The Design of Discrete N-Section and Continuously

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL.MTT-17,NO.8,AUGUST 1969
577
The Design of Discrete N-Section and Continuously
Tapered Symmetrical Microwave
TEM Directional Couplers
DAVID W.
AbstractA
method is presented for the computer-aided design of
either N-section discrete or continuously tapered symmetrical microwave
couplers.The coupling distribution function k(x) is parametrized in the
form J@,j),and a special optimization process (of the generalized
Remez type) is used to determine the set of parameters j which produce an
optimum power coupling response.Standard parametric forms based on
an approximate Fourier analysis as well as more general spline parametric
forms for k(x,j) are developed and illustrated.
INTRODUCTION
M
ETHODS FOR the synthesis of symmetrical TEM-
mode coupled transmission line directional cou-
plers have been available since 1965,when Cristal
and Young [1] published a complete design procedure and
fairly extensive tables of design parameters for discrete N-
section couplers (N= 3,5,7,9).Since these discrete couplers
have jump discontinuities in the coupling coefficient function
with corresponding jump discontinuities in the stripline
dimensions (see Fig.1),one might expect that the directivity
of a broad-band discrete coupler would be poor at the higher
microwave frequencies where the wavelength is comparable
to the transition region between adjacent sections of the
coupler.This has been found to be the case.
In an effort to remove the problems associated with the
jump discontinuities in the coupling distribution function,
one would naturally attempt to round off the corners and
thereby allow a more gradual transition from the region of
tight coupling to the region of zero coupling at the ends of
the coupler (see Fig.1),One method for doing this was
developed in 1966 by Tresselt [2] who made a clever use of
the Cristal-Young procedure and an approximate nonuni-
form line analysis developed by Orlov [3 ]and Sharpe [4].
Unfortunately,the Tresselt procedure does not always lead
to an optimum design.
Consequently,an entirely different computer-aided design
procedure has been developed.The end product of the anal-
ysis is a computer code which designs either an N-section or
a continuously tapered coupler of a specified length having a
specified nominal power coupling over a specified bandwidth
with the smallest possible ripple.The basic idea is to param-
etrize the coupling coefficient k(x) in the form k(x,~) so that
the resulting power coupling frequency response is of the
Manuscript received February 18,1969;revised April 28,1969.This
work was supported by the U.S.Naval Weapons Center,China Lake,
Calif.,under Contract NO0123-67-C-0714.
The author is with the Equipment Research and Development
Laboratory,Texas Instruments,Inc.,MS 244,P,O.Box 6015,Dallas,
Tex.75222.
KAMMLER
form C(U,#).A specialized optimization process (of the
generalized Remez type [5]) may then be used to determine
the optimum parameters ~,which produce the best fre-
quency response.With this objective in mind,we must now
determine an efficient means of computing the power cou-
pling response,of parametrizing k(x),and of carrying out
the optimization procedure.
Although the remainder of this paper will deal with the
application of this idea to the problem of designing sym-
metrical couplers having a flat power coupling response,it
should be noted that the methods used are quite general
and may obviously be extended to the design of a variety of
N-section discrete or continuously tapered ncmuniform line
components having specified frequency responses,
ANALYSIS OF THE POWER COUPLING RESPONSE
We shall first develop an efficient numerical means for
computing the power coupling and phase response of an
arbitrary quadrature coupler which is uniquely character-
ized by its coupling coefficient function k(x).Let us first con-
sider an elemental section of uniformly coupled line as in
Fig.2.In this case,the incident wave coefficients al,ai
at
ports 1,4,and the reflection coefficients
b,,b,,
at
ports
2,3,
are related by
b,
[1 [1
a4
= AT(k,AL)
al b,
where the elemental transfer matrix is given by
[
I/C
c/c
AZ(I%,AL) = _ _
c/c 1/(7
1
(1)
(2)
where
c= 
j%
sin
DAL
@  k cos PAL + j
sin ~~
(3)
is
the coupled transfer,
is the dc transfer,and we use a bar to denote a complex con-
jugate.The propagation constant is given by
where the symbols have their usual meaning.
578
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
CONVENTIONAL 5-SECTION
-9.343 d13 COUPLER
COMPARABLE CONTINUOUS
-S.343 dB COUPLER
I
I
PHYS lCAL
APPEARANCE
_ ._ - 
I
0.6.-
cOUPLING.4- -
COEFFICIENT
FUNCTION
.2- -
1
I
I
I

0
kc/4 x
/
I
I
I
1
k(x)
0.6
I

o
1./4
x
0,15 -
0.15 -
0.10 
0.10 -
0.05
0.05 -

FREQUENCY
%
RESPONSE
(B = 5)
Fig.1.Acomparison ofatiscrete andacontinuously tapered symmetrical microwave coupler.
Fig.
2.
An elemental seetion of a uniformly coupled transmission line.
-.
Fig.3.A number of elemental sections may be cascaded together.
Fig.4.A continuously tapered coupler can be approximated
by a number of elemental sections.
KAMMLER:DESIGN OF TEM DIRECTIONAL COUPLERS
579
It is now clear (see Fig.3) that we may cascade a number
of elemental sections and thus obtain the overall transfer
matrix
T = AT(kl,AL) AT(k,,AL)...
Al!(kN,,
AL)
(6)
for a multisection system where kl,kz,...,kN,are the
coupling coefficients for the respective sections.In the obvi-
ous manner,we have for the continuous coupler of Fig.4,
the overall transfer matrix
l?+.
