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The dispersion equation of magnetoinductive waveguides (MIWs) is traditionally solved directly for the propagation constant (<inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>) to obtain diagrams for the attenuation (<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>) and phase (<inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>) constants vs. angular frequency (<inline-formula> <tex-math notation="LaTeX">$\omega $ </tex-math></inline-formula>). Here, we introduce an indirect method of solving the equation, in two steps. By doing so, additional information and insights are provided in the intermediate step, not available via the traditional direct method. The first step of the reported approach splits <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> into attenuation constant in the propagating passband (<inline-formula> <tex-math notation="LaTeX">$\alpha '$ </tex-math></inline-formula>) and the evanescent stopband (<inline-formula> <tex-math notation="LaTeX">$\beta ''$ </tex-math></inline-formula>); and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> into phase constant in the propagating passband (<inline-formula> <tex-math notation="LaTeX">$\beta '$ </tex-math></inline-formula>) and the evanescent stopband (<inline-formula> <tex-math notation="LaTeX">$\alpha ''$ </tex-math></inline-formula>). The resulting four diagrams can then be suitably combined in the second step to obtain <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> diagrams that show excellent congruence with the direct method. The additional information obtained is fundamental in nature and can be utilized to aid in the understanding and design of MIWs. For instance, it can help formulate quantitative criteria to precisely segregate the propagating passband from the evanescent stopband, which further enables a theoretical bound for the propagating mode of operation. The reported indirect method is generic and can be used for any type of MIW structure and any order of coupling.

An Indirect Method for an Alternate Perspective to Dispersion Diagrams of Magnetoinductive Waveguides