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Received 2018 June 2; Accepted 2018 July 20; Accepted 2018 August 11.

An electromagnetic boundary-value problem of a tapered rectangular waveguide is rigorously solved based on eigenfunction expansion and the mode-matching method. Scattering parameters of the tapered rectangular waveguide are represented in a series form and calculated in terms of different rectangular waveguide combinations. Computation is performed to analyze reflection and transmission characteristics. Conductor loss by surface current density is also calculated and discussed.

Electromagnetic scattering from waveguide transition structures is a canonical problem. During the last few decades, there have been extensive studies on the scattering of waveguide transitions for many applications such as transformers [1], filters [2], and power dividers [3]. Moreover, a tapered waveguide can be used in a measurement system to characterize the electromagnetic properties of biaxial anisotropic materials [4].

The numerical method can be used to analyze electromagnetic scatterings from discontinuities in rectangular waveguides, but the more powerful method is the mode-matching method because it provides rigorous mode solutions in given regions. The analysis of the waveguide with one or more discontinuities is usually conducted using the mode-matching method in conjunction with the generalized scattering matrix technique [5–7]. In fact, the tapered structure can be analyzed using the mode-matching method by dividing the transition region into a number of sub-regions where the electromagnetic field can be defined in a series form of modes [2, 8, 9]. One study analyzed the rectangular waveguide transition using mode matching in X- to Ku-band applications to predict scattering parameters [2]. Studies on tapered waveguides have also been conducted using impedances [10], equivalent circuits, and the numerical method [11]. However, there has been no study on tapered waveguides based on the mode-matching method for V to W band applications, and much of the research has focused only on reflection and transmission characteristics to design the transition structure.

In this paper, we solve an electromagnetic boundary-value problem on a linearly tapered waveguide based on the mode-matching method. The eigenfunction expansion is used to represent the electromagnetic field in each region. The boundary conditions are enforced to obtain a set of simultaneous equations. Scattering parameters are represented in a series form and computed. To validate our formulation, simulation results of ANSYS High Frequency Structure Simulator (HFSS), a full-wave electromagnetic simulator, are compared with our results. Conduction loss in each rectangular waveguide is also calculated and discussed.

The geometry of a tapered rectangular waveguide is shown in Fig. 1. An incident wave is assumed to be *TE*_{10} mode, which is a dominant mode of rectangular waveguides. Region II should be divided into a number of rectangular waveguides to solve a boundary-value problem of a rectangular waveguide transition based on the mode-matching method. Each step waveguide has different widths and heights. A time convention of *e** ^{jωt}* is suppressed throughout the analysis. The permittivities and permeabilities in each region are

(1)
E y i ( x , y , z ) = sin [ π a ( x + a 2 ) ] e - j k 10 I z

(2)
H x i ( x , y , z ) = - k 10 I ω μ 1 sin [ π a ( x + a 2 ) ] e - j k 10 I z

(3)
F z I ( x , y , z ) = ∑ m = 0 ∞ ∑ n = 0 ∞ A m n - cos [ m π a ( x + a 2 ) ] × cos [ n π b ( y + b 2 ) ] e j k m n I z

(4)
A z I ( x , y , z ) = ∑ m = 1 ∞ ∑ n = 1 ∞ B m n - sin [ m π a ( x + a 2 ) ] × sin [ n π b ( y + b 2 ) ] e j k m n I z ,

where

(5)
F z I I - ( i ) ( x , y , z ) = ∑ s i = 0 ∞ ∑ p i = 0 ∞ [ C s i p i + e - j k s i p i I I ( z - d i ) + C s i p i - e j k s i p i I I ( z - d i ) ] × cos [ s i π x i ( x + x i 2 ) ] cos [ p i π y i ( y + y i 2 ) ]

(6)
A z I I - ( i ) ( x , y , z ) = ∑ s i = 1 ∞ ∑ p i = 1 ∞ [ D s i p i + e - j k s i p i I I ( z - d i ) + D s i p i - e j k s i p i I I ( z - d i ) ] × sin [ s i π x i ( x + x i 2 ) ] sin [ p i π y i ( y + y i 2 ) ] ,

where

(7)
F z I I I ( x , y , z ) = ∑ u = 0 ∞ ∑ v = 0 ∞ E u v + cos [ u π c ( x + c 2 ) ] × cos [ v π d ( y + d 2 ) ] e - j k u v I I I ( z - L )

(8)
A z I I I ( x , y , z ) = ∑ u = 1 ∞ ∑ v = 1 ∞ F u v + sin [ u π c ( x + c 2 ) ] × sin [ v π d ( y + d 2 ) ] e - j k u v I I I ( z - L ) ,

where

In order to predict the transmission characteristics of the linearly tapered rectangular transition structure, we calculate scattering parameters that can be represented as:

(9)
S 11 = P r e f P i n

(10)
S 21 = P t r a n s P i n ,

With calculated modal coefficients, the incident and reflected powers in Region I (*P** _{in}*,

(11)
P i n = k 10 I a b 4 ω μ 1

(12)
P r e f = 1 2 Re { 1 ω μ 1 2 ɛ 1 ∑ m = 1 ∞ ∑ n = 1 ∞ ∣ B m n - ∣ 2 × [ ( m π a ) 2 + ( n π b ) 2 ] k m n I a b 4 + 1 ω μ 1 ɛ 1 2 ∑ m = 0 ∞ ∑ n = 0 ∞ ∣ A m n - ∣ 2 × [ ( m π a ) 2 γ n + ( n π b ) 2 γ m ] ( k m n I ) * a b 4 }