{ATP(X+IATP(,)*I ,7)
T =
Iim
where xi is a point in the ith interval.In practice this limiting
process can be truncated with the size of N.depending upon
how rapidly the function k(x) varies.For most couplers
of practical interest,the use of N,= 100 to 500 gives a rea-
sonable approximation to the limiting transfer matrix for
frequencies in the practical range of interest.
Now in order to numerically compute a frequency re-
sponse for a given coupling distribution function k(x),it is
clear that we must numerically evaluate the matrix product
of (6) a Ial ge number of times,i.e.,once for each frequency
point which we desire.Then,since we are formulating our
design problem as an optimization process,it follows that
we must evaluate the frequency response for a number of
different coupling distributions k(x,j) corresponding to a
variation of the parameters ~.In all,we shall need to eval-
uate the matrix product of(6) several thousand times during
the course of a typical design problem,and as the numer-
ical evaluation of this matrix product is the most time con-
suming portion of our optimization process it is clear that
an expeditious means of numerically evaluating this product
is required.
Let us first consider the cost (in computer multiplications)
of a straightforward evaluation of this matrix product.
From (2) through (4),we see that typical AT matrix is of the
form
AT =
where
sin e sin 0
~k @  k cOsO+ J@ -k
J
2.fAL
O=~AL=.
v
We will now show that for a symmetrical ccupler,a simple
recursive relation may be developed which enables us to
evaluate this product at a cost of about 2* multiplications
per section.We first define
T.= AT(k,,AL) AT(k.-l,AL)...AT(k,,AL)
(lo)
.AT(kl,AL) AT(k,,AL)...AT(k8,AL)
so that
T,
represents the transfer matrix for the 2s 1 center
sections of the symmetrical coupler.We next use an induc-
tive argument to show that we may write T.in the form
T,=
[
p,
cos 0  ja.sin e
jr,sin e
1
(11)
 jr.sin O
p,
cos e + ju,sin e
where
p,,as,TS
are trigonometric polynomials in o with real
coefficients depending on kl,kz,...,k,.Specifically,we
have from (8) through (11)
pl=l
al = l/~1 
k12
(12)
K1= kl/dl  iilz.
Recursively then,for s= 2,3,4,...we may compute
T.= AT(k,,AL).T.-l.AT(k8,AL) (13)
and,after a straightforward matrix multiplication,we may
easily verify the recursive relations
{s
(da-
p,= P,I 
(2sin20)
p .l +
-(d&) T.-l}
8=8-l+(<A){PS+ PS-lJ
.s=2,3,....N.
(14)
where we assume that the fully developed coupler has 2N*  1
sections.It will be noted that by using (14) we may add a
pair of sections at a cost of only five multiplications.Hence,
(8) by using this recursive scheme we may evaluate the matrix
product of(6) at a cost of only 2* multiplications per section.
Now let PN3,aNs,WV.be computed for the fidly developed
coupler,which is assumed to have 2N, 1 elemental sections.
By comparing (2) and(11) we have
(9)
(?/(?= j7fN.
sin 6
(15)
l/c =,0.v8(X)s 8 + jTN,sin &
Assuming that cos 0,sin o are already computed and that u.
smg these equations along with the conservation of energy
1/<1  kz,k/<1  kz
are
computed and stored in an array condition
(for each of the sections),we see that we must expend two
multiplications to assemble a AT matrix.Direct multipli-
\c]+/c]2=1
(16)
cation of N.such
AT matrices then requires about 8N,com-
we have as the power coupling
plex multiplications or equivalently 32N$ real multiplica-
tions.Hence,a straightforward calculation of the matrix
Ic[z=
7W.2 Sinz d
product of (6) requires about 34 multiplications per section,
1 + TN,Z sirl~
(17)
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
K(U)
r
c(u))
2
TRANSFORMATION
u
u
Fig.5.The power coupling response for a continuously tapered symmetrical coupler can be rapidly computed by approximating k(x) by a
number of elemental sections as in Fig.4 and then using the recursive relation of (12) and (14) to compute the transfer matrix.
and we have as the phase of the coupled transfer
{}
7rAT,
sin O
phase {C } =  arctan
(18)
pN,COSd 
It will be noted that the symmetrical nature of the coupler
implies that
phase {C] = phase {C} + 90 degrees
(19)
i.e.,the coupler has the familiar quadrature property,Equa-
tions (12),(14),and (17) through (19) thus provide us with
an efficient means to compute a power coupling or phase
response for a symmetrical but otherwise arbitrary coupler
if we are given the coupling distribution function k(x) (see
Fig.5).
FO-ATION OF THE OPTIMIZATION PROBLEM
We would now like to use some optimization procedure to
find the coupling distribution function k(x) which produces
a flat power coupling response over some frequency band
W~i~~U~W~~x
so
that we have a bandwidth
B==.
(20)
~min
Now if the coupler is to be discrete with 2iV 1 sections,
each of which is a quarter wavelength long at the center fre-
quency,
LIJznin+ ~,~.~
tic =
2
(21)
i.e.,
AL=:=;:
(4
c
then we might use the coupling coefficients kl,kz,...,kN
as parameters.(Here kl is the center coupling coefficient
and kN is the coupling at the end sections of the coupler.)
Since in cases of practical interest N will be no larger than
perhaps 11,we see that the coupler is easily specified by a
relatively small number of parameters.
When we consider a continuously tapered coupler,which
we approximate by several hundred small elemental sections
(each of which is of length much less than a center frequency
quarter wavelength),then we have another type of problem
altogether.In this case,it is clear that we cannot let the cou-
pling coefficients on the elemental approximating sections
act as independent parameters since no currently available
nonlinear optimization process is capable of dealing with
several hundred parameters.There is no real need,however,
to use several hundred independent parameters to model a
coupling distribution of the type shown in Fig.5 because it is
clear that a small number of parameters will be sufficient.
