Progress In Electromagnetics Research B, Vol. 5, 35–48, 2008

A COMPACT H-PLANE MAGIC TEE DESIGNED AT WBAND

Z. X. Wang, W. B. Dou, and Z. L. Mei †

State Key Lab of Millimeter wavesSoutheast UniversityNanjing 210096, P. R. China

Abstract—Magic tee is a widely used component in microwavesystems; the four arms of a conventional magic tee direct at fourdifferent directions, which occupy much space and give inconveniencesto the assemblage of a system. In this paper, a waveguide narrow-wallslot directional coupler and an E-plane dielectric loaded waveguidephase shifter are used to make up of a magic tee with four arms in thesame H-plane. The narrow-wall slot directional coupler is analyzedwith mode matching method and optimized with simulated annealingmethod, and the dielectric loaded waveguide phase shifter is designedwith edge based finite element method. Numerical results of the magictee are presented, which show that the performance of the designedmagic tee is good.

1. INTRODUCTION

Magic tee is a widely used component in microwave systems and itcan be used to make up of many functional components such as sum-and-difference network in a mono-pulse antenna system, frequencydiscriminator and balanced mixer, etc. Conventional magic tee is amatched waveguide double-tee, and the four arms locate in differentplanes (E-plane and H-plane), which makes the magic tees occupymuch space and especially not suitable for planar circuits. As everyoneknows, a −3 dB directional coupler combining with a π/2 phase shiftergives the performance of magic tee, so in this paper, a planar magictee based on a −3 dB H-plane narrow-wall slot directional coupler(NWSDC) and a dielectric loaded rectangular waveguide phase shifter(DLWPS) is designed, and the four arms of this kind of magic tee† Also with Guangzhou Haige Communications Group, Guangzhou, P. R. China

36 Wang, Dou, and Mei

locate in the H-plane (called HPMT in the following). The NWSDCcontains many slots and wall sections, and it is to say that differentcombination of the slots and wall sections gives a different NWSDC, soa large number of NWSDC based on different combination of the slotsand wall sections should be simulated to find out the best one whichrealizes a −3 dB directional power, which is a time consuming course,and for this reason, fast computing method is needed to simulate thetransmission characteristic of the NWSDC. On the other hand, tosearch the best combination of the slots and wall sections, a multi-parameters optimization method has to be used. So in this paper modematching method (MMM) [1, 2] and the simulated annealing method(SAM) [3] are combined to design and optimize the NWSDC. ThoughMMM is efficient for designing the NWSDC, it is difficult to design theDLWPS using this method if the loaded dielectric block is irregular,so edge based finite element method (EBFEM) [4] is used to designand analyze the DLWPS. EBFEM not only has all the advantagesof the traditional FEM, but also overcomes the drawback of pseudo-solution which is often met in the traditional FEM [5]. In Section 2, thedesign of the NWSDC using MMM combined with SAM is discussed;in Section 3, the design of the DLWPS using the EBFEM is introduced;then in Section 4, two HPMT examples are given and the scatteringcharacteristics of the HPMT are presented.

2. DESIGN OF THE NWSDC

2.1. Theory

The structure of a NWSDC is shown in Fig. 1(a), the two waveguideare coupled through the slots in the narrow wall. The key to designthe coupler is to obtain the width of the slots and the width of thewalls between the slots.

See Fig. 1(b), the overall scattering matrix SD of the NWSDCmay be obtained by the direct combination of all the local scatteringmatrixes S1

c , S2c , . . . , S

Nc [6, 7]. According the odd and even mode

incident wave theory [8], in order to solve the local scattering matrixes,say S1

c , we may firstly solve a two-port scattering matrix S2p(e) of thehatched section (see Fig. 1(c)) in the case when w(x = 0) is magneticwall, and a two-port scattering matrix S2p(o) of the hatched section inthe case when w(x = 0) is electric wall, then the scattering matrixes

Progress In Electromagnetics Research B, Vol. 5, 2008 37

(a) Overall view

1cS 2

cS NcS

(b) Top view

IwS

)(fS III

wS)(IIwS )(’

fS

(c) Zoom-in view of the hatched section in (b)

ν ν ν

Figure 1. Configuration of the narrow-wall slot directional coupler.

