Analysis of a ridge waveguide using overlapping T-blocks

Page 1

Analysis of a ridge waveguide using overlapping T-blocks

Yong H. Cho and Hyo J. Eom

Department of Electrical Engineering

Korea Advanced Institute of Science and Technology

373-1, Kusong Dong, Yusung Gu, Taejon, Korea

Phone +82-42-869-3436 Fax +82-42-869-8036

E-mail : [email protected]

Abstract A T-block (TB) approach is proposed to analyze the dispersion relation of a

ridge waveguide. The �eld representations of a TB are obtained with the Green's function

and mode-matching technique. Rigorous yet simple dispersion equations for symmetric and

asymmetric ridge waveguides are presented using a superposition of overlapping T-blocks.

The rapid convergence characteristics of the dispersion equation are illustrated in terms

of the cuto� wavenumbers. A closed-form dispersion relation, based on a dominant-mode

approximation, is shown to be accurate for most practical applications such as coupler,

�lter, and polarizer designs.

Index Terms � Ridge waveguide, dispersion, Green's function, mode-matching technique,

superposition.

1 Introduction

The propagation and coupling characteristics of a ridge waveguide have been investi-

gated with various methods [1-7] because of its broad band, low cuto� frequency, and

low impedance characteristics compatible with a coaxial line. When the geometry of a

waveguide is complicated such as a ridge waveguide, the formulation and dispersion analy-

sis becomes naturally involved. It is therefore desirable to develop an analysis scheme with

fast CPU time, increased accuracy, simple applicability, and wide versatility. To that end,

we propose a new approach using the T-block and superposition. In [8], a simple and new

equivalent network for T-junction, which is similar to T-block, is presented to obtain the

closed-form expression for open and slit-coupled E-plane T-junctions. In this paper, the

approach of the T-block and superposition is employed to divide a total region into several

overlapping T-blocks. It is possible to represent the �eld within a T-block in simple and

numerically e�cient series based on the Green's function and mode-matching method. The

Green's function approach allows us to reduce the number of unknown modal coe�cients

and improve the convergence rate. Since the Green's function for the T-block is available,

the involved residue calculus as in [7] is unnecessary, thereby increasing computational ef-

�ciency. The advantage of the T-block approach lies in substantially reducing the amount

of computational e�ort. In this paper, we will analyze the propagation characteristics of a

1

Page 2

double ridge waveguide by utilizing the T-block approach and superposition principle. It is

important to note that our T-block approach is applicable to other waveguide structures,

whose geometry can be broken into several overlapping T-blocks such as the �nline, shielded

microstrip line, and multiconductor transmission line.

2 Analysis of T-Block

T

2a

b

dRegion (I)

Region (II)

xy

z

Figure 1: Geometry of a T-block.

Assume that a TE-wave propagates along the z-direction inside a T-block (TB) in

Fig. 1. The phase factor e

i(�z�!t)

is omitted throughout. In region (I) (�d < y < 0), we

represent the H

z

component as

H

I

z

(x; y) =

1

X

m=0

q

m

cos a

m

(x+ a) cos �

m

(y + d)

h

u(x+ a)� u(x� a)

i

(1)

where a

m

= m�=(2a), �

m

=

q

k

2

0

� �

2

� a

2

m

, k

0

= 2�=�

0

, and u(�) is a unit step function.

In order to represent the H

z

�eld in region (II) (0 < y < b), we divide region (II) into two

subregions such as Fig. 2(a) and 2(b). Based on the superposition, the �eld in region (II)

is given by

H

II

z

(x; y) =

1

X

m=0

s

m

h

H

m

(x; y) +R

H

m

(x; y)

i

(2)

where H

m

(x; y) and R

H

m

(x; y) are the �eld components within subregions of Fig. 2(a) and

2(b), respectively. Similar to the H

I

z

(x; y) component, we represent H

m

(x; y) as

H

m

(x; y) =

cos �

m

(y � b)

�

m

sin(�

m

b)

cos a

m

(x+ a)

h

u(x+ a)� u(x� a)

i

(3)

Multiplying the E

x

�eld continuity at y = 0 between regions (I) and (II) by cos a

l

(x + a)

and integrating over �a < x < a, we obtain

s

m

= �q

m

�

m

sin(�

m

d) (4)

2

Page 3

2a

b

dRegion (I)

Region (II)

xy

z

(a) Subregion for H

m

(x; y)

T

bRegion (II)

xy

z

2a

(b) Subregion for R

H

m

(x; y)

Figure 2: Subregions of a T-block.

