(1)

(2)

Using c₁ and c₂ to describe the two edge curves of one arm of the spiral requires and The width of the metal strip can then be defined as (3) and the resulting slot width can be obtained by noting that. The metallization ratio can then be defined by (4).

(3)

(4)

These parameters are well-defined in other works related to the spiral but they have been included here for continuity of nomenclature in later sections, to provide a quick reference for their physical meaning.

The feed region expanded in the bottom of Figure 1 has a gap length g with gap width W_g. More importantly, the design of this feed section remains critical to achieving the desired performance from the spiral antenna. It is discussed here since a simplified gap-fed excitation across g can provide an accurate prediction of radiation patterns (when compared to measured results), but discrepancies are often encountered when using this feed to predict the input resistance. This has historically been attributed to the presence of a feed cable and mutual coupling [19], which often shifts the measured impedance of the test antennas below the theoretically predicted impedance.

Hence, a theoretical self-complementary impedance can be obtained by numerical simulation (for example, in [19]) but difficulties are faced in measuring this same value—the experimental observations in [18,20] illustrate this shift. This suggests that the modeling of the feed structure must be included in the calculation in order to verify with the measured results for impedance [20], or careful consideration must be taken to minimize the capacitive loading it can create. In this work, the ratio of the gap width to gap length is made as small as possible in an effort to reduce the gap capacitance and reduce its effect on the antenna’s impedance.

Although there is no rigorous mathematical description attached to the Band Theory, the theory states that for a two-wire spiral antenna with negligible wire-width, the radiation occurs in annular regions where currents in the neighboring arms are in-phase. The active region where this radiation occurs is similar to a loop antenna whose circumference on the spiral plane corresponds to one wavelength at the operating frequency f in (5), where r_rad is the radius of the radiation region, λ_g is the guided wavelength propagating along the spiral arm, v_p is the guided phase velocity, c is the speed of light, and ε_reff is the effective dielectric constant that the propagating wave experiences (if a dielectric substrate is present).

(5)

3. Periodic Coplanar Waveguide Model

The Band Theory for two-wire spirals can be extended to include microstrip [17], stripline [11], and other printed antenna topologies where metal width can no longer be considered negligible. It is well-known that the current distribution resides on the edges of the metal strips in these transmission lines and a slot-line mode propagates between the arms. The spiral shape creates a path difference between the outer curve and inner curve of one spiral arm, so following the principles of the Band Theory, it can be deduced that the power radiates when the two neighboring current distributions on the same metal strip are in-phase. This is similar to common mode radiation. Figure 2 shows the cross-section of a nominal spiral (top) and the field distribution on these arms when the two neighboring currents on one spiral arm are in-phase (bottom).

Hence, when radiation occurs from the common slotline mode, both the field distribution and its physical structure are similar to the propagating TEM coplanar waveguide mode. With multiple turns, the interdigitated Arm 1 and Arm 2 shown in Figure 2 (bottom) appear as a periodic structure. This leads to the analysis of periodic coplanar waveguide (PCPW). Using the symmetry of the electric field distribution, a set of perfect electric conductor (PEC) walls and perfect magnetic conductor (PMC)

walls can be placed at the middle plane of the slot and the metal strip, respectively. The notional electric field distribution shown in Figure 2 is explicitly valid only for an infinite number of arms with equal current and field distributions. A similar distribution can be assumed for a finite number of arms over the frequencies where common mode radiation occurs.

3.1. Conformal Mapping Analysis in Free Space

The conformal mapping analysis is performed here to calculate the characteristic impedance of PCPW transmission line. The quasi-static assumption can provide an accurate prediction across a wide frequency band [21,22] where RF and microwave spiral antennas operate. The quasi-TEM wave propagation will be maintained by ensuring for Archimedean spiral and PCPW. The metal strips have negligible thickness and are perfect electrical conductors. The per-unit-length (P.U.L.) capacitance C₀ is the only unknown parameter required to obtain the quasi-static characteristics [23]. Conformal mapping techniques are used here since the potential functions and the capacitances between corresponding conductors are preserved after mapping to a simpler domain for which solutions are easily obtained. The general mapping processes used in this work can be found in [24,25]. The characteristic impedance Z₀ of the propagating wave in PCPW when the two neighboring currents are in-phase can be obtained by (6), where ε₀ is the permittivity in free space and is called a complete elliptic integral of first kind with the modulus k₀ expressed in (7).

