A New Technique for Broadband Matching of Open-Ended Rectangular Waveguide Radiator

The maximum reflection at an open end of a standard rectangular waveguide is about −10 dB in its operating frequency range. It is often used without matching. For critical applications, it is desirable to further reduce the reflection coefficient. In this paper, a new technique is presented for the broadband impedance matching of an open-ended rectangular waveguide. The proposed technique employs three thin capacitive matching elements placed at proper intervals via a low-loss dielectric material. The capacitance of, and distance between, the matching elements are optimized for broadband impedance matching using a simulation tool. Based on the proposed technique, two design examples are presented for the matching of a WR75 waveguide radiator. A reflection coefficient of less than −16 dB and −20 dB has been achieved over a ratio bandwidth of 2.13:1 and 1.62:1, respectively.

The aperture reflection of a rectangular waveguide open end strongly depends on the ratio of the narrow-wall height (b) to the broad-wall width (a) and weakly depends on the waveguide wall thickness (t).The smaller the b/a ratio, the larger the aperture reflection.In a standard rectangular waveguide ranging from the largest WR2300 (584.2 × 282.1 mm 2 ; 0.32−0.49GHz) to the smallest WR1(0.254× 0.127 mm 2 ; 750−1100 GHz), b/a and t/a range from 0.406 to 0.512 and from 0.0081 to 0.467, respectively.The maximum reflection is 0.313 (−10.1 dB) for b/a = 0.406 and 0.225 (−13.0 dB) for b/a = 0.512 [9].
The level of reflection in an unmatched open end of a rectangular waveguide might be acceptable in some applications.For other applications, it is desirable for an open-ended waveguide radiator to have a smaller reflection coefficient over a broad frequency range.With reduced aperture reflection, accuracy is improved in measurement applications and efficiency is increased in array antenna applications.
Broadband impedance matching of the open-ended rectangular waveguide (OEG) is of the highest importance in precision measurement applications, where it is a usual practice to use metrology-grade accessories and instruments.The OEG's improved impedance matching would be beneficial in antenna near-field measurements, the precise generation and measurement of electromagnetic fields and material constants measurements using the free-space method.Improved matching in measurement applications reduces errors arising from imperfections in calibration.
In a large phased array employing open-ended waveguide radiating elements, impedance matching reduces the reflected power, which leads to a significant improvement in power efficiency and a resultant saving on the device's cooling costs.
In the following, we will describe in some detail a need for better impedance matching of an OEG probe in the antenna near-field measurements.A commercial near-field waveguide probe by TTI Norte S.L. Co. uses a WR62 waveguide (15.80 × 7.90 mm 2 ) open end [28].The aperture walls are chamfered as shown in Figure 1a.An absorber collar is placed around the probe to reduce wave reflection and scattering from the probe fixture, as shown Figure 1b.The probe's input VSWR at the coaxial port is shown in Figure 1c and ranges from 1.3 to 2.0, or the reflection coefficient from −17.7 dB to −9.5 dB with multiple maxima and minima caused by the reflection between the probe aperture and the coaxial-to-waveguide transition.
Sensors 2023, 23, x FOR PEER REVIEW 2 Broadband impedance matching of the open-ended rectangular waveguide (OE of the highest importance in precision measurement applications, where it is a usual tice to use metrology-grade accessories and instruments.The OEG's improved imped matching would be beneficial in antenna near-field measurements, the precise gener and measurement of electromagnetic fields and material constants measurements the free-space method.Improved matching in measurement applications reduces e arising from imperfections in calibration.
In a large phased array employing open-ended waveguide radiating element pedance matching reduces the reflected power, which leads to a significant improve in power efficiency and a resultant saving on the deviceʹs cooling costs.
In the following, we will describe in some detail a need for better impedance m ing of an OEG probe in the antenna near-field measurements.A commercial near waveguide probe by TTI Norte S.L. Co. uses a WR62 waveguide (15.80 × 7.90 mm 2 ) end [28].The aperture walls are chamfered as shown in Figure 1a.An absorber co placed around the probe to reduce wave reflection and scattering from the probe fix as shown Figure 1b.The probe's input VSWR at the coaxial port is shown in Figure 1 ranges from 1.3 to 2.0, or the reflection coefficient from −17.7 dB to −9.5 dB with mu maxima and minima caused by the reflection between the probe aperture and the co to-waveguide transition.In near-field antenna measurements, it is important to reduce the wave reflectio scattering by the scanning probe in order to minimize the effects of multiple reflec between the probe and the antenna under test (AUT), which is complicated and diffi In near-field antenna measurements, it is important to reduce the wave reflection and scattering by the scanning probe in order to minimize the effects of multiple reflections between the probe and the antenna under test (AUT), which is complicated and difficult to calibrate out [29].Reflection from the probe aperture caused by impedance mismatch is another contributor to the multiple-reflection effects.The reflection coefficients of a probe and an AUT in antenna near-field measurements are accounted for in a power transfer equation via 1 − |Γ| 2 , where Γ is the reflection coefficient.The reflection coefficient is entered into a classical formula for the probe's boresight gain by Yaghijan [3], which reads Note that the complex value of the reflection coefficient Γ is used in (1), not just the magnitude.The near-field probe typically operates in the recommended operating frequency range (1.5:1 bandwidth) of a standard rectangular waveguide.Therefore, it is desirable to reduce the reflection coefficient of a waveguide open end as much as possible over a bandwidth greater than 1.5:1.
In view of the problems discussed in the above, this paper presents a new technique for the broadband impedance matching of an open-ended rectangular waveguide.A thorough literature survey reveals that this is still an open problem.By broadband, we mean the recommended frequency range (1.5:1) of a standard rectangular waveguide and beyond.A comprehensive review of existing works on this issue is presented below.
A simple method for the impedance matching of an open-ended rectangular waveguide is to load the aperture with a dielectric plug.Ivanchenko and co-workers used a Teflon ® (dielectric constant ε r = 2.04) plug in a WR90 waveguide aperture, with fourteen dielectric cylinders of ε r = 3.8 embedded in the plug [30].They obtained a reflection coefficient <−15 dB at 9.5−10.5GHz (1.11:1 bandwidth).Zhang and co-workers used a dielectric slab (ε r = 2.7, thickness t = 0.5 mm) inside a circular waveguide (diameter = 12 mm) and close to the aperture.The aperture was equipped with a slot and a choke that were employed for radiation pattern control.They reduced the reflection coefficient to −20 dB at 17.0−19.5GHz (1.15:1 bandwidth) [31].The unmatched reflection coefficient ranges from −11.5 to −7.5 dB in the same frequency band.
In a phased array application, it is necessary to obtain good impedance matching in a waveguide radiator operating over large scan angles.An inductive iris matching technique has been employed for this application [32,33].Van Schaik used an inductive iris on a 35.0 × 11.4 mm 2 aperture with a polythene dielectric sheet (ε r = 2.3, t = 5 mm) placed outside the waveguide at 5 mm distance from the aperture to obtain reflection <−10 dB at 5.4−5.9GHz for scan angles up to 60 • [32].
More recent studies on this topic have been concerned with the impedance matching of an evanescent waveguide aperture.Ludlow and Fusco placed a dielectric slab (ε r = 6.15, t = 6.36 mm) at 0.7-mm in front of a 55.0 × 27.5 mm 2 waveguide (f cTE10 = 2.73 GHz) aperture to obtain −10-dB reflection at 2.13−2.70GHz [34].Their radiator is more like a cavity antenna than a waveguide radiator.A coaxial probe is placed at 23 mm from a back short and at 24 mm from the aperture.The probe sets up an aperture field via proximity coupling, and the dielectric slab changes the aperture admittance so that aperture matching occurs below the waveguide cutoff.The method of placing a dielectric slab in front of an aperture has long been employed for wide-scan angle impedance matching in waveguide phased arrays [35].
Ludlow and co-workers presented an impedance matching method for an open-ended evanescent-mode rectangular waveguide, using a conducting post in the aperture which is coupled directly to a feeding coaxial probe [36], or via three conducting posts placed between a feeding probe and a conducting post in the aperture [37].The aperture in [36] works at 4.43−4.57GHz in a waveguide having a cutoff at 6.56 GHz.The aperture of [37] is realized in a waveguide with a cutoff at 2.72 GHz and has a reflection coefficient of less than −10 dB at 2.30−2.68GHz.
Ludlow and co-workers presented the concept of distributed coupled resonators for the impedance matching of an evanescent-mode rectangular waveguide radiator [38].An aperture containing a dielectric slab is excited by two dielectric plugs placed between a feeding coaxial probe and the dielectric slab in the aperture.The dielectric plugs and the waveguide section between them form two coupled resonators.They achieved a ratio bandwidth of 1.23 (2.00−2.45GHz) for a reflection coefficient <−10 dB using a waveguide with a cutoff at 2.58 GHz.
Other impedance matching methods for a below-cutoff aperture include the use of a negative permeability material [39], an electromagnetic metamaterial [40][41][42][43], a radiator-filter combined structure [44][45][46][47].Except for the radiator-filter combined structure, the input reflection coefficient is on the level of −10 dB, not −20 dB.These methods tend to yield a narrow-band matching.
A particular challenge in OEG impedance matching is to achieve a large matching bandwidth.In all of the previous studies, it has been difficult to obtain a good impedance matching over a large bandwidth, for example, over more than an octave bandwidth.In this paper, we present a technique for the broadband impedance matching of an open-ended rectangular waveguide radiator using capacitive elements.This is the first time that the capacitive matching technique has been applied to solve this problem.The novelty of the proposed technique is in using printed capacitive elements spaced by a low dielectric constant material for a simple, low-cost implementation of the broadband impedance matching of the rectangular waveguide open-end radiator.The following are the major contributions of this paper.

