A theory for decoupling lossless coupled tapered lines with applications to directional couplers

Journal tgh ghghghghghnstitute DEVOTED

TO SCIENCE AND THE MECHANIC

ARTS

June

Volume 299, Number 6

1975

A Theoryfor Decoupling Lossless Coupled Tapered Lines with Applications to Directional Couplers by

M. N. s. SWAMY,

ands.

B. B. BHATTACHARYYA

N. VERMA*

Department of Electrical Engineering Concordia Uniuersity,t Montreal, Canada ABSTRACT: A simple theory is presented for decoupling

Iranamission

lines (CNUTL),

.?QS&SS nonuniform

transmission

lines (NUTL).

symmetry conditions,

terminations,

etc.

decoupled

lines

CN U TL’a

as a four-port

aa two-ports.

aa directional

This

should

decoupled coupler couplers N U TL’s

line,

lines have to be dusk is found

while for

of each other.

to be periodic,

while

with frequency.

with hyperbolic response

the matrix para-

in lerma of those of the

to study

the application

tk

coupler,

contradirectional

of

The coupling

phase

shift

The coupling

response

between

response

each of the decoupled

coupler

solutions”, and

are studied in detail.

that the CNUTL’s

lines have the best response

the two

coupled

and transmitted

of various contradirectional lines are “basic

It is shown that all these couplers

with “hyperbolic

of all the CNUTL’s

action,

of the codirectional

with sntoolh transition at one of the en&, for which the decoupled

have a high-pass decoupled

to behave aa a codirectional

be a proportional

signals varies linearly

expressed

is then utilized

of the port

couplers.

It is shown that for CNUTL’a lines

waves, into two

relates the line parameters of

lines and vice versa; further,

are explicitly

theory

TEM

This theory is independent

The method directly

the lossless CN U T L’a to those of the decoupled metera of the CN U TL’s

a pair of lossless coupled nonuniform

with a common return and supporting

cosine squared lines”

as

considered.

I. Introduction

In the present decade, with the advancement of compact packing techniques and improved rise and fall times of semiconductor devices, the study of multiple parallel transmission lines has become increasingly important. * Present address : Western Union Telegraph Company, 1 Lake Street, Upper Saddle River, New Jersey 07458, U.S.A. t Incorporating Sir George Williams University ctnd Loyole of Montreal.

381

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma For example, the backplane wiring of a modern high-speed computer is best represented by multiple parallel transmission lines. At microwave frequencies lossless transmission lines with distributed electromagnetic coupling are also widely employed as directional couplers, Balun transformers, magic “T” and filters (1-8). The problem of analyzing coupled lossless transmission line networks is complicated due to the presence of electromagnetic coupling terms in the pertinent telegrapher’s equations. A method of analyzing coupled identical uniform transmission lines is described by Vlostovskiy (9). A simpler approach to this problem has been taken by several authors. Jones and Bollajhan (10) study this problem by using the concept of even and odd mode equivalent systems ; in this case it is necessary that the coupled line network including the terminations be symmetrical. The conditions of symmetry have been modified by Ozaki and Ishi (11)to allow a class of nonidentical uniform lines to be analyzed. Yamamoto et al. (12) have extended the even and odd mode approach to tapered coupled transmission lines; the lines, however, must be identical. Note that in the above methods, except in the case of identical uniform lines, it is not clear how the coupled line parameters such as self inductance per unit length at any point on the transmission line, etc. can be related to those of the even and odd mode lines. In this paper we first introduce a simple approa’ch whereby two coupled lines with a common return are reduced to t,wo pairs of decoupled lines. This converts the analysis problem of a coupled line four-port into one of analyzing two pairs of transmission lines as two two-ports. The method is independent of the port terminations, any symmetry conditions, etc. Another feature of this technique is that it directly relates the coupled line parameters to those of the corresponding decoupled lines, and vice versa. coupled line contradirectional and codirectional Using this theory, couplers having infinite directivity, and exhibiting impedance transformation or not, are analyzed and the corresponding constraints on port terminations and decoupled line distributions derived. Various contradirectional couplers exhibiting different characteristics are studied by properly choosing the distribution of the decoupled lines. The properties of contradirectional couplers using “basic lines with hyperbolic solutions” as decoupled lines are Finally, the coupling response of codirectional couplers is obtained. investigated. II.

