Page 1
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 1
EE247 Lecture 4
• Ladder type filters –For simplicity, will start with all pole ladder type filters
• Convert to integrator based form- example shown–Then will attend to high order ladder type filters
incorporating zeros• Implement the same 7th order elliptic filter in the form of
ladder RLC with zeros– Find level of sensitivity to component mismatch – Compare with cascade of biquads
• Convert to integrator based form utilizing SFG techniques–Effect of Integrator Non-Idealities on Filter
Frequency Characteristics
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 2
LC Ladder Filters
• Design:–Filter tables
• A. Zverev, Handbook of filter synthesis, Wiley, 1967.• A. B. Williams and F. J. Taylor, Electronic filter design, 3rd edition,
McGraw-Hill, 1995.–CAD tools
• Matlab• Spice
RsC1 C3
L2
C5
L4
inV RL
oV
Page 2
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 3
LC Ladder Filter Design Example
Design a LPF with maximally flat passband:f-3dB = 10MHz, fstop = 20MHzRs >27dB
From: Williams and Taylor, p. 2-37
Stopband A
ttenuation dB
Νοrmalized ω
• Maximally flat passband Butterworth• Find minimum filter order :
- Use of Matlab- or Tables
• Here tables used
fstop / f-3dB = 2Rs >27dB
Minimum Filter Order5th order Butterworth
1
-3dB
2
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 4
LC Ladder Filter Design Example
From: Williams and Taylor, p. 11.3
Find values for L & C from Table:Note L &C values normalized to
ω-3dB =1
Denormalization:Multiply all LNorm, CNorm by:
Lr = R/ω-3dBCr = 1/(RXω-3dB )
R is the value of the source and termination resistor (choose both 1Ω for now)
Then: L= Lr xLNorm
C= Cr xCNorm
Page 3
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 5
LC Ladder Filter Design Example
From: Williams and Taylor, p. 11.3
Find values for L & C from Table:Normalized values:C1Norm =C5Norm =0.618C3Norm = 2.0L2Norm = L4Norm =1.618
Denormalization:Since ω-3dB =2πx10MHz
Lr = R/ω-3dB = 15.9 nHCr = 1/(RXω-3dB )= 15.9 nF
R =1
C1=C5=9.836nF, C3=31.83nF
L2=L4=25.75nH
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 6
Last Lecture:Example: 5th Order Butterworth Filter
Frequency [MHz]
Mag
nitu
de (d
B)
0 10 20 30-50
-40
-30
-20
-10-50
30dB
Rs=1OhmC19.836nF
C331.83nF
L2=25.75nH
C59.836nF
L4=25.75nH
inV RL=1Ohm
oV
SPICE simulation Results
Specifications:f-3dB = 10MHz, fstop = 20MHzRs >27dB
Used filter tables to obtain Ls & Cs
Page 4
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 7
Low-Pass RLC Ladder FilterConversion to Integrator Based Active Filter
1I2V
RsC1 C3
L2
C5
L4
inV RL
4V 6V
3I 5I
2I4I 6I
7I
• Use KCL & KVL to derive equations:
1V+ − 3V+ − 5V+ −oV
1 in 2
1 31 3
25 6
5 74
I2V V V , V , V V V2 3 2 4sC1I I4 6V , V V V , V V V4 5 4 6 6 o 6sC sC3 5
V VI , I I I , I2 1 3Rs sL
V VI I I , I , I I I , I4 3 5 6 5 7sL RL
= − = = −
= = − = =
= = − =
= − = = − =
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 8
Low-Pass RLC Ladder FilterSignal Flowgraph
SFG
1Rs 1
1sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1sC 5
1sC2
1sL 4
1sL
1RL
1− 1− 1−1 1
1− 13V 4V 5V 6V
3I 5I4I 6I 7I
1 in 2
1 31 3
25 6
5 74
I2V V V , V , V V V2 3 2 4sC1I I4 6V , V V V , V V V4 5 4 6 6 o 6sC sC3 5
V VI , I I I , I2 1 3Rs sL
V VI I I , I , I I I , I4 3 5 6 5 7sL RL
= − = = −
= = − = =
= = − =
= − = = − =
Page 5
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 9
