Determining a Perturbation Factor to Design Tunable Resonant Cavities in SIW Technology

Abstract This paper presents an effective and novel design procedure employing a novelty named as Perturbation Factor to predict the resonance frequency of resonant cavities in SIW technology perturbed by a shape perturbation. The design procedure can be applied in tunable cavities and bandpass filters with shunt reactance of symmetrical inductive window, and regarding the application technologies, rectangular waveguide resonators and resonant cavities in SIW can be employed due to the equivalence of operation between both technologies. Its straightforward application allows a frequency variation of up to 20% using only one metal post which allows the reduction of the fabrication cost and the preservation of the frequency response across the considered bandwidth. To validate the design procedure, two resonant cavities were fabricated, a rectangular resonant cavity designed to operate at 5.0 GHz and 6.0 GHz and a square resonant cavity designed to operate at 6.0 GHz and 7.2 GHz, and frequency variations of 17.74% and 21.81% were obtained for the rectangular and square cavities, respectively. Still regarding the reached results, was verified a mean error of 1.75% for the predicted resonance frequency, which validates the design procedure. The experimental results are in good agreement with the full-wave computational results and electromagnetic theory.


I. INTRODUCTION
Nowadays, the needs exposed by the recent technological trends are stimulating the development of new ways and concepts of structures operating from lower frequencies to higher frequencies, changing profoundly people's lives.Therefore, an idea comes to our mind; the pursuit of higherperformance communication systems is never-ending.In this context, the Fifth Generation (5G) technology is being widely explored and applied worldwide to improve the performance and capacity of communication systems to be used in Industry 4.0 and its smart manufacturing, Internet of Things (IoT), and mobile communication.
The 5G aims to improve the transmission capacity data, broadcasting multiple and simultaneous links, and support Multiple Inputs and Multiple Outputs (MIMO) systems.Regarding its implementation, the global situation of spectrum for 5G technology can be divided into two spectrums: below 6 GHz and above 6 GHz.Various countries and regions from South America, North America,

