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EECS 247 Lecture 3: Filters © 2005 H.K. Page 1
EE247 Lecture 3
• Last week’s summary• Active Filters
– Active biquads• Sallen- Key & Tow-Thomas• Integrator based filters
– Signal flowgraph concept– First order integrator based filter– Second order integrator based filter & biquads
– High order & high Q filters• Cascaded biquads
– Cascaded biquad sensitivity
• Ladder type filters
EECS 247 Lecture 3: Filters © 2005 H.K. Page 2
SummaryLast Week
• Major success in CMOS technology scaling:à Inexpensive DSPs technology à Resulted in the need for high performance
Analog/Digital interface circuitry
• Main Analog/Digital interface building blocks includes– Analog filters– D/A converters– A/D converters
EECS 247 Lecture 3: Filters © 2005 H.K. Page 3
Monolithic Filters
• Monolithic inductor in CMOS tech. – Integrated L
EECS 247 Lecture 3: Filters © 2005 H.K. Page 5
Biquad Complex Poles
( )1412
s 2
21
−±−=
>
PP
P
P
QjQ
Q
ω
Distance from origin in s-plane:
( )2
2
2
2 1412
P
PP
P QQ
d
ω
ω
=
−+
=
EECS 247 Lecture 3: Filters © 2005 H.K. Page 6
s-Plane
poles
ωj
σ
Pω radius =
P2Q- part real P
ω=
PQ21
arccos
EECS 247 Lecture 3: Filters © 2005 H.K. Page 7
Implementation of Biquads
• Passive RC: only real poles- can’t implement complex conjugate poles
• Terminated LC– Low power, since it is passive– Only noise source à load and source resistance– As previously analyzed, not feasible in the monolithic form for f
EECS 247 Lecture 3: Filters © 2005 H.K. Page 9
Imaginary Axis Zeros
• Sharpen transition band• “notch out” interference• High-pass filter (HPF)• Band-reject filter
Note: Always represent transfer functions as a product of a gain term,poles, and zeros (pairs if complex). Then all coefficients have a physical meaning, reasonable magnitude, and easily checkable unit.
2
Z2
P P P
2P
Z
s1
H( s ) Ks s
1Q
H( j ) Kω
ω
ω ω
ωω
ω→∞
+
=
+ +
=
EECS 247 Lecture 3: Filters © 2005 H.K. Page 10
Imaginary Zeros• Zeros substantially sharpen transition band
• At the expense of reduced stop-band attenuation at high frequency
PZ
P
P
ffQ
kHzf
32100
===
Pole-Zero Map
Real Axis
Imag
Axi
s
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2x 106
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
6
104
105
106
107
-50
-40
-30
-20
-10
0
10
Frequency [Hz]
Mag
nitu
de [
dB]
With zerosNo zeros
EECS 247 Lecture 3: Filters © 2005 H.K. Page 11
Moving the Zeros
PZ
P
P
ffQ
kHzf
===
2100
Pole-Zero Map
Real AxisIm
ag A
xis
-6 -4 -2 0 2 4 6x 10
5
-6
-4
-2
0
2
4
6
x 105
104 105 106 107-50
-40
-30
-20
-10
0
10
20
Frequency [Hz]
Mag
nitu
de[d
B]
EECS 247 Lecture 3: Filters © 2005 H.K. Page 12
Tow-Thomas Active Biquad
Ref: P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973.
