L3 f10 2

EECS 247 Lecture 3: Filters© 2010 H.K. Page 1EE247Administrative
–Homework #1 will be posted on EE247 site and is due Sept. 9th
–Office hours held @ 201 Cory Hall:
•Tues. and Thurs.: 4 to 5pmEECS 247 Lecture 3: Filters
© 2010 H.K. Page 2
EE247 Lecture 3
•Active Filters
–Active biquads-
–How to build higher order filters?
•Integrator-based filters
–Signal flowgraph concept
–First order integrator-based filter
–Second order integrator-based filter & biquads
–High order & high Q filters
•Cascaded biquads & first order filters
–Cascaded biquad sensitivityto component mismatch
•Ladder type filters

EECS 247 Lecture 3: Filters © 2010 Page 3
Filters
2nd Order Transfer Functions (Biquads)
• Biquadratic (2nd order) transfer function:
P H j Q
H j
H j
P






 





( )
( ) 0
( ) 1
0
2
2
P P P
2 2 2
2
P P P
P 2
P
P
1
P 2
1
H( s )
s s
1
Q
1
H( j )
1
Q
Biquad poles @: s 1 1 4Q
2Q
Note: for Q poles are real , comple x otherwise
 

 
 


 

   
      
   
 
     
 

Biquad Complex Poles
Distance from origin in s-plane:
 
2
2
2
2 1 4 1
2
P
P
P
P Q
Q
d



   


 



1
2
2
Complex conjugate poles:
s 1 4 1
2
P
P
P
P
Q
j Q
Q

 
       
 
poles
j

d
S-plane

s-Plane
poles
j

P radius  
P 2Q
real part - P 

P 2Q
1
arccos
2 s 1 4 1
2
P
P
P
j Q
Q
        
 
Example
2nd Order Butterworth
j

45 deg
2 2
4 2 2
Since 2nd order Butterworth:
ree
Cos p Q

 
   
P 2Q
1
arccos

Implementation of Biquads
• Passive RC: only real poles  can’t implement complex conjugate
poles
• Terminated LC
– Low power, since it is passive
– Only fundamental noise sources  load and source resistance
– As previously analyzed, not feasible in the monolithic form for
f <350MHz
• Active Biquads
–Many topologies can be found in filter textbooks!
–Widely used topologies:
• Single-opamp biquad: Sallen-Key
• Multi-opamp biquad: Tow-Thomas
• Integrator based biquads
Active Biquad
Sallen-Key Low-Pass Filter
• Single gain element
• Can be implemented both in discrete & monolithic form
• “Parasitic sensitive”
• Versions for LPF, HPF, BP, …
 Advantage: Only one opamp used
 Disadvantage: Sensitive to parasitic – all pole no finite zeros
Vout
C1
Vin
R1 R2
C2
G
2
2
1 1 2 2
1 1 2 1 2 2
( )
1
1
1 1 1
P P P
P
P
P
G
H s
s s
Q
R C R C
Q
G
R C R C R C
 



 



 

Addition of Imaginary Axis Zeros
• Sharpen transition band
• Can “notch out” interference
– Band-reject filter
• High-pass filter (HPF)
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, and readily identifiable units.
2
Z
2
P P P
2
P
Z
s
1
H( s ) K
s s
1
Q
H( j ) K


 




 
 
  
 
  
 
 
  
 
Imaginary Zeros
• Zeros substantially sharpen transition band
• At the expense of reduced stop-band attenuation at
high frequencies
Z P
P
P
f f
Q
f kHz
3
2
100



10
4
10
5
10
6
10
7
-50
-40
-30
-20
-10
0
10
Frequency [Hz]
Magnitude [dB]
With zeros
No zeros
Real Axis
Imag Axis
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x 10
6
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
6
Pole-Zero Map

Moving the Zeros
Z P
P
P
f f
Q
f kHz



2
100
104 105 106 107 -50
-40
-30
-20
-10
0
10
20
Frequency [Hz]
Magnitude [dB]
Pole-Zero Map
Real Axis
Imag Axis
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x10
5
x10
5
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

Frequency Response
   
   
1 0
2
0 2 0 1 0 0 1
1 0
3
1 0
2
1 0
2
2 2
1 0
2
2 1 1 2 0 0
2
1
1
s a s a
b b a s a b a b
V k a
V
s a s a
b s b s b
V
V
s a s a
b a b s b a b
k
V
V
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
• Possible to use combination of 3 outputs
Component Values
8
7
2
3 7 1
2 8 2
1
1 1
1
2 3 7 1 2
8
0
6
8
2
4 7
1 8
6
8
1 1
1
3 5 7 1 2
8
0
1
1
R
R
k
R R C
R R C
k
R C
a
R R R C C
R
a
R
R
b
R R
R R
R
R
R C
b
R R R C C
R
b