•
OEG impedance matching over more than an octave bandwidth; • Achieving −20 dB reflection in the standard waveguide band; • OEG matching with a series of printed-circuit capacitive elements precisely placed inside a waveguide via a low-dielectric-constant material.
We will show the proposed technique using two design examples-a broadband design and a low-reflection design.The two examples have been obtained using a powerful modern electromagnetic simulation tool (CST Studio Suite TM V2023).The authors believe that the simulation tool used in this paper is accurate enough to fully demonstrate the proposed technique.All the details-the structure, technique, simulation and dimensions-are provided, so that anyone can easily reproduce the result.In the next section, we will present the proposed technique and design examples based on it.
Firstly, the reflection coefficient of an empty OEG is analyzed.Secondly, the geometry of the proposed matching structure is presented, along with its equivalent circuit representation, followed by the analysis of the circuit property of the capacitive matching element.Thirdly, two design examples of the OEG matching using the proposed technique are presented.Finally, this study is compared with previous research, followed by conclusions.

Aperture Matching of the Rectangular Waveguide Open End
Figure 2a shows an open-ended WR75 waveguide with broad-wall width a = 19.05mm, narrow-wall width b = 9.53 mm and wall thickness t = 1.27 mm.The cutoff frequency f c and cutoff wavelength λ c of the TE 10 mode in a rectangular waveguide filled with a material of dielectric constant ε r are given by where a is the broad-wall width in mm.In a WR75 waveguide with ε r = 1, the fundamental TE 10 mode cutoff is at 7.87 GHz. Figure 2b shows the simulated aperture reflection coefficient Γ A of a WR75 waveguide radiator.It decreases steadily from −10 dB at 8.17 GHz (≡ f 1 = 1.038 f c ) to −17.9 dB at 20 GHz (≡ f 2 = 2.54 f c ). Figure 2c shows the normalized aperture admittance y A (= g A + jb A ), whose real part g A and imaginary part b A range from 0.839 to 1.17 and from 0.253 to 0.587, respectively, at 8.17−20 GHz.The aperture has a positive susceptance b A presenting a capacitive load to the waveguide.Chamfering the walls of the open end for scattering reduction changes the aperture admittance only slightly.
1.17 and from 0.253 to 0.587, respectively, at 8.17 Figure 3a shows the proposed structure for the broadband matching of an o ended rectangular waveguide radiator, whose equivalent circuit representation is sh in Figure 3b.The proposed technique utilizes three matching elements M1, M2 and M a shunt capacitive type.D1 and D2 are a dielectric material of low dielectric constan employed to support and to precisely position the matching elements M1, M2 and M3 same dielectric material D3 fills the waveguide W. Without D3, the level of imped matching is reduced.Filling the space between the aperture and the first matching elem M1 with a dielectric material also reduces the level of impedance matching.The diele constant εr is small, so that a coaxial-to-waveguide transition can be designed using same technique as that for an air-filled waveguide.
In Figure 3b, YA is the aperture admittance of the matching elements M1 to M3, Y Y0d are the waveguide characteristic impedance, C1 to C3 are the capacitance, L1 is th erture-to-M1 distance, L2 and L3 are the inter-element distance.The same dielectric material D 3 fills the waveguide W. Without D 3 , the level of impedance matching is reduced.Filling the space between the aperture and the first matching element M 1 with a dielectric material also reduces the level of impedance matching.The dielectric constant ε r is small, so that a coaxial-to-waveguide transition can be designed using the same technique as that for an air-filled waveguide.
1.17 and from 0.253 to 0.587, respectively, at 8.17 Figure 3a shows the proposed structure for the broadband matching of an openended rectangular waveguide radiator, whose equivalent circuit representation is shown in Figure 3b.The proposed technique utilizes three matching elements M1, M2 and M3 of a shunt capacitive type.D1 and D2 are a dielectric material of low dielectric constant (εr) employed to support and to precisely position the matching elements M1, M2 and M3.The same dielectric material D3 fills the waveguide W. Without D3, the level of impedance matching is reduced.Filling the space between the aperture and the first matching element M1 with a dielectric material also reduces the level of impedance matching.The dielectric constant εr is small, so that a coaxial-to-waveguide transition can be designed using the same technique as that for an air-filled waveguide.
In Figure 3b, YA is the aperture admittance of the matching elements M1 to M3, Y0 and Y0d are the waveguide characteristic impedance, C1 to C3 are the capacitance, L1 is the aperture-to-M1 distance, L2 and L3 are the inter-element distance.In Figure 3b, Y A is the aperture admittance of the matching elements M 1 to M 3 , Y 0 and Y 0d are the waveguide characteristic impedance, C 1 to C 3 are the capacitance, L 1 is the aperture-to-M 1 distance, L 2 and L 3 are the inter-element distance.
The matching elements are implemented in a thin horizontal metal strip symmetrically placed in the waveguide's E plane.They can be constructed in a printed form on a film substrate [48] or on a thin reinforced PTFE laminate [49].By optimizing the matching element's capacitance and the distance between the matching elements, one can obtain a broadband impedance matching with a start frequency close to the TE 10 mode cutoff.
The mechanism behind the all-shunt-capacitive broadband matching of a waveguide aperture is not simple.The shunt capacitance, as well as the aperture admittance, is frequency-dependent.Thus, computer optimization is one of the efficient ways to find a desired solution for broadband matching.
Firstly, characteristics of the capacitive matching element are studied.Figure 4a shows a capacitive matching element placed inside a rectangular waveguide whose length is 2 L plus the matching element's thickness.Figure 4b shows its equivalent circuit.When the matching element's thickness is very small relative to the guided wavelength λ g , it can be approximated as a purely shunt element.An E-plane-centered horizontal strip of zero thickness in a rectangular waveguide can be represented by a shunt capacitance as shown in Figure 4b.The normalized susceptance B/Y 0 obtained using the variational method is given by [50] where B is the un-normalized shunt susceptance, Y 0 is the characteristic admittance in the equivalent circuit representation of the waveguide, H is the strip height, b is the waveguide narrow-wall height, λ g is the guided wavelength, λ is the wavelength in vacuum and a is the waveguide broad-wall width.
