Cascaded Lossless Commensurate Line Synthesis

(1)

Cascaded Lossless Commensurate Line Synthesis

Metin engül

Kadir Has University

Electronics Engineering Department

Cibali-Fatih, Istanbul, Turkey

msengul@khas.edu.tr

Abstract— In this paper, a synthesis algorithm for cascaded lossless commensurate lines is summarized. The algorithm is based on transfer matrix factorization. Firstly the characteristic impedance of the extracted element is calculated and then after extracting the element, the reflection factor of the remaining network is calculated. This process is repeated until the termination resistance is reached. An example is included, to illustrate the implementation of the algorithm.

Keywords-synthesis; commensurate lines; lossless networks

I. I

NTRODUCTION

Especially at microwave frequencies, distributed element

networks are preferred. In lots of microwave filter and

matching network designs, finite homogenous transmission

lines of commensurate lengths have been used [1, 2]. This

approach is based on that the distributed networks composed

of commensurate lengths of transmission lines could be

regarded as lumped element networks under the

transformation [3],

)

tanh(

τ

λ

=

p

(1)

where

p

=

σ

+

j

ω

is the complex frequency and

τ

is the

commensurate delay of the transmission line.

In literature, characteristic impedance of the extracted

commensurate line is calculated by using the following

formula [4], if

S

(

λ

)

is the given reflection factor,

)

1 (

1 )

1 (

1

S

Z

−

+

=

(2)

Then, the reflection factor of the remaining network is [4]

λ

=

₋

−

₋

+

1

1 )

1 (

)

(

1 )

1 (

)

(

)

(

S

_R

.

(3)

As can be seen from Eq. (3), to get a degree reduction, the

denominator must has a root at

λ

=

−

1 , and numerator at

1 +

=

λ

. But in the new method, there is no need to find roots

at 1

λ

=

±

, to get a degree reduction.

In literature, many researches have been worked on the

analysis [5-8] and synthesis problem [9-17] of

distributed-element networks. For instance in [18], a transformation is

proposed, in [19], input impedance function is utilized in

synthesis process. But in the synthesis method presented here,

the network is thought as a lossless, reciprocal two-port

expressed in Belevitch form [20], and directly the reflection

factor is used.

In [21], the synthesis algorithm presented here has been

proposed. But when the number of transmission lines is

increased, the coefficients used in the algorithm get larger. So

a coefficient normalization step must be inserted to get more

precise element values. In the updated algorithm presented in

this paper, this coefficient normalization step has been

included.

In the following section, transfer matrix factorization is

explained briefly. Then, the modified synthesis algorithm is

given. Finally, an example is presented, to illustrate the

implementation of the algorithm.

II. T

RANSFER

M

ATRIX

F

ACTORIZATION

Canonic form of the scattering transfer matrix

{ }

T of a

lossless, reciprocal two-port is defined as [1],

»

¼

º

«

¬

ª

=

g

h

g

f

T

* *

1 μ

μ

,

(4)

where

=

*

=

±

1 f

f

μ

is a unimodular constant, g is a strictly

Hurwitz real polynomial. These polynomials satisfy the

Feldtkeller equation,

gg

_*

=

hh

_*

+

ff

_*

(where “*” represents

paraconjugation).

Fig. 1. Cascade decomposition of a two-port.

It is desired to decompose the lossless, reciprocal two-port

{ }

N into two cascade connected lossless two-ports

{

N ,

_a

N

_b

}

which are also lossless and reciprocal (Fig. 1). This means to

factor the scattering transfer matrix

{ }

T into a product of two

scattering transfer matrices,

a

N

b

N

R

T

978-1-4244-3896-9/09/$25.00 ©2009 IEEE

295

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b a

T

=

⋅

,

(5a)

where

¸¸¹

· ¨¨©

§

=

a a a a a a a a

g

h

g

f

T

* *

1 μ

μ

and

¸¸¹

· ¨¨©

§

=

b b b b b b b b

g

h

g

f

T

* *

1 μ

μ

. (5b)

The polynomials

{

g

a

,

h

a

,

f

a

}

and

{

g

b

,

h

b

,

f

b

}

have the

same properties as

{

g

,

h

,

f

}

, and must satisfy the Feldtkeller

equation. Namely,

b a a b a

g

h

g

=

+

μ

_*

,

(6a)

b a a b a

g

h

=

+

μ

_*

,

(6b)

b a

f

=

,

(6c)

b a

μ

=

.

