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High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters (S&S 12.5)
2nd-order filter 2 1
2 1 02 1
1 0
( )a S a S a
T sS b S b
+ +=
+ +
22 1 0
2 20 0
( )( / )
a s a s aT s
s Q sω ω+ +
=+ +
2 1 2
1 2
( )( )( )( )
a s z s zs p s p
− −=
− −If p1, p2 are real,
Consider T(s) with complex poles
201 2 0, 1 (1 / 4 )
2p p j Q
Qω
ω= − ± −
Depending on zero locations, various filter characteristics!
(Q > ½)
2nd-order filters can be easily made by cascading two 1st-order filters
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
Second-order filter pole-zero diagrams
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
LP Filter
At ω=ω0, |T| = |a0/ω02| Q
Larger Q More peaking
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
How to realize passive second-order LP filter
Use LCR
Two zeros at s infinity
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
How to realize passive second-order LP filter
o
i
V( )
VT s ≡
2
1 /(1 / ) (1 / )
LCs s CR LC
=+ +
2
1 2
ZZ Z
=+
1
1 2
YY Y
=+
1/(1 / ) (1 / )
sLsL sC R
=+ +
01LC
ω = 0 1Q CRω
=0
CQ CR RL
ω= =
20
2 200s s
Q
ωω
ω=
+ +
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
HP Filter
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
Same pole characteristics as LP Same natural modes
o
i
V( )
VT s ≡ 2
1 2
ZZ Z
=+
1
1 2
YY Y
=+ (1 / ) (1 / )
sCsC R sL
=+ +
2
2 (1 / ) (1 / )s
s s CR LC=
+ +
Two zeros at s=0
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
BP Filter
Large Q Sharper response
3dB bandwidth
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
o 1
i 1 2
V 1/( )V (1/ ) (1 / )
Y RT sY Y R sC sL
= = =+ + +
2
(1 / )(1 / ) (1 / )
s RCs s RC LC
=+ +
0CQ CR RL
ω= =
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
Band-Rejection or Notch Filter
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
1 2
1
1 1
sL sLsCZs LCsL
sC
⋅= =
++
2o 2
2i 1 2
2
V (1 / )( )V (1 / ) (1 / )
1
Z R s LCT s sLZ Z s s RC LCRs LC
+= = = =
+ + +++
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
Low-Pass Notch Filter
21
2
1 2 1 2
(1 / )( ) 1 1
( ) ( )
s LCT s
s sC C R L C C
+=
+ ++ +
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
High-Pass Notch Filter
21
21 2
(1 / )( )
(1 / ) [1 /( ) ]s L C
T ss s CR L L C
+=
+ +
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
High-Speed Circuits and Systems LaboratoryElectronic Circuits 2
Lect. 14: Second-Order Passive Filters
2 20 0 0
2 2 2 20 0 0 0
( / ) 2( / )( ) 1
( / ) ( / )s s Q s Q
T ss s Q s s Q
ω ω ωω ω ω ω
− += = −
+ + + +
02 2
0 0
( / )122 ( / )
s Qs s Q
ωω ω
⎛ ⎞= −⎜ ⎟+ +⎝ ⎠