The Archimedean spiral continues to be a widely studied antenna topology thanks to its broadband impedance and radiation characteristics. These have been investigated experimentally and computationally since their initial development in the late 1950s [1,2], and a number of numerical methods have been developed and utilized in the decades following their introduction to model these broadband attributes. Examples of this include the method of moments based on a thin-wire assumption [3,4], finite-volume time-domain (FVTD) [5], finite-difference time-domain (FDTD) [6,7], finite element method (FEM) [8,9], time-domain finite-element method (TDFEM) [10], and similarly constructed commercial full-wave solvers [11-14].

While these methods have no doubt been collectively successful in their own right, the pursuit of a physically descriptive and rigorous analytical analysis of the Archimedean spiral antenna has in many ways received less attention. This is especially true with regards to the input resistance of the structure. A semi-circular model was first proposed in 1960 [2] for this purpose, and the solution for an infinite number of equiangular spirals was obtained in 1961 [15]. Since then, the development of analytical solutions has seemingly been limited by the geometric complexity presented by the spiral antenna and the many variations it can embody.

It is commonly accepted that the basic operation of the Archimedean spiral can be accurately explained using band theory [1]. This theory states that for the two-wire spiral transmission line with negligible wire width the radiation occurs in annular regions where currents in the neighboring arms are in-phase. The lossy transmissionline model in [16] applies this concept using the radiation resistance of loop antennas as a means to capture the impedance behavior. By extending this explanation to include microstrip [17], stripline [11], and other printed antenna topologies where wire width can no longer be considered negligible, the current distribution will reside on the edge of the conductor and the power will radiated when the two neighboring current distributions are inphase. This leads to the concept of common slot-line mode radiation for the spiral antenna. When the common slotline mode radiation occurs, the field distribution and its structure are similar to the propagating coplanar waveguide mode, for which a closed-form analytical solution can be obtained using conformal mapping. The purpose of this work is to therefore provide physical insight into the radiation mechanism of the two-arm Archimedean spiral antenna and derive an efficient analytical solution for quick results in the design phase.

In this work, a conformal mapping approach is proposed to derive quasi-static closed-form solutions for the characteristic impedances of PCPW, which is used to characterize the input resistance of the balanced two-arm Archimedean spiral antenna operating in its radiation region. For completeness, the radiating mechanism of spiral antenna is reviewed first along with its geometry. This leads to the development of a model for the PCPW assuming the conductor is of negligible thickness. The mapping between physical and finite image domains is discussed next as a more straightforward approach towards deriving the input resistance of the spiral. A comparison is then made with full-wave electromagnetic solutions to validate the accuracy of this approach over a wide range of design parameters. A brief summary on the conformal mapping and resulting characterization of the spiral concludes the discussion.

Figure 1 shows the two-arm planar gap-fed Archimedean spiral antenna considered in this work. It is center-fed with a tapered section based on the Dyson-style balun [18] to reduce the capacitive loading at the central feed location. The boundaries of the two metal arms are defined by four spiral curves using well-known Expression (1), where r is the radius of the spiral curve, θ is the winding angle in radians, is the radius change rate of the spiral, RC is the radius change for one turn of a spiral arm, and r_in is the inner radius of the spiral. The same values of a and r_in are applied on these four curves, and the outer taper of the arm into a point is given by a circular curve defined using an offset angle θ_off with the outer radius r_out in (2) and N turns of the spiral.