Current Reflection Coefficient Research Articles

Natural Frequencies of Long Symmetric Multiconductor Transmission Lines

The goal of this paper is to approximate the natural frequencies of symmetric multiconductor transmission lines (MTLs). The MTLs consist of multiple parallel thin wires above the ground with a long horizontal section in the middle. The wire trajectories at the terminals can be arbitrary as long as the wires are identical. Then, the vector containing the wire currents can be expressed as a function of the current reflection coefficient matrices from the two ports. The reflection coefficient matrices are analytically approximated with an iterative approach. The iterative approach includes high-frequency phenomena and respects the nonuniformity of the wires. These effects are not considered with the classical transmission line theory. In the end, the natural frequencies of the coupled wires are approximated via fixed-point iteration. Frequency splitting, which is common for coupled resonant systems, is observed. This phenomenon is explained using common and differential modes. All analytic results are compared to a numerical reference. Very good agreement is observed.

Transmission Line Reflection Coefficient Including High-Frequency Effects

This paper deals with the analytic determination of the current reflection coefficient for a wire above ground. The trajectory of the wire at the wire terminal is arbitrary and the wire is connected to the ground with an impedance. The reflection coefficient is approximated by an iterative approach for the solution of the so-called mixed potential integral equations. The iteration is initialized by the classical transmission line solution. The tangential electric field on the surface of the thin wire corresponding to the classical solution is then used as a distributed source to determine a correction term for the current. By this way, the scattering of the transverse electromagnetic modes at the discontinuity is included. Very good agreement between the analytic solution and a numerical one is observed. It is shown that broadband matching is not possible with a frequency-independent load. The importance of the incorporation of the high-frequency effects is highlighted by the evaluation of the natural frequencies for two specific wire configurations.

New face of Ramsauer–Townsend effect by using a Quaternionic double Dirac potential

In this article, we have studied scattering of non-relativistic particles from Quaternionic double Dirac delta potential. This scattering is investigated in Quaternionic version of quantum mechanics which is based of quaternions. Probability current density functions, reflection and transmission coefficients for different regions have been calculated analytically and conservation law of probability has been checked. Then an analogy is presented between derived results and Ramsauer–Townsend effect.

Design of a Wideband Circularly Polarized Strip-Helical Antenna With a Parasitic Patch

A wideband strip-helical antenna with a parasitic circular patch for circular polarization is designed in this paper. A strip-helix with 2.5 turns and a pitch angle of 5° is placed over a ground plane and coaxial probe-fed by a 50- $\Omega $ SMA connector. A parasitic patch is electromagnetic-coupled to the open end of the strip-helix with an extremely small distance. By combining the strip-helix with the parasitic patch, two nearby circularly polarized (CP) resonances are produced and a wide axial ratio (AR) bandwidth is obtained. The strip-helix can excite a CP resonance in axial mode, while the parasitic patch can generate another nearby one in fundamental mode. Radiation characteristics of the proposed design, including current distributions, reflection coefficient (S11), AR, and radiation patterns, are studied. Measured results shows that the proposed antenna has an impedance bandwidth (S11 ≤ − 10 dB) of 53%, an AR bandwidth (AR ≤ 3 dB) of 44%, and a peak gain of 9.4 dBi in its volume of $0.43\lambda _{0} \times 0.43\lambda _{0} \times 0.25\lambda _{0}$ , where $\lambda _{0}$ is the wavelength in free space at the center operation frequency.

