Filters and Tuned Amplifierscc.ee.nchu.edu.tw/~aiclab/teaching/Electronics3/lect12.pdf · Filters...
Transcript of Filters and Tuned Amplifierscc.ee.nchu.edu.tw/~aiclab/teaching/Electronics3/lect12.pdf · Filters...
Ching-Yuan Yang
National Chung-Hsing UniversityDepartment of Electrical Engineering
Filters and Tuned Amplifiers
Microelectronic Circuits
12-1 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Outline
Filter Transmission, Types, and Specification
The Filter Transfer Function
Butterworth Filters and Chebyshev Filters
First-Order and Second-Order Filter Functions
The Second-Order LCR Resonator
Second-Order Active Filters Based on Inductor Replacement
Second-Order Active Filters Based on the two-Integrator-Loop Topology
Single-Amplifier Biquadratic Active Filters
Sensitivity
Switch-Capacitor Filters
Tuned Amplifiers
12-2 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Filters
Electronic filters are an important building block of communication and instrumentation systems.
Filter design is one of the very few areas of engineering for which a complete design theory exists, starting from specification and ending with a circuit realization.
12-3 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Filter Types (Implementation)
Passive LC filters
Oldest technology
Work well at high frequency
For low frequency application (DC ~ 100kHz), L’s are large and impossible to fabricate in ICs.
Inductorless filter
Active-RC filters
Switched-capacitor filters
Transconductance-C filters
…
12-4 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Filter Transmission, Types, and Specification
12-5 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Two-Port Network
The filters studied in this chapter are linear circuits represented by the general two-port network.
)()()()()(
)( ωφωω j
i
o ejTjTsVsV
sT =⇒≡Transfer function:magnitude phase
Gain function: )(log20)( ωω jTG = dB
Attenuation function: )(log20)( ωω jTA −= dB
Output spectrum: )()()( ωωω jVjTjV io =
12-6 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Filter Specification
Ideal transmission characteristics of the four major filter types:
12-7 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Specification of the Transmission Characteristics of a Low-Pass Filter
ωp : the pass band edge
Amax : the max. allowed variation in passband transmission
ωs : the stop band edge
Amin : the min. required stopband attenuation
12-8 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Ideal LP:
Lower Amax
Higher Amin
Selectively ratio 1ωω
→s
p
12-9 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Transmission Specification for a Bandpass Filter
Note that this bandpass filter has a monotonically decreasing transmission in the passband on both sides of the peak frequency.
12-10 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The Filter Transfer Function
12-11 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Filter Transfer Function
Filter transfer function
Poles and zeros must be real or complex conjugate pairs.
To obtain zero or small stopband transmission, zeros are usually placed on
the jω-axis at stopband frequencies.
01
1
01
1)(bsbs
asasasT N
NN
MM
MM
++++++
= −−
−−
N : the filter order (the degree of the denominator)
)())(()())((
)(21
21
N
MM
pspspszszszsa
sT−−−−−−
= z1, z2, ⋅⋅⋅, zM : transfer function zeros
p1, p2, ⋅⋅⋅, pN : transfer function poles
12-12 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Pole-zero pattern
N = 5, fifth-order LPF01
22
33
44
5
22
221
24 ))((
)(bsbsbsbsbs
ssasT ll
+++++++
=ωω
12-13 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Pole-zero pattern for the bandpass filter (N = 6)
05
56
22
221
25 ))((
)(bsbs
sssasT ll
+++++
=ωω
12-14 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Pole-zero pattern for the low-pass filter (N = 5)
012
23
34
45
0)(bsbsbsbsbs
asT
+++++=
12-15 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Butterworth Filters
12-16 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Butterworth Filter
N-th order Butterworth filter:
N
p
jT2
21
1)(
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
ωωε
ω
Butterworth filter exhibits a monotonically decreasing transmission with all the transmission zeros at ω = ∞.
At ω = ω p,
ε determines the maximum variation in passband
transmission,
21
1)(
εω
+=pjT
ωp: passband edge
1101log20 102max
max −=⇒+= AA εε
At the edge of the stopband, ω = ωs, the attenuation of the Butterworth filter is
[ ][ ] min
22
22
)/(1log10
)/(11log20)(
A
AN
ps
Npss
≥+=
+−=
ωωε
ωωεω
Magnitude response:
12-17 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Maximally flat response
The first 2N−1 derivative of |T | relative to ω are zero at ω = 0.
Response is very flat near ω = 0.
Order N passband flatness
Attenuation at stopband edge ω =ωs
Filter order can usually be obtained from A(ωs ).
2 2
2 2
120
1
10 1
= −+
⎡ ⎤= + ≥⎣ ⎦
s Ns p
Ns p
A
A
ωε ω ω
ε ω ω min
( ) log( / )
log ( / )
12-18 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Magnitude response of Butterworth filters of various order with ε = 1
12-19 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Normalized Butterworth Polynomials
Find poles:
22 2
2 2
22
2
( ) ( )
1 1( ) ( ) ( )
1 1 ( 1)
1( )
1 ( 1)
=⎯⎯⎯→
− = = =⎛ ⎞ ⎛ ⎞
+ + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
=⎛ ⎞
+ − ⎜ ⎟⎜ ⎟⎝ ⎠
s j
N N
N
p p
N
N
p
T s T j
T j T j T j
T ss
ω ω
ω ω ωω ωε εω ω
εω
2 2
22
11
⎛ ⎞ ⎛ ⎞ −= − − = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
N NN
N
p p
s sεω ω ε
( )( )
; is even.poles where
; is odd.
