### I. Introduction

### II. Field Analysis

*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*

*is suppressed throughout the analysis. The permittivities and permeabilities in each region are*

^{jωt}*ε*

_{1},

*μ*

_{1},

*ε*

_{2},

*μ*

_{2},

*ε*

_{3}, and

*μ*

_{3}. In Region I, the incident and reflected fields based on vector potentials are:

##### (5)

##### (6)

### III. Numerical Results and Discussion

*P*

*,*

_{in}*P*

*) and the transmitted power in Region III (*

_{ref}*P*

*) can be obtained as follows:*

_{trans}##### (12)

##### (13)

*a*

*= 3.7592 mm,*

_{V}*b*

*= 1.8796 mm,*

_{V}*a*

*= 3.0988 mm,*

_{E}*b*

*= 1.5494 mm,*

_{E}*a*

*= 2.54 mm,*

_{W}*b*

*= 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*

_{W}*L*

_{2}= 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

*S*

_{21}, the results show sufficient convergence. Likewise,

*S*

_{11}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.

*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.

*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, 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.