Synthesis Two Ele

7/21/2019 Synthesis Two Ele

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Synthesis of one-port networks

with two kinds of elements

11.1 Properties of L-C immittance functions

11.2 Synthesis of L-C driving-point immittances



Contents

Properties of L-C immittance functions Properties

Examples of immittance and non-immittance

functions

Synthesis of L-C driving-point immittance Synthesis of L-C circuit

Examples of synthesis



Property 1. L-C immittance function

1. ZLC (s or !LC (s is the ratio of odd to even or even to odd

polynomials.

Consider the impedance Z(s of passive one-port net"or#.

($ is even % is odd

&s "e #no"' "hen the input current is I ' the average po"er

dissipated y one-port net"or# is )ero*

&verage Po"er = =0

(for pure reactive net"or#

1 1

2 2

( ) ( )( )( ) ( )

M s N s Z s M s N s

+=

+

21Re[ ( )]

2 Z j I ω



1 2 1 2( ) ( ) ( ) ( ) 0 M j M j N j N jω ω ω ω − =

1 2 1 2

2 2

2 2

( ) ( ) ( ) ( )( ) 0

( ) ( )

M s M s N s N s EvZ s

M s N s

−= =

+

1 20 M N = = OR 2 10 M N = =

1 1

2 2

( ) , ( ) M N

Z s Z s N M

= =

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



Property ". L-C immittance function

+.,he poles and )eros are simple and lie on the

axis.

Since oth $ and % are ur"it)' they have only

imaginary roots' and it follo"s that the poles and )eros

of Z(s or !(s are on the imaginary axis.

Consider the example

jω

1 1

2 2

( ) , ( ) M N

Z s Z s

N M

= =

4 2

4 2 0

5 3

5 3 1

( ) a s a s a

Z sb s b s b s

+ +=

+ +



Ex highest order of the numerator * +n - highest order of the

denominator can either e +n-1 (simple pole at s/ or

the order can e +n01 (simple )ero at s/ .

mpedance function cannot hae multiple poles or #eros on the

axis.

$n order for the impedance to e positive

real the coefficients must e real and

positive.

4 2

4 2 0

5 3

5 3 1

( ) a s a s a

Z sb s b s b s

+ +=

+ +

jω

∞

,he highest po"ers of the numerator and the denominator

polynomials can differ y' at most' unity.

∞



Property %. L-C immittance function

%.&he poles and #eros interlace on the a'is. jω

0 +1

+2

+3

+4

ω

(w)

i*hest power+ "n -, ne't hi*hest power must e "n-"

&hey cannot e missin* term. nless/

if s025

5 1 0b s b s+ = 1/ 4 ( 2 1) / 41

5

( ) i k

k

b s e

b

π −

=



2 2 2 2 2 2

1 3

2 2 2 2 2 2

2 4

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

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

i

j

K s s s Z s

s s s s

ω ω ω

ω ω ω

+ + +=

+ + +

3e can write a *eneral L-C impedance or admittance as

0 2 4

2 2 2 2

2 4

2 2( ) ...

K K s K s Z s K s

s s sω ω ∞

= + + + +

+ +

Since these poles are all on the 2" axis' the residues must e real andpositive in order for Z(s to e positive real .

S04w Z(4w)04(w) (no real part)



2 20 2 2

2 2 2

2

( )( )...

( )

K K dX K

d

ω ω ω

ω ω ω ω ∞

+= + + +

+

( )0

dX

d

ω

ω

≥

Since all the residues 3i are positive' it is eay

to see that for an L-C function

Ex)

2 2

3

2 2 2 2

2 4

( )( )

( )( )

Ks s Z s

s s

ω

ω ω

+=

+ +

2 2

3

2 2 2 2

2 4

( )( )

( )( )

K jX j

ω ω ω ω

ω ω ω ω

− += +

− + − +

w

X(w)

0w2

w3

w4



Properties 5 and 6. L-C immittance

function

,he highest po"ers of the numerator and

denominator must differ y unity4 the lo"est

po"ers also differ y unity.