Determining a Perturbation Factor to Design Tunable Resonant Cavities in SIW Technology
Ricardo Caranicola Caleffo , Fatima Salete Correra 232 considered shape perturbations and they are responsible for varying the resonance frequency [16]- [17], [19], [22], operating frequency.This section presents a novel design procedure and expression for the resonance frequency, as a function of the position of the metal post, when a resonant cavity in SIW technology is perturbed by a change in its shape, a single metal post inserted in its internal volume and permanently connected at the bottom and upper walls.To obtain an expression for the resonance frequency of an arbitrary cavity in SIW technology perturbed by a metal post connecting its bottom and upper walls, let us consider Fig. 1. diameter of the metal vias, copper conductors, that compose the SIW sidewalls � , length � , longitudinal spacing �, and height ℎ.The dielectric substrate is considered lossless with a dielectric constant � � and nonmagnetic with relative permeability ( � � ) equal to unity, a purely lossless dielectric material.Assuming harmonic fields, then Maxwell's curl equations [16]- [18] in the phasor form for both cases can be written as where � = 2휋� is the radian frequency.Reference [16] shows the mathematical procedure to obtain an exact expression for the new resonance frequency, related to a resonant cavity perturbed by shape perturbation, as follows However, ( 5) is not particularly useful since it is hard to find expressions for � � and � � .
Normally, we only have � � and � � .For weakly perturbed cavities, � � ≈ � � and � � ≈ � � (Perturbation Method [16]- [17]), and assuming ∆� is small, it is possible to obtain an expression for the approximate fractional change in resonance frequency as [16]- [17]: where ∆� � is the magnetic energy removed, ∆� � the electric energy removed, and � � + � � the total stored energy.Equation (6) shows that a shape perturbation affects the stored energy and causes a shift in the resonance frequency.It is important to cite that in the condition of resonance, the stored magnetic and electric energies are equal.If a shape perturbation is made, such as a metal post connecting upper and bottom walls, this will change one type of energy more than the other, for this case, the resonance frequency would then shift to again equalize the energies.Equation ( 6) cannot be applied in strongly perturbed cavities, � � ≠ � � and � � ≠ � � , however, it is useful to explain the phenomenon of resonance frequency variation ( ∆� ) in resonant cavities perturbed by shape perturbations.For a metal post connecting the bottom and upper walls, imposing the boundary condition � × � = 0 at the location and shifting the maximum electric field, there is an increase of the magnetic energy caused by the flow of a current density (� ) through its metal structure.
In general, to obtain an exact value for � � in strongly perturbed cavities using (5), it is crucial knowing expressions for � � and � � , and to get it, it is necessary to use Maxwell's equations, correctly apply the boundary conditions, and consider the position and physical dimension of the metal post in the investigation.Normally, this is a challenging task to perform.Aiming to make it easier, computational simulators, such as the High Frequency Structure Simulator (HFSS), has been widely used to obtain the field distributions in complex structures operating in the microwave spectrum.
Related to HFSS, there is a valuable tool named Fields Calculator that allows solving equations and expressions from the field distributions of complex structures without performing approximations.
The Fields Calculator was used during this investigation to obtain the Perturbation Factor (PF), a novelty developed in this work, aiming to predict the resonance frequency of resonant cavities in SIW technology perturbed by a single metal post inserted in its internal volume, connecting the bottom and upper metal walls.In order to present the PF, let us consider a resonant cavity perturbed by a single metal post, as shown in Fig. 2. In this case, the dielectric substrate has a thickness ℎ and a dielectric constant � � , � � and � � are the metal post position, and � � is the effective width and � � is the effective length of an ordinary resonant cavity in SIW technology [3]- [8].To obtain the PF, it is being considered that the original cavity, the unperturbed cavity, operates in TE101 mode and the metal post is moved along Line 1-3 or Line 2-4, due to its symmetry, in the perturbed cavity, allowing to rewrite (5) as where � is the speed of light in free-space, and defining the normalized distance �, see Fig. 2, as The first term on the right side of (7) will be written as a multiple of the second term, as follows The 푃 � is named as Perturbation Factor and depends on the fields of the original and perturbed cavity, the resonance frequency of the original cavity, and the dielectric constant.According to (9), if Aiming to fully understand the behavior of the PF, seven resonant cavities were designed on a RT/duroid 5880 and a RO4003C, being three square cavities and four rectangular cavities, in SIW technology perturbed by a single metal post each one of them.Fig. 3 shows the Seven-Step Procedure (SSP), a procedure developed in this work to obtain values for the PF.
where � is the width and � is the length of the SIW cavity.In other words, the PFR says that the � � must be 3% of the larger physical dimension of the resonant cavity.Fig. 4(a) shows the 푃 � considering the SSP and PFR.
In summary, Fig. 4(a) shows that the resonant cavity is strongly perturbed for a metal post located at its center, � = 0.50.For a metal post located close to the sidewalls, � = 0 or � = 1, the cavity is weakly perturbed and the PF is minimum, approximately zero.Furthermore, considering the PFR, the � � can be increased up to 20% of � � .For comparative purposes, [10] predicts a perturbed resonance frequency of 18% of � � for a metal post with a diameter of 1.00 mm located at the center of a square cavity.The obtained values for the PF are in good agreement with the field distributions and the boundary conditions imposed by the resonant cavity structure.For the original cavity operating in the TE101 mode, the electric field is maximum at its center and minimum close to the sidewalls; and for the perturbed cavity, the metal post imposes � × � = 0 , shifting the transverse electric field, letting its distribution with a "donut" shape, and changing the stored magnetic and electric energies, see Fig. 5.
Thus, there is an increase of the magnetic energy caused by the flow of a current density through the perturbation structure.
Therefore, applying ( 9) and ( 13) to (7) gives Equation ( 14) is strictly valid for the metal post being moved along Line 1-3, or Line 2-4 due to the symmetry, original cavity operating in TE101 mode, and using the PFR.It is important to cite that (14) can be applied and used in rectangular waveguide resonators and resonant cavities in SIW technology due to the equivalence of operation between both technologies.According to the performed investigation, it was verified that the resonant cavity in SIW technology is strongly perturbed by a metal post having its position varied.Moreover, the diameter of the metal post also influences the operation of the perturbed resonant cavity.So far it has been considered a fixed value for � � , 3% of the larger physical dimension of the resonant cavity, according to the PFR, however, it is straightforward to verify that other values also perturb and influence the operation of resonant cavities.
Aiming to fully understand the influence of the diameter of the metal post in perturbed resonant cavities, various electromagnetic simulations were performed to comprehend how the resonance 238 frequency varies as a function of the diameter of the metal post.The electromagnetic simulations consider a resonant cavity in SIW technology designed on a RO4003C with a thickness of 0.635 mm, width � and length � of 33.2 mm, longitudinal spacing � of 2.54 mm, diameter � of the metal vias that compose the SIW sidewalls of 1.00 mm, and maximum perturbation (� = 0.50).The diameter of the metal post varies from 0.60% up to 9.04% of the larger physical dimension of the resonant cavity, corresponding to diameters ranging from 0.20 mm up to 3.00 mm, respectively.giving the linear trend, dashed line, with Pearson Correlation Coefficient (R 2 ) of 0.9908.The resonance frequency varies from 94,5% up to 110% of 4.19 GHz, where 4.19 GHz is the resonance frequency related to the diameter of 1.00 mm of the metal post given by PFR.Concerning the frequency of 4.1936 GHz, a key point to observe is an error of 1.7% for � � of 1.00 mm, given by the PFR.Equation ( 14) predicts a resonance frequency of 4.12 GHz, demonstrating its effectiveness.
It is important to note that thin metal posts are not capable of strongly perturbing the field distributions, in contrast, thick metal posts change the original structure of the resonant cavity.
Furthermore, diameters of the metal post smaller than 0.60% and bigger than 9.00%, considering the larger physical dimension, create difficulties for manufacturing and should be avoided.Another approach to increase the perturbation is using various metal posts inside the cavity structure [11], [13]- [15], but for this approach, there is no equation to predict the perturbed resonance frequency as a function of the metal posts' location.

