# A New Observation in Spiral and Curl Antenna Configurations above the Ground Plane
## 2 Spiral Antenna and Analysis Method  
Figure 1 shows the configuration of a two-arm spiral antenna. The antenna is located at height \(h\) above the ground plane. The spiral arm is made of a wire of radius \(\rho\) and is defined by the Archimedean spiral function: \(r=a_{\mathrm{s}}\cdot\phi_{\mathrm{s}}\), where \(r\) is the distance from the spiral center \(o^{\prime}\) (\(x=y=0\), \(z=h\)) to a point on the arm, \(a_{\mathrm{s}}\) is a spiral constant, and \(\phi_{\mathrm{s}}\) is a winding angle starting at \(\phi_{\mathrm{st}}\) and ending at \(\phi_{\mathrm{end}}\). The spiral outer circumference and adjacent arm distance are designated as \(C\) (=2\(\pi a_{\mathrm{s}}\cdot\phi_{\mathrm{end}}\)) and \(d\) (=\(\pi a_{\mathrm{s}}\)), respectively. The antenna is excited at the spiral center \(o^{\prime}\), the center point of a straight part of length \(L_{\mathrm{f}}\) (=2 \(a_{\mathrm{s}}\cdot\phi_{\mathrm{st}}\)).  
Figure 1: Two-arm spiral antenna: (**a**) perspective and top views; (**b**) side view.  
The ground plane is assumed to be infinite, and image theory is applied to the antenna analysis [16]. The current distributions along the antenna arms are determined using the moment method [14; 16], where piecewise sinusoidal functions are used for both the expansion and weighting functions. Based on the obtained current distributions, the radiation characteristics are calculated.  
The wire radius is taken to be \(\rho=0.006\lambda_{0}\) (\(\lambda_{0}\): the free-space wavelength at a test frequency of \(f_{0}\)) [15]. The wire radius \(\rho\) is fixed throughout this paper. The other configuration parameters (\(C,d,L_{\ell}\)) are varied subject to the objectives of the analysis.  
In the analysis, self-developed software is used, where the wire radius is assumed to be small compared with \(\lambda_{0}\). This assumption ensures that the radiation characteristics can be calculated by only the current in the wire axis direction [17]. Note that image theory allows us to analyze the (real) antenna on the \(+z\) side and the image antenna on the \(-z\) side, removing the ground plane. Therefore, the antenna height reduction means that the real antenna approaches the image, resulting in stronger mutual coupling between the real and image antennas.  
The right side of Figure 2, denoted as \"spiral antenna\", shows the optimized parameters (\(C\), \(d\), \(L_{\ell}\)) versus the antenna height \(h\). The optimization is performed so that the antenna radiates a CP wave with an axial ratio of less than 0.1 dB (Section 3 will explain the left side of Figure 2, denoted as \"curl antenna\"). It is found that the adjacent arm distance \(d\) monotonously decreases, and the circumference \(C\) converges to \(1\lambda_{0}\) with a decrease in the antenna height \(h\). The optimized parameters, for example, are \(C\), \(d\), \(L_{\ell}=1.18\lambda_{0}\), \(0.02\lambda_{0}\), \(0.20\lambda_{0}\), respectively, at \(h=0.15\lambda_{0}\). Note that the antenna of \(h<0.25\lambda_{0}\) cannot radiate a CP wave without the straight part of proper length \(L_{\ell}\).  
The frequency bandwidth for a 3 dB axial ratio criterion against the antenna height \(h\) is presented on the right side of Figure 3. The optimized parameters (\(C\), \(d\), \(L_{\ell}\)) on the right side of Figure 2 are used. The axial ratio bandwidth decreases as the antenna height is reduced. A bandwidth of 24% at \(h=0.25\lambda_{0}\) decreases to 8% at \(h=0.15\lambda_{0}\).  
The right side of Figure 4 shows the input impedance \(Z_{\text{in}}=R_{\text{in}}+j\)\(X_{\text{in}}\) versus \(h\). It is observed that the input impedance is almost purely resistive, even when the antenna height \(h\) is reduced. The resistance \(R_{\text{in}}\) becomes higher as \(h\) decreases.  
Figure 2: Configuration parameters optimized for CP radiation versus antenna height \(h\).