,here must e either a )ero or a pole at the

origin and infinity.



Summary of properties

1. or !LC (s is the ratio of odd to even or even toodd polynomials.

+. ,he poles and )eros are simple and lie on the 2" axis

5. ,he poles and )eros interlace on the 2" axis.

6. ,he highest po"ers of the numerator and denominatormust differ y unity4 the lo"est po"ers also differ y unity.

7. ,here must e either a )ero or a pole at the origin andinfinity.

( ) LC Z s



Examples

2

2 2

( 4)( )

( 1)( 3)

Ks s Z s

s s

+=

+ +

5 3

4 2

4 5( )

3 6

s s s Z s

s s

+ +=

+

2 2

2 2

( 1)( 9)( )

( 2)( 10)

K s s Z s

s s

+ +=

+ +

2 2

2

2( 1)( 9)( )

( 4)

s s Z s

s s

+ +=

+



11." Synthesis of L-C 7riin* point

immittances

L-C immittance is a positive real function "ith

poles and )eros on the 2" axis only.

,he synthesis is accomplished directly from the partial

fraction .

8(s is impedance - then the term *

capacitor of 193 farads

the 3(infinites is an inductance of 3(infinite henrys.

0 2 4

2 2 2 22 4

2 2( ) ...

K K s K s Z s K s

s s sω ω ∞

= + + + +

+ +

0 / K s



s a parallel tan# capacitance and inductance.

'

8or Z(s) partial fraction

2 22 /( )i i K s s ω +

22 /

i i K ω 1/ 2 i K

2 2

2 2

9 152( 1)( 9) 2 2( ) 2

( 4) 4

s s s

Z s s s s s s

+ += = + +

+ +

2/15f

2/9 f 2h 15/8 h



8or Y(s) partial fraction

In admittance

2 2

2 2 2 2

1 3

( 2)( 4) 2 2

( ) ( 1)( 3) 3 1

s s s s s

Y s s s s s s

+ += = + +

+ + + +

(!)1f

3/2 f

2/3 h2h

1/" f



9nother methodolo*y

:sing property 6 ;,he highest po"ers if numerator and

denominator must differ y unity4 the lo"est po"ers also

differ y unity.< ,herefore' there is al"ays a )ero or a pole at s/infinite .

suppose Z(s numerator*+n 'denominator*+n-1

this net"or# has pole at infinite. - "e can remove this

pole y removing an impedance L 1 s

Z+ (s / Z(s - L 1 s

=egree of denominator * +n-1 numerator*+n-+ Z+ (s has )ero at s/infinite.

!+ (s /19 Z+ (s ' !5 (s / !+ (s - C+ s



This infinite term removing process continue until

the remainder is zero.

Each time e remove the pole! e remove aninductor or capacitor depending upon hether the

function is an impedance or an admittance.

"inal synthesized is a ladder hose series arms

are inductors and shunt arms are capacitors.



5 3

4 2

2 12 16( )

4 3

s s s Z s

s s

+ +=

+ +

5 3 3

2 4 2 4 2

2 12 16 4 10( ) 2

4 3 4 3

s s s s s Z s s

s s s s

+ + += − =

+ + + +

24 2

3 3 3

3 34 3 1 2( )

4 10 4 4 10

s s s

Y s s s s s s

++ +

= − =

+ +

4 32

8 2

( ) 333

2

s

Z s Z s s

= − =

+2h

2h

1/4 f 3/4 f

2/3 h

8/3 h



,his circuit (Ladder called as Cauer

ecause Cauer discovered the continues fraction

method.

>ithout going into the proof of the statement m

in can e said that oth the 8oster and Cauer

form gice the minimum numer of elements for aspecified L-C net"or#.



Exam#$e %f &a'e eth%d

2 2

2

( 1)( 3)( )( 2)

s s Z s

s s

+ +=

+

5/4 h 5 h

2/25 f 2/3 f

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Synthesis Two Ele