An analysis of the general asymmetric

directional coupler withnon-mode-converting terminations

D.J. Gunton and E.G.S. Paige

Indexing terms: Directional couplers

Abstract: It is shown that for a general asymmetric directional coupler formed by two electromagnetic trans-mission lines, resistive terminations can be chosen so that the normal modes of the coupled system are re-flected without conversion from one to the other. With such 'non-mode-converting' terminations thescattering matrix assumes a particularly simple form which can readily be interpreted physically. Practicalmeans of finding the value of non-mode-converting terminations are indicated and the behaviour of couplerswith non-mode-converting terminations is predicted.

1 Introduction

In recent years there have been several types of electro-magnetic directional coupler proposed and demonstratedwhich have non-TEM wave propagation and which may in-volve an asymmetry of transmission lines or of dielectric.Their coupling mechanism is bidirectional, i.e. the coupledpower from an elemental length of coupled structure isdirected both forwards and reverse. Such devices includethe slot-line coupler,1 certain modified edge-couplers2'3

and the combline directional coupler (c.d.c.).4 The last ofthese is experimentally the most versatile of the three, sincethe phase velocity and characteristic impedance of each linetogether with the coupling coefficient may all be varied in-dependently. A theoretical analysis of the behaviour ofsuch couplers obviously needs a treatment sufficientlygeneral to include all of the possible variations inasymmetry and coupling mechanism. The general problemhas been formulated by Krage and Haddad,5 while Grivet6

has given a review of theoretical methods. The solution forthe components of the scattering matrix of a particularsystem requires a knowledge of the boundary conditions atthe ports of the device, in particular the terminatingimpedances. With general terminations the expressions forthe S-parameters can be expected to be very complicated;indeed, explicit expressions have only been given when thegenerality has been removed by restriction to special coup-ling situations (insufficient for analysis of the coupler typesmentioned above), and even then the expressions are some-what intractable and difficult to interpret physically.

In this paper we shall present an analysis in which wepreserve the generality regarding asymmetry and couplingmechanism, only restricting its applicability to the use ofspecial resistive terminations which have the property thatthere is no conversion of one normal mode to another onreflection at the ports of the coupler. We refer to these asnon-mode-converting (n.m.c.) terminations. From a

Paper Tl 12 M, received 15th September 1977Crown copyright Dr. Gunton is, and Prof. Paige was formerly, withthe Royal Signals and Radar Establishment, St. Andrews Road,Great Malvern, Worcs. WR14 3PS, England. Prof. Paige is now withthe Department of Engineering Science, University of Oxford, ParksRoad, Oxford, England

practical point of view this is a not unreasonable restrictionsince we are able to treat any electromagnetic directionalcoupler type provided only that the externally appliedterminations are appropriate. We shall show, moreover, thatthe S-parameter expressions which result from the use ofn.m.c. terminations are not only exact but are relativelystraightforward, allowing coupler behaviour to be assessedby inspection and leading to a clear physical interpretationof the operation of a coupler so terminated.

In Section 2 we derive explicit expressions for the valueof the njn.c. terminations for a given pair of coupled linesand deduce the S-parameters for a general coupler termin-ated in an njn.c. manner. Section 3 outlines ways of de-terming expermintally the coupler parameters and thevalues of the n.m.c. terminations, while in Section 4 we in-vestigate the consequences of the use of n.m.c. terminationsas they affect the design of practical devices. We considerthe practical details and experimental results of the synthesisof broadband combline directional couplers, making use ofthe results derived herein, in a companion paper.7

2 Derivation of S-matrix for couplers with n.m.c.terminations

We wish to derive expressions which may be employed toanalyse the behaviour of planar transmission line directionalcouplers. Currents and voltages on the pair of lines arecoupled as though through an inductive coupling coefficient&X, and a capacitive coupling coefficient kc- The wave-length is assumed to be sufficiently long (frequency suffici-ently low) that any dispersion effects can be ignored. Lossin the lines is neglected. These are usual assumptions andapproximations made in dealing with this type of prob-lem;6 their introduction here enables us to extend in adirect way the work of other authors.

