network theory · 2014-09-29 · 2. the poles and zeros are simple and lie on the jw axis 3. the...

NETWORK THEORYNETWORK THEORY

LECTURE 4SECTION-D:TYPES OF FILTERS AND THEIR CHARACTERISTICSD:TYPES OF FILTERS AND THEIR CHARACTERISTICS

PROPERTY 3. L-C IMMITTANCE FUNCTION

• 3.The poles and zeros interlace on the axis.

0 +1

+2

+3

+4

X(w)

Highest power: 2n -> next highest power must be 2n

They cannot be missing term. Unless?

if s=0, 55 1 0b s b s

C IMMITTANCE FUNCTION

3.The poles and zeros interlace on the axis.j

+4

> next highest power must be 2n-2

They cannot be missing term. Unless?

s=0, 1/ 4 (2 1) / 41

5

( ) i kk

bs eb

2 2 2 2 2 21 3

2 2 2 2 2 22 4

( )( )...( )...( )( )( )...( )...

K s s sZ ss s s s

We can write a general L-C impedance or admittance as

0 2 42 2 2 2

2 4

2 2( ) ...K K s K sZ s K ss s s

Since these poles are all on the jw axis, the residues must be real and positive in order for Z(s) to be positive real .S=jw Z(jw)=jX(w) (no real part)

2 2 2 2 2 21 3

2 2 2 2 2 22 4

( )( )...( )...( )( )...( )...

i

j

K s s ss s s s

C impedance or admittance as

2 42 2 2 2

2 4

2 2( ) ...K s K sZ s K ss s s

Since these poles are all on the jw axis, the residues must be real and positive in order for Z(s) to be positive real .

Z(jw)=jX(w) (no real part)

2 20 2 22 2 2

2

( )( )( )

K KdX Kd

Since all the residues Ki are positive, it is eay to see that for an L-C function

Ex)

2 23

2 2 2 22 4

( )( )( )( )

Ks sZ ss s

( )jX j

X(w)

0w2

w3

w4

2 22 2

2 2 2

( ) ...( )

( ) 0d Xd

Since all the residues Ki are positive, it is eay

2 23

2 2 2 22 4

( )( )( )( )

KjX j

ww4

PROPERTIES 4 AND 5. LIMMITTANCE FUNCTION

• The highest powers of the numerator and denominator must differ by unity; the lowest powers also differ by unity.

• There must be either a zero or a pole at the origin and infinity.

PROPERTIES 4 AND 5. L-C IMMITTANCE FUNCTION

The highest powers of the numerator and denominator must differ by unity; the lowest powers also differ by unity.

There must be either a zero or a pole at the origin and infinity.

SUMMARY OF PROPERTIES

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

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

3. The poles and zeros interlace on the jw axis.

4. The highest powers of the numerator and denominator must differ by unity; the lowest powers also differ by unity.

5. There must be either a zero or a pole at the origin and infinity.

( )LCZ s

SUMMARY OF PROPERTIES

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

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

3. The poles and zeros interlace on the jw axis.

4. The highest powers of the numerator and denominator must differ by unity; the lowest powers also differ by unity.

5. There must be either a zero or a pole at the origin and

EXAMPLES

2

2 2

( 4)( )( 1)( 3)

Ks sZ ss s

Z s

2 2

2 2

( 1)( 9)( )( 2)( 10)K s sZ ss s

2 2

2

2( 1)( 9)( )( 4)

s sZ ss s

5 3

4 2

4 5( )3 6

s s sZ ss s