Synthesis of directional couplers based on ridged waveguides coupled by a system of circular apertures

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2011, Vol. 56, No. 7, pp. 805–811. © Pleiades Publishing, Inc., 2011.Original Russian Text © G.F. Zargano, V.V. Zemlyakov, R.V. Peletskii, 2011, published in Radiotekhnika i Elektronika, 2011, Vol. 56, No. 7, pp. 789–795.

805

INTRODUCTION

Waveguide directional couplers (DCs) are widelyused in radar, radioastronomical, and measuringequipment and in various broadband systems. Amongthe most widely spread directional couplers, there aredevices with a system of small coupling aperturescharacterized by sufficiently high (usually over 10 dB)and weak interation between the primary and second�ary waveguides [1, 2].

Coupling between waveguides through small aper�tures can be calculated using variational methods [3],a mode matching method [4], and a method ofmoments [5], which is especially suitable for solvingproblems with apertures of an arbitrary shape, since itmakes allowance for higher order modes and thethickness of the common wall by choosing an appro�priate basis function. These methods, however, lead toconsiderable difficulties, both analytical and compu�tational; therefore, a classical theory for calculatingdirectional couplers with small apertures is quasi�static Bethe’s theory [6] expanded by several authors[7, 8]. The advantages of this theory are the simplicityof the analysis and the calculation accuracy (3–5%)good for practical application, although this methodmakes no allowance for the effect of higher ordermodes in the coupling area.

As is known [9–11], the use of waveguides withcomplex cross sections, e.g., single� and double�shaped waveguides, makes it possible to substantiallyimprove the broadbandness and mass�dimensionalcharacteristics of microwave and extremely�high�fre�quency elements and units. The use of waveguideswith a complex cross section in directional couplersallows improving directivity, broadening an operatingfrequency band, and reducing linear dimensions. Cal�culation accuracy of the parameters of such a direc�tional coupler is directly related to that of the electro�

dynamic parameters of the waveguides used. Forexample, an error of about 1% made in the calculationof critical wavenumbers and components of the elec�tromagnetic fields may yield an error of about 50% inthe final parameters. Moreover, when building a direc�tional coupler with the use of a system of small aper�tures, one should not only know the parameters of asingle aperture with good accuracy but also take intoaccount the interaction between the apertures.

In the available literature [11, 12], there are a fewworks on calculation of the parameters of single cou�pling apertures of waveguides of a complex cross sec�tion and even a directional coupler. However, thewaveguide eigenfunctions used there were determinedwith considerable errors, which substantially degradedthe results of the calculation.

As is known [9], the partial domain methodwithallowance for the features of the electromagnetic fieldon the edge makes it possible to calculate the electro�magnetic fields and critical frequencies of waveguideswith complex cross sections with high accuracy. In thisstudy, using the fields determined by this method,directional couplers based on Π�shaped waveguidescoupled by a system of small circular apertures in thecommon wide wall are analyzed and synthesized usingquasi�static Bethe’s theory.

1. SCATTERING MATRIXOF A DIRECTIONAL COUPLER

To calculate a directional coupler, it is necessary toknow the dependence of the parameters of an individ�ual aperture on its dimensions, displacement relativeto the waveguide axis, wall thickness, and frequency.In designing directional couplers, coupling elementsin the form of circular apertures, longitudinal andtransversal gaps, and cross�shaped and elliptical aper�tures are most frequently used.

Synthesis of Directional Couplers Based on Ridged Waveguides Coupled by a System of Circular Apertures

G. F. Zargano, V. V. Zemlyakov, and R. V. PeletskiiReceived July 27, 2010

Abstract—A technique of electrodynamic analysis and synthesis of directional couplers based on Π�shaapedwaveguides with a system of small coupling apertures is reported. Parameters of coupling between twowaveguides through a small aperture are calculated within quasi�static Bethe’s theory. Critical wavenumbersand components of the electromagnetic fields of ridged waveguides are calculated with allowance for the fea�tures of the electromagnetic field on the edge. The characteristics of synthesized directional couplers basedon Π�shaped waveguides with coupling along the wide wall of 20 and 10 dB are reported.

DOI: 10.1134/S106422691107014X

ELECTRODYNAMICSAND WAVE PROPAGATION

806

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 56 No. 7 2011

ZARGANO et al.