The first part of this section sets up the mathematicalframework of the problem, restates several previouslyderived results which we require and serves to introducethe notation. The second part introduces the n.m.c. ter-mination concept, establishes the values of the n.m.c. ter-minations and derives the 5-matrix of a general directionalcoupler with n.m.c. terminations.

MICROWA VES, OPTICS AND ACOUSTICS, JANUAR Y1978, Vol. 2, No. 1 31

0308-6976/78/112M-0031 $1-50(0

The differential equations describing a lossless pair ofcoupled electromagnetic transmission lines in this approxi-mation are given by8

dz

Mv2

h

Wdt

ooCy

ALm

0

0

L2

0

0 /

v2h

lr\= 0

where Vt and /,• are the voltage and current on line / (/ = 1or 2)

Li and C,- are the self-inductance and self-capacitance perunit length of line / in the presence of the other

Lm and Cm are the mutual inductance and capacitanceper unit length. Using these parameters we define

Z,- = y/(Li/Ci) as the impedance characteristic of line i

0,- = w V ^ i Q ) as a wave vector associated with line/, where to is the angular frequency

KL = Lm fy/(L i L2) as the inductive couplingcoefficient

^c = Cm I\/(C\ C2 ) as the capacitive couplingcoefficient.

An alternative formulation of eqn. 1 may be written interms of four parameters which are linear combinations ofV\, ^2,h,andI2. We define

b+ =1

(V2±I2Z2)

Within the coupler these represent waves travelling eitherforwards (+) or reverse (—) on lines 1 and 2 and are re-ferred to as the coupled modes. Eqn. 1 then becomess

bz

fa:a-

b++ 1

0

- 0 !

\

VCPlft)*

02 o

o - 0 2

where d = \{kc — kL), s = \{kc + kL) and a solution ofthe form e ^ u t " ^ ) has been assumed. The advantage ofthis method of describing the coupled lines is that itshows clearly that coupling between waves travelling in thesame direction (a+ and b+ or a_ and b-) depends only on d,while that between waves in opposite directions (a+ andb- or a_ and b+) depends only on s. The situation in whichcoupling is exclusively between two waves travelling in thesame direction (s = 0) is the well known codirectionalcoupler; the other case ( J = 0) is that of contradirectionalcoupling. The couplers in which we are primarily interestedhere have s=£0 and d&O. Solution of eqns. 1 or 2 foreigenvalues and eigenvectors yields identical results, thoughexpressed in a different form. Assuming a propagating wavesolution as above, the results are well known and we givethem below in a form convenient for this work. Fournormal modes of the coupler result, two with positiveeigenvalues and two with negative, corresponding to apair of modes propagating in each direction. Of the modeswithin each pair, one has voltage (and current) vectors in

phase and the other has them in antiphase. The formerwill be referred to as the even mode (subscript e) and thelatter as the odd mode (subscript 6). The eigenvalues are

(1)

± IPel. ± IftJ, where | |3e| and ||3O| are given2 by

\[(D-Af + 4BC)V2f2

andA =

(r = e or o) (3)

~LmCm) = 0? + (d2 -

B = u2(LmC2 -LxCm) =

C =

D =

-L2Cm) •= -

-LmCm) =

The ratio of voltages for each mode on each line at adjacentpoints is given by V2r/Vlr = mr, and of currents byhrlhr = «r for r = e or o

(4)

(5)

AA= o (2)

The assignment of the labels e and o is made self-consist-ently in eqns. 3, 4 and 5 in accordance with the definitiongiven above. Mode impedances associated with each line,Zir = Vir/Iir (i=l,2;r = e,o)are given by

dPr-sQrZlr = Z,

z 2 r —dPr-sRr

dPr + sRr

where

Pr = (01 +0rX02 +0r)

Qr = (01 "0rX02 +0r)

Rr = (01 +0rX02 - 0 r )

(6a)

(6b)

32 MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1978, Vol. 2, No. 1

The eigenvectors may be written as

w= V \r (7)

mr{±\Z2r\

and the eigenvalue ± ||3r| is associated with the eigenvectorlabelled ±r. There are two useful relations which followfrom eqns. 4 and 5

(a) meno = mone = - 1

(b) me/ne = mo/no = C/B.