S1c can be expressed as:

S1c =

(Sc111) (Sc1

21) (Sc131) (Sc1

41)(Sc1

21) (Sc122) (Sc1

32) (Sc131)

(Sc131) (Sc1

32) (Sc122) (Sc1

21)(Sc1

41) (Sc131) (Sc1

21) (Sc111)

(1)

38 Wang, Dou, and Mei

where(Sc1

11

)=

12

[(S2p

11 (e))

+(S2p

11 (o))]

(2a)

(Sc1

21

)=

12

[(S2p

21 (e))

+(S2p

21 (o))]

(2b)

(Sc1

22

)=

12

[(S2p

22 (e))

+(S2p

22 (o))]

(2c)

(Sc1

31

)=

12

[(S2p

21 (e))−

(S2p

21 (o))]

(2d)

(Sc1

32

)=

12

[(S2p

22 (e))−

(S2p

22 (o))]

(2e)

(Sc1

41

)=

12

[(S2p

11 (e))−

(S2p

11 (o))]

(2f)

The two-port scattering matrix S2p(ν) can be obtained by directcombination of five two-port scattering matrixes: SI

w, Sνf , S

II(ν)w ,

S′νf and SIII

w (ν = e or o when w is magnetic wall or electric wall

respectively ). SIw, S

II(ν)w and SIII

w are the scattering matrix of thewaveguide I, II and III respectively, Sν

f is the scattering matrix ofthe discontinuity at z = l1, and S

′νf is the scattering matrix of the

discontinuity at z = l2, however, S′νf can be obtained directly from Sν

f

by exchanging the ports of the two-port scattering matrix. Consideringthe case when the incident wave is TE 10 mode, the waves excited inregion I can be expressed as

EIyn =

N∑n=1

(AI

nejkI

znz + DIne

−jkIznz

)sin kxn (x− t/2)

HIxn =

N∑n=1

kIzn

wµ

(AI

nejkI

znz −DIne

−jkIznz

)sin kxn (x− t/2)

(3)

where

kIxn = nπ/a.

kIzn =

√k2

0 − (kIxn)2.

AIn and DI

n represent the amplitudes of the scattering wave and theincident wave in region I.

Progress In Electromagnetics Research B, Vol. 5, 2008 39

The waves excited in region II can be expressed as

EIIyn =

N∑n=1

(AII

n e−jkIIznz + DII

n ejkIIznz

)sin kII

xnx

HIIxn =

N∑n=1

kIIzn

wµ

(−AII

n e−jkIIznz + DII

n ejkIIznz

)sin kII

xnx

(4)

where kIIxn = nπ/a1, when w is electric wall, or kII

xn = (2n+ 1)π/(2a1),when w is magnetic wall.

kIIzn =

√k2

0 − (kIIxn)2.

The continuous conditions at the interface z = l1 are{EI

y = EIIy |z=l1

t/2 ≤ x ≤ a1

EIIy = 0 x ≤ t/2

(5a)

HIx = HII

x |z=l1 t/2 ≤ x ≤ a1 (5b)

Matching the tangential field component [9, 10] at z = l1 accordingto (5a) and (5b), the coefficients AI

n and AIIn can be related to each

other by a group of equations, solving these equations to get AIn and

AIIn , and then Sν

f can be written as

Sνf =

[ (AI(I)

) (AI(II)

)(AII(I)

) (AII(II)

)]

(6)

where(AI(II)

)is the coefficient matrix of the scattering wave in

region I when incident by(DII

), and other symbols can be understood

similarly.The scattering matrix of waveguide I, II and III can be easily

obtained, and can be written as

Spw = Diag

{e−jkp

znL}p = I, II or III. (7)

As there are many slots and wall sections in the NWSDC, theSAM is used to find the optimized combination of the sizes of the slotsand wall sections.

40 Wang, Dou, and Mei

2.2. Numerical Results

Based on the theory described in 2.1, a-3 dB NWSDC using eight slotsis designed at w band with the width of waveguide a = 2.54 mm, theconfiguration (d1 = d8 = 1.41 mm, d2 = d7 = 0.3 mm, d3 = d6 =1.58 mm, d4 = d5 = 1.46 mm, t = 0.6 mm, L1 = L7 = 0.77 mm,L2 = L6 = 2.25 mm, L3 = L5 = 1.22 mm, L4 = 1.49 mm) andits scattering parameters are shown in Fig. 2. The NWSDC is alsocomputed using HFSS 9.0, and the results agree well with that obtainedby MMM in this paper (see Fig. 2(b)).

