Theory of a Class of Planar Frequency-Independent Omnidirectional Traveling-Wave Antennas

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2 Theory of a Cass of Panar Frequency-Independent Omnidirectiona Traveing-Wave Antennas Johnson J. H. Wang Wang Eectro-Opto Corporation (WEO) Marietta, Georgia 367 USA Emai: jjhwang@weo.com I. Introduction There is a growing interest in utrawideband (UWB) antennas due to expanding appications in wireess communications, networking, detection, sensing, etc. Athough UWB is defined by IEEE as having a fractiona bandwidth wider than 2%, to many UWB means DC to dayight. In this context the frequency-independent (FI) traveing-wave (TW) antenna is an idea candidate. For many practica appications the antenna must have an omnidirectiona pattern, and be panar in shape with ow-profie and patform-conformabe features. Recenty, such an antenna with a : gain bandwidth (- GHz) was reported []. It was a mode- SMM (spira-mode microstrip) antenna 5.7-inch in diameter and.6-inch in height [2]. The genera theory for this antenna as a TW antenna has been deveoped, and its soution by the method of stationary phase discussed in [3, 4]. This paper presents detais on this theory for the omnidirectiona type (the mode-). II. Formuation of a Panar FI Omnidirectiona TW Antenna The new cass of FI omnidirectiona TW antenna is depicted in Fig.. Panar TW surface S Feed cabe z Figure. Mode- SMM antenna in transmit operation. The panar broadband TW surface S can be a mutiarm spira, and is of a finite and preferaby sma diameter. The ground pane aso has a finite diameter dictated by the θ Matching structure Ground pane x mounting patform. Both panar structures are conforma to the surface of the patform. The use of spherica, cyindrica and rectanguar systems with (r, θ, φ ), (ρ, φ, z) and (x, y, z) coordinates, respectivey, is impicit, with the z-axis being norma to the ground pane. Without oss of generaity, and in ight of the reciprocity theorem, we consider ony the transmit case. It is assumed that a TW wave, specificay a mode- SMM wave [2], has been successfuy aunched. The mode- SMM wave corresponds to the case with spira mode number n, in which a the spira arms are excited in equa ampitude and phase. It is assumed that a traveing wave is argey supported and confined between the panar broadband structure and the ground pane. The far-zone radiation can be readiy derived from a simpe TW theory in cosed form since the source and fieds are sufficienty decouped. We wi formuate the probem in terms of a magnetic current M over the nonconducting part of the antenna surface, S, which is the sot region, instead of an equivaent eectrica current for other FI antennas of non-zero modes [4]. By the image theory, the magnetic fied in the far zone due to M with the conducting surface removed is jkr jke jkrˆ r' H(r) e ds r 2M ( r' ) 4πη S () where k 2π/λ, λ is the waveength of the TW, and η is the free-space wave impedance equa to µ or 2π. The primed and unprimed o /ε o position vectors and coordinates refer to source and fied points, respectivey. The equivaent magnetic current M is given by M ẑ E over the sot region r (2) The integra in Eq. () can be evauated as a Riemann-Stietjes integra by noting that S N f ( r; ρ, φ ) ρ dρ dφ F( θ, φ )S (3)

3 where f denotes an arbitrary but integrabe function, and we define + ρ S g( r ; ρ ' )dρ' (4) ρ S in Eq. (4) is proportiona to the far-zone radiated fied of the we-known thin circuar annuar sot with uniform aperture excitation. This is both physicay and mathematicay significant, and we wi take advantage of this reevance by examining the annuar sot first. III. The Annuar Sot Antenna as a Buiding Bock Fig. 2 depicts a thin circuar annuar sot on a ground pane in the x-y pane, which is a buiding bock of this theory. The thin circuar sot has a mean radius a, excited by a uniform radia eectric fied parae to ρ, with a resuting votage V across the sot aperture. Fig. 2. A thin annuar sot on x-y pane. It can be shown that the far-zone radiation of this annuar sot is fuy represented by a magnetic fied having ony a φ component as foows: jav exp( jkr) H φ ( φ) 2πλr 2π cos ( φ -φ' ) exp[ jka sinθ cos( φ -φ' )] dφ' (5) The integra in Eq. (5) can be evauated exacty as H φ av exp( jkr) J( kasinθ ) (6) 6λr where J denotes a Besse function of the first kind of order. For a sma sot (a λ/(2π)), we have V exp( jkr) A