This Lecture - EECS Instructional Support Group Home Pageee247/fa06/lectures/L4_f... · 2006-09-07 · EECS 247 Lecture 4: Filters © 2006 H.K. Page 1 EE247 Lecture 4 • Last lecture

EECS 247 Lecture 4: Filters © 2006 H.K. Page 1

EE247 Lecture 4

• Last lecture– 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• Cascade of biquads & 1st order sections


This Lecture• Sensitivity of high order high-Q filters built with cascade of

biquads to component matching– Build a 7th order elliptic filter

• 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• Example shown


Higher Order FiltersCascade of Biquads

Example: LPF filter for baseband portion of a CDMA phone receiver path

• LPF with the following specifications:– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 45 dB– Assumption: Can compensate for phase distortion in the digital domain

• Choice filter type requiring lowest order 7th order Elliptic filter• Implementation with cascaded biquads

– Pair poles and zeros & implement with one biquad– Highest Q poles with closest zeros is a good starting point, but not

necessarily optimum– Ordering: Lowest Q poles first is a good start


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

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 X

107

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 axis zero

1st orderFilter Biquad2 Biquad4 Biquad3


Individual Section Magnitude 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

-1


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

-1


Magnitude Response of 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 mismatch in Biquad 4 (highest pole Q):

– 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)

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


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 mismatch • Compare with cascade of biquads

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


Ladder Type FiltersLow-Pass RLC Filter

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

Doubly terminated LC ladder filters Lowest sensitivity to component mismatch when RS=RL

RsC1 C3

L2

C5

L4

inV RL

oV


Ladder FiltersLow-Pass RLC Filter

• 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


RLC Ladder Filter Design ExampleDesign 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 Butterworth•Determine minimum filter order :

-Use of Matlab-or tables & graphs from books

•Here tables & graphs used

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

Minimum Filter Order5th order Butterworth

1

-3dB

2


RLC Ladder Filter Design Example

From: Williams and Taylor, p. 11.3

Find values for L & C from Table:Normalized values:C1Norm =C5Norm =0.618C3Norm = 2.0L2Norm = L4Norm =1.618

Note L &C values normalized to:

ω-3dB =1



Denormalization Example:Since ω-3dB =2πx10MHz

R=50 ΩLr = R/ω-3dB = 796 nHCr = 1/(RXω-3dB )= 318.3 pF

L2= Lr xL2Norm= 1.288mHL2=L4= 1.288mHC1= Cr xC1Norm=196.7pFC1=C5= 196.7pF, C3= Cr xC3Norm= 636.6pF

Denormalization Recipe:Multiply all LNorm, CNorm by:

Lr = R/ω-3dBCr = 1/(RXω-3dB )

Then: L= Lr xLNorm

C= Cr xCNorm

R is the value of the source and termination resistor (choose both 50Ω for now)


Low-Pass RLC FilterMagnitude Response Simulation

Frequency [MHz]

Mag

nitu

de (d

B)

0 10 20 30-50

-40

-30

-20

-10-50

Note:-6 dB passband attenuationdue to double termination

30dB

Rs=50 ΩC1196.7pF

L4= 1.288mΗ

inV

oV

SPICE simulation Results

L2= 1.288mΗ

RL=50 ΩC5196.7pF

C5636.6pF


Low-Pass RLC 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

= − = = −

= = − = =

= = − =

= − = = − =


Low-Pass RLC Ladder FilterSignal Flowgraph

SFG

1Rs 1

1sC

2I1I

2VinV 1−1

1

1V oV1− 11

3

1sC 5

1sC2

1sL 4

1sL

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

= − = = −

= = − = =

= = − =

= − = = − =


Low-Pass RLC Ladder FilterSignal Flowgraph

SFG

1Rs 1

1sC

2I1I

2VinV 1−1

1

1V oV1− 11

3

1sC 5

1sC2

1sL 4

1sL

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+ −


Low-Pass RLC Ladder FilterNormalize

1

1

*RRs *

1

1sC R

'1V

2VinV 1−1 1V oV1− 1

*

2

RsL

1− 1− 1−1 1

1− 13V 4V 5V 6V

'3V'2V '