Low-Pass RLC Ladder FilterSignal Flowgraph
SFG
1Rs 1
1sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1sC 5
1sC2
1sL 4
1sL
1RL
1− 1− 1−1 1
1− 13V 4V 5V 6V
3I 5I4I 6I 7I
1I2V
RsC1 C3
L2
C5
L4
inV RL
4V 6V
3I 5I
2I4I 6I
7I1V+ − 3V+ − 5V+ −
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 10
Low-Pass RLC Ladder FilterNormalize
1
1
*RRs *
1
1sC R
'1V
2VinV 1−1 1V oV1− 1
*
2
RsL
1− 1− 1−1 1
1− 13V 4V 5V 6V
'3V'2V '
4V '5V '
6V '7V
*3
1sC R
*
4
RsL
*5
1sC R
*RRL
1Rs 1
1sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1sC 5
1sC2
1sL 4
1sL
1RL
1− 1− 1−1 1
1− 13V 4V 5V 6V
3I 5I4I 6I 7I
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EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 11
Low-Pass RLC Ladder FilterSynthesize
1
1
1
*RRs
*1
1sC R
'1V
2VinV 1−1 1V oV1− 1
*
2
RsL
1− 1− 1−1 1
1− 13V 4V 5V 6V
'3V'2V '
4V '5V '
6V '7V
*3
1sC R
*
4
RsL
*5
1sC R
*RRL
inV
1+ -
-+ -+
+ - + -
*R Rs−
*R RL21
sτ 31
sτ 41
sτ 51
sτ11
sτ
oV2V 4V 6V
'3V '
5V
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 12
Low-Pass RLC Ladder FilterIntegrator Based Implementation
* * * * *2* *
L L4C C C C.R , .R , .R , .R , C .R11 2 2 3 3 4 4 5 5R R
τ τ τ τ τ= = = = = = =
Building Block:RC Integrator
V 12V sRC1
= −
inV
1+ -
-+ -+
+ - + -
*R Rs−
*R RL21
sτ 31
sτ 41
sτ 51
sτ11
sτ
oV2V 4V 6V
'3V '
5V
Page 7
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 13
Negative Resistors
V1-
V2-
V2+
V1+
Vo+
Vo-
V1-
V2+ Vo+
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 14
Integrator Based Implementation of LP Ladder FilterSynthesize
oV
4V
'3V
'5V
2V
Page 8
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 15
Frequency Response
oV
4V
'3V
'5V
2V
0.5
1
0.1
1
0.5
10MHz 10MHz
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 16
Scale Node Voltages
Scale Vo by factor “s”
Page 9
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 17
Node Scaling
VO X 2
X1.8/2
V4 X 1.6
V3’ X 1.2
X 1.2
X 1.8/1.6
X 1.2/1.6 X 1.6/1.2
X 1/1.2
X 1.6/1.8
X 2/1.8
V5’ X 1.8
V2
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 18
Maximizing Signal Handling by Node Voltage Scaling
Scale Vo by factor “s”
Before Node Scaling After Node Scaling
Page 10
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 19
Filter Noise
Total noise @ the output: 1.4 μV rms(noiseless opamps)
That’s excellent, but the capacitors are very large (and the resistors small high power dissipation). Not possible to integrate.
Suppose our application allows higher noise in the order of 140 μV rms …
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 20
Scale to Meet Noise TargetScale capacitors and resistors to meet noise objective
s = 10-4
Noise: 141 μV rms (noiseless opamps)
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EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 21
Completed Design
5th order ladder filterFinal design utilizing:
-Node scaling -Final R & C scaling based on noise considerations
oV
4V
'3V
'5V
2V
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 22
Sensitivity
• C1 made (arbitrarily) 50% (!) larger than its nominal value
• 0.5 dB error at band edge
• 3.5 dB error in stopband
• Looks like very low sensitivity
Page 12
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 23
inVDifferential 5th Order Lowpass Filter
• Since each signal and its inverse readily available, eliminates the need for negative resistors!
