This Lectureee247/fa05/lectures/L4_f05.pdfEECS 247 Lecture 4: Filters © 2005 H.K. Page 19...

EECS 247 Lecture 4: Filters © 2005 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• Cascaded biquads

– Sensitivity of cascaded biquad


This Lecture

• 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

RLC with zeros– Find level of sensitivity to component variations – Compare with cascade of biquads

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

• Effect of Integrator Non-Idealities on Filter Performance


LC Ladder 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 variations when RS=RL

RsC1 C3

L2

C5

L4

inV RL

oV


LC 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


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


Low-Pass RLC FilterMagnitude 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


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

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

= − = = −

= = − = =

= = − =

= − = = − =


Low-Pass RLC 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+ −


Low-Pass RLC 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


Low-Pass RLC Ladder FilterSynthesize

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

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


Scale Node Voltages

Scale Vo by factor “s”

oV

4V

'3V

'5V

2V


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 à 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 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


RLC Ladder Filter Design Example

• 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

– 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 industry case


RLC Ladder Filter Design ExampleCDMA IS95 Receive Filter Frequency Mask

+10

-1

Frequency [Hz]

Mag

nit

ud

e (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

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

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

due to component & temperature variationsà Passband ripple 0.2dB

• For stopband rejection add a few dB margin 44+4=48dB• Design to spec.:

– fpass = 650 kHz Rpass = 0.2 dB– fstop = 750 kHz Rstop = 48 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=0.2dBà 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

nit

ud

e (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

nit

ud

e (d

B)


RLC Ladder Filter Sensitivity

• The design has the same spec.s 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 à most sensitive to L4 variations

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


RCL Ladder Filter Sensitivity

Component variation 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 1200

Frequency [kHz]

Mag

nit

ud

e (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 (

dB)

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 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


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 designed integrator based ladder filters without transmission zerosàQuestion:

How do we implement the transmission zeros in the integrator-based version?


Integrator Based Ladder FiltersHow Do We Implement Transmission zeros?

• Use KCL & KVL to derive :

1I 2V

RsC1 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 capacitor with voltage controlled voltage sources:

+-

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

CaV4 C C1 a+CaV2 C C3 a+

+-


LC Ladder FiltersTransmission zeros

1I 2V

Rs 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

1

s +

2I1I

2VinV 1−1

1

1V oV1− 11

( )3 a

1

s C C+2

1

sL1

RL

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

R

sL

*RRL

1− 1−1

3V 4V

CaC C1 a+

CaC C3 a+

1−*

4

R

sL

1

5V 6V

RL1I

2V

Rs 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− = − + ++


5th Order Butterworth Filter

From:Lecture 4page 22

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


Effect of Integrator Non-Idealities on Filter Performance

Ideal Intg.

opamp DC gain

oH( s )s

1/ RCo

ω

ω

=−

=

=

∞

oV

C

inV

-

+R


Ideal Integrator Quality Factor

o oH( s )s j

ω ω

ω

− −= =

( ) ( ) ( )( )( )

1H jR jX

XQ

R

ωω ω

ωω

=+

=

Ideal Intg.

Since Q is defined as:

Then: int g.Qideal =∞


Effect of Integrator Non-Idealities on Filter PerformanceIdeal Intg. Real Intg.

a

( )( )( )o

os sa

p2 p3

aH( s ) H( s )

1 11 ss . . .ω

ω− −= ≈

+ ++


Effect of Integrator Finite DC Gain on Q

-90

-89.5

ωoω

o( i n r a d i a n )

P1

P1 o

Arctan2 o

Phase lead @ω

π

ω

ω− +

→

∠

Example: P1/ ω0 =1/100

0P1 aω=

0ω

a

-90o

( )log H s

ψ


Effect of Integrator Finite DC Gain on Q

• Phase lead @ ω0

àDroop in the passband

Normalized Frequency

Mag

nitu

de (

dB)

1

Droop in the passband


Effect of Integrator Non-Dominant Poles

-90

-90.5

ωoω

oi

oi

pi 2

opi 2( in rad ian )

Arctan2

Phase lag @

ω

ω

π

ω

∞

=

∞

=

− −

→

∑

∑

∠

Example: ω0 /P2 =1/100

0ω

-90o

( )log H s

ψ

P2P3


Effect of Integrator Non-Dominant Poles

Normalized Frequency

Mag

nitu

de (

dB)

1

• Phase lag @ ω0

àPeaking in the passbandIn extreme cases could result in oscillation!

Peaking in the passband


Effect of Integrator Non-Dominant Poles & Finite DC Gain on Q

-90

ωoω

P1Arctan2 o

oArctan pii 2

πω

ω

∠ − +

∞− ∑

=

0ω

a

-90o

( )log H s

ψ

P2P30P1 a

ω=

-90o

( )log H s

ψ

Note that the two terms have different signs à Can cancel each other’s effect


Integrator Quality Factor

( )( )( )os sa

p2 p3

aH( s )

1 11 s . . .ω

−≈

+ ++Real Intg.

o 1 & a 1p2,3,. . . . .

int g. 1Qreal1 1

oa pii 2

ω

ω

<< >>

≈ ∞− ∑

=

Based on the definition of Q and assuming that:

It can be shown that in the vicinity of unity-gain-frequency:

Phase lead @ ω0 Phase lag @ ω0


Example:Effect of Integrator Finite Q on Bandpass Filter Behavior

Integrator DC gain=100 Integrator P2 @ 100.ωo

IdealIdeal


Example:Effect of Integrator Finite Q on Filter Behavior

Integrator DC gain=100 & P2 @ 100. ωο

Ideal


SummaryEffect of Integrator Non-Idealities on Q

• Amplifier DC gain reduces the overall Q in the same manner as series/parallel resistance associated with passive elements

• Amplifier poles located above integrator unity-gain frequency enhance the Q! – If non-dominant poles close to unity-gain freq. à Oscillation

• Depending on the location of unity-gain-frequency, the two terms can cancel each other out!

i1 1o pi 2

int g.ideal

int g. 1real

Q

Q

a ω∞

=

=

≈−

∞

∑

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