Sensors 2023, 23, x FOR PEER REVIEW The matching elements are implemented in a thin horizontal metal strip sym cally placed in the waveguide's E plane.They can be constructed in a printed form film substrate [48] or on a thin reinforced PTFE laminate [49].By optimizing the ma element's capacitance and the distance between the matching elements, one can ob broadband impedance matching with a start frequency close to the TE10 mode cuto The mechanism behind the all-shunt-capacitive broadband matching of a wave aperture is not simple.The shunt capacitance, as well as the aperture admittance, quency-dependent.Thus, computer optimization is one of the efficient ways to find sired solution for broadband matching.
Firstly, characteristics of the capacitive matching element are studied.Figu shows a capacitive matching element placed inside a rectangular waveguide whose is 2 L plus the matching element's thickness.Figure 4b shows its equivalent circuit.the matching element's thickness is very small relative to the guided wavelength λg be approximated as a purely shunt element.An E-plane-centered horizontal strip o thickness in a rectangular waveguide can be represented by a shunt capacitance as s in Figure 4b.The normalized susceptance B/Y0 obtained using the variational met given by [50] ( ) where B is the un-normalized shunt susceptance, Y0 is the characteristic admittance equivalent circuit representation of the waveguide, H is the strip height, b waveguide narrow-wall height, λg is the guided wavelength, λ is the wavelen vacuum and a is the waveguide broad-wall width.We calculate the scattering parameters S11′ (=S22′) and S21′ (=S12′) of the structure ure 4a and then move the port reference plane to the surface of the matching elem obtain the de-embedded scattering parameters Sij, i.e., where λg is the guided wavelength given in Equation ( 4).We calculate the scattering parameters S 11 (=S 22 ) and S 21 (=S 12 ) of the structure of Figure 4a and then move the port reference plane to the surface of the matching element to obtain the de-embedded scattering parameters S ij , i.e., where λ g is the guided wavelength given in Equation ( 4).From the de-embedded scattering parameters, we calculate the normalized susceptance B/Y 0 of the capacitive element.The reflection coefficient S 11 at the relocated reference plane of Port 1 shown in Figure 4b is now given by since Port 2 is matched when calculating the scattering parameters.In Equation ( 6), Y 0 is the equivalent characteristic impedance of the TE 10 -mode wave.Note that there is no need to explicitly calculate Y 0 , since b c can be obtained from S 11 .For the capacitance calculation, however, the value of Y 0 is required.The un-normalized susceptance B is calculated using Equation ( 6) and the capacitance C at frequency f is now given by the following equation.
The computed capacitance will be frequency-dependent [50].In the proposed technique, it is not necessary to calculate the capacitance of the matching elements.Figure 5 shows the loci at 8−16 GHz of the simulated reflection coefficient S 11 (shown in Figure 4b) of a matched load in parallel with a capacitive matching element (with width a, height H) printed on a film substrate Parylux ® TAHS124500 by DuPont TM [48] (substrate: ε r = 3.4, tanδ = 0.0045 at 10 GHz, h = 0.0045 mm; strip: copper, t = 0.0012) for H/b ratios 0.2, 0.4, 0.6 and 0.8.
since Port 2 is matched when calculating the scattering parameters.In Equation ( 6), the equivalent characteristic impedance of the TE10-mode wave.Note that there is no to explicitly calculate Y0, since bc can be obtained from S11.For the capacitance calcula however, the value of Y0 is required.The un-normalized susceptance B is calculated u Equation ( 6) and the capacitance C at frequency f is now given by the following equa The computed capacitance will be frequency-dependent [50].In the proposed nique, it is not necessary to calculate the capacitance of the matching elements.One can identify the normalized susceptance bc of the capacitive element dir from the admittance Smith chart in Figure 5. Table 1 shows the range of the norma susceptance bc versus the H/b ratio at 8−16 GHz.The range of susceptance values is enough to cover the aperture susceptance shown in Figure 1c of the WR75 waveg open-end radiator.We have not used the theoretical Equation (3) to verify the suscep of the capacitive strip, since our design is not based on the theory but on the com simulation.A capacitive element in a dielectric-filled waveguide behaves in a similar and can be analyzed using the same method as described above.One can identify the normalized susceptance b c of the capacitive element directly from the admittance Smith chart in Figure 5. Table 1 shows the range of the normalized susceptance b c versus the H/b ratio at 8−16 GHz.The range of susceptance values is large enough to cover the aperture susceptance shown in Figure 1c of the WR75 waveguide open-end radiator.We have not used the theoretical Equation (3) to verify the susceptance of the capacitive strip, since our design is not based on the theory but on the computer simulation.A capacitive element in a dielectric-filled waveguide behaves in a similar way and can be analyzed using the same method as described above.Figure 6 shows the dimensional parameters of the proposed matching structure, whose meanings are explained in Table 2.The proposed impedance matching structure can be optimized for |S 11 | < −20 dB over the recommended operating frequency range of a standard rectangular waveguide (b = a/2), which we call a 'Standard-Band Design'.Alternatively, a matching structure can be designed for the widest possible bandwidth over which |S 11 | is less than, for example, −16 dB, which we call a 'Broadband Design'.In Figure 6, P 0 to P 3 refer to the reference planes to be used in the progressive impedance matching analysis.
Figure 6 shows the dimensional parameters of the proposed matching structure, whose meanings are explained in Table 2.The proposed impedance matching structure can be optimized for |S11| < −20 dB over the recommended operating frequency range of a standard rectangular waveguide (b = a/2), which we call a 'Standard-Band Design'.Alternatively, a matching structure can be designed for the widest possible bandwidth over which |S11| is less than, for example, −16 dB, which we call a 'Broadband Design'.In Figure 6, P0 to P3 refer to the reference planes to be used in the progressive impedance matching analysis.
Dimensional parameters of the proposed impedance matching structure.
Table 2. Meaning of the dimensional parameters in the proposed impedance matching structure.