• Parasitic insensitive• Multiple outputs
EECS 247 Lecture 3: Filters © 2005 H.K. Page 13
Frequency Response
( ) ( )
( ) ( )01
21001020
01
3
012
012
22
012
0021122
1
1asas
babasabbakV
V
asasbsbsb
VV
asasbabsbab
kVV
in
o
in
o
in
o
++−+−
−=
++++
=
++−+−
−=
• Vo2 implements a general biquad section with arbitrary poles and zeros
• Vo1 and Vo3 realize the same poles but are limited to at most one finite zero
EECS 247 Lecture 3: Filters © 2005 H.K. Page 14
Component Values
8
72
173
2821
111
21732
80
6
82
74
81
6
8
111
21753
80
1
1
RR
k
CRRCRR
k
CRa
CCRRRR
a
RR
b
RRRR
RR
CRb
CCRRRR
b
=
=
=
=
=
−=
=
827
2
86
20
015
112124
10213
20
12
111
111
11
1
RkR
bR
R
Cbak
R
CbbakR
CakkR
Cak
R
CaR
=
=
=
−=
=
=
=
821 and , ,,, given RCCkba iii
thatfollowsit
11
21732
8
CRQ
CCRRRR
PP
P
ω
ω
=
=
EECS 247 Lecture 3: Filters © 2005 H.K. Page 15
Integrator Based Filters
• Main building block for this category of filters àintegrator
• By using signal flowgraph techniques àconventional filter topologies can be converted to integrator based type filters
• Next few pages:– Signal flowgraph techniques– 1st order integrator based filter– 2nd order integrator based filter– High order and high Q filters
EECS 247 Lecture 3: Filters © 2005 H.K. Page 16
What is a Signal Flowgraph (SFG)?
• SFG à Topological network representation consisting of nodes & branches- used to convert one form of network to a more suitable form (e.g. passive RLC filters to integrator based filters)
• Any network described by a set of linear differential equations can be expressed in SFG form.
• For a given network, many different SFGs exists. • Choice of a particular SFG is based on practical
considerations such as type of available components.
*Ref: W.Heinlein & W. Holmes, “Active Filters for Integrated Circuits”, Prentice Hall, Chap. 8, 1974.
EECS 247 Lecture 3: Filters © 2005 H.K. Page 17
What is a Signal Flowgraph (SFG)?• SFG nodes represent variables (V & I in our case), branches represent transfer
functions (we will call these transfer functions branch multiplication factor BMF)• To convert a network to its SFG form, KCL & KVL is used to derive state space
description:• Example:
1SL
Circuit State-space SFG description
R
L
R
oV
C 1SC
Vin
Vo
oI
VoinI
inI
inI
inI Vo
Vin oI
R I Vin o
1V Iin o
SL
1I Vin o
SC
× =
× =
× =
EECS 247 Lecture 3: Filters © 2005 H.K. Page 18
Signal Flowgraph (SFG) Rules• Two parallel branches can be replaced by a single branch with overall BMF equal to sum of two BMFs
•A node with only one incoming branch & one outgoing branch can be replaced by a single branch with BMF equal to the product of the two BMFs
•An intermediate node can be multiplied by a factor (x). BMFs for incoming branches have to be multiplied by x and outgoing branches divided by x
1V
a
2Vb
a+b1V 2V
a.b1V 2V3V
a b1V 2V
3Va b1V 2
V x.a b/x1V 2V
3x.V
EECS 247 Lecture 3: Filters © 2005 H.K. Page 19
Signal Flowgraph (SFG) Rules
• Simplifications can often be achieved by shifting or eliminating nodes