2
2 2
2
2 2
1
0 1 2 2 11
+ ⋅
⋅
⎧− =⎪⎪= = −⎨⎪ =⎪⎩
j l
j l
eN
l Ne
N
π π
πε ε
ε ε
( )
( ), , , ,( )
12-20 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
E.g. N = 2, l = 1
;
poles
4
2
24 4
41
4 22
43
34 2
4
1
0 1 2 3
1 12 2
1 12 2
1 12 2
1 12 2
+ ⋅
⎛ ⎞+ ⋅⎜ ⎟⎝ ⎠
−
⎛ ⎞− +⎜ ⎟⎝ ⎠
⎛ ⎞− +⎜ ⎟⎝ ⎠
⎛ ⎞− +⎜ ⎟⎝ ⎠
⎛ ⎞= − =⎜ ⎟⎜ ⎟
⎝ ⎠
= =
⎧ ⎛ ⎞= = +⎪ ⎜ ⎟⎝ ⎠⎪
⎪ ⎛ ⎞= = − +⎜ ⎟⎝ ⎠
= ⎨⎛ ⎞= = − −⎜ ⎟⎝ ⎠
⎛ ⎞= = −⎜ ⎟⎝ ⎠
j l
p
lj
p
j
p p
j
p p
j
p p
j
p p
se
s e l
s e j
s e j
s e j
s e j
π π
π π
π
π π
π π
π π
ω
ω
ω ω
ω ω
ω ω
ω ω
( )
, , ,
⎪⎪
⎪⎪⎪⎪⎪⎩
12-21 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Normalized polynomials for N = 2
Take left plane poles
Let ωp = 1 frequency normalization
Normalized Butterworth polynomials
for 2
2 3
22
1 12
1 21 1
1 2
= = =⎛ ⎞⎛ ⎞ + ++ +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠= + +
T s Ns ss s
s s
B s s s
( ) .
( )
N
1
2
3
4
5
6
7
8
(s + 1)
(s2 + 1.414s + 1)
(s + 1)(s2 + s + 1)
(s2 + 0.765s + 1)(s2 + 1.848s + 1)
(s + 1)(s2 + 0.618s + 1)(s2 + 1.618s + 1)
(s2 + 0.518s + 1)(s2 + 1.414s + 1)(s2 + 1.932s + 1)
(s + 1)(s2 + 0.445s + 1)(s2 + 1.247s + 1)(s2 + 1.802s + 1)
(s2 + 0.390s + 1)(s2 + 1.111s + 1)(s2 + 1.663s + 1)(s2 + 1.962s + 1)
Factors of polynomial Bn(s)
12-22 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Graphical Construction of Butterworth Filters
)())(()(
21
0
N
N
pspsps
KsT
−−−=
ω
Transfer function:
where K is a constant equal to the required dc gain of the filter.
12-23 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Graphical Construction of Butterworth Filters
General case: Graphical construction for determining Butterworth filter of order N.
All the poles lie in the left half of the s-plane on a circle of radius ω0 = ωp(1/ε )1/N ,
where ε is the passband deviation parameter. ( )
Transfer function:)())((
)(21
0
N
N
pspsps
KsT
−−−=
ω
where K is a constant equal to the required dc gain of the filter.
110 10max −= Aε
12-24 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
12-25 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
How to design a Butterworth filter
Determine ε.
Determine the required filter order N.
Determine the N natural modes.
Determine T(s).
1101log20 102max
max −=⇒+= AA εε
[ ] min22 )/(1log10)( AA N
pss ≥+= ωωεω
)())(()(
21
0
N
N
pspspsK
sT−−−
=ω
12-26 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Example: Find the Butterworth Transfer Function
Low-pass filter specifications: fp = 10 kHz, Amax = 1 dB, fs = 15 kHz,
Amin = 25 dB, dc gain = 1.
Determine ε :
Determine N :
Determine the N natural modes:
Combining p1 with its complex conjugate p9 yields the factor (s2 + s0.3472ω0 + ω02 )
in the denominator of the transfer function. The same can be done for the other
complex poles.
[ ] min22 )/(1log10)( AA N
pss ≥+= ωωεω
5088.0110110 10110max =−=−= Aε
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
≥⇒
p
s
A
N
ωω
ε
log
110log
21 2
10min
76.8≥∴ N Select N = 9
The poles all have the same frequency:
rad/s 10773.6)5088.0/1(10102)/1( 49/1310 ×=××== πεωω N
p
The first pole: )9848.01736.0()80sin80cos( 001 jjp +−=+−= ωω
12-27 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Determine T(s):
Poles of the ninth-order Butterworth filter
)3472.0)()(5321.1)(8794.1)(()( 2
0022
0022
0022
002
0
90
ωωωωωωωωωω
+++++++++=
ssssssssssT
12-28 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Chebyshev Filters
Equiripple in the passband
Monotonically decreasing in the stopband
Odd-order filter exhibits |T(0)| = 1
Even-order filter exhibits |T(0)| = |T(ωp )| = Amax
The total no. of passband maxima and minima equals the order of the filter, N. All the transmission zeros at ω = ∞, making it an all-pole filter.