(a) Top view

88 90 92 94 96 98 100-35

-30

-25

-20

-15

-10

-5

0

S11 MMM S11 HFSS S21 MMM S21 HFSS S31 MMM S31 HFSS S41 MMM S41 HFSS

|Sij|

dB

Frequency / GHz

S11

S21

S31S41

(b) Scattering parameters

Figure 2. Configuration of the eight-slot narrow-wall directionalcoupler and its scattering parameters.

In order to design a NWSDC with better performance, more slotshave to be used, and this means more parameters may be used in theoptimization procedure, which will add difficulties to the designing.Heinz Schmiedel and Fritz Arndt have given a solution in [11], wherethey tandem connect two −8.34 dB directional coupler to get a −3 dBdirectional coupler. This solution is adopted in this paper. Wedesigned a −8.34 dB NWSDC using six slots first (the scattering

Progress In Electromagnetics Research B, Vol. 5, 2008 41

parameter is shown in Fig. 3(a)), then connect two such directionalcouplers to get a −3 dB directional coupler (twelve slots), and thescattering parameter of the −3 dB NWSDC is shown in Fig. 3(b). Itcan be seen that the reflection of port 1 and the isolation between port1 and port 2 is less than −30 dB in a wide bandwidth of 6 GHz (from91 GHz to 97 GHz).

88 90 92 94 96 98 100-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-8.34

S21

S11

S41

S31

|Sij|

(dB

)

Frequency / GHz

88 90 92 94 96 98 100-45

-40

-35

-30

-25

-20

-15

-10

-5

0

S21

S11

S31

S41

|Sij|

(dB

)

Frequency / GHz

(a) (b)

Figure 3. (a) S-parameters of the −8.34 dB NWSDC using six slots,(b) S-parameters of the −3 dB NWSDC constructed by two −8.34 dBNWSDCs.

3. DESIGN OF THE DLWPS

In this paper, dielectric is loaded to the waveguide to achieve a phaseshift of π/2, as the dielectric block may be irregular, and it is difficultto analyze it with MMM, so the EBFEM is used to analyze and designthe phase shifter.

3.1. Theory of the EBFEM for Solving General ScatteringParameters of Waveguide Discontinuity Problems

A typical waveguide discontinuity problem is shown in Fig. 4, theelectric field �E in the region V enclosed by the port face S1, S2 andthe conductor boundary satisfies

∇×(

1µr

∇× E

)− k2εrE = 0 (8)

Assume the incident wave of unit amplitude transmits along zdirection, then the total field at the port face S1 can be expressed by

42 Wang, Dou, and Mei

x

yz

a

b

S1 S2

V

PEC

Figure 4. Waveguide loaded with dielectric block.

the sum of the incident field and the scattered field [4], and the electricfield can be expressed as

E(x, y, z) =∞∑

m=0

∞∑n=0

amn�eTEmn(x, y)eγmnz

+∞∑

m=1

∞∑n=1

bmn

[�eTM

tmn(x, y) + �z�eTM

zmnx, y)]eγmnz (9)

At the port face S2 the scattered electric field can be expressed by

E(x, y, z) =∞∑

m=0

∞∑n=0

cmn�eTEmn(x, y)e−γmnz

+∞∑

m=1

∞∑n=1

dmn

[�eTM

tmn(x, y) − �z�eTM

zmn(x, y)]e−γmnz (10)

where

�eTEmn(x, y) = Nmn

(�xn

bcos

mπx

asin

nπy

b− y

m

asin

mπx

acos

nπy

b

)(11)

�eTMtmn(x, y) = Nmn

(�xm

acos

mπx

asin

nπy

b+ y

n

bsin

mπx

acos

nπy

b

)(12)

�eTMzmn(x, y) =

Nmn

πγmn

[(mπ

a

)2+

(nπ

b

)2]

sinmπx

asin

nπy

b(13)

Nmn =√νmνn

/√n2

a

b+ m2

b

a, νm =

{1 m = 02 m �= 0 (14)

In expression (9) and (10), amn, bmn, cmn, dmn are unknownconstant coefficients; amn and bm represent the reflection coefficient of

