ee 508 lecture 4 - iowa state universityclass.ece.iastate.edu/ee508/lectures/ee 508 lect 19 fall...

EE 508

Lecture 19

Sensitivity Functions • Comparison of Filter Structures

• Performance Prediction

• Design Characterization

Basic Biquadratic Active Filters

Filter Design/Synthesis Considerations Most odd-ordered designs today use one of the following three basic architectures

T1(s)

Biquad

T2(s)

Biquad

Tk(s)

Biquad

VOUTVIN Tm(s)

Biquad

1 2 mT s T T T

Tm+1(s)First

Order

I1(s)

Integrator

I2(s)

Integrator

I3(s)

Integrator

I4(s)

Integrator

Ik(s)

Integrator

VIN

VOUTIk-1(s)

Integrator

a2a1

T1(s)

Biquad

T2(s)

Biquad

Tm(s)

Biquad

αF

XOUTXIN

α1α2 αm

α0 Tm+1(s)

First

Order

αm+1

Cascaded Biquads

Leapfrog

Multiple-loop Feedback (less popular)

What’s unique in all of these approaches?

Review from last time

Filter Design/Synthesis Considerations

T(s)

Biquad

XOUTXIN

α1

α2αk

α0

What’s unique in all of these approaches?

I(s)

Integrator

• Most effort on synthesis can focus on synthesizing these four blocks

(the summing function is often incorporated into the Biquad or Integrator)

• Some issues associated with their interconnections

• And, in many integrated structures, the biquads are made with integrators

(thus, much filter design work simply focuses on the design of integrators)

2

2 1 0

2

1 0

a s +a s+aT s =

s +b s+b

Tm+1(s)First

Order

0II s = s

1 0

0

a s+aT s =

s+b

(the first-order block is much less challenging to design than the biquad)

k

OUT i

i=0

X = α


Filter Design/Synthesis Considerations

There are many different filter architectures that can realize a given transfer

function

Will first consider second-order Bandpass filter structures

XIN XOUT T s

H

0

2 200

ωs

QT s

ωs + s + ω

Q

0B A

ωBW = ω -ω =

Q

PEAK 0ω = ω


Example 1:

OUT

2IN

V 1 sT s

1 1V RCs +s +

RC LC

R

LCVIN

VOUT

0

1ω

LC

CQ = R

L

1BW =

RC

Can realize an arbitrary 2nd order bandpass function within a gain factor

Simple design process (sequential but not independent control of ω0 and Q)


Example 2:

C

C

RR

VIN

VOUTK

R

OUT

2IN2

V K sT s

4-K 2V RCs +s +

RC RC

3 degrees of freedom (effectively 2 since dimensionless)

Equal R, Equal C Realization

0

2ω

RC 2

Q = 4-K

Can satisfy arbitrary 2nd=order BP constraints within a gain factor with this circuit

Very simple circuit structure

4-KBW =

RC

Can actually move poles in RHP by making K >4

Independent control of ω0 and Q but requires tuning more than one component


Example 3:

3 degrees of freedom (2 effective since dimensionless)

C

CR

RVIN

VOUT-K

OUT

IN 2

2

V K sT s

V 1+K RC 3 1 1s +s +

RC 1+K 1+K RC

Equal R, Equal C Realization

0

1ω

RC 1+K 1+K

Q = 3

Can satisfy arbitrary 2nd=order BP constraints within a gain factor with this circuit

Very simple circuit structure

3

BW = RC 1+K

Simple design process (sequential but not independent control of ω0 and Q,

requires tuning of more than 1 component if Rs used)


Example 4:

VOUT

VIN

R1

R2

CC

VOUT

VIN

R1

R2

C1C2

K

VOUT

VIN

R1 R2

C1

C2

VOUTVIN

R1 R2

C1

C2

K

Some variants of the bridged-T feedback structure


Example 5:

OUT

IN 0 2 2

Q 2 1 2 1 2

V 1 sT s

V R C 1 1s +s +

R C R R C C

Second-order Bandpass Filter


Denote as a two-integrator-loop structure

R0

R1RQ

RA

RAR2

C1 C2

VOUT

VIN

Example 5:

2

OUT

IN 2

Q

V 1 sT s

V RC R 1 1s +s +

R RC RC


R

RRQ

RA

RARC C

VOUT

VINEqual R Equal C

(except RQ)

0 ?ω Q = ? BW = ?

