An endlaunched adapter combining a coaxial loop within a rectangular waveguide has been analyzed and designed. The accurate electromagnetic modeling of input impedance for the adapter design has been developed using a modal analysis method. Also, design parameters to improve the endlaunched adapter have been investigated. Numerical and experimental results of the proposed model have been compared with the results of previous works, and verified by an electromagnetic simulator of High Frequency Structure Simulator (HFSS). The input voltage standing wave ratios (VSWRs) are smaller than 2 over the wide frequency band from 7.5 GHz to 10.5 GHz.
I. INTRODUCTION
An endlaunched rectangular waveguide adapter is used as an excitation element to transmit a modulated radio frequency (RF) signal to microwave devices such as a cavity filter and a horn antenna. The endlaunched adapter consists of a rectangular waveguide and a coaxial loop shorted on the broadwall of the waveguide. This adapter converts a coaxial mode to a waveguide dominant mode. This is known as a coaxialtowaveguide transition and has been heavily researched. The basic theoretical procedure for the design of a coaxialtowaveguide transition has been reported in
[1

5]
. However, these works could not obtain exact solutions because they applied approximation techniques to derive an expression of the input impedance.
This paper develops electromagnetic modeling of the input impedance to design an endlaunched rectangular waveguide adapter by using the modal analysis method. This method gives a more accurate expression that does not use any approximation scheme for the input impedance, which is considered the dominant mode along with the higherorder modes inside a waveguide. Also, the result of the input impedance expression is analyzed and compared with the results of the previous work in
[2
,
4]
. The validity of the proposed analytical model of input impedance is verified by an electromagnetic tool within High Frequency Structure Simulator (HFSS; ANSYS Inc, Canonsburg, PA, USA) based on a numerical method.
II. LOOP COUPLING GEOMETRY OF ENDLAUNCHED ADAPTER
Fig. 1
shows the geometry of the endlaunched rectangular waveguide adapter, where an inner conductor of the coaxial line is collinearly interconnected with the coupling loop shorted on the waveguide bottom. This type of a rectangular waveguide adapter converts the coaxial transverse electromagnetic (TEM) mode into the waveguide propagating mode. The radius of the inner conductor and the radius of the outer conductor of the coaxial line are c
_{i}
and c
_{o}
, respectively, and the inside of the coaxial line is filled with a material with a dielectric constant of
ε _{r}
. For the coupling loop, the zdirectional current element is placed on the zaxis (0 < z < L
_{1}
), while the ydirectional current element is placed on the yaxis (0 < y < L
_{2}
). The grounded position of the loop that is shorted on the waveguide broadwall bottom is located at the point (x = x
_{o}
, y = 0, z = L
_{1}
). The modes inside a waveguide, which are excited by a coupling loop, are formed by two current elements. The transverse electric (TE) modes are only excited by the ydirectional current element, while the transverse magnetic (TM) modes are excited by the zdirectional current element of the loop. Therefore, hybrid modes exist in the endlaunched adapter.The coupling loop dimension should be appropriately selected in order to convert the TEM mode of the coaxial line to the dominant TE
_{10}
mode inside the rectangular waveguide. The loop geometry has an especially large effect on the signal transmission, bandwidth, and return loss. To find the input impedance expression of the adapter, an analytical procedure is outlined in the following four steps
[6]
: First, sinusoidal trial current densities, which are the current distribution in the zdirection and the current distribution in the ydirection, are defined. Second, the magnetic vector potentials due to the current densities are derived. Third, the expression of the electric field without any approximation scheme inside a rectangular waveguide is obtained from the magnetic vector potentials using the Lorentz gauge condition
[7]
. Finally, the exact solution of the input impedance for the coaxialtowaveguide endlaunched adapter is derived by applying the energy reciprocity theorem from electromagnetic field theory
[8]
.
Geometry of endlaunched rectangular waveguide adapter.
The modal analysis expression for the input impedance can be obtained as
where
k
and
γ
are the propagation constant in free space and inside the rectangular waveguide, respectively.
ab
is a waveguide crosssection.
ε_{n}
= 1 for n=0 and
ε _{n}
= 2 otherwise. A
_{mn}
, B
_{mn,1}
, B
_{mn,2}
, C
_{mn}
, D
_{mn}
, E
_{mn}
, F
_{mn,1}
, F
_{mn,2}
, and F
_{mn,3}
are mentioned in Appendix A1 to A9.
The real component of the input impedance of the endlaunched adapter is attributed to the dominant mode; the higher order modes only contribute to the reactive component because of evanescent modes. The shorted perpendicular arm correlates with the longitudinal arm of the loop as shown in
Fig. 1
, and from this, the real component of input impedance can be matched to the characteristic impedance of the coaxial line and the reactive component can be canceled out. The best performance of the endlaunched adapter is achieved when the reactance cancellation occurs over the broadband operating frequency and the resistance is matched to the input impedance of the adapter connected by the coaxial line.
