as ω continues to increase as
ωC
seen in Fig. 39-2b
¾ At some frequency in between the two passbands, the center frequency ω0 , the two impedances
cancel out causing the output voltage to be zero Vo = 0 . At ω0 the series combination of L and C
appears as a short circuit.
¾ The plot of the voltage magnitude ratio is shown in Fig. 39-2a. Note the ideal bandstop filter plot
is also overlaid on the plot of the RLC circuit.
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EE 205 Dr. A. Zidouri
|H(jw)|
1
0.707
Actual
Ideal
0
C1
C2
o
o/
(a)
Q}
θ(j
C1
o
C2
+90o
-90o
(b)
Fig. 39-2 The frequency response plot for the series RLC
bandstop filter circuit in Fig. 39-1
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EE 205 Dr. A. Zidouri
Bandstop Series RLC Circuit Analysis Cont.
Let’s now examine the circuit quantitatively. Consider the s-domain equivalent circuit for the series RLC
shown in Fig. 39-3 below
R
1
sC
Vi(s)
sL
Vo(s)
Fig. 39-3 s-domain equivalent
of the Circuit of Fig. 39-1a
Transfer function of the circuit:
s2 +1 LC
sL
+
1
sC
H ( s) =
=
R + sL +1 sC 2 R
1
s + s+
L LC
(39-5)
This equation gives magnitude and phase as before if we express it in polar form and substituting
s = jω we get:
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EE 205 Dr. A. Zidouri
Bandstop Series RLC Circuit Analysis Cont.
H ( jω ) =
1
LC
−ω 2
(39-6)
2
2
⎛1
⎞
2
−ω ⎟ + ω R L
⎜
⎝ LC
⎠
⎛
⎞
R
ω
⎜
L ⎟⎟
θ jω = − tan−1 ⎜
(39-7)
2
⎜⎜ 1
⎟
−ω ⎟
⎝ LC
⎠
This confirms the frequency response shape pictured in Fig. 39-2 from qualitative analysis.
We now calculate the five parameters that characterize this RLC bandstop filter:
(
)
( )
Center Frequency: The TF in (39-5) will be purely real when
solving for
ω0 we get:
ω0 =
Cutoff Frequencies ω
C1
and
1
LC
ωC 2 :
j ω0L +
1
j ω0C
=0
(39-8)
(39-9)
Substituting (39-9) into (39-6) and solving in exactly the same way
as for the bandpass filter we get:
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EE 205 Dr. A. Zidouri
Bandstop Series RLC Circuit Analysis Cont.
2
⎛
⎛ R ⎞
R
ωC 1 = − + ⎜ ⎟ + ⎜ 1
2L ⎝ 2L ⎠ ⎝ LC
⎞
⎟
⎠
(39-10)
2
⎛
⎞
R
R
1
⎜
⎟
ωC 2 =
+
+⎜
⎜
2L
2L ⎝ LC ⎟⎟⎠
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
We can use these last two equations to confirm that
frequencies
ω0 = ωC1ωC 2 =
1
LC
We can compute also the bandwidth
ω0 is the geometric mean of the two cutoff
(39-12)
β = ωC 2 − ωC1 = R
1
ω0
And the Quality factor Q =
= LC
β ⎛R ⎞
⎜
⎟
⎝
(39-11)
(39-13)
L
=
L⎠
L
(39-14)
CR2
Again we can express the cutoff frequencies in terms of center frequency and bandwidth as:
⎛ ⎞
ωC1 = − β + ⎜⎜ β ⎟⎟ + ω02
2
2
⎝
2⎠
⎛ ⎞
ωC 2 = β + ⎜⎜ β ⎟⎟ + ω02
2
(39-15)
2
⎝
2⎠
(39-16)
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EE 205 Dr. A. Zidouri
Example 39-1
Using the series RLC circuit in Fig. 39-1a compute the component values that yield a bandreject filter
with β = 250Hz a center frequency ω = 750Hz , using 100nF capacitor. Compute values of R, L,
0
the cutoff frequencies
ωC 1 and ωC 2 , and the Quality factor Q.
Solution:
We begin by Quality factor Q, consider the s-domain circuit in Fig. 39-3:
1
sC
Q = ω0 β = 3
To compute L we convert
=
To compute R
⎡
⎢ 2π
⎣
Vi(s)
ω0 to radians per second and
L = 12
ω0 C
1
(750)⎤⎥⎦
2
R
sL
Vo(s)
Fig. 39-3 s-domain equivalent
of the Circuit of Fig. 39-1a
= 450mH
⎛100 ×10−9 ⎞
⎜
⎟
⎝
⎠
R=βL
Or
R = 2π ( 250) ⎛⎜ 450 ×10−3 ⎞⎟ = 707Ω
⎝
⎠
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EE 205 Dr. A. Zidouri
Cutoff frequencies: we can use the values of the
⎛ ⎞
ωC1 = − β + ⎜⎜ β ⎟⎟ + ω02 = 3992.0 rad s
2
2
⎝
2⎠
Check
ω0 and
β to compute ω
C1
and
ωC 2 :
⎛ ⎞
ωC 2 = β + ⎜⎜ β ⎟⎟ + ω02 = 5562.8 rad s
2
2
⎝
2⎠
ωC 2 − ωC1 = 885.3 − 635.3 = 250Hz
ω = ωC 2 ×ωC1 = 885.3×635.3 = 750Hz
0
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EE 205 Dr. A. Zidouri
Self Test 39:
If the bandstop filter in Fig. 39-1a, is to reject a 200Hz sinusoid while passing other frequencies,
calculate the values of L and C. Take R = 150Ω , and the bandwidth β = 100Hz .
Answer:
L = 0.2387 H , C = 2.66µ F
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