(6d)

So if one writes

T

_b

=

T

_a−1

T

, two equations can be reached

as,

* a a a a a b

f

gh

hg

h

μ

−

=

,

(7a)

* * * a a a a b

f

hh

gg

g

=

−

.

(7b)

Now, the problem is to solve (7) in the unknown

polynomials

{

g

_a

,

h

_a

,

g

_b

,

h

_b

}

subject to the Feldtkeller

equation with

g

_a

and

g

_b

being strictly Hurwitz polynomials.

The problem has been solved by Fettweis by using a

modified formulation of the factorization problem [22, 23]. In

[23], a different set of equations are chosen as the basis for the

solution instead of solving (7).

III. A

LGORITHM

Fig. 2 Commensurate line extraction.

Consider the circuit shown in Fig. 2.

g

(λ

)

,

h

(λ

)

and

)

(

λ

f

polynomials are given as follows,

n n g g g g g(

λ

)= 0+ 1

λ

+ 2

λ

2+!+

λ

, (8a)

n n

h

(

λ

)

=

0

+

1

λ

+

2

λ

2

+

!

+

λ

,

(8b)

2 / 2

₎

1 (

)

(

n

f

λ

=

−

λ

.

(8c)

Characteristic impedance

Z

₁

of the first commensurate line

can be calculated as,

)

1 (

)

1 (

)

1 (

)

1 (

1

h

g

h

g

Z

−

+

=

.

(9)

Then,

g

(

λ

)

,

h

(

λ

)

and

f

(

λ

)

polynomials of the remaining

network are obtained as,

¦

₌ −

=

n j j j

D

g

1 1

)

(

λ

,

(10a)

¦

₌ −

=

n j j j

N

h

1 1

)

(

λ

, (10b)

2 / ) 1 ( 2

)

1 (

)

(

=

−

n−

f

λ

,

(10c)

where

¦

₌

₋

+

=

j i i j i j

y

D

1

)

1 (

,

j=1,2,!,n

,

(11a)

¦

=

j i i j

x

N

1

,

j=1,2,!,n

,

(11b)

where

)

1 (

)

1 (

₁ 1

g

h

x

_i

=

_i₋

−

_i₋

,

i=1,2,!,n

,

(12a)

)

1 (

)

1 (

₁ 1

g

h

g

y

_i

=

_i₋

−

_i₋

,

i=1,2,!,n

. (12b)

The coefficients of the polynomials found in (10) may be

very large after a few elements are extracted. In the next steps

this cause to work with large numbers. As a result, element

values may not be calculated. So at this step coefficients must

be normalized.

The extraction of commensurate lines is implemented in a

similar fashion until the termination resistance (

R ) is reached.

IV. E

XAMPLE

Example given in [18] is solved, to illustrate the

implementation of the proposed algorithm. The given input

reflection factor is

)

(

)

(

)

(

λ

g

h

S

=

where

0211 . 0 316 . 0 29 . 2 31 . 10 76 . 33 44 . 79 6 . 152 7 . 206 8 . 248 9 . 167 7 . 121 ) ( 0105 . 0 165 . 0 298 . 1 072 . 6 75 . 21 45 . 52 8 . 111 8 . 151 2 . 209 2 . 136 7 . 121 ) ( 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 + + + + + + + + + + = + − + − + − + − + − = λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ g h

Step 1

:

)

(

)

(

) 1 (

_λ

h

=

and

g

(1)

(

λ

)

=

g

(

λ

)

.

Step 2 :

2632 . 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 = − + = h g h g Z 1

Z

2

Z

n

R

)

(

),

(

),

(

λ

h

λ

f

λ

g

296

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Step 3 :