Consider the coupling between two Π�shapedwaveguides through a small aperture. Let there is aninfinite Π�shaped waveguide coupled parallel with theother, analogous, Π�shaped waveguide through asmall aperture in the common wide wall. Assume thatarms 1 and 3 are the input and output ports of the firstwaveguide, respectively, and ports 2 and 4 are the aux�iliary ports of the second waveguide. The lossless prob�lem is solved. Then, we write the elements of the scat�tering matrix of the four�port as S11 and S31 for the firstwaveguide and S21 and S41 for the second waveguide,respectively.

Scalar functions describing the distribution ofthe field of the fundamental mode in partial domainsof the Π�shaped waveguide (Fig. 1) were determinedfrom the relations obtained in [9] using the partialdomain method with allowance for the edge singularity:

(1)

where and are the undetermined decompositioncoefficients; ;

,iϕ

( ) ( )

( )[ ] ( )

=

=

ϕ = =

ϕ =

= −

∑

∑

1 1 1 1

0

2 2

2 2

0

( , ) ( , ) cos cos ,

( , ) ( , )

sin cos ,

N

z n x y

n

z

N

n x y

n

x y H x y A k x k y

x y H x y

B k l x k y

nA nB

1 ;yk n h= π 2 ;yk n c= π 0, 1, 2, 3...n =

and is the cutoffwavenumber of the propagating mode.

According to Bethe’s theory, we use formulas from[1, 10] and calculate the coupling between twowaveguides operating in the single�mode regime througha small aperture in the common wide wall (the ridges arelocated on the outer walls). The elements of the scatter�ing matrix of the corresponding four�port are

(2)

2 2 21 1 ;x c yk k k= −

2 2 22 2 ;x c yk k k= − ck

( )

( )

( )

( )

⎧ ⎡ ⎤ ⎫γ += − Φ + Φ⎨ ⎬⎢ ⎥γ γ⎩ ⎣ ⎦ ⎭

⎧⎛ ⎞γ= − Φ − Φ⎨⎜ ⎟γ⎩⎝ ⎠

⎫− Φ ⎬

γ ⎭

⎧ ⎡ ⎤ ⎫γ −= − Φ + Φ⎨ ⎬⎢ ⎥γ γ⎩ ⎣ ⎦ ⎭

γ= − Φ + Φγ

2 2 22 22

11 1 1 0 2 02

22 2

21 1 1 0 2 2 02

222 02

2 2 22 22

31 1 1 0 2 02

22 2

41 1 1 0 2 22

( ) ;2

'( )2

( ) ;

1 ( ) ;2

2

c

c

cn n

c

e

c

c

c

cn

c

k M PkjS M x x

k

kjS M x F M x F

k

kP x Fk

k M PkjS M x x

k

kjS M x F M

k

⎧⎛ ⎞⎨⎜ ⎟⎩⎝ ⎠

⎫− Φ ⎬

γ ⎭

0

222 02

'( )

( ) ;

n

e

c

x F

kP x Fk

y

l

g

1

2

h

c

x

Fig. 1. Cross section of the П�shaped waveguide with partial domains 1 and 2; l, h, g, c are the dimensions of the partial domainsof the waveguide.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 56 No. 7 2011

SYNTHESIS OF DIRECTIONAL COUPLERS BASED ON RIDGED WAVEGUIDES 807

(3)

(4)

Here, , , and are the coefficients of mag�netic and electric polarizability, respectively;

is the phase constant; k is the wave�number of free space; and are the coefficientstaking into account the effect of thickness t of thecommon wall of the waveguides that are determineddepending on the shape of the aperture. Usually, asmall aperture coupling two waveguides is consideredas a very short waveguide section with length t operat�ing in the evanescent mode. The microwave power isexponentially attenuated along such a waveguide.As was shown in study [13], coefficients canalso be determined using empirical formulas, which,in certain cases, improve the accuracy of calculation ofthe parameters of coupling apertures.

Let us determine the values of the magnetic andelectric moments for a circular aperture with radius R[9, 10] as

(5)

To take into account the closeness of the aperture tothe resonance, the corresponding corrections are rec�ommended [14].

The results of calculation of the elements of scatter�ing matrix (2) show that, in order to calculate parameter

more accurate, it is more effective to use the energyconservation law for an ideal lossless four�port.

The characteristics of a directional coupler consist�ing of a series of coupling apertures were found usingthe theory of cascade connection of multiports.

