Overlapping T-block analysis of axially grooved rectangular waveguide

1. 1
Overlapping T-block analysis of an axially
grooved rectangular waveguide
Yong H. Cho and Hyo J. Eom
Division of Information Communication and Radio Engineering
Mokwon University
800 Doan-dong, Seo-gu, Daejeon, 302-729, Republic of Korea
Phone: +82-42-829-7675 Fax: +82-42-825-5449
Email: yhcho@mokwon.ac.kr
Department of Electrical Engineering and Computer Science
Korea Advanced Institute of Science and Technology
373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea
August 13, 2003 DRAFT

2. 2
Abstract
The dispersion relations of an axially grooved rectangular (AGR) waveguide are presented using the
overlapping T-block approach. The elds within an AGR waveguide are divided into those of three
overlapping asymmetric T-blocks. The convergence characteristics of the dispersion relations are shown
in terms of the cuto frequency and compared with other results.
I. Introduction
An axially grooved rectangular (AGR) waveguide in Fig. 1 is a rectangular waveguide
with axially-spaced grooves on the broad side of a rectangular waveguide. The AGR
waveguide has been used for the interaction region of a high harmonic rectangular gyrotron
1]. The dispersion characteristics of the AGR waveguide and gyrotron cavity are analyzed
with the eld approximation 2] and mode-matching technique 3], respectively. In this
letter, the overlapping T-block approach is applied to the TE and TM dispersion analyses
of an AGR waveguide. The structure of an AGR waveguide in Fig. 1 is divided into three
overlapping asymmetric T-blocks in Fig. 2. The eld representations of an asymmetric
T-block in Fig. 3 are obtained using a Green's function and mode-matching technique 4].
II. Field Analysis of an AGR Waveguide
Consider the TE-wave in the AGR waveguide propagating along the z-direction with the
time convention ei( z;!t) shown in Fig. 1. In order to determine a propagation constant
of an AGR waveguide, we utilize the overlapping T-block approach in 4]. The AGR
waveguide is divided into three overlapping T-blocks as shown in Fig. 2. The Hz- eld is
represented with respect to the superposition of three overlapping T-blocks. Therefore,
the Hz- eld representations of an asymmetric T-block in Fig. 3 should be determined.
Utilizing the same method in 4], we obtain
HI
z(x y) =
1X
m=0
qm cosam(x + a)cos m(y + d)
h
u(x + a) ;u(x ;a)
i
(1)
HII
z (x y) = ;
1X
m=0
qm m sin( md)
h
Hm(x y) + RH
m(x y)
i
(2)
where am = m =(2a), m =
q
k2
0 ; 2 ;a2
m, k0 = !p
0 0,
Hm(x y) = cos m(y ;b)
m sin( mb)
cosam(x + a)
h
u(x + a) ;u(x ;a)
i
(3)
August 13, 2003 DRAFT

3. 3
RH
m(x y) = ;
Z
Hm(r0) @
@n
h
GH(r r0)
i
dr0
= ;1
b
1X
v=0
cos( vy)
v( 2
v ;a2
m)
h
fH(x ;a v) ;(;1)mfH(x a v)
i
(4)
GH(r r0) = 2
b
1X
v=0
cos( vy)cos( vy0)
v
g(x x0
v) (5)
g(x x0 ) = sin (x< + T1)sin (T2 ;x>)
sin( T) (6)
fH(x x0 ) = sgn(x ;x0)
h
ei jx;x0
j ;(;1)mei (2T0
;jx;x0
j)
i
h
1 + (;1)mei2 (T;T0
)
i
1 ;ei2 T (7)
T0 =
8
><
>:
T1 (x < x0)
T2 (x > x0)
(8)
0 = 2, v = 1 (v = 1 2 ), u( ) is a unit step function, v = v =b, v =
q
k2
0 ; 2 ; 2v,
sgn( ) = 2u( ) ; 1, T = T1 + T2, x> the greater of x or x0, and x< the lesser of x or
x0. A discussion on the Green's function is available in 5]. It is interesting to note that
(7) is identical to (9) in 4] when T1 = T2. We then obtain the total longitudinal magnetic
eld as
TH(x y) = HI
z(x y) + HII
z (x y) (9)
Using the superposition of the Hz- eld representation (9), we present the Hz- eld of an
AGR waveguide as
Hz(x y) = T(1)
H (x + T=3 y)+ T(2)
H (x y) + T(3)
H (x ;T=3 y) (10)
where T( )
H (x y) denotes the Hz- eld within the T( )(x y) block in Fig. 2. By enforcing the
Hz(x 0) eld continuities for ;a < x < a, T=3 ;a < x < T=3 + a, and ;T=3 ;a < x <
;T=3 + a, it is possible to determine the TE dispersion relation of an AGR waveguide.
The Hz- eld matching procedures are similar to those in 4]. Note that the TE dispersion
equation obtained by the Hz- eld matching is an analytic series solution. Although the
optimum interaction of a high harmonic rectangular gyrotron is obtained with the TE-
modes of the cavity 2], it is of theoretical interest to obtain the TM dispersion relation of
an AGR waveguide. We apply the overlapping T-block approach to the dispersion analysis
August 13, 2003 DRAFT