159440 0715 . 119 9 . 167 8 . 1023 2 . 136 ) 1 ( ) 1 ( 184560 0715 . 119 8 . 248 8 . 1023 2 . 209 ) 1 ( ) 1 ( 180030 0715 . 119 7 . 206 8 . 1023 8 . 151 ) 1 ( ) 1 ( 96295 0715 . 119 6 . 152 8 . 1023 8 . 111 ) 1 ( ) 1 ( 63159 0715 . 119 44 . 79 8 . 1023 45 . 52 ) 1 ( ) 1 ( 18249 0715 . 119 76 . 33 8 . 1023 75 . 21 ) 1 ( ) 1 ( 4 . 7444 0715 . 119 31 . 10 8 . 1023 072 . 6 ) 1 ( ) 1 ( 3 . 1056 0715 . 119 29 . 2 8 . 1023 298 . 1 ) 1 ( ) 1 ( 5597 . 206 0715 . 119 316 . 0 8 . 1023 165 . 0 ) 1 ( ) 1 ( 2379 . 8 0715 . 119 0211 . 0 8 . 1023 0105 . 0 ) 1 ( ) 1 ( ) 1 ( 9 ) 1 ( 9 10 ) 1 ( 8 ) 1 ( 8 9 ) 1 ( 7 ) 1 ( 7 8 ) 1 ( 6 ) 1 ( 6 7 ) 1 ( 5 ) 1 ( 5 6 ) 1 ( 4 ) 1 ( 4 5 ) 1 ( 3 ) 1 ( 3 4 ) 1 ( 2 ) 1 ( 2 3 ) 1 ( 1 ) 1 ( 1 2 ) 1 ( 0 ) 1 ( 0 1 − = ⋅ − ⋅ − = − = = ⋅ − ⋅ = − = − = ⋅ − ⋅ − = − = = ⋅ − ⋅ = − = − = ⋅ − ⋅ − = − = = ⋅ − ⋅ = − = − = ⋅ − ⋅ − = − = = ⋅ − ⋅ = − = − = ⋅ − ⋅ − = − = = ⋅ − ⋅ = − = h g g h x h g g h x h g g h x h g g h x h g g h x h g g h x h g g h x h g g h x h g g h x h g g h x 188120 0715 . 119 2 . 136 8 . 1023 9 . 167 ) 1 ( ) 1 ( 229820 0715 . 119 2 . 209 8 . 1023 8 . 248 ) 1 ( ) 1 ( 229700 0715 . 119 8 . 151 8 . 1023 7 . 206 ) 1 ( ) 1 ( 142930 0715 . 119 8 . 111 8 . 1023 6 . 152 ) 1 ( ) 1 ( 87579 0715 . 119 45 . 52 8 . 1023 44 . 79 ) 1 ( ) 1 ( 31975 0715 . 119 75 . 21 8 . 1023 76 . 33 ) 1 ( ) 1 ( 11279 0715 . 119 072 . 6 8 . 1023 31 . 10 ) 1 ( ) 1 ( 2190 0715 . 119 298 . 1 8 . 1023 29 . 2 ) 1 ( ) 1 ( 1793 . 343 0715 . 119 165 . 0 8 . 1023 316 . 0 ) 1 ( ) 1 ( 3527 . 20 0715 . 119 0105 . 0 8 . 1023 0211 . 0 ) 1 ( ) 1 ( ) 1 ( 9 ) 1 ( 9 10 ) 1 ( 8 ) 1 ( 8 9 ) 1 ( 7 ) 1 ( 7 8 ) 1 ( 6 ) 1 ( 6 7 ) 1 ( 5 ) 1 ( 5 6 ) 1 ( 4 ) 1 ( 4 5 ) 1 ( 3 ) 1 ( 3 4 ) 1 ( 2 ) 1 ( 2 3 ) 1 ( 1 ) 1 ( 1 2 ) 1 ( 0 ) 1 ( 0 1 = ⋅ + ⋅ = − = = ⋅ − ⋅ = − = = ⋅ + ⋅ = − = = ⋅ − ⋅ = − = = ⋅ + ⋅ = − = = ⋅ − ⋅ = − = = ⋅ + ⋅ = − = = ⋅ − ⋅ = − = = ⋅ + ⋅ = − = = ⋅ − ⋅ = − = h h g g y h h g g y h h g g y h h g g y h h g g y h h g g y h h g g y h h g g y h h g g y h h g g y

Step 4 :