12-29 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
N-th order Chebyshev filter ωp: passband edge
At ω = ω p,
ε determines the passband ripple,
p
p
p
p
T jN
T jN
ω ω ωε ω ω
ω ω ωε ω ω
−
−
= ≤+
= ≥+
for
for
( )cos [ cos ( / )]
( )cosh [ cosh ( / )]
2 2 1
2 2 1
1
1
1
1
( )pT jωε
=+ 2
1
1
maxmax log AA ε ε= + ⇒ = −10220 1 10 1
12-30 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The attenuation achieved by the Chebyshev filter at the stopband edge (ω = ωs ) is
The poles of the Chebyshev filter:
The transfer function of the Chebyshev filter:
where K is a constant equal to the required dc gain of the filter.
min( ) log[ cosh ( cosh ( / ))]s s pA N Aω ε ω ω−= + ≥2 2 110 1
sin sinh sinh cos cosh sinh
, , ,
k p pk k
p jN N N N
k N
π πω ωε ε
− −− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
=
1 12 1 1 1 2 1 1 12 2
1 2
( )( )( ) ( )
Np
NN
KT s
s p s p s p
ωε −=
− − −11 22
12-31 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
How to design a Chebyshev filter
Determine ε from Amax.
Determine N, the number of the order, from given A(ωs ).
Determine the poles, pk, using Chebyshev equation.
Determine the transfer function, T(s).
where K is the dc gain.
sin sinh sinh cos cosh sinh
, , ,
k p pk k
p jN N N N
k N
π πω ωε ε
− −− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
=
1 12 1 1 1 2 1 1 12 2
1 2
( )( )( ) ( )
Np
NN
KT s
s p s p s p
ωε −=
− − −11 22
12-32 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Example: Find the Chebyshev Transfer Function
Low-pass filter specifications: fp = 10 kHz, Amax = 1 dB, fs = 15 kHz,
Amin = 25 dB, dc gain = 1.
Determine ε :
Determine N :
To meet identical specifications, one requires a lower order for the Chebyshev than
the Butterworth filter. Alternatively, for the same order and the same Amax, the
Chebyshev filter provides greater stopband attenuation than the Butterworth filter.
Determine the N natural modes:
Determine T(s):
max .Aε = − = − =10 11010 1 10 1 0 5088
min( ) log[ cosh ( cosh ( / ))]s s pA N Aω ε ω ω−= + ≥2 2 110 1
Select N = 5
dB
dB
( ) .
( ) .s
s
N A
N A
ωω
= ⇒ == ⇒ =
4 21 6
5 29 9
, ( . . )
, ( . . ) ( . )p
p p
p p j
p p j p
ω
ω ω
= − ±
= − ± = −1 5
2 4 3
0 0895 0 9901
0 2342 0 6119 0 2895
( ). ( . )( . )( . . )
p
p p p p p
T ss s s s s
ωω ω ω ω ω
=+ + + + +
5
2 2 2 28 1408 0 2895 0 4684 0 1789 0 9883where ωp = 2π×104 rad/s
12-33 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
First-Order and Second-Order Filter Functions
12-34 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
First-Order filters
Transfer function:
The bilinear transfer function characterizes:
a pole (natural mode) at s = −ω0,
a transmission zero at s = −a0 /a1,
a high-frequency gain that approaches a1, i.e., T( j∞) = a1.
The numerator coefficients, a0 and a1, determine the type of filter (e.g., low-
pass, high-pass, etc.)
0
01)(ω++
=s
asasT
12-35 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Low-pass (LP) filters
( )a
T ss ω
=+
0
0
12-36 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
High-pass (HP) filters
( )a s
T ss ω
=+1
0
12-37 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
General filters
0
01)(ω++
=s
asasT
12-38 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
First-Order All-Pass Filters
Special case of first-order filters
Transmission zero and pole are symmetrically located relative to jω-axis
Transmission is constant at all frequencies
Phase is not constant at all frequencies
Most often used in the design of delay equations
12-39 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Second-Order Filters (Biquadratic Filters)
Transfer function:
where ω0 and Q determine the natural modes (poles) according to
We are usually interested in the case of complex-conjugate natural modes,
obtained for Q > 0.5.
The numerator coefficients (a0, a1, a2) determine the type of filter function
(i.e., LP, HP, etc.).
20
02
012
2)(ωω +⎟
⎠⎞
⎜⎝⎛+
++=
sQ
s
asasasT
200
21 41
12
,Q
jQ
pp −±−= ωω
12-40 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Definition of the parameter ω0 and Q of a pair of complex conjugate poles:
Pole frequency ω0 : distance of pole (from origin).
Pole quality factor Q : distance of the poles from the jω-axis
Higher Q higher selectivity (pole is closer to jω-axis)
Q = ∞ poles are on the jω-axis
can yield sustained oscillation
Q is negative poles are in the right half of s-plane
unstable
12-41 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
2nd-order Low-pass Filters
Two transmission zeros are at s = ∞.