Progress In Electromagnetics Research B, Vol. 5, 2008 43

the mn-th TE mode and the mn-th TM mode at S1 respectively; cmn

and dmn represent the transmission coefficient of the mn-th TE modeand the mn-th TM mode at S2 respectively. Discretize the region Vwith tetrahedron elements and set up the FEM linear equations set asthat in [4], the electric field at S1 and S2 can be calculated, then thecoefficients amn, bmn, cmn and dmn can be determined by

amn = e−γmnz1

∫ a

0

∫ b

0�eTE

mn ·[�E − �Einc

]z=z1

dxdy (15)

bmn = e−γmnz1

∫ a

0

∫ b

0�eTM

tmn ·[�E − �Einc

]z=z1

dxdy (16)

cmn = eγmnz2

∫ a

0

∫ b

0�eTE

mn · �E|z=z2dxdy (17)

dmn = eγmnz2

∫ a

0

∫ b

0�eTE

tmn · �E|z=z2dxdy (18)

After amn, bmn, cmn and dmn are obtained, the general scatteringparameter of the DLWPS can be given directly as

S =[

(amn, bmn) (cmn, dmn)(cmn, dmn) (amn, bmn)

](19)

For the H-plane problem, expression (19) can be simplified as

S =[

(amn) (cmn)(cmn) (dmn)

](20)

3.2. Numerical Results of the Designed DLWPS

The EBFEM was implemented with C program language; then thescattered parameters of unmatched double tee was calculated to testifyour program, numerical results obtained by our program agree wellwith the results in [12], see Fig. 5.

Then the program was used to design the DLWPS. In order torealize the characteristic of the magic tee, the phase of the transmittedwave at the exit port of the DLWPS should be π/2 ahead of thestandard waveguide. Assuming the transmitting coefficient of theDLWPS is Sp

21, and the transmitting coefficient of the standardwaveguide is Sw

43 (see Fig. 6(a)), then the following relation shouldbe satisfied,

re (Sp21) − im (Sw

43) = 0, im (Sp21) + re (Sw

43) = 0 (21)

44 Wang, Dou, and Mei

(a) (b)

Figure 5. Scattering parameters of the unmatched tee.

where re (Sp21) and im (Sp

21) are the real part and imaginary part ofSp

21 respectively, re (Sw43) and im (Sw

43) are the real part and imaginarypart of Sw

43 respectively.

1 2

(a) Standard waveguide and the DLWPS (b) Cut plane of the DLWPS

Figure 6. Designed DLWPS.

A designed DLWPS is shown in Fig. 6. The DLWPS is realizedby waveguide loaded with dielectric slab which is full height and haswedge-shaped end and front; the characteristic of the DLWPS is shownin Fig. 7, both the scattering characteristic and the phase shiftingcharacteristic are very good.

4. CHARACTERISTIC OF THE MAGIC TEE

After the S-parameter of the NWSDC (SD), S-parameter of theDLWSP (SDLWSP) and S-parameter of the standard waveguide (SSW)are obtained, the S-parameter of the HPMT can be obtained by direct

Progress In Electromagnetics Research B, Vol. 5, 2008 45

90 92 94 96 98-50

-40

-30

-20

-10

0|S

ij| / d

B

Frequency / GHz

S11 S21

90 92 94 96 98-55

-50

-45

-40

-35

-30

-25

dB

Frequency / GHz

Re(Sp21)-Im(S

w21)

Im(Sp21)+Re(S

w21)

(a) Scattered parameters (b) Phase shifting characteristic

Figure 7. Characteristic of the DLWPS.

combination of these three S-parameter matrixes, and the sketch mapof the matrix combination is shown in Fig. 8.

SD

SDLWSP

SSW

Figure 8. S-parameter matrix combination sketch.

In a −3 dB NWSDC, the phase of the output wave from port 3leads π/2 phase ahead of port 4 when incident at port 1. If a phaseshifter is linked to port 3 and a waveguide of the same length is linkedto the port 4, when the phase shifter leads π/2 phase ahead of thewaveguide, the output wave from port 3 leads π phase ahead of thatfrom port 4, and this gives the performance of a magic tee. TwoHPMT have been designed using eight-slot and twelve-slot NWSDCrespectively with the DLWSP. The configuration of the twelve-slotHPMT is shown in Fig. 9, and configuration of the eight-slot HPMTis similar. Numerical results of the two HPMT are given in Fig. 10and Fig. 11. The reflection and isolation of the twelve-slot HPMT (see

2

1 4

3

Figure 9. Configuration of the twelve-slot HPMT.