The second of these may be written more fully and we seethat

mt Vie he

Vle J2e

m

no

_ _— I7lel7lo — 7/

l oThe above results regarding the properties of coupled lines

are well known, although we have recast certain of them ina form convenient for this work. We are now in a positionto investigate the behaviour of coupled lines whenterminated so as to form a directional coupler. Here wedepart from previous authors by introducing the non-mode-converting concept and we first discuss its impli-cations as they affect the choice of terminations. Con-sider, by way of example and with reference to Fig. 1, thecoupler with a single voltage generator Vgl (i.e. Vg2 = 0).In general, the voltage generator Vgl will launch both oddand even modes into the coupled system. They will travelto ports 3 and 4 (z = /) where they will each in part under-go transmission into the loads and reflection with and with-out mode conversion. This process is repeated for thereflected waves at ports 1 and 2 and at all subsequent inter-actions with the terminations. If instead of this generalbehaviour, terminations can be chosen so that both odd andeven modes are reflected without mode conversion both atports 1 and 2 and at ports 3 and 4 then the situation isgreatly simplified; a normal mode of the coupled lines,once launched, continues to be reflected between the endsof the coupler as that particular normal mode: the normalmodes of the coupled lines are not coupled further by theterminations.

We shall now show that in the most general directionalcoupler it is possible to select terminations which satisfythe n.m.c. condition. This follows from the property (eqn.8) that Z2e/Zie = JZ2o/Zlo = 7?, for now, if the coupled

port 1 port l*

port 2

ZrO 2=1'

Fig. 1 Example of directional coupler sources and terminations

3 7

lines are terminated at each end in impedances ZlT online1 and Z2T on line 2 such that Z2T/ZlT = 77, then the re-flection coefficients are equal for those waves on each linewhich together form a normal mode, i.e. the voltage re-flection coefficients (for example) satisfy

Zi

Plr =l r Z lr

Zjy ~r Z, l r "T Z>- P2r = eoio.

2r

Under these conditions a normal mode of the coupled linesincident on the terminations at either pair of ports 1 and 2or 3 and 4 is partially transmitted and partially reflectedwith the property that the voltage and current ratios areunchanged from those incident, i.e. there is no mode con-version. Thus the condition for n.m.c. terminations issimply

Z2T/Z1T — Z2e/Zie — Z2o/Zlo — 77 (9)

We now examine some of the ways in which the use of suchterminations simplifies the analysis of a general coupler andalso leads to results of practical significance. The analyticalaspect follows, while the practical considerations are dis-cussed in Section 4.

We have pointed out that with n.m.c. terminations eachmode of the coupled lines can be treated separately. In thespecial case of symmetric lines the customary use ofidentical terminating impedances equal to the characteristicimpedances of the lines, which leads to a straightforwardanalysis of the behaviour of each mode,8 may now be seento be included in the range of n.m.c. terminations for suchlines (the condition is ZlT=Z2T). We are now in aposition to consider the general asymmetrical coupler inthe same way, through use of n.m.c. terminations. Summ-ing the series of reflections of each normal mode of thecoupled lines leads directly to expressions for the normalmodes of the complete system (including now the termin-ations) in terms of the electrical length ft,/ of the evenmode and 0O/ for the odd mode on lines of length / termin-ated as discussed above. With the arrangement shown inFig. 1 either a pure even or a pure odd mode may belaunched through proper choice of generator voltage ratios,i.e. Vgi = meVgi or Vg2 = moVgl. For excitation of moder we find the voltages at / ' on lines 1 and 2 to be given by

(10)

= mT I in

where pr = Pi r = P2r and VUn = VgJ2 is the voltageamplitude of the input wave to port 1. (The form of thecurrent modes is obtained by replacing mr by nr and pr by

The components of the scattering matrix may be foundby setting Vgl and V& in turn to zero (and end-for-endsymmetry makes the coupler behaviour for inputs to ports3 and 4 identical to that for inputs to ports 2 and 1,respectively). Since the normal modes of the completesystem (eqn. 10) are expressed in terms of the input volt-ages (or currents) on the feed lines the behaviour of thecoupler for an input to port 1 only or to port 2 only maybe obtained from the appropriate linear combinations ofthese modes. The S-parameters are then obtained

