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In this contribution, an electric field integral equation (EFIE) formulation is proposed, for the analysis of microwave circuits based on rectangular waveguides with an unlimited number of arbitrarily 3-D-shaped conducting elements. For this purpose, the Lorenz gauge rectangular waveguide Green’s functions are used. Moreover, the Ewald method has been employed to significantly speed up the evaluation of these rectangular waveguide Green’s functions. Strategies are also proposed to switch between different ways of calculating the Green’s functions depending on the source-observation distance along the propagation direction. In addition, the method of moments (MoM) has been applied to solve the EFIE. Following the application of this technique, the impedance matrix resulting from the MoM has been divided into dynamic and static parts, thus reducing the computational time required to obtain the frequency response of practical 3-D microwave circuits by up to a factor of 3 compared with the traditional formulation. On the other hand, a generic expression is derived to evaluate the multimode scattering parameters of rectangular waveguide circuits, independently on the mode used to excite the structure. The evaluation of the electromagnetic fields inside the rectangular waveguides has also been carried out. The proposed technique has been validated by comparison with results provided by commercial full-wave software, such as ANSYS HFSS and CST Studio Suite, showing good agreement and good numerical efficiency.

In this contribution, a novel quart-torus cavity for the design of waveguide filters is presented. A filter breadboard based on this kind of cavity is designed at $Ku$ -band, and then fabricated using additive manufacturing (AM) techniques. The objective is to optimize the footprint distribution of cavity filter structures without compromising their electrical performance. The proposed resonator is built by applying a circular revolution process to a 2-D surface generator. The presented study is performed for circular and elliptical cross-sectional quart-torus cavities. The resonances found in the proposed cavities show similar theoretical unloaded- $Q$ factors ( $Q_{U}$ ) to their equivalent circular waveguide cavities (around $Q_{U}=5000$ ), while exhibiting interesting advantages in terms of physical layout disposition, due to the bending applied. To demonstrate the feasibility of the proposed idea, a tenth-order in-line bandpass filter is designed with elliptical quart-torus (EQT) cavities. Due to the geometrical complexity, a prototype is manufactured using a selective laser melting (SLM) technique, and its measured response has shown very good agreement with respect to the full-wave simulations from the ideal designs. Measured results show a fractional bandwidth (FBW) of 5.3%, with return losses (RLs) better than 20 dB, and unloaded quality factor ( $Q_{U}$ ) above 700. The footprint optimization and flexibility offered by the new concept can make these AM filters a very attractive option for the microwave industry.