Applications of right/left handed and resonant left handed transmission lines for microwave circuit design, In: Proceedings of 36th European Microwave Conference, Manchester, UK, 2006, pp. 955-958. 12
for electric fields with a non-zero z-component, i.e., the TM mode. It is clear that the expression in (12) is independent of , and equal to (13) for ϭ 0. A graph of the expression in square brackets in (13) is shown in Figure 2.
CONCLUSIONSWe have demonstrated that it is possible to define effective material parameters, which do not vary in space, modeling the macroscopic behavior of heterogeneous problems varying on a small scale. Our reasoning is based on the idea that both the heterogeneous and the homogenized problem should have the same dispersion relations. This leads to identifying the effective material parameters from the phase velocity of each mode. The group velocity, which is concerned with the propagation time through a slab of the medium, will still be the same in the effective medium if the dispersion is properly taken into account (say, through a stationary phase analysis). Thus, when interpreting material measurements, the material properties are more easily extracted from measurements involving relatively thin substrates, say shorter than a wavelength, since the interpretation would then be primarily based on the phase velocity.At the present level of understanding, we can only define temporally dispersive effective material parameters in very simple cases, typically for a fixed propagation direction. Thus, the spatial dispersion seems fundamentally inherent in the problem, and probably only vanishes in the extreme homogenization limit k30. , 1166-1198. 8. R.C. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media
Metamaterial transmission lines can be realized as a combination of transmission line sections with positive and negative dispersion. Such lines, which are also known as right-and left-handed transmission lines, exhibit different dispersion characteristics. Compared to homogeneous networks, the use of cascaded line sections gives additional degrees of freedom for improving the performance of microwave devices. For the design process, LC-equivalents of the right-and left-handed transmission line sections are used. This facilitates the consideration of tunable capacitors, e.g., varactor diodes, to provide frequency tuning of the devices. At the same time, the geometrical dimensions of the devices can be drastically reduced. This paper presents the results of simulation and experimental investigation of a miniature rat-race-ring and a dual-band filter free of spurious response, both manufactured as fully-integrated ceramic multilayer modules based on Low-Temperature Co-fired Ceramics. The design of tunable versions of these devices is also discussed.
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