## 2 Transmission Line Equivalent Circuit Model of a Flexible Antenna
#### Bending Impacts on the Parameters and Radiation Characteristics of Flexible Antenna  
Figure 2 illustrates the bending characteristic of the substrate at various bending levels. The bending of the antenna will change the antenna geometry and shape. Generally, bending causes a change in the effective length, thus introducing a change in the impedance characteristics of the antenna (i.e., mainly capacitance), causing a shift in the resonance frequency and a potential shift in the radiation pattern. Discontinuity in the shape or geometry of the antenna mainly leads to a distortion of the electric field between the patch and ground plane [11; 12; 13; 14]. The model in Figure 2 describes the bending levels with additional capacitance, where the length L of the patch is broken up into \(n\) equal segments. This capacitance \(C_{bend}\) is an additional capacitance from the bend of the antenna and varies with the bending curvature, and \(\gamma\) is the propagation constant.  
Previous studies have investigated the bending and crumbling effects of different antennas, but not necessarily polymer-based ones. The effects of bending on electromagnetic radiation performance have been addressed in the literature [6; 15; 16; 17]. At its desired frequency, an antenna should, ideally, absorb all available energy when receiving and purge all its energy when transmitting. Under bent and flat conditions, according to [15], very small differences between input return losses are observed. The radiation pattern, the  
Figure 1: The TL model of a flexible microstrip antenna.  
distribution of the electrical field that propagates from the radiating element of the antenna, is an important characteristic that is mostly dependent on its geography. The crumpling of the antenna may adversely affect the radiation pattern. In [16], the authors present a dual-band coplanar waveguide antenna fabricated on a 48 \(\times\) 33 mm polyimide substrate that operates at the frequencies of 2.45 and 5.75 GHz. Results showed that crumpling demonstrated a negligible impact on the antenna's performance, particularly regarding resonant frequency and gain, and illustrated that up to 5.3 mm crumpling curvature causes only minor changes in the resonant frequency while yielding a high gain. Moreover, this antenna can be integrated into flexible electronic devices that operate in multiple frequency bands.  
Different flexible materials that improve a bent antenna's efficiency and radiation characteristics have been considered in various studies and approaches [18]. Essentially, a bent antenna's performance is dependent on the electrical, physical, mechanical, and chemical properties of the flexible substrate material, which we stated comprehensively in [1]. These properties can be adjusted by the combination of different flexible materials [19]. In [6], the proposal of a UWB flexible antenna for flexible applications observed that a flexible substrate should consequently have high flexibility and robustness, and exhibit high tolerance against twisting and bending [6]. For such flexible antennas, the fabrication process may be a challenge because the properties of the flexible materials are significantly different from the materials used in the fabrication of traditional antennas. In the following section, the fabrication techniques that have been used extensively for flexible antenna designs are described.