Reactance Theorem Research Articles

Input Admittance, Directivity, and Quality Factor of Biconical Antenna of Arbitrary Cone Angle

New analytical expressions and numerical results for the mode coefficients, the directivity, and the quality factor and computationally convenient expressions for the input admittance of a symmetrical biconical antenna of arbitrary length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> and cone angle <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta _{0}$ </tex-math></inline-formula> are presented. The quality factor for a wide-angle biconical antenna is evaluated using three alternative formulations: 1) the evanescent energy stored outside the circumscribing sphere; 2) the total evanescent energy stored in all space; and 3) by equivalent circuit model, and these are all compared with Chu’s lower limit for an ideal antenna. Numerical calculations based on the analytical formula for antenna admittance confirm the conjecture that Foster’s reactance theorem remains invalid even for perfectly conducting antennas. Furthermore, the variation of directivity of a wide-angle biconical antenna is a slowly varying function of its electrical length and is shown to depart significantly from that of a thin cylindrical dipole. Finally, the ratio of directivity to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> of an electrically small biconical antenna is shown to approach 78% of the value of an ideal omnidirectional antenna.

An Alternative Path to Foster's Reactance Theorem and Its Relation to Narrow-Band Equivalent Circuits

Foster's seminal treatise on lossless networks has been published almost 100 years ago and a particularly notable conclusion drawn therein, i.e. that the reactance (and susceptance) functions are always monotonically increasing with frequency, is frequently referred to as Foster?s theorem. In this paper we present two variants for an alternative simple derivation of a stronger form of this theorem, which holds for the driving point reactance (susceptance) of general lossless devices, i.e. also configurations without lumped elements. One version introduces a realizable lumped element equivalent circuit approximating the considered circuit in a narrow band around a particularly considered frequency. It turns out that this avenue of proof also facilitates an alternative validation of the realizability of the so called Foster 1 and 2 realizations.

Design of Dual-Band Frequency-Selective Rasorber

A novel frequency-selective rasorber (FSR) with dual-band transmission and wideband absorption property is proposed in this letter. The FSR is a two-layer structure composed of a lossy resistive sheet at the top and a lossless bandpass frequency-selective surface (FSS) at the bottom, separated by an air spacer. The element of the resistive sheet is a lumped-resistor-loaded metallic dipole with a dual-resonance structure inserted in the center. The dual-resonance structure is realized by three different series LC structures in parallel, based on the Foster reactance theorem. There are two impedance poles between the three impedance zeros of the series LC circuits, where two transparent windows can be obtained for the lossy resistive sheet. The bandpass FSS is a simple dual-band FSS by etching two slots with different shapes and dimensions in a metallic plane. The FSR has realized two transmission bands at 7.7 and 12.6 GHz, and the band with | S 11| < −10 dB is 3.7–11.1 GHz. The design has been validated by full-wave simulation and experimental measurement.

High-Order Modes Analysis of Complex Plasmonic Surface Using the Field-Network Joint Solution

The plasmonic surface is made of metallic surface with deep sub-wavelength decorations, which are designed to mimic the surface plasmon polaritons at lower frequency. Recently, high-order modes in some special structures are analyzed. In order to analyze the dispersion problem of high-order modes of complex lossless structures faster and simpler, a convenient and simple method using the average input wave impedance obtained by a field-network joint solution is proposed. Meanwhile, based on Forster's reactance theorem, high-order modes of the complex structure always exist. Then, the dispersion curves of the high-order modes calculated using proposed method are in perfect match with the simulated results, which can verify the effectiveness and correctness of the proposed method. Additionally, the simulated and experimentally measured electric field distribution of high-order modes have great agreement. Using the proposed method, one can quickly obtain dispersion curve of a plasmonic structure to verify or estimate structure design. Hence, the proposed method may make a big step forward for obtaining the special dispersion relations.

Bandwidth and size limits of achromatic printed-circuit metasurfaces.