Parameter Meaning a, b, t
Waveguide broad-wall width, narrow-wall height and wall thickness εr Dielectric constant of the material supporting the matching elements and filling the waveguide H1, H2, H3 Strip width of the capacitive matching elements M1, M2 and M3, respectively L1 Distance between the matching element M1 and the waveguide aperture L2, L3 Distances between the matching elements M1 and M2, M2 and M3, respectively P0 Plane of the aperture P1, P2, P3 Plane right after the matching elements M1, M2 and M3, respectively Before optimization via a simulation tool, we carried out a series of parametric analyses on the reflection coefficient of the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.The results are shown in Figure 7.The dielectric constant εr supporting the capacitive elements is 1.13 in the 'Broadband Design', so that the TE10mode cutoff is at 7.40.The reflection coefficient rises above −10 dB from 16.74 GHz onwards.
In a WR75 waveguide filled with a material of εr = 1.13, the first six TE modes in order of increasing cutoff frequency fc are TE10, TE20, TE01, TE11, TE21 and TE30, with fc of 7.40, 14.80, 14.80, 16.55, 20.94 and 22.21 GHz, respectively.Of these modes, the TE20, TE01 and TE21 modes can be suppressed by using structures that are symmetric in the H-plane of the waveguide.The TE11 and TE30 modes, however, can be excited along with the fundamental TE10 mode.Therefore, the waveguide's operating frequency will be greater than the TE10-mode cutoff (7.40 GHz) and less than the TE11-mode cutoff (16.55 GHz), resulting  Before optimization via a simulation tool, we carried out a series of parametric analyses on the reflection coefficient of the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.The results are shown in Figure 7.The dielectric constant ε r supporting the capacitive elements is 1.13 in the 'Broadband Design', so that the TE 10 -mode cutoff is at 7.40.The reflection coefficient rises above −10 dB from 16.74 GHz onwards.
In a WR75 waveguide filled with a material of ε r = 1.13, the first six TE modes in order of increasing cutoff frequency f c are TE 10 , TE 20 , TE 01 , TE 11 , TE 21 and TE 30 , with f c of 7. 40, 14.80, 14.80, 16.55, 20.94 and 22.21 GHz, respectively.Of these modes, the TE 20 , TE 01 and TE 21 modes can be suppressed by using structures that are symmetric in the H-plane of the waveguide.The TE 11 and TE 30 modes, however, can be excited along with the fundamental TE 10 mode.Therefore, the waveguide's operating frequency will be greater than the TE 10 -mode cutoff (7.40 GHz) and less than the TE 11 -mode cutoff (16.55 GHz), resulting in a ratio bandwidth of 2.24 (16.55/7.40).In Figure 7, the frequency range of the analysis is set from 6 GHz to 18 GHz. in a ratio bandwidth of 2.24 (16.55/7.40).In Figure 7, the frequency range of the analysis is set from 6 GHz to 18 GHz.
In Figure 7, we note that the matching element heights H1, H2, H3 and distances L1, L2, L3 between the matching elements have a sensitive effect on the reflection coefficient.The final 'Broadband Design' is obtained with an optimum combination of all of these parameters.Next, an automatic optimization process is employed to find optimum values of H1, H2, H3, L1, L2, L3 and εr.The CST Studio Suite TM 2023 offers a set of built-in optimization algorithms, among which, the "Trust Region Framework" has been used in this study.The ranges of frequency and parameter values in the optimization have been obtained from the parametric analysis carried out in the previous step.
Figure 8 shows the reflection coefficient variation during an optimization of the 'Broadband Design'.The curve in red is the reflection coefficient of the final design of the impedance-matched WR75 waveguide radiator, while the curves in gray are intermediate reflection coefficients.As will be shown later in the 'Standard-Band Design', one can find In Figure 7, we note that the matching element heights H 1 , H 2 , H 3 and distances L 1 , L 2 , L 3 between the matching elements have a sensitive effect on the reflection coefficient.The final 'Broadband Design' is obtained with an optimum combination of all of these parameters.
Next, an automatic optimization process is employed to find optimum values of H 1 , H 2 , H 3 , L 1 , L 2 , L 3 and ε r .The CST Studio Suite TM 2023 offers a set of built-in optimization algorithms, among which, the "Trust Region Framework" has been used in this study.The ranges of frequency and parameter values in the optimization have been obtained from the parametric analysis carried out in the previous step.
Figure 8 shows the reflection coefficient variation during an optimization of the 'Broadband Design'.The curve in red is the reflection coefficient of the final design of the impedance-matched WR75 waveguide radiator, while the curves in gray are intermediate reflection coefficients.As will be shown later in the 'Standard-Band Design', one can find an optimum design for a specific frequency range over which the reflection coefficient is less than a specified value.With a given number of the matching elements, the bandwidth will be smaller for a smaller reflection coefficient.an optimum design for a specific frequency range over which the reflection coefficient is less than a specified value.With a given number of the matching elements, the bandwidth will be smaller for a smaller reflection coefficient.Following the aforementioned procedures, we obtained a second design examplethe 'Broadband Design' whose dimensions are listed in Table 3.In Table 3, h is the substrate thickness and t is the metal-trace thickness of the capacitive matching elements.
Figure 9a shows the structure of the 'Broadband Design', and Figure 9b compares the reflection coefficients of an unmatched radiator and the 'Broadband Design'.In the unmatched case, the reflection coefficient ranges from −10.0 dB at 8.17 GHz to −18.0 dB at 20 GHz.In the 'Broadband Design', the reflection coefficient is less than −16.0 dB at 7.53−16.01GHz (ratio bandwidth 2.13:1).Filling the waveguide with a material of εr = 1.13 lowers the cutoff frequency from 7.87 GHz of the air-filled WR75 waveguide to 7.40 GHz.Following the aforementioned procedures, we obtained a second design example-the 'Broadband Design' whose dimensions are listed in Table 3.In Table 3, h is the substrate thickness and t is the metal-trace thickness of the capacitive matching elements.Figure 9a shows the structure of the 'Broadband Design', and Figure 9b compares the reflection coefficients of an unmatched radiator and the 'Broadband Design'.In the unmatched case, the reflection coefficient ranges from −10.0 dB at 8.17 GHz to −18.0 dB at 20 GHz.In the 'Broadband Design', the reflection coefficient is less than −16.0 dB at 7.53−16.01GHz (ratio bandwidth 2.13:1).Filling the waveguide with a material of ε r = 1.13 lowers the cutoff frequency from 7.87 GHz of the air-filled WR75 waveguide to 7.40 GHz. an optimum design for a specific frequency range over which the reflection coefficient is less than a specified value.With a given number of the matching elements, the bandwidth will be smaller for a smaller reflection coefficient.Following the aforementioned procedures, we obtained a second design examplethe 'Broadband Design' whose dimensions are listed in Table 3.In Table 3, h is the substrate thickness and t is the metal-trace thickness of the capacitive matching elements.