 


 


 

7 2 8
2
8
6
0 2
1 0
5
2 1 2 1 1
4
1 2 0 1
3
0 2
1
2
1 1
1
1 1 1
1 1
1
R k R
b
R
R
b C
k a
R
k a b b C
R
k k a C
R
a C
k
R
a C
R








1 2 8 given a ,b ,k , C , C and R i i i
it follows that
1 1
2 3 7 1 2
8
Q RC
R R R C C
R
P P
P





EECS 247 Lecture 3: Filters© 2010 H.K. Page 15Higher-Order Filters in the Integrated Form
•One way of building higher-order filters (n>2) is via cascade of 2ndorder biquads & 1storder , e.g. Sallen-Key,or Tow-Thomas, & RC2ndorderFilter
……
Nx 2ndorder sections Filter order: n=2N
1 2 NCascade of 1stand 2ndorder filters:
 Easy to implement
 Highly sensitive to component mismatch -good for low Q filters only For high Q applications good alternative: Integrator-based ladder filters2ndorderFilter 1stor 2ndorderFilter EECS 247 Lecture 3: Filters© 2010 H.K. Page 16
Integrator Based Filters
•Main building block for this category of filters Integrator
•By using signal flowgraphtechniques Conventional RLC filter topologies can be converted to integrator based type filters
•How to design integrator based filters?
–Introduction to signal flowgraphtechniques
–1st order integrator based filter
–2nd order integrator based filter
–High order and high Q filters

EECS 247 Lecture 3: Filters © 2010 H.K. Page 17
What is a Signal Flowgraph (SFG)?
• SFG  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.
What is a Signal Flowgraph (SFG)?
• Signal flowgraph technique consist of nodes & branches:
– Nodes represent variables (V & I in our case)
– Branches represent transfer functions (we will call the
transfer function branch multiplication factor or BMF)
• To convert a network to its SFG form, KCL & KVL is used to
derive state space description
• Simple example:
Circuit State-space description SFG
Z
Z
Iin Vo
Iin
Vo
IinZ Vo

Signal Flowgraph (SFG)
Examples
1
SL
Circuit State-space description SFG
R
L
R
Vo
C 1
SC
Vin
Vo
Io
Iin Vo
Iin
Iin
Iin Vo
Vin
Io
Iin R Vo
1
Vin Io
SL
1
Iin Vo
SC
 
 
 
Useful Signal Flowgraph (SFG) Rules
V1
a
V2
b
a+b
V1
V2
a.b
V1
V2
a V3 b V1 2 V
a.V1+b.V1=V2 (a+b).V1=V2
a.V1=V3 (1)
b.V3=V2 (2)
Substituting for V3 from (1) in (2) (a.b).V1=V2
• 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 eliminated &
replaced by a single branch with BMF equal to the product of the two BMFs

•An intermediate node can be multiplied by a factor (k). BMFs for incoming
branches have to be multiplied by k and outgoing branches divided by k
V3
a b
V1 2 V k.a b/k
V1 V2
k.V3
a.V1=V3 (1)
b.V3=V2 (2)
Multiply both sides of (1) by k
(a.k) . V1= k.V3 (1)
Divide & multiply left side of (2) by k
(b/k) . k.V3 = V2 (2)
h Vi
V2 a
Vo
b
-b g
V3
h Vi
V2 a/(1+b)
Vo
b
g
V3
V3
c Vi
V2 a
Vo
b
c -b d Vi
V2 a
Vo
-1 -b
1
d
V3
• Simplifications can often be achieved by shifting or eliminating nodes
• Example: eliminating node V4
• A self-loop branch with BMF y can be eliminated by multiplying the BMF
of incoming branches by 1/(1-y)
V4

1st Order LPF
• Conversion of simple lowpass RC filter to integrator-based
type by using signal flowgraph techniques
in
Vo 1
V 1 s RC


Vo
Rs
C Vin
What is an Integrator?
Example: Single-Ended Opamp-RC Integrator
Vo
C
Vin
-
+
R a  
in
osC , o in o in
V 1 1
V V V , V V dt
R sRC RC
       
• Node x: since opamp has high gain Vx=-Vo /a 0
• Node x is at “virtual ground”
 No voltage swing at Vx combined with high opamp input impedance
 No input opamp current
Vx
IR
IC