•A self-loop branch with BMF y can be eliminated by multiplying the BMF of incoming branches by 1/(1-y)
1iV2V a
oV
1/b
1-1/b
3V1iV
2V a/(1+1/b)oV
1/b
1
3V
3V1iV
2V aoV
1/b
1-1/b
4V
1iV2V a
oV-1 -1/b
1 1
3V
EECS 247 Lecture 3: Filters © 2005 H.K. Page 20
Integrator Based Filters1st Order LPF
• Start from RC prototype• Use KCL & KVL to derive state space description:
• Use state space description to draw signal flowgraph (SFG)
in
V 1os CV 1 R
=+
1IoV
Rs
CinV2I
1V+ −
CV
+
−
EECS 247 Lecture 3: Filters © 2005 H.K. Page 21
Integrator Based FiltersFirst Order LPF
1IoV
1
Rs
CinV2I
•KCL & KVL to derive state space description:
2C
1 in C
11
2 1
o C
IV
sC
V V VV
IRs
I IV V
=
=
= −
==
1V+ −
1Rs
1
sC
2I1I
CVinV 1−1 1V
• Use state space description to draw signal flowgraph (SFG)
SFG
CV
+
−
oV1
EECS 247 Lecture 3: Filters © 2005 H.K. Page 22
Normalize• Since integrators - the main building blocks- require in & out signals in
the voltage form (not current)
à Convert all currents to voltages by multiplying current nodes by a scaling resistance R*
à Corresponding BMFs should then be scaled accordingly
1 in o
11
2o
2 1
V V VV
IRsI
VsC
I I
=
=
= −
=
1 in o*
*1 1
*2
o *
* *2 1
V V V
RI R V
Rs
I RV
sC R
I R I R
=
=
= −
=
* 'x xI R V=
1 in o*
'1 1
'2
o *
' '2 1
V V V
RV V
Rs
VV
sC R
V V
=
=
= −
=
EECS 247 Lecture 3: Filters © 2005 H.K. Page 23
Normalize
*1
sCR
oVinV 1−1 1V
1
*RRs
1Rs
1
sC
2I1I
oVinV 1−1
1
1V
'1V '2V
oVinV 1−1 1V
1
*RRs
1I R∗× 2I R ∗×
*1
sCR
EECS 247 Lecture 3: Filters © 2005 H.K. Page 24
Synthesis
'1V
1*1
sC R
oVinV 1−1 1V
'2V1
*RRs
* * CR Rs , Rτ= = ×
'1V
1
sτ ×
oVinV 1−1 1V
'2V1
1
sτ ×
oVinV 1−1 1V
'2V
1
Consolidate two branches
EECS 247 Lecture 3: Filters © 2005 H.K. Page 25
First Order Integrator Based Filter
oV
inV
-+ ( )
1H s
sτ=
×
∫+
1
sτ ×
oVinV 1−1 1V
'2V
1
EECS 247 Lecture 3: Filters © 2005 H.K. Page 26
Opamp-RC Single-Ended Integrator
oo in
in
1 V 1V V dt ,
RC V sRC= − = −∫
oV
C
inV
-
+R
RCτ =
- ∫
EECS 247 Lecture 3: Filters © 2005 H.K. Page 27
1st Order Filter Built with Opamp-RC Integrator
oV
inV
-+
∫+
• Single-ended Opamp-RC integrator has a sign inversion from input to output
à Convert SFG accordingly by modifying BMF
oV
'in inV V= −
+
∫+
-
oV
'inV
∫--
EECS 247 Lecture 3: Filters © 2005 H.K. Page 28
1st Order Filter Built with Opamp-RC Integrator (continued)
o'
in
V 1
V 1 sRC= −
+
oV
C
in'V
-
+R
R
--
oV'
inV
∫
EECS 247 Lecture 3: Filters © 2005 H.K. Page 29
Opamp-RC 1st Order Filter Noise
2n1v
( )
n 22o im0m 1i
2 21 2 2
2 2n1 n2
2o
v S ( f ) dfH ( f )
S ( f ) Input referred noise spectral densi ty1H ( f ) H ( f )
1 2 fRC
v v 4KTR f
k Tv 2 C2
π
α
∞
==
→
= =+
= = ∆
=
=
∑ ∫
oV
C
-
+R
R
Typically, α increases as filter order increases
Identify noise sources (here it is resistors & opamp)Find transfer function from each noise sourceto the output (opamp noise next page)
2n2v
α
EECS 247 Lecture 3: Filters © 2005 H.K. Page 30
Opamp-RC Filter NoiseOpamp Contribution
2n1v
2opampv oV
C
-
+R
R
So far only the fundamental noise sources are considered.In reality, noise associated with the opamp increases the overall noise.The bandwidth of the opamp affects the opamp noise contribution to the total noise