Figure 9a shows the structure of the 'Broadband Design', and Figure 9b compares the reflection coefficients of an unmatched radiator and the 'Broadband Design'.In the unmatched case, the reflection coefficient ranges from −10.0 dB at 8.17 GHz to −18.0 dB at 20 GHz.In the 'Broadband Design', the reflection coefficient is less than −16.0 dB at 7.53−16.01GHz (ratio bandwidth 2.13:1).Filling the waveguide with a material of εr = 1.13 lowers the cutoff frequency from 7.87 GHz of the air-filled WR75 waveguide to 7.40 GHz.To see the mechanism of the capacitive matching, the complex reflection coefficient of the 'Broadband Design' is calculated at 7.53−16.01GHz in the aperture (on the plane P 0 in Figure 6) and just after each matching element M 1 , M 2 and M 3 (planes P 1 , P 2 and P 3 in Figure 6, respectively) and is shown in Figure 10.We observe that the range of variation in the reflection coefficient magnitude is progressively reduced after each matching element.Figure 11 shows the change in the reflection coefficient magnitude in dB with the addition of each matching element.The reduction in the reflection coefficient magnitude is not monotonic.To see the mechanism of the capacitive matching, the complex reflection coefficient of the 'Broadband Design' is calculated at 7.53−16.01GHz in the aperture (on the plane P0 in Figure 6) and just after each matching element M1, M2 and M3 (planes P1, P2 and P3 in Figure 6, respectively) and is shown in Figure 10.We observe that the range of variation in the reflection coefficient magnitude is progressively reduced after each matching element.Figure 11 shows the change in the reflection coefficient magnitude in dB with the addition of each matching element.The reduction in the reflection coefficient magnitude is not monotonic.To see the mechanism of the capacitive matching, the complex reflection coefficient of the 'Broadband Design' is calculated at 7.53−16.01GHz in the aperture (on the plane P0 in Figure 6) and just after each matching element M1, M2 and M3 (planes P1, P2 and P3 in Figure 6, respectively) and is shown in Figure 10.We observe that the range of variation in the reflection coefficient magnitude is progressively reduced after each matching element.Figure 11 shows the change in the reflection coefficient magnitude in dB with the addition of each matching element.The reduction in the reflection coefficient magnitude is not monotonic.Following the same procedures, we obtained a 'Standard-Band Design' shown in Figure 12a.The reflection coefficient is shown in Figure 12b.The frequency range of optimization is set from 10 GHz to 15 GHz, which is the recommended operating frequency range of the WR75 waveguide.The target reflection coefficient is set at −20 dB.The smaller reflection coefficient can be set as a target if the required bandwidth is smaller.ure 12a.The reflection coefficient is shown in Figure 12b.The frequency range of optimization is set from 10 GHz to 15 GHz, which is the recommended operating frequency range of the WR75 waveguide.The target reflection coefficient is set at −20 dB.The smaller reflection coefficient can be set as a target if the required bandwidth is smaller.
Table 3 lists the dimensions of the 'Standard-Band Design' also.The matching elements are implemented on I-Tera MT40 ® substrate by Isola Group (substrate: εr = 3.45, tanδ = 0.0031 @ 10 GHz, h = 0.51 mm; strip: copper, t = 0.0018) [49].In the 'Standard-Band Design', the reflection coefficient is less than −20 dB at 9.89-15.99GHz (1.62:1 bandwidth).Figure 13a shows the complex reflection coefficient loci and Figure 13b the reflection coefficient magnitude at planes P0 to P3, with the frequency from 9.89 GHz to 15.99 GHz.In Figure 13b, we observe that the magnitude of the reflection coefficient even increases before the addition of the final matching element M3.After the final matching element, the reflection coefficient is significantly reduced.Table 3 lists the dimensions of the 'Standard-Band Design' also.The matching elements are implemented on I-Tera MT40 ® substrate by Isola Group (substrate: ε r = 3.45, tanδ = 0.0031 @ 10 GHz, h = 0.51 mm; strip: copper, t = 0.0018) [49].In the 'Standard-Band Design', the reflection coefficient is less than −20 dB at 9.89-15.99GHz (1.62:1 bandwidth).Figure 13a shows the complex reflection coefficient loci and Figure 13b the reflection coefficient magnitude at planes P 0 to P 3, with the frequency from 9.89 GHz to 15.99 GHz.In Figure 13b, we observe that the magnitude of the reflection coefficient even increases before the addition of the final matching element M 3 .After the final matching element, the reflection coefficient is significantly reduced.Table 3 lists the dimensions of the 'Standard-Band Design' also.The matching elements are implemented on I-Tera MT40 ® substrate by Isola Group (substrate: εr = 3.45, tanδ = 0.0031 @ 10 GHz, h = 0.51 mm; strip: copper, t = 0.0018) [49].In the 'Standard-Band Design', the reflection coefficient is less than −20 dB at 9.89-15.99GHz (1.62:1 bandwidth).Figure 13a shows the complex reflection coefficient loci and Figure 13b the reflection coefficient magnitude at planes P0 to P3, with the frequency from 9.89 GHz to 15.99 GHz.In Figure 13b, we observe that the magnitude of the reflection coefficient even increases before the addition of the final matching element M3.After the final matching element, the reflection coefficient is significantly reduced.In Table 4, our design is compared with previous studies.As can be seen in Table 4, the new impedance matching technique proposed in this paper delivers a 1. erating frequency range (10−15 GHz) of the WR75 rectangular waveguide.This enables a realization of a near-field measurement probe with improved characteristics.With reflection <−16 dB, the achieved bandwidth is 2.13:1 (7.53−16.01GHz), offering more than an octave bandwidth which can be useful for wideband/multi-band RF applications and materials measurements.