(Specific parametric forms will be developed later on.)
Hence,from this point on,we shall assume that the coupling
coefficient function may be characterized in the form
k(x,PI,P2,.0
.,p.),or more compactly k(x,p),where the
number of parameters is relatively small,e.g.,less than 20.
Then,using k(x,j),we may use the numerical procedure of
the previous section to compute the frequency response
G(O,~) = 10
log,,] c(m)/
(23)
which now depends upon the parameters pl,
pz,.-.p..
Our
optimization problem is now to find the parameters p such
that
Maximum I P(CO,~)  Q.I = Minimum,
(24)
%lin s.swmax
Thus,we wish to find the parameters which make the power
coupling response as close as possible to the desired nominal
value @.(in decibels) over the specified frequency band.
Now rather than dealing with the complete frequency
band u~i.< w< W~&x
in our optimization process,we shall
choose to work with a number of equally spaced frequency
points
Wmin=Wl<Wz<WS<..< WN,= W,,,ax
(25)
within the passband.Since @(w,#) is obviously a continuous
function of a,it follows that if Nf is large enough (e.g.,sev-
eral hundred),we may safely replace the complete interval
[Wmin,
w~~.] by the finite point set { w,}.This being the case,
we may initially compute and store the values of
( )
wiAL
Sinz e,= sinz 
~=1,2,...,~J (26)
v
which are needed in order to compute @(w~,p) by means of
(14),(17),and (23).(Note that these expressions are indepen-
dent of the parameters used to specify the coupling distribu-
tion.) Then,when we are given a set of parameters p,we may
compute and store the constants 1/<1  k,z,k./~l  k,z for
each of the elemental sections of length AL.(These con-
KAMMLER:DESIGN OF TEM DIRECTIONAL COUPLERS
A
c(kl,~)
.
...
..
,
..<
.....
..
I I
I I
I
I I
G!4

01 .Q2..Q3..
L?5
!dl.j
!-27
I.J
.
....
....
Fig.6.From among the points w,UZ,...
~~~ we
select the points w,%,..,
~n
where the error curve C(U,~) takes on an extremum.
stants,of course,are independent of the frequency points.) Now with!21,W,...,% fixed,we may regard the m func-
Once these arrays are available,we may use (14),(17),and
tions ~(~i,}) as functions of the parameters
PI+
PZ,.,P..
(23) to rapidly compute P(W,~),i= 1,2., o ,Nf.It will be
We would like to determine the effect of slight changes of
noted that by making use of computer storage,this compu-
the parameters F on the extremal errors 6(W,$),and hence
tational procedure avoids the repeated computation of ex-
we numerically compute the approximate Jacobian matrix
pressions like sinz (6;) and <1  k,g which are used repeat- D where
4%
PI,m, 
Di,=
,
Pj1,Pi + @j,Pj+l;  j Pn)  ~(QiJ PIJ P2,   j Pn).
(30)
edly in obtaining the power coupling response,and since
these are basically one-dimensional arrays the amount of
storage required is not at all prohibitive.
THE OPTIMIZATION PROCESS
Now in an effort to solve nonlinear Chebyshev optimiza-
tion problems of the type just discussed,we have developed
a general optimization procedure,TFIT,which is described
in detail in [6].For completeness (and since the method has
a wide potential application in the design of microwave com-
ponents) we shall briefly summarize the main features of this
optimization plan.It must first be noted that the TFIT algo-
rithm is based upon the assumption that the error curve
(27)
E(U,p) = P(Q,P)  n
will have a minimal maximum deviation from zero whenever
the n parameters
PI,P2,.0
.,
p.
are chosen in such a fashion
that the error curve has n+ 1 equal maximum deviations
from zero which alternate in algebraic sign.
With this in mind,we shall now describe the basic iterative
step,i.e.,the procedure used to find parameter increments
Al such that C(O,}+ A#) is a better error curve (i.e.,has a
smaller maximum deviation) than c(co,I).Given a set of new
parameters ~,we first compute 6(w,j5),i= 1,2,...Nf using
the computational procedure which has already been dis-
cussed.We then examine the error curve and pick out the
extremal points CL oz,  -,fi~ (from among the points
ml,
@2>   
Wf)
at which the error curve has a local max-
imum or a local minimum,e.g.,see Fig.6.We define
{
+1 if W corresponds to a local maximum
~i =
 1 if Q corresponds to a local minimum
(28)
and we define the average extremal deviation by
:g
14%10I
davg=
(29)
tipj
We might now note that to first order in the parameter incre-
ments Apl,Apz.,...,Apn we have
Ae,= c(%,F + Afl)  6(% P)
(31)
j=
1
so that we have a system of linear equations relating small
changes in the parameters to the resulting changes in the
error curve at the extremal points.
We would like to use (31) to determine parameter incre-
ments,A#,which will improve the error curve.Now ideally,
we would like to have one more extremal point than we have
parameters (i.e.,m= n+ 1),since this is the normal situa-
tion for best Chebyshev approximation,Unfortunately,in
nonlinear approximation problems this is often not the case
for our initial estimates of the parameters.Consequently,
our procedure breaks down into three cases depending on
whether m< n+ 1 (under determined case),m= n+ 1
(exactly determined case),or m>n+ 1 (over determined
case).We shall now discuss each of these three cases in turn.