Lightning strikes to tall objects: A study of wave interactions at the return‐stroke front using a nonlinear transmission line model

AbstractA theoretical study is presented to investigate lightning strikes to towers, with focus on wave interactions occurring at the return‐stroke front due to the arrival of current pulses that propagate upward on the channel after being transmitted from the tower. The lightning channel is represented as a transmission line including corona and nonlinear losses. Analyses for the hypothetical case of a lossless channel considering matched tower and channel impedances show that the arrival of current pulses at the upward moving return‐stroke front leads to an increase in corona currents leaving the channel. This transient process generates current pulses whose arrival at the tower top can be interpreted as the effect of a current reflection at the upward moving front, even though no impedance discontinuity exists at that point if return‐stroke and leader channel properties are assumed the same. In the more realistic case of a lossy channel considering unmatched channel and tower impedances, the nonlinear channel resistance modifies the current pulses that propagate along the channel so that they merge smoothly with the return‐stroke front. In this case, the interaction of the current pulses transmitted from the tower to the channel with the upward moving return‐stroke front does not lead to features that can be clearly interpreted as the result of current reflections at that point in the evaluated conditions. Finally, it is argued that the current reflection coefficient at the tower top should be viewed as a current dependent parameter as opposed to a constant, linear value.

The Influence of CNT Contents on the Electrical and Electromagnetic Properties of CNT/Vinylester

Characterizing electrical and electromagnetic properties of carbon nanotube (CNT)-based nanocomposites is conducted as an important issue in their development process for the space applications. The properties of CNT/Vinylester are experimentally investigated including alternative current and direct current electrical conductivities, permittivity, transmission and reflection coefficients, loss tangent, and skin depth in the Ku frequency band (12.4–18 GHz). The investigated nanocomposites were fabricated using an in␣situ polymerization method. A vector network analyzer is employed for measuring electrical properties. The wt.% of incorporated CNT are chosen between 0.1% and 3% evaluating the influence of CNT contents on the investigated properties. Finally, the equivalent circuit model is used to describe the observed behavior on the basis of a semi-empirical approach.

Transient current pulses in rocket‐extended wires used to trigger lightning

We analyze current, electric field, and optical signatures of the sudden electrical breakdown processes (precursors) that occur at the top of the upward extending, grounded, Kevlar‐covered copper wires used to artificially trigger lightning. For one launch, before the sustained upward positive leader initiated, we estimate that up to 10,000 precursors occurred (one every few hundred microseconds) with peak currents from 1 to more than 100 A. Luminosity at the wire tip was observed for 339 of 410 precursors examined in detail and, in seven cases, discharge channels developed to lengths of 3 to 8 m over times of several hundred microseconds. The measured propagation speeds of current pulses on the triggering wires were less than the speed of light, and decreased from about 2.8 × 108 m s−1 to about 2.3 × 108 m s−1with increasing wire‐top heights from about 80 m to about 340 m. The triggering wire and its grounding system are modeled as uniform transmission lines with model predictions that are consistent with the measured wire‐base precursor current signatures. The modeling shows that (1) the characteristic impedance of the triggering wire, the ratio of the propagating precursor voltage pulse to its associated current pulse, is between 600 and 800 Ω; (2) the 25 m ground rod grounding impedance for the peak precursor current is about 100 Ω, while the DC grounding resistance is 20 Ω; and (3) the current reflection coefficient at ground for peak precursor current is ∼0.9.

Measurements and simulation of the M-component current and simultaneous electromagnetic fields at 60 m and 550 m

Simultaneous measurements of the M-component current (surges superimposed on lightning continuing currents) and the corresponding electromagnetic fields at 60m and 550m from the lightning channel are analyzed and simulated with a two-wave model. The measured results reveal that the M-component current at the bottom of the channel exhibits a V-shape character with a leading edge of 78μs and a trailing edge of 194μs, while the electric field pulses at 60m and 550m have trailing edges faster than leading edges. The peak of the M-component current lags behind the electric field peak by tens of microseconds, when the distance increases to 550m, the disparity of the time shift increases as well. However, the waveshape of the M-component current is similar to that of the magnetic field pulse. The M-component electric fields at 60m and 550m are 1.16kV/m and 0.17kV/m, respectively, and exhibit a logarithmic distance dependence which implies that the M-component charge density increases with height. Additionally, a two-wave model is used to examine the sensitivity of the predicted electric and magnetic fields to the speed and current reflection coefficient variations of the M-component. The simulated results show that the effects are different for the electric and magnetic fields. The M-component speed essentially controls the electric field, but has little effect on the magnetic field. Larger reflection coefficient results in a larger magnetic field, but a smaller electric field.