Magnitude response
The peak occurs only for .
The response obtained for is the Butterworth, or maximally flat, response.
2/1>Q
2/1=Q
12-42 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
2nd-order High-pass Filters
Two transmission zeros are at s = 0 (dc).
Magnitude response shows a peak for .
Duality between the LP and HP response.
2/1>Q
12-43 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
2nd-order Bandpass Filters
One transmission zero is at s = 0 (dc), and the other is at s = ∞.
Magnitude response peaks at ω = ω0. The center frequency is equal to the pole frequency ω0.
The selectivity of 2nd-order bandpass filter is measured by 3-dB bandwidth.
As Q increases, BW decreases and the bandpass filters become more selective.
QQ 24
11, 0
2021
ωωωω ±+=Q
BW 012
ωωω =−≡
12-44 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Notch Filters
Transmission zeros are located on the jω-axis. (ωz = ±jωn )A notch in the magnitude response occurs at ω = ωn. (ωn : notch frequency)
3 cases of 2nd-order notch filters:Regular notch (ωn = ω0 )Low-pass notch (ωn > ω0 )High-pass notch (ωn < ω0 )
The transmission at dc and at s = ∞ is finite. (Because there are no transmission zeros at either s = 0 or s = ∞.)
12-45 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
12-46 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
2nd-order All-pass Filters
Two transmission zeros are in the right half of the s-plane, at the mirror-image locations of the poles.
Magnitude response is constant over all frequencies; the flat gain is equal to |a2|.
The frequency selectivity of the all-pass function is its phase response.
12-47 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The Second-Order LCR Resonator
12-48 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Second-Order LCR Resonator
Use LCR resonator to derive various second-order filter functions.
Replacing inductor L by a simulated inductance obtained using an OPAMP-
RC circuit results in an OPAMP-RC resonator.
The second-order parallel LCR resonator:
(without changing poles)
12-49 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Two ways for exciting the resonator without change its natural structure:
The resonator is excited with a current source I connected in parallel.
The resonator is excited with a voltage source Vi.
LCCRss
Cs
RsC
sLYI
Vo
111111
2 ++=
++==
CRQLCo11 02 ==
ωω
CRQLC 001 ωω ==
The resonator poles are poles of Vo /I.
The resonator poles are poles of Vo /Vi.
LCCRss
LC
sCRsL
sCR
VV
i
o
11
1
1
1
2 ++=
⎟⎠⎞
⎜⎝⎛+
=
CRQLCo11 02 ==
ωω
CRQLC 001 ωω ==
12-50 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Various Second-Order Function Using Resonator
Transmission zeros:
The values of s are at which Z2(s) is zero, provided that Z1(s) is not simultaneously zero.
The values of s are at which Z1(s) is infinite, provided that Z2(s) is not simultaneously infinite.
)()()(
)()(
)(21
2
sZsZsZ
sVsV
sTi
o
+==
12-51 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the Low-Pass Function
LCCRss
LC
RsC
sL
sLYY
YZZ
ZVV
sTi
o
11
1
11
1
)(
2
21
1
21
2
++=
++=
+=
+=≡
12-52 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the High-Pass Function
20
02
22)(
ωω ++=≡
Qss
saVV
sTi
o
12-53 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the Bandpass Function
LCCRss
CRs
sCsLR
RYYY
YsT
CLR
R
11
1
11
1
)(2 ++
=++
=++
=
12-54 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the Notch Functions
20
02
20
2
2)(ωω
ω
++
+=≡
Qss
sa
VV
sTi
o
General notch function:
2111
n
CLω
= The L1C1 tank circuit introduces a pair of transmission zeros at ±jωn.
LLLCCC ==+ 2121
12-55 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the low-pass notch (LPN) function 0ωω >n
))(( 212111 CCLLCL +<
Satisfied with L2 = ∝, L1 = L.
2 2
22 20
0
( ) o n
i
V sT s a
V s sQ
ωω ω
+≡ =
+ +
where ωn2 =1/LC1, ω0
2 = 1/L(C1 + C2), ω0 /Q = 1/CR,
and the high frequency a2= C1 /(C1 + C2).
12-56 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the High-Pass Notch (HPN) Function 0ωω <n
Satisfied with C2 = 0, C1 = C.
CLLCRss
CLs
VV
sTi
o
)(11
1
)(
21
2
1
2
++
+=≡
))(( 212111 CCLLCL +>
12-57 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of the All-Pass Function
20
02
20
02
)(ωω
ωω
++
+−=≡
Qss
Qss
VV
sTi
o
20
02
02
1)(ωω
ω
++−=⇒
Qss
Qs
sT
Bandpass filter with a center-frequency
gain of 2.
Allpass filter with a flat gain of 0.5:
20
02
0
5.0)(ωω
ω
++−=
Qss
Qs
sT
12-58 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realization of Various Second-Order Filter Functions Using LCR Resonator
12-59 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Second-Order Active Filters Based on Inductor Replacement
12-60 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The Antoniou Inductance-Simulation Circuit (1969)
Many inductor replacement circuit exists – Antoniou inductance simulation circuit is the best, i.e., it is very tolerant to the nonideal properties of the OPAMP, gain and bandwidth.
