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 TE₁₀ 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 ε₁, μ₁, ε₂, μ₂, ε₃, and μ₃. In Region I, the incident and reflected fields based on vector potentials are:

where kmnI=k12-(mπa)2-(nπb)2 and k1=ωμ1ɛ1. In Region II, the transmitted and reflected fields can be represented by vector potentials, as follows:

where ksipiII-(i)=k22-(siπxi)2-(piπyi)2 and k2=ωμ2ɛ2. The vector potentials in Region III are:

where kuvIII=k32-(uπc)2-(vπd)2 and k3=ωμ3ɛ3. In all regions, electromagnetic fields can be obtained by using the above vector potentials [12]. The boundary conditions are enforced to obtain a set of simultaneous equations for modal coefficients Amn-, Bmn-, Csipi+, Csipi-, Dsipi+, Dsipi-, Euv+, and Fuv+.

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

With calculated modal coefficients, the incident and reflected powers in Region I (P_in, P_ref) and the transmitted power in Region III (P_trans) can be obtained as follows:

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, b_V = 1.8796 mm, a_E = 3.0988 mm, b_E = 1.5494 mm, a_W = 2.54 mm, b_W = 1.27 mm. Subscript represents the frequency band name at which rectangular waveguides operate. The tapered rectangular waveguide is assumed to have a linearly changing transition in Region II, and the length is L₂ = 1 mm. Before dividing the transition region (Region II) into sub-waveguides, we check the convergence of our formulation. Fig. 2 illustrates scattering parameters against frequency in the case of V to W transition. Regarding S₂₁, the results show sufficient convergence. Likewise, S₁₁ results also converge. Therefore, the transition region is divided by 10 rectangular waveguides (M = 10). The number of modes used in our computation is N = 6 to achieve convergence to within 0.5 dB. These numbers of modes are used throughout the analysis unless specified. Fig. 3 shows a simulation model of the linearly tapered rectangular waveguide in ANSYS HFSS and illustrates the comparison of computed scattering parameters based on our formulation with those of the HFSS simulation. The comparison between our results and the simulation results shows good agreement. From the results, it is verified that our formulation based on the mode-matching method is valid. Cut-off frequencies of rectangular waveguides operating at the V, E, and W bands are 39.9, 48.4, and 59 GHz, respectively. Therefore, the reflection of V to W (Fig. 3(b)) or E to W (Fig. 3(c)) near 60 GHz is larger than the reflection of the V to E (Fig. 3(d)) transition.

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 TE₁₀. Moreover, the surface current density on the inner walls is similar to that of TE₁₀. 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]:

where Rs=ωμ/2σ, the surface resistance of a metal. In most cases, the perturbation method is used to estimate conductor loss in the rectangular waveguide [13]. However, in the transition region, it is hard to calculate conductor loss because the electromagnetic field cannot be defined easily in Region II. For the mode-matching method, the transition structure in Region II is divided into a number of small rectangular waveguides, and therefore conductor loss can be estimated approximately by using the electromagnetic field in the stepped waveguides. Fig. 5 illustrates the calculated conductor loss of the TE₁₀ 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 TE₁₀ because this mode dominates over the proposed structure and there is rarely loss by conductor. In Table 1, 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.