MICROWA VES, OPTICS AND ACOUSTICS, JANUAR Y1978, Vol. 2, No. 1 33

immediately as

Sn = S44 =PoAo

ie mo

peAe poAo

m. mt

S 2 i — Si2 — S34 — 543 —

53 1 = 5,3 = 524 = S42 = 0tsJr]{Bo-Be)

Bn Bp

_ = s23 = an I — - —

where1 -

A =

01/)

Br =

l-pr2exp(-2/0r/)'

(l-pr2)exp(-//y)

(/• = e, o)

and

<* = (tno-me)~\ T? = ~memo.

Here we have taken the usual definition of S-parameters asrelating the input and output waves for an jV-port network:

N

bn =

where* an = \/2y/ZnT(VnT+ZnTInT)mdbn =(VnT — ZnTInT) are the input and output waves, respect-ively, and VnT, InT and ZnT are as shown in Fig. 1. (Forn.m.c. terminations Z 4 T = Z1 T, Z 3 T = Z 2 T = T?ZIT.)

The relative simplicity of these expressions stemsdirectly from the choice of n.m.c. terminations. They areexact within the stated assumptions of the theory andsatisfy the power conserving condition, e.g. 2 \Sln I2 = 1.

They apply to any coupler whose operation is described byeqns. 1 or 2 for a general coupling mechanism and for anydegree of asymmetry. They seduce to the well known ex-pressions for certain special cases, such as codirectionalcoupling (5 = 0), contradictional coupling (d = 0) or sym-metrical lines. These cases will be mentioned again inSection 4 in the context of coupler performance. However,when using eqn. 11 for the case of the contradirectionallimit care must be taken in the definition of certain of thequantities involved, a point further discussed in Section 4.

Physically the coupler with n.m.c. terminations is equiv-alent to a Fabry-Perot interferometer or a resonant cavitythat is able to support two different modes which retaintheir identity, as may be seen from a comparison of eqn.10 with the expressions describing the operation of aFabry-Perot. To obtain eqn. 10 it will be recalled wedeliberately excited each mode of the cavity separately,giving this equation a direct identity with that of a normalmode of a resonant cavity. From eqns. 11 we see clearly theexpressions of multiply internally reflected waves in the re-flection |p,>4r| and transmission i£r|terms of theS-matrix.

an and bn are distinct from a+ and ft±; an and bn refer to wavestravelling in the circuit external to the coupler while a± and 6+ referto the coupled waves of the device itself.

It follows that we can visualise the performance of thecoupler for an arbitrary input as firstly the appropriatedegree of excitation of the modes of the coupled lines,secondly their separate multiple reflections within thecoupler and finally their emergence at the ports of thecoupler. The output from a given port is then obtained asthe result of a straightforward interference analysis of twomultiply reflected modes viewed in reflection or trans-mission as appropriate.

3 Determination of coupler parameters and the n.m.c.condition from experiment

There are six basic parameters required for a completedescription of a directional coupler as analysed in Section2 and for which the scattering matrix components withn.m.c. terminations are given in eqn. 11. From eqn. 1 theseare seen to be C1} C2, Lx, L2, Cm and Lm; all otherquantities, such as normal-mode velocities, impedances andvoltage and current ratios may be derived from them. Thevalue of the ratio of terminating impedances necessary toachieve the n.m.c. condition may also be found from them.We outline here procedures suitable for obtaining experi-mentally the six basic parameters which are sufficient tocharacterise any coupler type. We make use of the relationsbetween the properties of the transmission lines before andafter coupling.6'9

— C{ + Cm; C2 — C"

— Li; L2 = L2

(12)

where the superscript u denotes a quantity referred toeither line in the absence of the other. We also have

/ = 1,2 (13)

and

Z? = y/(Lf/Cn0", 0", Z", and Z" may each be measured using convent-ional techniques (e.g. time-domain reflectometry) givingvalues of £" ,£" , C" and C"; combining eqns. 13 with eqns.12 leaves two unknowns Lm and (C, — Cm). Substitutioninto eqn. 3 gives two equations in Lmi Cm and the normalmode wave vectors of the coupler, j3e and |3O.

