SEE2043 - C6Note Filter Design.pdf (683k)

Signals and Networks

(SEE 2043)

Why Analog Passive Filter?

Filter deal with

analog

signal.

Make up of

passive

components (capacitor, inductor, and

resistor).

The Design objectives:

To obtain the transfer function, and

To obtain the passive filter circuit that meet the design

specification.

Analog Passive Filter

The design specification:

1. Passband, Transition band, and Stopband.

2. Gain and Attenuation.

3. Type of Filter response.

The Design Specification

• Passband, Transition band, and Stopband. • Gain and Attenuation.

• Type of Filter response.

Analog Passive Filter

Ideal filter response

Analog Passive Filter

Analog Passive Filter

passband stopband

( )

j

ω

(dB)

H

ω

0

Practical filter response

passband stopband

( )

j

ω

(dB)

H

ω

0

Transition band

The Design Specification

• Passband, Transition band, and Stopband.

• Gain and Attenuation. • Type of Filter response.

Analog Passive Filter

Analog Passive Filter

The Design Specification

• Passband, Transition band, and Stopband.

• Gain and Attenuation.

• Type of Filter response. Gain is denoted by

α

.

Attenuation is denoted by

A

.

Practical Filter Response

They are opposite of each other.

Example:

α

A

( )

j

ω

(dB)

H

ω

α

Transition band

α

ω

ω

The passband edge frequency is known as

ω

.

2. Sketch a practical filter response with following specification:

(i) The passband gain is between 0 dB and -2.5 dB for the

frequency range of (0<

ω

<15) rad/s.

(ii) The stopband gain must not exceed -22 dB for frequency

(

ω

>30) rad/s.

Analog Passive Filter

Analog Passive Filter

The Design Specification

• Passband, Transition band, and Stopband.

• Gain and Attenuation.

• Type of Filter response.

1. Sketch a practical filter response with following specification:

(i) The passband attenuation must not exceed 2.5 dB, and the

stopband gain should less than -22 dB for the frequency

ω

= 100

rad/s

and

ω

= 100 rad/s

.

Analog Passive Filter

Analog Passive Filter

The Design Specification

• Passband, Transition band, and Stopband. • Gain and Attenuation.

• Type of Filter response.

α

α

( )

j

ω

(dB)

H

ω

ω

ω

0

ω

( )

j

ω

(dB)

H

A

α

ω

ω

0

Signals and Networks

(SEE 2043)

First described by British engineer, Stephen Butterworth in his paper, “On the

Theory of Filter Amplifiers” appeared in

Wireless Engineer

in 1930.

It is designed to have a magnitude plot which is as flat as possible in the

passband but slower roll-off towards the stopband. Also known as ‘

maximally flat

magnitude

’ filters

Maximal flat magnitude in the

passband

slower roll-off

Butterworth Filter

Butterworth Filter

α

( )

j

ω

(dB)

H

ω

ω

α

ω

The design procedure of Butterworth filter are to:

1. Determine the order, n

of the filter.

2. Determine the actual cut-off frequency,

_ω

_c

.

3. Determine the transfer function, H(s) of the filter.

Design of Butterworth Filter

• Determine the order of the filter based on specifications given. • Determine the actual cut-off frequency.

• Determine the transfer function of the filter.

LP-BF is used as the reference in this lecture. High Pass and Band Pass

Butterworth can be modified from this LP design.

( )

n

c

j

H

2

1

1

ω

ω

ω

ω_c: 3dB cutoff frequency

n : order of the filter

Design of Butterworth Filter

Design of Butterworth Filter

passband

stopband

α

( )

j

ω

(dB)

H

ω

ω

α

ω

transition band

0

-3

ω

The magnitude response of nth_{-order LP-BF:}

Design of Butterworth Filter

• Determine the order of the filter based on specifications given.

• Determine the actual cut-off frequency.

Design of Butterworth Filter

• Determine the order of the filter based on specifications given.

• Determine the actual cut-off frequency.

• Determine the transfer function of the filter.

Design of Butterworth Filter

Design of Butterworth Filter

(

)

(

)

( )

A_s p

n

ω

log

2

1

10

/

1

10

log

Note:n must be integer (1,2,3,…) ω_sncan be obtained from Table F.1

Example:

For LPF, ω_sn= ω_s/ ω_p

Filter order, n

passband

stopband

α

( )

j

ω

(dB)

H

Design of Butterworth Filter

• Determine the order of the filter based on specifications given.

• Determine the actual cut-off frequency.

• Determine the transfer function of the filter.

Design of Butterworth Filter

Design of Butterworth Filter

n A

p c

p 2

1

10

₁

10

1

ω

ω

Cut-off Frequency,ω_C

passband

stopband

α

( )

j

ω

(dB)

H

ω

ω

α

ω

transition band

0

-3

Design of Butterworth Filter

• Determine the order of the filter based on specifications given. • Determine the actual cut-off frequency.