Decoupling

of Tapered

Coupled

Lines

Consider the pair of coupled nonuniform transmission lines as shown in Fig. l(a); the lines A and B may be similar, or dissimilar. Let the per unit length parameters be self inductance

of line A,

(la)

-&?A4= -LJ2(2)>

self inductance

of line B,

(lb)

CnW

capacitance

4,(z)

382

= &Pl@), = C,,,G&L

of line A to ground,

Journal

of The Franklin

(lc)

Inst,itote

A Theory for Decoupling Lossless Coupled Tapered Lines

C,,(x)= G,Q’,@),

capacitance

j%4

= J&l JW+

mutual inductance

Q&4

= CL, G&),

mutual capacitance

I-

x4r

of line B to ground, between A and B, between A and B.

A(s ’ x)

IO

Line

+

(Id) (le) w

03

A

V,‘(s,x) 04

20 x=0

11, (58 XI Line B vB,(S,X)

X=1

(a) J, (s,x)_

0 E,

I

(5

Line

0

I

.d

0

(b)

FIG. 1. (a) General lossless tapered coupled lines. (b) The corresponding decoupled lines.

It is known that for TEM mode propagation

(13)

GZ, [ul

ru41 r@41 = where w(z) is a positive function,

(2)

7

[U] is a unit matrix and

LWI = [ f?Q il: 1, KY41 = [ ‘~~~’

ci:i12

]

Pa)

having -&j(Z) 2 0,

L,&)

>&j(X),

C&(s) > 0.

(3b)

From (2) and (3)

L,,(x)

L,,(x) -k?,(x) c,,(x)= G2(4 + c,,cx,= C,,(x) +C12(x)

Vol.

299,

No. 6, June

1975

(4a)

383

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma Thus a pair of coupled lines supporting TEM waves have to satisfy the requirements (4). Telegrapher’s equations for these lines are [V’(s,

41 = - MS, 41 [I@,@I, [I’(%@I = - Ewt 41 EV@, 41,

where

(54 (5b) (5c)

WI = @@)I,

ry1 = &Y~)l

and s is the complex frequency. In order to decouple the lines, we introduce where

lY(%41 = [&Ib% 41,

(54

the linear transformations

[k 41 = [Ml [Jb, 41,

(64 (6b)

a, b, c, d, e, f, 9 and h being arbitrary constants. Then the equations of the coupled line reduce to W’(% where

41 = - [&-lZw [J(s, 41 = - [-q&, 41 CJ(4q1, LJ’h 41 = - [Jf-l WI VW,41 = - rus, 41 VW, 41, [.%I = wlZm,

[Y,] = [Jf-l WI.

(7) (8) (9)

To achieve decoupling [Z,] and [Y,] should be diagonal. Setting the offdiagonal elements to zero and using (4a) we have the following two independent conditions to satisfy : W,,

+ J4,)

-W&z

WW

+ h-%,1 = 0,

It can be shown t,hat c = + pa, satisfy the conditions

d = T pb,

g = + e/p,

h = ~f/p

(loa) and (lob) where &2 C11+G P2=~=ce2+c12.

(11)

Thus the CNUTL parameters should satisfy (11) in order to be decoupled. Now choosing a = e = f = 1, [Q] and [M] may be written as

[&I= [ :,

19[HI=[ &:ip Gip 1.

f,

(12)

Then Eqs. (7) and (8) reduce to [E’(s, x)] = -8

384

0

Lll -t GalP) 0

&I

T (b!lP)

[J@,

41~

(134

1 Journal of The Franklin Institute

A Theory for Decoupling Lossless Coupled Tapered Lines

[J’(s,x)]

= -s

Cll+PTP)% 0

0

GAz+P+P)~1,1

~~(s,41.

Equations (13) may be interpreted as those of two pairs of decoupled say lines 1 and 2 with per unit length parameters given by and

(13b) lines,

Ll=JG1+wl,/P)~

~l=cll+(l-P,%

(14a)

-%! = JL-

Q, = c22+ (1+ P)%

(14b)

(&Z/P),

where the upper signs have been taken into consideration in (13). This method of decoupling neither requires the coupled lines to be symmetrical about an axis nor depends on port terminations. In fact, the method is quite general except that (11) has to be satisfied, since the coupled telegrapher’s equations have been directly decoupled without any assumptions. Knowing the distributions of the decoupled lines, the matrix parameters of these two pairs of lines may be found by using any one of the known techniques (14-17). Since [V] and [I] of the coupled lines are related to [A’] and [J] of the decoupled lines through (6), the matrix parameters of the original coupled lines may be obtained in terms of those of the decoupled lines. Using (6a) the impedance matrix [Z] of the coupled lines can now be expressed in terms of the impedance matrices of decoupled lines [ZJ and [Z,] in the form :

where [Q] and [M] are given by (12) with the upper signs taken into consideration,

