## COURSE OF THE GREAT CIRCLE PATH

The basic path for airways is a direct path between two fixes, which may be a VOR station, an NDB station, or simply a geographical location. In terminal area procedures the most common path is defined by an inbound course to a fix. The RNAV equipment approximates such paths as segments of a great circle. Considering the case of a path defined as a radial of a VOR, the actual true course depends upon the alignment of the VOR transmitter antenna with respect to true north. The angular difference between the zero degree radial of the VOR and true north is called the VOR declination. When the VOR station is installed, the 0° VOR radial is aligned with the magnetic north so the VOR declination is the same as the magnetic variation at the station at the time of installation.

Magnetic variation is the difference between the direction of north as indicated by a magnetic compass and true north defined by the reference ellipsoid. As such, it is subject to the local anomalies of the magnetic field of the earth. The mag­netic field of the earth varies in a systematic manner over the surface of the earth. It is much too complex to be defined as a simple bar magnet. The magnetic field is also slowly chang­ing with time in a manner that has some random characteris­tics. Every 5 years a model, both spatial and temporal, is de­fined by international agreement using worldwide data. A drift of magnetic variation of 1° every 10 years is not uncom­mon on the earth. The model is defined in terms of spherical harmonic coefficients. Data from this model are used by sen­sors and RNAV systems to calculate the magnetic variation at any location on the earth. In particular, inertial navigation systems are references to true north and produce magneti­cally referenced data by including the local magnetic varia­tion as computed from a magnetic variation model.

Because the magnetic variation of the earth is slowly changing, a VOR whose 0° radial is initially aligned with the magnetic north will lose this quality after a period of time. This discrepancy between the VOR declination and the local magnetic variation is one reason for ambiguity in course values.

As one progresses from along the great circle path, the de­sired track changes due to the convergence of the longitude lines and due to the magnetic variation. Figure 6 shows the effect of position on true and magnetic courses. The true course at the fix, CT, is different from the true course, CT, at the aircraft because the longitude lines are not parallel. The difference in the magnetic courses is the result of the differ­ence in the true courses together with the difference in mag­netic variation at the two locations.

For the pilot, an important piece of information is the mag­netic course to be flown to stay on the great circle path. With no wind, when on track, the current magnetic heading of the aircraft should agree with the displayed magnetic course. To achieve this goal, the RNAV equipment first computes the true course of the desired path and then adjusts it for local magnetic variation.

On the aeronautical charts, the magnetic course of the path is defined as the termination point of the path. When the aircraft is some distance from the termination point, both the true course and the magnetic variation are different. This causes the FMS to display a magnetic course at the aircraft that is different than that of the chart. As explained above,

 True

this difference is necessary to provide a display of course that is consistent with the magnetic heading of the aircraft as it progresses along the path.

## LEAKY-WAVE ANTENNAS

Leaky-wave antennas (LWAs) constitute a type of radiators whose behavior can be described by an electromag­netic wave (leaky wave) that propagates in guiding structures that do not completely confine the field, thus allowing a continuous loss of power to the external environment (leakage).

According to IEEE Standard 145-1983, a leaky-wave antenna is “an antenna that couples power in small increments per unit length either continuously or discretely, from a traveling wave structure to free space.”

## Transmembrane Potential

 First post shock beat

For the shock to cause either a new action potential to be triggered or to prolong an action potential, it must alter the transmembrane potential. It has been estimated that only about one quarter of the total current traversing the heart crosses the membrane to enter the cells (72). Since the defi­brillation electrodes are located extracellularly, current from the shock that enters myocardial cells in some regions must exit the cells in other regions. These currents, which flow through the cell membrane, will introduce changes in the transmembrane potential that include depolarization or hy­perpolarization during the shock pulse. Several mathematical formulations have been proposed to describe which regions of the heart are depolarized and which are hyperpolarized dur­ing shocks from a particular defibrillation electrode configu­ration. These formulations include the cable equations, the sawtooth model (73,74), the bidomain model (75,76), and the secondary source model (77). In their simplest form, these for­mulations incorporate the extracellular and intracellular spaces as low resistance media and the membrane as a high resistance in parallel with a capacitance. Therefore, these simple case models incorporate only passive myocardial prop­erties.

 S1-S2 (MS)

8.4 V/cm, 2 ms

I

100 ms

(b)

Figure 11. (a) Recordings that illustrate the response to an S2 stimulus of 1.6 V/cm oriented along the fibers. The S1-S2 stimulus intervals for each of the responses are indicated to the right of the recordings. The responses are markedly different even though the change in S2 timing was only 3 ms. An S1-S2 interval of 222 ms caused almost no response, whereas an interval of 225 ms produced a new action potential. (b) A range of action potential extensions produced by an S2 stimulus generating a potential gradient of 8.4 V/cm oriented along the long axis of the myofibers. The recordings were obtained from the same cell as (a). The action potential recordings, obtained from one cellular impairment, are aligned with the S2 time. An S1 stimulus was applied 3 ms before phase-zero of each recording. The longest and shortest S1-S2 intervals tested, 230 ms and 90 ms respectively, are indicated beneath their respective phase-zero depolar­izations. The S1-S2 intervals for each response after S2 are indicated to the right. Reproduced with permission from the American Heart Association (59).