What is an Integrator?
Example: Single-Ended Opamp-RC Integrator
Vin - 
Note: Practical integrator in CMOS technology has input & output both in the
form of voltage and not current  Consideration for SFG derivation
Vo
  RC
o
in
V 1
V sRC
 
0
1
RC
 
-90o
20log H

0dB
Phase


Ideal Magnitude &
phase response
1st Order LPF
Convert RC Prototype to Integrator Based Version
1. Start from circuit prototype-
Name voltages & currents for all components
2. Use KCL & KVL to derive state space description in such a way to have
BMFs in the integrator form:
 Capacitor voltage expressed as function of its current VCap.=f(ICap.)
 Inductor current as a function of its voltage IInd.=f(VInd.)
3. Use state space description to draw signal flowgraph (SFG) (see next
page)
I1
Vo
Rs
C Vin
I2
V1 
VC



First Order LPF
I1
Vo
1
Rs
Vin C
I2
V1 
1
Rs
1
sC
I1 I2
Vin 1 V1 1 VC
• All voltages & currents  nodes of SFG
• Voltage nodes on top, corresponding
current nodes below each voltage node
SFG
VC


1 Vo
V1 Vin VC
1
VC I2
sC
Vo VC
1
I1 V1 Rs
I2 I1
 
 

 

Normalize
• Since integrators are the main building blocks require in & out signals
in the form of voltage (not current)
 Convert all currents to voltages by multiplying current nodes by a
scaling resistance R*
 Corresponding BMFs should then be scaled accordingly
1 in o
1
1
2
o
2 1
V V V
V
I
Rs
I
V
sC
I I


 

1 in o
*
*
1 1
*
2
o *
* *
2 1
V V V
R
I R V
Rs
I R
V
sC R
I R I R


 

* '
IxR Vx
1 in o
*
'
1 1
'
2
o *
' '
2 1
V V V
R
V V
Rs
V
V
sC R
V V


 


1st Order Lowpass Filter SGF
Normalize
'
V2
*
1
sCR
Vo Vin 1 V1 1
1
* R
Rs
1
Rs
1
sC
I1 I2
Vin 1 1 Vo
1
V1
'
V1
Vin 1 V1 1 Vo
* R
Rs
I1 R   I2 R  
*
1
sCR
*
*
R
R
1st Order Lowpass Filter SGF
Synthesis
'
V1
1
*
1
sC R
Vin 1 V1 1 Vo
'
1 V2
* R
Rs
* *
R  Rs ,   R C
'
V1
1
 s
Vin 1 V1 1 Vo
'
1 V2
1
 s
Vo Vin 1 V1 1
'
V2
1
Consolidate two branches
*
Choosing
R  Rs

First Order Integrator Based Filter
Vo
Vin
-
+
 
1
H s
 s



+
1
 s
Vo Vin 1 V1 1
'
V2
1
1st Order Filter
Built with Opamp-RC Integrator
Vo
Vin
-
+

+
• Single-ended Opamp-RC integrator has a sign inversion from input to
output
 Convert SFG accordingly by modifying BMF
Vo
Vin
-

+
-
Vo
'
Vin  Vin
+

+
-

1st Order Filter
Built with Opamp-RC Integrator
• To avoid requiring an additional opamp to perform summation at the
input node:
Vo
'
Vin  Vin
+

+
-
Vo
'
Vin

- -
1st Order Filter
Built with Opamp-RC Integrator (continued)
o
'
in
V 1
V 1 sRC
 

Vo
C
in
'
V
-
+
R
R
- -
Vo
'
Vin


Opamp-RC 1st Order Filter
Noise
2
vn1
 
k
2 2
o 0 m m
m 1
th
i
2 2
1 2 2
2 2
n1 n2
2
o
v H ( f ) S ( f ) df
S ( f ) Noise spectral densi ty of i noise sou rce
1
H ( f ) H ( f )
1 2 fRC
v v 4KTR f
k T
v 2
C
2






 

  


 
Vo
C
-
+
R
R
Typically,  increases as filter order increases
Identify noise sources (here it is resistors & opamp)
Find transfer function from each noise source
to the output (opamp noise next page)
2
vn2

Opamp-RC Filter Noise
Opamp Contribution
2
vn1
2
vopamp Vo
C
-
+
R
R
• So far only the fundamental noise
sources are considered
• In reality, noise associated with the
opamp increases the overall noise
• For a well-designed filter opamp is
designed such that noise contribution
of opamp is negligible compared to
other noise sources
• The bandwidth of the opamp affects
the opamp noise contribution to the
total noise
2
vn2