Conclusions
This paper proposes, for the first time in a published work, a capacitive impedance matching technique for an open-ended rectangular waveguide radiator.The proposed technique uses a three-section capacitive matching circuit of a shunt type.For easy implementation, the capacitive elements are printed on a film or on a laminated substrate, which are supported and precisely positioned by a low-loss dielectric material.The capacitance of the matching elements is frequency-dependent, and computer-based optimization has been employed to achieve a broadband matching.For improved matching performance, it is necessary to fill the space between the aperture and the first matching element with air, and to fill the space between the matching elements and the waveguide interior with a low-dielectric-constant material.Two design examples based on the proposed method show that an open-ended rectangular waveguide radiator can be matched with a 2.13:1 bandwidth for a reflection coefficient <−16 dB and with a 1.62:1 bandwidth for a reflection coefficient <−20 dB, which is more than enough to cover the recommended operating frequency range of a standard rectangular waveguide.Areas of further improvements in the proposed technique are (1) achieving a reflection coefficient <−20 dB with a ratio bandwidth greater than 2:1 and (2) reducing the reflection coefficient to less than −30 dB in the operating frequency range of a standard rectangular waveguide.Nonetheless, considering the wide-ranging applications of the open-ended rectangular waveguide radiator, we expect that the proposed technique will significantly contribute to the art of the related engineering disciplines.