2n2v
EECS 247 Lecture 3: Filters © 2005 H.K. Page 31
Integrator Based Filter2nd Order RLC Filter
oV
1
1
sL
R CinI•State space description:
R L C o
CC
RR
LL
C in R L
V V V VI
VsC
VI
RV
IsL
I I I I
=
=
= = =
=
= − −
• Draw signal flowgraph (SFG)
SFG
L
1R
1
sC
CIRI
CV
inI
1−1
RV 1
1−
LV
LI
CV
LI
+
-CI
LV+
-
Integrator form
+
-RV
RI
EECS 247 Lecture 3: Filters © 2005 H.K. Page 32
1 1
*RR *
1
sCR
'1V
2V
inV
1−1
1V
*R
sL
1−
oV
'3V
Normalize
1
1R
1
sC
CIRI
CV
inI
1−1
RV 1
1−
LV
LI
1
sL
• Convert currents to voltages by multiplying all current nodes by the scaling resistance R*
* 'x xI R V=
'2V
EECS 247 Lecture 3: Filters © 2005 H.K. Page 33
inV
1*R R 21
sτ11
sτ1 1
*RR *
1sCR
'1V
2V
inV
1−1
1V
*R
sL
1−
oV
'3V
Synthesis
∑
oV
--
* *1 2R C L Rτ τ= × =
EECS 247 Lecture 3: Filters © 2005 H.K. Page 34
-20
-15
-10
-5
0
0.1 1 10
Second Order Integrator Based Filter
Normalized Frequency [Hz]
Mag
nit
ud
e (d
B)
inV
1*R R 21
sτ11
sτ
BPV
-- LPVHPV
∑
Filter Magnitude Response
EECS 247 Lecture 3: Filters © 2005 H.K. Page 35
Second Order Integrator Based Filter
inV
1*R R 21
sτ11
sτ
BPV
-- LPVHPV
∑
BP 22in 1 2 2
LP2in 1 2 2
2HP 1 2
2in 1 2 2* *
1 2*
0 1 2
1 2
1 2 *
V sV s s 1V 1V s s 1
V sV s s 1
R C L R
R R
11
Q 1
Frommatch ing pointof v iewdes irable:RQ
R
L C
β
β
β
β
β
ττ τ τ
τ τ ττ τ
τ τ τ
ω τ τ
τ τ
τ τ
τ τ
=+ +
=+ +
=+ +
= × =
=
= =
= ×
= → =
EECS 247 Lecture 3: Filters © 2005 H.K. Page 36
Second Order Bandpass Filter Noise
2 2n1 n2
2o
v v 4KTRdf
k Tv 2 Q C
= =
=
inV
1*R R2
1sτ
11
sτ
BPV
--
2n1v
2n2v
•Find transfer function of each noise source to the output•Integrate contribution of all noise sources•Here it is assumed that opamps are noise free (not usually the case!)
n 22o im0m 1
v S ( f ) dfH ( f )∞=
= ∑ ∫
Typically, α increases as filter order increasesNote the noise power is directly proportion to Qα
∑
EECS 247 Lecture 3: Filters © 2005 H.K. Page 37
Second Order Integrator Based FilterBiquad
inV
1*R R 21
sτ11
sτ
BPV
--
21 1 2 2 2 30
2in 1 2 2
a s a s aVV s s 1β
τ τ ττ τ τ
+ +=+ +
∑oV
∑
a1 a2 a3
HPV
LPV
• By combining outputs can generate general biquad function:
s-planejω
σ
EECS 247 Lecture 3: Filters © 2005 H.K. Page 38
SummaryIntegrator Based Monolithic Filters
• Signal flowgraph techniques utilized to convert RLC networks to integrator based active filters
• Each reactive element (L& C) replaced by an integrator• Fundamental noise limitation determined by integrating capacitor:
– For lowpass filter:
– Bandpass filter:
where α is a function of filter order and topology
2o
k Tv Q Cα=
2o
k Tv Cα=
EECS 247 Lecture 3: Filters © 2005 H.K. Page 39
Higher Order Filters
• How do we build higher order filters?– Cascade of biquads and 1st order sections
• Each complex conjugate pole built with a biquad and real pole with 1st order section
• Easy to implement• In the case of high order high Q filters à highly sensitive to
component variations– Direct conversion of high order ladder type RLC filters
