# Bent and Branched Microstrip-Line Antennas for Circular Polarization
## 2 Bent Line Antenna  
Figure 1a shows an antenna configuration. The antenna has \(N\) periodical bends in a feedline \(F\)-\(T\) located above the ground plane at height \(h\). Each periodical bend consists of a line of length \(d\), a square loop of perimeter \(P\) with a gap \(\delta\), and two segments of length \(L_{\mathrm{s}}\), as shown in Figure 1b. Feedline F-T is excited at the left terminal \(F\) via a vertical wire \(F\)-\(F^{\prime}\) using a coaxial line (see Figure 1e), and the right terminal \(T\) is open-circuited. A matched termination [1,2] is not used to avoid decreased radiation efficiency. The antenna is made of wires with a radius \(\rho\)[4,11].  
It is known that a wire is regarded as a strip having the equivalent width of \(w=4\rho\)[12]. Therefore, a wire analysis implies its equivalent strip one. In our analysis, the strip is located on an air-dielectric substrate to avoid decreased radiation efficiency [4]. An antenna printed on a conventional dielectric substrate may be designed based on the antenna on an air-dielectric substrate, i.e., by changing the configuration parameters in the free-space wavelength to those in the wavelength guided on the conventional dielectric substrate [2,4].  
Using the moment method [10], we design the antenna to radiate a CP beam in a direction normal to the antenna plane in the \(+z\)-axis direction. For this, we chose the loop perimeter \(P\) and path length along the periodical bend (\(d+2L_{\mathrm{s}}+P-\delta\)) to be \(1\lambda_{0}\) and \(2\lambda_{0}\), respectively, where \(\lambda_{0}\) is the free-space wavelength at a test frequency \(f_{0}\). The other configuration parameters are fixed to be \((h,\rho)=(\lambda_{0}/8,\lambda_{0}/200)\)[9,11] throughout this paper. Note that the ground plane is assumed to be perfectly conducting and of infinite extent, and image theory [13] is used in the analysis. Moreover, note that the line length in a periodical bend of #1 is arbitrary (set to be \(d_{1}=\lambda_{0}/8\)).  
We use self-developed software, where wire radius \(\rho\) is assumed to be small compared with \(\lambda_{0}\) (\(\rho<<\lambda_{0}\)). This assumption ensures that the antenna characteristics can be calculated by only the current flowing in the wire axis direction. The current is expanded using piecewise sinusoidal functions. A segment length of approximately \(\lambda_{0}/50\) is used in the analysis.  
Calculations for \(N=12\)[4] show that one of the critical parameters for CP beam formation is segment length \(L_{\mathrm{s}}\). The simulated beam direction and axial ratio versus \(L_{\mathrm{s}}\) are shown in Figure 3, where the other parameters are (\(d\), \(P\), \(\delta\)) = (0.96\(\lambda_{0}\), 1.12\(\lambda_{0}\), 0.06\(\lambda_{0}\)). It is observed that the beam direction becomes \(\theta=0^{\circ}\) for \(L_{\mathrm{s}}=0.06\lambda_{0}\), together with an axial ratio of less than 3 dB. Note that \(L_{\mathrm{s}}\) affects both the beam direction and axial ratio.  
Since the antenna feedline \(F\)-\(T\) has open-circuited termination \(T\), there should be a minimum number of periodical bends for CP radiation. This is shown with small circles in Figure 4. The axial ratio for \(L_{\mathrm{s}}=0.06\lambda_{0}\) is evaluated as a function of \(N\) with the other parameters being fixed. It is found that the antenna radiates a CP wave with an axial ratio of less than 3 dB for \(N>6\).  
Figure 3: Simulated beam direction and axial ratio versus segment length \(L_{\mathrm{s}}\) for a bent line antenna with \(N=12\).  
The increased axial ratios for a few periodical bends are due to increased current reflected from the terminal \(T\)[14]. The reflected current gives rise to the opposite rotational sense of CP radiation to that of traveling current, deteriorating the axial ratio. In the next section, we investigate a line antenna that travels and reflects currents that contribute to the same rotational sense of CP radiation.