Hφ sinθ (7) 2 6r λ as a where A πa 2. To our knowedge these equations have appeared ony in [5, 6], but with some errors. We beieve references 5 and 6 are in error since Eq. (6) can be verified independenty by invoking duaity from the case of an eectric circuar oop antenna based on Maxwe equations having fufedged presentation of magnetic sources [7]. Other errors in annuar sots in the st edition of the widey used antenna handbook [5] incude Figs. 8- (b) and 27-43, which were argey eiminated in its 3 rd edition. Corrections for these are aso ong overdue. An eectricay sma annuar sot has a desired eevation pattern, sin θ, according to Eq. (7). However, the beam becomes tited and narrowed, with additiona beams emerging, as the frequency increases. Thus, its usefuness as a wideband omnidirectiona antenna is imited. Fig. 3 shows the cacuated ange of the eevation beam peak as a function of ka using Eq. (6). (Ony two beams are incuded in Fig. 3.) As can be seen, for annuar sots of a sma diameter a, the pattern is simpy sin θ as given by Eq. (7), peaked at θ 9 o, which is idea for omnidirectiona coverage. Note that around ka 2 the first beam tits up rapidy as ka increases. It is aso worth mentioning that omnidirectiona antennas, such as a monopoe or annuar sots, mounted on a finite ground pane of radius b have an increasing beam tit in eevation as kb decreases. These two beam tit mechanisms were miseadingy presented in Fig in the st ed. of [5]. Beam peak ange Beam peak θ Fig. 3. Beam peaks versus ka for an annuar sot. ka Beam peak θ 2

4 IV. The TW Antenna as an Array of Annuar Sots Pus Edge Sot Based on the discussions above, the farzone radiated fieds of the panar FI omnidirectiona antenna in Fig. is given by H H φ N N H ( θ, φ) av exp( jkr) J 6λr θ e ( ka sin ) jψ (8) where ψ and V denote, respectivey, the phase and ampitude of the votage of the equivaent annuar sot eement. Thus the far-zone radiation of the TW surface S can be fuy represented by its magnetic fied, which ony consists of a φ component, and which is the superposition of the fieds from the concentric annuar sots of varying ampitude and phase, V and ψ, respectivey, aong ρ. Therefore, the far-zone radiated fied of this TW antenna is the superposition of the fieds due to the eements of a concentric array of equay spaced annuar sots, pus a circuar edge sot at the rim of the spira. V. Radiation Zones and Radiated Fieds Consider the case in which the panar structure in Fig. is a sef-compementary 2- arm Archimedean spira. The center ines of a two-arm Archimedean spira on the S pane are ρ bφ φ ε [φ, φ t ] (9a) ρ 2 b(φ π) φ ε [φ + π, φ t + π] (9b) The two feed points are at φ φ and (φ + π) for arms # and #2, respectivey, with equa inphase votages V. The arc engths L aong spira arm # from its feed point to (ρ, φ), and L 2 aong the adjacent spira arm #2 from its feed point to (ρ 2, φ), respectivey, are given by z (Zenith) b 2 φ + + sinh φ 2 (a) L [ φ ] φ φ L 2 b 2 φ π [ φ + φ + sinh φ] 2 φ (b) Aso, between adjacent arms, ρ bφ b(φ π/2) bπ/2 (a) L (L L 2 ) ~ ρ ρ/b ~ πρ as a (b) The phase change ψ J of the TW fieds between adjacent spira arms is given by ψ J (2π/λ) L (2) The series in Eq. (8) can be approximatey evauated by incuding ony a few terms that have significant in-phase contribution. Physicay, this means incuding ony sots at the radiation zones. For a 2-arm spira, the radiation zones are at circumferences where ψ J π/2 between adjacent arms, so that an equivaent annuar sot is formed over three adjacent arms (or two adjacent annuar sots), with a resuting votage V. For a 4- arm spira, at radiation zones ψ J π/4 between five adjacent arms or four adjacent sots. Thus, radiation zones are at radia distances ρ r given by ρ r λ/(4π) + nλ/π for 2-arm spira (3a) ρ r λ/(8π) + nλ/π for 4-arm spira (3b) where n,, 2, 3, and the edge of the spira. Fig. 4 shows measured eevation patterns for a 4-arm mode- SMM antenna of Fig., 5.7-inch in diameter and.6-inch in height operating over.5- GHz, described in []. As can be seen, the patterns are consistent with Eq. (8) and Fig. 3 with regard to the beam tit and beam peaks as a/λ increases from.2 to 2.42 (or ka from.75 to 5.8). For exampe, at 2.5 GHz, ka 3.75, the beam peaked at θ ~ 42 o can be cacuated from two terms in Eq. (8) the fieds from two radiation zones n, 2 in Eq. (3b). 5 db/div θ.5 GHz 2.5 GHz. GHz Fig. 4. Measured eevation patterns.

