High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad (S&S 12.7)

Two-integrator-loop biquad (biquad)

2hp

2 20 0

VV ( / )i

Kss s Qω ω

=+ +

Consider

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

= − −

One circuit configuration for several different second-order filters?

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

20 0

hp hp hp2

1V V V ViKQ s sω ω

= − −

Express the above equation graphically block diagram

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

2hp

2 20 0

V,

V ( / )i

Kss s Qω ω

=+ +

Since

hp 00 2 2

0 0

V( / )

V ( /Q)+i

K ss

s sω

ωω ω

− = −+ BP

2hp2 2 0

0 2 20 0

V( / )

V ( /Q)+i

Ks

s sω

ωω ω

=+

LP

Biquad Universal Active Filter

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

How to implement biquad?

Vi Vo

0

0

1 -

1- -

1

io

o

i

VV

R sCVV RCs s

RC

ω

ω

= ⋅

= =

=

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

Use superposition (Vi, Vbp, Vlp)

13hp

2 3 1

V Vfi

R RRR R R

+⎛ ⎞= ⎜ ⎟+ ⎝ ⎠

2bp

2 3 1

1 VfRRR R R

⎛ ⎞+ +⎜ ⎟+ ⎝ ⎠

lp1

VfRR

−

Weighted sum of Vi, Vbp, Vlp

Kerwin-Huelsman-Newcomb (KHN) biquad

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

13 2hp bp lp

2 3 1 2 3 1 1

V V 1 V V

f f fi

R R R RR RR R R R R R R

+⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

KHN

20 0

hp hp hp2

1V V V ViKQ s sω ω

= − −

13 2

2 3 1 2 3

20 0

hp hp21 1

V1 VVf f fi

R R R RR RR R R R R R sRs

ω ω+⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎠

1/ 1fR R∴ = 2 32

2 3 2

2 1 2

R RRQ

R R Q R+

= ∴ =+

3

2 3

2 12R

KR R Q

= = −+

ω0=?

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

How about Notch, All-Pass Filters?

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

How about Notch, All-Pass Filters?

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

− +−

+ +

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

How about Notch, All-Pass Filters?

2 20 0

2 20 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− +

−+ +

Notch Filter: 1,F

H

RR

= 1F

L

RR

=,BR = ∞

All-Pass Filter 1,F

H

RR

=1 ,F

B

RR Q

= 1F

L

RR

=

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

KHN Biquad

Other types of biquads?

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

Tow-ThomasBiquad

No Vbp

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

Vbp with Tow-Thomas biquad

V1 V2

2

11

1( // )1//

io

V VV QR

R sCRsC

⎛ ⎞⎜ ⎟⎜ ⎟= − +⎜ ⎟⎜ ⎟⎝ ⎠

12

3

iV VV r

R r⎛ ⎞

= − +⎜ ⎟⎝ ⎠

12

1o iV VV

R R sC⎛ ⎞

= − +⎜ ⎟⎝ ⎠

High-Speed Circuits and Systems LaboratoryElectronic Circuits 2

Lect. 16: Two-Integrator-Loop Biquad

Vbp with Tow-Thomas biquad

V1

V2

2 12

1 3 2

22 2

1 1 1V

= 1 1Vo

i

C rs sC C R RR C RR

s sQCR C R

⎛ ⎞⎛ ⎞ + − +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠−

+ +

0CR ω=

1 20, C R= = ∞

1 3

1 1DC Gain: rC R RR⎛ ⎞

−⎜ ⎟⎝ ⎠

All other filter functions arepossible with proper setting of R,C values (Table 12.2 in S&S)