• Differential design has the advantage of even order harmonic distortion components and common mode spurious pickup automatically cancels
• Disadvantage: Double resistor and capacitor area!
+
+
--+
+
--
+
+--+
+--
+
+
--
inV
oV
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 24
RLC Ladder FiltersIncluding Transmission Zeros
RsC1 C3
L2
C5
L4
inV RLC7
L6
C2 C4 C6
oV
RsC1 C3
L2
C5
L4
inV RL
oVAll poles
Poles & Zeros
Page 13
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 25
RLC Ladder Filter Design Example
• Design a baseband filter for CDMA IS95 cellular phone receive path with the following specs.
– Filter frequency mask shown on the next page– Allow enough margin for manufacturing variations
• Assume pass-band magnitude variation of 1.8dB• Assume the -3dB frequency can vary by +-8% due to
manufacturing tolerances & circuit inaccuracies– Assume any phase impairment can be compensated in the
digital domain
* Note this is the same example as for cascade of biquad while the specifications are given closer to a real product case
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 26
RLC Ladder Filter Design ExampleCDMA IS95 Receive Filter Frequency Mask
+10
-1
Frequency [Hz]
Mag
nitu
de (d
B)
-44
-46
600k 700k 900k 1.2M
Page 14
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 27
RLC Ladder Filter DesignExample: CDMA IS95 Receive Filter
• Since phase impairment can be corrected for, use filter type with max. roll-off slope/pole
Filter type Elliptic• Design filter freq. response to fall well within the freq. mask
– Allow margin for component variations & mismatches• For the passband ripple, allow enough margin for ripple change
due to component & temperature variationsDesign nominal passband ripple of 0.2dB
• For stopband rejection add a few dB margin 44+5=49dB• Final design specifications:
– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 49 dB
• Use Matlab or filter tables to decide the min. order for the filter (same as cascaded biquad example)– 7th Order Elliptic
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 28
RLC Low-Pass Ladder Filter DesignExample: CDMA IS95 Receive Filter
RsC1 C3
L2
C5
L4
inV RLC7
L6
C2 C4 C6
oV
7th order Elliptic
• Use filter tables to determine LC values
Page 15
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 29
RLC Ladder Filter DesignExample: CDMA IS95 Receive Filter
• Specifications– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 49 dB
• Use filter tables to determine LC values – Table from: A. Zverev, Handbook of filter synthesis, Wiley,
1967– Elliptic filters tabulated wrt “reflection coeficient ρ”
– Since Rpass=0.2dB ρ =20%– Use table accordingly
( )2Rpass 10 log 1 ρ= − × −
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 30
RLC Ladder Filter DesignExample: CDMA IS95 Receive Filter
• Table from Zverev book page #281 & 282:
• Since our spec. is Amin=44dB add 5dB margin & design for Amin=49dB
Page 16
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 31
• Table from Zverev page #281 & 282:
• Normalized component values:
C1=1.17677C2=0.19393L2=1.19467C3=1.51134C4=1.01098L4=0.72398C5=1.27776C6=0.71211L6=0.80165C7=0.83597
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 32
-65
-55
-45
-35
-25
-15
-5
200 300 400 500 600 700 800 900 1000 1100 1200
RLC Filter Frequency Response
• Frequency mask superimposed
• Frequency response well within spec.
Frequency [kHz]
Mag
nitu
de (d
B)
Page 17
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 33
Passband Detail
-7.5
-7
-6.5
-6
-5.5
-5
200 300 400 500 600 700 800
• Passband well within spec.
