# Implications of Metamaterial on Ultra-Wide Band Microstrip Antenna Performance
## 3 Metamaterials Design  
The new idea of structured materials known as metamaterials was studied to benefit from its functionality and behavior. This material can produce new types of materials with negative electric permittivity, negative magnetic permeability and a negative refractive index [27]. Metamaterials are classified according to the properties of electric permittivity \(\varepsilon\) and magnetic permeability \(\mu\) into four groups [28]: (1) double-positive (DPS) materials are materials with positive permittivity \(\varepsilon>0\) and positive permeability \(\mu>0\). Most of the materials in nature such as dielectrics belong to DPS; (2) epsilon negative (ENG) materials are materials that have negative permittivity \(\varepsilon<0\) and positive permeability \(\mu>0\) like plasmas. For microwave applications, thin metal wires are used to produce ENG material; (3) Mu-negative (MNG) materials are materials with positive permittivity \(\varepsilon>0\) and negative permeability \(\mu<0\). Gyrotropic material is an example of a material that has negative permeability, in addition, split-ring resonator (SRR) is the most widely used MNG material; (4) double negative (DNG) materials are materials that are not available in nature so it should be produced artificially. In this class both permittivity and permeability are negative \(\varepsilon<0\) and \(\mu<0\). SRR material with MNG properties was used extensively in antenna and microwave applications. In 1999, [PERSON] proposed the double slotted ring resonator structure, where both rings are split [29]. SRRs have many geometric shapes, but in this study, the analysis will be concentrated on circular and square SRR.  
A unit cell of double SRR structure consists of two concentric metallic rings with slots (gaps) at opposite sides; the two rings have a small separation between them and are made of nonmagnetic metal such as copper [30; 31]. Exciting the SRR by a time-varying electromagnetic wave, with a magnetic field (H-field) is perpendicular to the plane of the SRR, this induces a current in the two rings. The two gaps on the opposite side prevent current flow around the rings but allow an induced current to flow on the ring surface and pass from one ring to the other as a displacement current through the capacitive split between the two rings. The high value of distributed capacitance on the gap between the two rings, which store the same amount of charges, but an opposite sign at both sides of the gap, prompts strong coupling between the two rings and lower the resonant frequency. The total capacitance of an SRR structure is the series distributed capacitance on the two rings and the two gaps capacitance [31; 32; 33; 34; 35; 36]. Moreover, SRR inductance is specified according to the length of the rings and it is becoming larger when ring length increased; due to SRR capacitance and inductance, it behaves as a resonant LC circuit [34; 35; 37]. Figure 8 shows how the current flow on circular and square SRR.  
SRR, circular and square, behaves as an LC resonator that can be excited by an electromagnetic field [38]; the equivalent LC circuit model is displayed in Figure 9; L\({}_{\text{s}}\) is the resonator self-inductance and the two SRR rings have the capacitance in halves, C\({}_{0}\)/2 [39]. SRR structure's total capacitance is the series capacitance of its two rings [35].  
Figure 8: Electric current flow on split-ring resonators (SRRs).  
From the equivalent circuit of the SRR, it can be noticed that the structure acts as an LC circuit with the resonant frequency specified by the total inductance and capacitance of the SRR structure [40]. In the next analysis, the designing equations and HFSS analysis for circular and square SRR are presented.