C m ) -2C m L m ]co 2

(14fl)

Cm)

- [Lm(C2 + Cm)~LlCm]

Introduction of an impulse into one port of a pair ofcoupled lines, arbitrarily terminated, results in its decom-position into two pairs of impulses propagating at thenormal mode velocities.t Detection after such a distanceas to give time resolution enables 0e and j3o to be found.The solution of eqns. 14 gives four pairs of values of Cm

and Lm; the additional physical restriction that for thegeometry of two conductors and an earth plane in anisotropic dielectric medium both Cm and Lm are positive

t In Section 2 dispersion has been assumed absent. Here, by impli-cation, we assume that the main frequency components of theexciting 'impulse' are sufficiently far removed from dispersiveregions of the real coupler that dispersion can legitimately beneglected.

34 MICROWA VES, OPTICS AND ACOUSTICS, JANUAR Y1978,. Vol. 2, No. 1

means that only one pair is suitable. If the device under con-sideration were a pure contradirectional coupler (d = 0)this would be recognised experimentally by the equalityof the normal mode velocities both with each other andwith the uncoupled velocities.

An alternative procedure, but one which only givesvalues of me and mo (though sufficient to determine then.m.c. condition), consists of introducing an impulse toone port only of the pair of arbitrarily terminated coupledlines and detecting the time resolved normal-mode impulseamplitudes on each line using a high-impedance probe.This method, however, is not applicable for contradirect-ionally coupled lines since the impulses will then all travelat the same velocity.

4 Predicted behaviour of a directional coupler withn.m.c. terminations

We have already observed that the use of n.m.c. termin-ations allows the propagation of each normal mode of ageneral directional coupler to be considered independentlyand, moreover, that a useful analogue for understandingthe frequency response of each mode is the well known be-haviour of a Fabry-Perot resonator. The resonator analogymay be heightened by going to the extreme of highlyreflecting terminations — still satisfying the njn.c. con-dition (eqn. 9) — when narrow band, high transmission,performance of the coupler may be achieved. In this sit-uation the dissipative loss, which we have ignored, willbecome important and will limit the achievable Q. Usually,however, broad-band performance is required which impliesthat relatively low reflection coefficients are needed andthat loss in the coupler is not normally a significant factor.

We now turn to a discussion of various types of couplers.The first two are well known — the purely codirectionaland the purely contradirectional coupler — but are men-tioned here so that their behaviour may be consideredwithin the context of the n.m.c. termination concept. Thethird case is more general and we shall treat it in somedetail.

4.1. =*The codirectional coupler (ki_ = —kc, s = 0)

The codirectional coupler is characterized by Z l e = Z l o =Zi , Z2c = Z2 o = Z2 and me mo = — 1 as may be shownfrom eqns. 4 and 6. In this case, from the range of possiblen.m.c. terminations, it is possible to choose a set whichmatches the impedance of the coupled line giving rise tozero reflection of each mode, i.e. Z 1 T = Z1} Z 2 T = Z2,hence pXe = pXo - 0. Under this condition Sn - S22 =^12 = S2i'= 0 and there is perfect isolation and zero inputreflection at all frequencies. We see then that in this casethe ideal terminations are, in fact, a particular choice ofn.m.c. terminations. The well known behaviour of this typeof coupler10 in which all power is divided between ports 3and 4 with the asymmetry of the structure appearing in thevalues of me and mo, follows from eqn. 11.