Frequency [kHz]
Mag
nitu
de (d
B)
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 34
RLC Ladder Filter Sensitivity
• The design has the same specifications as the previous example implemented with cascaded biquads
• To compare the sensitivity of RLC ladder versus cascaded-biquads:
– Changed all Ls &Cs one by one by 2% in order to change the pole/zeros by 1% (similar test as for cascaded biquad)
– Found frequency response most sensitive to L4 variations – Note that by varying L4 both poles & zeros are varied
Page 18
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 35
RCL Ladder Filter Sensitivity
Component mismatch in RLC filter:– Increase L4 from its nominal value by 2%– Decrease L4 by 2%
-65
-55
-45
-35
-25
-15
-5
200 300 400 500 600 700 800 900 1000 1100 1200Frequency [kHz]
Mag
nitu
de (d
B)
L4 nomL4 lowL4 high
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 36
RCL Ladder Filter Sensitivity
-6.5
-6.3
-6.1
-5.9
-5.7
200 300 400 500 600 700
-65
-60
-55
-50
600 700 800 900 1000 1100 1200
Frequency [kHz]
Mag
nitu
de (d
B)
1.7dB
0.2dB
-65
-55
-45
-35
-25
-15
-5
200 300 400 500 600 700 800 900 1000 1100 1200
Page 19
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 37
-10
Sensitivity of Cascade of BiquadsComponent mismatch in Biquad 4 (highest Q pole):
– Increase ωp4 by 1%– Decrease ωz4 by 1%
High Q poles High sensitivityin Biquad realizations
Frequency [Hz]1MHz
Mag
nitu
de (d
B)
-30
-40
-20
0
200kHz
3dB
600kHz
-50
2.2dB
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 38
Sensitivity Comparison for Cascaded-Biquads versus RLC Ladder
• 7th Order elliptic filter – 1% change in pole & zero pair
1.7dB(21%)
3dB(40%)
Stopband deviation
0.2dB(2%)
2.2dB (29%)
Passband deviation
RLC LadderCascadedBiquad
Doubly terminated LC ladder filters Significantly lower sensitivity compared to cascaded-biquads particularly within the passband
Page 20
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 39
RLC Ladder Filter DesignExample: CDMA IS95 Receive Filter
RsC1 C3
L2
C5
L4
inV RLC7
L6
C2 C4 C6
oV
7th order Elliptic
• Previously learned to design integrator based ladder filters without transmission zeros
Question: o How do we implement the transmission zeros in the integrator-
based version? o Preferred method no extra power dissipation no active
elements
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 40
Integrator Based Ladder FiltersHow Do to Implement Transmission zeros?
• Use KCL & KVL to derive :
1I 2VRs
C1 C3
L2
inV RL
4V
3I5I
2I4I
1V+ − 3V+ −
Ca
oV
( )
( )
I I IC1 3I a2CI I I I , I V V s , V , VC C2 1 3 2 4 a 2 2a a sC sC1 1Substituting for I and rearranging :Ca
CI I1 3 aV V2 4C Cs C C1 1a a
− −= − − = − = =
−= + ×
+ +
Page 21
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 41
Integrator Based Ladder FiltersHow Do to Implement Transmission zeros?
• Use KCL & KVL to derive :
1I 2VRs
C1 C3
L2
inV RL
4V
3I5I
2I4I
1V+ − 3V+ −
Ca
oV
( )
( )
CI I1 3 aV V2 4C Cs C C1 1a a
CI I3 5 aV V4 2C Cs C C3 a 3 a
−= + ×
+ +
−= + ×
+ +
Frequency independent constantsCan be substituted by:
Voltage-Controlled Voltage Source
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 42
Integrator Based Ladder FiltersTransmission zeros
1I 2V
( )
( )
CI I1 3 aV V2 4C Cs C C1 1a aCI I3 5 aV V4 2C Cs C C3 a 3 a
−= +
+ +
−= +
+ +
Rs L2
inV RL
4V
3I5I
2I 4I
1V+ − 3V+ −
Ca