First of all,we consider the under determined case where
m <n + 1 so that we have too few extremal deviations.Our
approach will be to attempt to both level and to com-
press
the error curve,and thereby in a natural manner
introduce additional extrema.Consequently,we define
Aei = $u&~  E(Q;) ~),
~=1,2,....m (32)
and then determine Aj5 such that
AZ = DAfi
(33)
by using the Penrose pseudo-inverse,i.e.,
A@= DT(DDT)lA;(34)
where the superscript T denotes a matrix transpose.In so
doing,we assume that them Xm matrix DDT is nonsingular.
582
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
We might note that this choice of AZ attempts to produce a
new error curve having extremal deviations which are all
equal in magnitude to half of the old average extremal devi-
ation,and that the choice of the Penrose pseudo-inverse
enables us to solve (33) for A~ in such a manner as to make
~ (Ap,)2 = Minimum
(35)
i=l
so that none of the parameters are changed by an excessive
amount.Thus we have a means of determining A~ in the
under determined case.
We turn next to the ideal exactly determined case where
m= n+ 1 so that we have one more extremal point than we
have parameters.In this case our approach is to attempt to
level the error curve so that all of the n+ 1 extrema are
equal in magnitude.Accordingly,we introduce a new
dummy parameter,d,and set
A~i = O-jd  ~(~jl @)j
i=l,2,....m.
(36)
Using this expression for Aci we may rewrite (33) so that the
parameter d appears as an unknown along with the param-
eter increments Apl,Apz,, s,
Apn.Hence,we have
.
 C(G,
p)
 @2,p)
 6(Q.,
p)

6(LL,
p)
(37)
and we may solve this m Xm system of linear equations to
obtain the parameter increments ApI,Apz,  ,
Apn.(The
dummy parameter
d is then discarded.) Thus we have a
means of computing A#in the exactly determined case.It
will be noted that this procedure for dealing with the exactly
determined case is simply the ordinary generalized Remez
method [5].
Finally,for the over determined case where m> n+ 1,we
again use the compress and level approach.Hence,we
again define A= using (32) and then solve (33) to obtain Aj5
by using the least squares pseudo-inverse,i.e.,
Ap = (DTD)lDTAz
(38)
assuming,of course,that the n Xn matrix DTD is nonsingu-
lar.If this process fails to produce a useable A~ (due to a
singular matrix D*D,etc.),we delete some of the extrema,
and thereby reduce the over determined system to an under
or exactly determined one.
NOW once the new parameters $+ A$ are computed by
means of the appropriate under,exactly,or over determined
procedure,we test to make sure the new parameters satisfy
any imposed constraints of the form
ai <
pi + Apt < bi,
~=1,2...,n
>
(39)
with parameter increments being halved,etc.if these con-
straints are violated.The new error curve C(CO,~+ A#) is then
computed at each test point W,i=
1,
2,  ciV~ with the
new maximum error being computed.If the maximum error
is reduced by at least some specified tolerance then the new
parameters are accepted and a new iteration is begun.
Otherwise,the parameter increments are cut in half,and the
error curve is recomputed,etc.If after ten halvings the error
curve has still not been improved by the specified tolerance,
then the iterative process is terminated.
To illustrate the behavior of the optimization process we
have prepared Fig.7 which shows the successive iterations in
the design of a 5-section discrete coupler having a nominal
power coupling of  8.343 dB and a bandwidth
B=
5.The
standard parametrization for Ic(x,j) (which is developed in
the next section) is used with the coupling coefficients kl,kz,
kt (center to end) being given in terms of the parameters
pl,p~,ps by
Ziz _ 1
h,=
Ziz+l
i=l,2,3
where
=ex:[:+3
(40)
(41)
It will be noted that for a discrete section coupler,the fre-
quency response ~(u,~) is symmetric about the center fre-
quency.Hence in Fig,7 we need only optimize over the fre-
quency range ti~i.< u< u.rather than over the complete
band from ~rnin to ~~.x.
The behavior shown in Fig.7 is
quite typical of the optimization process when the parametri-
zation has been reasonably chosen.
SIMPLE FOURIER ANALYSIS AND STANDARD
PARAMETRIC FORMS
We turn now to the problem of developing a specific
parametric form for lc(x,~).We shall consider first a para-
metric form which gives rise to coupling distributions analo-
gous to those obtained by Tresselt [2].Accordingly,we
define an auxiliary function p(x) by
p(z) = +-
~ln
ZO(Z)
(42)
where ZO(X) is related to the coupling distribution by
o(x)=[:
nll2
(43)
Now Sharpe [4] has shown that under conditions of light
coupling (where k2<<1),we may obtain the coupled transfer
KAMMLER:DESIGN OF TEM DIRECTIONAL COUPLERS
583
COUPLING
!TERATION
PARAMETERS
COEFFICIENTS
o
-0.9605
0.6334
-0,9605
0.2542
-0.9605
0.0971
2
-0.9335
0.6113
-1.1089
0.2073
1.2871
0.0702
o,9450
;.::;;
1
.2167
-1.7557
0:0440
3 -0.9463
0.5755
-1.2501
0.1601
1.8538
C.0399
4
-0.9464
0.5753
-1.2514
0.159@
-1.8562
0.0398
MAX
P,
FREQUENCY RESPONSE
RIPPLE(DB)
-8 -
n
1.384 _,._
-12
o
1
2 lJ/ld,
P.
8 -
0.595 -10

12
1
0
P
8
!
0.239
-lo
1
~
-12
0
P
I
8
0,18!..,~,
.-12i_
o
P
-8
0.170
lo
t
12-
!
n
L...-.
1
2LJ/1J<
L
2 I.J/lJc
L..+
1
2 u/lJc
G
o
1
2 w/w,
Fig.
7.
