High Order OA-RC Filters Design x ELE3

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Prof. D. ManstrettaAA2012-13

LEZIONI DI

FILTRI ANALOGICIDanilo Manstretta

AA 2012-13



Prof. D. ManstrettaAA2012-13 2

High Order OA-RC Filters

The goal of this lecture is to learn how to design high order

OA-RC filters using real OTAs.

1 2

1 2 1 0

1 2

1 2 1 0

...( )

...

n

n

n n

n

a s a s a s a H s

s b s b s b s b




Ideal Operational Amplifier

3

In an ideal op-amp we assume:

input resistance R i approaches infinity, thus i 1 = 0

output resistance R o approaches zero

amplifier gain A approaches infinity

R i

R o

A(e+

-e-)

e+

e-

e2

i 1

equivalent circuitsymbol

e+

e-

e2




Operational Transconductance Amplifier

4

In a real Operational Transconductance Amplifier (OTA) we assume:

input resistance R i approaches infinity, thus i 1 = 0

Finite DC gain (A0=GmR o )

Finite gain-bandwidth product (Gm/C0)

Gm has a finite value

R oGm(e+-e-)

e+

e-

e2

i 1

C o

equivalent circuitsymbol

e+

e-

e2




Cacaded Biquads Implementation

H ( s) ( s s z ,1)( s s z ,1

*)

( s s p,1)( s s

p ,1

* )( s s z ,2)( s s z ,2

*)

( s s p ,2

)( s s p ,2

* )( s s z ,3)( s s z ,3

*)

( s s p ,3

)( s s p ,3

* )

s or 1/s

B1 B2 B3

biquads




SINGLE-OTA BIQUADSSallen-Key, Rauch




Low-Pass Rauch Filter

Q C 1

C 2

1

R2 R3

R1

R3

R2

R2

R3

0 1

R2 R3C 1C 2

G R2

R1

() = −2 11 + 2 2 + 3 +

231 + 21223




Rauch Low-Pass with Real OTA

8

• In order to simplify the equations,

consider a specific implementation:

• C1=4C2=4C

• R1=R2=2R3=2R

• What is the effect of the real OTA

on the transfer function?

() =−1

1 + 4 + 2822

G = -1Q=1/√20 =1

2 2

Ideal transfer function:




Rauch Low-Pass with Real OTA (ii)

9




Rauch Low-Pass with Real OTA (iii)

10

Zero is located in the right half-plane (RHP) zero

close to gm /C

Complex poles frequency is shifted and Q is

modified: limited gm and GBW lead to poles Q

enhancement, finite gain leads to Q degradation

Additional (real) pole: around the OTA unity gain

frequency (- gm/C0)




Sallen-Key Biquad (ii)

0

1

R1 R

2C

1C

2

Q

1

R1 R

2C

1C

2

1

R1C 1

1

R2C 11

R2C 2

G () = 121211212

+ 111

+ 121

+1 − 22

+ 2

02+s0/Q+s2

C1

C2

VIN R1 R2

Rb

Ra




Sallen-Key Design Strategies

12

Design 1. R1=R2=R, C1=C2=C

0=1/(RC)

G=3-1/Q

Note: Q is independent of R,C

Design 2. G=2 (Ra=Rb), C1=C2=C

02=1/(R1R2C

2)

R1=Q/0C

Note: R 1 /R 2 =Q2

Design 3. G=1, R1=R

2=R

02=1/(R2C1C2)

C1=2Q/0C ; C2=1/2Q0C

Note: C 1 /C 2 =4Q2

0

1

R1 R

2C

1C

2

Q

1

R1 R

2C

1C

2

1

R1C 1

1

R2C 1

1

R2C 2

G

Five design elements, two

main properties (G is less

important): several design

degrees of freedom




Sallen-Key: finite op-amp gain

• The inverting and non-inverting terminals are not virtually shorted

C1

C2

R1 R2

Rb Ra

E1 E2

E3 E4

E2=A0(E4-V-)

E 4 E

2

Ra R

b

Ra E

2

A0

=0

1 +0

0

0 = 1 +

4 2

0 0

1 1 E E

A

2 4 E E

1 /




Sallen-Key with OTA

• In order to simplify the equations, consider a specificimplementation:

• C1=2C2=2C

• R1=R2=R

• =1

2C

C

R RE1 E2

E3 E4

0 =1

2

G=1Q=1/√2

() =1

1 + 2 + 2222




Sallen-Key with real op-amp (OTA)

1 =2

2 =2

() = 1 + 1 + 22

0 + 1 + 22 + 33

2 zeros and 3 poles

0 = 1 +1

0

1 = 40 − 2 +2 + 0

2 = 2022 + 4 + 20

3 = 220

2




Sallen-Key with real op-amp (OTA) (ii)

• The complex zeros have Q=(gmR) ½/2 >>1

• zeros are close to being pure imaginary: notch in H(s) at the zero

frequency!