Example 5:

2

OUT

IN 2

Q

V 1 sT s

V RC R 1 1s +s +

R RC RC


R

RRQ

RA

RARC C

VOUT

VINEqual R Equal C

(except RQ)

0

1ω

RC QR

Q = R Q

R 1BW =

R RC

Simple design process (sequential but not independent control of ω0 and Q with Rs,

requires more tuning more than one R if Cs fixed )

Modest component spread even for large Q

Example 5:

R0

R1RQ

R3

R3R2

C1 C2

VOUT

VIN

INT1 INT2

R0

R1RQ

RA

RAR2

C1 C2

VOUT

VIN

Two Integrator Loop Representation

Example 5:

R0

R1RQ

R3

R3R2

C1 C2

VOUT

VIN

INT1 INT2

Two Integrator Loop Representation

0I

s

0I

sXIN

XOUT

α

0I

s+

0I

sXIN

XOUT

0I

s+

0I

s

XIN

XOUT

Integrator and Lossy Integrator Loop

Inverting and Noninverting Integrator Loop

Integrator and Lossy Integrator Loop

How does the performance of these bandpass filters compare?

R

LCVIN

VOUT

R

RRQ

RA

RARC C

VOUT

VIN

VOUT

VIN

R1

R2

CC

C

CR

RVIN

VOUT-K

C

C

RR

VIN

VOUTK

R

Ideally, all give same performance (within a gain factor)

How does the performance of these bandpass filters compare?

R

LCVIN

VOUT

R

RRQ

RA

RARC C

VOUT

VIN

VOUT

VIN

R1

R2

CC

C

CR

RVIN

VOUT-K

C

C

RR

VIN

VOUTK

R

• Component Spread

• Number of Op Amps

• Is the performance strongly dependent upon how DOF are used?

• Ease of tunability/calibration (but practical structures often are not calibrated)

