Bode Plot for a First-Order Low-Pass. Filter. A B d l t h th it d f. t k. A Bode plot
shows the magnitude of a network function in decibels versus frequency using a.
Lecture 19 Bode Plot,, High g Pass Filter,, Series Resonance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
First-Order Low Pass Filter
1 H( f )= = 1 + j( f f B )
H(f ) =
1
1+ (f
2 ) fB
1∠0 o 1+ ( f fB )
2
⎛ f ⎞ ⎟⎟ ∠ arctan⎜⎜ ⎝ fB ⎠
⎛ f ⎞ ∠H ( f ) = − arctan ⎜⎜ ⎟⎟ ⎝ fB ⎠
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Bode Plot for a First-Order Low-Pass Filter AB Bode d plot l t shows h the th magnitude it d off a network t k function in decibels versus frequency using a l logarithmic ith i scale l for f frequency. f
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Bode Plot for a First-Order Low-Pass Filter H(f ) = H ( f ) dB
1 1 + ( f f B )2
⎛ 1 ⎜ = 20 log H ( f ) = 20 log⎜ ⎜ 1 + ( f f )2 B ⎝
⎞ ⎟ ⎟ ⎟ ⎠
= 20 log( l (1) − 20 log l ⎛⎜ 1 + ( f f B )2 ⎞⎟ ⎝ ⎠ = −20 log⎛⎜ 1 + ( f f B )2 ⎞⎟ ⎝ ⎠
(
2 1/ 2
= −20 log 1 + ( f f B )
)
(
)
= −10 log 1 + ( f f B )2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Asymptotic Behavior of Magnitude for Low and High Frequencies ⎡ ⎛ f ⎞2 ⎤ ⎟⎟ ⎥ H ( f ) dB = −10 log ⎢1 + ⎜⎜ ⎢ ⎝ fB ⎠ ⎥ ⎣ ⎦ For f > f B
H ( f ) dB
2
⎛ f ⎞ ⎛ f ⎞ ⎟⎟ = −20 log⎜⎜ ⎟⎟ ≈ −10 log⎜⎜ ⎝ fB ⎠ ⎝ fB ⎠
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnitude Bode Plot for First-Order Low Pass Filter
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Asymptotic Behavior of Phase for Low and High Frequencies 1 1∠0 o H(f )= = 1 + j( f f B ) 1 + ( f f B )2 ∠ arctan( t ( f fB )
∠H ( f ) = − arctan( f f B ) =0
f > f B
= −45
f = fB
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1. A horizontal line at zero for f < fB /10. 2. A sloping line from zero phase at fB /10 to –90° at 10fB. 3 A hhorizontal 3. i t l line li att –90° 90° for f f > 10fB. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.11
Sketch the Bode plots for the low-pass filter shown above:
1 fB = = 2πRC
1 = 1000 Hz ⎛ 1000 ⎞ −6 2π ⎜ ⎟(1x10 ) ⎝ 2π ⎠
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.11 H(f ) =
1 1 + ( f 1000 )2
⎛ f ⎞ ∠H ( f ) = − arctan⎜ ⎟ ⎝ 1000 ⎠
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
First-Order High-Pass g Filter
Vout = Vout Vin
R
Vin =
1 R− j 2πfC j( f / f B ) = 1+ j ( f / f B )
1 1 1− j 2πfRC
j 2πfRC Vin = Vin j 2πfRC + 1
1 whe e f B = where 2πfRC
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
First-Order High-Pass g Filter Vout ( j( f f B ) f f B )∠90 o H(f )= = = Vin 1 + j ( f f B ) 1 + ( f f B )2 ∠ arctan( f f B ) H(f ) =
( f fB ) 1 + ( f f B )2
∠90 o = 90 o − arctan( f f B ) ∠H ( f ) = ∠ arctan( f f B )
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
First-Order High-Pass g Filter H(f ) =
( f fB ) 1 + ( f f B )2
o ( ) ∠H f = 90 − arctan( f
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
fB )
First-Order High-Pass g Filter H(f ) = H ( f ) dB
f fB 1 + ( f f B )2
⎛ ⎞ f fB ⎜ ⎟ = 20 log H ( f ) = 20 log⎜ ⎟ ⎜ 1 + ( f f )2 ⎟ B ⎝ ⎠ = 20 log( f f B ) − 20 log⎛⎜ 1 + ( f f B )2 ⎟⎞ ⎠ ⎝
(
= 20 log( l ( f f B ) − 20 log l 1+ ( f fB )
(
)
2 1/ 2
= 20 log( l ( f f B ) − 10 log l 1 + ( f f B )2
)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Asymptotic Behavior of First-Order High-Pass Filter
(
H ( f ) = 20 log( l ( f f B ) − 10 log l 1 + ( f f B )2
For f > f B H ( f ) ≈ 20 log( f f B ) − 10 log( f f B )2 = 0 ⎛ f ⎞ ⎟⎟ ∠H ( f ) = 90 − arctan⎜⎜ ⎝ fB ⎠ o
For f > f B ∠H ( f ) ≈ 0 o
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Bode Plots for the First-Order HighPass Filter (
H ( f ) = 20 log( f f B ) − 10 log 1 + ( f f B )2
For f > f B H ( f ) ≈ 20 log( f f B ) − 10 log( f f B )2 = 0
For f > f B H ( f ) ≈ 20 log( f f B ) − 10 log( f f B )2 = 0
⎛ f ⎞ ⎟⎟ ∠H ( f ) = 90 o − arctan⎜⎜ ⎝ fB ⎠ For f > f B ∠H ( f ) ≈ 0 o
(
H ( f ) = 20 log( f f B ) − 10 log 1 + ( f f B )2
)
For f > f B
For f >> f B ∠H ( f ) ≈ 0 o
H ( f ) ≈ 20 log( f f B ) H ( f ) ≈ 20 log( f f B ) − 10 log( f f B )2 = 0
For f
