ZVVRZ = + ( ) 1 wR C Av ww RRCC wR C wR C

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FILTRO PASABANDA: BANDA ANCHA
a. Hallar
Av ( w) = f ( R, C , w)
Z c1
Vx =
Vi
R1 + Z c1
Vy =
R2
Vx
R2 + Z c 2
→
→
1
jwC
Vx =
V
1 i
R1 +
jwC
Vo =
R2
R2 +
1
jwC2
Vx
→
Vx =
1
Vi (1)
jwR1C1 + 1
→
Vo =
jwR2C2
Vx (2)
jwR2C2 + 1
Reemplazo 1 en 2
Vo =
⎤
jwR2C2 ⎡
1
V
i⎥
⎢
jwR2C2 + 1 ⎣ jwR1C1 + 1 ⎦
Vo =
→
jwR2C2
Vi
( jwR2C2 + 1)( jwR1C1 +1)
Vo
jwR2C2
=
2
Vi − w R1 R2C1C2 + jwR2C2 + jwR1C1 + 1
Av ( w) =
wR2C2
(1 − w R R C C ) + ( wR C
2
2
1
b. Hallar
wc y f c
2
1
2
2
2
+ wR1C1 )
2
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1
=
2
wR2C2
(1 − w R R C C ) + ( wR C
2
2
1
⎛
⎜
⎝
2
1
2
2
2
+ wR1C1 )
(1 − w R R C C ) + ( wR C
2
2
1
2
1
2
2
(1 − w R R C C ) + ( wR C
2
2
1
2
1
2
2
2 + wR1C1 )
2
2
2
(
⎞
⎟ = wR2C2 2
⎠
)
2
+ wR1C1 ) = 2w2 R2 2C2 2
2
2
1 − 2 w 2 R1 R2C1C2 + w 4 R12 R2 2C12C2 2 + w 2 R2 2C2 2 + 2 w 2 R1 R2C1C2 + w 2 R12C12 = 2 w 2 R2 2 C2 2
1 + w4 R12 R2 2C12C2 2 − w2 R2 2C2 2 + w2 R12C12 = 0
Como: f H = 10 f L , entonces:
fL =
1
2π R2C2
fH =
10
1
=
2π R2C2 2π R1C1
R2 C 2 = 10 R1C1
Lo reemplazo en la ecuación anterior:
1 + w 4 R12 C12 (10 R2 C2 ) 2 − w 2 (10 R1C1 ) 2 + w 2 R12C12 = 0
1 + 100 w4 R14C14 − 100 w2 R12C12 + w2 R12C12 = 0
100 w4 R14C14 − 99 w2 R12C12 + 1 = 0
w2 (100w2 R14C14 − 99 R12C12 ) + 1 = 0
w2 + 1 = 0
100w2 R14C14 − 99 R12C12 = 0
w = −1
99 R12C12
w =
100 R14C14
2
99
2π f H =
10 R1C1
w=
→
99
fH =
20π R1C1
0.1583
RC
fL = 1 1
10
fL =
0.01583
R1C1
c. Hallar
99
100 R12C12
→
Av ( f ) =f ( f , f c )
→
→
w=
99
10 R1C1
fH =
0.1583
R1C1
→
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⎛ 0.1583 ⎞ ⎛ 0.01583 ⎞ 0.05
fr = ⎜
⎟⎜
⎟=
R
C
R
C
⎝ 1 1 ⎠ ⎝ 1 1 ⎠ R1C1
Av ( w) =
R1C1 ==
→
0.05
fr
⎛ 0.05 ⎞
10w ⎜
⎟
⎝ fr ⎠
2
2
2
⎛
⎞ ⎛
⎛ 0.05 ⎞ ⎞
2 ⎛ 0.05 ⎞
⎜1 − 10 w ⎜
⎟ ⎟⎟ + ⎜11w ⎜
⎟⎟
⎜
f
⎝ r ⎠ ⎠ ⎝
⎝ fr ⎠ ⎠
⎝
Av (w) =
d. Dibujar
⎛ f ⎞
3.1415 ⎜ ⎟
⎝ fr ⎠
2
2
2
⎛
⎛ f ⎞ ⎞ ⎛
⎛ f ⎞⎞
⎜1 − 0.986 ⎜ ⎟ ⎟ + ⎜ 3.4557 ⎜ ⎟ ⎟
⎜
⎝ f r ⎠ ⎠⎟ ⎝
⎝ fr ⎠ ⎠
⎝
Av ( f ) Vs f
Av (w) =
⎛ f ⎞
3.1415 ⎜ ⎟
⎝ fr ⎠
2
⎛
⎛ f ⎞ ⎞ ⎛
⎛ f ⎞⎞
⎜1 − 0.986 ⎜ ⎟ ⎟ + ⎜ 3.4557 ⎜ ⎟ ⎟
⎜
⎝ f r ⎠ ⎠⎟ ⎝
⎝ fr ⎠ ⎠
⎝
2
2
f
Av ( f )
Av ( f ) db
0
0.1fc
0.5fc
0.8 fc
fc
2 fc
4 fc
6 fc
8 fc
10 fc
0
0.3
0.8
0.9
0.9
0.8
0.6
0.5
0.4
0.3
0
-10
-2
-1
-1
-2
-4
-7
-9
-10
⎛V ⎞
Av = 20 log ⎜ 0 ⎟
⎝ Vi ⎠