Metasurfaces can implement a wide variety of wave-manipulation functions with sub-wavelength layers. They are typically created from resonant elements, thus their refraction properties depend strongly on frequency. The resulting chromatic aberration is undesirable for most applications, motivating recent efforts in the development of achromatic metasurfaces. However, it remains unclear whether there are any physical limits on the achievable operating bandwidth of achromatic metasurfaces. Here we address this question, considering a common microwave metasurface geometry based on three metallic layers, separated by dielectric substrates. Since each of these metallic layers is modeled as an impedance, we apply Foster's reactance theorem to determine the bandwidth over which they are physically realizable using passive, causal and lossless structures. We derive limits for the bandwidth and total size of the metasurface, showing that there is a trade-off between these two parameters. A higher angle of refraction, corresponding to a larger numerical aperture for a lens, further limits the realizable bandwidth. We consider both Huygens' and Omega-bianisotropic metasurface types, and show that the limit is more severe for bianisotropic metasurfaces, making them less suitable for broadband achromatic designs.

High-Impedance Surface Loaded With Graphene Non-Foster Circuits for Low-Profile Antennas

In this letter, a broadband circuit based on the negative differential resistance characteristic of a graphene field-effect transistor (GFET) is proposed. The device disobeys Foster's reactance theorem and performs as a tunable negative capacitor/inductor depending on the load condition. Compared with Linvill's model, this non-Foster circuit (NFC) is simple and contains no positive feedback loop. The simulation results demonstrate negative impedance behavior up to 2 GHz. As the diode-connected GFET used in this design requires no additional gate bias, the circuit diagram is further simplified. This miniaturized circuit is extremely suitable for bandwidth enhancement of high-impedance surfaces (HIS), where limited space is allowed for device placement. An HIS bandwidth of more than 50% has been achieved with the proposed NFCs, enabling the realization of broadband and highly efficient low-profile antennas.

On the propagation of electromagnetic waves in isotropic media that are both electrically and magnetically dispersive

The response of a material to an electromagnetic field is governed by the frequency-dependent dielectric and magnetic parameters ε and μ. An analysis is presented of the storage and transport of energy in an electromagnetic wave passing through an isotropic non-dissipative dispersive medium. This is achieved by the use of electric circuits as analogues of the actual mechanisms within the medium that account for the dispersive behaviour. These analogue circuits are treated in a general way with the aid of Foster’s Reactance Theorem. The energy within a polarized material consists of two parts: the strain energy, made up of the field and elastic energies, and the kinetic energy of the constituent particles. For an electric field e=Ecosωt it is shown that the mean, total, stored energy density is 14ε0E2d(ωε)/dω, an exact relation involving no approximations. Furthermore, the difference between the mean strain energy and the mean kinetic energy densities is 14ε0E2ε. Similar results hold in the magnetic case. The early results obtained by Abraham, and by Brillouin, are confirmed and extended. The models proposed for ε and μ admit of the possibility that either, or both, of these parameters may become negative at some frequencies. It is shown that the Poynting vector and the propagation vector are always parallel to one another provided that ε and μ have the same sign. The speed of energy transmission is calculated and is shown to differ from the group velocity. No support is found for the possibility of negative refraction by materials with negative ε and μ, however it is found that such materials would display unusual refractive properties.

Dispersion requirements of low-loss negative refractive index materials and their realisability

A necessary dispersion requirement is given for the effective relative permeability μr(ω) and permittivity εr(ω) as a function of frequency for low-loss homogeneous dispersive materials and negative refractive index materials, in particular, if they exist at some dimensional scale D. The requirements are valid both for isotropic and uniaxial materials, the latter is defined by material properties that are invariant under a rotation about a principal axis. Using asymptotic requirements at zero frequency and Foster&apos;s reactance theorem, the general bandgap structure is given for low-loss materials. It is shown that such materials can only be achieved if the material exhibits finite bandgaps separating regions of positive and/or negative refractive index regions. In degenerate cases, the bandgaps shrink to zero but low-loss propagation remains impossible in the neighbourhood of the critical frequencies. The effect of small losses are considered and compared with Stockman&apos;s criterion. Losses may be arbitrarily small within a band provided there are no bounds to the size of the real parts of the relative permittivity and permeability at the band edges. A numerical example is given showing the effect of small losses.

