This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY 1 Determining the Stopband of a Periodic Bed of Nails From the Dispersion Relation Measurements Prediction Shoukry I. Shams, Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE Abstract—It is useful to determine the stopband of a bed of nails that can be used for packaging applications. The traditional methodology to identify the cell characteristics is to use the eigenmode solver, which is a numerical method that cannot be validated using a measurement setup. Here, we introduce a mathematical procedure to extract the dispersion relation out of the scattering parameters. The scattering parameters express the transmission and the reflection at the ports, which are functions of the phase constant of the propagating modes inside the device under test. A measurement setup is established by placing several successive cell rows inside a Ku-band rectangular waveguide. The proposed algorithm is validated through examples of well- known dispersion relations. The extracted dispersion relation with the introduced methodology is in good agreement with the one obtained from the eigenmode solver. The ride gap waveguide is used as an application example. Index Terms—Dispersion relation, periodic cells, ridge gap waveguide (RGW). I. I NTRODUCTION R ECENTLY, the ridge gap waveguide (RGW) has been evolving as one of the promising technologies to transfer the electromagnetic signals in high-frequency bands, espe- cially for millimeter- and submillimeter-wave applications. This guiding structure is mainly formed from two parallel plates. The upper plate is a perfect electrical conductor (PEC) plate, while the lower plate is the ridge, which is used to guide the signal in the required path. The ridge is surrounded by an artificial magnetic conductor (AMC) surface. The signal is able to propagate inside the parallel-plate PEC–PEC in the form of quasi-TEM mode. On the other hand, the PEC–AMC boundary conditions outside the ridge are preventing the leakage of the signal. The idea of this structure is proposed for the first time in [1]. This idea is initially developed from the concepts of hard and soft surfaces presented three decades ago [2], [3]. Many advantages are associated with this type of guiding structure as it carries the signal in the form of quasi-TEM mode in an air gap. This leads to having minimal disper- sion and attenuation as there are no dielectric losses inside Manuscript received December 7, 2016; revised January 27, 2017; accepted February 12, 2017. Recommended for publication by Associate Editor A. Orlandi upon evaluation of reviewers’ comments. The authors are with the Department of Electrical and Computer Engi- neering, Concordia University, Montreal, QC H4B 1R6, Canada (e-mail: shoukry.shams@ieee.org). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCPMT.2017.2671518 the structure. The configuration also requires no electrical contacts between its lower and upper parts, which is a major problem in other structures like the rectangular waveguides. These advantages are utilized in many applications such as antennas and antenna arrays [4], [5]. The first step in the design procedure of such a configuration is the cell analysis and design. The cell analysis is performed by the eigenmode solver, which is available in many simula- tion tools. In this simulation technique, the infinite periodic boundary conditions are assumed, and the final solution is the valid values for the propagation constant corresponding to dif- ferent values of frequencies to specify the dispersion relation for various modes inside the required structure. Although this methodology is well established, no experimental setup can be configured to implement these boundary conditions. The RGW measurement is introduced before in many articles [6], [7], but no measurement setup is presented to characterize the unit cell alone. Some work also is introduced to study the effect of these cells in the packaging of other technologies such as the microstrip line packaging [8]–[10]. In the literature, the main concern about the RGW unit cell is always the stopband of the cell and the possible techniques to widen this band [11], [12], which is directly related to the usable frequency band for the whole circuit. Not much attention is given to the cell analysis accuracy and validation. Theoretical approaches are presented in the literature to obtain expressions for the dispersion rela- tions of the bed of nail unit cells (BNUC). Despite neglecting some boundary conditions in these trials, some of these expressions’ results are pretty close to the eigenmode solution results [13], [14], where the fields are solved inside the real BNUC structure. In other presented papers, equivalent surface impedance is assumed to have more simple mathematical manipulations [15]. Some trials are presented in many articles to obtain the mode dispersion relation for the given fields inside the struc- ture [16], [17]. This method was presented to utilize the finite-difference time-domain solution to extract the dispersion relation of all modes, which can never be experimentally implemented as it depends on having full knowledge of the fields inside the device under test (DUT). In practice, this might be possible if we can probe the field without having leakage. This would be limited to cases such as a slotted waveguide or slotted coaxial line. This would be impossible to achieve with the presence of the periodic structure. There- fore, we must depend on deembedding the DUT through the 2156-3950 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.