The scattering matrix of a combined multiport iscalculated from the known scattering matrices of itscomponents SI and SII. If cross sections α and β areinput and output ones for multiport SI, respectively,and β and γ are those for multiport SII, the, matricesSI and SII can be presented in the block form

(6)

The blocks in expressions (6) describe the interac�tion of waves at all the inputs in cross sections α, β, andγ. The multiports are connected in cross section β. Theresulting scattering matrix also has a block structure:

(7)

For the blocks of scattering matrix S, the followingexpressions are valid:

(8)

where Е is the unit matrix.The scattering matrix of the four�port describing

coupling aperture SI is written with allowance for sym�metry of the device as

(9)

and the regular section of the waveguides with lengthΔl between two neighboring coupling apertures isdescribed by the scattering matrix

(10)

( )

( )[ ]

=

=

Φ

⎧< =⎪

⎪= ⎨⎪

− < ≤ =⎪⎩

∑

∑

, when

, when

1 0

1 0 0

0

2 0 0

0

( )

cos , 1;

sin , 2,

N

n x

n

N

n x

n

x

A k x x g i

B k l x g x l i

( )

[ ]

=

=

Φ

⎧− < =⎪⎪⎪⎪

= − −⎨⎪⎪ < ≤ =⎪⎪⎩

∑

∑

sin when

when

2 0

1 1 0 0

0

2 2 0

0

0

( )

, , 1.

cos ( )

, 2.

N

n x x

n

N

n x x

n

x

A k k x x g i

B k k l x

g x l i

1M 2M P

γ = −2 2 1 2( )ck k

', ,h h eF F F

', ,h h eF F F

3 31 2 4 3, 2 3.M M R P R= = =

31S

, .II

I III

S S S S

S S S S

αα αβββ βγ

βα ββ γβ γγ

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

S S

.S S

S S

Σαα αγ

Σ

γα γγ

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

S

( )

( )

( )

( )

−Σαα αα αβ ββ ββ ββ βα

−

αγ αβ ββ ββ βγ

−

γα γβ ββ ββ βα

−Σγγ γγ γβ ββ ββ ββ βγ

= + −

= −

= −

= + −

I 1

1

1

1

,

,

,

,

II II

II I

II I

II I I

S S S E S S S S

S S E S S S

S S E S S S

S S S E S S S S

⎡⎛ ⎞ ⎛ ⎞⎤⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥=⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎣⎝ ⎠ ⎝ ⎠⎦

11 21 31 41

21 11 41 31

31 41 11 21

41 31 21 11

,I

S S S S

S S S S

S S S S

S S S S

S

( )

( )

( )

( )

− γ Δ⎡ ⎛ ⎞⎤⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥− γ Δ⎝ ⎠ ⎝ ⎠⎢ ⎥=

− γ Δ⎢ ⎥⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥− γ Δ ⎝ ⎠⎣⎝ ⎠ ⎦

exp 00 0

0 exp0 0.

exp 0 0 0

0 exp 0 0

I

IIII

I

II

j l

j l

j l

j l

S

808


ZARGANO et al.

In this case, if several coupling apertures arelocated in one cross section of the waveguides, then,the resulting scattering matrix is obtained through

direct cascade connection of the four�ports describingthese apertures. If the directional coupler contains papertures, then, relations (8) should be applied

times.

2. CALCULATED PARAMETERSOF THE DIRECTIONAL COUPLERS

Scattering characteristics of single coupling aper�tures in waveguides with complex cross sections werethoroughly analyzed in [10]. In the mentioned study,using the above expressions, the authors synthesizeddirectional couplers with a system of circular apertureswith a coupling of 20 and 10 dB for the Π�shapedwaveguides with the dimensions h/l = 0.9, g/l = 0.845,and c/l = 0.345 by minimization of the root�mean�square deviation of the coupling from the specifiedvalues. In the calculation of the critical wave numbersand components of the electromagnetic fields, thethird approximation of the method and 50 N terms inthe series of the matrix elements were used.

Figure 2 shows the appearance of directional cou�plers with a coupling of 20 dB (x0 = 0.3551, Δl =0.7090) and 10 dB (x0 = 0.3636, Δl = 0.8602). Thedimensions of the coupling apertures are given in thetable. Figure 3 presents the dependences of the direc�tivity and the coupling C =

of the synthesized directional couplers.It can be seen from the figure that the directional cou�plers possess good broadbandness (∼50%) with a cou�pling drop of no more than ±0.5 dB and directivity ofno less than 30 dB. Both directional couplers have verylow VSWR (Fig. 4), which is a considerable advantageof directional couplers with coupling through smallapertures. The relative longitudinal size of the synthe�sized directional couplers was L1/l = 7.8 and L2/l =19.8 or L1/λ = 2.5 and L2/λ = 6.3, where λ is the cen�tral wavelength of the operating range.