• The frequency of the zeros is

• Assuming the OTA DC gain gmR0>>1 and isolating the terms

corresponding to the ideal transfer function from the parasitic

pole, the denominator can be approximated as:

• The complex poles change in frequency and Q

• A parasitic real pole has appeared around gm/C0

2

1 + 2 +

1

+ 22

2

2

1 +

2

1 +

0

1




TWO-OTA BIQUADSFleisher-Tow, KHN, Tow-Thomas

17

P f D M t ttAA2012 13




Fleischer-Tow Biquad

2 1

5 1 1 5 4 3 6 1 22

21

1 1 2 3 1 2

1

1

R R s R s

R R C R R R R C C E

s E s

R C R R C C

2

6

RG

R

0

2 3 1 2

1

R R C C

1 1

22 3

R C Q C R R

P f D M t ttAA2012 13 19




Kerwin-Huelsman-Newcomb (KHN)

19

21215

60

1

C C R R R

R

22

11

5

6

65

43

3

5

C R

C R

R

R

R R

R R

R

RQ

43

65

6

4

R R

R R

R

RG

C

E2

R5

C

R3E1

R4

R6

R1 R2

2

02

2 2010

G E

E s s

Q

P f D M t ttAA2012 13 20




Two-Integrators Biquad Design

S -1/s

b 1

-1/s S

-b0

a0 E 1 E 2

02

2

1 1 0

a E

E s b s b

C

RE2

Rb0

C

Ra0E1

R

Rb1

R

01

P

bQ

2

0 0 0a b

P f D M t ttAA2012 13 21




Tow-Thomas

C

E2

C

R1

R4

E1

R3

R

R

R2

2

02

2 2010

G E

E s s

Q

2

2 42

212

1 2 3

1

1

R R C E

s E s

R C R R C

3

4

RG

R

0

2 3

1

R R C 1

2 3

RQ

R R

Prof D ManstrettaAA2012 13 22




Filter Design Using Real OPAMP

22

What happens if we replace the ideal OPAMP with a real

circuit?

Let’s consider the impact of circuit limitations on the active

integrator , the basic building block of high order filters

designs.





OTA-RC Integrator

23

Ideal integrator:VIN VO

C

R

gm(V--V+) RO CO

• What happens if the ideal OA is replaced with a real OTA?

• Compute the transfer function as a function of the OTA

parameters

• Derive design guidelines for the OTA

0 IN v

v sRC





OTA-RC Integrator Analysis

24

To keep equations simple lets consider different effects

separately:

• Finite DC gain

• Finite gm

• Output capacitance

Study effect on poles Q:• define Integrator quality factor (QINT)





Finite DC Gain

25

Neglect output/load capacitance (C0=0), Gm is very large

but the DC gain (GmR0) is finite and equal to A0.

R oGm(e+-e-)

e+ v 0

i IN

v IN

R

C

IN O IN O m

O

v v vi v v sC G v

R R

1

O m

O

v sC G sC v

R

1m

OOG

O m O

sC vv v

A G A

11 IN

O

vv v

sRC sRC

0

01 1O

IN

v A

v sRC A

Prof D ManstrettaAA2012-13 26




Finite DC Gain (2)

26

The effect of a finite OTA DC gain is to move the pole from

the origin to = 0 /(1+A0).

0

01 1

O

IN

v Av sRC A

AMPLITUDE PHASE





Finite DC Gain (3)

27

The effect of a finite OTA DC gain is to move the pole from

the origin to = 0 /(1+A0). The phase shift at 0 is slightly

larger than 90° ( phase lead )

0

01 1

O

IN

v A

v sRC A

0

0

180 arctan 1O

IN

v A

v





Finite Gm

28

Neglect output/load capacitance (C0=0) and conductance:

DC gain is infinite but Gm has a finite value.