• Replace shunt capacitors with voltage controlled voltage sources:
+-
( )C C1 a+ ( )C C3 a+
CaV4 C C1 a+CaV2 C C3 a+
+-
Page 22
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 43
3rd Order Lowpass FilterAll Poles & No Zeros
1I 2VRs L2
inV RL
4V
3I5I
2I 4I
1V+ − 3V+ −
1Rs
1sC1
2I1I
2VinV 1−1
1
1V oV1− 11
1sC32
1sL
1RL
1− 1−1
3V 4V
3I 4I
oV
C3C1
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 44
Transmission Zero ImplementationW/O Use of Active Elements
1I 2VRs L2
inVRL
4V
3I5I
2I 4I
1V+ − 3V+ −
( )C C1 a+ ( )C C3 a+
CaV4 C C1 a+CaV2 C C3 a+
1Rs ( )1 aC C
1s +
2I1I
2VinV 1−1
1
1V oV1− 11
( )3 a
1s C C+2
1sL
1RL
1− 1−1
3V 4V
3I 4I
oV
CaC C1 a+
CaC C3 a+
+- +-
Page 23
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 45
Integrator Based Ladder FiltersHigher Order Transmission zeros
C1
2V 4V
C3
Ca6VCb
2V 4V
+- +-
( )C C1 a+ ( )C C C3 a b+ +
CaV4 C C1 a+
CaV2 C C3 a+
6V
+-
( )C C5 b+
CbV4 C C3 b++-CbV6 C C3 b+
C5Convert zero generating Cs in C loops to voltage-controlled voltage sources
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 46
Higher Order Transmission zeros
*RRs ( )1 aC C
1*s R+
2VinV 1−1
1
1V oV1− 11
( )3 a b
1*s C C CR + +
*
2
RsL
*RRL
1− 1−1
3V 4V
CaC C1 a+
CaC C3 a+
1−*
4
RsL
1
5V 6V
RL1I
2VRs L2
inV
4V
3I5I
2I
4I1V+ − 3V+ −
+-+-
( )C C1 a+ ( )C C C3 a b+ +
CaV4 C C1 a+
CaV2 C C3 a+
oVL4
6V 7I
6I
5V+ −
+-
( )C C5 b+
CbV4 C C5 b++- CbV6 C C3 b+
( )5 bC C
1*sR +
CbC C3 b+ Cb
C C5 b+
1
1−'1V '
3V'2V '4V
'5V '
6V '7V
Page 24
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 47
Example:5th Order Chebyshev II Filter
• 5th order Chebyshev II
• Table from: Williams & Taylor book, p. 11.112
• 50dB stopband attenuation• f-3dB =10MHz
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 48
Realization with Integrator
( )i 1 a2
1 3*s a 1a 1
V V CV1V VR C Cs C C R−⎡ ⎤= − +⎢ ⎥ ++ ⎣ ⎦
-Rs
R*
Rs
Page 25
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 49
5th Order Butterworth Filter
From:Lecture 4page 14
oV
4V
'3V
'5V
2V
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 50
5th Order Chebyshev II Filter Opamp-RC Simulation
oV
4V
'3V
'5V
2V
Page 26
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 51
7th Order Differential Lowpass Filter Including Transmission Zeros
+
+
--
+
+
--
+
+--
+
+
--
+
+--
+
+
--
+
+--
inV
oV
Transmission zeros implemented with pair of coupling capacitors
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 52
Effect of Integrator Non-Idealities on FilterFrequency Characteristics
• In the passive filter design (RLC filters) section:–Reactive element (L & C) non-idealities expressed in the
form of Quality Factor (Q)–Filter impairments due to component non-idealities explained
in terms of component Q
• In the context of active filter design (integrator-based filters)
–Integrator non-idealities Translated to have form of Quality Factor (Q)
–Filter impairments due to integrator non-idealities explained in terms of integrator Q
Page 27
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 53
Effect of Integrator Non-Idealities on Filter Performance
• Ideal integrator characteristics
• Real integrator characteristics:– Effect of opamp finite DC gain– Effect of integrator non-dominant poles
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 54
Effect of Integrator Non-Idealities on Filter PerformanceIdeal Integrator
Ideal Intg.
oV
C
inV
-
+R
Ideal Intg.