• Total capacitance or total resistance

• Power Dissipation

• Sensitivity

• Effects of Op Amps

C

C

RR

VIN

VOUTK

R

0

2ω

RC 2

Q = 4-K

Consider effects of Op Amp on +KRC Bandpass with Equal R, Equal C

RX (1+K0)RX

Amplifier with gain H

VINVOUT

Assume K realized with standard Op Amp Circuit

0

0

KK s

K1+ s

GB

• Significant shift in peak frequency

• BW does not change very much

• Some drop in gain at peak frequency

Practically, GB/ω0 must be must less than 100

0

+ -

V GBA s =

V -V s

OV+V

-V

Consider 2nd Order Lowpass Biquads

H20

2 200

ωT s

ωs + s + ω

Q

0B A

ωBW = ω -ω

Q

PEAK 0ω ω

XIN XOUT T s

Consider 2nd Order Lowpass Biquads

C1

C2

R2R1

VIN

VOUTK

Example: 2nd Order +KRC Lowpass

C

C

RR

VIN

VOUTK

0

1ω

RC 1

Q3-K

Equal R, Equal C

1 2 1 2

2

1 1 2 1 2 2 1 2 1 2

1

R R C CT s =K

1 1 1-K 1s +s + + +

R C R C R C R R C C

2 2

2

2 2

1

R CT s =K3-K 1

s +s +RC R C

C1

C2

R2R1

VIN

VOUTK


C1

C2

RR

VIN

VOUTK=1

0

1 2

1ω

R C C 1

2

1

2

CQ

C

Equal R, K=1

RX (1+K0)RX

Amplifier with gain H

VINVOUT

0

0

KK s

K1+ s

GB

0

+ -

V GBA s =

V -V s

OV+V

-V

2

1 2

2

2

1 1 2

1

R C CT s =K

2 1s +s +

RC R C C


C1

C2

RR

VIN

VOUTK=1

C

C

RR

VIN

VOUTK

Equal R, K=1

Equal R, Equal C

3%p

6%p

n

0

GBGB = = .01

ω

consider

C1 C2

R2R1

VIN

VOUT-K

R3

R4

C C

RR

VIN

VOUT-K

R

R

0

5+Kω =

RC

5+KQ =

5

Example: 2nd Order -KRC Lowpass

Equal R, Equal C

1 2 1 2

1 3 1 4 2 3 2 12 1 2

1 1 3 4 2 2 2 1 1 2 1 2

1

R R C CT s = -K

1+ R R 1+K + R R 1+ R R + R RR C1 1 1s +s 1+ + + 1+ +

R C R R C R C C R R C C

2 2

2

2 2

1

R CT s = -K5 5+K

s +s +RC R C

C C

RR

VIN

VOUT-K

R

R

0

5+Kω =

RC

5+KQ =

5

Example: 2nd Order -KRC Lowpass

n

0

GBGB = = .01

ω

consider

70%p

Even very large values of GB will cause instability

Poles “move” towards RHP as GB degrades

VIN VOUT

RB

RA

-KVOUTVIN

2

1

RK = -

R

0

+ -

V GBA s =

V -V s

OV+V

-V

0

0

KK s = -

1+K s1+

GB

VOUT

VIN

R3 R1 R2

C2

C1

VOUT

VIN

R R R

C2

C1

0

1 2

1ω

R C C 1

2

1

2

CQ

C

Example: 2nd Bridged-T FB Lowpass

Equal R

2 3 1 2

2

1 1 2 3 1 2 1 2

1

R R C CT s = -

1 1 1 1 1s +s + + +

C R R R R R C C

2

1 2

2

2

1 1 2

1

R C CT s = -

3 1s +s +

RC R C C

VOUT

VIN

R R R

C2

C1

0

1 2

1ω

R C C 1

2

1

2

CQ

C

Example: 2nd Bridged-T FB Lowpass

n

0

GBGB = = .01

ω

consider

8.5%p

0

+ -

V GBA s =

V -V s

OV+V

-V

R0

R1 RQ

RA2

RA1R2

C1 C2

VOUT

VIN

R

R RQ

RA

RARC C

VOUT

VIN

0

1ω

RC QR

QR

Example: 2nd Two-Integrator-Loop Lowpass

Equal R, Equal C

(except RQ)

0 2 1 2

2 A2 A1

2 Q 1 2 1 2

1

R R C CT s = -

R /R1s +s +

C R R R C C

2 2

2

2 2

Q

1

R CT s = - 1 1

s +s +CR R C

R

R RQ

RA

RARC C

VOUT

VIN

0

1ω

RC QR

QR

Example: 2nd Two-Integrator-Loop Lowpass

n

0

GBGB = = .01

ω

consider

1%p

Poles “move” towards RHP as GB degrades

0

+ -

V GBA s =

V -V s

OV+V

-V

VOUT

VIN

R R R

C2

C1

C C

RR

VIN

VOUT-K

R

R

C

C

RR

VIN

VOUTK

R

R RQ

RA

RARC C

VOUT

VIN

n

0

GBGB = = .01

ω

consider

8.5%p

1%p

3%p

70%p

Comparison of 4 second-order LP filters

Some Observations

• Seemingly similar structures have dramatically different sensitivity to frequency response of the Op Amp

• Critical to have enough GB if filter is to perform as desired

• Performance strongly affected by both magnitude and direction of pole movement

• Some structures appear to be totally impractical – at least for larger Q

• Different use of the Degrees of Freedom produces significantly different results

Sensitivity analysis is useful for analytical characterization of

the performance of a filter

ee 508 lecture 4 - iowa state universityclass.ece.iastate.edu/ee508/lectures/ee 508 lect 19 fall...

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