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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)