Figure 1 .
Figure 1.Commercial near-field antenna measurement probe by TTI Norte S.L. Co. [28]: (a) view, (b) side view and (c) probe's input VSWR at the coaxial input port.

Figure 1 .
Figure 1.Commercial near-field antenna measurement probe by TTI Norte S.L. Co. [28]: (a) front view, (b) side view and (c) probe's input VSWR at the coaxial input port.

Figure 3a shows the
Figure 3a shows the proposed structure for the broadband matching of an openended rectangular waveguide radiator, whose equivalent circuit representation is shown in Figure 3b.The proposed technique utilizes three matching elements M 1 , M 2 and M 3 of a shunt capacitive type.D 1 and D 2 are a dielectric material of low dielectric constant (ε r ) employed to support and to precisely position the matching elements M 1 , M 2 and M 3 .The same dielectric material D 3 fills the waveguide W. Without D 3 , the level of impedance matching is reduced.Filling the space between the aperture and the first matching element M 1 with a dielectric material also reduces the level of impedance matching.The dielectric constant ε r is small, so that a coaxial-to-waveguide transition can be designed using the same technique as that for an air-filled waveguide.

Figure 4 .
Figure 4. Capacitive matching element in a rectangular waveguide (a) and its equivalent circu resentation (b).
From the de-embedded s ing parameters, we calculate the normalized susceptance B/Y0 of the capacitive ele The reflection coefficient S11 at the relocated reference plane of Port 1 shown in Fig is now given by

Figure 4 .
Figure 4. Capacitive matching element in a rectangular waveguide (a) and its equivalent circuit representation (b).