Compact intracloud lightning discharges: 1. Mechanism of electromagnetic radiation and modeling

On the basis of experimental evidence of multiple reflections and modeling, we infer that, from the electromagnetic point of view, the so‐called compact intracloud lightning discharge (CID) is essentially a bouncing‐wave phenomenon. The shortest radiating‐channel length appears to be about 100 m. Reflections, occurring at CID channel ends, influence the magnitude of the overall CID electric field waveform and are responsible for its fine structure, as well as, by inference, for “noisiness” of dE/dt waveforms and for accompanying HF‐VHF bursts. From modeling the CID as a wave traveling on an elevated vertical transmission line and comparing model‐predicted electric fields with measurements, we estimated that the effective current reflection coefficients at channel ends (additionally accounting for current attenuation along the channel) should be in the range of 0 to −0.5, that the wave propagation speed ranges from 0.3 to 3 × 108 m/s, and the channel length is less than 1000 m. In these calculations, we assumed that the current wave had a zero‐to‐peak risetime of 6 μs and a total duration of 30 μs. Influence of current risetime on field waveforms was also examined, and it was found to be typically in the range from about 2 to 8.5 μs.

Lightning strikes to tall objects: Currents inferred from far electromagnetic fields versus directly measured currents

We have derived far‐field‐to‐current conversion factors for lightning strikes to tall objects for (1) the initial peak current at the object top, (2) the largest peak current at the object top, and (3) the peak current at the object bottom. These far‐field‐to‐current conversion factors are needed for proper interpretation of peak currents reported by lightning detection networks and are each expressed here as the product of (a) the far‐field‐to‐current conversion factor for lightning strikes to flat ground based on the transmission line model and (b) an appropriate correction factor to account for the transient process in the strike object. The correction factors for the three considered cases are: (1) ftall_ini.top = v/(v + c), (2) ftall_top = [1 + ρbot(1 + ρtop)] v/(v + c), and (3) ftall_bot = (1 + ρbot) v/(v + c), where ρbot and ρtop are current reflection coefficients at the object bottom and at the object top for upward‐propagating waves, respectively.

Return stroke current profiles and electromagnetic fields associated with lightning strikes to tall towers: Comparison of engineering models

In this paper, we compare five engineering return stroke models, extended to include the presence of an elevated strike object, namely the Bruce–Golde (BG) model, the transmission line (TL) model, the traveling current source (TCS) model, and the two modified TL models (MTLL and MTLE). The model-predicted current profile along the channel and along the strike object, as well as radiated electric and magnetic fields at different distances are presented and discussed. The comparison is made assuming the same ‘undisturbed current’ (that is the idealized current that would be measured at the tower top if the current reflection coefficients at its both extremities were equal to zero) and the same return-stroke speed. Except for the case of very close (50 m) electric field, it is found that the computed electromagnetic fields associated with a strike to a 168-m tall tower are less model-dependent than those corresponding to a strike to flat ground.

Voltages Induced on an Overhead Wire by Lightning Strikes to a Nearby Tall Grounded Object

The aim of this study was to identify conditions under which the presence of tall strike object can serve to increase or decrease lightning-induced voltages on a nearby overhead wire. We examined the ratios of magnitudes of lightning-induced voltages on the overhead wire for the cases of strikes to a tall object and to flat ground as a function of distance from the lightning channel d, current reflection coefficients at the top of the strike object rhotop and at the bottom of the strike object rhobot, the current reflection coefficient at the channel base (in the case of strikes to flat ground) rhogr, and the return stroke speed v. Lightning-induced voltages were computed using the finite-difference time-domain (FDTD) method. The transmission line (TL) model was used to find the distribution of current along the lightning channel and the strike object. The ratio of magnitudes of lightning-induced voltages for tall-object and flat-ground cases increases with increasing d (ranging from 40--200 m), decreasing rhobot(<1), decreasing rho top (<0, except for the case of rhobot=0), and decreasing v (<c, speed of light). Also, the ratio increases with decreasing the lightning current rise time. Under realistic (expected) conditions such as rhobot=1,rhotop=-0.5, and v=c/3, the ratio is larger than unity (the tall strike object serves to enhance lightning-induced voltages), but it becomes smaller than unity (the tall object serves to decrease lightning-induced voltages) under some special conditions, such as rhobot=1,rhotop =0, and v=c

