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Butterworth Filter Characteristics: A Comprehensive Guide for ELEC 302 - Prof. Ernest Kim, Lecture notes of Electronics

A detailed explanation of butterworth filters, often referred to as maximally flat filters. It covers key characteristics, including the butterworth polynomial, damping coefficients, and filter order determination. The document also explores the relationship between the resonant frequency, passband edge frequency, and filter order. Additionally, it delves into the high-pass butterworth filter characterization, highlighting the interchange of quantities for high-pass applications.

Typology: Lecture notes

2023/2024

Uploaded on 12/11/2024

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ELEC 302 Butterworth Characteristics 1
BUTTERWORTH FILTERS
Butterworth filters, often called maximally flat filters, are smooth filters that transition
monotonically from the passband to the stop band. The polynomials that characterize the
Butterworth response are functions of only two parameters: the order of the filter, n, and the
3 dB frequency,
ω
o.
The response of an nth order low-pass Butterworth filter is an all-pole response characterized
by the nth Butterworth polynomial, Bn(
ω
):
AV
ω
( )
=AV0
Bn
ω
( )
. (9.3-1)
The magnitude of the nth Butterworth polynomial applied to low-pass filters is given by:
Bn
ω
( )
=1+
ω
ω
o
!
"
#$
%
&
2n
(9.3-2)
At ω = ωo the magnitude of every Butterworth polynomial is:
Bn
ω
o
( )
=1+1
( )
2n=2
, (9.3-3)
with half-power frequency, here ωo, is also commonly called the 3 dB frequency:
20 log 2
( )
=3.01030 dB 3dB
. (9.3-4)
Since all Butterworth filters have a response that is reduced by 3 dB at the resonant
frequency, it is often common to specify Butterworth filters with γmax = 3 dB.
The slope of the filter magnitude response at ω = 0 is zero, and at high frequencies, ω 〉〉 ωo,
is -20n dB/decade.
pf3
pf4
pf5

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BUTTERWORTH FILTERS

  • Butterworth filters, often called maximally flat filters, are monotonically from the passband to the stop band. The polynomials that characterize the smooth filters that transition Butterworth response are functions of only two parameters: the order of the filter, 3 dB frequency, ω (^) o. n, and the
  • The response of an by the n th Butterworth polynomial, n th order low-pass Butterworth filter Bn ( ω ): is an all-pole response characterized

AV ( ω ) = BAnV ( ω^ 0 ). (9.3-1)

  • The magnitude of the n th Butterworth polynomial applied to low-pass filters is given by:

Bn ( ω ) = 1 +^! " # ω^ ωo $ % &^2 n (9.3-2)

  • At ω = ωo the magnitude of every Butterworth polynomial is:

Bn ( ω o) = 1 + ( 1 ) 2 n = 2 , (9.3-3)

with half-power frequency, here ωo, is also commonly called the 3 dB frequency: 20 log (^) ( 2 ) = 3. 01030 dB ≈ 3 dB. (9.3-4)

  • Since all Butterworth filters have a response that is reduced by 3 dB at the resonant frequency, it is often common to specify Butterworth filters with γmax = 3 dB.
  • The slope o is - 20 n dB/decade.f the filter magnitude response at ω = 0 is zero, and at high frequencies, ω 〉〉 ωo,

(^45) 0.1 1 10

40

35

30

25

20

15

10

5

0

5

ω

n = 1 n = 2

n = 3 n = n = 6 n = 5

Figure 9.3-1 Butterworth Filter Frequency Response.

  • For a Butterworth filter, the roots of an derivatives of the magnitude of the response are zero at n th order polynomial are chosen so that the first 2 ω = 0. These roots can easily be n - 1 derived to be: rk , n = ωo e^ "^ #^ $^^ j^ (^2 k +^2 nn −^1 )^ π%^ &^ ' , k = 1, 2, ..., n , (9.3-5) where complex plane on a circle of radius, rk,n is the k th root of the n th order Butterworth polynomial. These roots lie in the ω Butterworth polynomials have one real root ato, and are separat - ωo and (ed by an angle of by n - 1)/2 complex conjugate pairs.^ π^ / n.^ Odd order Even order polynomials have n /2 complex conjugate pairs and no real roots.
  • There are many cases where the edge of the passband must be defined by a variation in the gain, γmax. In those situations a slight variation in definition is necessary. The Butterworth polynomials can be defined with a third degree of freedom:

Bn ( ω ) = 1 + ε^2! " # ω^ ωc $ % &^2 n (9.3-7)

Here polynomials is: ωc is the edge frequency of the passband. At ω = ωc, the magnitude of all Butterworth

Bn ( ω ) = γmax = 1 + ε^2 (9.3-8)

  • Notice that previously. ε = 1 relates to the standard definition of Butterworth polynomials given
  • Once n and ε are known the resonant frequency of the filter can be determined: (9.3-9)
  • The order of the filter can now be obtained by solving Equation 9.3-7:

Bn ( ω ) = 1 + ε^2! " # ω^ ωc $ % &^2 n = 10 λm^20 in

  • The resonant frequency of the filter is given by Equation 9.3-9:

High-pass Butterworth Characterization:

  • High quantities-pass Butterworth filters are characterized by Butterworth polynomials with the ω and ωo interchanged. This interchange has the effect of adding two zero- frequency zeroes to the transfer function.

Bn ( ω ) = 1 +^! " # ω ωo $ % &^2 n = 1 + ε^2! " # ω ωc $ % &^2 n (9.3-10)

where (9.3-11)