• Determine the transfer function of the filter.

Design of Butterworth Filter

Design of Butterworth Filter

Obtain the normalized transfer function,

H

(

s

) based on the filter order,

n

obtained earlier.

Obtain the actual transfer function,

H

(

s

) of the filter by replacing

s

in the

polynomial

H

(

s

)with its transformation pairs (refer Table F.5 Teaching Module)

depend on the type of filters (e.i. LPF, HPF, BPF, BSF).

( )

_{( )}

1

1

1

1 1

1

s

...

a

s

a

s

s

B

s

H

n n n

n

B

(s) can be obtained from

Table F.6(a)

or

Table F.6(b)

(Teaching Module).

For example:

Determine the order of the Butterworth filter with following

specifications.

ω

_p

=100,

ω

_s

=200,

_p

=0.5 dB and

_s

=20 dB.

Solution:

(

)

(

)

( )

5

8

.

4

log

2

1

10

/

1

10

log

10

10 / 10

/ 10

n

n

sn p s

ω

Design a LP Butterworth filter with the following specification:

(i) The passband gain is between 0 dB and -2.5 dB for the

frequency range of (0<

ω

<15)

(ii) The stopband gain must not exceed -22 dB for frequency

(

ω

>30)

p

( )

j

ω

(

dB

)

H

ω

ω

s

ω

Step 1: Determine the filter order, n.

n=4

Step 2: Determine the cut-off frequency,

ω

.

ω

=15.477

Step 3: Determine the normalized transfer function, H

(s).

Step 4: Determine the actual transfer function H(s).

1

6131

.

2

4142

.

3

6131

.

2

1

)

(

1

)

(

Signals and Networks

(SEE 2043)

The filter characteristics are derived from Chebyshev polynomials

,

named after a

Russian mathematician, Pafnuty Chebyshev (1821-1894).

Better roll-off (steeper) compared to Butterworth filter but more passband

ripples.

Consist of Type I and Type II (Type I will be discussed)

Ripple in the passband

Steeper roll-off

Chebyshev

Chebyshev

Filter

Filter

ω

( )

j

ω

(dB)

H

A

α

ω

ω

The design of Chebyshev filter will involve the following:

•Circuit design

•Transfer function computation

Chebyshev

Chebyshev

Filter

Filter

R

=1

C

L

V

+

-v

C

L

C

L

R

( )

_{( )}

0 1

1

1

s

...

a

s

a

a

s

K

s

T

K

s

H

n n

n

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

The design procedure of Chebyshev filter circuit are to:

1. Determine the order and the cut-off frequency of the filter.

2. Determine the value for normalized components (C, L and R

).

3. Determine the actual value of the components.

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter. • Determine the value of normalized components.

The magnitude response of nth_{-order LP-CF:}

( )

p n n

T

j

H

ω

ω

ω

2 2

1

1

: ripple factor

ω_p : passband edge frequency

T_n : Chebyshev polynomial

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

transition band

ω

stopband

( )

j

ω

(dB)

H

passband

-3

A

α

ω

ω

ω

0

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter.

Chebyshev polynomial, T_n

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

p p n

n

T

ω

ω

ω

ω

cos

cos

1

p

T

ω

ω

T

ω

ω

ω

ω

1

T

2

T

T

,

n

1

p n p n p p n

_ω

ω

ω

ω

ω

ω

ω

ω

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter.

• Determine the value of normalized components. • Determine the actual value of the components.

ω

( )

j

ω

(dB)

H

α

ω

0

T_n fluctuates between 0 and 1 for ω< ω_p therefore,

( )

j

ω

1

or

H

( )

j

ω

(

dB

)

0

H

( )

( )

n

j

H

j

H

ω

ω

2 2

₁

1

log

20

)

dB

(

or

1

1

,

,

,

T_n= 0 :

Ripple factor,

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter.

• Determine the value of normalized components. • Determine the actual value of the components.

1

10

max

A

(

)

10

2 max

1

log

10

1

1

log

20

p

A

Maximum attenuation, A_max

Note:A_max will be given in the circuit design specification (0.1dB or 1dB). Refer Table F.2(a) or Table F.2(b)respectively.

transition band

ω

stopband

( )

j

ω

(dB)

H

passband

-3

A

α

ω

ω

ω

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

( )

1

cosh

10

1

10

1

cosh

1

sn

s

n

ω

Note:n must be integer (1,2,3,…)

ω_sncan be obtained from Table F.1

Example:

For LPF, ω_sn= ω_s/ ω_p

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter.

• Determine the value of normalized components. • Determine the actual value of the components.

ω

ω

1

cosh

1

cosh

n

Cut-off Frequency,ω_C

Filter order, n

transition band

ω

stopband

( )

j

ω

(dB)

H

passband

-3

A

α

ω

ω

ω

The design is based on the normalized circuit of Low Pass Chebyshev Filter (LP-CF) as illustrated below.