12-61 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The design is usually based on selecting R1 = R2 = R3 = R5 = R and C4 = C, which leads to L = CR2 .
.2
5314
2
5314
1
1
RRRRC
LL
RRRRC
sIV
Zin
=
=≡
by given inductance an of that is h whic
is impedance input The
12-62 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The Op Amp-RC Resonator
An LCR resonator
An op amp-RC resonator obtained by replacing the inductor L in the LCR resonator with a simulated inductance realized by the Antoiou circuit.
531
2
4
66660
25316460 /
11
RRRR
CC
RRCQ
RRRRCCLC
==
==
ω
ω
Selecting R1 = R2 = R3 = R5 = R and C4 = C6 = C, then
RR
Q
CR
6
01
=
=ω
12-63 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Implementation of the buffer amplifier K
Note that not only does the amplifier K buffer the output of the filter, but it also allows the designer to set the filter gain to any desired value by appropriately selecting the value of K.
12-64 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Realizations for the Various Second-Order Filter Functions Using the Op-Amp-RC Resonator
12-65 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
12-66 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
12-67 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
12-68 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Homework
HW1
Problems 8, 12, 19, 35, 41
12-69 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Second-Order Active Filters Based on the two-Integrator-Loop Topology
12-70 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Two-Integrator-Loop Biquad
Derivation
High-pass:
Bandpass:
Low-pass:
Block diagram:
hp
i
V KsV s s
Qω ω
=+ +
2
2 200
hp hp hp
hp hp hp
i
i
V V V KVQ s s
V KV V VQ s s
ω ω
ω ω
⎛ ⎞⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞= − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
20 0
2
20 0
2
1
1
bp hp( / )
i i
V s VK sV Vs s
Q
ωωω ω
−= =
+ +
00
2 200
lp hp( / )
i i
V s VKV Vs s
Q
ωωω ω
= =+ +
2 2200
2 200
12-71 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Universal active filter realizes LP, BP, and HP, simultaneously.
Versatile
Easy to design
Very popular
Block-diagram realization:
Performance limited by the finite BW of OPAMP.
200
20
bp )/()(
ωωω
++=
Qss
sKsT
200
2
20
lp )/()(
ωωω
++=
Qss
KsT
12-72 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Integrator
Miller integrator
Ideal OPAMP
Actual OPAMP (one-pole example)
o
i
V sT s
V s sRC= = −
( )( )
( )1
OPAMP transfer function
, , ,
(i.e., unity-gain BW )
( )/
/
/
( )( )
i o o
u
AA s
s s
R R A A RC s
s RC
AT s
ssRCA
A s
ω
=+
→∞ = >> >>
= >>−
=⎛ ⎞+ +⎜ ⎟
⎝ ⎠
0
1
0 1
1
0
00 1
1
0 1 1
1
1 1
12-73 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
log f
dB
Open-loop gain A0
Real integrator
Ideal integrator
f11
RCA0
20log A0
20dB/dec
40dB/dec
( )
( )
uu
T s sRCsRC
A s
RC ss s s
RC RCω
ω
= − ++
= −⎡ ⎤⎛ ⎞+ + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
2 11
11
11
Since
( )( )
u
u u u
uu
A s s
RC
A RC
sRC A RC
T sRC
s sA RC
ω
ω ω ω
ωω
= >>
>>
>>
+ + ≈ ≈ +
= −⎛ ⎞
+ +⎜ ⎟⎝ ⎠
0 1 1
0
10
0
1
1
1 1
11
12-74 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Circuit Implementation of Universal Filter
Kerwin-Huelsman-Newcomb (KHN) biquad
Arbitrarily and practically choose R1, Rf , R2, and R3 to meet the above relationship.
fR
R
RQ
R
⎧ =⎪⎪⎨⎪ = −⎪⎩
1
2
3
1
2 1
12-75 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
To obtain notch and all-pass functions, the three outputs are assumed with appropriate weight.
E.g. Notch filter: choose
hp bp lp hp bp lpF F F F F F
o iH B L H B L
R R R R R RV V V V V T T T
R R R R R R⎛ ⎞ ⎛ ⎞
= − + + = − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
( / ) ( / ) ( / )( / )
o F H F B F L
i
V R R s s R R R RK
V s s Qω ω
ω ω− +
= −+ +
2 20 0
2 20 0
, H nB
L
RR
Rωω⎛ ⎞
= ∞ = ⎜ ⎟⎝ ⎠
2
0
12-76 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Alternative Two-Integrator-Loop Biquad
Two-Thomas biquad (Single-ended fashion)
Universal filter
12-77 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Two-Thomas biquad with input feedward paths
Realize all special second-order function
Design table
222
22
31
12
11
111
RCQCRss
RRCRRr
RCs
CC
s
V
V
i
o
++
+⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟
⎠⎞
⎜⎝⎛
−=
12-78 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Single-Amplifier Biquadratic Active Filters
12-79 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Single-OPAMP Biquad (SAB) – compared with two-integrator biquad
Economic
require 1 OPAMP instead of 3 or 4.
