SEARCH · Search NASA

Results for “Rectangular Wave”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 271 records · Page 15

All sources NASA NTRS DOE OSTI

NASA NTRS1988

Vacillations in a coupled ocean-atmosphere model

Results are presented from a 35-year integration of a coupled ocean-atmosphere model. Both ocean and atmosphere are two-level, nonlinear primitive equation models. The global atmospheric model is forced by a steady, zonally symmetric Newtonian heating. The ocean model is solved in a rectangular tropical basin. Heat fluxes between ocean and atmosphere are linear in air-sea temperature differences, and the interfacial stress is proportional to lower-level atmospheric winds. The coupled models produce ENSO-like variability on time scales of 3 to 5 years. Since there is no external time-dependent forcing, these are self-sustained vacillations of the nonlinear system. It is argued that the energetics of the vacillations is that of unstable coupled modes and that the time scale is crucially dependent on the effects of ocean waves propagating in a closed basin.

Schopf, Paul S.↗

DOE OSTI2022

Preshot Report for Carbonara: Dynamics of a Boron Carbide Sandwich

The main purpose of the Carbonara experiment is to provide much-needed data to inform physics modeling of boron carbide (B 4 C) subjected to a sweeping high explosive (HE) detonation. The proposed experiment will include three separate shots, each employing a rectangular slab of HE (4” × 3” × 1”) sandwiched between two wedges of B 4 C, see Fig.1. In each case the top B 4 C wedge tapers from a height of 2.5 cm to 0.5 cm, while the bottom wedge tapers from 1.5 cm to 0.5 cm. The wedges are 4” in the direction of the proton beam, and 3” in the direction transverse to the beam. Three shots are currently planned, each one using a different HE: PBX 9501, PBX 9502, and PBX 9701. The HE will be initiated using a line wave generator (LWG) to preserve planar symmetry, in conjunction with a Comp. B booster to smooth out the profile for the initial condition on the main charge and to aid in initiation of the PBX 9502 shot. In each case the booster spans the entire length and height of the main HE slab and extends 0.5” in toward the LWG.

36 MATERIALS SCIENCE↗

NASA NTRS1995

Rectangular Waveguides with Two Conventional and Two Superconducting Walls

The propagation properties of transverse electric TE(sup pm) modes and their dispersion relations in rectangular waveguides with two conventional and two superconducting walls, derived by using the Meissner boundary conditions on the superconducting walls, are presented. In addition to recovering some previously known results, some novel results have been obtained: the cut-off wavelength of the dominant TE(sup 10) mode is greater than that of the conventional TE(sub 10) mode, and the tangential electric field and normal magnetic field for the dominant mode TE(sup 10) exist on the superconducting surfaces. Expressions for electromagnetic components, surface currents, attenuation coefficient, maximum transmitted power, dispersion and wave impedance are also presented.

Yalamanchili, Raj↗

NASA NTRS1995

Microwave Resonator Measurements of Atmospheric Absorption Coefficients: A Preliminary Design Study

A preliminary design study examined the feasibility of using microwave resonator measurements to improve the accuracy of atmospheric absorption coefficients and refractivity between 18 and 35 GHz. Increased accuracies would improve the capability of water vapor radiometers to correct for radio signal delays caused by Earth's atmosphere. Calibration of delays incurred by radio signals traversing the atmosphere has applications to both deep space tracking and planetary radio science experiments. Currently, the Cassini gravity wave search requires 0.8-1.0% absorption coefficient accuracy. This study examined current atmospheric absorption models and estimated that current model accuracy ranges from 5% to 7%. The refractivity of water vapor is known to 1% accuracy, while the refractivity of many dry gases (oxygen, nitrogen, etc.) are known to better than 0.1%. Improvements to the current generation of models will require that both the functional form and absolute absorption of the water vapor spectrum be calibrated and validated. Several laboratory techniques for measuring atmospheric absorption and refractivity were investigated, including absorption cells, single and multimode rectangular cavity resonators, and Fabry-Perot resonators. Semi-confocal Fabry-Perot resonators were shown to provide the most cost-effective and accurate method of measuring atmospheric gas refractivity. The need for accurate environmental measurement and control was also addressed. A preliminary design for the environmental control and measurement system was developed to aid in identifying significant design issues. The analysis indicated that overall measurement accuracy will be limited by measurement errors and imprecise control of the gas sample's thermodynamic state, thermal expansion and vibration- induced deformation of the resonator structure, and electronic measurement error. The central problem is to identify systematic errors because random errors can be reduced by averaging. Calibrating the resonator measurements by checking the refractivity of dry gases which are known to better than 0.1% provides a method of controlling the systematic errors to 0.1%. The primary source of error in absorptivity and refractivity measurements is thus the ability to measure the concentration of water vapor in the resonator path. Over the whole thermodynamic range of interest the accuracy of water vapor measurement is 1.5%. However, over the range responsible for most of the radio delay (i.e. conditions in the bottom two kilometers of the atmosphere) the accuracy of water vapor measurements ranges from 0.5% to 1.0%. Therefore the precision of the resonator measurements could be held to 0.3% and the overall absolute accuracy of resonator-based absorption and refractivity measurements will range from 0.6% to 1.