4.2. The contradirectional coupler (k(_ = kc, d — 0)

The purely contradirectional coupler only exists under con-ditions in which the waves are synchronous6'8 (fix =j32).Under these conditions eqns. 4, 5 and 6 become indeter-minate when d = 0, for a coupling mechanism which onlyrelates waves travelling in opposite directions will not giverise to any specific relationships between waves in the same

direction. However, each of the quantities given by eqns. 4,5 and 6 may be expressed in terms of the single indeter-minate ratio d/d. Once a value has been assigned (arbitrarily)to that ratio all the quantities are defined self-consistently.It has been customary in the literature concerned with con-tradirectional couplers to assume limiting values for thesequantities as d -• 0 (i.e. to take d/d - 1), so that mr reflectsdie symmetry of the coupled lines, but there is no a priorireason for doing so other than analytical convenience. Forcontradirectional couplers it follows from eqns. 3—6 that0e = j3o and me/mo = ne/no = — 1 while the general con-dition for n.m.c. terminations becomes Z2T/ZXT =me/ne = C/B. Thus eqn. 11 may be used to treat the con-tradirectional coupler arbitrarily terminated (though main-taining end-for-end symmetry) since a value of me mayalways be chosen for which the terminations are non-mode-converting. However, advantageous features in performanceresult if we set Z1T = Zt (which are examples of n.m.c.terminations for d/d = 1) for then pie = — piOi Ae=Ao

and Be -Bo so that \S3X \2 is zero at all frequencies givingperfect isolation. For the symmetrical coupler (in whichZi = Z2) with ZXT = Z2T = Zx = Z2, the condition of nomode conversion at the terminations requires me = 1, sothat then in addition to the property \S3X |2 = 0 we findl^n I2 = |52 2 |

2 = 0 which represents zero input reflection.

4.3. The bidirectional coupler (ki_ ^kc)

Here we discuss the general coupler and include allsituations in which the coupling mechanism is such as todirect power in both forward and reverse directions. Norestriction is placed on the degree of asymmetry or on thevalues of kL and kc except that their magnitudes are notequal. The normal mode velocities are different and thereare in general four distinct values of the mode impedancesZ\e, Z\o> %2e a nd Z2o. Simultaneous matching of bothmodes is not possible and the quantities |5"n |2 , |52212

and |52i |2 can only be zero at spot frequencies depend-ent on the coupler parameters. Nevertheless in certainspecial cases the use of n.m.c. terminations gives resultsof significant practical value.

First we consider n.m.c. terminations which lead to thecondition pie = —pio 0 = 1 , 2 ) ; they are Z 1 T = \/(ZXeZXo)and Z 2 T = y/(Z2eZ2o). We see from eqn. 11 that at thecentre frequency, defined by the condition |j3e/ — |3O/| = nthe coupler behaves symmetrically in the sense thatSu ==S22 and S41 =Sr

32. This can be a useful property;it is, of course, always true of a symmetrical coupler withidentical terminations.

A second potentially useful situation arises when oneor other of the modes is matched. The power emergingfrom ports 1 and 2 now arises solely from the reflections ofthe unmatched mode and the frequency dependence is verystraightforward. If mode r is matched (pir = pr = 0,/ = 1, 2) then the use of n.m.c. terminations gives thefollowing expressions for the power emerging at ports 1 and2 for an input to port 1 or to port 2 only:

\Sn\2 = ot2r)2p2\Ar>\2/m2'

\S22\2 = a2

V2p2\Ar'\

2/m2

1̂ 21 I2 = 1̂ 12 I2 = OL2p2T]\Ar'\2

where r denotes the unmatched mode and p2 = {(ZXe —Z\o)/(Zie + Zio)}2. These each have zeros when (irl = Nrr

MICRO WA VES, OPTICS AND ACOUSTICS, JANUARY 1978, Vol. 2, No. 1 35

and maxima given by

4a 2 r?V 4c*V and(1+P2)2\2>

respectively,

when j3r/ = (2N + l)7r/2. This behaviour may be contrastedwith the presence of principal and secondary maxima foundin each of the port outputs when both modes contributeand mode interference effects are seen. A comparison ofexpressions for IS2112 a t t n e centre frequency shows thatn.m.c. terminations which match one mode only result in asmaller value (and hence better isolation in the case offorward couplers) than those which give equal matching forboth modes, i.e. pe = — po. Also, for conditions of largeasymmetry the value of either \Sn |2 or of |522 I2 may bereduced at centre frequency by single-mode matching(which parameter is reduced depends on which mode ismatched). The condition that there is a reduction is

_(Vzlo-Vzle)2 (zlo+zlefmc

where me has been taken to be greater than unity.Any type of asymmetric, bidirectional coupler having

n.m.c. terminations may be designed so that Sn , S22 an^S21 are all zero at centre frequency. This occurs when theadditional requirements &el = NeK and (lol = Noir are met,(Ne and No are integers), and may be effected through con-trol of the normal mode velocities. In the Fabry-Perot re-sonator analogy this condition corresponds to superpositionin frequency of a mode from each of the two sets ofresonator modes. In general the periodic spacing (in fre-quency) between modes of each set (even and odd) willnot be the same, hence Ne ^No and superposition willoccur at frequencies widely separated compared with themode spacing.