More sensitive to R and C variations
Greater dependence on limited gain and bandwidth
Single-amplifier biquads (SABs) are therefore limited to the less strigent filter
spec. E.g., pole Q < 10.
Biquads can be cascaded to construct high-order filters.
12-80 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
SAB Synthesis
Two-step procedure
Synthesis of a feedback loop that realizes a pair of complex conjugate
poles characterized by ω0 and Q.
Injecting the input signal in a way that realizes the desired transmission
zeros.
Synthesis of the feedback loop
12-81 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The produce of the op amp gain A and t(s):
The characteristic equation: 1 + L(s) = 0
The poles sP of the close-loop circuit obtain as solutions to the equation
In the ideal case, A = ∞ and the pole are obtained from
The circuit poles are identical to the zeros of the RC network.
N(s): zeros of the RC network
D(s): poles of the RC network
( )( ) ( )
( )AN s
L s At sD s
= =
( )Pt sA
= −1
( )PN s = 0
12-82 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Bridged-T RC Networks
open-circuited
open-circuited
b
a
V
Vst =)(
12-83 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The pole polynomial of the active-filter circuit will equal to the numerator polynomial of the Bridged-T network.
s s s sQ C C R C C R Rω ω
⎛ ⎞+ + = + + +⎜ ⎟
⎝ ⎠2 2 20
01 2 3 1 2 3 4
1 1 1 1
C C R R
C C R RQ
R C C
ω
−
=
⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
01 2 3 4
1
1 2 3 4
3 1 2
1
1 1
and
Let
QC C C m Q RC
R R
RR
m
ω⎧ = = ⇒ = =⎪⎪
=⎨⎪⎪ =⎩
21 2
0
3
4
24
Q and ω0 can be used to determine the component values.
12-84 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Injecting the input signal
To obtain transmission zero
Example: A bandpass filter
o
i
sV C RV
s sC C R C C R R
α−=
⎛ ⎞+ + +⎜ ⎟
⎝ ⎠
1 4
2
1 2 3 1 2 3 4
1 1 1 1The denominator polynomial is identical to the numerator polynomial of t(s).
12-85 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Complementary Transformation
Complement of transfer function – interchanging input and ground
Note ac a c
bc b c
V V Vt
V V V−
= =−
12-86 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Two-step procedure
Nodes of the feedback network and any of the op amp inputs that are connected to ground should be disconnected from ground and connected to the op amp output. Conversely, those nodes that were connected to the op amp output should be now connected to ground. That is, we simply interchange the op amp output terminal with ground.
The two input terminals of the op amp should be interchanged.
Example
01 =+ At0)1(
11 =−
+− t
A
ACharacteristic equation:
01 =+ Atsame poles
12-87 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Ex. Appling the complementary transformation to the feedback loop
Sallen-and-Key SAB circuit: (HP function)
C1 = C2 = C, R3 = R, R4 = R/4Q2, CR = 2Q/ω0, and the value of C is arbitrarily chosen to be practically convenient.
12-88 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Another example: LP filter obtained by injecting
1
214
2143
21430
11
1
−
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
=
RRC
RRCCQ
RRCCω
Selecting R1 = R2 = R, C4 = C, C3 = C/m,
02 /24 ωQCRQm ==
Complementary Trans.
Injecting Vi through R1:
12-89 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Sensitivity
12-90 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Sensitivity
Real components deviate from their designed values
initially inaccurate due to fabrication tolerances.
drift due to environmental effects such as temperature and
humidity.
chemical changes which occurs as the circuit ages.
inaccuracies in modeling the passive and active devices, e.g.,
nonideal OPAMP and parasitics.
All coefficients, and therefore poles and zeros of H(s), depend on
circuit element.
The size of H(s) error depends on how large the component
tolerances are and how sensitive the circuit’s performance is to
these tolerances.
12-91 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Sensitivity calculation, allow the designer
to select the better circuit from those in the literature.
to determine whether a chosen filter circuit satisfies and will keep satisfying the given specifications.
Component χ
Performance criterion p(χ), such as
Quality factor
Pole frequency
Zero frequency
or p(s, χ), if p is also a function of frequency and stands for
H(s),
magnitude of H(s),
phase of H(s).
12-92 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Sensitivity
Taylor series
pSχ( , ) ( , )
( , ) ( , ) ( )P s P s
P s P s d dχ χ
χ χχ χ χ χχ χ
∂ ∂= + + +
∂ ∂0 0
22
0 212
P
P
d dPd
P sP s P s d P s d
P s P s dP s P s
P s d PP PS
P s d
P PS
χ
χ
χχ χ χ
χ
χχ χχ χ
χχ χ χ χ χχ
χ χχ χχ χ χ χ
χχχ χ χ χ χ
χχ
<<=
⎧ ∂Δ = + − ≈⎪ ∂⎪⎨
Δ ∂⎪ ≈⎪ ∂⎩
∂ ∂= = =
∂ ∂
Δ ∂<< ≈
∂
if and is small
if , then
( , )( , ) ( , ) ( , )
( , ) ( , )( , ) ( , )
( , ) (ln )/( , ) / (ln )
/
0
0
0 0 0
00
0 0 0
0 0
0 0 0
0
0
0
1
1χ χ/
12-93 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Sensitivity is to predict the deviations from
the tolerances in component values
the finite op-amp gain
Definition
x denotes the value of a component (a resistor, a capacitor, or an amplifier gain) and y denotes a circuit parameter of interest (ω0 or Q).