Integrator Based Filter
2nd Order RLC Filter
Vo
1
1
sL
Iin R C
•State space description:
R L C o
C
C
R
R
L
L
C in R L
V V V V
I
V
sC
V
I
R
V
I
sL
I I I I


  

  
• Draw signal flowgraph (SFG)
SFG
L
1
R
1
sC
IR IC
VC
Iin
1
1
VR 1
1
VL
IL
VC
IL
+
-
IC
VL
+
-
Integrator form
+
-
VR
IR
1 1
* R
R
*
1
sCR
'
V1
V2
Vin
1
1
V1
* R
sL
1
Vo
'
V3
2nd Order RLC Filter SGF
Normalize
1
1
R
1
sC
IR IC
VC
Iin
1
1
VR 1
1
VL
IL
1
sL
• Convert currents to voltages by multiplying all current nodes by the scaling
resistance R*
* '
IxR Vx
'
V2

Vin
1 * R
R
2
1
s 1
1
s
1 1
* R
R
*
1
sCR
'
V1
V2
Vin
1
1
V1
* R
sL
1
Vo
'
V3
2nd Order RLC Filter SGF
Synthesis

Vo
- -
* *
1  R C  2  L R
-20
-15
-10
-5
0
0.1 1 10
Second Order Integrator Based Filter
Normalized Frequency [Hz]
Magnitude (dB)
Vin
1 * R
R
2
1
s 1
1
s
VBP
- - VLP
VHP

Filter Magnitude Response

Vin
1 * R
R
2
1
s 1
1
s
VBP
- - VLP
VHP

BP 2
in 2 1 2 2
LP
in 2 1 2 2
2
HP 1 2
in 2 1 2 2
* *
1 2
*
0 1 2
1 2
1 2 *
V s
V s s 1
V 1
V s s 1
V s
V s s 1
R C L R
R R
1 1
Q 1
From matching pointof viewdes i rable :
Q R
R
L C






  
  
 
  
  
 
 
 

 

 

 
  

 
 
  
Second Order Bandpass Filter Noise
2 2
n1 n2
2
o
v v 4KTRdf
k T
v 2 Q
C
 

Vin
1 * R
R
2
1
s
1
1
s
VBP
- -
2
vn1
2
vn2
• 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!)
k
2 2
o 0 m m
m 1
v H ( f ) S ( f ) df


  
Typically,  increases as filter order increases
 Note the noise power is directly proportion to Q


Biquad
Vin
1 * R
R
2
1
s 1
1
s
VBP
- -
2
0 1 1 2 2 2 3
in 2 1 2 2
V a s a s a
V s  s 1
  
  
 

 

Vo

a1 a2 a3
VHP
VLP
• By combining outputs can generate
general biquad function:
s-plane
j

Summary
Integrator 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 value:
– For lowpass filter:
– Bandpass filter:
where  is a function of filter order and topology
2
o
k T
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EECS 247 Lecture 3: Filters© 2010 H.K. Page 45Higher Order Filters
•How do we build higher order filters?
–Cascade of biquads and 1storder sections
•Each complex conjugate pole built with a biquad and real pole with 1storder section
•Easy to implement
•In the case of high order high Q filters highly sensitive to component mismatch
–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 mismatch compared to cascade of biquadsEECS 247 Lecture 3: Filters© 2010 H.K. Page 46Higher Order FiltersCascade of Biquads
Example: LPF filter for CDMA cell phone baseband receiver
•LPF with
–fpass = 650 kHzRpass = 0.2 dB
–fstop = 750 kHzRstop = 45 dB
–Assumption: Can compensate for phase distortion in the digital domain
•Matlab used to find minimum order required 7th order Elliptic Filter
•Implementation with cascaded BiquadsGoal: Maximize dynamic range
–Pair poles and zeros
–In the cascade chain place lowest Q poles first and progress to higher Q poles moving towards the output node

EECS 247 Lecture 3: Filters© 2010 H.K. Page 47Overall Filter Frequency ResponseBode Diagram Phase (deg)Magnitude (dB)
-80
-60-40-200-540-360-1800
Frequency [Hz]
300kHz1MHz Mag. (dB) -0.203MHzEECS 247 Lecture 3: Filters© 2010 H.K. Page 48Pole-Zero Map (pzmap in Matlab)
Qpole fpole[kHz]
16.7902659.496
3.6590 611.744
1.1026 473.643
319.568
fzero [kHz]
1297.5
836.6
744.0
Pole-Zero Map-2-1.5-1
-0.50-1-0.500.51
Imag Axis X107
Real Axis x107s-Plane