The filter order, n determines the number of passive components (L and C) need to be included in the circuit.

[e.g. 1st _{order: C}

1; 2nd order: C1 and L2; 3rd order: C1, L2 and C3; etc….]

The value of normalized components is obtained using Table F.2(a) for A_max = 0.1dB and

Table F.2(b) for A_max = 1dB.

Normalized value of R_p, C and L depends on the value of n obtained earlier.

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

R

=1

C

L

v

+

-v

C

L

C

L

R

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter.

• Determine the value of normalized components.

• Determine the actual value of the components.

Chebyshev

Chebyshev

Filter

Filter

–

–

Circuit Design

Circuit Design

The actual value of the components is obtained using Table F.3 where,

Normalized factor, R = R_L/R_P; R_S(actual) = R ; and ω_c as obtained earlier.

The transformation pairs depend on the type of filters.

Design of Chebyshev Filter Circuit

• Determine the order and the cut-off frequency of the filter. • Determine the value of normalized components.

• Determine the actual value of the components.

R

=

R

C

L

v

+

-v

C

L

C

L

R

RL

L

ω

R

C

C

ω

_C

R

L

ω

C

_R

_ω

_L

1

LPF

HPF

1. Below are the specifications of a Low Pass Chebyshev Filter:

•

Maximum passband attenuation is 0.1 dB.

•

Minimum stopband attenuation is 18 dB at

ω

= 2.5 rad/s.

•

ω

= 1 rad/s

•

Load resistance, R

= 100

Design the filter circuit and obtain the components actual value.

Circuit Design

Circuit Design

–

–

Practice

Practice

2. Design a High Pass Chebyshev Filter which meet these specifications:

•

Passband attenuation must not exceed 0.1 dB for

ω

> 200 krad/s.

•

Minimum stopband attenuation is 20 dB for

ω

< 100 krad/s.

The design of Chebyshev transfer function are to:

1. Determine the order, n of the filter.

2. Determine the normalized transfer function,

H

(

s

).

3. Determine the actual transfer function,

H

(

s

).

Chebyshev

Chebyshev

Filter

Filter

–

–

Transfer Function

Transfer Function

Design of Chebyshev Transfer Function

• Determine the order of the filter.

Chebyshev

Chebyshev

Filter

Filter

–

–

Transfer Function

Transfer Function

Design of Chebyshev Transfer Function

• Determine the order of the filter.

• Determine the normalized transfer function. • Determine the actual transfer function.

( )

1

cosh

10

1

10

1

cosh

1

sn

s

n

ω

nmust be integer (1,2,3,…)

ω_sncan be obtained from table F.1

Example:

For LPF, ω_sn= ω_s/ω_p

Maximum Attenuation,

A

(

)

10 2

10

10

1

1

1

1

20

log

log

A

1

10

max

A

Ripple Factor,

ε

Filter Order,

n

Note:A_max will be given in the transfer function design

Chebyshev

Chebyshev

Filter

Filter

–

–

Transfer Function

Transfer Function

Design of Chebyshev Transfer Function

• Determine the order of the filter.

• Determine the normalized transfer function.

• Determine the actual transfer function.

The normalized transfer function,

H

(

s

) for Low Pass Chebyshev

Filter (LP-CF) of n

_{order can be written as:}

( )

_{( )}

0 1

1

1

s

...

a

s

a

a

s

K

s

T

K

s

H

The Chebyshev polynomial,

T

(

s

) can be obtained from Table F.4.

K

is a constant which is determined by the value of

A

and filter

order, n can be written as:

Chebyshev

Chebyshev

Filter

Filter

–

–

Transfer Function

Transfer Function

Design of Chebyshev Transfer Function

• Determine the order of the filter.

• Determine the normalized transfer function.

• Determine the actual transfer function.

The actual transfer function,

H

(

s

) of the filter is obtained by

replacing

s

in the polynomial

H

(

s

)with its transformation pairs

(refer Table F.5 Teaching Module) depend on the type of filters

(e.i. LPF, HPF, BPF, BSF).

For example:

Low pass filter, LPF –

s

→

s

/

ω

Determine the transfer function of LP-CF with specifications:

•

A

= 3 dB in the passband region, 0

ω

12

•

Maximum stopband gain,

= -23 dB for

ω

> 18 rad/s.

Transfer Function

Step 1: Determine filter order, n

n = 3.47

n = 4.

Step 2: Determine

H

(

s

)

From Table F.4,

for n = 4: a

= 0.1769, a

= 0.4047, a

= 1.1691, a

= 0.5815.

since n is even,

Therefore,

Step 3: Determine

H

(

s

)

As for LPF – s

s/

ω

;

ω

= 12 rad/s, therefore,

1252

0

4125

1

1769

0

10

_.

.

.

a

K

Solution

Solution

( )

1769

0

4047

0

1691

1

5815

0

1252

0

4

_.

_s

_.

_s

_.

_s

_.

Thank for your attention

Wish all of you