For small changes
/lim
/yx
x
y y y xS
x x x yΔ →
Δ ∂≡ =
Δ ∂0
//
yx
y yS
x xΔ
≈Δ
12-94 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Example: find the sensitivities of ω0 and Q relative to all passive components
Note that: A 10% increase in R3 results in a 5% decrease in the value of ω0 and a 5% increase in the value of Q.
C C R RS S S Sω ω ω ω= = = = −0 0 0 01 2 3 4
12
, ,Q Q Q QC C R RS S S S= = = = −
1 2 3 4
1 10
2 2
Assume C1 = C2,
s s s sQ C C R C C R Rω ω
⎛ ⎞+ + = + + +⎜ ⎟
⎝ ⎠2 2 20
01 2 3 1 2 3 4
1 1 1 1
C C R R
C C R RQ
R C C
ω
−
=
⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
01 2 3 4
1
1 2 3 4
3 1 2
1
1 1
1
2
1
1
2
2
1
1
2
2
11
−
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
C
C
C
C
C
C
C
CS Q
C
/lim
/yx
x
y y y xS
x x x yΔ →
Δ ∂≡ =
Δ ∂0
12-95 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
- The sensitivities relative to the amplifier gain
Assume the op amp have a finite gain A, the characteristic equation for the loop
1 + At(s) = 0
Using the obtained design, C1 = C2 = C, R3 = R, R4 = R/4Q2, and CR = 2Q/ω0,
1 + At(s) = 0
20
20
2
200
2
)12)(/(
)/()(
ωωωω
+++++
=QQss
Qssst
( ) 0)/()12)(/( 200
220
20
2 =++++++ ωωωω QssAQQss
01
21 2
0
202 =+⎟⎟
⎠
⎞⎜⎜⎝
⎛+
++∴ ωωA
Q
Qss
⎪⎩
⎪⎨⎧
++=
=
)1/(21 2
00
AQ
QQa
a ωω
)1/(21
)1/(2
1 0 2
20
+++
⋅+
==AQ
AQ
A
ASS aa Q
AAω .12
2 22
>>>>≈ AQAA
QS aQ
A and for :Note
12-96 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Switch-Capacitor Filters
12-97 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Switched-Capacitor Filters
Switched capacitor performing as a simulated resistor
large area resistor smaller area capacitor
VLSI technology requirements
Good switch
Well-defined capacitor
OPAMP
First discussion on the equivalent resistance of a periodically switched capacitor methods in 1946.
Rapid evolution of practical SC is due to the implementation in MOS technology.
Followed by a rapid development and implementation of analog signal processing techniques.
SC circuits are sampled-data analog MOS integrated circuits.
12-98 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
RC Time Constant
Active-RC integrator
If τ = R1C2, then
The absolute accuracies of R and C is very poor.
dCd dRR C
ττ
= + 21
1 2
12-99 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Basic Principle of the Switched-Capacitor Filter Technique
Switched-capacitor integrator
Operation
Nonoverlapping two-phase clock
iC vCq 11=
c
Cav T
qi 1=
1CT
iv
R c
av
ieq =≡
An equivalent time-constant for the integrator:
1
22 C
CTRC ceq ==τ determined by Tc C2 /C1 accuracy
12-100 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Accurate RC time constant
SC as a resistor
If R1 is replaced by a switched capacitor resistance realization with a C1, then
If clock frequency is assumed to be constant, then
Relative accuracy is good for two capacitors fabricated in the same integrated circuit.
Equivalent C
C C C
Q CV V TI I R
T T R C f C= = = = = =1 2
1
CC
C C
C C dCd dT dCT
f C C T C Cτττ
= = = + −2 2 2 1
1 1 2 1
1
dCd dCC C
ττ
= − →2 1
2 1
0
12-101 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Z-Transform
12-102 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
At cycle n, i.e., t = nTs, we have Q1(n) = C1vi (n) and Q2(n) = C2vo(n)At cycle n + 1/2, i.e., t = (n + 1/2)Ts,
Q1(n + 1/2) = 0 Q2(n + 1/2) = Q2(n) − Q1(n) = C2vo(n) − C1vi(n)At cycle n + 1, i.e., t = (n + 1)Ts,
Q1(n + 1) = C1vi (n + 1) Q2(n + 1) = C2vo(n + 1) = Q2(n + 1/2) = C2vo(n) − C1vi(n)
Thus, the time-domain difference equation is
C2vo(n + 1) = C2vo(n) − C1vi(n)In the z-domain
-1
1 12 2 1 -1
2 2
( ) 1( ) ( )- ( ) - -
( ) -1 1-o
o o ii
v z C C zzC v z C v z C v z
v z C z C z= ⇒ = × = ×
12-103 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Stray Capacitance in SC Integrators
11
12
( )( ) 1
o S
i
v z C C zv z C z
−
−
⎛ ⎞+= − ×⎜ ⎟ −⎝ ⎠
12-104 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Practical Circuits of the Switched-Capacitor integrators
Noninverting switched-capacitor integrator
Inverting switched-capacitor integrator
( )( )
( )o
i
V z C zH z
V z C z
−
−= =−
11
12 1
( )( )
( )o
i
V z CH z
V z C z −= = −−
11
2
11
12-105 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Practical Circuits of the Switched-Capacitor Filters
1
43210 RRCC=ω
12-106 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Substituting for R2 and R4 by their SC equivalent values, that is
Give ω0 of the SC biquad as
If and C1 = C2 = C , then C3 = C4 = KC, where K = ω0Tc
(from Eq. 12.93).
