Exact Design of Stepped-Impedance Transformers - IEEE Xplore

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exact solutions of the design of stepped-im- pedance transformers presented until now have been tedious and subject to computational error. We present here ...
620

IEEE

Exact

Design

of Stepped-Impedance

TRANSACTIONS

ON MICROWAVE

From

Transformers

order FENG-CHENG

CHANGAND

HAROLD

MOTT

(8), Z,-I (s) appears

than

Absfracf—The exact solutions of the design of stepped-impedance transformers presented until now have been tedious and subject to computational error. We present here an exact synthesis procedure which requires less effort than other exact procedures. In addition, two recurrence formulas are given for determining the characteristic impedances of each section. The coefficients so obtained can be compared to eliminate computational errors. Stepped-impedance in

graphical

the

transformers literature.

solutions

[5],

The

susceptible

to computational

exact than

exact

as well

published.

synthesis

The

SIT

electrical

with

put impedance

tables thus

far

short

paper

error.

In this

which

Z,n,,–l(@)

isthe

characteristic

Ifweintroduce

W

+

impedance

and

we present with

Let

us write

where

we wish

to determine

coefficients

denominator

~k’

form

the

and denomina-

of

(9) by

we

(s’ – 1) and

(9)

Lo

the coefficients N~’.

~,F’S’

~ /

lf~’-l

and N~’–l

multiply

compare

the the

from

the

numerator

result

to

and

(8),

and

obtain Nh-2~1

– Nk’_l

= f,ikfk_~’

(lOa)

– N~’

– M~’–l = N~_L” – ~,ikfk’.

dfw-l

RL.

each of The

(lOb)

fik’-l

=

(Nk’

~fk=l

=

rr(~kr

may be solved

h7~_f +

+

N~_4’

+

~k_2’

[I]and

+

. . . )



i_,(Mk-f

+

~k-4’

(1)

section

to give

in-

and Z,

where

the series are continued

OU1-statement

different

define

ML3’

-.

. )

(8).

~fh_#

iv,-.,’+

over positive

following

+

We

+

subscripts

may

. . . )

Nk-,r +

also

)

.0.

only,

obtain

(ila)

(Ilb)

in accord

from

(1 O) a

form,

.– (Nk+f + Nk+4r + (N~+lr + N~+# + N~+# +

=

the

series

continued

only

Nk+6’

+

. “ . )

(12a)

J’fk+6’

+

. . “ )

(1.2b)

. . . )

– ~r(~f~+z’ with

of (1) as

+ +

– (N)+-,’+

iVkF1

a normalized

use

are zero. Thus

.— 1

and

(3)

write

of higher (6) and

numerator

N,’-’sk

b=o

(2)

we may

both

r— 1

= ~

is

tan@

from

1972

as

z,-,(s)

with

s=j

polynomials

s = t 1 into

and denominator

be removed

Z,-I(S)

an

section.

variable

numerator

SEPTEMBER

less effort

sections,

(~—l)

to contain

if we substitute

tan @

of the

of ther

the frequency

been

tan O

Z, + fZ,n,,_I(@)

impedance

have

Rg and

loads



is the input

[6],

and

TECHNIQUES,

tor of zr_l(s).

These equations

to the left,

Zin,r-l(@)

where

[~],

are tedious

out

transmission-line

looking

Zirl,r(+)=zr

discussed

procedures.

in resistive

of the ~ section,

[3],

can be ca,rried

of nlossless

+, terminated

been

approximate

published

some other

consists

length

have

[1]–[3],

as design

solutions

procedure

is necessary

(SIT’s)

Exact

both

AND

at first

However,

(s2 – 1) may

known frequently

z,(s).

(5), we see that factor

THEORY

+

for

Jfk+4’

+

subscripts

less than

or equal

to

superscripts.

2,4(S) + z,(s)

from

which

we

note

=

= Z.+l/Zo.

(4)

r=

1,2,

!ATe may solve

(5)

–.,(–1).

. . . . n, Zfi~l =RL,

and from

~, for

know all

Z.(s),

we

r. Further,

successively

see from

to find

from

(5)

and

(6)

that

we

of Z.+,(

a knowledge

the characteristic

=

impedances

may

find

z,(s)

(3) maybe

~L),

and

used

Z, of all sections

of

the SIT. The functions method.

use of

is not For

Equations

(6)

convenient,

a lossless SIT

to determine and we will we may

the

normalized

develop

express

a more

impedance

they

or

(14)—or

the order

are applied.

equal

Thus,

powers

useful

a coefficient of equal

(6), which

z,(s) as

In general

the

may be written

power

may

reflection

all of the

Note

powers

that

if of

a coefficient

ofs.