nels in the membrane. Because the extracellular space throughout the body is primarily resistive, with very little re­active components, the defibrillation shock appears in the ex­tracellular space of the heart almost immediately and without significant distortion. For example, a shock in the form of a square wave given across the defibrillation electrodes will ap­pear almost immediately as a square wave in the extracellu­lar space of the heart. Because of the capacitance and the ion channels of the membrane, however, phase delays and alter­ations of the appearance of the shock wave occur in the trans­membrane potential. For example, a square wave shock may appear as an exponential change in the transmembrane po­tential that reaches an asymptote (Fig. 12). Because of the nonlinear behavior of the membrane introduced by the ion channels, reversing defibrillation shock polarity does not just reverse the sign of the change in the transmembrane poten­tial but also alters the magnitude and time-course of the change in transmembrane potential (Fig. 12).

 _ hvn kT

Proximity Effect. When a superconductor and normal metal are brought into contact, Cooper pairs from the superconductor can diffuse into the normal metal. Due to phonon-induced pair breaking, the pair amplitude (also known as the superconducting order parameter or wavefunction) in the normal metal decays exponentially over a decay length defined as the normal metal coherence length, fn. In the clean limit where ln, the mean free path in N, is much greater than fn, the coherence length is given by

(4)

while in the dirty limit with ln ^ fn, the coherence length is

where the diffusion constant, Dn equals vnln/d and d is the dimensionality. Theories describing the details of the superconductor-normal-metal “proximity-effect” interaction have been developed for a variety of cases including back-to-back SN contacts—that is, the SNS weak link (24,25). In the SNS Josephson junction,

 Fig. 10. Critical current versus temperature for a YBCO/Co-YBCO/YBCO edge SNS junction. The solid line is a fit to the proximity effect theory of DeGennes. Despite the fact that the junction is nonideal, in that it exhibits a large interface resistance, it still appears to exhibit behavior consistent with the proximity effect.

pairs from each superconducting electrode “leak” into the normal metal interlayer and the overlap of the exponentially decaying pair amplitudes determines the strength of interaction between the superconductors. Consequently, the magnitude of the Josephson critical current scales as exp[-L/fn(T)], where L is the normal metal bridge length. More specifically, in the dirty limit for long SNS bridges (L > fn) relatively close to Tc (T > 0.3Tc), it is found that

where Ai is the superconducting gap at the superconductor-normal-metal interface. This equation indicates that the critical current of an SNS weak link should also vary exponentially with temperature, because of the (T)-1/2 temperature dependence of the dirty-limit normal-metal coherence length. The exponential length and temperature dependence of the critical current are the distinguishing signatures of true proximity effect devices. Indeed, there are a number of examples of HTS SNS devices which are largely consistent with proximity effect theory, most notably the junctions using Co – or Ca-doped YBCO as the normal metal layer. An example of exponential critical current dependence on temperature for a Co-doped YBCO SNS edge junction is shown in Fig. 10, along with a proximity theory fit to the data (26). However, it is often found that HTS devices with a nominal SNS configuration do not show an exponential critical current dependence on temperature. In fact, such devices commonly exhibit a quasilinear temperature dependence, which may indicate that pinhole conduction through the normal metal is dominating the electrical characteristics (25).

Control of Resistance in SNS Devices. The normal state resistance of an SNS weak link is given by the sum of the normal metal resistance plus the resistance of each of the two SN interfaces. In the “ideal” SNS device the interface resistances are zero and the total device resistance is just Rn = pnL/A, where pn is the normal metal resistivity, L is the normal metal thickness, and A is the cross-sectional area. For typical values of these parameters in an SNS edge junction with a YBa2Cu2.8Co02O7 normal metal layer at 65 K (pn = 250 дй-cm, L = 100 A, and A = 4 x 0.2 ^m2) we find Rn = 0.03 Й. In practice, such low values of resistance are often undesirable. For example, for SFQ digital applications at 65 K junctions are biased at a fixed current of order 500 дА so that the available IcRn product with Rn = 0.03 Й is only 15 дУ, far less than the required value of approximately 300 дУ. Increasing SNS device resistances to a practical level requires adding interface resistance without degrading the inherent IcRn product. In principle, this can be done in at least two ways: (1) by incorporating an inhomogeneous interface resistance to reduce the effective device area or (2) by producing a thin insulator at one SN interface to form an SINS structure. In practice, different groups have seen widely varying values of SNS resistance, ranging from the ideal but impractical case of very low RnA (16) to the more technologically interesting case of high-RnA devices (27).