Diffraction of electromagnetic waves by periodic arrays of rectangular cylinders

Reflection, transmission, and absorption of electromagnetic waves by periodic arrays of conducting or dielectric rectangular cylinders are studied by a finite-difference time-domain technique. Truncated gratings made of lossless and lossy conducting and dielectric elements are considered. Results for surface current density, transmission, and reflection coefficients are calculated and compared with corresponding results in the literature, which are obtained by approximate or rigorous methods applicable only to idealized infinite models. An excellent agreement is observed in all cases, which demonstrates the accuracy and efficacy of our proposed analysis technique. Additionally, this numerical method easily analyzes practical gratings that contain a finite number of elements made of lossless, lossy, or even inhomogeneous materials. The results rapidly approach those for the idealized infinite arrays as the number of elements is increased. The method can also solve nested gratings, stacked gratings, and holographic gratings with little analytical or computational effort.

On the Interpretation of Ground Reflections Observed in Small-Scale Experiments Simulating Lightning Strikes to Towers

Using the finite-difference time-domain (FDTD) method for solving Maxwell's equations, we have simulated small-scale experiments intended to study the interaction of lightning with towers. In these experiments, employing the time-domain reflectometry (TDR), the tower was represented by a conical conductor placed between two horizontal conducting planes, and a relatively high grounding impedance (about 60 /spl Omega/, constant or decreasing with time) of the bottom plane was inferred, based on the assumption that a conical conductor could support propagation of unattenuated waves in either direction. We have shown, using the FDTD simulations, that a current pulse suffers no attenuation when it propagates downward from the apex of the conical conductor to its base, but it attenuates significantly when it propagates upward from the base of the conical conductor to its apex. We show that the current reflection coefficient at the base of the conical conductor is close to 1, so that the equivalent grounding impedance of the conducting plane is close to zero. Our analysis suggests that the relatively high grounding impedance of conducting plane inferred from the small-scale experiments is an engineering approximation to the neglected attenuation of upward propagating waves. When the dependence of cone's waveguiding properties on the direction of propagation is taken into account, the results of small-scale experiments simulating lightning strikes to towers can be interpreted without invoking the fictitious grounding impedance of conducting plane. Representation of a vertical strike object by a uniform transmission line terminated in a fictitious grounding impedance appears to be justified in computing lightning-generated magnetic fields and relatively distant electric fields, but may be inadequate for calculating electric fields in the immediate vicinity of the object. This study was motivated by the growing interest in extending lightning return stroke models to include a tall strike object and calculating associated electric and magnetic fields.

Lightning electromagnetic environment in the presence of a tall grounded strike object