The behavior of the optimization urocess is illustrated by
following through the 4 iterations required to design a.5-section 8.343-dB
.
coupler having a bandwidth B= 5.The c&pling coefficients are related to the parameters by (40) and (41).The initial estimates of the param-
eters are obtained from (64).The computer time required for the calculation was 0.1 second on the IBM 360/50.
C(o) from p(x) by means of the approximate Fourier trans-
form relation
J
LIz
C(fd)
=
exp [ jc.LL,/2]
exp [
j2cJz/v]p(z)dz.(44)
Lf2
Here we choose our coordinate system in the coupler so
that x= Ocorresponds to the center of the coupler.Since we
are dealing with a symmetrical coupler this ensures that k(x)
has even parity,and thus from (42) and (43),we see that
p(x) has odd parity.Now by taking the inverse Fourier
transform of (44) we obtain
03
p(z) = +
s
exp
[j2u/v (z + L/2) C(u)OkJ.
(45)
m) ~
If we now ~~troduce
@(cd)=  j
exp [jLOIJ/V]C(OJ)
(46)
and make use of the odd parity of p(x),we obtain the real
transform pair
(47)
s
L/2
2c0x
e(co) = 2
sin  p (Z)CZZ
o
v
2WZ
p(z) =;
s
.
sin  C(ti)dw
(48)
o
v
We might note at this point that the light coupling approx-
imation has,in effect,linearized our prc)blem as can be
seen from (47) and (48).
Let us now find the form of p(x) corresponding to a cou-
pling amplitude
(-A,-2u.<w<O
C3(6J)=\A,0<w<2cu.(49)
[ O otherwise
584
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
A
C( (L))
A.
CONTINUOUS:
I

u
c
w
DISCRETE:
I
I
f
P(x)
e.g.,see Fig.8.[Note that ~(u) corresponds to a coupler
which passes all frequencies u < 2~.and thus has an infinite
bandwidth B as defined by (20).] The amplitude A is related
to the nominal power coupling by
(P.=  10 loglo A~.
(.50)
Using (48) and (49) we now have
4
p(x)
4
-2A
T
t
T
At/4
1
i
1

x
+
Fig.8,
where
and
2
A
sinz m

n-  (kc/4)  u
AJ4
27W
A,=
u,
(51)
(52)
(53)
so that & is a center frequency wavelength and u measures
distance along the coupler in units of a center frequency
quarter wavelength.
Analogously,we may obtain the p(x) corresponding to a
discrete section coupler where the coupling amplitude is
periodic with period 4% i.e.,we take
(
A,
(2(0) ==
26JC<(.J<O
(
A,
O< W<2W,
(?(cd + 4W.) = e(w)
(54)
or equivalently
{
sin (2n  1) ~ ~
@
}
e(w) = ~ ~
L
.
2?-1
(55)
r
n-l
We then have from (48) and (55)
{
}
.
sin
(2n  1) ~ ~ dw (56)
.,
2AM

-x
J
m
(At/4)
.=l
2n 
1
where 6 is the Dirac delta function.Graphs of t!(u) and p(x)
are shown in Fig,8 for both the continuous and discrete
cases.
Now since the coupling amplitudes of (49) and (55) for the
continuous and discrete cases,respectively,correspond to an
infinite bandwidth,it is not surprising that a coupler which is
infinitely long is required in each case,e.g.,see(51) and (56).
From Fig.8,however,we see that we might obtain a coupler
of finite length by truncating the p(x) distribution after an
integral number of center frequency quarter wavelengths.
Naturally the corresponding c?(o) as given by (47) would
then have a finite bandwidth and definite ripples in the pass-
band.Accordingly,we suppose that p(x) vanishes for
KAMMLER:DE-SIGN OF TEM DIRECTIONAL COUPLERS
x> NlJ4.
We may then use (42) and the truncated p(x) dis-
tributions corresponding to (51) and (56) to obtain
{s
2A
N ~i~2 ~u~
Z~(u) = exp 
du
}
(57)
Tu
u
for the continuous case and
z,(u)
{s
2A  1
= exp 
z
if U.=I 2n  1
[
.6 lu1 ~]du},
U>o
(.58)
for the discrete case,By inverting (43) we may thus obtain
the coupling distribution
2(?(U) 
1
k(u) =
z,(u) + 1
(59)
corresponding to an arbitrary power coupling amplitude for
either the continuous or discrete cases.
Unfortunately,the mere truncation of the p(x) distribu-
tion does not produce a k(x) which gives rise to a useable
power coupling.However,examination of (57) and (58)
along with Fig,8 does suggest a very useful parametric form
for the coupling distribution.Accordingly,when we truncate
a p(x) distribution after N center frequency quarter wave-
lengths,we define the weighting function
1
@
for O<u<l
w(u,13
=/2
fOrl<~<2
1:
(60)
\
ef~~
for Nl<u<N
and then modify (57) and (58) such that
.ZO(U,PI,P2,  ,pN)
2N
{J
sin2 7ru
= exp 
W(u,.Z)
)
,U>o 61)
du _
n-,( u
for the continuous case and
Zo(u,
pl,pz,  ,Rv)
2N
H
1
=exp
W(21,p) ~ :
TU
n=l zn 
[
2n1
1)
(62)
.6 ]U] 
du
2
for the discrete case.In either case,we then use (59) to obtain
what we shall call the standard parametric form for the
respective continuous or discrete coupler.
Now in deriving (61) and (62),we have replaced the cou-
pling amplitude A by the weight function W(U,@.Hence,to
the extent that the light coupling approximation (upon
585
which the whole Fourier analysis is based) is valid,we may
expect
W(U,P) =
A
(63)
and thus,upon comparing (50),
the initial parameter estimates
ln10
Pi
=G.