EECS 247 Lecture 3: Filters
© 2010 H.K. Page 49CDMA FilterBuilt with Cascade of 1stand 2ndOrder Sections
•1storder filter implements the single real pole
•Each biquad implements a pair of complex conjugate poles and a pair of imaginary axis zeros1storderFilter
Biquad2Biquad4 Biquad3EECS 247 Lecture 3: Filters© 2010 H.K. Page 50Biquad Response10
4105
10
6107-40
-200LPF1
-0.50-0.500.5
1104
10
510610
7-40-20
0Biquad 210410
5106
10
7-40-200
Biquad 3104
10
510610
7-40-20
0Biquad 4

EECS 247 Lecture 3: Filters© 2010 H.K. Page 51
Individual Stage Magnitude Response
Frequency [Hz] Magnitude (dB)
10
410510
6107
-50-40
-30-20-10
010LPF1
Biquad 2
Biquad 3Biquad 4-0.50
-0.500.5
1
EECS 247 Lecture 3: Filters
© 2010 H.K. Page 52Intermediate Outputs
Frequency [Hz]
Magnitude (dB) -80
-60
-40-20 Magnitude (dB) LPF1 +Biquad 2
Magnitude (dB)
Biquads 1, 2, 3, & 4 Magnitude (dB)
-80
-60
-40
-200
LPF1010kHz
10
6-80-60-40
-20
0100kHz
1MHz
10MHz
567
-80
-60-40
-20
0456
Frequency [Hz]
10kHz10
100kHz
1MHz
10MHz
LPF1 +Biquads 2,3LPF1 +Biquads 2,3,4

EECS 247 Lecture 3: Filters© 2010 H.K. Page 53
-10Sensitivity to Relative Component MismatchComponent variation in Biquad 4 relative to the rest (highest Q poles):
–Increase p4by 1%
–Decrease z4by 1% High Q poles High sensitivityin Biquad realizationsFrequency [Hz]
1MHz
Magnitude (dB) -30
-40
-20
0
200kHz3dB
600kHz
-502.2dBEECS 247 Lecture 3: Filters© 2010 H.K. Page 54High Q & High Order Filters
•Cascade of biquads
–Highly sensitive to component mismatch not suitable for implementation of high Q & high order filters
–Cascade of biquads only used in cases where required Q for all biquads <4 (e.g. filters for disk drives)
•Ladder type filters more appropriate for high Q & high order filters (next topic)
–Will show later Less sensitive to component mismatch

Ladder Type Filters
• Active 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
RLC Ladder Filters
Example: 5th Order Lowpass Filter
•Made of resistors, inductors, and capacitors
• Doubly terminated or singly terminated (with or w/o RL)
Rs
C1 C3
L2
C5
L4
Vin RL
Vo
Doubly terminated LC ladder filters  Lowest sensitivity to
component mismatch

LC Ladder Filters
• First step in the design process is to find values for Ls and Cs
based on specifications:
– Filter graphs & tables found in:
• 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
Rs
C1 C3
L2
C5
L4
Vin RL
Vo
LC Ladder Filter Design Example
Design a LPF with maximally flat passband:
f-3dB = 10MHz, fstop = 20MHz
Rs >27dB @ fstop
• Maximally flat passband  Butterworth
From: Williams and Taylor, p. 2-37
Stopband Attenuation
Normalized 
• Find minimum filter order
:
• Here standard graphs
from filter books are
used
fstop / f-3dB = 2
Rs >27dB
Minimum Filter Order
c5th order Butterworth
1
-3dB
2
-30dB
Passband Attenuation

EECS 247 Lecture 3: Filters© 2010 H.K. Page 59LC Ladder Filter Design ExampleFrom: Williams and Taylor, p. 11.3Find values for L & C from Table: Note L &C values normalized to -3dB=1Denormalization: Multiply all LNorm, CNormby: Lr= R/-3dBCr= 1/(RX-3dB) Ris the value of the source and termination resistor (choose both 1Wfor now) Then: L= LrxLNormC= CrxCNormEECS 247 Lecture 3: Filters© 2010 H.K. Page 60LC Ladder Filter Design ExampleFrom: Williams and Taylor, p. 11.3Find values for L & C from Table: Normalized values: C1Norm=C5Norm=0.618C3Norm= 2.0L2Norm= L4Norm=1.618Denormalization: Since -3dB =2x10MHzLr= R/-3dB= 15.9 nHCr= 1/(RX-3dB)= 15.9 nFR=1cC1=C5=9.836nF, C3=31.83nFcL2=L4=25.75nH

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