5 VI. Impedance Matching Under the condition that higher-order modes are suppressed, the panar TW surface S can be considered a oaded surface consisting of both a reactive component and a resistive component, the atter accounting for possibe radiation through the nonconducting (sot) region. At the edge of the surface S, there is a circuar sot from which the residua power is radiated. In the region where the panar surface structure S is a soid conductor, the TW antenna can be viewed as a circuar radia waveguide of height h, and its characteristic impedance Z at ρ for the m n mode is given by Z 6h/ρ (4) Note that Z changes with ρ, the distance from the center of the radia waveguide, but is independent of frequency. In regions where the surface S is sefcompementary, haf metaic and haf sot, its characteristic impedance Z c can be obtained, by invoking the principe of superposition and duaity in the context of Maxwe equations formuated with fu-fedged presentation of magnetic sources [7], as Z c ~ 2 Z (5) Each annuar sot can be represented by a radiation resistance pus a sma capacitance. Various techniques are avaiabe to suppress higher-order modes to ensure that the simpe transmission ine mode is an adequate representation for the radia waveguide with generay reactive TW surface S. Eqs. (4) and (5) are consistent with our experimenta observations. Indeed, utrawideband impedance matching over : bandwidth has been demonstrated, with SWR <.3 mosty, rising to ~ 2. at high and ow frequencies, over a : bandwidth (at - GHz and other frequency ranges) with breadboard and brassboard modes. Some of the resuts have been presented in []. It is worth commenting that the sefcompementary geometry of the TW surface S, the frequency-independent impedance of Z, and the fu-fedged duaity formuation of the Maxwe equations, a contributed to the insight that utra-wideband impedance matching for the panar FI TW antenna in Fig. is feasibe. VII. Concusions The theory for a new cass of panar frequency-independent omnidirectiona antenna has been deveoped and found to be consistent with measured data, with better agreement than many of those obtained by brute-force numerica computation. More importanty, the theory is usefu for design and synthesis because of its simpe cosed-form soution, which has direct and cose reevance to the physica parameters and performance of the panar frequency-independent omnidirectiona antenna. A buiding bock for the theory is that for the annuar sots, which has some errors in the iterature. These errors in the equations and figures were aso discussed and corrected in this paper. References. J. J. H. Wang, C. J. Stevens, and D. J. Tripett, Utrawideband Omnidirectiona Conformabe Low-Profie Mode- Spira-Mode Microstrip (SMM) Antenna, 25 IEEE Antennas and Prop. Symp., Washington, DC, Juy J. J. H. Wang, The Spira as a Traveing Wave Structure for Broadband Antenna Appications, Eectromagnetics, 2-4, Juy- August 2; aso U.S. Patents #5,58,7, Apri 6, 996, and #5,62,422, Apri 5, J. J. H. Wang, The Physica Foundation, Deveopmenta History, and Utra-wideband Performance of SMM (Spira-Mode Microstrip) Antennas, 25 IEEE Antennas and Prop. Symp., Washington, DC, Juy J. J. H. Wang, Theory of Frequency- Independent Antennas as Traveing-Wave Antennas and Their Asymptotic Soution by Method of Stationary Phase, 25 Internationa Symposium on Antennas and Propagation (ISAP), Seou, Korea, August 3-5, H. Jasik, editor; Antenna Engineering Handbook, McGraw-Hi, New York, st ed., 96; aso 3 rd edition, R. C. Johnson, editor, A. A. Pistokors, Theory of the Circuar Diffraction Antenna, Proc. IRE, pp. 56-6, January J. J. H. Wang, Generaized Moment Methods in Eectromagnetics Formuation and Computer Soution of Integra Equations, pp. 6-7, 5-7, Wiey, New York, 99.