Walter, Steven J.↗

NASA NTRS1984

Analysis of an infinite array of rectangular microstrip patches with idealized probe feeds

A solution is presented to the problem of an infinite array of microstrip patches fed by idealized current probes. The input reflection coefficient is calculated versus scan angle in an arbitrary scan plane, and the effects of substrate parameters and grid spacing are considered. It is pointed out that even when a Galerkin method is used the impedance matrix is not symmetric due to phasing through a unit cell, as required for scanning. The mechanism by which scan blindness can occur is discussed. Measurement results are presented for the reflection coefficient magnitude variation with angle for E-plane, H-plane, and D-plane scans, for various substrate parameters. Measured results from waveguide simulators are also presented, and the scan blindness phenomenon is observed and discussed in terms of forced surface waves and a modified grating lobe diagram.

Pozar, D. M.↗

NASA NTRS1984

Input-output characterization of an ultrasonic testing system by digital signal analysis

The input/output characteristics of an ultrasonic testing system used for stress wave factor measurements were studied. The fundamentals of digital signal processing are summarized. The inputs and outputs are digitized and processed in a microcomputer using digital signal processing techniques. The entire ultrasonic test system, including transducers and all electronic components, is modeled as a discrete-time linear shift-invariant system. Then the impulse response and frequency response of the continuous time ultrasonic test system are estimated by interpolating the defining points in the unit sample response and frequency response of the discrete time system. It is found that the ultrasonic test system behaves as a linear phase bandpass filter. Good results were obtained for rectangular pulse inputs of various amplitudes and durations and for tone burst inputs whose center frequencies are within the passband of the test system and for single cycle inputs of various amplitudes. The input/output limits on the linearity of the system are determined.

Karaguelle, H.↗

DOE OSTI2022

Mooring System Cost Estimates for Wave Energy Farms in Shared Mooring Arrays

As wave energy converters (WECs) become more advanced and cost-efficient, so too must their mooring systems. A key question in the development plans for WECs is the cost of the mooring system, particularly for large wave farms. WEC devices deployed in a WEC farm array, where each device can be connected by shared mooring lines in various array layouts, have potential to reduce mooring system costs significantly. This paper presents the modeling and designing of mooring systems for large WEC arrays and the calculation of the cost of each mooring system to determine the change in cost as the number of WECs in a farm increases. A baseline mooring system for a single floating oscillating water column (OWC) WEC was developed for this analysis. The mooring system utilizes four anchored mooring lines connected to a square assembly of wire rope mooring lines, supported by four floating buoys, and attached to the floating WEC by four polyester rope mooring lines. This assembly, referred to as a floating cell, can be tiled to form various rectangular WEC arrays. The objective of this analysis is to determine how the mooring system cost changes as more WECs are added to an array layout, each with their own interconnected floating cell. To do this, complete mooring systems need to be designed for each WEC array layout. To narrow down the design space of a WEC array mooring system, a couple assumptions were made. It was assumed that the floating cell parameters of the baseline design were to stay constant across all floating cells in the WEC array. It was also assumed that the anchored mooring lines would be of the baseline configuration, a predominantly chain mooring line with a short section of polyester rope near the fairlead, and a drag-embedment anchor. These anchored lines were assumed to extend from the outer edges of the WEC array, inline with the headings of the wire rope mooring lines of the floating cells, or diagonal if extending from a corner of the WEC array. Full mooring systems were designed for 2xN and NxN WEC array layouts and efficiently simulated in the mooring dynamics simulation tool, MoorDyn, to ensure all dynamic constraints were met. The system costs were calculated and then refined by shortening unnecessary chain line lengths and reducing the chain diameters of the downstream anchored mooring lines. It was found that, in general, mooring system costs per WEC decrease when WECs are installed in an array. Compared to the baseline mooring system for a single WEC, the 2x3 mooring system array had the lowest mooring system cost per WEC, reducing the cost per WEC by 59%. The 3x3 and 4x4 array mooring systems also saw significant reductions in cost per WEC but had negligible cost savings between the two designs, primarily because the larger 4x4 mooring system requires larger chain diameters, which increases cost. These results provide an interesting glimpse into modeling, designing, and calculating the cost of mooring systems for large WEC arrays.

cost↗

NASA NTRS1973

Acoustic radiation from plates excited by flow noise.