IN IN O m

v vi v v sC G v

R

O mv sC G sC v

1 mO

Gv v

sC

1 IN

m

vv G

R R

1

1

mO

IN m

sC Gv

v sC R G

Gm(e+-e-)

e+ v 0

i IN

v IN

R

C




Prof. D. ManstrettaAA2012 13

Finite Gm (2)

29

The effect of a finite Gm is to introduce a RHP zero at Gm /C

and to move the unity gain frequency from 1/RC to

1/(R+1/Gm)C.

AMPLITUDE PHASE

11

mO

IN m

sC Gvv sC R G

z = 100 0





Finite Gm (3)

30

The zero must be at a much higher frequency than 0 . The

phase shift at 0 is slightly smaller than 90° ( phase lag )

90 arctanO

IN z

v

v

1

1

z O

IN m

sv

v sC R G

z = Gm/C

z = 100 0





Finite Gain-Bandwidth Product (GBW)

31

Now consider a finite output/load capacitance C0 and a

finite Gm (GBW= GmC0). Neglect output conductance

(infinite DC gain).

IN IN O m O O

v vi v v sC G v v sC

R

O TOT mv sC G sC v

11 IN

O

vv v

sRC sRC

1 1 /

1 /O m

IN O m

v sC G

v sR C sC G

C oGm(e+-e-)

e+ v 0

i IN

v IN

R

C TOT OC C C

11

IN TOT O

m

v C sRC v

sRC G sC sRC

1 /1 O

m

C C

G R





Finite GBW (2)

The effect of a finite GBW is to introduce an additional

pole at approximately -Gm /CO. The unity gain frequency

is also slightly increased.

AMPLITUDE PHASE 1 1 /

1 /O m

IN O m

v sC Gv sR C sC G

z = 100 0

p = 500 0




Finite GBW (3)

The phase shift at 0 is slightly smaller than 90° ( phase lag ).

The pole and the RHP zero both contribute a phase lag

90 arctan arctanO

IN z p

v

v

z = Gm/C

1 1 /

1 /O z

IN p

v s

v sR C s

p = Gm/CO

z = 100 0

p = 500 0




Integrator Quality Factor (QINT)

The cumulative effect of integrator non-idealities on the

poles can be summarized in a single number: QINT

QINT is a measure of the integrator phase deviations from

90° at the unity gain frequency.

DEFINITION

1tan

tan 90 INT Q

0 H j

H. Khorramabadi and P. R. Gray, “High-frequency CMOS continuous time filters,” JSSC Dec 1984.




Integrator Quality Factor: QINT

Ideal integrator

0

1 H j

j

INT Q

1 H j

R jX

0

0

1tantan 90

INT

X Q R

j H j H j e

arctan

X

R

Real integrator

2

This alternative definition may be easier to remember.




Summary

0

0

0

1 INT

X Q A

R

0

01 1

O

IN

v A

v sRC A

• Finite DC Gain

• Finite Gm (RHP zero z=Gm/C)

• Finite GBW (additional pole)

0

0 z

INT Q

1

1

z O

IN m

sv

v sC R G

0 0

0

1 /

1 / z

X

R

0 1

1 /

O

IN p

v

v s s

0

0 0

1

/ p

X

R

0

0 p

INT Q




Summary (2)

0 0

0

1

1

1

INT

z p

Q

A

• Finite DC Gain gives a positive Q. Right half-plane zero

(such as introduced by the finite Gm) and additional poles

(such as introduced by load capacitance) introduce a

negative Q.

• The two effects partially cancel each other but ensuringcomplete cancellation across process variations is not

trivial.

• Design guidelines: large DC gain, ensure that zeros and

poles are well above the poles frequency




Biquad with Finite Integrator Q

• If a biquad is realized using integrators having finite QINT, the poles QP will be modified as follows:

'2

1

P

P

P

INT

QQ

Q

Q

Re(s)

Im(s)

Re(s)

Im(s)

QINT >0 QINT <0

QINT must be much higher than the poles Q to preserve

the filter shape




Biquad with Finite Integrator Q (2)

• If a biquad is realized using integrators having finite QINT, the biquadtransfer function will be modified as follows:

• If the error on the gain of the biquad at the pole frequency is to be

lower than ERR , a specification on the minimum QINT is derived:

• Notice that the above considerations apply to any pair of po les of the filter

transfer function, independent of the f i l ter imp lementat ion (i.e. even if the

filter is implemented as a ladder )

0

0'2

1

LP

LP

P

INT

H H

Q

Q

, 20

2

10 1ERR dB

P

INT

QQ

W.J.A. De Heij, E. Seevinck, and K. Hoen, “Practical Formulation of the Relation Between

Filter Specifications and the Requirements for Integrator Circuits,” TCAS Aug 1989.