4V '5V '

6V '7V

*3

1sC R

*

4

RsL

*5

1sC R

*RRL

1Rs 1

1sC

2I1I

2VinV 1−1

1

1V oV1− 11

3

1sC 5

1sC2

1sL 4

1sL

1RL

1− 1− 1−1 1

1− 13V 4V 5V 6V

3I 5I4I 6I 7I


Low-Pass RLC Ladder FilterSynthesize

1

1

1

*RRs

*1

1sC R

'1V

2VinV 1−1 1V oV1− 1

*

2

RsL

1− 1− 1−1 1

1− 13V 4V 5V 6V

'3V'2V '

4V '5V '

6V '7V

*3

1sC R

*

4

RsL

*5

1sC R

*RRL

inV

1+ -

-+ -+

+ - + -

*R Rs−

*R RL21

sτ 31

sτ 41

sτ 51

sτ11

sτ

oV2V 4V 6V

'3V '

5V


Low-Pass RLC 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= = = = = = =

Building Block:RC Integrator

V 12V sRC1

= −

inV

1+ -

-+ -+

+ - + -

*R Rs−

*R RL21

sτ 31

sτ 41

sτ 51

sτ11

sτ

oV2V 4V 6V

'3V '

5V


Negative Resistors

V1-

V2-

V2+

V1+

Vo+

Vo-

V1-

V2+ Vo+


Synthesize

oV

4V

'3V

'5V

2V


Frequency Response

oV

4V

'3V

'5V

2V

0.5

1

0.1

1


Scale Node Voltages

Scale Vo by factor “s”


Node Scaling

VO X 2

X1.8/2

V4 X 1.6

V3’ X 1.2

X 1.2

X 1.8/1.6

X 1.2/1.6 X 1.6/1.2

X 1/1.2

X 1.6/1.8

X 2/1.8

V5’ X 1.8

V2


Maximizing Signal Handling by Node Voltage Scaling

Scale Vo by factor “s”


Filter Noise

Total noise @ the output: 1.4 μV rms(noiseless opamps)

That’s excellent, but the capacitors are very large (and the resistors small high power dissipation). 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

5th order ladder filterFinal design utilizing:

-Node scaling -Final R & C scaling based on noise considerations

oV

4V

'3V

'5V

2V


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


inVDifferential 5th Order Lowpass Filter

• Since each signal and its inverse readily available, eliminates the need for negative resistors!

• Differential design has the advantage of even order harmonic distortion components and common mode spurious pickup automatically cancels

• Disadvantage: Double resistor and capacitor area!

+

+

--+

+

--

+

+--+

+--

+

+

--

inV

oV


RLC Ladder FiltersIncluding Transmission Zeros

RsC1 C3

L2

C5

L4

inV RLC7

L6

C2 C4 C6

oV

RsC1 C3

L2

C5

L4

inV RL

oVAll poles

Poles & Zeros



• Design a baseband filter for CDMA IS95 receiver with the following specs.– Filter frequency mask shown on the next page– Allow enough margin for manufacturing variations

• Assume pass-band magnitude variation of 1.8dB• Assume the -3dB frequency can vary by +-8% due to

manufacturing tolerances & circuit inaccuracies– Assume any phase impairment can be compensated in the

digital domain

* Note this is the same example as for cascade of biquad while the specifications are given closer to a real product cases


RLC Ladder Filter Design ExampleCDMA IS95 Receive Filter Frequency Mask

+10

-1

Frequency [Hz]

Mag

nitu

de (d

B)

-44

-46

600k 700k 900k 1.2M


RLC Ladder Filter DesignExample: CDMA IS95 Receive Filter

• Since phase impairment can be corrected for, use filter type with max. cut-off slope/pole

Elliptic• Design filter freq. response to fall well within the freq. mask

– Allow margin for component variations & mismatches• For the passband ripple, allow enough margin for ripple change

due to component & temperature variationsPassband ripple 0.2dB

• For stopband rejection add a few dB margin 44+5=49dB• Design to spec.:

– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 49 dB

• Use Matlab or filter tables to decide the min. order for the filter (same as cascaded biquad example)– 7th Order Elliptic


RLC Low-Pass Ladder Filter DesignExample: CDMA IS95 Receive Filter

RsC1 C3

L2

C5

L4

inV RLC7

L6

C2 C4 C6

oV

7th order Elliptic

• Use filter tables to determine LC values



• Spec.– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 49 dB

• Use filter tables to determine LC values – Table from: A. Zverev, Handbook of filter synthesis, Wiley,

1967– Elliptic filters tabulated wrt “reflection coeficient ρ”

– Since Rpass=0.2dB ρ =20%– Use table accordingly

( )2Rpass 10 log 1 ρ= − × −



• Table from Zverev book page #281 & 282:

• Since our spec. is Amin=44dB add 5dB margin & design for Amin=49dB


• Table from Zverev page #281 & 282:

• Normalized component values:

C1=1.17677C2=0.19393L2=1.19467C3=1.51134C4=1.01098L4=0.72398C5=1.27776C6=0.71211L6=0.80165C7=0.83597


-65

-55

-45

-35

-25

-15

-5

200 300 400 500 600 700 800 900 1000 1100 1200

RLC Filter Frequency Response

•Frequency mask superimposed•Frequency response well within spec.

Frequency [kHz]

Mag

nitu

de (d

B)


Passband Detail

-7.5

-7

-6.5

-6

-5.5

-5

200 300 400 500 600 700 800

• Passband well within spec.

Frequency [kHz]

Mag

nitu

de (d

B)


RLC Ladder Filter Sensitivity

• The design has the same specifications as the previous example implemented with cascaded biquads

• To compare the sensitivity of RLC ladder versus cascaded-biquads:– Changed all Ls &Cs one by one by 2% in order to

change the pole/zeros by 1% (similar test as for cascaded biquad)

– Found frequency response most sensitive to L4 variations

– Note that by varying L4 both poles & zeros are varied


RCL Ladder Filter Sensitivity

Component mismatch in RLC filter:– Increase L4 by 2%– Decrease L4 by 2%

-65

-55

-45

-35

-25

-15

-5

200 300 400 500 600 700 800 900 1000 1100 1200Frequency [kHz]

Mag

nitu

de (d

B)

L4 nomL4 lowL4 high


RCL Ladder Filter Sensitivity

-6.5

-6.3

-6.1

-5.9

-5.7

200 300 400 500 600 700

-65

-60

-55

-50

600 700 800 900 1000 1100 1200

Frequency [kHz]

Mag

nitu

de (d

B)

1.7dB

0.2dB

-65

-55

-45

-35

-25

-15

-5

200 300 400 500 600 700 800 900 1000 1100 1200


-10

Cascade of Biquads SensitivityComponent mismatch 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


Sensitivity Comparison for Cascaded-Biquads versus RLC Ladder

• 7th Order elliptic filter – 1% change in pole & zero pair

1.7dB(21%)

3dB(40%)

Stopband deviation

0.2dB(2%)

2.2dB (29%)

Passband deviation

RLC LadderCascadedBiquad

Doubly terminated LC ladder filters Significantly lower sensitivity compared to cascaded-biquads particularly within the passband



RsC1 C3

L2

C5

L4

inV RLC7

L6

C2 C4 C6

oV

7th order Elliptic

• Previously learned to design integrator based ladder filters without transmission zeros

Question: o How do we implement the transmission zeros in the integrator-

based version? o Preferred method no extra power dissipation no active

elements


Integrator Based Ladder FiltersHow Do to Implement Transmission zeros?

• Use KCL & KVL to derive :

1I 2VRs

C1 C3

L2

inV RL

4V

3I5I

2I4I

1V+ − 3V+ −

Ca

oV

( )

( )

CI I1 3 aV V2 4C Cs C C1 1a a

CI I3 5 aV V4 2C Cs C C3 a 3 a

−= + ×

+ +

−= + ×

+ +

Frequency independent constants


Integrator Based Ladder FiltersHow Do to Implement Transmission zeros?