(16) and Zij and Z:j are the impedance parameters of lines 1 and 2, respectively. The other matrix parameters may be obtained in a similar way and are given in Table 1. Thus the problem of analyzing a coupled transmission line as a four-port reduces to that of analyzing two sets of transmission lines as two ports. The theory of decoupling presented here may be applied to formulate design procedures for various microwave components such as directional couplers, all pass networks, filters, etc. using coupled lines. In the next section we apply this decoupling theory to study coupled tapered lines as directional couplers. ZZZ. Tapered

Line

Contradirectional

Couplers

Consider a lossless reciprocal four-port with terminating resistors as shown in Fig. 2. This four-port will behave as a contradirectional coupler if, when terminated in prescribed resistances, it is matched at all ports with ports (1) and (4), and (2) and (3) mutually isolated (18). There may or may not be

Vol. 299, No. 6, June 1975

385

,I = [ t:

il ]

Decoupled line two-port

matrices

[a] = f

bl = f ,

P

CCl+qC2) (01 -DA (G-C,) P

(G+C*)

P

wh+m

@,+A,) (G-C,) p

&%--A,)

pP,-JM

(B,+B,)

@yQ 4)

P2

(4

P

(Yil + Yil)

(YL -Y&)

P

(Y:n-Y%

P

P

(Yil +pY:l)

matrices

(Y:, - Y&J

(Y:l+Y;J

YL)

(YLP

YL)

(Y:l+

Coupled line four-port

Coupled line parameters in terms of those of decoupled lines

TABLE I

_I

(0, + De)

pm-m

pW,+Bz)

p(B,-23,)

pa

(Yi2 +?/%‘A

P2

(YA + Y:2)

(Y:a-Y:n) p

A Theory for Decoupling Lossless Coupled Tapered Lines Loss!ess tapered coupied lines

FIG.

2.

Lossless

tapered

coupled

lines as contradirectional

or codirectional

coupler.

impedance transformation depending on the values of t,he terminating resistors. The scattering matrix for the contradirectional coupler is (18)

.

(17)

Let us consider the tapered line given by (1) to be the four-port in Fig. 2. In terms of the transfer scattering matrix parameters corresponding to (17) we have for a contradirectional coupler t,, = t,, = t,, = t,, = t,, = t,, = t,, = t,, = 0 and t,,fO,

t,,#o,

t,,+o,

t,lfO,

hl f ',

t21 # '2

tllt44 #t14t419

t,,fO,

t,,#o, (18)

t33t22#t32t23.

Expressing these parameters in terms of the “chain matrix” parameters of the coupled lines, which themselves may now be expressed in terms of those of the corresponding decoupled lines using Table 1, we obtain 4t

=

(Al-A2b4+(Dl-D2)f_(Bl-B2)~_(cl-c2)f-'4rl=0

12

> P

4t

=

P4

P4 r1

P

(Al-A2)~f'3+(D1-D2)~2_(B1-B2)~_(cl-C2)~2f-'3~l=0 9

21 P2

PP3

P2 P3 r1

P

(Dl+D2)+(Bl+B2)_(C+C)p 4tl3

=

(Al

+ A,) ~3 -

=O 9

P3 r1

(Q+D2) 4t31 =

r 1231

Pa

(&+B2)+(C+C)P3T

(Al+A2)~3-

=O 19

12 P3

-

P3rl

(19) 4t

=(Al+A2)~4_(Dl+D2)f2+(Bl+B2)~2_(~+c2)~2~4rl

o

24 P2 4t

P4

P2 Pa r1

p2

=2

p2

=j

=(A~+A2)~4_(Dl+D2)~2_(B~+B2)~2+(c~+c2)~2~4rl

o

42 P2 4t

=

P4

P2 P4 9.1

(Al-A2)P4+(Dl-D2)P+(B1-B2)P+(cl-c2)P4T1

=o

34 P 4t

=

P4

Pa r1

P

(Al-A2h3+(Dl-D2)f2+(Rl-B2)~+(cl-c2)~2~3Tl=0

43

9 P2

Vol. 299, No. 8, June 1975

PP3

P2 P3 r1

P

387

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Vemuz. where Pa= J($

Ps=J($.

/%=J($

P”

J&.

(20)

Since for a lossless line the parameters A and D are real while B and C are imaginary at s = Jo, by considering the real and imaginary parts of (19) we get

(A, +A,)

rv-$+C,)

(D,+D,)

= (B,+B,)

(4 -4

wi--~l)

4

p4”= $$

= p2_

(B,-B,)

(DI-D,)=

1 =& =z

p;-n-=*

Pi

_

P2 Pn Pa

(214 @lb)

From (21), we get P: P4.