20
(6o),and (63),we have as
(64)
= o.1151@n,
i=l,2,....N
where ~.is the desired nominal power coupling in decibels.
Thus we now have standard parametric forms lc(x,~) for
both continuous and discrete section symmetrical couplers
with good initial parameter estimates so that at this point
our optimization problem is completely formulated.
NUMERICAL EVALUATION OF S(U)
In using the standard parametric form for the continuous
coupler,it is useful to have a means of comp uting the func-
tion
s
 sin2 irx
A(u) =  dz
o
x
(65)
[see (61)],and we shall briefly discuss a means of evaluating
this integral.
For small values of the argument,we may expand the
integrand in a Taylor series about the origin and thereby
obtain
u 1  Cos27rx
JS(u) = f - d~
(66)
J()
2X
= 4.93480220u2 8.11742425u4 +7.12140143u6
 3.76529009u 8 + 1.32131284t~10  0.32931402u12
+0.06122824u14
0.00881269u1G +0.00101063u1S
 0.00009450uz0 +0.00000734u2~  0.00000048u24
+0.00000003U26 ....
Hence,for OS u< 1 we can easily compute S(u) from (66).
Next for u> 1 we have

 l/2{Y + in (%m)  Ci (%-u)]
(67)
where
Y = 0.57721566...(68)
is Eulers constant,and Ci(z) is the cosine integral,e.g.,see
[7].We then have
where for z> 1 we have
586
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
(70)
f(2!) =
Jo ~
d
1
{
Z8+ 38.027264#+ 265.187033z4 + 335.677320z2 + 38.102495
=
z Z8 + 40.021433#+ 322.624911Z4 + 570.236280z2 + 157.105423
}
s
m
Cos t
g(.2) ==

dt
0 t+z
1.Z8+ 42.242855zG + 302.757865z4 + 352.018498e2 + 21.821899
{
}
(71)
~
.Z2 ZS+ 48.196927z6 + 482.485984z4 + 1114.978885z2 + 449.6903%
(see [7]).Equations (67) through (71) may now be used to
compute S(u) to six place accuracy when u> l/(27r) = 0.16.
Finally,we shall note that if An is the area under the nth
node of the sitf (rx)/x curve,i.e.,
s
 sin2 TX
An =  dx,
n=l,2,3,...
(72)
n 1
x
then we have
A,=
1.218825 A,= 0.111291
A,=
0.058849
A,=
0.338325 A6 = 0.091007
Alo =
0.052651
Ag =
0.201059 A,= 0.076983
All = 0.047633
(73)
A,= 0,143240 As = 0.066706 A,,= 0.043489.
These constants are useful in the numerical evaluation of
(61) since we have
s
N
sin2 7ru
W(U,@) -j--
du
u
(

sin2 ru
= ep
du
J.u
(74)
sin2 7i-LL
+?e
s
n
 du
rb=-+
1 n 1
u
n=;,+
1
where v is the smallest integer greater than u.
A
COMPARISON WITH THE TRSSSELT PROCEDURE
It will be noted from (61) and (62) that there is a good deal
of similarity in the standard parametric forms for the con-
tinuous and discrete section couplers.In both cases the use
of the weight function w)(x,~) enables us to overcome the
inaccuracies of the low k approximation which are inherent
in the Fourier analysis,and thus,by using an optimization
procedure,shape the coupling coefficient function in such a
manner as to obtain a coupler of the desired physical
length which has a desired nominal power coupling and a
specified bandwidth.Now Tresselt [2] observed the simi-
larity between the discrete and continuous parametric forms
and made use of it in the following manner.He first used
the Cristal-Young procedure to design a discrete (21V l)-
section coupler having a desired bandwidth and nominal
power coupling.This then determined the set of weighting
parameters which are applied to the p(x) distribution,i.e.,
determined the weighting function W(W ~) of (60).Tresselt
then took this weighting function derived for the discrete
coupler and used it for the corresponding continuously
tapered coupler,which is 2N center frequency quarter wave-
lengths long.
It may not be clear why the weighting function used to
design a discrete section coupler should produce a useful
continuously tapered coupler,and consequently,a comment
or two at this point might be helpful.We might observe that
the respective areas An of (73) are closely related to the
areas 1/(2n  1) under the corresponding delta functions for
the discrete case,Indeed,we may compute (2n 1).An and
thereby,obtain the sequence of numbers 1.218,1.015,1.005,
1.002,
1,001,...
which rapidly converge to unity.Hence,
from a closer examination of (61) and (62),it is clear that
when the same weighting function w(u,~) is used,the cor-
responding discrete and continuously tapered couplers will
have almost the same coupling at the points u= 1,
2,3,...
on the
k(u,p)
distributions.In effect,the Tresselt procedure
provides a means of smoothing out a discrete section coupler
with the principle variation of the two designs existing in the
center of the k
distribution where  1<u< 1.
It must be noted,however,that the Tresselt procedure
does not always give rise to a desirable continuously tapered
coupler.To illustrate this point,we have prepared Fig.9,
which compares three corresponding  8.343 dB
couplers
having a common bandwidth of 2.61129 (see Table A3 from
[1]).The first coupler is a 3-section discrete coupler,the sec-
ond is the corresponding Tresselt design,and the third is the
optimized design based on the standard parametric form.