110110 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 63159 96295 180030 184560 159440 49329 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 63159 96295 180030 184560 135230 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 63159 96295 180030 44798 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 63159 96295 51497 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 63159 11662 2379 . 8 5597 . 206 3 . 1056 4 . 7444 18249 4 . 6586 2379 . 8 5597 . 206 3 . 1056 4 . 7444 9450 . 857 2379 . 8 5597 . 206 3 . 1056 3218 . 198 2379 . 8 5597 . 206 2379 . 8 1 2 3 4 5 6 7 8 9 10 10 1 2 3 4 5 6 7 8 9 9 1 2 3 4 5 6 7 8 8 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 1 2 3 4 5 5 1 2 3 4 4 1 2 3 3 1 2 2 1 1 − = + − + − + − + − + − = + + + + + + + + + = = + − + − + − + − = + + + + + + + + = − = + − + − + − + − = + + + + + + + = = + − + − + − = + + + + + + = − = + − + − + − = + + + + + = = + − + − = + + + + = − = + − + − = + + + = = + − = + + = − = + − = + = = = x x x x x x x x x x N x x x x x x x x x N x x x x x x x x N x x x x x x x N x x x x x x N x x x x x N x x x x N x x x N x x N x N 110090 3527 . 20 1793 . 343 2190 11279 31975 87579 142930 229700 229820 188120 78029 3527 . 20 1793 . 343 2190 11279 31975 87579 142930 229700 229820 151790 3527 . 20 1793 . 343 2190 11279 31975 87579 142930 229700 77910 3527 . 20 1793 . 343 2190 11279 31975 87579 142930 65016 3527 . 20 1793 . 343 2190 11279 31975 87579 22563 3527 . 20 1793 . 343 2190 11279 31975 6 . 9411 3527 . 20 1793 . 343 2190 11279 2 . 1867 3527 . 20 1793 . 343 2190 8266 . 322 3527 . 20 1793 . 343 3527 . 20 1 2 3 4 5 6 7 8 9 10 10 1 2 3 4 5 6 7 8 9 9 1 2 3 4 5 6 7 8 8 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 1 2 3 4 5 5 1 2 3 4 4 1 2 3 3 1 2 2 1 1 = − + − + − + − + − = − + − + − + − + − = = + − + − + − + − = + − + − + − + − = = − + − + − + − = − + − + − + − = = + − + − + − = + − + − + − = = − + − + − = − + − + − = = + − + − = + − + − = = − + − = − + − = = + − = + − = = − = − = = = y y y y y y y y y y D y y y y y y y y y D y y y y y y y y D y y y y y y y D y y y y y y D y y y y y D y y y y D y y y D y y D y D

297

(4)

Step 5 :

9 8 7 6 5 4 3 2 9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2 1 ) 2 ( 110090 78029 151790 77910 65016 22563 6 . 9411 2 . 1867 8266 . 322 2327 . 20 ) ( λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ + + + + + + + + + = + + + + + + + + + = D D D D D D D D D D g 9 8 7 6 5 4 3 2 9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2 1 ) 2 ( 110110 49329 135230 44798 51497 11662 4 . 6586 9450 . 857 3218 . 198 2379 . 8 ) ( λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ − + − + − + − + − = + + + + + + + + + = N N N N N N N N N N h

As can be seen from the expressions of

g

(2)

(

λ

)

and

)

(

) 2 (

_λ

h

, the coefficients are very large even at the first step. If

the process is repeated, the coefficients will be much larger

and it will be impossible to work with these larger numbers.

Only seven elements can be extracted in this manner. So at

this step, the coefficients must be normalized. After

normalizing and applying the algorithm until the termination

resistance (

R ) is reached, the following impedance values are

obtained

.

9811

.

2 3431

.

0 ,

9743

.

2 ,

3453

.

0 ,

9046

.

2 ,

3564

.

0 7783

.

2 ,

3876

.

0 ,

3295

.

2 ,

5662

.

0 ,

2632

.

1

10 9 8 7 6 5 4 3 2 1

=

R

Z

Characteristic impedances found in [18] are

.

9779

.

2

343 .

0 ,

97 .

2 ,

346 .

0 ,

92 .

2 ,

355 .

0

79 .

2 ,

387 .

0 ,

33 .

2 ,

566 .

0 ,

26 .

1

10 9 8 7 6 5 4 3 2 1

=

R

Z

V. C

ONCLUSION

The proposed algorithm in [21] is a simple and powerful

method, but can cause large polynomial coefficients, as the

number of commensurate lines increases. So it needs a

modification; at each step the coefficients must be normalized

to be able to calculate all characteristic impedances. As a

result, a very simple to implement commensurate line synthesis

algorithm is obtained.

A

CKNOWLEDGMENT

Fruitful discussions with S.B. Yarman (Istanbul) are

gratefully acknowledged.

R

EFERENCES

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