(224 Pb)

P3P=Pa*

(23)

Pt = P”z Pi, PIi Pi We

get from

=

(22a) and (22b) Pz=PJ

Now using (23) and (21) we obtain P29 =

(24)

Dd42*

Since pi = rs/rl is independent of frequency we may get its value by evaluating it at w = 0. Using the fact that for lossless lines A, and D, are unity at o = 0, we have ps= 1. (25) Thus from (20), (23) and (25) we have r2 = r4 = p?,.

f.1 = r3,

(26)

Equation (26) puts constraints on the terminating resistors. If the lines are identical, p = 1; hence all the terminating resistors are equal, thus providing no impedance transformation. However, an impedance transformation of p2 between ports (1) and (2) and (3) and (4) may be achieved by using nonidentical lines. Thus if a contradirectional coupler is designed using coupled lines, the port pair (1,3) should be terminated with equal resistances. A similar statement holds for the port pair (2,4), while there may be an impedance transformation between the port pairs (1,3) and (2,4). Finally, using (21) and (26) we may interrelate the parameters of decoupled lines 1 and 2 as A, = D,,

D, = A,,

B, = r:C,,

C, = B&f.

(27)

It is shown in the Appendix that two lossless lines, whose parameters are interrelated as in (27) should have “dual distributions” (19), that is, if line 1 has distributions L, = L,,F(x),

C, = C,,G(x)

(O
Wa)

(O
(28b)

then line 2 will have for its distributions L, = L,,G(z),

388

C, = C,,F(x)

Journal of The Franklin

Institute

A Theory for Decoupling Lossless Coupled Tapered Lines where (29) rl” = &J/c,, = L,IGW Thus a coupled line will behave as a contradirectional coupler, provided the corresponding decoupled lines have dual distributions as given by (28) and (29). Clearly, parameters of the coupled line may be obtained from those of the decoupled lines using (14). Relations (27) may be used to express the scattering matrix (17) of the contradirectional coupler in terms of the chain parameters of the decoupled lines as 2 0

0a

010 S=$ [

where

0

2

2

0

0

a

0

a

ci

0

1 ’

((4 -RI + W%lrJ- Clrll~~ P = {(A, +&I + [Pllrl) +44. a=

(304

(3ob) (304

coupler It is noted from (30) that S,, = S,,. Hence, a contradirectional formed by a pair of lossless reciprocal coupled lines always has the property that Sr3 = S,,. Thus, the conditions S,, = Sz4, as well as the interrelations (27) between the matrix parameters of the two decoupled lines, apply to a more general case of coupled line contradirectional coupler than that considered by Sharp (13), since his results were confined to the case of CNUTL’s having an absolutely continuous characteristic impedance matrix. The various characteristics of the contradirectional coupler, namely, the directivity 3, the coupling Y and the phase shift &r between coupled and transmitted signals may be obtained from (30) as @=

co,

(3Ia)

Y- = 2Olog,

(A,+D,)+[(B,/r,)+C,r,l’ O (Al-u,)+[(~,jrl)-C,r,II’

+CT= tan-1 ‘+2rzFf 1

where 9, 9

@lb) (3lc)

1

and +oT are defined as (23): 9 = 20 log,, I s21/fL/9 9- = 201%,,l

W,,l,

4CT = G%?l/fM~ From (31~) it is seen that in general the phase shift between voltages at ports (2) and (3) is not constant but depends on frequency. However, by choosing suitable line distributions for the decoupled lines we may get contradirectional couplers having frequency independent phase characteristics. It is found that for symmetric tapered lines (20) or proportional lines (21) as decoupled lines, the contradirectional couplers exhibit 90’ phase shift between voltages at ports (2) and (3) at all frequencies, a feature displayed

Vol. ZQQ,No.6,June 1975

389

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma by uniform coupled line contradirectional couplers. The phase difference may be made zero for all frequencies if the two decoupled lines are selected to have J’(x) = emx,

G(x) = e-m(z-s),

rf = (L,,/C,,)e””

with L,, = L,,,

i.e. the two decoupled lines are exponential lines of length I, one looking from left to right while the other from right to left. Using “basic lines” with hyperbolic solutions (22) as decoupled lines and normalizing the terminating resistances to 1 !A, the coupling characteristics of the various contradirectional couplers were then investigated. It was found that these couplers exhibit a high pass response as has been noted earlier by Yamamoto et al. (12) for exponential lines. Further, when the decoupled lines are “hyperbolic cosine squared” or “trigonometric”, that is L, = L,,cosh-2mx,

C, = C,,cosh2mx

L, = L,, cosh2 mx,

C, = C,, cosh-2 mx

L, = L,, cos-2 mx,

C, = C,, (308~mx

L, = L,, cos2 mx,

C, = C,, cos2 mx

(O
(32)

or (O
(33)

the corresponding coupled line couplers have the best coupling response. For the sake of comparison the coupling characteristics for line (32) as well as when decoupled lines are exponential (12) are given in Fig. 3. These cosine squared couplers have the advantage of smooth transition in electromagnetic coupling at the input port thus providing better matching and isolation ; in addition, they require a shorter length for the same range of frequencies. -----

Normalized

frequency,

radslsec.