Two-Integrators Biquad Design

S -1/s

b 1

-1/s S

-b0

a0 E 1 E 2

02

2

1 1 0

a E

E s b s b

C

RE2

Rb0

C

Ra0E1

R

Rb1

R

01

P

bQ

2

0 0 0a b




Integrator Non-Idealitities

S HINT(s)

b 1

HINT(s) S

-b0

a0 E 1 E 2

The effect of integrator non-idealities will be evaluated using the modified

integrator transfer function HINT(s)

HINT(s)

• Finite DC Gain

• RHP zero

• Finite GBW

(additional pole)

1

1 z

s s

0

0

1

1 s

A

1 1

1 / p

s s

2

0

2

1 01

INT

INT INT

a H s H s

b H s b H s

2

0 0 0a b

01

P

bQ




Finite DC Gain

2

0

21 01

INT

INT INT

a H s H s

b H s b H s

2

0 0b 01

P

bQ

2 20 0

0 2

0 0 0

1

2 1 11

1 1 1 P P

H s

s sQ A Q A A

00 0 2

0 0

0

1 1' 1

11 1 1

2 1

P

P

Q A A

Q A

Assuming A0 >>1

0

'2 2

1 11

P P P

P P

INT

Q QQ

Q Q

A Q

0

1 INT

Q A

0 0

1

1 / 1 / INT

H A s




RHP Zero

1

1 INT z

H s s s

2

0

21 01

INT

INT INT

a H s H s

b H s b H s

2

0 0b 01

P

bQ

22

0

2 22 20 0 0 002

1 /

1 2

z

P z z P z

s H s

s sQ Q

0 00 2

00 0

2

'

11 2 P z

P z z QQ

Assuming

z >>0

0

'2 2

1 1

P P P

P P

z INT

Q QQ

Q Q

Q

0

z INT Q




Finite GBW

2

02

1 01

INT

INT INT

a H s H s

b H s b H s

2

0 0b 01

P

bQ

0

p

INT Q

Assuming

p >>0

1

1 INT

p

H s s s

2

02

2 200

1

1 /'

p

P

H s s s s

Q

0

'2 2

1 1

P P P

P P

p INT

Q QQ

Q Q

Q




Summary

S HINT

(s)

b 1

HINT(s) S

-b0

a0 E 1 E 2

The effect of integrator non-idealities on the poles Q is:

• Finite DC Gain

• Finite Gm (RHP zero)

• Finite GBW (additional pole)

1

1 INT

p

H s s s

1

1 INT z H s s s

'2

1

P P

P

INT

QQ

Q

Q

0

p

INT Q

0

z INT Q

0 0

1

1 / 1 / INT H

A s

01 INT Q A




HIGHER ORDER

OPAMP-RC FILTERSCanonic Synthesis




Multiple-Loop Feedback Architectures

• As the filter order increases, the Q of the poles increases

and biquad based implementations become too sensitive

to components variations and mismatches.

• Multiple-loop feedback architectures are to be

preferred due to lower sensitivity

• The state-variable synthesis method allows the

synthesis of an nth order filter (low-pass, band-pass or

high-pass) in the form

1 2

1 2 1 0

1 2

1 1 2 1 0

...

...

n

n n

n n

n

E a s a s a s a

E s b s b s b s b




State-Variable Method:

All-pole filters

E1 E2

E1 E

2 E3

a0

E3

E2 E4

0

1 2

1 1 2 1 0...

n

n n

n

E a

E s b s b s b s b

1 2

1 2 1 0 2 0 1...n n

n s b s b s b s b E a E

1 2

1 2 1 2 0 2 3...n n

n s b s b s b s E b E E

1 2

1 2 1 2 3 0 2...

n n

n s b s b s b s E E b E

S

-b0

1 0

1

...n s b s b

1 2

1 2 1 2 4...n n

n s s b s b s b E E

1 2

1 2 1 0 2 3...n n

n s b s b s b s b E E

1

1

1n s b

1

s




Controller Canonic Form Realization

E1s

1

s

1

s

1

-bn-1

-bn-2

-b1

-b0

E2

a0

sE2s

n-2E2s

n-1E2s

nE2

0

1 2

1 1 2 1 0...

