Abstract-An initial-boundary value problem for the system of Maxwell's equations with time derivative is formulated and solved rigorously for transient modes in a hollow waveguide. It is supposed that the latter has perfectly conducting surface. Cross section, S, is bounded by a closed singly-connected contour of arbitrary but smooth enough shape. Hence, the T E and T M modes are under study. Every modal field is a product of a vector function of transverse coordinates and a scalar amplitude dependent on time, t, and axial coordinate, z. It has been established that the study comes down to, eventually, solving two autonomous problems: i) A modal basis problem. Final result of this step is definition of complete (in Hilbert space, L 2 (S)) set of functions dependent on transverse coordinates which originates a basis. ii) A modal amplitude problem. The amplitudes are generated by the solutions to Klein-Gordon equation (KGE), derived from Maxwell's equations directly, with t and z as independent variables. The solutions to KGE are invariant under relativistic Lorentz transforms and subjected to the causality principle. Special attention is paid to various ways that lead to analytical solutions to KGE. As an example, one case (among eleven others) is considered in detail. The modal amplitudes are found out explicitly and expressed via products of Airy functions with arguments dependent on t and z.

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The relativistic invariance of a complete set of time-domain modes under Lorentz transformation is proved. The time-domain modes are exhibited as the particular solutions to the system of Maxwell's equations with time derivative. The waveguide surface has the properties of perfect electric conductor. The modal …elds are presented via transverse-longitudinal decompositions. Every …eld component is a product of a modal amplitude depending on longitudinal coordinate and time, and an element of modal basis in the waveguide cross section. The modal basis is speci…ed in a general form. Exact explicit solutions for the modal amplitudes are presented.

In electromagnetics, power flow and energy densities are associated with time‐varying electromagnetic fields. For time‐harmonic waves, such quadratic characteristics are derived using the complex‐valued spatial forms of field vectors resulting in time‐averaged power flow and energy densities. Hence, the opportunity to study dynamic processes of the quadratic characteristics is lost in the time‐harmonic electromagnetics. To overcome this drawback, the authors consider the problem of a waveform propagation in a hollow waveguide via solving the system of Maxwell's equations with time derivative in an energetic space of ‘real‐valued’ functions. The transverse electric and transverse magnetic modal field components are presented as the products of modal basis elements and the modal amplitudes. The vector basis elements are obtained with required physical dimensions, volt per metre and ampere per metre. Thereby, the modal amplitudes are dimension‐free scalar functions. The system of evolutionary equations for the modal amplitudes is derived and solved explicitly. The Klein–Gordon equation (KGE) plays a central role herein. A novel set of real‐valued solutions to the KGE is obtained and applied for analysis of the energetic processes pertinent to the time‐domain signal propagation. The velocity of transportation of the modal field energy is obtained and energetic exchange between the transverse and longitudinal field components is discussed.

ABSTRACT:Capacitively loaded coplanar waveguide (CPW) Key words: coplanar waveguide resonators; open-loop resonators; filters INTRODUCTIONMiniaturization of microwave components is one of the most important requirements in the design of a microwave circuit. Conventional planar transmission line circuits cannot reduce the guide wavelength by more than the factor ͌ r (where r is the relative permittivity of the substrate) from the free-space wavelength. A reduction in size requires a change in the geometry of the microwave component. There have been some reports of miniaturization based on curving conventional transmission lines into coplanar meander lines [1,2]. In addition, a capacitively loaded microstrip loop resonator has been investigated by Hong and Lancaster [3]. They have also used the microstrip open-loop resonators (MOLRs) for cross-coupled planar microwave filter applications [4], and described a microstrip slow-wave OLR [5]. Yu and Chang [6] have proposed compact elliptic-function bandpass filters using MOLRs. Subsequently, Görür et al. [7,8] discussed the effect of the width of the strips of both MOLR and CPW OLR on their resonance characteristics, and showed that these resonators are not only much simpler in structure, but also have a larger slow-wave effect compared to conventional half-wavelength resonators and conventional loop resonators.On the other hand, uniplanar transmission lines such as coplanar waveguide (CPW), coplanar strips (CPS), and slot line (SL) have become preferable over the conventional microstrip lines with their increased use in monolithic or hybrid integrated circuit applications at microwave frequencies, due to their small dispersion, low radiation, easy integration with lumped elements or active devices, high circuit density, and lack of via holes. In addition, the fabrication process of uniplanar circuits is simpler than those of microstrip or double-sided circuits, because only one side of the substrate is used. Consequently, many attractive components using uniplanar structures have been developed. However, in spite of the advantages mentioned above, the uniplanar types of microwave components have not been sufficiently considered in microwave literature, as compared with microstrip ones. In this paper, we investigate the influence of the capacitive fingers between the arms of coplanar OLRs on the resonance characteristics. CAPACITIVELY LOADED CPW OPEN-LOOP RESONATORA conventional CPW rectangular OLR is shown in Figure 1 The extending parts of the open arms, whose lengths are equal to the length l f of a finger, increase the circumference of the OLR and, consequently, the resonance frequency of the OLR shift down because the guided wavelength is increasing. On the other hand, the CPW variable gap OLR (CPW VGOLR) discussed in [7] has similar resonance characteristics. The authors of [7] have shown that the CPW VGOLR, illustrated in Figure 1(c), has a significant slow-wave effect as well as a higher performance, as compared with the conventional half-wavelength CPW an...