We have analyzed and compared distance dependences of electric and magnetic fields due to a lightning strike to a tall object and due to the same lightning strike to flat ground. In both cases, lightning was represented by a transmission line energized by a lumped voltage source connected at the channel attachment point. The resultant total charge transfer to ground was the same regardless of the presence of strike object. The electric field for the strike‐object case is reduced relative to the flat‐ground case at closer distances from the object. If we assume, in an idealized case, that the return stroke wave front speed is equal to the speed of light, v = c, the current reflection coefficient at the bottom of the strike object ρbot = 1 (grounding impedance Zgr = 0), and that at the top of the object for upward‐propagating waves ρtop = 0 (characteristic impedance of the object is equal to that of the channel Zob = Zch), the ratio of the vertical electric fields on ground for the strike‐object and flat‐ground cases (electric field attenuation factor) will be d/√(d2 + h2), where h is the height of the strike object and d is the horizontal distance from the object. The corresponding ratio for the azimuthal magnetic field is equal to unity. We show that the ratio for either electric or magnetic field increases with decreasing ρbot (ρbot &lt; 1), decreasing ρtop (ρtop &lt; 0 except for the case of ρbot = 0), and decreasing v (v &lt; c), and at larger distances can become greater than unity. We additionally show that the ratio of the far fields for the strike‐object and flat‐ground cases is given by (1 − ρtop) (c/v + 1)/(1 + ρgr), where ρgr is the current reflection coefficient at the lightning channel base when the channel terminates directly on ground. For realistic values of ρtop = −0.5, ρgr = 1, and v = 0.5c, this ratio (far field enhancement factor) is equal to 2.3.

Far-Field–Current Relationship Based on the TL Model for Lightning Return Strokes to Elevated Strike Objects

New general expressions relating lightning return stroke currents and far radiated electric and magnetic fields are proposed, taking into account the effect of an elevated strike object, whose presence is included as an extension to the transmission line (TL) model. Specific equations are derived for the case of tall and electrically short objects. The derived expressions show that, for tall structures (when the round-trip propagation time from top to bottom within the tower is greater than the current zero-to-peak risetime), the far field is enhanced through a factor with respect to an ideal return stroke initiated at ground level. The enhancement factor can be expressed in terms of the return stroke wavefront speed v, the speed of light in vacuum c, and the current reflection coefficient at the top of the elevated strike object. For typically negative values of this top reflection coefficient, lightning strikes to tall towers result in a significant enhancement of the far electromagnetic field. Expressions relating the far electromagnetic field and the return stroke current are also presented for electrically short towers and for very long return stroke current wavefronts. For the case of return strokes initiated at ground level (h=0), these expressions represent a generalization of the classical TL model, in which the reflections at the ground are now taken into account. We describe also simultaneous measurements of return stroke current and its associated electric and magnetic fields at two distances related with lightning strikes to the 553-m-high Toronto Canadian National (CN) Tower performed during 2000 and 2001. The derived expressions for tall strike objects are tested versus obtained sets of simultaneously measured currents and fields associated with lightning strikes to the CN Tower, and a reasonable agreement is found. Additionally, it is shown that the peak of the electromagnetic field radiated by a lightning strike to a 553-m-high structure is relatively insensitive to the value of the return stroke velocity, in contrast to the lightning strikes to ground.

SISTEMA PARA MEDIR PÉRDIDA DE TRANSMISIÓN DE SILENCIADORES ACÚSTICOS EN UN TUBO CON TERMINACIÓN ABSORBENTE ACTIVA

An experimental arrangement for measuring transmission loss of acoustical filters in a tube with active non-reflecting terminal is implemented. A loudspeaker is used as a secondary source to suppress the reflected wave by means of a opposite phase radiation. Primary and secondary sources are fed with pure tones which excite only plane waves. The signals for the primary and secondary sources are generated by a computer, at the same time the computer calculates the current reflection coefficient from the microphone signals. The amplitude and phase of secondary signal are changed by means of an iterative method until a minimal reflection coefficient result. The experimental arrangement is tested by measurements on simple silencers (expansion chambers).

Effect of vertically extended strike object on the distribution of current along the lightning channel

On the basis of a distributed‐source representation of the lightning channel, the mathematical formulations of the so‐called engineering lightning return stroke models are generalized to take into account the presence of a vertically extended strike object. The strike object is modeled as a lossless uniform transmission line, and the reflection coefficients are all assumed to be constant. The distribution of current along the lightning channel for each model is expressed in terms of the “undisturbed” current, object height, and current reflection coefficients at the top and the bottom of the object. The undisturbed current is defined as the current that would flow in the channel if the current reflection coefficients at the extremities of the strike object were equal to zero, that is, the characteristic impedances of the lightning channel and the strike object were equal to each other and equal to the grounding impedance of the strike object. The distributed‐source representation of the lightning channel adopted in this study allows for a more general and straightforward formulations of the generalized return‐stroke models than the traditional representations implying a lumped current source at the bottom of the channel, including a self‐consistent treatment of the impedance discontinuity at the tower top.