(6)

(7)

3.2. Conformal Mapping Analysis in a Dielectric Slab

The spiral antenna embedded in a dielectric slab is examined because of the author’s previous research on stripline-fed Archimedean spiral antennas [11]. For the sake of clarity, a single-layered substrate-embedded PCPW is sufficient to explain conformal mapping steps, while the theory presented below is applicable to the general case of multilayered structures or PCPW on top of substrates. Note that the CPW have less field overlap with surface wave modes at low frequencies where the phase velocity of the surface wave mode is much higher than the CPW mode [26].

Figure 3 shows the cross sectional view of spiral antenna embedded in a dielectric slab with thickness of 2h and dielectric constant of ε_r. The total P.U.L. capacitance C of substrate-embedded PCPW can be evaluated by the partial capacitance approximation [27] and modeling the air-dielectric interface as a PMC. The accuracy of this approximation [28] is limited to ε_r being greater than that of background medium. The P.U.L. capacitance C₁ can then be obtained using (8) and the modulus in (9), where is the Jacobian elliptic function with a variable z and a modulus k.

(8)

(9)

The effective dielectric constant ε_reff and the characteristic impedance Z₀ for the quasi-TEM propagating mode in PCPW can then be derived in (10) and (11), respectively.

(10)

(11)

4. Measurement and Discussion

The two-arm gap-fed Archimedean spiral antenna in Figure 4 was fabricated and measured for further verification of the PCPW model and conformal mapping results. The prototype antenna has metallization ratio of with the radius change of RC = 35 mm per turn. The outer radius of spiral is 75.9 mm and consists of two turns. The antenna was fabricated on a square substrate with and a thickness of h = 1.5748 mm [29]. The backside of fabriccated antenna has no conductor plane. The infinite balun design proposed in [18] is adopted as a feed, which requires a large metal arm to carry the feeding cable with less impact on antenna properties. The dummy cable on the other arm is necessary to remain the symmetry of spiral structure, as shown in Figure 4. The length of coaxial cable is 914.4 mm for de-embedding the measured point to the input terminal of spiral antenna.

The conformal mapping analysis presented in previous section can be easily extended to the case of spiral antenna on the top of substrate, whose P.U.L. capacitance C₁ becomes half of (8) and effective dielectric constant is

(12)

The characteristic impedance of spiral antenna can be obtained by substituting (12) into (11). The calculated impedance for the fabricated antenna is 75.7854 Ω.

Figure 5 shows the measured and simulated impedances from 500 MHz to 8 GHz, both of which are around the calculated impedance of 75.7854 Ω (green point) above 1 GHz. The average measured and simulated impedances from 1 GHz to 8 GHz are 74.707 Ω and 72.157 Ω, respectively. As expected, the deviation of impedance curve is attributed primarily to the differences between the ideal model of spiral antenna and fabrication of the prototype (soldering of cables, adhesion of tape to foam, ragged surface of copper, and etc.). The experimental result is however in reasonable good agreement with the calculated results.

5. Conclusion

A simple model and analytical closed-form expressions for the modal resistance of an Archimedean spiral antenna have been obtained. This is the first time that quasi-static analysis is performed on spiral antenna to predict its input resistance operating in its radiation region, allowing physical insight when the spiral arm width

is non-negligible and non-self-complementary. These expressions provide an accurate prediction for the impedance in free space or the band-average impedance with present substrates. The accuracy of the expressions has been verified by measurement and numerical simulation.

REFERENCES