n

n n

n

E a

E s b s b s b s b

Follow-the-leader feedback architecture

K. Laker, M. Ghausi, "Synthesis of a low-sensitivity multiloop feedback active RC

filter," IEEE Transactions on Circuits and Systems, Mar 1974




Inverting Integrators Realization

E1

bn-1

-bn-2

(+/-) b1

(+/-) b0

E2

a0

-sE2(-s)n-2

E2(-s)n-1

E2(-s)nE2

s

1

s

1

s

1

0

1 2

1 1 2 1 0...

n

n n

n

E a

E s b s b s b s b




Inverse Follow-the-Leader Feedback




Circuit Realization Example

E1

b3

b1

E2

a0

s

1

s

1

s

1

s

1

b0

b2

-1

2

4 3 2

1

0.2756

0.9528 1.4539 0.7426 0.2756

E

E s s s s




Circuit Realization Example (ii)

Ri=1/bi

2

4 3 2

1

0.2756

0.9528 1.4539 0.7426 0.2756

E

E s s s s

E1

b3

b1

E2

a0

s

1

s

1

s

1

s

1

b0

b2

-1 The feedback resistors are

inversely proportional to the

corresponding coefficient:




General Realization

• Realization of a generic transfer function with number of zerosstrictly lower than the number of poles

• The realization with equal number of poles and zeros is only slightly

more complicated

1 2

1 2 1 0

1 2

1 1 2 1 0

...

...

n

n n

n n

n

E a s a s a s a

E s b s b s b s b




Filter Design Steps

Designspecification

Normalization

Approximation

Network

Function

Realization

Denormalizati

on

p

A0

PASS-BAND

STOP-BAND

|H|

s

IN-BAND

RIPPLE

Min

ATTEN

.

FILTER MASK




Example 3

• Design a 5th order all-pole low-pass filter using follow-the-

leader feedback architecture

• DC gain = 0dB

• Chebyshev, 5th order, e=0.509, 0=1MHz

0

5 4 3 2

4 3 2 1 0

OUT

IN

V a

V s b s b s b s b s b




Design Example of a Chebyshev Filter

• 5th order, e=0.5088

• n = (1/5) sinh-1(1/0.5088) = 0.2856

• Sinh n = 0.2895 ; Cosh n = 1.041• Normalized poles:

• k=1 s1= sin(/10) x 0.2895 = 0.0895; 1= cos(/10) x 1.041 = 0.99

• P1 = (s12+ 1

2)1/2 =0.994; Q1 = P1 /(2 s1) = 5.556

• k=2 s2= sin(3/10) x 0.2895 = 0.234; 1= cos(3/10) x 1.041 = 0.612

• P2 = (s22

+ 22

)1/2

=0.655; Q2 = P2 /(2 s2) = 1.4• k=3 s1= sin(5/10) x 0.2895 = 0.2895; 1= cos(5/10) x 1.041 = 0

• P3 = 0.2894

u (2k 1)

2nv

1

nsinh

1 1

e

sk s k j k sin

(2k 1)

2n

sinh v j cos

(2k 1)

2n

cosh v

2 2 2 21 21 2 3

1 2

( ) P P P P P

P P

s s D s s s s

Q Q




Design Example of a Chebyshev Filter

• Calculate the normalized polynomial coefficients:

5 4 3 24 3 2 1 0( ) D s s b s b s b s b s b

1 24 3

1 2

P P P

P P

bQ Q

2 2 1 2 3 1 3 23 1 2

1 2 1 2

P P P P P P P P

P P P P

bQ Q Q Q

2 22 2 1 2 2 1 1 2

2 3 1 2

1 2 2 1

P P P P P P P P P

P P P P

bQ Q Q Q

2 22 21 2 2 1

1 3 1 2

1 2

P P P P P P P

P P

bQ Q

2 2

0 3 1 2 P P P b

2 2 2 21 21 2 3

1 2

( ) P P P P P

P P

s s D s s s sQ Q




Flowgraph

E1

b4

b2

a0

s

1

s

1

s

1

s

1

b0

b3

-1

E2s

1

b1

To de-normalize the coefficients:

b’k= bk 05-k

If we substitute the integrators with 1/(sRC), the feedback

coefficients are changed as follows: b’’k=bk(RC0)5-k




Circuit Implementation

R

VOUT

Rb4

Rb0

VIN

Rb2

CC

R

C

R

C

R

C

R

R

Rb1 Rb0Rb3

0

1

RC bk

k

R R

b''k k b b




BACKUP SLIDESMATERIALE INTEGRATIVO




Pole Quality Factor

0

sP

P

Im(s)

Re(s)

The solutions of D(s)=0 (i.e. the poles of thenetwork function) are complex numbers,

typically represented in the S plane.

sp = sp+jp

Solutions sk can be real or complex.Complex solutions always appear in

conjugate pairs.