Theoretical formulation of noise radiation from thin, rectangular, flexible plates excited by turbulent wall pressure fluctuations. An extensive parametric study including surface roughness and both plate and flow parameters is presented. From the results, presented graphically, it is possible to predict the spectrum of the total radiated power provided that the power spectral density of the wall-pressure fluctuations, the flow velocity, the plate dimensions and the plate modal damping parameters are known. Along with the parametric study, an estimate of the dependence of the noise radiation on plate boundary conditions based on a consideration of the wavenumber-frequency selectivity of the plate is presented. The parametric study is based on the assumption that the wave number/frequency spectrum of the wall-pressure fluctuations can be described by a Fourier transform of a Corcos model of the cross spectral density. The results indicate that at frequencies above the hydrodynamic-coincidence frequency the radiated power is dependent on the low-wave-number spectrum of the excitation. The dependence on the high-wave-number spectrum, however, is limited to a narrow band of frequencies below the hydrodynamic-coincidence frequency.

Aupperle, F. A.↗

NASA NTRS2024

‘Trapped Wave’ Resonance Tones Radiated in the Forward Arc of High-Speed Jets

An experimental study is conducted investigating the characteristics of ‘trapped wave’ resonance tones in the forward arc of jet noise radiation fields. While these spectral peaks disappear in the far acoustic field in the aft and sideline directions, which is why they went unnoticed in decades of jet noise measurements, the present study clearly shows that they radiate in the forward arc (shallow upstream polar locations in the approximate range of θ < 45°). Far-field jet noise spectra in the forward arc, data on which had been lacking in the literature, are not smooth but are characterized by these peaks. This is found for high subsonic to supersonic jets up to the highest jet Mach number covered in the experiment (MJ ≈ 1.9), for round, rectangular as well as convergent-divergent nozzles. With heated jets these spectral peaks are weakened especially around transonic conditions, however, they persist in supersonic conditions. Note that digital files containing all spectral data shown in the following are in a supplemental file to this NASA Technical Memorandum (TM-20240009873-SUPPL). These files can be obtained from the NASA Technical Report Server (NTRS) as a separate download from this report.

jets↗

NASA NTRS1996

On the Two Components of Turbulent Mixing Noise from Supersonic Jets

It is argued that because of the lack of intrinsic length and time scales in the core part of the jet flow, the radiated noise spectrum of a high-speed jet should exhibit similarity. A careful analysis of all the axisymmetric supersonic jet noise spectra in the data-bank of the Jet Noise Laboratory of the NASA Langley Research Center has been carried out. Two similarity spectra, one for the noise from the large turbulence structures/instability waves of the jet flow, the other for the noise from the fine-scale turbulence, are identified. The two similarity spectra appear to be universal spectra for axisymmetric jets. They fit all the measured data including those from subsonic jets. Experimental evidence are presented showing that regardless of whether a jet is supersonic or subsonic the noise characteristics and generation mechanisms are the same. There is large turbulence structures/instability waves noise from subsonic jets. This noise component can be seen prominently inside the cone of silence of the fine-scale turbulence noise near the jet axis. For imperfectly expanded supersonic jets, a shock cell structure is formed inside the jet plume. Measured spectra are provided to demonstrate that the presence of a shock cell structure has little effect on the radiated turbulent mixing noise. The shape of the noise spectrum as well as the noise intensity remain practically the same as those of a fully expanded jet. However, for jets undergoing strong screeching, there is broadband noise amplification for both turbulent mixing noise components. It is discovered through a pilot study of the noise spectrum of rectangular and elliptic supersonic jets that the turbulent mixing noise of these jets is also made up of the same two noise components found in axisymmetric jets. The spectrum of each individual noise component also fits the corresponding similarity spectrum of axisymmetric jets.

Tam, Christopher K. W.↗

NASA NTRS1986

Propagation characteristics of some novel coplanar waveguide transmission lines on GaAs at MM-wave frequencies

Three new Coplanar Waveguide (CPW) transmission lines, namely, Suspended CPW (SCPW), Stripline-like Suspended CPW (SSCPW) and Inverted CPW (ICPW), are proposed and also analyzed for their propagation characteristics. The substrate thickness, permittivity and dimensions of housing are assumed to be arbitrary. These structures have the following advantages over conventional CPW. Firstly, the ratio of guide wavelength to free space wavelength is closer to unity which results in larger dimensions and hence lower tolerances. Secondly, the effective dielectric constant is lower and hence the electromagnetic field energies are concentrated more in the air regions which should reduce attenuation. Thirdly, for a prescribed impedance level, the above structures have a wider slot width for identical strip width. Thus, low impedance lines can be achieved with reasonable slot dimensions. Fourthly, in an inverted CPW shunt mounting of active devices, such as Gunn and IMPATT diodes, between the strip and the metal trough is possible. This feature further enhances the attractiveness of the above structures. Lastly, an E-plane probe type transition from a rectangular waveguide to suspended CPW can also be easily realized. The computed results for GaAs at Ka-band illustrate the variation of normalized guide wavelength, effective dielectric constant and the characteristic impedance as a function of the: (1) frequency; (2) distance of separation between the trough side walls; (3) normalized strip and slot widths; and (4) normalized air gap.