4. 4
of the TM-mode. Then,
EI
z(x y) =
1X
m=1
pm sinam(x + a)sin m(y + d)
h
u(x + a) ;u(x ;a)
i
(11)
EII
z (x y) =
1X
m=1
pm sin( md)
h
Em(x y) + RE
m(x y)
i
(12)
where
Em(x y) = sin m(b ;y)
sin( mb)
sinam(x + a)
h
u(x + a) ;u(x ;a)
i
(13)
RE
m(x y) = ;
Z
GE(r r0) @
@n0
h
Em(r0)
i
dr0
= ;ami
b
1X
v=1
v sin( vy)
v( 2
v ;a2
m)
h
fE(x ;a v) ;(;1)mfE(x a v)
i
(14)
GE(r r0) = 2
b
1X
v=1
sin( vy)sin( vy0)g(x x0
v) (15)
fE(x x0 ) =
h
ei jx;x0
j + (;1)mei (2T0
;jx;x0
j)
i
h
1 + (;1)mei2 (T;T0
)
i
1 ;ei2 T (16)
Similar to (10), we obtain the Ez- eld of an AGR waveguide
Ez(x y) = T(1)
E (x + T=3 y)+ T(2)
E (x y) + T(3)
E (x ;T=3 y) (17)
where
TE(x y) = EI
z(x y) + EII
z (x y) (18)
Similar to the TE-mode analysis, we enforce the Hx(x 0) eld continuities for ;a < x < a,
T=3;a < x < T=3+a, and ;T=3;a < x < ;T=3+a in order to obtain the TM dispersion
relation of an AGR waveguide.
III. Numerical Computation
Table I shows the cuto frequency of the TEpq mode for an AGR waveguide, con rming
that our solutions agree well with those in 2] except for the TE22 and TE02 modes.
Note that p and q in the TEpq mode signify the number of half-wave variations of the Hz
component along the x-axis and the qth cuto frequency, respectively 2]. The convergence
characteristics of our TE solutions in Table I are excellent with respect to the mode index
m. This indicates that a dominant-mode solution (m = 0) is useful for most practical
August 13, 2003 DRAFT

5. 5
applications. Table II shows the cuto frequency of the TMpq mode for an AGR waveguide.
Table II also indicates that our series solutions for the TM-mode converge rapidly.
References
1] A. M. Ferendeci, Rectangular cavity high harmonic gyrotron ampli er," IEEE MTT-S International Mi-
crowave Symposium Digest, pp. 430-431, 1983.
2] K. P. Ericksen and A. M. Ferendeci, TE modes of an axially multiple-grooved rectangular waveguide," IEEE
Trans. Microwave Theory Tech., vol. 43, no. 9, pp. 2001-2006, Sept. 1995.
3] V. Kasiphotla and A. H. McCurdy, A mode-matching technique for mode coupling in a gyrotron cavity,"
IEEE Trans. Microwave Theory Tech., vol. 44, no. 2, pp. 225-232, Feb. 1996.
4] Y. H. Cho and H. J. Eom, Analysis of a ridge waveguide using overlapping T-blocks," IEEE Trans. Microwave
Theory Tech., vol. 50, no. 10, pp. 2368-2373, Oct. 2002.
5] R. E. Collin, Field Theory of Guided Waves, 2nd ed., New York: IEEE Press, pp. 78-86, 1991.
August 13, 2003 DRAFT

6. 6
;;;;;;;;;;
;;;;;
d
T
b
a2
T/3
T/3
T/3
d d
a2 a2
PEC
x
y
z
Fig. 1. Geometry of an axially grooved rectangular waveguide.
;;;;;;;;;;;;;;;;;;;;;;T/2T/2 T/65T/6T/6 5T/6
+ +
T(1)
(x, y) T(2)
(x, y) T(3)
(x, y)
Fig. 2. Superposition of three asymmetric T-blocks.
August 13, 2003 DRAFT

7. 7
;;;;;
;;;;;
;;;;;
d
T
b
a2
PEC
x
y
z
T2T1
Fig. 3. Geometry of an asymmetric T-block.
August 13, 2003 DRAFT

8. 8
TABLE I
Cutoff frequency GHz] of the TEpq mode for an axially grooved rectangular
waveguide with a = 0:17 cm, b = 2:71 cm, d = 0:84 cm, and T = 5:73 cm.
p q 2] m = 0 m = 1 m = 2
1 1 2.5327 2.5358 2.5398 2.5398
2 1 5.0165 4.9307 4.9378 4.9391
0 1 5.1666 5.0512 5.0512 5.0523
1 2 5.6927 5.5038 5.5045 5.5069
2 2 7.0163 6.4533 6.4533 6.4575
0 2 9.2994 7.6988 7.6988 7.7077
TABLE II
Cutoff frequency GHz] of the TMpq mode for an axially grooved rectangular
waveguide with a = 0:17 cm, b = 2:71 cm, d = 0:84 cm, and T = 5:73 cm.
p q m = 1 m = 2 m = 3
0 1 6.1029 6.1028 6.1022
1 1 7.6009 7.6009 7.6003
2 1 9.5809 9.5809 9.5801
0 2 11.3324 11.3323 11.3309
1 2 12.2051 12.2050 12.2037
2 2 13.5042 13.5042 13.5018
August 13, 2003 DRAFT