# Equivalent Circuits for Microwave Metamaterial Planar Components
## 1 Introduction  
Metamaterial components and antennas for high-frequency applications have been widely studied during the last two decades for their unusual properties suitable for novel applications, and several papers and books are available on this topic. See, for instance, [1; 2; 3; 4]. As is well known, the seminal theoretical paper from [PERSON] inspired this topic [5], and when the technology was mature enough to create structures suitable for the predicted properties of the dielectric constant and the magnetic permeability, a significant increase in scientific contributions and potential applications happened. The main characteristic of the special materials introduced by [5] was the simultaneous negative sign of the dielectric constant and magnetic permeability, originating the excitation of backward waves in the propagation medium. The frequency dispersion is a fundamental requirement to accept a negative sign for both quantities, while still assuring a positive value for the involved energy. In this way, energy flows away from the source, but the wavefront is coming back in the opposite direction. That scientific contribution was followed by experimental attempts to demonstrate such a prediction with media and devices properly designed and excited. Controlling wave propagation in the forward or backward direction allows for additional telecommunication solutions and was considered an advantage for controlling the beam direction of an antenna. Focusing on a beam was also an exciting result for [5], where it is discussed how a negative refractive index modifies the wave path. This discussion was followed by additional theoretical considerations in [6] to predict the possibility of focusing a wave with a flat lens for better imaging resolution. Experimental proof and the practical application of such a lens were first demonstrated in [7]. Any frequency range can benefit from the above-cited effects due to the simultaneously negative value of the dielectric constant and magnetic permeability, and planar components for microwave signal processing based on the metamaterial solution can be improved compared with the classical ones. Looking at the dispersion characteristics offered by metamaterials, additional frequencies can be excited, and zeroth order resonators can be designed. The last concept is helpfulin exciting a zero-value of the wavevector even at a non-zero frequency, and it was also applied to the miniaturization of microwave patch antennas [8; 9]. The circuital modeling of metamaterial devices for high-frequency components and antennas was studied initially to identify the origin of the difference with respect to traditional configurations. It has to be stressed that circuits are familiar to researchers dealing with devices and applied electromagnetics. At the same time, scientists working on material science are usually interested in something other than an equivalent circuit; in propagation characteristics, for example, they are interested in electromagnetic materials. Moreover, an equivalent circuit is only sometimes possible for high frequencies because a distributed structure is rigorously modeled when additional components, namely parasitic contributions, are present. On the other hand, for many configurations and sometimes also for heuristic purposes, the lumped element modeling can be considered valid up to the beginning of the millimeter wave range, i.e., for frequencies not higher than 30 GHz. Most initial papers focused immediately on the lumped inductive or capacitive contributions of specific circuit branches, correlating them to their classical or novel responses, utilizing a transmission line (TL) approach [10]. To model simple TLs, we have two possibilities, neglecting losses and parasitic components and using the so-called T-model: (i) an inductor in line with a shunt capacitor and (ii) a capacitor in line with a shunt inductor.