For SNS weak links using Co-YBCO as the normal metal, it has been found that the interface resistance is sensitive to a variety of factors including the base electrode material and the normal metal and counterelectrode deposition conditions (17). For example, SNS devices using YBa2Cu3O7 base electrodes grown by pulsed laser deposition (PLD) exhibit more than an order of magnitude lower resistance than devices with La-doped YBCO base electrodes (YBa1.95La005Cu3O7), or GdBa2Cu3O7 or NdBa2Cu3O7 base electrodes. Varying the normal metal and counterelectrode growth parameters can also have a dramatic effect on device resistance: High – pressure PLD growth in an Ar-O2 atmosphere results in RnA products over a factor of 10 smaller than for devices produced using the more conventional PLD deposition conditions in a pure oxygen background. While the detailed nature of the interface resistance is not understood at this point, the base electrode material dependence suggests that cation disorder (e. g., Y and Ba exchange) is affecting device resistance. The fact that the growth conditions of the normal metal also have a strong effect on SNS resistance indicates that defects “frozen in” in the early stages of normal metal growth may also play an important role in determining interface resistances. Because SNS interface resistances are strongly affected by a number of material and fabrication parameters, it is possible to control SNS device resistances over two to three orders of magnitude, with RnA products ranging from 0.03 Й-дш2 to more than 10 й-дш2. Importantly, even in the relatively high RnA limit required for SFQ applications (0.5 Й-дш2 to 2 Й-дш2), Co-doped YBCO SNS devices incorporating significant interface resistance still behave like true proximity effect devices (see, for example, Fig. 10) with parameter uniformity suitable for small-scale SFQ circuits.

Limits on Reproducibility. Speculation on the origins of parameter spreads have led to many experi­ments in fabrication of edge SNS junctions. Surprisingly, the fabrication parameters which result in junction resistances greater than Rn = pnL/A do not appear to systematically contribute to larger spreads in critical currents or other junction parameters. Similarly, the uniformity of current flow through a junction which can be inferred from Ic(B) indicates that junctions with significant interface resistance maintain uniform current distributions.

Poor control over many fabrication parameters will certainly result in junction spreads worse than the state of the art. A good example is the roughness of YBCO base electrodes which gets transferred into the edge by patterning with Ar ion milling. While improvements from 10 nm to 2 nm rms surface roughness provide a measurable benefit for junction reproducibility, further improvements in smoothness have had a negligible effect. A second example is that junctions facing in all four in-plane directions sometimes exhibit a distribution of critical currents that is direction-dependent. However, when all processing steps are made isotropic, state – of-the-art junction uniformity can be achieved as easily in a set of junctions facing in four directions as in a set facing just one way.

These results have led us to examine defects that are intrinsic to YBCO. The role of oxygen disorder in YBCO has been investigated in several types of edge junction experiments. Decreasing the number of oxygen vacancies by plasma oxidation or annealing in ozone simply scales Ic for all treated junctions by a constant factor as large as five. Experiments in which orthorhombic YBCO electrodes were replaced by doped YBCO compounds which were tetragonal have been inconclusive in determining the possible role that twinning in YBCO might have on parameter spreads. Junctions were fabricated to face in (110) in-plane directions instead of the standard (100) directions to minimize the effects attributable to twinning, but no improvement injunction uniformity was observed.

Finally, the fact that similar best-case critical current spreads are observed for different junction fab­rication processes using the same base electrode materials suggests that microstructural defects in the base electrode or base electrode edge are limiting Ic spreads. Further improvements in materials quality and edge formation techniques are expected to lead to improved junction spreads.

Conclusions

Josephson junctions based on YBCO are the fundamental building blocks for a variety of superconducting electronics applications operating at temperatures >50 K. The properties of individual junctions fabricated in a variety of configurations are sufficiently close to ideal Josephson behavior to meet application require­ments. However, integration of junctions into multilayer circuits and demands on reproducibility of junction parameters when higher junction counts are needed have narrowed development efforts to a few promising configurations. Most of the current HTS circuit fabrication effort in industrial laboratories is based on edge SNS junctions which have been used for the most sophisticated and extendible digital circuit demonstra­tions. Further incremental improvements in the uniformity of these junctions to 1-а Ic spreads less than 10% will permit medium-scale integrated circuit fabrication. A parallel effort, mainly by university researchers, is exploring higher-risk alternative junction configurations intended to circumvent some of the limitations to junction uniformity that may exist for edge junctions.

## THE FILTERING PROBLEM

Here we describe a variety of nonlinear filters and identify some current areas of research on nonlinear methods. An ex­tensive treatment of nonlinear filters can be found in the books by Astola and Kuosmanen (1) and by Pitas and Venet – sanopoulos (2). The fact that nonlinear methods lack a unify­ing framework makes presenting a general overview difficult. However, we organize the presented filters into two general methodologies:

1. Weighted Sum Filters. The output of a linear filter is formed as a linear combination, or weighted sum, of ob­served samples. Nonlinearities can be introduced into this general filtering methodology by transforming the observation samples, through reordering or nonlinear warping for instance, prior to the weighting and sum-

The goal in many filtering applications is to transform an ob­served signal into an approximation of a desired signal, where the transformation is designed to optimize some fidelity crite­rion. This scenario is illustrated in Fig. 1, where {x(n)} and {y(n)} represent the observation (input) and approximation (output) sequences, respectively. In this representation