A very important design parameter is the

pole quality factor Q.

The pole quality factor is given by the ratio

between the pole frequency (distance fromthe origin in the s-plane) and the real part

(distance from the imaginary axis in the s-

plane)

S1

S2

=0

−2




Biquad Magnitude Response

H(j0)=-jQ

=1

1 + 0 + 0 2

Frequency response at 0

Two conjugate poles at0

Notice how the network function has been written, with explicit reference to 0

and Q. This is usually done in biquad-based filter design since it greatly

simplifies the design procedure.




Biquad Phase Response

=1

1 + 0 + 0 2

= − 002 −2

f(j0)=-/2

Notice how, in low Q poles, the phase starts to move about a decade before the

pole (i.e. much earlier than the magnitude response.

As we shall see, low Q poles are desirable to lower power consumption and to

minimize the sensitivity of the filter reponse to parameter variations.




Appendix material (use only if needed)

• Poles and poles Q

• Multi-loop feedback architectures (summary)

• Ladder filter synthesis (summary)

• Frequency normalization and de-normalization




Example 1

Design of a low-pass biquad:

• DC gain=1

• 0=1MHz

• QP=0.7

2 0

21 1 0

E a

E s b s b

2

0 0 0a b 0

1 P b Q




Flowgraph Implementation

S -1/s

b 1

-1/s S

b0

a0 E 1 E 2

-1

S S E 2 1

sRC

1

sRC

2

0b RC

1b RC

2

0a RC

E1 E3 E

4

Modify flowgraph using flowgraph rules to simplify circuit

implementation: divide and multiply by RC.




Flowgraph (2)

S S E 2 1

sRC 1 sRC E1

E3 E4

1

-1

1/QP

0

1

RC

To get a 1:1 equivalence with the circuit, let:

C

E2

QPR

RE1

R

R

R

R

C

Circuit

implementation:

bk

k

R R

b

2 E 3 E

2 E




Generic Biquad Implementation

2

2 1 02

2

1 1 0

a s a s a E

E s b s b

'2

2

1 1 0

1 E E s b s b

' 2

2 2 2 1 0 E E a s a s a

2 '

6 2

E s E

+a2 -a1

a0

'

3 2 E sE

Low-pass output

Band-pass

output

High-pass

output

6 E




Example 2

Design of a low-pass biquad with transmission zero

• DC gain=1

• 0=1MHz

• Q=0.7

• z=2MHz

22 2 0

2

1 1 0

E a s a

E s b s b

2

0 0 0a b 01

P

bQ

2

0

2 2

z

a




Flowgraph

+a2 a0

E2

2

2 0 2

2

5 1 0

E a a s

E s b s b




Example 2

S S E 2 1

sRC

1

sRC

02

2

1 1 0

a E

E s b s b

2

0b RC

1b RC

Start with the poles.Modify flowgraph using flowgraph rules to simplify circuit

implementation: divide and multiply by RC

2

0 0 0a b 01

P

bQ

2

0a RC

E1

E3 E4




Example 2

S S E 2 1

sRC

1

sRC

If we let 0

1

RC the flowgraph is simply:

E1

E3 E4

1

-1

1/QP




Example 2

Adder circuit implementation:3

4 1 2 P

E

E E E Q

RGE1

R

E4

E2

E3

Rb1

Rb0

4

3 1 0

1G

b G b

E R R

E R R R

4

2 0b

E R

E R

1

1 2 G

P b G

R

Q R R

1 2 1b P G R Q R

4

1

24

P

E

E Q

0b R R

With this implementation we cannot set

the gain independent of the poles Q!




Zeros Implementation

The zeros are now implemented with another adder:

2 2

0 z

2

02 42OUT

z

V E E

S S E 2 1 sRC

1 sRC

E1 E3 E4

1

-1

1/QP

VOUT

1




Circuit Implementation

E2

R

Ra2

VOUT

R

E4

2

2 2

0

z a R R

2

0

2 42OUT z

V E E




Final Circuit Implementation

E2

R

C

RE1

R

R R

C

R

VOUT

R

E3

E4

2 1 P Q R

2

2

0

z R

Documents

High Order OA-RC Filters Design x ELE3