To check the reliability of the results obtained, wecompared them to the results of the computer simula�tion performed by mesh�type numerical methods thatsolve the strictly set problem. The results of computersimulation are presented in Fig. 3 (dots). It should benoted that even now, with powerful computers avail�able, the use of the mesh�type methods remainsextremely laborious and lengthy. Note for comparisonthat, while the calculation of the directional couplerswith fixed dimensions with the use of the developedalgorithm lasts several seconds, the obtainment of ananalogous accuracy of the mesh�type methods withthe use of the same computer requires over 30 min.Moreover, taking into account that, during the synthe�sis of directional couplers, the optimization programperforms over 1000 iterations, it seems fairly difficultto synthesize a multielement directional coupler bythe mesh�type methods.

( )2 1p −

41 2120 logN S S=

4120 log 1 S

(a)

(b)

Fig. 2. Directional couplers 1 and 2 with a coupling of (a)20 and (b) 10 dB, respectively.

Dimensions of coupling apertures of synthesized direction�al couplers 1 and 2

Directional couplers

1 2

i Ri i Ri i Ri

1 0.2500 1 0.2655 7 0.28492 0.2800 2 0.2781 8 0.28543 0.2800 3 0.2799 9 0.28554 0.2800 4 0.2807 10 0.28655 0.2800 5 0.2847 11 0.28706 0.2800 6 0.2848 12 0.2870

Note: i is the number of the aperture in coupling.



50

01.5 2.3

40

30

20

10

1.7 1.9 2.1

C, N, dB

k l

(b)

50

1.5 2.3

40

30

20

101.7 1.9 2.1

C, N, dB

k l

(а)

Fig. 3. Dependences of (N, dashed curve) directivity and (С, solid curve) coupling of directional couplers (a) 1 and (b) 2 on k l.Dots correspond to the simulation.

810


ZARGANO et al.

Thus, the presented algorithm of the analysis andsynthesis of the multielement directional couplersbased on Π�shaped waveguides is shown to be effectivein both calculation speed and accuracy of the results

obtained. The developed technique allows synthesiz�ing directional couplers with coupling through notonly circular but also rectangular and cross�shapedsmall apertures and building directional couplers on

1.03

1.001.5 2.3

1.02

1.01

1.7 1.9 2.1

VSWR

k l

(b)

1.015

1.0001.5 2.3

1.010

1.005

1.7 1.9 2.1

VSWR

k l

(а)

Fig. 4. Dependence of the VSWR of directional couplers (a) 1 and (b) 2 on k l.



the basis of other waveguides with complex cross sec�tion (H�shaped, T�shaped, cross�shaped, etc.) whose�electrodynamic characteristics can be calculated bythe partial domain method with allowance for the edgesingularities.

REFERENCES

1. V. A. Sosunov and A. A. Shibaev, Microwave DirectionalCouplers (Privolzh. Knizh. Izd., Saratov, 1964) [in Rus�sian].

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3. A. J. Sangster, Proc. IEE 112, 2171 (1965).4. H. Jia, K. Yoshitomi, and K. Yasumoto, Prog. Electro�

magn. Res. (PIER) 46, 245 (2004).5. V. K. Thong, IEEE Trans. Microwave Theory Tech. 20,

416 (1972).

6. H. A. Bethe, Phys. Rev. 66 (7�8), 163 (1944).

7. S. B. Cohn, Proc. IRE 40, 696 (1952).

8. R. Levy, IEEE Trans. Microwave Theory Tech. 16, 995(1968).

9. G. F. Zargano, V. P. Lyapin, V. S. Mikhalevskii, et al.,Waveguides of Complex Sections (Radio i Svyaz’, Mos�cow, 1986) [in Russian].

10. G. F. Zargano, V. V. Zemlyakov, and R. V. Peletskii,Elektromagn. Volny Elektron. Sist., No. 2, 48 (2010).

11. N. A. Gal’chenko, V. S. Mikhalevskii, and G. P. Sin�yavskii, Waveguides of Complex Sections and Strip Lines(Rostov. Gos. Univ., Rostov�on�Don, 1978) [in Rus�sian].

12. W. J. Getsinger, IRE Trans. Microwave Theory Tech.10, 41 (1962).

13. L. N. Akhiezer, Izmer. Tekh., No. 3, 50 (1963).

14. N. A. McDonald, IEEE Trans. Microwave TheoryTech. 20, 689 (1972).

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Synthesis of directional couplers based on ridged waveguides coupled by a system of circular apertures