Determine C5 :
The center-frequency gain of the bandpass function:
12
430
43210
1
1CCCC
TRRCC c
== ωω
44
33
CT
RCT
R cc == and
14
23
CCT
CCT cc =
(12.93)
QC
TQKC
QC
CCTCT
Q cc
c0
45
4
5 // ω====
CTC
QCC
c0
6
5
6 ω
==gainfrequency -Center
12-107 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Summary of Filters
Continuous-time filter
RLC passive
RC active
Sampled-Data filter
Switched-Capacitor Filter
Digital Filter
OperationsMultiplyDelayAdd
12-108 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Tuned Amplifiers
12-109 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Tuned Amplifiers
A special kind of frequency-selective network
RF and IF application
Response is similar to that of bandpass filters
Center frequency ranges from a few hundred kHz to a few hundred MHz
Skirt selectivity SB⎛ ⎞⎜ ⎟⎝ ⎠
Center frequency
dB bandwidthdB bandwidth
SB
−=
−303
12-110 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Single-Tuned amplifier
(Transistor operates in class A mode)
Equivalent circuit
( )
( )
L o FET
L FET
R R r
C C C
=
= +
12-111 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
( / ) /o m
i
V g sV C s s CR LC
= −+ +2 1 1
2nd-order bandpass function
Center frequency gain om
i
BCRLC
Q CRB
Vg R
V ω ω
ω
ω ω
=
= =
≡ =
= −0
0
00
1 1
m i m io
L
g V g VV
Y sCR sL
− −= =
+ +1 1
12-112 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Inductor Losses
Inductor equivalent circuits
The inductor Q factor: (typically 50 ~ 200)
Relationship between Q and Rp:
For Q >> 1,
s
LQ
rω
≡ 00
( / )( )
( / )s
j QY j
r j L j L Qω
ω ω+
= =+ +
00 2
0 0 0
1 1 1 11 1
( )p
Y j jj L Q j L LQ j L R
ωω ω ω ω
⎛ ⎞≈ + = + = +⎜ ⎟
⎝ ⎠0
0 0 0 0 0 0
1 1 1 1 1 11
p pL L L R LQQ
ω⎛ ⎞
= + ≈ =⎜ ⎟⎝ ⎠
0 020
11 Fig.(a) Fig.(b)
12-113 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Use of Transformers
Impedance transformer
Examples
Allow using a high inductance and a smaller capacitance.
Equivalent circuit:
12-114 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Increase the effective input impedance of the second stage.
Equivalent circuit:
12-115 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Amplifiers with Multiple Tuned Circuits
Greater selectivity can be obtained.
Double-tuned amplifier
Miller capacitance will cause
1. detuning of the input circuit.
2. skewing of the response of the input circuit.
12-116 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Cascode and CC-CB Cascade
Amplifier configurations do not suffer from the Miller effect.
Cascode
Common-collector common-base cascade
12-117 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Synchronous Tuning
N identical tuned circuits (do not interact)
Bandwidth shrinkage
where is known as the bandwidth-shrinkage factor.
-NBQω
=1
0 2 1
/ -N12 1
12-118 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Stagger-Tuning
Maximal flatness around ω0 (Center frequency)
12-119 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
The flat response can be obtained by transforming the response of a maximally flat (Butterworth) low-pass filter up the frequency axis to ω0.
⎟⎟⎠
⎞⎜⎜⎝
⎛−++⎟⎟
⎠
⎞⎜⎜⎝
⎛−−+
=
200
200
1
41
124
11
2
)(
Qj
Qs
Qj
Qs
sasT
ωωωω
12-120 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
For a narrow-band filter, Q >> 1, and s in the neighborhood of +jω0, the transfer function of 2nd-order BP filter can be expressed in terms of its poles as
The magnitude response has a peak value of a1Q /ω0 at ω = ω0.
LP to BP transformation for narrow-band filter:
Qjs
a
jQs
asT
2/)(
2/
2/
2/)(
00
1
00
1
ωωωω +−=
−+≈ (Narrow-band approximation)
LP BP
02/
)(0 ωω
jspQp
KsT
−=+
=
12-121 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Transform a maximally flat 2nd-order LP filter (Q = 1/ ) to obtain a maximally flat BP filter:
2
BQ
BB
B
BQ
BB
B
011002
011001
2
222
2
222
ωωω
ωωω
≈=−=
≈=+=
(4th-order)
12-122 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Magnitude response:
12-123 Ching-Yuan Yang / EE, NCHUMicroelectrics (III)
Homework
HW2
Problems 49, 56, 65, 67,74