Lecture Notes Session 3

EECS 247 Lecture 3: Filters © 2005 H.K.

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


Monolithic Filters

• Monolithic inductor in CMOS tech. – Integrated L<10nH with Q<10 combined with max. cap.

10pF

àLC filters in the monolithic form feasible: freq>500MHz

• Analog/Digital interface circuitry require fully integrated filters with critical frequencies << 500MHz

• Good alternative:

cActive filters built without the need for inductors


2nd Order Transfer Functions (Biquads)

• Biquadratic (2nd order) transfer function:

PQjH

jH

jH

P=

=

=

=

∞→

=

ωω

ω

ω

ω

ω

ω

)(

0)(

1)(0

2

2P P P

P 2P

P

1P 2

1H( s )

s s1

Q

Biquad poles @: s 1 1 4Q2Q

for Q poles are real , complex otherwise

ω ω

ω

=

+ +

= − ± −

≤


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

=

−+

=


s-Plane

poles

ωj

σ

Pω radius =

P2Q- part real Pω

=

PQ21

arccos


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

<500MHz

• 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, …

à Advantage: Only one opamp used à Disadvantage: Sensitive to parasitics

221211

2211

2

2

111

1

1)(

CRG

CRCR

Q

CRCR

sQsG

sH

PP

P

PPP

−++=

=

++=

ω

ω

ωω

outV

C1

inV

R2R1

C2

G


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→∞

+

=

+ +

=


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


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]


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

( ) ( )

( ) ( )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


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

=

=


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


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.


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

× =

× =

× =


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 2V x.a b/x1V 2V

3x.V


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


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

+

−


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


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

=

=

= −

=


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


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


First Order Integrator Based Filter

oV

inV

-+ ( )

1H s

sτ=

×

∫+

1

sτ ×

oVinV 1−1 1V

'2V

1


Opamp-RC Single-Ended Integrator

oo in

in

1 V 1V V dt ,

RC V sRC= − = −∫

oV

C

inV

-

+R

RCτ =

- ∫


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

∫--


1st Order Filter Built with Opamp-RC Integrator (continued)

o'

in

V 1

V 1 sRC= −

+

oV

C

in'V

-

+R

R

--

oV'

inV

∫


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


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


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


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


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= × =


-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

-- LPV

HPV

∑

Filter Magnitude Response


Second Order Integrator Based Filter

inV

1*R R 21

sτ11

sτ

BPV

-- LPV

HPV

∑

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

=+ +

=+ +

=+ +

= × =

=

= =

= ×

= → =


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

∑


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


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=


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


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


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


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


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


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


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


-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


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 <4 (e.g. filters for disk drives)

• LC ladder filters more appropriate for high Q & high order filters (next topic)– Less sensitive to component variations


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 incorporatingzeros– Implement the same 7th order elliptic filter in the form of

ladder type• Find level of sensitivity to component variations • Compare with cascade of biquads

– Convert to integrator based form utilizing SFG techniques– Example shown


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


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


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



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-3dB

Cr = 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



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 nH

Cr = 1/(RXω-3dB )= 15.9 nF

R =1

cC1=C5=9.836nF, C3=31.83nF

cL2=L4=25.75nH


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


LC 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

= − = = −

= = − = =

= = − =

= − = = − =


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

= − = = −

= = − = =

= = − =

= − = = − =


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

2I4I 6I

7I1V+ − 3V+ − 5V+ −


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


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


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

= −


Negative Resistors

V1-

V2-

V2+

V1+

Vo+

Vo-

V1-

V2+Vo+


Synthesize


Frequency Response


Scale Node Voltages

Scale Vo by factor “s”


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 …


Scale to Meet Noise TargetScale capacitors and resistors to meet noise objective

s = 10-4

Noise: 141 µV rms (noiseless opamps)


Completed Design


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

Lecture Notes Session 3

Documents

Transcript of Lecture Notes Session 3