THE SPLINE PARAMETRIC FORM
We turn now to the development of another parametric
form for
k(u,p),which has been found to be quite useful in
the practical design of continuously tapered couplers.In
the most natural manner,we wish to let our parameters
pl,p2, 
.
represent ordinates of the coupling distribution
function at a sequence of equally spaced abscissas.Now
from our work with the standard parametric form,we know
that if we have a continuously tapered coupler which is 2N
center frequency quarter wavelengths long,then in the fre-
quency passband we may expect to have 2N+ 1 equal maxi-
mum deviations from the desired nominal power coupling
with these deviations alternating in algebraic sign (see Fig.
9).Next,we also know from our discussion of the optimiza-
tion procedure that it is highly desirable to have one more
deviation in the passband than we have parameters,and
hence we choose to use n= 2N parameters.Finally,we wish
KAMMLER:DESIGN OF TEM
DIRECTIONAL
COUPLERS
?
w)
.,.1
t
-,,2
I
587
DISCRETE COUPLER
TRESSELT DESIGN:
-.,
-8A
2
-8 6 -
1 I [ 1
L
-,6.i
-2 -,
0
2.0
5
,0!,5.4/.),
t
F(W,
t
K
,,
I
-8.
Z
1
A
-s,4
-8,.
-8 8
2 -9
0 I
,.
5,0,.0
,.5
OPTIMIZED DESIGN
(STANDARD PARAMETRIZATION)
A:Im
~-u
U__lL_
-,
-,
0
2
s
!0
,,s
d%
Fig.9.Three 8.343-dB couplers having abandwidth of B=2.61129are shown.Thek(u) distribut~on may reobtained from (59)to (62) by using
the parameters p,=0.9971,p,=l.7786 for the discrete and corresponding Tresselt design and by using theparameters pl=O.9869,
P2=  1.7571 for the optimized standard design.The maximum deviations from the  8.343 nominal value are 0.100 dB,0.164 dB,and O.O93
dB,respectively.
Fig.10.Thespline parametrization uses anatural splineto interpolate the4N+l points given by(75).This illustration corresponds tothecme
where N=3 so that wehavea continuously tapered coupler which is 6 center frequency quarter wavelengths long,Note that 6 parameters are
used to specify k(u,j).
to ensure that k(u) is an even function of u corresponding to
a symmetrical coupler.Taking all of these requirements into
consideration,we are led to a function k(u,j),which passes
through the 4N+ 1 points
[u,
k(u)] =
(N,
o),(N + +,p,),(N+
1,
p,),

 ,
(+,
IA-1),(0) %),(+,FL-l),  ,
(75)
(N  +,
n),(~,0)
where n= 2N (see Fig.10).
We have now
determined that
k(u,$) must pass through
a number of discrete points,and we would
like
some means
of interpolating between these points to complete the deter-
mination of a continuous coupling distribution function.If
we were given such a set of points and asked to draw a
smooth curve connecting them,we would probably make use
of a French curve or a draftsmans spline.We shall choose
to use the natural mathematical spline (which we shall dis-
cuss in the next section) in order to connect these points and
thereby complete the determination of
k(u,}).
588
IEEE TRANSACTIONS ON MICROWAVE THEORY AND
t
K,Ul
TECHNIQUES,
AUGUST 1969
COUPLING
DISTRIBUTION
FUNCTION
,2 -
L
I
I
I
-6 4
-2
0
2
4
6U
P(w)
A
-8.0 -
8 4.
-
-8.8
POWER
COUPLING
~dEbS,PON S E
-9.2 -
-9.G 
-,0.0
i
I
b
u,
26+ (J
Fig.11.
A continuously tapered 8.343-dB coupler having a 14:1 bandwidth has been designed using the standard parametrization.The maximum
value of k(u) is 0.7531 and the maximum ripple in the passband is O.331 dB.The physical length of the coupler is 12 center frequency quarter
wavelengths.The 6 parameters used to specify k(u) are 0.91826, 0.98728, 1.09078, 1.25054, 1.47631,and  1.37331.
A
.8
.()
COUPLING
DISTRIBUTION
FuNCTION
POWER
,(,4)
T
COUPLING
RESPONSE
[rfiA*wA
(m)
-8.0
8,4
-8.8
9.2
9.6
10.0
I
1
+
fJc
al=
w
Fig.12.A continuously tapered  8.343-dB couple!having a 14:1 bandwidth has been de~igned using the spline parametrization.The maximum
value of k(u) is 0.7822 and the maximum ripple m the passband
IS 0.286 & The
physical length of the coupler is 12 centel frequency quar-
ter wavelengths.The 12 parameters used to specify k(u) are 0.023980,0.030866,0.049965,0.064106,0.091635,0.117418,0.157327,0.204514,
0.265438,0.361863,0.490237,0.782224.
KAMMLER:DESIGN OF TEM DIRECTIONAL COUPLERS
Finally,we might indicate one means for choosing initial
estimates for the parameters pl,pz,- -.pfi which are used in
the spline parameterization for Ic(u,j).A 2N 1 section
discrete coupler having the desired bandwidth and nominal
power coupling is first designed using the standard paramet-
rization.The corresponding continuously tapered coupling
distribution k(u) is then obtained by Tresselts procedure.
Then from this approximate k(u),we may easily obtain
initial estimates of the spline parameters using (75).
In concluding this section,we might compare two con-
tinuously tapered  8.343 dB couplers having a common
bandwidth B= 14 and a common length of 12 center fre-
quency quarter wavelengths.The first has been designed
using the standard parametrization and the second using
the spline parametrization.The optimized results are shown
in Figs.11 and 12.The spline parametric form has resulted
in a somewhat smaller ripple and a somewhat larger maxi-
mum coupling coefficient than the standard parametric
form (which is almost identical to the corresponding
Tresselt design).In both cases N,= 360 elemental sections
were used in the numerical computations.