On the dispersion characteristics of metamaterial transmission lines

In this paper, a detailed analysis of the dispersion characteristics of metamaterial transmission lines, based on the lumped element T-circuit model is carried out. One of the main relevant characteristics of these artificial lines is the possibility to tailor the phase response. This leads to unique properties which are of interest for microwave circuit design, such as bandwidth enhancement or multiband (dual-band) operation, among others. However, it is shown in this paper that, in spite of the larger number of circuit parameters (as compared to conventional lines), there exist intrinsic limitations that may limit the performance of such metamaterial transmission lines under certain conditions. In this paper these limitations are pointed out from an accurate analysis of the phase response and the Foster’s reactance theorem [Bell Syst. Tech. 3, 259 (1924)]. From the results of this paper, important guidelines for the design of microwave components based on metamaterial transmission lines are inferred. The fabrication and characterization of different metamaterial transmission lines will corroborate the theoretical results.

Characterization and modelling of a microstrip line loaded with complementary split-ring resonators (CSRRs) and its application to highpass filters

A method to characterize and model a microstrip line coupled with complementary split-ring resonators (CSRRs) is investigated. The detailed parameter extraction approach based on three characteristic frequencies is presented. Good agreement between the results of the equivalent circuit model and the full wave simulations supports the effectiveness of the proposed modelling methodology. In particular, it is found that the shunt capacitance in the equivalent circuit has a negative value which appears to contradict the general physical perception. The physical rationality of the problem is discussed and justified. It is found that the negative capacitance is a natural part required to approximate more closely the distributed nature of the CSRR-loaded microstrip line and the whole equivalent circuit still satisfies Foster's reactance theorem. To extract the effective permittivity of the CSRR-loaded microstrip, the dielectric window concept and the effective medium theory are both applied. Both their results show the negative permittivity at the vicinity of the resonance. Finally, the application of the CSRRs in microstip highpass filters is presented to highlight the unique features of the CSRRs and the validity of their equivalent circuit descriptions. Compared with conventional structures, the proposed highpass filters not only have via free structure but also exhibit extremely steep out-of-band rejection. This may lead to useful applications.

Comments on "The Foster Reactance Theorem for Antennas and Radiation Q

For original paper by Geyi et al., see IEEE Trans. Antennas Propag., vol.48, no.3, p.401-8 (March 2000)

Reply to "Comments on 'The Foster Reactance Theorem for Antennas and Radiation Q'"

Presents a reply to the comment by Andersen and Berntsen (IEEE Trans. Antennas Propag., vol.55, no.3, pt.II, p.1013-14, March 2007) on the original paper by Geyi et al. (IEEE Trans. Antennas Propag., vol.48, p.401-8, 2000)

A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab

A method is presented, for the efficient derivation of the dispersion equation associated with electromagnetic bandgap (EBG) structures composed by lossless frequency selective surfaces (FSS) printed on stratified dielectric media. The method, valid for the range of frequency where a single propagating Floquet mode occurs, is based on Foster's reactance theorem applied to an equivalent transmission line network. This theorem implies that the admittance functions of frequency which represent the FSS satisfy the pole-zero analytical properties of the driving point LC admittance functions. By these basic properties and by the full-wave identification of the FSS resonances, an analytical form of the dispersion equation is obtained. This equation is next solved for both surface wave and leaky wave modes by a conventional numerical technique. The results are successfully compared with those from a full-wave analysis.

The Foster reactance theorem and quality factor for antennas

The validity of the Foster reactance theorem for the general lossless or lossy antenna is considered for the straight-wire monopole and half-loop antennas. The finite diameter, lossy or lossless monopole, and half-loop behave as either a series RLC circuit near resonance or a parallel RLC circuit near antiresonance. For the Foster reactance theorem to be valid for the general antenna, the frequency derivative of the antenna's feed point reactance, X'(/spl omega/), must be positive for all values of frequency. This behavior is not consistent with that of a parallel RLC circuit near antiresonance and, therefore, the Foster reactance theorem is not valid for either the lossy or the lossless finite diameter antenna. Simulated and measured reactances of the finite diameter monopole and half-loop are shown to be in good agreement and are shown not to obey the Foster reactance theorem in frequency ranges near antiresonance. It is shown that the commonly used relationship between quality factor and |X'(/spl omega/)| is not valid in frequency ranges near antiresonance.