• SFG techniques used to perform exact conversion of ladder type filters to integrator based filters
• More complicated conversion process• Much less sensitive to component variations compared to
cascade of biquads
EECS 247 Lecture 3: Filters © 2005 H.K. Page 40
Higher Order FiltersCascade of Biquads
Example: LPF filter for CDMA baseband receiver • LPF with
– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 45 dB– Assumption: Can compensate for phase distortion in the digital domain
• 7th order Elliptic Filter• Implementation with Biquads
Goal: Maximize dynamic range– Pair poles and zeros
highest Q poles with closest zeros is a good starting point, but not necessarily optimum
– Ordering:Lowest Q poles first is a good start
EECS 247 Lecture 3: Filters © 2005 H.K. Page 41
Filter Frequency Response
Bode Diagram
Pha
se (
deg)
Mag
nitu
de (
dB)
-80
-60
-40
-20
0
-540
-360
-180
0
Frequency [Hz]
300kHz 1MHz
Mag
. (dB
)
-0.2
0
3MHz
EECS 247 Lecture 3: Filters © 2005 H.K. Page 42
Pole-Zero Map
Qpole fpole [kHz]
16.7902 659.496
3.6590 611.7441.1026 473.643
319.568
fzero [kHz]1297.5
836.6744.0
Pole-Zero Map
-2 -1.5 -1 -0.5 0-1
-0.5
0
0.5
1
Imag
Axi
s X
107
Real Axis x107
s-Plane
EECS 247 Lecture 3: Filters © 2005 H.K. Page 43
Biquad Response
104
105
106
107
-40
-20
0
LPF1
-0.5 0
-0.5
0
0.5
1
104
105
106
107
-40
-20
0
Biquad 2
104
105
106
107
-40
-20
0
Biquad 3
104
105
106
107
-40
-20
0
Biquad 4
EECS 247 Lecture 3: Filters © 2005 H.K. Page 44
Biquad ResponseBode Magnitude Diagram
Frequency [Hz]
Mag
nit
ud
e (d
B)
104
105
106
107
-50
-40
-30
-20
-10
0
10
LPF1Biquad 2Biquad 3Biquad 4
-0.5 0
-0.5
0
0.5
1
EECS 247 Lecture 3: Filters © 2005 H.K. Page 45
Intermediate Outputs
Frequency [Hz]
Mag
nitu
de (
dB)
-80
-60
-40
-20
Mag
nitu
de (
dB)
LPF1 +Biquad 2M
agni
tude
(dB
)
Biquads 1, 2, 3, & 4
Mag
nitu
de (
dB)
-80
-60
-40
-20
0
LPF10
10kHz10
6
-80
-60
-40
-20
0
100kHz 1MHz 10MHz
5 6 7
-80
-60
-40
-20
0
4 5 6
Frequency [Hz]
10kHz10
100kHz 1MHz 10MHz
LPF1 +Biquads 2,3 LPF1 +Biquads 2,3,4
EECS 247 Lecture 3: Filters © 2005 H.K. Page 46
-10
Sensitivity Component variation in Biquad 4 (highest Q pole):
– Increase ωp4 by 1%– Decrease ωz4 by 1%
High Q poles à High sensitivityin Biquad realizations
Frequency [Hz]1MHz
Mag
nitu
de (
dB)
-30
-40
-20
0
200kHz
3dB
600kHz
-50
2.2dB
EECS 247 Lecture 3: Filters © 2005 H.K. Page 47
High Q & High Order Filters
• Cascade of biquads– Highly sensitive to component variations à not suitable
for implementation of high Q & high order filters– Cascade of biquads only used in cases where required
Q for all biquads
EECS 247 Lecture 3: Filters © 2005 H.K. Page 49
LC Ladder Filters
• Made of resistors, inductors, and capacitors• Doubly terminated or singly terminated (with or w/o RL)
Doubly terminated LC ladder filters _ Lowest sensitivity to component variations
RsC1 C3
L2
C5
L4
inV RL
oV
EECS 247 Lecture 3: Filters © 2005 H.K. Page 50
LC Ladder Filters
• Design:– CAD tools
• Matlab• Spice
– 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.