0ω
ψ
-90o
( )log H s
ψIdeal opamp DC gainSingle pole @ DC no non-dominant poles
oH( s )s
1/ RCo
ω
ω
=
→−
=
=
∞
0dB
Page 28
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 55
Ideal Integrator Quality Factor
( ) ( ) ( )( )( )
1H j R jXXQ R
ω ω ωωω
=+
=Since component Q is defined as::
Then: in t g.Qideal =∞
1o oH( s )s j j
o
ω ωωωω
− −= = = −Ideal intg. transfer function:
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 56
Real Integrator Non-Idealities
Ideal Intg. Real Intg.
( )( )( )o
os sa
p2 p3
aH( s ) H( s )1 11 ss . . .ω
ω− −= ≈
+ ++
0ω
a
-90o
ψ
P2P30P1 a
ω=
-90o
( )log H s
ψ
0ω
ψ
-90o
( )log H s
ψ
Page 29
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 57
Effect of Integrator Finite DC Gain on Q
-90
-89.5
ωoω
o
P1
P1 o
( in radian )
Arctan2 o
Phase lead @ω
π
ω
ω− +
→
∠
Example: P1/ ω0 = 1/100phase error ≅ +0.5degree
0P1 aω=
0ω
a
-90o
( )log H s
ψ
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 58
Effect of Integrator Finite DC Gain on Q
• Phase lead @ ω0Droop in the passband
Normalized Frequency
Mag
nitu
de (d
B)
1
Droop in the passband
Ideal intgIntg with finite DC gain
Page 30
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 59
Effect of Integrator Non-Dominant Poles
-90
-90.5
ωoω
oi
oi
pi 2
opi 2( in rad ian )
Arctan2
Phase lag @
ω
ω
π
ω
∞
=
∞
=
− −
→
∑
∑
∠
Example: ω0 /P2 =1/100 phase error ≅ −0.5degree
0ω
-90o
( )log H s
ψ
P2P3
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 60
Effect of Integrator Non-Dominant Poles
Normalized Frequency
Mag
nitu
de (d
B)
1
• Phase lag @ ω0Peaking in the passbandIn extreme cases could result in oscillation!Peaking in the passband
Ideal intgOpamp with finite bandwidth
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EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 61
Effect of Integrator Non-Dominant Poles & Finite DC Gain on Q
-90
ωoω
P1Arctan2 ooArctan pii 2
πω
ω
∠ − +
∞− ∑
=
0ω
a
-90o
( )log H s
ψ
P2P30P1 a
ω=
-90o
( )log H s
ψ
Note that the two terms have different signs Can cancel each other’s effect!
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 62
Integrator Quality Factor
( )( )( )os sa
p2 p3
aH( s )
1 11 s . . .ω
−≈
+ ++Real intg. transfer function:
o 1 & a 1p2,3,. . . . .
int g. 1Qreal 1 1oa pii 2
ω
ω
<< >>
≈ ∞− ∑
=
Based on the definition of Q and assuming that:
It can be shown that in the vicinity of unity-gain-frequency:
Phase lead @ ω0 Phase lag @ ω0
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EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 63
Example:Effect of Integrator Finite Q on Bandpass Filter Behavior
Integrator DC gain=100 Integrator P2 @ 100.ωo
IdealIdeal
0.5ο φlead @ ωointg 0.5ο φexcess @ ωo
intg
EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 64
Example:Effect of Integrator Q on Filter Behavior
Integrator DC gain=100 & P2 @ 100. ωο
Ideal
( 0.5ο φlead −0.5ο φexcess ) @ ωointg
φerror @ ωointg ~ 0
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EECS 247 Lecture 4: Active Filters © 2007 H.K. Page 65
SummaryEffect of Integrator Non-Idealities on Q
• Amplifier DC gain reduces the overall Q in the same manner as series/parallel resistance associated with passive elements
• Amplifier poles located above integrator unity-gain frequency enhance the Q! – If non-dominant poles close to unity-gain freq. Oscillation
• Depending on the location of unity-gain-frequency, the two terms can cancel each other out!
i1 1o pi 2
int g.ideal
int g. 1real
Q
Qa ω
∞
=
=
≈−
∞
∑