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EECS 247 Lecture 3: Filters © 2008 H.K.

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EECS 247 Lecture 3: Filters © 2008 H.K. Page 2

EE247 Lecture 3

• 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 & first order filters

– Cascaded biquad sensitivity to component mismatch• Ladder type filters


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

2 22

2P PP

P 2P

P

1P 2

1H( s )

s s1

Q

1H( j )

1Q

Biquad poles @: s 1 1 4Q2Q

Note: for Q poles are real , complex otherwise

ω ω

ωω ωω ω

ω

=

+ +

=⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞= − ± −⎜ ⎟⎝ ⎠

≤


s-Plane12

2

Complex conjugate poles:

s 1 4 12

P

PP

P

Q

j QQ

ω

> →

⎛ ⎞= − ± −⎜ ⎟⎝ ⎠

P2Q- part real Pω=

PQ21arccos

poles

ωj

σ

Pω radius =


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

<a few 100s of MHz

• 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 to obtain 2polesDisadvantage: 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

02

P

Z

s1

H( s ) Ks s

1Q

H( j ) K

H( j ) K

ω

ω

ω

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

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

filterrejectBandffQ

kHzf

PZ

P

P

−→===

2100


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

=

=

=

−=

=

=

=

thatfollowsit

11

21732

8

CRQCCRRR

R

PP

P

ω

ω

=

=

zerospolesdesiredthefromexampleforRCCkba iii / and , ,,,given 821


Higher-Order Filters in the Integrated Form

• One way of building higher-order filters (n>2) is via cascade of 2nd

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

2nd orderFilter ………

Nx 2nd order sections Filter order: n=2N

1 2 Ν

Cascade of 2nd order biquads:☺ Easy to implement

Highly sensitive to component mismatch -good for low Q filters only

Good alternative: Integrator-based ladder type filters

2nd orderFilter

2nd orderFilter


Integrator Based Filters• Main building block for this category of filters

Integrator• By using signal flowgraph techniques

Conventional RLC filter topologies can be converted to integrator based type filters

• Next few pages:– Introduction to 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 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

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

× =

× =

× =


Useful Signal Flowgraph (SFG) Rules

1Va

2Vb

a+b1V

2V

a.b1V

2V3Va b1V 2V

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 incomingbranches have to be multiplied by k and outgoing branches divided by k

3V

a b1V 2V k.a b/k

1V 2V

3.Vk

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)



hiV2V a

oV

b

g-b

3VhiV

2V a/(1+b)oV

b

g

3V

3VciV

2V aoV

b

d-bciV2V a

oV-1 -b

1 d

3V

• 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 BMFof incoming branches by 1/(1-y)

V4


Integrator Based Filters1st Order LPF

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

in

V 1os CV 1 R

=+

oV

RsCinV


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

∫inV

o in o in1 1V V , V V dt

sRC 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

oV

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 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)

1IoV

RsCinV

2I

1V+ −

CV

+

−


Integrator Based FiltersFirst Order LPF

1IoV

1

RsCinV

2I

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 formV V V1 in C

1V IC 2 sCV Vo C

1I V1 1 RsI I2 1

= −

= ×

=

= ×

=


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

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

=

=

= −

=


1st Order Lowpass Filter SGFNormalize

'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


1st Order Lowpass Filter SGFSynthesis

'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 ty of 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 is negligible compared to 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

2nd Order RLC Filter SGFNormalize

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

2nd Order RLC Filter SGFSynthesis

∑

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 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 biquads


Higher Order FiltersCascade of Biquads

Example: LPF filter for CDMA cell phone 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

• 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


Overall Filter 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 (pzmap in Matlab)

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 poles and a

pair of imaginary axis zeros

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


Individual Biquad Magnitude Response

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 to Relative Component MismatchComponent variation in Biquad 4 relative to the rest

(highest Q poles):

– 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


High 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

• For simplicity, will start with all pole (no finite zero) ladder type filters

– Start with LC ladder type and find values for Ls & Cs– Convert to integrator based form – example shown

• Next 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


LC Ladder Filters

• Made of resistors, inductors, and capacitors• Doubly terminated or singly terminated (with or w/o RL)

RsC1 C3

L2

C5

L4

inV RL

oV

Doubly terminated LC ladder filters Lowest sensitivity to component mismatch


LC Ladder Filters

• First step in the design process find values for Ls & Cs– 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


LC Ladder Filter Design Example

Design a LPF with maximally flat passband:f-3dB = 10MHz, fstop = 20MHzRs >27dB @ fstop

• Maximally flat passband Butterworth

From: Williams and Taylor, p. 2-37

Stopband A

ttenuation

Νοrmalized ω

• Find minimum filter order :

• Here standard graphs from filter books are used

fstop / f-3dB = 2Rs >27dB

Minimum Filter Order5th order Butterworth

1

-3dB

2

-30dBP

assb

and

Atte

nuat

ion



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



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

R =1 Lr = R/ω-3dB = 15.9 nHCr = 1/(RXω-3dB )= 15.9 nF

C1=C5=9.836nF, C3=31.83nF

L2=L4=25.75nH


Magnitude Response Simulation

Frequency [MHz]

Mag

nitu

de (d

B)

0 10 20 30-50

-40

-30

-20

-10-50

-6 dB passband attenuationdue to double termination

30dB

Rs=1OhmC19.836nF

C331.83nF

L2=25.75nH

C59.836nF

L4=25.75nH

inV RL=1Ohm

oV

SPICE simulation Results

ee247 administrative - eecs instructional support group...

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