RsC1 C3
L2
C5
L4
inV RL
oV
EECS 247 Lecture 3: Filters © 2005 H.K. Page 51
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 c Butterworth•Determine minimum filter order :
-Use of Matlab-or Tables
•Here tables used
fstop / f-3dB = 2Rs >27dB
Minimum Filter Orderc5th order Butterworth 1
-3dB
2
EECS 247 Lecture 3: Filters © 2005 H.K. Page 52
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 =1Denormalization:
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 xLNormC= Cr xCNorm
EECS 247 Lecture 3: Filters © 2005 H.K. Page 53
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 = 2.0
L2Norm = L4Norm =1.618
Denormalization:
Since ω-3dB =2πx10MHzLr = R/ω-3dB = 15.9 nHCr = 1/(RXω-3dB )= 15.9 nF
R =1
cC1=C5=9.836nF, C3=31.83nFcL2=L4=25.75nH
EECS 247 Lecture 3: Filters © 2005 H.K. Page 54
Magnitude Response Simulation
Frequency [MHz]
Mag
nitu
de (
dB)
0 10 20 30
-50
-40
-30
-20
-10-50
-6 dB passband attenuationdue to double termination
30dB
Rs=1Ohm
C19.836nF
C331.83nF
L2=25.75nH
C59.836nF
L4=25.75nH
inV RL=1Ohm
oV
SPICE simulation Results
EECS 247 Lecture 3: Filters © 2005 H.K. Page 55
LC Ladder FilterConversion to Integrator Based Active Filter
1I2V
RsC1 C3
L2
C5
L4
inV RL
4V 6V
3I 5I
2I 4I 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 3: Filters © 2005 H.K. Page 56
LC Ladder FilterSignal Flowgraph
SFG
1Rs 1
1
sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1
sC 5
1
sC2
1
sL 4
1
sL
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
= − = = −
= = − = =
= = − =
= − = = − =
EECS 247 Lecture 3: Filters © 2005 H.K. Page 57
LC Ladder FilterSignal Flowgraph
SFG
1Rs 1
1
sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1
sC 5
1
sC2
1
sL 4
1
sL
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
2I 4I 6I
7I1V+ − 3V+ − 5V+ −
EECS 247 Lecture 3: Filters © 2005 H.K. Page 58
LC Ladder FilterNormalize
1
1
*RRs
*1
1
sC R
'1V
2VinV 1−1 1V oV1− 1
*
2
R
sL
1− 1− 1−1 1
1− 13V 4V 5V 6V
'3V
'2V '4V
'5V
'6V
'7V
*3
1
sC R
*
4
R
sL*
5
1
sC R
*RRL
1Rs 1
1
sC
2I1I
2VinV 1−1
1
1V oV1− 11
3
1
sC 5
1
sC2
1
sL 4
1
sL
1RL
1− 1− 1−1 1
1− 13V 4V 5V 6V
3I 5I4I 6I 7I
EECS 247 Lecture 3: Filters © 2005 H.K. Page 59
LC Ladder FilterSynthesize
inV
1+ -
-+ -+
+ - + -
*R Rs−
*R RL21
sτ 31
sτ 41
sτ 51
sτ11
sτ
oV
1
1
1
*RRs
*1
1
sC R
'1V
2VinV 1−1 1V oV1− 1
*
2
R
sL
1− 1− 1−1 1
1− 13V 4V 5V 6V
'3V
'2V '4V
'5V
'6V
'7V
*3
1
sC R
*
4
R
sL*
5
1
sC R
*RRL
EECS 247 Lecture 3: Filters © 2005 H.K. Page 60
LC 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= = = = = = =
inV
1+ -
-+ -+
+ - + -
*R Rs−
*R RL21
sτ 31
sτ 41
sτ 51
sτ11
sτ
oV
Building Block:RC Integrator
V 12V sRC1
= −
EECS 247 Lecture 3: Filters © 2005 H.K. Page 61
Negative Resistors
V1-
V2-
V2+
V1+
Vo+
Vo-
V1-
V2+Vo+
EECS 247 Lecture 3: Filters © 2005 H.K. Page 62
Synthesize
EECS 247 Lecture 3: Filters © 2005 H.K. Page 63
Frequency Response
EECS 247 Lecture 3: Filters © 2005 H.K. Page 64
Scale Node Voltages
Scale Vo by factor “s”
EECS 247 Lecture 3: Filters © 2005 H.K. Page 65
Noise
Total noise: 1.4 µV rms(noiseless opamps)
That’s excellent, but the capacitors are very large (and the resistors small). Not possible to integrate.
Suppose our application allows higher noise in the order of 140 µV rms …
EECS 247 Lecture 3: Filters © 2005 H.K. Page 66
Scale to Meet Noise TargetScale capacitors and resistors to meet noise objective
s = 10-4Noise: 141 µV rms (noiseless opamps)
EECS 247 Lecture 3: Filters © 2005 H.K. Page 67
Completed Design
EECS 247 Lecture 3: Filters © 2005 H.K. Page 68
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