EE247 Lecture 3ee247/fa06/lectures/L3_f... · 2006-09-05 · EECS 247 Lecture 3: Filters © 2006 H.K. Page 1 EE247 Lecture 3 • Last lecture’s summary • Active Filters – Active

EECS 247 Lecture 3: Filters © 2006 H.K. Page 1

EE247 Lecture 3

• Last lecture’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 to component variations• Ladder type filters


Summary of Last Lecture

– Nomenclature– Filter specifications

• Quality factor• Frequency characteristics• Group delay

– Filter types• Butterworth• Chebyshev I• Chebyshev II• Elliptic• Bessel

– Group delay comparison example– RLC filters


Integrated Filters

• Implementation of RLC filters in CMOS technologies requires on-chip inductors– Integrated L<10nH with Q<10 – Combined with max. cap. 10pF

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

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

• Hence:

Need to build active filters built without inductors


Filters2nd 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

Distance from origin in s-plane:

( )2

22

2 1412

P

PP

P QQ

d

ω

ω

=

−+⎟⎟⎠

⎞⎜⎜⎝

⎛=

12

2

Complex conjugate poles:

s 1 4 12

P

PP

P

Q

j QQ

ω

> →

⎛ ⎞= − ± −⎜ ⎟⎝ ⎠

poles

ωj

σ

d

S-plane


s-Plane

poles

ωj

σ

Pω radius =

P2Q- part real Pω=

PQ21arccos

2s 1 4 12

PP

Pj Q

Qω ⎛ ⎞= − ± −⎜ ⎟

⎝ ⎠


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 <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, …

Advantage: Only one opamp used Disadvantage: Sensitive to parasitic – all pole no zeros

outV

C1

inV

R2R1

C2

G

2

2

1 1 2 2

1 1 2 1 2 2

( )1

1

1 1 1

P P P

P

PP

GH ss sQ

R C R C

Q GR C R C R C

ω ω

ω

ω

=+ +

=

= −+ +


Addition of Imaginary Axis Zeros

• Sharpen transition band• Can “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, and readily identifiable units.

2

Z2

P P P

2P

Z

s1

H( s ) Ks s

1Q

H( j ) Kω

ω

ω ω

ωωω→∞

⎛ ⎞+ ⎜ ⎟⎝ ⎠=

⎛ ⎞+ + ⎜ ⎟

⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎝ ⎠


Imaginary Zeros• Zeros substantially sharpen transition band• At the expense of reduced stop-band attenuation at

high frequencyPZ

P

P

ffQ

kHzf

32100

===

104 105 106 107-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Mag

nitu

de [

dB]

With zerosNo zeros

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

Pole-Zero Map


Moving the Zeros

PZ

P

P

ffQ

kHzf

===

2100

104 105 106 107-50

-40

-30

-20

-10

0

10

20

Frequency [Hz]

Mag

nitu

de[d

B]

Pole-Zero Map

Real AxisIm

ag A

xis

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

x105

x105


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

Vasasbsbsb

VV

asasbabsbabk

VV

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

RRk

CRRCRRk

CRa

CCRRRRa

RRb

RRRR

RR

CRb

CCRRRRb

=

=

=

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

827

2

86

20

015

112124

10213

20

12

111

111

11

1

RkRbRR

Cbak

R

CbbakR

CakkR

CakR

CaR

=

=

=

−=

=

=

=

821 and , ,,, given RCCkba iii

thatfollowsit

11

21732

8

CRQCCRRR

R

PP

P

ω

ω

=

=


Higher-Order Filters• Higher-order filters (N>2) can be built with cascade of 2nd order

biquads, e.g. Sallen-Key,or Tow-Thomas

2nd orderFilter

2nd orderFilter

2nd orderFilter

………

Nx 2nd order sections Filter with 2N order

1 2 Ν

As will be shown later:High-Q high-order filters built with cascade of 2nd order sections

Highly sensitive to component variationsGood alternative: Integrator-based ladder type filters


Integrator Based Filters

• Main building block for this category of filters integrator

• By using signal flowgraph techniques conventional 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 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 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

ZZ

VoinIinI

VoI Z Vin o× =


Signal Flowgraph (SFG)Examples

1SL

Circuit State-space description SFG

R

L

R

oV

C 1SC

Vin

Vo

oI

VoinI

inI

inI

inI Vo

Vin oI

I R 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 (k). BMFs for incoming branches have to be multiplied by k and outgoing branches divided by k