(13)
P t r a n s = 1 2 Re { 1 ω μ 3 2 ɛ 3 ∑ u = 1 ∞ ∑ v = 1 ∞ ∣ F u v + ∣ 2 × [ ( u π c ) 2 + ( v π d ) 2 ] k u v I I I c d 4 + 1 ω μ 3 ɛ 3 2 ∑ u = 0 ∞ ∑ v = 0 ∞ ∣ E u v + ∣ 2 × [ ( u π c ) 2 γ v + ( v π d ) 2 γ u ] ( k u v I I I ) * c d 4 }

(14)
γ k = { 2 k = 0 1 otherwise ,

To validate our formulation and analyze the transmission characteristic of the tapered rectangular waveguide shown in Fig. 1, our computation results are compared with the simulation results from ANSYS HFSS. We take into account different combinations of rectangular waveguides designed for the V, E, and W bands in the analysis. The rectangular waveguides operating at each band have the following dimensions: *a** _{V}* = 3.7592 mm,

In order to check that the dominant mode is sustained through the transition structure, the electric field distribution and surface current density in a receiving waveguide (Region III) are plotted in Fig. 4 in a case of V to W transition at 75 GHz. The electric field distribution is calculated at the center of the waveguide parallel to the wider plane and all truncated modes are considered. The electric field distribution is similar to that of *TE*_{10}. Moreover, the surface current density on the inner walls is similar to that of *TE*_{10}. In addition, higher modes contribute to purely reactive powers and are evanescent modes produced by the junctions. The mode-matching method can provide the physical meaning, including the effects of each mode on the scattering characteristics, if the higher propagating modes exist in structures such as the W to V junction. As a result, the tapered rectangular waveguide with a linearly changing slope can transmit most of the incident energy to the receiving port without a significant change of mode.

Because of the existence of surface current densities on the inner walls of the rectangular waveguide, conductor losses must exist. Conductor loss is defined as in [13]:

(15)
P l = R s 2 ∫ C | J s → | 2 d l

where
*TE*_{10} mode with respect to the lengths of each waveguide in the linearly tapered rectangular waveguide. Note that our results are very similar to the theoretical results of *TE*_{10} because this mode dominates over the proposed structure and there is rarely loss by conductor. In Table 1大发体育, we calculate the total percentage of conductor loss in three cases (no transition, single step transition, and tapered transition) when the total length is assumed to be 3 mm. Compared to the result of WR15, conductor loss increases due to the discontinuity of the transition structure. In addition, the conductor loss of the single step transition structure is larger than that of the linearly tapered waveguide because the reflection increases.

Total length is 3 mm, and the length of the waveguide for V band is 1 mm in the transition structure.

We have solved the electromagnetic boundary-value problem of the tapered rectangular waveguide based on eigenfunction expansion and the mode-matching method. Scattering parameters are represented in a series form and computed under different combinations of rectangular waveguides operating at the V, E, and W bands. Our formulation can also be applied to various transition structures and can be used researching a measurement system using waveguide transition structures.

This research was supported by Physical Metrology for National Strategic Needs funded by Korea Research Institute of Standards and Science (No. KRISS-2018-GP2018-0005).

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**Sangsu Lee** received a BS degree from the Department of Electrical and Computer Engineering at Ajou University, Suwon, Korea in 2017. He is currently working towards an M.S. in Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea. His research interests include electromagnetic field scattering analysis and metamaterials.

**Jae-Yong Kwon** received a B.S. degree in Electronics from Kyungpook National University, Daegu in 1995 and M.S. and Ph.D. degrees in Electrical Engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea in 1998 and 2002, respectively. In 2001, he was a Visiting Scientist at the Department of High-Frequency and Semiconductor System Technologies, Technical University of Berlin, Berlin, Germany. From 2002 to 2005, he was a Senior Research Engineer at the Devices and Materials Laboratory, LG Electronics Institute of Technology, Seoul, South Korea. Since 2005, he has been a Principal Research Scientist with the Division of Physical Metrology, Center for Electromagnetic Metrology, Korea Research Institute of Standards and Science, Daejeon. Since 2013, he has been a Professor of the Science of Measurement at the University of Science and Technology, Daejeon. His current research interests include electromagnetic power, impedance, and antenna measurement. He is an IEEE Senior Member.

**Dongchan Son** received a B.S. degree from the Department of Electrical and Computer Engineering at Ajou University, Suwon, Korea in 2017. He is currently working towards an M.S. in the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea. His research interests include electromagnetic field scattering analysis and frequency selective surfaces.

**Yong Bae Park** received B.S., M.S., and Ph.D. degrees in Electrical Engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1998, 2000, and 2003, respectively. From 2003 to 2006, he was with the Korea Telecom Laboratory, Seoul, Korea. In 2006, he joined the School of Electrical and Computer Engineering, Ajou University, Suwon, Korea, where he is now a Professor. His research interests are electromagnetic field analysis and electromagnetic interference and compatibility.

© Copyright The Korean Institute of Electromagnetic Engineering and Science

Conductor loss (%) | |
---|---|

WR15 (no transition) | 0.0917 |

Single step transition | 0.1868 |

Tapered transition (Fig. 4) | 0.1559 |