Exponential

W” = [w

line

\/(

L,,

C,,)ll

FIG. 3. Coupling characteristics of CNUTL contradirectional couplers with exponential and hyperbolic cosine squared lines as decoupled lines.

390

Journal of The Franklin Institute

A Theory for Decoupling Lossless Coupled Tapered Lines IV.

Tapered

Line

Codirectionul

Coupler

The lossless reciprocal four-port with terminating resistors, as shown in Fig. 2, behaves as a codirectional coupler if, when terminated in prescribed resistances, it is matched at all ports and also ports (1) and (2), and (3) and (4) are mutually isolated (24). Such a coupler may or may not exhibit impedance transformation depending upon the values of the terminating resistors. The scattering matrix of a codirectional coupler is (24)

-

[fll =

(34)

Let us again consider the coupled line given by (1) to be the four-port shown in Fig. 2. In terms of the transfer scattering matrix parameters, conditions (34) become t,, = t,, = t,, = t,1 = t,, = t,, = t,, = taz = 0 and

Following

t,,#O,

&,#O,

&,#O,

t,, # 0,

t,,#o,

&$t44f.t34t43.

&fO,

&,#O,

(A,+&)P,-

4t31=

(Al

(D1p+,D,) +@;;tlp,) - (Cl + C,) (Bl+B2)+(C

(Q+D2) + A,) ~3 P3

-

p3 r1

+c)p 1231

r

=

0,

=o 7

P3f.l - @l-c2b4rl=~

=(Al-A2b4_(Dl-D2)f)+(Bl-B2h

,

14 P 4t

=

(35) I

the procedure used in the previous section, we obtain

%, =

4t

t,,fO,

P4

Pa Tl

P

(~l-~2)~f3_(D1-D2)~2_(~1-~2)~+(~~-C2)~2~3~l

=

o

41 P2 4t

=

PP3

(36) - (%c2b2P3rl=o

(A1-A2)pp3_(D,-D2)p2+(B1-B2)P

,

23 P2 4t

=

PP3

P2 P3 r1

P

(Al-A2)P4_p(D1--D2)_(B,-B2)p+(Cl-Q2)P4rl

=

32 P 4t

P4

P4 r1

=(A,+A2)p4_(D,+D2)p2+(Bl+B2)p2_(C,+C2)p2p4rl=0

4t

P4

P2 P4 f.1

o

,

P

,

24 Pz

,

P

P2 P3 r1

P2

= (Al+A2)~4_~2(D,+D2)_(Bl+B2)~2+@l+c2b2~4F1=0 42 P2

Vol. 299, No. 0, June 1975

Pa

P2 P4f.l

P2

391

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma Separating the real and imaginary parts of (36) for s = j,,

&+A,)

G+C,)rT

(Dl+D2)

(Cl-C,)r; =

P4

=z p;=a’

= (B,+B,)

b&-A,) (Dl__D2)

1=&

P2

=z

(B,-B,)

=&2

r3,

r2 =

r4 =

(374

P2

W’b)

=x2. P2 P3

Following the same approach as for contradirectional solving (37), r1 =

we obtain

couplers, we have, on

p2r,

(38)

and A, = D,,

B, = rt C,,

A, = D,,

B, = rf C,.

(39)

From (38) we note that the port terminating conditions in this case are the same as in that of the contradirectional coupler, that is, an impedance transformation of p2 between ports (1) and (2), and (3) and (4), may be obtained by using nonidentical coupled lines. Relations (39) may be used to express the scattering matrix (34) of the codirectional coupler in terms of the chain parameters of the decoupled lines as

where

(bob) 8= Hence, always It is related

(A, - A,) - (B, - B,)/r,.

(4Oc)

codirectional couplers formed from a pair of lossless coupled lines have the property that S,, = X2, and S,, = X2,. shown in the Appendix that a lossless line whose parameters are as A=D,

with r independent

B=r2C,

(41a)

of s and x, should have proportional L = LOP(X),

distributions,

c = COP(X),

that is (41b)

with (4lc)

r = J(W%).

