Analysis for dispersion characteristics of elliptic- and parabolic-groove waveguides

International Journal of lnf~ared and Millimeter Waves. Vol, 1Z No. 5, 1996

ANALYSIS FOR DISPERSION CHARACTE~STICS OF

ELLIPTIC- AND PARABOLIC-GROOVE

WAVEGUIDES

Keyu Zhao, Fuyong Xu, and Mai Lu

Department of Electronics and Information Science Lanzhou University, Lanzhou 730000, Gansu, People's Republic of China

Received March 15, 1996

Abs t r ac t

In this paper, the dispersion characteristics of elliptic- and parabolic-groove guides are analysed by using the equiv- aJent network method with rectangular step approximation. The results have important values in the studying of the trans- mission characteristics of curvilineal-groove guides as well as its manufacture and application.

K e y Words . : elliptic- and parabolic-groove waveguides; dispersion characteristics; equivalent network method

*author for correspondence

887

0195-9271/96/0800-0887509.50/00 1996 Plenum Publishing Corporation

888 Zh8o et a t

I. Introduction The groove guide was first proposed by FA.Tischer in

1952 for use at millimeter and submiUimeter wavebands [1]. Ever since, the field distribution and transmission character- istics of groove waveguldes have been analysed by many au- thors with different methods. The groove guide has many advantages, such as simple constructure, easy manufacture, lower toss, less dispersion and higher transmission power and so on. Some of the groove guides have been used in practi- ca/applications. So far as the groove shape is concerned, it can be divided into two kinds i.e. broken line groove guide and curvilineal-groove guide. The rectangular -[2's], V- and trapezoidal- groove guides can be considered as broken line groove guides [4'5], and the circular- [6], elliptic- and parabohc- groove guides can be considered as curvilineal-groove guides. In this paper, we analysed the dispersion characteristics of the elliptic-and parabohc-groove guides by using the equiv- alent network method with the rectangular step approxima- tion. The method is effective and precise, it has sufficient high reliability3 and it can be especially used in the CAD technique with shorter computer time. The paper consummated study for the dispersion characteristics of curvilineal-groove guides, and it has important values for the estimating influences of the transmission characteristics, especially when a circular- groove guide is deformed by compressing in applications or there are errors in its manufacture.

II. Theoretical Analyses It is well known, the step in rectangular-groove guide can

be eqmva~ent to a paratiel network with the iumped param- eters which comprised of a shunt susceptance B and a ideal transformer with turn ratio n. The step and its equivalent

pa~boU©-Groove Waveguide$ 8S9

[ ] Ycl hi h2 Y,a

(a) Rectangular step

A1 A~

~i Y,2 BI

K,1 Kz2

C1 rtl : 1 C~

(b) Equivalent network

Fig.1 The step in the rectangular-groove guide and its equivalent network

network are shown in Fig.(1), and the relationships between the dimensions of the step and the parameters of the equiva- lent network are Iv]

2h2 B1 = 0.55Yd h~l ~ cot2(0.5~rh~l); (1)

.- 4 cos(O.5, h,1) (2) where Y,1 and Y,2 are the characteristic admittances shown in Fig.l(a), Kz and Kz axe the wave numbers along the z di- rection and x direction, h,1 is the relative height ratio of the step, they are given by

Yc] -- go2 -- K~ (3) w#0Kzl '

890 Z h a o a' aL

2 2 IQ~ = K~ - K~. w#og,'---'-~' (4)

K, z = K g - K , z , - ( ) ' - - - - K g - K ~ , - (5)

K0 2 -- w2eo/~o; (6)

h2 h,., = E ," (7)

First, we analyse the elliptic-groove guide. The groove region is approximated by n steps as shown in Fig.2(a), be- cause at the plane x=0, an electric wall is introduced for the main mode, its equivalent network is shown in Fig.2(b).

Y

1 0

~' I ~ c

(a) Elliptic-groove guide

c, ct c, c; c. c~ n l : 1 n~ : 1 n . : 1

f ~ Equivalent ne+,,.~r 1.

K , ,

Fig.2 The elliptic-groove guide and its equivalent network

PanaboJJc..Groove Waveguides 891

Given the elliptic equation is

~2 y2 ~ + V - - - - - 1 . (8)

where b=d+~, d is the depth of the groove, and a is the width of the parallel plates.

By using the above equation, we can calculate the di- mensions for each step, then we can get the admittance at the plane A~C* (see left from the plane A*C~) i.e. Y/s. Because the parallel plate region can be considered as the infinite uni- form transmission lines, so the input admittance (see right from the plane A~,C*) is Yi, = Yi~o.

At the plane A'C*, according to the transverse resonant condition, we have

v . + = 0. (9)

Hence we can get the dispersion characteristics of the elliptic-groove guide.

Secondly, given the parabolic equation is

- 00) y=d+

By using the same approach with that of the elliptic- groove guide, we can get the dispersion characteristics of the parabolic-groove guide.

In order to get the exact result, theoretically, we should choose the step numbers n as infinity, but in practice, as long as the step numbers are sufficient large, we can get a higher precision for ~he calculated results.

892 Zhao ~ aL

III . N u m e r i c a l Resu l t s a n d Discussions Based on the above theoretical analyses, we first calcu-

lated the guide wavelengths of the circular-groove guide shown in Table I along with the calculated and measured values given in the reference [6], their agreements are excellent; thus the accuracy of the present method is verified.