46 Wang, Dou, and Mei

Fig. 11(a)) is nearly −30 dB in a frequency band of 7 GHz, which isbetter than that of the eight-slot HPMT (see Fig. 10(a)). When TE 10

mode is incident at port 1, output waves of port 3 and port 4 shouldbe out phase and carry equal power, this could be represented by thefollowing expression:

|S31 + S41| → 0 (22)

88 90 92 94 96 98 100

-35

-30

-25

-20

-15

-10

-5 S11 S21

dB

Frequency / GHz

88 90 92 94 96 98 100-40

-35

-30

-25

-20

-15

-10

-5

0|S31+S41|2

dB

Frequency / GHz

(a) Reflection and isolation (b) Out phase characteristic

Figure 10. Characteristic curves of the eight-slot HPMT.

88 90 92 94 96 98 100

-45

-40

-35

-30

-25

-20

-15

-10

-5

S11 S21

dB

Frequency / GHz88 90 92 94 96 98 100

-30

-25

-20

-15

-10

-5

0

|S31+S41|2

dB

Frequency / GHz

(a) Reflection and isolation (b) Out phase characteristic

Figure 11. Characteristic curves of the twelve-slot HPMT.

The numerical results of |S31 + S41| are shown in Fig. 10(b) andFig. 11(b) in dB form for the two HPMT respectively. The −20 dBbandwidth of the twelve-slot HPMT is nearly 3 GHz, better than thatof the eight-slot HPMT (only 2 GHz).

Progress In Electromagnetics Research B, Vol. 5, 2008 47

5. CONCLUSION

In this paper, two NWSDCs are designed using MMM combinedwith the SAM and a DLWPS is designed using EBFEM, then bycombination of the DLWPS with the NWSDCs respectively, twoHPMTs are presented. With the four arms located in the same H-plane, it is convenient for this kind of magic tee to be used in planarcircuits and be assembled in a system. Numerical analysis shows thatthe presented HPMTs have good performances.

REFERENCES

1. Riabi, M. L., R. Thabet, and M. Belmeguenai, “Rigorousdesign and efficient optimizattion of quarter-wave transformers inmetallic circular waveguides using the mode-matching method andthe genetic algorithm,” Progress In Electromagnetics Research,PIER 68, 15–33, 2007.

2. Huang, R. and D. Zhang, “Application of mode matching methodto analysis of axisymmetric coaxial discontinuity structures usedin permeability and/or permittivity measurement,” Progress InElectromagnetics Research, PIER 67, 205–230, 2007.

3. Press, W. H., S. A. Teukolsky, W. T. Vetterling, andB. P. Flannery, Numerical Recipes in C, 2nd edition, CambridgeUniversity Press, 1995.

4. Jin, J. M., The Finite-element Method in Electromagnetics, JohnWileys & Sons, Inc., New York, 1993.

5. Hernandez-Lopez, M. A. and M. Quintillan-Gonzalez, “A finiteelement method code to analyse waveguide dispersion,” Journalof Electromagnetic Waves and Applications, Vol. 21, No. 3, 397–408, 2007.

6. Arndt, F., B. Koch, H.-J. Orlok, and N. Schroder, “Fieldtheory design of rectangular waveguide broad-wall metal-insertslot couplers for millimeter-wave applications,” IEEE Trans. onMicrowave Theory and Techniques, Vol. 33, No. 2, 95–104, 1985.

7. Partzelt, H. and F. Arndt, “Double-plane steps in rectangularwaveguides and their application for transformers, irises, andfilters,” IEEE Trans. on Microwave Theory and Techniques,Vol. 30, No. 5, 771–776, 1982.

8. Reed, J. and G. J. Wheeler, “A method of analysis of symmetricalfour-port networks,” IRE Trans. Microwave Theory Tech., Vol. 4,246–252, 1956.

48 Wang, Dou, and Mei

9. Uher, J., J. Bornemann, and U. Rosenberg, WaveguideComponents for Antenna Feed Systems: Theory and CAD, ArtechHouse Publishers, May 1, 1993.

10. Park, J. K., D. H. Shin, J. N. Lee, and H. J. Eom, “A full-wave analysis of a coaxial waveguide slot bridge using the Fouriertransform technique,” Journal of Electromagnetic Waves andApplications, Vol. 20, No. 2, 143–158, 2006.

11. Schmiedel, H. and F. Arndt, “Field theory design of rectangularwaveguide multiple-slot narrow-wall couplers,” IEEE Trans. onMicrowave Theory and Techniques, Vol. 34, No. 7, 791–797, 1986.

12. Ritter, J. and F. Arndt, “Efficient FDTD/matrix-pencil methodfor the full-wave scattering parameter analysis of waveguidingstructures,” IEEE Trans. on Microwave Theory and Techniques,Vol. 44, No. 12, 2450–2456, 1996.