The input reflection coefficient and the input voltage standing wave ratio (VSWR) from (1) are defined as
where Z
_{0}
is the characteristic impedance of the input coaxial line. Re(Z
_{in}
) and Im(Z
_{in}
) are the real component and the imaginary component of the input impedance, respectively.
III. NUMERICAL ANALYSIS AND RESULTS
In order to analyze the adapter proposed in this paper, the physical dimensions of the rectangular waveguide are fixed on the specific waveguide WR90, but the lengths and radius of the coupling loop are controlled in order to achieve a resistance of 50 ohms through the computer simulation of (1). Therefore, the input impedance of the endlaunched adapter has to be matched to the characteristic impedance of the coaxial line filled with a material with a dielectric constant of 2.1.
To find the behavior of the input impedance, a commercial WR90 waveguide is selected and the lengths and radius are chosen to be in the ranges of L
_{1}
= 11.415.4 mm, L
_{2}
= 4.04.4 mm, c
_{i}
= 0.51.5 mm, and x
_{o}
= 5.76.3 mm, respectively. According to procedure of
Fig. 2
and
Table 1
, the number of modes is extended up to the 9th higher order mode to ensure the convergence rate of the input impedance.
From the perspective of the above physical parameters of the adapter, the input impedance of the endlaunched adapter can be matched from the above physical parameters of the adapter by properly choosing the length of the loop (L
_{1}
, L
_{2}
), the radius of c
_{i}
, and the input feeding point of x
_{o}
. It is the ideal case that the resistive component of input impedance equals the characteristic impedance of the coaxial line and the reactive component equals zero to ensure the optimum design of the adapter. However, it is difficult in practice to ensure that the reactive component is zero over the whole waveguide band. Therefore, we have to have a tradeoff in which the resistive component approaches the characteristic impedance of the coaxial line and also the reactive component approaches zero over the operating frequency band. This is done by controlling the physical parameters (L
_{1}
, L
_{2}
, c
_{i}
, x
_{o}
) of the coupling loop of the endlaunched adapter.
Figs. 3
and
4
show the evaluation of the input impedance and the input VSWR, respectively. The results of the proposed method, previous work, the HFSS, and the measured data have been compared with each other. The physical dimensions of the endlaunched adapter have been determined to be L
_{1}
= 13.4 mm, L
_{2}
= 4.2 mm, c
_{i}
= 1 mm and x
_{o}
= 5.9 mm under the bestmatched condition below the VSWR = 3. The comparison shows that the proposed result based on this letter for the input VSWR of the endlaunched adapter agree well with the measured data and the HFSS (Δ
s
= 0.01) data over the wideband frequency from 7.5 GHz to 10.6 GHz. However, in the results of
[2
,
4]
, the minimum VSWR are shown at 9.825 GHz and 9.3 GHz with bandwidths of 350 MHz and 1.05 GHz, respectively. Alternatively, the results of the previous studies have a narrow bandwidth and also have discrepancies with the others.
Fig. 5
shows the photograph of the endfed waveguide adapter combined with the WR90 waveguide with a coupling loop. This adapter is implemented by the copper metal and coaxial feeder line with the physical parameter dimensions of
Fig. 4
.
Sorting flow chart of cutoff frequency (fcmn) and propagation mode number (m,n) in waveguide.
Higher order mode and cutoff frequency in waveguide
Higher order mode and cutoff frequency in waveguide
Input impedance of endlaunched adapter. (a) Resistive component (b) reactive component. HFSS: high frequency structure simulator.
Voltage standing wave ratio (VSWR) of endlaunched adapter. HFSS: high frequency structure simulator.
Fabricated photograph of endlaunched adapter.
IV. CONCLUSIONS
The accurate expression of the input impedance for an analysis of the endlaunched rectangular waveguide adapter has been presented. It is a more accurate formula that does not use an approximation scheme considering the input impedance as the dominant mode together with the higher order modes inside a waveguide adapter. The variation of the input impedance according to the design parameters of the coupling loop has also been described. The results obtained from numerical calculation through the use of proposed method were compared with previous work, and the validity of these results is verified by a commercial electromagnetic tool in HFSS. There is a good agreement between the proposed analytical results and the results in HFSS for the input VSWR in the wideband frequency range of 7.5 GHz to 10.5 GHz. Therefore, the novel theoretical formula has more accuracy than the formulas derived in previous studies and allows the design to be based on a dependable quantitative criterion.
 APPENDIX
where
J_{n}
and
I_{n}
are the n
^{th}
order Bessel function of the first kind and the n
^{th}
order modified Bessel function of the first kind, respectively.
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