一、带通滤波器(BandpassFilter)

Basic Filter Circuits
1. High-Pass Filter （高通濾波電路）
in
in C out
v C j R R v X R R v ⎪⎪⎪⎪
⎭
⎫ ⎝⎛+=⎪⎪⎭⎫ ⎝
⎛+=ω1
Voltage Gain(out
in
v v ⇒=
=電壓放大倍數)
(1) 3dB Frequency
in
out
v v dB log
20= 2
1
log
203=⇒
=-⇒in out in
out
v v v v RC dB 13=
⇒ω 或 RC
f dB π213=
6dB/
-0
ω31dB RC
=
C
.0v out
v in
(2) Roll-off Attenuation （衰減斜率） At low frequencies, dB RC
31
ωω=<<
,
dB
in out RC v v 3ωω
ω=
≈ ⎪⎪⎭⎫ ⎝⎛=⇒dB in out
dB v v 3log 20)(log
20ωω or ()dB 3log log 20ωω-
At dB 32
1
ωω=
（called “Octave ” (八度音)), )(62
1
log 20log 20log
203dB -===dB in out v v ωω ⇒Roll-off Attenuation (衰減斜率) 6dB/octave =
0.1 1
10
)
/log(3dB ωωdB /6+or dB /20+
At dB 310
1
ωω=
(called “decade ” ), )(2010
1
log 20log
20dB -==in out v v ⇒
Roll-off Attenuation (衰減斜率) 20dB/decade =
(3) Phase Shift （相移）
22111⎪⎭
⎫ ⎝⎛+⎪
⎭⎫ ⎝⎛
+=+=C R C j R R C j R R v v in
out ωωω
The output voltage, v out , leads ahead of the input voltage, v in , by the angle θ, where 31/tan dB
C R ωωθω
==
. At 31
10
dB ωω=, 8410tan ≈⇒=θθ
At 310dB ωω=, 61.0tan ≈⇒=θθ
dB 3
θ9004500
2. Low-Pass Filter （低通濾波電路）
in in C C out
v C j R C j v X R X v ⎪⎪⎪⎪
⎭
⎫ ⎝⎛+=⎪⎪⎭⎫ ⎝
⎛+=ωω11
Voltage Gain out in v v ⇒==
(1) 3dB Frequency
in
out
v v dB log
20= 2
1
log
203=⇒
=-⇒in out in
out
v v v v RC dB 13=
⇒ω or RC
f dB π213=
ω3dB RC
=
v in v out 0.01μF
(2) Roll-off Attenuation （衰減斜率） At high frequencies, dB RC
31
ωω=>>
,
ω
ωωdB in out RC v v 31
=≈ ⎪⎭
⎫
⎝⎛=⇒ωωdB in out dB v v 3log 20)(log
20 or ()ωωlog log 203-dB
At dB 32ωω=（Octave(八度音),
)(62
1
log 20log 20log
203dB -===ωωdB in out v v ⇒Roll-off Attenuation(衰減斜率)6dB/octave =-
At dB 310ωω=,
)(2010
1
log 20log
20dB -==in out v v ⇒Roll-off Attenuation (衰減斜率) 20dB/decade =-
20log out
in v v 0
-10 -20 -30
0.1 1
10
102 )
/log(3dB ωω
(3) Phase Shift （相移）
2
211111⎪⎭
⎫ ⎝⎛+⎪⎭⎫
⎝⎛-=+=C R jR C C C j R C j v v in
out ωωωωω
The output voltage, v out , lags behind the input voltage, v in , by the angle θ,
where 3tan 1/dB
R C ω
θωω=-=-.
dB 310
1
ωω=
當， 61.0tan -≈⇒-=θθ
dB 310ωω=當， 8410
tan -≈⇒-=θθ
θ
-90-45000
dB 3ω
3. Band-Pass Filter (帶通濾波器)
C
L LC Z Z Z 111+= C
j L
j ωω111+
=
⎪⎭
⎫ ⎝⎛
-=L C j ωω1
111
111)(+⎪⎭⎫ ⎝⎛
-=
+⎪⎪⎭
⎫ ⎝⎛=+==
L C jR R Z Z R Z v v T LC LC LC in out ωωω 1
11
)(2
2+⎪⎭
⎫ ⎝⎛
-=⇒L C R T ωωω
Resonant Frequency (共振頻率): LC
f π210=
，or L
C 001
ωω=
∞=⇒LC Z ，
1=in
out
v v
Quality factor (品質因數)dB
f f 30
∆≡
(sharpness of the bell-shape curve) v out
v in
R
(ωT 1
ω
1
2
ωo =
LC
1
At 3dB point ，2
1)(=
ωT
112
2=⎪⎭⎫ ⎝⎛
-⇒L C R ωω
R
L C 1
1±=-⇒ωω
RC
1
20±=-⇒ωωω
There are two 3dB points: ω1 and ω2
⇒ RC 1
1201-
=-ωωω )(10ωω> RC
1
2202=
-ωωω )(20ωω<
⇒ RC
112=
-ωω 2
021ωωω=
⇒ L
C
R
RC f f Q dB ==-=∆=012030ωωωω
4. Band-Reject Filter (or called Notch Filter) (帶拒濾波器)
in
out v v T =
)(ω⎪
⎭
⎫ ⎝⎛
-+⎪
⎭⎫ ⎝⎛
-=C L j R C L j ωωωω11
2
211)(⎪
⎭⎫ ⎝
⎛
-+-
=
⇒
C L R C
L T ωωωωω
2021ωωω=
L
R =
-12ωω C
L
R R L f f Q dB 1012030=
=-=∆=⇒ωωωω
)(ωT
0.70
ω
1 2
v out
v in。