Analog Filter Explained

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26-1

26.1 Introduction

Analog lters are essential in many dierent systems that electrical engineers are required to design

in their engineering career. Filters are widely used in communication technology as well as in other

applications. Although we discuss and talk a lot about digital systems nowadays, these systems alwayscontain one or more analog lters internally or as the interace with the analog world [SV01].

Tere are many dierent types o lters such as Butterworth lter, Chebyshev lter, inverse Chebyshev

lter, Cauer elliptic lter, etc. Te characteristic responses o these lters are dierent. Te Butterworth

lter is at in the stop-band but does not have a sharp transition rom the pass-band to the stop-band

while the Chebyshev lter has a sharp transition rom the pass-band to the stop-band but it has the

ripples in the pass-band. Oppositely, the inverse Chebyshev lter works almost the same way as the

Chebyshev lter but it does have the ripple in the stop-band instead o the pass-band. Te Cauer lter

has ripples in both pass-band and stop-band; however, it has lower order [W02, KAS89]. Te analog

lter is a broad topic and this chapter will ocus more on the methodology o synthesizing analog lters

only (Figures 26.1 and 26.2).

Section 26.2 will present methods to synthesize our dierent types o these low-pass lters. Tenwe will go through design example o a low-pass lter that has 3 dB attenuation in the pass-band, 30 dB

attenuation in the stop-band, the pass-band requency at 1 kHz, and the stop-band requency at 3 kHz

to see our dierent results corresponding to our dierent synthesizing methods.

26.2 Methods to Synthesize Low-Pass Filter

26.2.1 Butterworth Low-Pass Filter

ppass-band requency

sstop-band requency

pattenuation in pass-band

sattenuation in stop-band

26Analog Filter Synthesis26.1 Introduction ....................................................................................26-126.2 Methods to Synthesize Low-Pass Filter .......................................26-1

Butterworth Low-Pass Filter Chebyshev Low-Pass Filter Inverse

Chebyshev Low-Pass Filter Cauer Elliptic Low-Pass Filter

26.3 Frequency ransormations ........................................................26-10Frequency Transformat ions; Low-Pass to High-Pass Frequency

Transformat ions; Low-Pass to Band-Pass Frequency

Transformat ions; Low-Pass to Band-Stop Frequency

ransormation; Low-Pass to Multiple Band-Pass

26.4 Summary and Conclusion ...........................................................26-13Reerences ..................................................................................................26-13

Nam PhamAuburn University

Bogdan M.Wilamowski

Auburn University

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26-2 Fundamentals of Industrial Electronics

Butterworth response (Figure 26.3):

T j

n n( )

/

2

20

2

1

1=

+ ( )

Tere are three basic steps to synthesize any type o low-pass lters. Te rst step is calculating the

order o a low-pass lter. Te second step is calculating poles and zeros o a low-pass lter. Te third step

is design circuits to meet pole and zero locations; however, this part is another topic o analog lters, so

it will be not be covered in this work [W90, WG05, WLS92].

All steps to design Butterworth low-pass lter.

Step 1: Calculate order o lter:

n ns p

s p

= log[( )( )]

log( / )

/ / /10 1 10 11010 1 2

( needs to be rooundup to integer value)

[dB]

20

40

Magnitude

[dB]

20

40

Magnitude

FIGURE 26.1 Butterworth lter (lef), Chebyshev lter (right). AQ1

[dB]

20

40

Magnitude

[dB]

20

40

Magnitude

FIGURE 26.2 Inverse Chebyshev lter (lef), Cauer elliptic lter (right).

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Analog Filter Synthesis 26-3

Step 2: Calculate pole and zero locations:

Angle in is odd:

=

=

k

nk

n1800 1

1

2; , , ,

Angle in is even:

= +

=

0 5 180 0 12

2. ; , , ,

k

nk

n

Normalized pole locations:

a b

k k= = =

cos( ); sin( ); ( )

0 1

0

1 2

10 10 1 410 1 10 1

1

2=

=( )

[( ) / ( )];

/

/ / /( )

p s

n k

ks p

Qa

Step 3: Design circuits to meet pole and zero locations (not covered in this work) (Figure 26.4).

Example:

Step 1: Calculate order o flter:

n n=

= =log[(10 1)(10 1)]

log(3000 1000)3.1456 4

30 /10 3/10 1/ 2

/

Step 2: Calculate pole and zero locations

Normalized values o poles and 0 and Q:

0.38291 + 0.92443i 1.00059 1.30656

0.38291 0.92443i 1.00059 1.30656

0.92443 + 0.38291i 1.00059 0.54120

0.92443 0.38291i 1.00059 0.54120

Normalized values o zeros none.

0 dB

s

s

p

p

FIGURE 26.3 Butterworth lter characteristic.

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26.2.2 Chebyshev Low-Pass Filter

ppass-band requency

sstop-band requency



Chebyshev response (Figure 26.5):

T j Cn( ) / ( ( ))

2 2 21 1= +

Step 1: Calculate order o lter:

ns p

s p s p

=

+

ln[ * ( ) / ( )]

log[( (( )

/ / /

/ ) /

4 10 1 10 1

1

10 10 1 2

2 2

)) ]/1 2( needs to be roundup to integer value)n

[dB]

20

40

Magnitude

Phase

s -plane

90

180

270

FIGURE 26.4 Pole-zero locations, magnitude response, and phase o Butterworth lter.

Frequencies at whichCn= 0

Frequencies at which|Cn| = 1

|T6(j)|

Is here

Is here 1

00

1/1+2

FIGURE 26.5 Chebyshev lter characteristic.

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Analog Filter Synthesis 26-5


= +

+

90

90 1 180

n

k

n

( )

=

=

10 1110 1 2

1p

n

//

;sinh ( / )

a b a b Qa

k k k k k K k

k

= = = + =sinh( )cos( ); cosh( )sin( ); ;

2 2

2

Step 3: Design circuits to meet pole and zero locations (not covered in this work) (Figure 26.6).

Example:

Step 1: Calculate order o flter:

n =

+

ln[4 *(10 1) (10 1)]

log[(3000 1000) ((3000 1

30 /10 3/10 1/ 2

2

/

// 0000 ) 1) ]2.3535 3

2

= =1/2

n

Step 2: Calculate pole and zero locations

Normalized values o poles and 0 and Q:

0.14931 + 0.90381i 0.91606 3.06766

0.14931 0.90381i 0.91606 3.06766

0.29862

Normalized values o zeros none.

[dB]

Magnitude

x

x

s-plane

Phase

30

40

90

180

FIGURE 26.6 Pole-zero locations, magnitude response, and phase o Chebyshev lter.

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26.2.3 Inverse Chebyshev Low-Pass Filter

ppass-band requency

sstop-band requency



Inverse Chebyshev response (Figure 26.7):

T j

C

CIC

n

n

( )( / )

( / )

22 2

2 2

1

1 1=

+

Te method to design the inverse Chebyshev low-pass lter is almost the same as the Chebyshev low-

pass lter. It is just slightly dierent.

Step 1: Calculate order o lter

n = order o the Chebyshev lter


Pa b i n

i k npick k

i=+

= =

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Analog Filter Explained