Transient response of a tall object to lightning

Experimental data showing the transient behavior of tall objects struck by lightning are reviewed. The influence of this transient behavior, illustrated by simple calculations, on measured lightning current and measured remote electromagnetic fields is discussed. The estimated equivalent impedance of the lightning channel at the time of the initial current peak is appreciably higher than the characteristic impedance of an ordinary tall object (a factor of 3 or so for both the Ostankino and Peissenberg towers and about a factor of 2 for the CN tower). The grounding impedance of a tower is typically lower than its characteristic impedance. Thus, the current reflection coefficient is negative at the top and positive at the bottom of the tower. The similarity of the statistical distributions of subsequent-return-stroke peak currents in: 1) natural downward lightning; 2) natural upward (object-initiated) lightning; and 3) rocket-triggered lightning measured at objects with heights ranging from 4.5 to 540 m suggests that current peaks are not significantly influenced by the presence of a tall object, provided that measurements are taken at the top of the object. This inference is consistent with modeling results of Melander (1984) who showed that the current peaks measured in Switzerland and Italy at the top of 70-m and 40-m towers, respectively, are essentially unaffected by the presence of the towers. If lightning current could be represented by an ideal current source, current at the top of the object would be equal to the source current at all times. The peak current measured at the bottom of a tall object is usually more strongly influenced by the transient process in the object than the peak current at the top. For example, peak currents measured in the lower part of the 540-m Ostankino tower are about a factor of two higher than the peak currents measured near the tower top because the current reflection coefficient at the bottom of the tower is near +1. Observations and modeling suggest that a tall metallic strike object replacing the lower part of lightning channel serves to enhance the lightning-radiated electromagnetic fields relative to the fields due to similar lightning discharges attached directly to ground, this effect being more pronounced for the sharper lightning current pulses.

Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return‐stroke models

We derive exact expressions for remote electric and magnetic fields as a function of the time‐ and height‐varying charge density on the lightning channel for both leader and return‐stroke processes. Further, we determine the charge density distributions for six return‐stroke models. The charge density during the return‐stroke process is expressed as the sum of two components, one component being associated with the return‐stroke charge transferred through a given channel section and the other component with the charge deposited by the return stroke on this channel section. After the return‐stroke process has been completed, the total charge density on the channel is equal to the deposited charge density component. The charge density distribution along the channel corresponding to the original transmission line (TL) model has only a transferred charge density component so that the charge density is everywhere zero after the wave has traversed the channel. For the Bruce‐Golde (BG) model there is no transferred, only a deposited, charge density component. The total charge density distribution for the version of the modified transmission line model that is characterized by an exponential current decay with height (MTLE) is unrealistically skewed toward the bottom of the channel, as evidenced by field calculations using this distribution that yield (1) a large electric field ramp at ranges of the order of some tens of meters not observed in the measured electric fields from triggered‐lightning return strokes and (2) a ratio of leader‐to‐return‐stroke electric field at far distances that is about 3 times larger than typically observed. The BG model, the traveling current source (TCS) model, the version of the modified transmission line model that is characterized by a linear current decay with height (MTLL), and the Diendorfer‐Uman (DU) model appear to be consistent with the available experimental data on very close electric fields from triggered‐lightning return strokes and predict a distant leader‐to‐return‐stroke electric field ratio not far from unity, in keeping with the observations. In the TCS and DU models the distribution of total charge density along the channel during the return‐stroke process is influenced by the inherent assumption that the current reflection coefficient at ground is equal to zero, the latter condition being invalid for the case of a lightning strike to a well‐grounded object where an appreciable reflection is expected from ground.