Using this result and conditions (39), it is seen that for a codirectional coupler, the distributions of the corresponding lines are required to be of the form

392

L, = L,,~(x),

Cl = fwv4,

L, = -&G(x),

C, = G,G(x),

I Journal

(424

of The Franklin

Institute

A Theory

forDecoupling

Lossless Coupled Tapered Lines

with L

L

c 10

GO

LO = 20.

,).a =

1

(42b)

Thus a pair of coupled lines will behave as a codirectional coupler when each of the corresponding decoupled lines is a proportional line described by (42). The coupled line parameters may now be obtained from those of the decoupled lines using ( 14). The expressions for the various characteristics of the contradirectional coupler may be obtained from the scattering matrix (40) as $3 = co,

MaI 2

r

= 2o log”

[A, - (BI/rI)]

(43b)

- [A, - (Bz/rl)] ’

(43c)

where (23) 9 = coupler directivity F

= 201og,, 1S,,/X,,

I,

(444

= coupling of the coupler = 20 log,, 1l/S,,

I,

+cT = phase shift between coupled and transmitted =

(44b) signals (44c)

~fJ41/&0,*

For decoupled

line distributions

(42), it is well known that (21, 25)

A, = costi,,

B, = Z,,(O) j sin el,

A, = cos ke,,

B, = Z,,(O) j sin kfl,, 1

(Qa)

where e1 = W~(~lo~lo)

b(r) s0

dx,

e2 = w J(L,,c,,)

(45b

1

(45c)

It is assumed that k # 1, since otherwise the original lines A and B will no longer be coupled. Substituting (45) in (43), we have ZT = lOlog,

1

O l-cos(k-1)8,

&vT = 742 + (k + 1) el.

’ (46b)

Thus, for lossless coupled line codirectional coupler, the coupling response is periodic with frequency, while the phase shift between coupled and transmitted signals varies linearly with frequency.

Vol.

299, No.

6, June

1975

393

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma V. Conclusions

A simple method of decoupling a pair of lossless coupled transmission lines with a common return, supporting TEM waves, into two pairs of decoupled lines is introduced. This method does not require any symmetry conditions to be satisfied by the coupled system nor does it depend on port terminations, since the pertinent telegrapher’s equations for the coupled lines are directly decoupled without any assumptions. The method also enables us to obtain the line as well as the matrix parameters of the coupled pair from those of the decoupled ones. As applications of this theory, properties of the contradirectional and codirectional couplers exhibiting impedance transformation or not and having infinite directivity are investigated. It has been shown that for coupled lines to behave as contradirectional coupler the decoupled lines should have dual distributions, while for codirectional action each of the decoupled lines should have proportional distributions. Along with directional coupler action impedance transformation between certain ports can be achieved if the coupled lines are selected to be nonidentical. It has been found that by properly choosing the decoupled line distributions the phase shift between coupled and transmitted signals can be made 0 or 90’ for all frequencies. The characteristics of contradirectional couplers for which the decoupled lines are “basic lines with hyperbolic solutions” have been studied in detail. It has been found that when the decoupled lines are “hyperbolic the corresponding contradirectional cosine squared” or “trigonometric”, couplers provide the highly desirable high pass coupling response in the frequency range of interest. Finally, it is shown that the “coupling” response of a codirectional coupler with infinite directivity is always periodic with frequency, while the phase shift between the coupled and transmitted signals varies linearly with frequency.

Acknowledgements This work was support,ed by the National Research Council of Canada under Grant Nos. A-7739 and A-7740 and is based on a doctoral thesis (26) submitted by S. N. Verma to Sir George Williams University, Montreal, Canada.

References

(1) W. E. Caswell and R. F. Schwartz, “Directional

couplers 1966”, IEEE Trans. Microwave Theory Tech., Vol. MTT-16, pp. 120-123, Feb. 1967. (2) R. J. Wynzel, “Exact design of TEM microwave networks using quarter wave lines”, IEEE Trans. Microwave Theory Tech., Vol. MTT-12, pp. 94-111, Jan. 1964. (3) W. J. D. Steenart, “The synthesis of coupled transmission lines all pass networks in cascade 1 to n”, IEEE Trans. Microwave Theory Tech., Vol. MTT-11, pp. 23-29, Jan. 1963. (4) N. Nagai and A. Matsumoto, “Application of distributed constant network theory to Balun transformers”, J. Elect. Comm. Engng, Jap., Vol. 50, pp. 114-121, May 1967.

