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.