The component of H

m

(�a; y) results in the H

z

discontinuity at x = �a. In order to enforce

the H

z

continuity at x = �a, we next consider a contribution from R

H

m

(x; y). Utilizing the

Green's function relation [8] gives

R

H

m

(x; y) =

1

�

0

r�

�

A

�

�

�

z component

= �r�

Z

G

H

(r; r

0

)

�

J(r

0

) dr

0

�

�

�

z component

= �

Z

@

@n

h

G

H

(r; r

0

)

i

H

m

(r

0

) dr

0

(5)

where

�

A is a magnetic vector potential,

�

J(r

0

) = H

m

(r

0

)a

z

� n, n is the outward normal

direction to r

0

in Fig. 2(a),

G

H

(r; r

0

) =

2

b

1

X

v=0

cos(�

v

y) cos(�

v

y

0

)

�

v

g(x; x

0

; �

v

) (6)

g(x; x

0

; �) =

sin �(x

<

+ T=2) sin �(T=2� x

>

)

� sin(�T )

(7)

x

>

� the greater of x or x

0

, x

<

� the lesser of x or x

0

, �

0

= 2; �

m

= 1 (m = 1; 2; � � �),

�

v

= v�=b, �

v

=

q

k

2

0

� �

2

� �

2

v

, G

H

(r; r

0

) is shown in [9], and g(x; x

0

; �

v

) is a one dimensional

Green's function with an electric wall at x = �T=2. Note that r denotes an observation point

(x; y) and r

0

is a source point (x

0

; y

0

) = (�a; 0 < y

0

< b). The scattered component, R

H

m

(x; y)

is thought of as the �eld produced to eliminate the surface current,

�

J(r

0

) = H

m

(r

0

)a

z

� n.

This means that R

H

m

(�a; y) produces the H

z

(�a; y) discontinuity at x = �a, which is an

inverse sign to H

m

(�a; y

0

). Integrating (5) with respect to y

0

from 0 to b, we obtain

R

H

m

(x; y) =

Z

b

0

@G

H

(r; r

0

)

@x

cos �

m

(y

0

� b)

�

m

sin(�

m

b)

�

�

�

x

0

=�a

dy

0

� (�1)

m

Z

b

0

@G

H

(r; r

0

)

@x

cos �

m

(y

0

� b)

�

m

sin(�

m

b)

�

�

�

x

0

=a

dy

0

= �

1

b

1

X

v=0

cos(�

v

y)

�

v

(�

2

v

� a

2

m

)

h

f

H

(x;�a; �

v

)� (�1)

m

f

H

(x; a; �

v

)

i

(8)

3

Page 4

where

f

H

(x; x

0

; �) =

sgn(x� x

0

)

h

e

i�jx�x

0

j

� (�1)

m

e

i�(T�jx�x

0

j)

i

1� (�1)

m

e

i�T

(9)

and sgn(x) = 2u(x) � 1. The total longitudinal magnetic �eld is therefore given as

T

H

(x; y) = H

I

z

(x; y) +H

II

z

(x; y) (10)

Note that the E

x

(x; 0) �eld, which is produced by R

m

(x; 0), is zero. Hence, the E

x

and H

z

�eld continuities are satis�ed except for the H

z

�eld discontinuity at y = 0 and �a < x < a.

By enforcing the continuity of T

H

(x; 0) for �a < x < a, it is possible to determine the

dispersion relation for a TB. If a waveguide structure can be divided into a number of TB,

it is possible to use (10) directly in the derivation of dispersion relation. The advantage of

our TB approach lies in its computational simplicity since (10) can be repeatedly used for

each TB. In the next section, we will show how to obtain the dispersion relation for a ridge

waveguide from (10).