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Journal of The Franklin

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A Theory for Decoupling Lossless Coupled Tapered Lines (5) N. Nagai and A. Matsumoto, “Application of multiwire networks to a distributed parameter hybrid networks”, J. Elect. Comm. Engng, Jap., Vol. 51, pp. 11-19, March 1968. strip line magic T”, IEEE Trans. Microwave (6) E. M. T. Jones, “Wide-band Theory Tech., Vol. MTT-8, pp. 160-168, March 1960. (7) D. I. Kraker, “Asymmetric coupled transmission line magic T”, IEEE Trans. Microwave Theory Tech., Vol. MTT-12, pp. 595-599, NOV. 1964. (8) A. Matsumoto, “Recent Advances in Microwave Filters and Circuits”, Academic Press, New York, 1970. “Theory of coupled transmission lines”, Telecomm. Radio (9) E. G. Vlostovskiy, Engng, Vol. 18, pp. 87-93, Apr. 1967. (10) E. M. T. Jones and J. T. Bolljahan, “Coupled strip transmission line filters and Trans. Microwaoe Theory Tech., Vol. MTT-4, directional couplers”, IEEE pp. 75-81, Apr. 1956. (11) H. Ozaki and J. Ishi, “Synthesis of a class of strip line filters”, IEEE Trans. Circuit Theory, Vol. CT-5, pp. 104-109, June 1958. (12) 8. Yamamoto, T. Azakami and K. Itakura, “Coupled nonuniform transmission line and its applications”, IEEE Trans. Microwave Theory Tech., Vol. MTT-15, pp. 220-231, Apr. 1967. (13) C. B. Sharpe, “An equivalence principle for nonuniform transmission line directional couplers”, IEEE Trans. Microwave Theory Tech., Vol. MTT-15, pp. 308405, July 1967. (14) E. N. Protonotarios and 0. Wing, “Delay and rise time of arbitrarily tapered RC transmission lines”, IEEE Int. Conw. Rec., Pt. 7, pp. 1-6, 1955. (15) M, N. S. Swamy, “Matrix parameters of an arbitrarily tapered RC-line”, Proc. Mid Am, Electron. Conf., pp. 72-82, 1965. (16) S. C. Dutta Roy, “Matrix parameters of nonuniform transmission lines”, IEEE Trans. Circuit Theory (Corresp.), Vol. CT-12, pp. 142-143, March 1965. (17) B. B. Bhattacharyya and M. N. S. Swamy, “Interrelationships among the chain matrix parameters of a nonuniform transmission line”, Proc. IEEE, Vol. 55, pp. 1763-1764, Oct. 1967. (18) R. N. Ghose, “Microwave Theory and Analysis”, McGraw-Hill, New York, 1963. (19) B. B. Bhattacharyya and M. N. S. Swamy, “Dual distributions of solvable nonuniform lines”, Proc. IEEE, Vol. 54, pp. 1979-1980, Dec. 1966. (20) B. B. Bhattacharyya, R. T. Pederson and M. N. S. Swamy, “On basic symmetric lossless lines with hyperbolic solutions”, Proc. 13th Midwest Symp. on Circuit Theory, pp. x.2.1-x.2.11, May 1970. (21) M. J. Helstrom, “Symmetrical RC distributed networks”, Proc. IRE (Corresp.), Vol. 50, pp. 97-98, Jan. 1962. (22) M. N. S. Swamy, J. Walsh, J. C. Grignere and B. B. Bhattacharyya, ‘Basic nonuniform lines?, Proc. IEEE, Vol. 116, No. 5, pp. 710-712, May 1969. (23) 0. L. Matthaei, L. Young and E. M. Jones, “Microwave Filters, Impedance Matching Networks and Coupling Structures”, McGraw-Hill, New York, 1963. (24) L. Young (Ed.), “Advances in Microwaves”, Vol. 1, Academic Press, New York, 1966. (25) M. N. S. Swamy and B. B. Bhattacharyya, “Generalized nonuniform lines and their equivalent circuits”, Proc. 10th Midwest Symp. on Circuit Theory, pp. 11-4.1-11-4.10, May 1967. (26) S. N. Verma, “Lossless coupled nonuniform lines and their applications to directional couplers and all-pass networks”, D.Engg. thesis, Sir George Williams Univ., Oct. 1972.

Vol.

299, No.