• Use KCL & KVL to derive :

1I 2VRs

C1 C3

L2

inV RL

4V

3I5I

2I4I

1V+ − 3V+ −

Ca

oV

( )

( )

CI I1 3 aV V2 4C Cs C C1 1a a

CI I3 5 aV V4 2C Cs C C3 a 3 a

−= + ×

+ +

−= + ×

+ +

Voltage Controlled Voltage Source!


Integrator Based Ladder FiltersTransmission zeros

1I 2V

( )

( )

CI I1 3 aV V2 4C Cs C C1 1a aCI I3 5 aV V4 2C Cs C C3 a 3 a

−= +

+ +

−= +

+ +

Rs L2

inV RL

4V

3I5I

2I 4I

1V+ − 3V+ −

Ca

• Replace shunt capacitors with voltage controlled voltage sources:

+-

( )C C1 a+ ( )C C3 a+

CaV4 C C1 a+CaV2 C C3 a+

+-


LC Ladder FiltersTransmission zeros

1I 2VRs L2

inVRL

4V

3I5I

2I 4I

1V+ − 3V+ −

( )C C1 a+ ( )C C3 a+

CaV4 C C1 a+CaV2 C C3 a+

1Rs ( )1 aC C

1s +

2I1I

2VinV 1−1

1

1V oV1− 11

( )3 a

1s C C+2

1sL

1RL

1− 1−1

3V 4V

3I 4I

oV

CaC C1 a+

CaC C3 a+

+- +-


Integrator Based Ladder FiltersHigher Order Transmission zeros

C1

2V 4V

C3

Ca6VCb

2V 4V

+- +-

( )C C1 a+ ( )C C C3 a b+ +

CaV4 C C1 a+

CaV2 C C3 a+

6V

+-

( )C C5 b+

CbV4 C C3 b++-CbV6 C C3 b+

C5Convert zero generating Cs in C loops to voltage-controlled voltage sources


Higher Order Transmission zeros

*RRs ( )1 aC C

1*s R+

2VinV 1−1

1

1V oV1− 11

( )3 a b

1*s C C CR + +

*

2

RsL

*RRL

1− 1−1

3V 4V

CaC C1 a+

CaC C3 a+

1−*

4

RsL

1

5V 6V

RL1I

2VRs L2

inV

4V

3I5I

2I

4I1V+ − 3V+ −

+-+-

( )C C1 a+ ( )C C C3 a b+ +

CaV4 C C1 a+

CaV2 C C3 a+

oVL4

6V 7I

6I

5V+ −

+-

( )C C5 b+

CbV4 C C5 b++- CbV6 C C3 b+

( )5 bC C

1*sR +

CbC C3 b+ Cb

C C5 b+

1

1−'1V '

3V'2V '4V

'5V '

6V '7V


Example:5th Order Chebyshev II Filter

• 5th order Chebyshev II

• Table from: Williams & Taylor book, p. 11.112

• 50dB stopband attenuation• f-3dB =10MHz


Realization with Integrator

( )i 1 a2

1 3*s a 1a 1

V V CV1V VR C Cs C C R−⎡ ⎤= − +⎢ ⎥ ++ ⎣ ⎦

-Rs

R*

Rs


5th Order Butterworth Filter

From:Lecture 4page 26

oV

4V

'3V

'5V

2V


Opamp-RC Simulation

oV

4V

'3V

'5V

2V


Seventh Order Differential Low-Pass Filter Including Transmission Zeros

+

+

--

+

+

--

+

+--

+

+

--

+

+--

+

+

--

+

+--

inV

oV

Transmission zeros implemented with coupling capacitors

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This Lecture - EECS Instructional Support Group Home Pageee247/fa06/lectures/L4_f... · 2006-09-07 · EECS 247 Lecture 4: Filters © 2006 H.K. Page 1 EE247 Lecture 4 • Last lecture