Similarly from the analysis of the TE mode, the E

z

components for the TM mode are

represented as

E

I

z

(x; y) =

1

X

m=1

p

m

sina

m

(x+ a) sin �

m

(y + d)

h

u(x+ a)� u(x� a)

i

(11)

E

II

z

(x; y) =

1

X

m=1

p

m

sin(�

m

d)

h

E

m

(x; y) +R

E

m

(x; y)

i

(12)

where

E

m

(x; y) =

sin �

m

(b� y)

sin(�

m

b)

sina

m

(x+ a)

h

u(x+ a)� u(x� a)

i

(13)

R

E

m

(x; y) = �

Z

G

E

(r; r

0

)

@

@n

0

h

E

m

(r

0

)

i

dr

0

= �

a

m

i

b

1

X

v=1

�

v

sin(�

v

y)

�

v

(�

2

v

� a

2

m

)

h

f

E

(x;�a; �

v

)� (�1)

m

f

E

(x; a; �

v

)

i

(14)

G

E

(r; r

0

) =

2

b

1

X

v=1

sin(�

v

y) sin(�

v

y

0

)g(x; x

0

; �

v

) (15)

f

E

(x; x

0

; �) =

e

i�jx�x

0

j

+ (�1)

m

e

i�(T�jx�x

0

j)

1� (�1)

m

e

i�T

(16)

Then, we obtain the total longitudinal electric �eld as

T

E

(x; y) = E

I

z

(x; y) +E

II

z

(x; y) (17)

Contrary to the TE mode, the total �eld, T

E

(x; y) is continuous at y = 0, while the H

x

�eld is discontinuous. The dispersion relation can be obtained by enforcing the H

x

�eld

continuity at y = 0 for �a < x < a. Although R

H

m

and R

E

m

are represented in series

forms, their convergence rate is very fast, thus e�cient for numerical computations. Note

that the convergence characteristics of the H

II

z

and E

II

z

are independent of T in Fig. 1.

These characteristics are di�erent from the standard mode-matching technique, where its

convergence rate is mainly determined by T . When T !1, our solutions converge to those

of a rectangular groove guide with an electric conductor cover at y = b in [10].

4

Page 5

3 Analysis of Ridge Waveguide Using Two TB

It is possible to apply the T-block approach to the dispersion analysis for a double ridge

waveguide. We �rst divide a double ridge waveguide in Fig. 3 into two overlapping

T2a

b

d

x

y z

(1)

(1)

b (2)

T(2)

T(2)

/

(x, y)/

T(1) /

(x, y)/

Figure 3: Geometry of a double ridge waveguide.

T2a

b

d

x

y z

(1)

(1)b (2)

T(2)

T(1) /

(x, y)/

/

/ 2a

d

T(2) /

(x, y)/

x

yz /

/

+

Figure 4: Superposition of two T-blocks.

T-blocks as shown in Fig. 4. The H

z

�eld in a ridge waveguide is represented as

H

z

(x; y) = T

(1)

H

(x; y) + T

(2)

H

(�x;�y � d) (18)

Note that the (x; y) coordinate in Fig. 3 is the global coordinate system for a double ridge

waveguide and the (x

0

; y

0

) in Fig. 4 is the local coordinate system for subregions of T-

blocks. The enforcement of the boundary conditions on the H

z

�eld continuities is required

to determine the relations of modal coe�cients, q

(1)

m

and q

(2)

m

. In matching the H

z

continuity,

it is expedient to introduce a general integration form as

I(x; y; c) =

Z

x+c

x�c

h

H

I

z

(x

0

; y)

�

�

�

T

(1)

+T

(2)

�H

II

z

(x

0

; y)

�

�

�

T

(1)

+T

(2)

i

cos c

l

(x

0

� x+ c) dx

0

5

Page 6

=

1

X

m=0

h

q

(1)

m

I

(1)

H

(x; y; c) + q

(2)

m

I

(2)

H

(�x;�y � d; c)

i

(19)

where c

l

= l�=(2c) and

I

H

(x; y; c) =

Z

x+c

x�c

n

cos a

m

(x

0

+ a) cos �

m

(y + d)[u(x

0

+ a)� u(x

0

� a)][u(y + d)� u(y)]

+�

m

sin(�

m

d)[H

m

(x

0

; y) +R

H

m

(x

0

; y)][u(y) � u(y � b)]

o

cos c

l

(x

0

� x+ c) dx

0

(20)

Since the integrand of I

H

(x; y; c) is composed of elementary functions, the evaluation of

(20) is trivial. In order to satisfy the H

z

continuity at (�a < x < a; y = 0) and (�a <

x < a; y = �d), we put (x; y) = (0; 0) and (0;�d) into (19), respectively, and obtain the

dispersion relation for a double ridge waveguide as

1

X

m=0

h

q

(1)

m

I

(1)

H

(0; 0; a) + q

(2)

m

I

(2)

H

(0;�d; a)

i

= 0 (21)

1

X

m=0

h

q

(1)

m

I

(1)

H

(0;�d; a) + q

(2)

m

I

(2)

H

(0; 0; a)

i

= 0 (22)

When a double ridge waveguide is symmetric with respect to the y direction, q

(1)

m

= �q

(2)

m

.