6,

June1975

395

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma Appendix The following two theorems tributional will now be proved.

concerning NUTL’s

with dual and proportional

dis-

Theorem I The necessary and sticient conditions for the chain parameters of two lossless lines -Ep and S?* of equal but arbitrary length 1 to be interrelated as A,

= D,,

D, = A,,

B, = P(Z) C,,

C, = B,/r2(1),

T being independent of 8, is that _PI and Pz be dual lines; that is, if the distribution of PI are L, = L,,P(z),

C, = C,,G(z)

(O
C, = C,,p(4

(O
then the distributions of _Yz are L, = L,,G(z), where

Necessity Let L,(s) and C,(z) be the distributions for line PI and L,(x) and C,(z) for line Ip,, where L,, C,, L, and C, are positive integrable functions of x. Then the chain parameters of lines PI and _!Z’*may be expressed in terms of the uniformly convergent series in the form (14, 15) A, = 1+&v+‘, 1

(Al)

a, = ;

(A2)

b,vs2n+1,

c, = 5 c,, 0

P+l,

(-43)

D, = l+5dnys2”, 1

(A4)

where

(A7)

(A8) and v = 1,2, respectively, corresponds to the line 9, and Ipz. For a length of ((0~ .$< I), let the total inductance of _YI be related to the total capacitance of Sz in the form

396

Journal of The Franklin

Institute

A Theory for Decoupling Lossless Coupled Tapered Lines while the total inductance of _P.. and total capacitance of Pi be related as ‘U!) s0 Since A, = D,, A, = D,, (AS) that

dt = P&)

s

>,(O

(AlO)

df.

B, = rz(Z)C, and B, = ~~(2)C,, it is necessary from (Al)

to

on = d 121

(All)

a 12 -

dw

(AW

b,, = rV) ciJ2,

(A13)

b,, = G(Z) col.

(A14)

From (A5) to (AS) and the above equations, it is seen that

s‘L,(q) 0

dz, = r2(Z) zC,(z,) dz,, s0

(A17)

dz, = ~~(1) zC,(z,) dz,, s0

(Al8)

zL,(2,) s 0

where x2 ranges from 0 to 1. Substituting zW%) s0

or

PLl(“z) %,(z,) s0 S’PU!) 0

(A9) in (A15)

dz, dz, =

CL&)

j$(x,)

ho%

-~.(&lj)M4

h,

dx,

dz, d5 = o.

Since the above expression should hold for any arbitrary length 2 C,(E)&)

or

-k(5)

= 0,

(OG 5GZ) (Al9)

U&

= /&)

C,(‘%

I

Similarly, using (A9) in (A16) it can be shown that (A20)

L,(E) = Z%(E) C,(5). Integrating

(A19) from 0 to 2 and using (A18) for any arbitrary I, it is obtained that

szWt)

dt =

0

Hence

j)(t) C,(t) d5 = Wj$~ dS. (A21)

Z%(5) = TYZ). Similarly, from (A20) and (A17), it can be shown that Pz(S) = r2(Z).

(A22)

Thus, from (A21) and (A22)

= ?-2(Z) and hence both p1 and pz are independent of 5. Therefore, from (A19), (A20) and (A21) WE)

Hence,

= TYZ) C,(S)

if the distributions of _!$‘rare L,(z)

Vol. 299, No. 8. June 1975

= J&OWr),

C,(z)

= C,, G(z),

(O
(A23a)

397

M. N. S. Swamy, B. B. Bhattacharyya and S. N. Verma then the distributions of _Y2 are &,(4

= J&, p(x),

C,(z) = C,, G(z),

(A23b)

(O
where r = &OI%l)

(A23c)

= ~(WC,cJ.

Thus the necessity is proved. Suficiency Let the lines Z’, and P’S have distributions as given by (A23). coefficients in the different series (Al)-(A4) are

4, = G,, Cd and ajz = (L, C,,)j

I

I

oE’h)

s

majG(x,j_l). .

0

jth

s

z2G(x,)dx, . . . dxzj+,

0

‘F(z,~) z*‘G(z,k_,) . . . zzG(z,) dz, j0 j0 j0

dj, = (L,, C,,)j

Then the various

‘G(zzj) %(z+~) j0 j0

...

%(sJ

j II

dz,

. . . dqj,

. . . dzzj.

From the above expressions, it is readily seen that ojr = djz,

djr = ojzt

bjr = (l&/C,,)

Hence, A,

= D,,

~$1 = (Cr,ILz,) bjz*

cjz,

D, = A,,

B, = r2 C,,

C, = B,/+.

Thus, the sutlkiency is established. Theorem II 9

The necessary and sufficient conditions for the chain parameters of a lossless NUTL of an arbitrary length 1 to be interrelated as A = D, B = r*(Z)C,

r being a constant independent distribution of L(x) is given by

of s, is that 5? be a proportional L(x)

then, the distribution

line, that is, if the

= LOP(Z)

of C(z) is C(x)

= (Lo/r2) P(x)

= C, F(x).

This theorem can be established by adopting a procedure similar to the one used for Theorem I.

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Journal of The Franklin Institute