III. RESONANT CAVITIES FABRICATION AND EXPERIMENTAL RESULTS
Regarding its applications, two resonant cavities in SIW technology, a rectangular cavity and a square cavity, perturbed by a single metal post each of them, were designed on a RT/duroid 5880 with a thickness of 0.508 mm employing the novel design procedure and then fabricated.Their performances were verified computationally, using the HFSS, and experimentally.A Vector Network Analyzer HP8722 was used to perform the electrical characterization.The rectangular resonant cavity was designed to operate at 5.0 GHz and 6.0 GHz, and the square resonant cavity was designed to operate at 6.0 GHz and 7.2 GHz; the lower resonance frequency is � � and the higher resonance frequency is � � for � = 0.50 and considering the PFR, (14).Concerning the electrical characterization, two steps were considered: the first step was the original cavity, without perturbation, and the second step was the cavity perturbed by a change in shape.For the perturbed cavity, the dielectric substrate was drilled, and the metal post was soldered at the bottom and upper walls.Fig. 7 shows the frequency response of both resonant cavities.
The frequency response of the rectangular cavity is presented in Fig. 7 The experimental results show that the resonance frequencies were tuned by 17.74% and 21.81% for the rectangular and square cavities, respectively.These frequency variations are compatible with the 20% for � = 0.50 predicted by (14).Still regarding the experimental results, was verified an arithmetic mean error of 1.75% for the predicted � � , validating the proposed design procedure.In many practical applications, microwave switches can provide electronic and mechanical frequency tuning [10], [12]- [16], however, the tuning range is reduced by the no-ideal behavior and parasitic effects [10], [16].Thus, the experimental resonance frequency value will be lower than the one predicted by (14).Aiming to compare this work with other previous works, a performance summary is presented in Table I.According to the published works presented in Table I, the novel design procedure allowed a frequency variation (∆�) up to 20% employing only one metal post.Generally, to obtain a ∆� about 20% is used more than one metal post which leads to an increase in the manufacturing cost due to the use of microwave switches and biasing networks.It is important to note that the use of various microwave switches to connect and disconnect the upper and bottom walls of the cavity reduces the frequency tuning and deteriorates the frequency response.Since an open metal post into the cavity does not significantly change its field distributions [10], [16]- [17], two or more metal posts can be used to achieve different operation states using the proposed design procedure, connecting and disconnecting the upper and the bottom walls, allowing various states in the frequency range � � ≤ � � ≤ 1.2� � .Moreover, this work is the first one that presents an investigation proposing an equation to predict the perturbed resonance frequency as a function of the position and considering the physical dimensions of the metal post.

IV. CONCLUSION
A design procedure employing a novelty developed in this work and named as Perturbation Factor is presented to predict the resonance frequency of resonant cavities in SIW technology perturbed by a single metal post inserted in its internal volume, connecting the bottom and upper metal walls.
Another novelty is that the physical dimensions of the metal post were also considered during the investigation.Due to the equivalence of operation between rectangular waveguide resonators and

Fig. 1 .
Fig.1.An original resonant cavity in SIW technology (on the left) and the same cavity perturbed by a metal post (on the right).According to Fig. 1, � � and � � are the electric and magnetic fields, respectively, and � � is the resonance frequency of the original cavity, and � � and � � are the electric and magnetic fields, respectively, and � � is the resonance frequency of the perturbed cavity.The original cavity has a volume � � and surface � � and the perturbed cavity has a volume � and surface �; the metal post has a volume ∆� = � � − � and surface ∆� = � � − �.Its physical dimensions are the width �,

Fig. 2 .
Fig.2.A resonant cavity in SIW technology perturbed by a metal post having its position varied along Line 1 -3.

Fig. 3 .
Fig. 3. Flowchart of the Seven-Step Procedure (SSP).After performing the SSP several times to obtain values for the PF, was observed a relation involving the metal post diameter and physical dimensions of the resonant cavity, such relation is named in this work as Perturbation Factor Rule (PFR), allowing a common PF for all the seven designed cavities.The PFR can be written as � > � → � � � = 0.03,

Fig. 4 .
Fig. 4. (a) The graph of the 푃 � and (b) The graph of the � � 푀푃 � � , in this case, is considered the resonant cavity on the condition of maximum perturbation, or � = 0.50.

237Fig. 5 .
Fig. 5. Field distributions of a resonant cavity in SIW technology perturbed by a metal post through its internal volume, connecting the bottom and upper walls.

Fig. 6 (
photographs of the fabricated rectangular cavity and square cavity, respectively, and Fig.6(c)shows the measurement setup used to perform the electrical characterization.

Fig. 7 .
Fig. 7. (a) Frequency response of the rectangular cavity and (b) Frequency response of the square cavity.

TABLE I .
PERFORMANCE SUMMARY AND COMPARISON WITH OTHER PUBLISHED WORKS.