Note that � sign denotes even and odd modes with respect to the y direction, respectively.

Then, the dispersion equations, (21) and (22) reduce to a simpli�ed one as

1

X

m=0

q

(1)

m

h

I

(1)

H

(0; 0; a) � I

(2)

H

(0;�d; a)

i

= 0 (23)

In a dominant mode approximation (m = 0; l = 0), (23) further simpli�es to

I

(1)

H

(0; 0; a) � I

(2)

H

(0;�d; a)

�

�

�

m=0; l=0

= 2a

h

cos(k

c

d)� 1

i

+ k

c

sin(k

c

d)

"

2a

k

c

tan(k

c

b)

�

2i

b

1

X

v=0

1� e

i�

v

2a

+ e

i�

v

T

� e

i�

v

(T�2a)

�

v

�

3

v

(1� e

i�

v

T

)

#

= 0 (24)

where k

c

=

q

k

2

0

� �

2

and �

v

=

p

k

2

c

� (v�=b)

2

. Similarly, utilizing the approach of T-block

and superposition, we obtain the TM dispersion relation of a double ridge waveguide as

1

X

m=1

h

p

(1)

m

I

(1)

E

(0; 0; a) � p

(2)

m

I

(2)

E

(0;�d; a)

i

= 0 (25)

1

X

m=1

h

p

(1)

m

I

(1)

E

(0;�d; a) � p

(2)

m

I

(2)

E

(0; 0; a)

i

= 0 (26)

where

I(x; y; c) =

Z

x+c

x�c

h

@

@y

E

I

z

(x

0

; y)

�

�

�

T

(1)

+T

(2)

�

@

@y

E

II

z

(x

0

; y)

�

�

�

T

(1)

+T

(2)

i

sin c

l

(x

0

� x+ c) dx

0

6

Page 7

Table 1: Cuto� wavenumbers (rad=m) of the �rst eight TE-modes for a single ridge waveg-

uide.

Mode number 1 2 3 4 5 6 7 8

m = 0 91.34 333.1 379.9 524.7 664.7 690.5 744.8 828.0

m = 0; 2 92.40 333.2 380.7 525.8 665.3 692.1 745.4 828.9

m = 0; 2; 4 92.60 333.2 380.8 526.0 665.3 692.1 745.5 829.2

m = 0; 2; 4; 6 92.66 333.2 380.9 526.1 665.3 692.1 745.5 829.2

[5] 92.6 333.2 381.1 526.3 665.3 691.6 745.3 829.5

Parameters : a = 1:7 mm; T

(1)

= T

(2)

= 19 mm; d = 0:3 mm; b

(1)

= b

(2)

= 9:35 mm

Table 2: Cuto� wavenumbers (rad=m) of the �rst eight TM -modes for a single ridge waveg-

uide.

Mode number 1 2 3 4 5 6 7 8

m = 2 471.2 471.4 741.1 741.7 748.3 748.8 940.5 942.4

m = 2; 4 471.1 471.4 741.0 741.7 748.2 748.8 940.3 942.3

m = 2; 4; 6 471.1 471.4 741.0 741.6 748.2 748.7 940.2 942.3

m = 2; 4; 6; 8 471.1 471.4 741.0 741.6 748.2 748.7 940.2 942.3

[5] 471.1 471.4 741.0 741.6 748.1 748.7 940.0 942.2

Parameters : a = 1:7 mm; T

(1)

= T

(2)

= 19 mm; d = 0:3 mm; b

(1)

= b

(2)

= 9:35 mm

=

1

X

m=0

h

p

(1)

m

I

(1)

E

(x; y; c) � p

(2)

m

I

(2)

E

(�x;�y � d; c)

i

(27)

I

E

(x; y; c) =

Z

x+c

x�c

n

sina

m

(x

0

+ a) sin �

m

(y + d)[u(x

0

+ a)� u(x

0

� a)][u(y + d)� u(y)]

� sin(�

m

d)[E

m

(x

0

; y) +R

E

m

(x

0

; y)][u(y) � u(y � b)]

o

sin c

l

(x

0

� x+ c) dx

0

(28)