5 Conclusions and implications for the design ofpractical couplers

In this paper we have applied the concept of the non-mode-converting terminations to an analysis of the behaviour ofdirectional couplers. Our main purpose has been to showthat the introduction of n.m.c. terminations leads torelatively simple expressions for the behaviour of very gen-eral directional couplers. Furthermore these expressionshave a clear physical interpretation as a combination ofeven and odd modes of the 'resonator' formed by thedirectional coupler. Experimental means of determining therelevant parameters which appear in the theory have beengiven.

We have shown that the terminations frequently em-ployed in the use of specific couplers, such as the sym-metrical contradirectional coupler or the codirectionalcouplers, are in fact examples of n.m.c. terminations, butthe main motivation of this work has been to provide aframework for the analysis of general, bidirectional couplerssuch as were mentioned in Section 1. The present analysisshows how the operation and behaviour of such couplersmay be treated exactly, through the use of n.m.c. termin-ations, enabling devices to be designed for specific applic-ations. In particular, we have seen how the output from the'isolated' port and the input reflections at all ports may bereduced at centre frequency through proper choice ofn.m.c. terminations from the range available or made equalto zero through control of the normal-mode velocities.Broadband improvement in both isolation and input re-flection becomes possible if Z2e/Zle (and Z 2 o /Z l o ) is closeto unity, for then pie and pio can be made very small andSn, S12 and S22 tend to zero. This point is developedfurther in the companion paper7 on the design of wide-band combline directional couplers.

Though the analysis we have given is based on thecoupling between a lossless pair of transmission lines, theadvantageous features of terminations which are non-mode-converting can obviously be extended to all types ofdirectional coupler.

6 References

1 MARIANI, E.A., and AGRIOS, J.P.: 'Slot-line filters andcouplers', IEEE Trans. 1970, MTT-18, pp. 1089-1095

2 SPECIALE, R.A.: 'Even and odd-mode waves for nonsym-metrical coupled lines in nonhomogeneous media', ibid. 1975MTT-23,pp. 897r908

3 DE RONDE, F.C.: 'Recent developments in broadband direct-ional couplers on microstrip', Proceedings of the IEEE MTTSymposium, Chicago, 1972, pp. 215-217

4 GUNTON, D.J., and PAIGE, E.G.S.: 'Directional Coupler forgigahertz frequencies, based on the coupling properties of twoplanar comb transmission lines', Electron. Lett., 1975, 11,pp. 406-8

5 KRAGE, M.K., and HADDAD, G.I.: 'Characteristics of coupledmicrostrip transmission lines - I: Coupled-mode formulation ofinhomogeneous lines', IEEE Trans., 1970, MTT-18,4, pp. 217-22

6 GRIVET, P.: 'The physics of transmission lines at high and veryhigh frequencies'. Vol. II: Microwave circuits and amplifiers(Academic Press, 1976), pp. 623-728

7 GUNTON, D.J.: 'The design of wideband codirectional couplersand their realisation at microwave frequencies using coupledcomblines', IEE J. Microwave Opt. & Acoust., 1978, 2, pp. 19-30

8 OLIVER, B.M.: 'Directional electromagnetic couplers', Proc.IRE, 1954,42, pp. 1686-1692

9 SCHELKUNOFF, S.A.: 'Electromagnetic waves' (Van Nostrand,1943)

10 LOUISELL, W.H.: 'Coupled mode and parametric electronics'(Wiley, 1960)

36 MICROWA VES, OPTICS AND ACOUSTICS, JANUARY 1978, Vol. 2, No. 1