1V

a

2Vb

a+b1V 2V

a.b1V

2V3V

a b1V 2V

3Va b1V 2V k.a b/k1V 2V

3k.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

13V

3V1iV

2V aoV

1/b

1-1/b

4V

1iV2V a

oV-1 -1/b

1 1

3V


Integrator Based Filters1st Order LPF

• Conversion of simple lowpass RC filter to integrator-based type by using signal flowgraph technique

in

V 1os CV 1 R

=+

oV

RsCinV


What is an Integrator?Example: Single-Ended Opamp-RC Integrator

o in o in1 1

V V , V V dtsRC RC

= − × = − ∫

oV

C

inV

-

+R

-

Note: Practical integrator in CMOS technology has input & output both in the form of voltage and not current Consideration for SFG derivation

oVinV ∫

RCτ =


Integrator Based Filters1st Order LPF

1. Start from circuit prototype-• Name voltages & currents for all components

2. Use KCL & KVL to derive state space description- 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)

1IoV

RsCinV

2I

1V+ −

CV

+

−


Integrator Based FiltersFirst Order LPF

1IoV

1

RsCinV

2I

V V V1 in C1

V IC 2 sC

V Vo C1

I V1 1 RsI I2 1

= −

= ×

=

= ×

=

1V+ −

1Rs

1sC

2I1I

CVinV 1−1 1V

• All voltages & currents nodes of SFG

• Voltage nodes on top, corresponding current nodes below each voltage node

SFG

CV

+

−

oV1

Integrator form


Normalize• Since integrators - the main building blocks- require in & out signals in

the voltage form (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

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

'2V

*1

sCR

oVinV 1−1 1V

1

*RRs

1Rs

1sC

2I1I

oVinV 1−1

1

1V

'1V

oVinV 1−1 1V

*RRs

1I R ∗× 2I R ∗×

*1

sCR

*

*R

R


Synthesis

'1V

1*1

sC R

oVinV 1−1 1V

'2V1

*RRs

* * CR Rs , Rτ= = ×

'1V

1sτ ×

oVinV 1−1 1V

'2V1

1sτ ×

oVinV 1−1 1V

'2V

1

Consolidate two branches

*Choosing

R Rs=


First Order Integrator Based Filter

oV

inV

-+ ( )1

H ssτ

=×

∫+

1sτ ×

oVinV 1−1 1V

'2V

1


1st Order Filter Built with Opamp-RC Integrator

oV

inV

-+

∫+

• Single-ended Opamp-RC integrator has a sign inversion from input to output

Convert SFG accordingly by modifying BMF

oV

inV-

∫+

-

oV

'in inV V= −

+

∫+

-


1st Order Filter Built with Opamp-RC Integrator

• To avoid requiring an additional opamp to perform summation at the input node:

oV

'in inV V= −

+

∫+

-

oV

'inV

∫--


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

o'

in

V 11 sRCV

= −+

oV

C

in'V

-

+R

R

--

oV'

inV

∫


Opamp-RC 1st Order Filter Noise

2n1v

( )

k 22o mm0m 1

thi

2 21 2 2

2 2n1 n2

2o

v S ( f ) dfH ( f )

S ( f ) Noise spectral densi tyof m noise source1H ( f ) H ( f )

1 2 fRC

v v 4KTR f

k Tv 2 C2

π

α

∞=

=

→

= =+

= = Δ

=

=

∑ ∫

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.

• For a well-designed filter opamp is designed such that noise contribution of opamp << contribution of other noise sources

• The bandwidth of the opamp affects the opamp noise contribution to the total noise

2n2v


Integrator Based Filter2nd Order RLC Filter

oV

1

1sL

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

1sC

CIRI

CV

inI

1−1

RV 1

1−

LV

LI

CV

LI

+

- CILV

+

-

Integrator form

+

-RV

RI


1 1

*RR *

1sCR

'1V

2V

inV

1−1

1V

*RsL

1−

oV

'3V

Normalize

1

1R

1sC

CIRI

CV

inI

1−1

RV 1

1−

LV

LI

1sL

• 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

*RsL

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

nitu

de (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 1V sV s s 1

R C L R

R R11

Q 1

From matching pointof v iew desirable :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!)

k 22o mm0m 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

value:

– 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 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


• Lecture 3 ended here


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 cascaded 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 Overall Frequency Response

Bode Diagram

Pha

se (d

eg)

Mag

nitu

de (d

B)

-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 659.4963.6590 611.7441.1026 473.643

319.568

fzero [kHz]

1297.5836.6744.0

Pole-Zero Map

-2 -1.5 -1 -0.5 0-1

-0.5

0

0.5

1

Imag

Axi

s X1

07

Real Axis x107

s-Plane


CDMA FilterBuilt with Cascade of 1st and 2nd Order Sections

• 1st order filter implements the single real pole• Each biquad implements a pair of complex conjugate pole and a

pair of imaginary zero

1st orderFilter Biquad2 Biquad4 Biquad3


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

nitu

de (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 (d

B)

-80

-60

-40

-20

Mag

nitu

de (d

B)

LPF1 +Biquad 2

Mag

nitu

de (d

B)

Biquads 1, 2, 3, & 4

Mag

nitu

de (d

B)

-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 High sensitivityin Biquad realizations

Frequency [Hz]1MHz

Mag

nitu

de (d

B)

-30

-40

-20

0

200kHz

3dB

600kHz

-50

2.2dB

Documents

EE247 Lecture 3ee247/fa06/lectures/L3_f... · 2006-09-05 · EECS 247 Lecture 3: Filters © 2006 H.K. Page 1 EE247 Lecture 3 • Last lecture’s summary • Active Filters – Active