Is Foster's reactance theorem satisfied in double‐negative and single‐negative media?

AbstractIn this paper, we address the issue of Foster's reactance theorem for the material media in which one or both of the material parameters ε and μ may possess negative real parts. We demonstrate that this theorem is indeed satisfied for a one‐port termination filled with a lossless metamaterial with negative real permittivity and permeability, known as a double‐negative (DNG) medium, or with a lossless metamaterial with either negative real permittivity or negative real permeability, which can be called a single‐negative (SNG) medium. However, in the case of DNG media when the reactive input impedance of such a termination is compared with that of its counterpart filled with a conventional lossless double‐positive (DPS) material, it is found that the two reactances have opposite signs. Similar conclusions can be made for a termination filled with a lossless epsilon‐negative (ENG) material when it is compared with that of its mu‐negative (MNG) counterpart. Therefore, if a one‐port termination filled with a lossless DPS or ENG material possesses inductive input reactance, when the same termination is filled instead with a lossless DNG or MNG material, its input reactance may be capacitive. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 39: 11–14, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11111

The Foster reactance theorem for antennas and radiation Q

The calculation of antenna Q has been an interesting and a controversial topic for years. We first give a rigorous study of antenna Q by introducing a complete description of the complex power balance relation for an antenna system. Using the complex Poynting theorem, we have shown that the antenna is essentially equivalent to a one port lossy network. The Foster reactance theorem is usually stated for a lossless network. The main purpose of this paper is to determine whether the Foster reactance theorem holds for antennas. By making use of a complex frequency domain version of the Poynting theorem, we have shown that the Foster reactance theorem is valid for an antenna. Finally, the Foster reactance theorem for the antenna has been applied to demonstrate the widely held assumption Q/spl ap/1/B, provided Q/spl Gt/1, where B stands for the fractional bandwidth of an arbitrary antenna.

Reply to 'Counterexample and correction to a recent result on robust stability of a diamond of complex polynomials' by H. I. Kang et al. (Nov. 1991 1370-3)

The authors show how a modification of their approach (see ibid., vol.36, p.1168-74, 1989) leads to the stability edges, correctly identified in the above-named work (see ibid., vol.38, p.1370-3, 1991) by K.I. Kang et al., for a diamond of complex polynomials. The feasibility of reduction from 32 to 16 distinguished edges is facilitated via use of Foster's (complex) reactance theorem in network theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comments on "Energy in lossless and low-loss networks and Foster's reactance theorem

For the original article see ibid., vol.36, no.4, p.561-7 (1989). The commenter points out that a general relation for the energy stored in a reactive network driven by a sinusoidal source provided in the above-mentioned paper appears in a book by H.W. Bode (1945). The author agrees that the result of equation (17) in his paper is contained implicitly in equation (9.16) in Bode's book, noting that he was not aware of this result. On the other hand, he points out that the derivation in his paper is more general and applies to both lumped and distributed networks. It is also general in a sense that it applies to any lossless physical system, provided a certain condition is met. He maintains that the derivation in his paper is much simpler and more lucid than the one based on network energy functions methods. >

Energy in lossless and low-loss networks, and Foster's reactance theorem

Energy stored in a lossless network driven by a voltage (current) source is related to a frequency derivative of the network susceptance (reactance). This relation is proven to be valid also for low-loss networks at frequencies far from the network resonance frequencies. Close to the resonance frequency, the stored energy is expressed in terms of the network susceptance (reactance) and the bandwidth of the resonance. A proof of Foster's reactance theorem follows from this consideration. It is also proven that Foster's theorem is a direct consequence of causality. >

Schur stability and stability domain construction

Abstract A discrete version of Foster's reactance theorem is developed and, subsequently, used to delineate necessary and sufficient conditions for a given polynomial with complex or real coefficients to be of the Schur type. These conditions, obtained from the decomposition of a polynomial into its circularly symmetric and anti-circularly symmetric components, facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. These results provide the complete discrete counterpart of recent results for a family of polynomials which are required to be tested for the Hurwitz property.