In a dominant mode approximation (m = 2; l = 2) for a symmetric single ridge waveguide,

(25) and (26) reduce to

I

(1)

E

(0; 0; a) � I

(2)

E

(0;�d; a)

�

�

�

m=2; l=2

= a�

2

h

cos(�

2

d)� 1

i

+ sin(�

2

d)

"

a�

2

tan(�

2

b)

�

2a

2

2

i

b

1

X

v=1

�

2

v

h

1� e

i�

v

2a

+ e

i�

v

T

� e

i�

v

(T�2a)

i

�

v

(�

2

v

� a

2

2

)

2

(1� e

i�

v

T

)

#

= 0 (29)

where �

2

=

q

k

2

0

� �

2

� a

2

2

. Tables 1 and 2 show the convergence characteristics of the TE

and TM modes in a single ridge waveguide, respectively. The agreement of our solution

7

Page 8

with [5] is excellent, and even a dominant-mode solution (m = 0) is accurate within 1%

error. This means that the dominant-mode dispersion relations, (24) and (29), are useful for

most practical applications. Fig. 5 represents the cuto� wavenumber characteristics of the

0.5 0.6 0.7 0.8 0.9 180

100

120

140

160

180

d = 0.3 mm

b / (1)

(T - d)(1)

Cu

toff

wa

ven

um

ber

[ra

d/

]m

m

d = 1.2 mm

m = 0

m = 0,2

m = 0,2,4

m = 0,2,4,6

Figure 5: Cuto� wavenumber of the TE

10

mode in a double ridge waveguide versus the

position of the ridge. ( a = 1:7 mm; T

(1)

= T

(2)

= 19 mm; and b

(1)

+ b

(2)

+ d = T

(1)

)

TE

10

mode in a single ridge waveguide versus the position of a ridge. As the width of the

ridge, d, increases, the cuto� wavenumber of the TE

10

mode decreases. This means that the

bandwidth of a single-mode operation increases. The dominant-mode solution agrees well

with the higher-mode solutions, thus con�rming the fast convergence. As b

(1)

=(T

(1)

� d)

approaches one, the solution converges to that of a single TB with b

(1)

= T

(1)

� d.

4 Analysis of Ridge Waveguide Using Four LB

For the analysis of an asymmetric double ridge waveguide of Fig. 6, it is convenient to

use a L-block (LB) as shown in Fig. 7. When the mode number, m, is even, the TE and

TM mode characteristics of a TB represent those of a LB. An asymmetric double ridge

waveguide in Fig. 6 is divided into four overlapping LB as in Fig. 8. The H

z

�eld in an

asymmetric double ridge waveguide is represented as the superposition of four LB. Then,

H

z

(x; y) = L

(1)

H

(x; y) + L

(2)

H

(x� T + a

(1)

+ a

(2)

; y)

+ L

(3)

H

(a

(3)

� a

(1)

� x; b� y) + L

(4)

H

(T � a

(1)

� a

(4)

� x; b� y) (30)

where L

H

(x; y) = T

H

(x+a; y)

�

�

�

m = even

. Matching points, (x; y) for the H

z

�eld continuities

are (0; 0), (T �a

(1)

�a

(2)

; 0), (a

(3)

�a

(1)

; b), and (T �a

(1)

�a

(4)

; b) in the global coordinates

8

Page 9

xy

z

L(2) /

(x, y)/

L(1) /

(x, y)/

L(3) /

(x, y)/

L(4) /

(x, y)/

T

b

Figure 6: Geometry of an asymmetric double ridge waveguide.

T

2a

b

d Region (I)

Region (II)

xy

z

Figure 7: Geometry of a L-block.

of Fig. 6. Using the same procedure as in Sect. 2, the dispersion relations of an asymmetric

double ridge waveguide are obtained. For instance, the dispersion equation corresponding

to the matching point (0; 0) is obtained as

I(0; 0; a

(1)

) =

Z

a

(1)

�a

(1)

h

H

I

z

(x

0

; 0)

�

�

�

L

(1)

+L

(2)

+L

(3)

+L

(4)

� H

II

z

(x

0

; 0)

�

�

�

L

(1)

+L

(2)

+L

(3)

+L

(4)

i

cos a

(1)

l

(x

0

+ a

(1)

) dx

0

=

1

X

m=0

h

q

(1)

m

I

(1)