NUMERICAL COMPUTATION OF A NATURAL SPLINE
We turn now to a brief discussion of the natural spline.
We say that the function y(x) is the natural spline interpolat-
ing the points (x,,y,),i= 1,2   ,n where
x1<x1< o 0.x.
if
l)v(3i) =Yi,i=l,2,,n
2) v(%),v(z) are continuous for $1 S x S z.
(76)
s
%
3)
y(z) 2dx = minimum,subject to 1) and 2).
21
It might be noted that if we force a draftsmans wooden
spline to pass through a series of points (x;,Y,),it will,sub-
ject to these constraints,assume a shape which will minimize
its stored potential energy.Thus conditions 1),2),and 3)
correspond to a mathematical model of a draftsmans
spline and thereby guarantee that y(x) will be a smooth and
otherwise well behaved interpolating function.Now it can
be shown that the function y(x) satisfying (76) may also be
characterized by
1)
~)
3)
4)
(see
zJ(zL)= yi,i=l,2,..)n.
g(x) is a cubic polynomial on each of the intervals
[%.ti+l),i= l, ,n-l
(77)
y(x),y(z),y(z) are continuous for zl 5 x 5 x.
y($h) = U(r.) = o
[8]).Thus on each subinterval [xi,X,+l) y(x) is a
(possibly different) cubic polynomial,and these poly-
nomials are smoothly joined at the knot points x2,
X3,...
,x~_l in such a manner that y,y,and y are con-
tinuous.
Now with a bit of straightforward algebra,we may show
that the function y(x) satisfying (77) may be written in the
form
y(z) = y,%(x) + y,%(z) +    + u.%(x)
(78)
589
where the basis function o,(x) are themselves natural splines
passing through the points
{
1
ifi=j
@,(x,) = atj =
()
i,j=l,2,....n
(79)
otherwise
Thus the basis functions are completely determined by the
abscissas xl,x2,o 0.,x~.Fig.13 illustrates the basis func-
tions for a natural spline having six equally spaced abscissas.
When the abscissas xl,x.2,...,X.are eclually spaced
with
we may develop an explicit means of calculating y(x).On
each subinterval [xi,Xi+l),y(x) is a cubic polynomial,and
hence we have
where yi,y,,yi
are the derivatives of y(x) at x=x;+.
Thus y(x) is determined if we can compute yt,yin,y: for
i=l,2,...,
n  1.By a bit of straightforward but tedious
algebra it can be shown that for i= 1,
2,...,n
_#ln-i + ~2n-i
hll(x,) =  3atl  <3
I.J1-l /J2n-1
1
~b _ ~ (/-1-+,u,~-) (pp  ~2.-*)
%8
>
/Jl-l  P21
iss,s=2,3,.c.,nl
(82)
M,(G) =
I
_38,+
s
(w- +
/J,-)
(pl-l - ~2,-,)
t.
(m- /J2-)  
i~s,s=2,3,....l l
\
_
~li-l + ~,i-l
h@J(Zi) =
38jti + V3
P1l /.@l
where
pl=2+lL3
p2=2 &l
Using (78) and (82) we may now compute
(83)
ZJ =
yl%(d + W5z(zi) +    +!-h%(d,
(84)
~=l,z,...,~t.
Finally,we may usc y,,y,,
i= 1,2,  ,rl to obtain
590
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,AUGUST 1969
A
.~.
1
X2
X3
X4 X5
X6
A
1.@2
,
x,
X2
X3
X4
X,cj X6
4
1.53
x,
2
X3
X4
Xs
X6
L
I?26
xl X2
x3
x4
x5
X6
*
1.*5

xl
X2
X3 X4
X5
xG
.
1 -*4

xl
X*
X3
X4 X5 X6
Fig,13.The natural spline y(x) passing through the points (x,,y,),(x2,y,)..
.
(X6,
~6)
Can
be written in the form of
(78) with the basis functions @,,@z,....@,being shown above.
Equations (81) through (85) then serve to explicitly deter-
which are of current technological interest,A number of sig-
mine the natural spline y(x).It will be noted that an implicit
nificant applications of these techniques will be discussed in
iterative method for computing the natural spline in the case
a companion paper [9].
where the abscission xl,x2,..0,X3 are arbitrarily spaced
is given in [8].
RIWERENCES
CONCLUSIONS
A frequently recurring problem of microwave engineering
is to find an input function, e.g.,k(x),which results in an
output function,
e.g.,o(u),having specified characteris-
tics.By using a parametric form k(x,}) and the nonlinear
optimization procedure TFIT (which belongs to the gen-
eralized Remez family of optimization techniques but in-
cludes special provisions for under and over deter-
mined cases which often occur) one can often solve such
problems in a relatively direct and efficient manner,This
method has been illustrated by discussing in detail the tech-
niques involved in obtaining coupling distributions char-
acterizing both discrete and continuously tapered
directional quadrature couplers.
In carrying out this procedure one is often led to a specific
parametric form by physical considerations,e.g.,the ap-
proximate Fourier analysis led to the standard parametric
form in the above analysis,but any number of reasonable
parametric forms will serve just as well if good initial
estimates of the parameters are available.The spline
parametric form is a perfectly general one which promises
to be especially useful in the practical design of a variety of
continuously tapered broadband microwave components
[1] E.G.Cristal and L.Young,Theory and tables of optimum sym-
metrical TEM-mode couded-transmission-line directional cou-
[2]
[3]
[4]
[5]
[6]
[7]
piers,
IEEE Trans.Micro;ave Theory and Techniques,vol.
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