Figure 5 .
Figure 5. Simulated admittance of a capacitive element in parallel with a matched load in the waveguide versus frequency.H is the capacitive strip height and b is the waveguide narrow height: (a) for H/b = 0.4 and 0.8; (b) for H/b = 0.2 and 0.6.

Figure 5 .
Figure 5. Simulated admittance of a capacitive element in parallel with a matched load in the WR75 waveguide versus frequency.H is the capacitive strip height and b is the waveguide narrow-wall height: (a) for H/b = 0.4 and 0.8; (b) for H/b = 0.2 and 0.6.

Figure 6 .
Figure 6.Dimensional parameters of the proposed impedance matching structure.

Figure 7 .
Figure 7. Reflection coefficient versus the capacitive strip heights H1, H2, H3 and the inter-element distances L1, L2, L3 of the first (a,b), second (c,d) and third (e,f) matching elements in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.

Figure 7 .
Figure 7. Reflection coefficient versus the capacitive strip heights H 1 , H 2 , H 3 and the inter-element distances L 1 , L 2 , L 3 of the first (a,b), second (c,d) and third (e,f) matching elements in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.

Figure 8 .
Figure 8. Reflection coefficient variation during an optimization of the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.The curve in red is the final converged result.

Figure 8 .
Figure 8. Reflection coefficient variation during an optimization of the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.The curve in red is the final converged result.

Figure 8 .
Figure 8. Reflection coefficient variation during an optimization of the 'Broadband Design' of an impedance-matched WR75 waveguide radiator.The curve in red is the final converged result.

Figure 10 .
Figure 10.Reflection coefficient loci in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (a), just after the first (b), the second (c) and the third (d) matching elements.The start frequency (7.53 GHz) is marked with an open circle and the end frequency (16.01 GHz) with a filled circle.

Figure 11 .
Figure 11.Reflection coefficient magnitude in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (P0), just after the first (P1), the second (P2) and the third (P3) matching elements.

Figure
Figure Reflection coefficient loci in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (a), just after the first (b), the second (c) and the third (d) matching elements.The start frequency (7.53 GHz) is marked with an open circle and the end frequency (16.01 GHz) with a filled circle.

Figure 10 .
Figure 10.Reflection coefficient loci in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (a), just after the first (b), the second (c) and the third (d) matching elements.The start frequency (7.53 GHz) is marked with an open circle and the end frequency (16.01 GHz) with a filled circle.

Figure 11 .
Figure 11.Reflection coefficient magnitude in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (P0), just after the first (P1), the second (P2) and the third (P3) matching elements.

Figure 11 .
Figure 11.Reflection coefficient magnitude in the 'Broadband Design' of an impedance-matched WR75 waveguide radiator: in the aperture (P0), just after the first (P1), the second (P2) and the third (P3) matching elements.

Figure 12 .
Figure 12. 'Standard-Band Design' of an impedance-matched WR75 waveguide radiator: (a) structure; (b) reflection coefficients of an unmatched radiator (in blue) and the 'Standard-Band Design' (in red).

Sensors 2023 ,
23, x FOR PEER REVIEW 12 of 15 Following the same procedures, we obtained a 'Standard-Band Design' shown in Figure 12a.The reflection coefficient is shown in Figure 12b.The frequency range of optimization is set from 10 GHz to 15 GHz, which is the recommended operating frequency range of the WR75 waveguide.The target reflection coefficient is set at −20 dB.The smaller reflection coefficient can be set as a target if the required bandwidth is smaller.

Figure 12 .
Figure 12. 'Standard-Band Design' of an impedance-matched WR75 waveguide radiator: (a) structure; (b) reflection coefficients of an unmatched radiator (in blue) and the 'Standard-Band Design' (in red).

Figure 13 .
Figure 13.Impedance loci (a) and reflection coefficient (b) in the 'Standard-Band Design' of an impedance-matched WR75 waveguide radiator: in the aperture (P0), just after the first (P1), the second (P2) and the third (P3) matching elements.In (a), the start frequency (7.53 GHz) is marked with an open circle and the end frequency (16.01 GHz) with a filled circle.

Figure 13 .
Figure 13.Impedance loci (a) and reflection coefficient (b) in the 'Standard-Band Design' of an impedance-matched WR75 waveguide radiator: in the aperture (P0), just after the first (P1), the second (P2) and the third (P3) matching elements.In (a), the start frequency (7.53 GHz) is marked with an open circle and the end frequency (16.01 GHz) with a filled circle.

Table 1 .
Range of the normalized admittance B/Y0 of the capacitive element at 8−16 GHz.

Table 1 .
Range of the normalized admittance B/Y 0 of the capacitive element at 8−16 GHz.

Table 2 .
Meaning of the dimensional parameters in the proposed impedance matching structure.
r Dielectric constant of the material supporting the matching elements and filling the waveguide H 1 , H 2 , H 3 Strip width of the capacitive matching elements M 1 , M 2 and M 3 , respectively L 1 Distance between the matching element M 1 and the waveguide aperture L 2 , L 3 Distances between the matching elements M 1 and M 2 , M 2 and M 3 , respectively P 0 Plane of the aperture P 1 , P 2 , P 3 Plane right after the matching elements M 1 , M 2 and M 3 , respectively

Table 3 .
Dimensions (mm) of the proposed impedance matching structure.

Table 4 .
Comparison with previous studies.