This

gives

in determining

the

L2~ (COSI#J) is an even s =j

M–IT = N_lT = M,+lT = N,+lT = ().

for a Iossless

polynomial

tan o we may impedance

L2.(COS

write

function

Zn(s) =

(15)

95)

of degree

the voltage

2n. By

reflection

the

trans-

coefficient

as Q.(s)

P“(s) of

n-section

_9”@ @)

1,=

~(s) _

University

each

of the coefficients

errors

coefficient

1 +

and the input

Manuscript received October 18, 1971, revised January 7, 1972. The authors are with the Electrical En~imering Department, Alabama, University, Ala. 35486.

is given,

be found.

in terms

be

function

as [4]

becomes formation

that

(12)—may

coefficients.

where

If we note

or

(12) or (14) gives

on computational

I,(.) (7) into

(11)

and higher

check

(7) substitute

relations

impedance

use, if z.(s)

of the SIT

of s, whereas,

of the coefficients

highly

a

by their

impedances

and lower

alternatively

of the normalized

(11) or (13) is used, we find

us

convenient

(13)

n characteristic

SIT

we

recurrence

(12) we get

used to reduce

in terms direct

the compact

and R =RL/Rg

Zo=Rg,

time we

obtain

(4) to obtain

(6)

If

(11) we may

that fr=zr(l)=

In these equations,

From

s

i-. ———— 1 + ZL,(s)s

(16)

Pm(s) + Q.(s) Pm(s) – Qm(~

(17)

SHORT

PAPERs

where

Q.(s)

passive

621

and .Pn (s) are positive

networks,

For

p(s) may

a Chebyshev

real polynomials

be found

transformer,

uniquely

we know

n. For

I p(s) I ‘ [7],

with

[8].

Ak = ~/Xh

that

__

Cos 41

1P(4) 1’=

I IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp. 379-3S6, July 1969. [4] S. B. Cohn, “Optimum design of stepped transmission-line transformers, ” IRE Trans. M~ “crowaue Theory Tech., vol. MTT-3, PP. l&21, Apr. 1955. [5] F. C. Chaqa H. Y. Yee, and N. F. Audeh, “A graphical method for the design of stepped Impedance transformers, ” 19?1 IEEE G.MTT Int. Microwaw SynaO. Dig., IEEE Cat. 71, C-25-M. UU. 4-5, Mav 1971. [6] L.-Young, “Stepped-impedance transform&s and filter prototypes, ” IRE Trans. Microwave Theory Tech., vol. MTT-10, Pp. 339-359, Sept. 1962.

(22)

H w 2k–1 cosh2 — + coshz — + COS’— n ?2 n

2k–1 w cosh — + COS—— n ?2’

(

1 + cosh ~

Yk=l–

1 + cosh y )(

\n

k=l,

r-zodd. xk and

The

2, . . ., (7t/2) zeros

may

~k

– yk become

The zeros are given

forneven

and k=l,

be found

by allowing

equal by

to the same value,

~k =

kjd~

Uk = ————— 1+

to find

in the form

Q.(s)

interested

2k–1

we denote

as Ub.

(24)

T 2k–1 COS—— 9a

.

2k–1 —7m n

)

,.

2 )

[7] L. Young, “Unit real functions in transmission line circuit theory, CG’czsitTheo?y, vol. CT-7, PP. 247-250, Sept. 1960. [s] P. I. Richards, “Resistor-transmission-line circuit, n Proc. IRE, 2 17–220, Feb. 1948.

+(1–

Dielectric

T

VAN

R)

the ~k and @ but rather

in p(s).

We therefore

write

(19)

by

‘H (1 +

Losses

in

Rectangular

(26)



in finding

Q,, and P. given ==

for

C-O.Then

’Os ——. n

the real polynomials

of (16) with

~- +’Os

2h–1

cosh — — COS——— It n

we wish

(ti+l/2)

w to go to which

1



IRE Tram. vol.

36,

PP.

where

H

We are not primarily

2, 0 ...

1/2

m—l

[’45)

~c~h

where

2k–1

HW

7r — 2 cosh — cosh — COs—— n n n

RE

BUI

an H-Plane-Loaded Waveguide

AND RfiAL

R.

J. GAGNfi

Absfracf—The attenuation constant due to dielectric losses in a rectangular waveguide loaded with a dielectric slab in the H-plane is calculated. Results for the dominant LSMU mode are presented graphically.

(27)

C,s’)

k=]

P.(s) = ~(1 + R) ‘~ k=l

(1 + Bks + AM2)

{;l



~ ~dn +

A(.+,,,,s),

(28)

Manuscript received October 26, 197 1; revised January 24, 1972. This work was supported in part by the National Research Council of Canada under Grant A-2S70. The authors are with the Department of Electrical Engineering, Laval University, Quebec 10, Que., Canada.