Table I The Calculated Results of the Guide Wavelengths for the Circular-Groove Guides

Op*r t t ia l V t l . • i. Vtluet ia I R e l t t l n *rmm wLv~|wttgtha thl* p.pe~ eef~r*l~*t [e] I

A0(mm ) Al l ( ram) CLlcu|ttad A l I ( m ~ ) EIxper. Aj~(mm) I IAJ 2 - A m l l / A f 2 e.09 ° e.210 e.lSS ease O.lS

i ~ e * e.x=s t.=oa e.lee O.ll 19 . t a ' " 21.011 2o.g2e 21.0le o.14

Annotations: ,Guide dimension: radius c=10mm for the circular groove,

distance a=13mm for the pax~llel plates. ** Guide dimension: ~--19.45mm, a=26mm .

The calculated curves for the elliptic- and parabolic- groove guides axe shown in Fig.3 to Fig.7.

2 . 0 0 ellipse a -=13m m c = l S m m d = 3 . S m m /

p a r a b o l a a = 1 0 m m m = 2 d = 4 m m /

I..50 ellipse * : : -" i /

,~1.00 f -

~ - 0 . 5 0 //~/

0 , 0 0

Fig.3

Ii|i ii iIi iii iii ~ii| i i'i,ii i,IIIII 111 III l'llllIili~

0 . 0 2 0 . 0 4 0 . 0 6 0 , 0 BO.O t 0 0 . 0 f(GHz)

The dispersion curves for the elliptic- and paxabolic-groove guides

Parabolic-Groove Waveguides 893

In Fig.3, the relationships between phase constants and frequency f for the elliptic- and parabolic-groove guides are given, from this figure, we find that the dispersion curves are almost straight lines, they mean that the dispersions of the elliptic- and parabolic-groove guides are both very little. Given a and f are constants, with the changing of axises for the elliptic-groove guides i.e. (i) major axis changes while minor axis is constant, Fig.(4), and (ii)minor axis changes while major axis is constant , Fig.(5), the guide wavelengths will not change very much, and they indicate that the groove shape has little effects on the guide wavelengths. Fig.6 shows that index m of the parabolic-groove guides has very little influence on the guide wavelengths, this also verifies that the shapes of grooves have little effects on the guide wavelengths. Fig.7 shows that the guide wavelengths will decrease slowly with increasing of the plate separation a.

7.00 1

/ - % E E 6.00 v

a=13mm d=4mm f=5OGH,

7.0 9 .0 1~'.5 . . . . . . 1~.o 15 .o ~(==)

Fig.4 The relationship between )~g and c for the elliptic-groove guide

894 Zhao ~' aL

7.00

E E 6 .00

v

5 .00

Fig.5

ellipm a=-13mm c=lSmm f=50GH.

pare, bole, a-..~-13mm m = 2 f=50GHz oll ipse : ; e -

paratbols . . . .

• 2.0 s . o 4 . o . . . . . i b d(n'~m)

The relationship between Ag and d for the elliptic-and parabolic-groove guides

7.0.0

E E 6 .00

v

5 .00

o~-13mm d=4mm f=f0GH,

i l l , l | , l l [ ~ l , , * l ' l "l

1 . o 2 . 0 . / .~ . . . . . . . . . . . . . . . . . . ' 4.0 5.0

Fig.6 The relationship between ,~g and index m for the parabolic-groove guide

Parabolic-Groove Waveguides 89S

7 . 0 0

E E 6 .00

5.00

Fig.7

el l ipN c ~ 1 5 m m d----4mm f = 5 0 G H .

p a r a b o l a m = 2 dff i4mm f=$OGHs ellilmm ,,: : ; z

parabola . . . .

7 . 0 . . . . . . '9 .0 . . . . . . 1 . . . . . . . . . . . 1 .0 13,' 0 . . . . . . . . . 15.0 ' . . . . . ~(mm)

The relationship between Ag and a for the elliptic-and paxabolic-groove guides

IV. Conclusions In this paper, the dispersion characteristics of the elliptic-

and parabolic-groove guides axe analysed by using the equiv- alent network method with rectangular step approximation. The results indicate the shapes of curvilineal-groove guides have very little influence on the dispersion characteristics. The results have important values in the studying of the trans- mission characteristics of curvilinea]-groove guides as well as the manufacture and application of the circulax-groove guide when it is deformed because of compressing in application or there are errors in its manufacture.

R.eferences

1. F.A.Benson and F.J.Tischer. Some Guiding Structure for Millimeter Waves. IE~. Proc. pt.A,131(7):,t29-449

896 Zhao ~ aL

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.

.

.

.

.

Xu Shanjia and Yin Lujin. Analysis of Dispersion Char- acteristics of Groove Guide with Mode-matching Method. International Journal of Infrared and Millimeter Waves, 1991,12(6):611-629

T.Nakahara and N.Kurauchi. Transmission Modes in the Grooved Guide. J.Inst. Electron. Commun. Eng.Jap.,

1964,47(7):43-51

Choi Yat Man. V-shaped Groove Guide and Compo- nents for 100GH~ Operation. IEEE MTT-S Interna- tional Microwaves Symposium Digest, 1987:165-167

Xu Fuyong, Zhao Keyu and Lu Mai. Analysis for Dis- persion Characteristics of Trapezoidal-Groove Waveg- uide. International Journal of Infrared and Millimeter Waves, 1996, 17(2)

Hong-Sheng Y~ng. Circular Groove Guide for Short Millimeter and Submil~meter Waves IEEE Trans. IVfi- crowave Theory Tech.,1995, 43(2):324-330

Oliner A and Lampariello P. The Dominant Mode Prop- erties of Open Groove Guide: An Improved Solution. IEEE Trans. Microwave Theory Tech., 1985, 33(9):755- 763.