NETWORK THEORYNETWORK THEORY

LECTURE 3SECTION-D:TYPES OF FILTERS AND THEIR CHARACTERISTICSD:TYPES OF FILTERS AND THEIR CHARACTERISTICS

Example - Passive Second Order Filter Function

T(dB)

0

01

0)(

22

2

sasjsasQ

sas

sQ

ssT o

oo

o

0 dBQ

3

* Low pass filter

Passive Second Order Filter Function

20

02

12

2

)()()(

sQ

s

asasasVsVsT o

i

o

General form of transfer function

20012

22

2

22

0a,0a

thfilter wiorder second a of form thehas This

1

11

1

1

1

11

1

1

1)(

111

111

)()()(

a

LCRRCQand

LCwhere

sQ

sLCRC

ss

LC

LCsRLs

sRCRsL

RsRC

sRCRsL

sRCR

ZRZRZ

sT

sosRCR

RsC

RZ

RZwhere

ZRZRZ

sVsVsT

oo

oo

o

LC

C

C

C

LC

C

i

o

Example - Passive Second Order Filter Function

00

1

0

)(

2

2

22

0

sassjsat

sasss

sQ

s

Qs

sT

o

o

oo

T(dB)

0

0 dB

-3 dB

Qo

4

Passive Second Order Filter Function* Bandpass filter

01a,0a

thfilter wiorder second a of form thehas This

1

1

11

1

11

1)(

111

111

)()()(

012

222

2

2

2

2

aQRC

LCRRCQand

LCwhere

sQ

s

sRC

LCRCss

RCs

LCsRsLsL

LCssLR

LCssL

RZZZZ

sT

soLCs

sL

sLsC

ZZ

ZZwhere

RZZZZ

sVsVsT

o

oo

oo

LC

LC

LC

LC

LC

LC

i

o

20

02

12

2

)()()(

sQ

s

asasasVsVsT o

i

o

General form of transfer function

Low Pass Butterworth Filter Design

CRRCss

CRVVT(s)

i

o

12

1

222

22

o CRso

RC 11

* Given the filter specification (0), we can determine the R and C.* One specification, two parameters – R and C * Pick a convenient value, say C = 5 nF.* Calculate R from C and ωo.

Hurwitz Polynomials

5

Low Pass Butterworth Filter Design

sasjsasj

sas

o

02/

01

T(dB)

0

0 dBQ(dB)

o1

), we can determine the R and C.R and C

NOTE 40 dB/dec

16sec)/1026.1)(105(

11 ,5

sec/1026.122Given :

79

70

radxFxCR

thennFCChooseradxMHzExample

o

Hurwitz Polynomials

PROPERTY 1. L-C IMMITTANCE FUNCTION

• 1. ZLC (s) or YLC (s) is the ratio of odd to even or even to odd

polynomials.

• Consider the impedance Z(s) of passive one

As we know, when the input current is dissipated by one-port network is zero:

Average Power= =0

1 1

2 2

( ) ( )( )( ) ( )

M s N sZ sM s N s

1 Re[ ( )]2

C IMMITTANCE FUNCTION

(s) is the ratio of odd to even or even to odd

Consider the impedance Z(s) of passive one-port network.

(M is even N is odd)

As we know, when the input current is I, the average power port network is zero:

= =0

( ) ( )( ) ( )

M s N sM s N s

21 Re[ ( )]2

Z j I

1 2 1 2( ) ( ) ( ) ( ) 0M j M j N j N j

1 2 1 22 2

2 2

( ) ( ) ( ) ( )( ) 0( ) ( )

M s M s N s N sEvZ sM s N s

1 20M N OR

1 1

2 2

( ) , ( )M NZ s Z sN M

Z(s) or Y(s) is the ratio of even to odd or odd to even!!

1 2 1 2( ) ( ) ( ) ( ) 0M j M j N j N j

1 2 1 22 2

2 2

( ) ( ) ( ) ( )( ) 0( ) ( )

M s M s N s N sM s N s

2 10M N

1 1

2 2

( ) , ( )M NZ s Z sN M

Z(s) or Y(s) is the ratio of even to odd or odd to even!!

PROPERTY 2. L-C IMMITTANCE FUNCTION

• 2.The poles and zeros are simple and lie on the axis.

• Since both M and N are Hurwitz, they have only imaginary roots, and it follows that the poles and zeros of Z(s) or Y(s) are on the imaginary axis.

• Consider the example

1 1

2 2

( ) , ( )M NZ s Z sN M

C IMMITTANCE FUNCTION

2.The poles and zeros are simple and lie on the

Since both M and N are Hurwitz, they have only imaginary roots, and it follows that the poles and zeros of Z(s) or Y(s) are on the

j

1 1

2 2

( ) , ( )M NZ s Z sN M

4 24 2 0

5 35 3 1

( ) a s a s aZ sb s b s b s

Ex) highest order of the numerator : 2n denominator can either be 2n-1 (simple pole at s= ) or

the order can be 2n+1 (simple zero at s= ).

Impedance function cannot have multiple poles or zeros

axis.

In order for the impedance to be positive real positive.

4 24 2 0

5 35 3 1

( ) a s a s aZ sb s b s b s

The highest powers of the numerator and the denominator polynomials can differ by, at most, unity.

Ex) highest order of the numerator : 2n -> highest order of the 1 (simple pole at s= ) or

the order can be 2n+1 (simple zero at s= ).

cannot have multiple poles or zeros on the

n order for the impedance to be positive the coefficients must be real and

positive.

j

The highest powers of the numerator and the denominator polynomials can differ by, at most, unity.