Lect. 17: Two-Integrator-Loop Biquad (S&S 11.7) 2nd-order...

Lect. 17: Two-Integrator-Loop Biquad (S&S 11.7)

2nd-order active filters

LP HP

BP BR

Electronic Circuits 2 (09/1) W.-Y. Choi

Lect. 17: Two-Integrator-Loop Biquad

One circuit configuration for several different second-order filters?

Two-integrator-loop biquad

2V Kshp2 2

0 0

VV ( / )i

Kss s Qω ω

=+ +

Consider

2 2 2hp hp 0 hp 0V V ( / ) V = Vis s Q K sω ω+ +

20 0

hp hp hp2

1V V V = ViKQ s s

ω ω⎛ ⎞⎛ ⎞+ + ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

20 0

hp hp hp2

1V V V ViKQ s sω ω

= − −



20 0

hp hp hp2

1V V V ViKQω ω

= − −hp hp hp2i Q s s

Express the above equation using a block diagram



2hV K sKs ωhp 0

0 0 2 2 2 20 0 0 0

V( / ) ( / )

V ( / ) ( /Q)+i

K sKss ss s Q s s

ωω ω

ω ω ω ω− = − = −

+ + +BP

22hp2 2 2 2 0

0 0 2 2 2 20 0 0 0

V( / ) ( / )

V ( / ) ( /Q)+i

KKss ss s Q s s

ωω ω

ω ω ω ω= =

+ + +LP

0 0 0 0( ) ( Q)i Q

Biquad Universal Active Filter


Biquad Universal Active Filter


How to implement biquad?1iV

V =

V V

0

-

1- -

o

o

VR sC

VV RCs s

ω

= ⋅

= =Vi Vo

01

iV RCs s

RCω =

(Ignoring the op-amp offset)



Kerwin-Huelsman-Newcomb (KHN) biquad

Use superposition (Vi, Vbp, Vlp)

13V VfR RR +⎛ ⎞= ⎜ ⎟

12 VfR RR +⎛ ⎞+ ⎜ ⎟ VfR

−hp2 3 1

V ViR R R= ⎜ ⎟+ ⎝ ⎠

bp2 3 1

VR R R

+ ⎜ ⎟+ ⎝ ⎠lp

1

VR

−

Weighted sum of V V V


Weighted sum of Vi, Vbp, Vlp


KHN

13 2hp bp lp

2 3 1 2 3 1 1

V V 1 V Vf f fi

R R R RR RR R R R R R R

+⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

20 0

hp hp hp2

1V V V ViKQ s sω ω

= − −2 3 1 2 3 1 1

R R R R R R R+ +⎝ ⎠ ⎝ ⎠ Q s s

213 2 0 0

hp hp2V VV 1f f fi s

R R R RR RR R R R R R R s

ω ω+⎛ ⎞ ⎛ ⎞= + +

⎛ ⎞⎛ ⎞− ⎜ ⎟⎜ ⎟⎝

−⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎠ ⎠ ⎝2 3 1 2 3 1 1sR R R R R R R s⎝+ +⎝ ⎠ ⎝ ⎠⎠ ⎠ ⎝

2 1 R RR 2 1R1/ 1fR R∴ = 2 32

2 3 2

2 1 2

R RRQ

R R Q R+

= ∴ =+

3

2 3

2 12R

KR R Q

= = −+

ω =?


ω0=?


Th b li d b li bi ti f V V V

How about Notch, All-Pass Filters?

These can be realized by linear combination of Vhp, Vbp, Vlp

KHN




hp bp lpV V V VF F Fo

H B L

R R RR R R

⎛ ⎞= − + +⎜ ⎟

⎝ ⎠

2

hp i2 20 0

V V( / )

Kss s Qω ω

=+ +

0 0bp hp i2 2

0 0

V ( )V V( / )K s

s s s Qω ω

ω ω−

= − =+ +

2 20 0

lp hp i2 2 20 0

V ( )V V( / )K

s s s Qω ω

ω ω= =

+ +

2 20 0

2 20 0

V ( / ) ( / ) ( / )=

V ( / )o F H F B F L

i

R R Ks R R K s R R Ks s Q

ω ωω ω

− +−

+ +




2 2V ( / ) ( / ) ( / )R R s R R s R Rω ω+0 02 2

0 0

V ( / ) ( / ) ( / )=

V ( / )o F H F B F L

i

R R s R R s R RK

s s Qω ω

ω ω− +

−+ +

N t h Filt 1FR 1FRRNotch Filter: 1,F

HR= 1F

LR=,BR = ∞

All-Pass Filter 1,F

H

RR

=1 ,F

B

RR Q

= 1F

L

RR

=



There are several Biquads

Tow Thomas biquad:Tow-Thomas biquad:

V1 V2

1V V⎛ ⎞⎜ ⎟⎜ ⎟ V V⎛ ⎞1V V⎛ ⎞2

11

1( // )1//

io

V VV QR

R sCRsC

⎜ ⎟⎜ ⎟= − +⎜ ⎟⎜ ⎟⎝ ⎠

12

3

iV VV r

R r⎛ ⎞

= − +⎜ ⎟⎝ ⎠

12

1o iV VV

R R sC⎛ ⎞

= − +⎜ ⎟⎝ ⎠



Tow-Thomas biquad

V1

V21

2 12

1 3 2

1 1 1V

=o

C rs sC C R RR C RR

⎛ ⎞⎛ ⎞ + − +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ 1ω =

22 2

= 1 1Vi s sQCR C R

−+ +

0 CRω =



For LP

1 0C = 1 3, R R= ∞ = ∞

2

DC Gain: RR

V1

V2

2 1 1 1 1C r⎛ ⎞⎛ ⎞ + +⎜ ⎟⎜ ⎟

For HP2 1

21 3 2

22 2

V= 1 1V

o

i

s sC C R RR C RR

s s

+ − +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠−

+ +1 2 3, , R R R= ∞ = ∞ = ∞

2 2QCR C R1HF Gain:

CC


C


For BP

2R = ∞1 0C =

V1

V2

11 3

1 1 raC R RR⎛ ⎞

= −⎜ ⎟⎝ ⎠

2 1 1 1 1C r⎛ ⎞⎛ ⎞ + +⎜ ⎟⎜ ⎟For BR

1 3⎝ ⎠

2 12

1 3 2

22 2

V= 1 1V

o

i

s sC C R RR C RR

s s

+ − +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠−

+ + 1 3, R R= ∞ = ∞2 2QCR C R

12

Ca

C= 2

2 0 22

1aC RR

ω =



For AP

V1

V2

1 2C a C=

2 1 1 1 1C r⎛ ⎞⎛ ⎞ + +⎜ ⎟⎜ ⎟

1R = ∞

1 2

3 2/R Qr a=2 1

21 3 2

22 2

V= 1 1V

o

i

s sC C R RR C RR

s s

+ − +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠−

+ +

2 2/R R a=

2 2QCR C R


Lect. 17: Two-Integrator-Loop Biquad (S&S 11.7) 2nd-order...

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