LH

(0; 0; a

(1)

) + q

(2)

m

I

(2)

LH

(a

(1)

+ a

(2)

� T; 0; a

(1)

)

+ q

(3)

m

I

(3)

LH

(a

(3)

� a

(1)

; b; a

(1)

) + q

(4)

m

I

(4)

LH

(T � a

(1)

� a

(4)

; b; a

(1)

)

i

= 0 (31)

where I

LH

(x; y; c) = I

H

(x+ a; y; c)

�

�

�

m = even

. It is straightforward to obtain the remaining

three dispersion equations corresponding to (x; y) = (T � a

(1)

� a

(2)

; 0), (a

(3)

� a

(1)

; b), and

(T �a

(1)

�a

(4)

; b). In order to verify the validity of the approach based on LB, in Table 3 we

show the cuto� wavenumber of an asymmetric ridge waveguide of Fig. 3. For comparison

purpose, we also show the result with the TB approach for the same geometry in Table

4. Since the width of LB, 2a, is twice that of TB in our computation, the convergence

characteristics of TB is better. This is because more higher modes are needed to obtain

9

Page 10

d(1)

2a(1)

L(1) /

(x, y)/

xy

z

/

/

L(2) /

(x, y)/

xy

z

/

/

d(2)

2a(2)

+

+

+ +

L(3) /

(x, y)/ L

(4) /

(x, y)/

x

y

z

/

/x

y

z

/

/d

(4)d(3)

2a(3) 2a

(4)

Figure 8: Superposition of four L-blocks.

convergence as the width 2a in region (I) becomes wider. It is interesting to note that

the cuto� wavenumbers of the second and third modes in Table 4 approximately agree

with those of a rectangular waveguide, 78.54 and 104.7, respectively. In Fig. 9(a), the

H

z

�eld distribution of the second mode in Table 4 is illustrated. The H

z

�eld is almost

concentrated within the left cavity of Fig. 3, whose �eld distribution is very similar to that

of a rectangular waveguide. Since the propagation constant for an asymmetric double ridge

waveguide is quite di�erent from the rectangular waveguide similar to the right cavity, the

wave within the right cavity becomes evanescent. The H

z

�eld plot of a symmetric ridge

waveguide is shown in Fig. 9(b). The H

z

�eld is distributed in two cavities as it should be.

Table 3: Cuto� wavenumbers (rad=m) of the three TE-modes for an asymmetric double

ridge waveguide using two LB.

Number of modes used 1 4 5

1 37.25 125.4 164.8

3 40.79 137.8 165.2

5 41.06 138.1 165.3

7 41.14 138.2 165.3

Parameters : a

(1)

= 10 mm; d

(1)

= 10 mm; a

(2)

= 7:5 mm; d

(2)

= 15 mm; b = 5 mm; T =

45 mm

10

Page 11

Table 4: Cuto� wavenumbers (rad=m) of the �rst �ve TE-modes for an asymmetric double

ridge waveguide using two TB.

Number of modes used 1 2 3 4 5

1 40.96 80.36 107.0 137.7 165.3

2 41.15 80.46 107.1 138.2 165.3

3 41.19 80.49 107.2 138.3 165.4

[6] 41.31 80.56 107.2 138.5 165.5

Parameters : a = 5 mm; d = 10 mm; T

(1)

= 30 mm; b

(1)

= 20 mm; T

(2)

= 40 mm; b

(2)

=

15 mm

5 Conclusion

A novel T-block approach is proposed for analyzing symmetric and asymmetric ridge waveg-

uides. Simple closed-form dispersion relations for ridge waveguides are expressed in rapidly-

convergent series. Computed results indicate that our method, based on a superposition of

overlapping T-blocks, is accurate and numerically-e�cient. A dominant-mode approxima-

tion for a ridge waveguide is shown to be valid and useful for most practical cases. It is

possible to extend our theory to the analysis of other complex waveguide structures that

can be divided into a superposition of overlapping T-blocks. For example, the shielded

microstrip line, �nline, nonradiative dielectric guide, etc. are some typical waveguides that

our T-block approach can be applied to obtain their dispersion relations.

11

Page 12

xy

zH

(a) Asymmetric double ridge waveguide

xy

zH

(b) Symmetric double ridge waveguide

Figure 9: H

z

�eld distributions of TE modes using two TB.

12

Page 13

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13