Simons, Rainee N.↗

NASA NTRS1995

The 3D Navier-Stokes analysis of a Mach 2.68 bifurcated rectangular mixed-compression inlet

The supersonic diffuser of a Mach 2.68 bifurcated, rectangular, mixed-compression inlet was analyzed using a three-dimensional (3D) Navier-Stokes flow solver. A two-equation turbulence model, and a porous bleed model based on unchoked bleed hole discharge coefficients were used. Comparisons were made with experimental data, inviscid theory, and two-dimensional Navier-Stokes analyses. The main objective was to gain insight into the inlet fluid dynamics. Examination of the computational results along with the experimental data suggest that the cowl shock-sidewall boundary layer interaction near the leading edge caused a substantial separation in the wind tunnel inlet model. As a result, the inlet performance may have been compromised by increased spillage and higher bleed mass flow requirements. The internal flow contained substantial waves that were not in the original inviscid design. 3D effects were fairly minor for this inlet at on-design conditions. Navier-Stokes analysis appears to be an useful tool for gaining insight into the inlet fluid dynamics. It provides a higher fidelity simulation of the flowfield than the original inviscid design, by taking into account boundary layers, porous bleed, and their interactions with shock waves.

Mizukami, M.↗

NASA NTRS1985

On the electromagnetic scattering from infinite rectangular conducting grids

The study and development of two numerical techniques for the analysis of electromagnetic scattering from a rectangular wire mesh are described. Both techniques follow from one basic formulation and they are both solved in the spectral domain. These techniques were developed as a result of an investigation towards more efficient numerical computation for mesh scattering. These techniques are efficient for the following reasons: (a1) make use of the Fast Fourier Transform; (b2) they avoid any convolution problems by converting integrodifferential equations into algebraic equations; and (c3) they do not require inversions of any matrices. The first method, the SIT or Spectral Iteration Technique, is applied for regions where the spacing between wires is not less than two wavelengths. The second method, the SDCG or Spectral Domain Conjugate Gradient approach, can be used for any spacing between adjacent wires. A study of electromagnetic wave properties, such as reflection coefficient, induced currents and aperture fields, as functions of frequency, angle of incidence, polarization and thickness of wires is presented. Examples and comparisons or results with other methods are also included to support the validity of the new algorithms.

Christodoulou, C.↗

NASA NTRS2009

Advanced Computational and Experimental Techniques for Nacelle Liner Performance Evaluation

The Curved Duct Test Rig (CDTR) has been developed to investigate sound propagation through a duct of size comparable to the aft bypass duct of typical aircraft engines. The axial dimension of the bypass duct is often curved and this geometric characteristic is captured in the CDTR. The semiannular bypass duct is simulated by a rectangular test section in which the height corresponds to the circumferential dimension and the width corresponds to the radial dimension. The liner samples are perforate over honeycomb core and are installed on the side walls of the test section. The top and bottom surfaces of the test section are acoustically rigid to simulate a hard wall bifurcation or pylon. A unique feature of the CDTR is the control system that generates sound incident on the liner test section in specific modes. Uniform air flow, at ambient temperature and flow speed Mach 0.275, is introduced through the duct. Experiments to investigate configuration effects such as curvature along the flow path on the acoustic performance of a sample liner are performed in the CDTR and reported in this paper. Combinations of treated and acoustically rigid side walls are investigated. The scattering of modes of the incident wave, both by the curvature and by the asymmetry of wall treatment, is demonstrated in the experimental results. The effect that mode scattering has on total acoustic effectiveness of the liner treatment is also shown. Comparisons of measured liner attenuation with numerical results predicted by an analytic model based on the parabolic approximation to the convected Helmholtz equation are reported. The spectra of attenuation produced by the analytic model are similar to experimental results for both walls treated, straight and curved flow path, with plane wave and higher order modes incident. The numerical model is used to define the optimized resistance and reactance of a liner that significantly improves liner attenuation in the frequency range 1900-2400 Hz. A liner impedance descriptor is used to determine the liner parameters that achieve the optimum impedance.

Gerhold, Carl H.↗

NASA NTRS1993

Asynchronous timing and Doppler recovery in DSP based DPSK modems for fixed and mobile satellite applications

While conventional analog modems employ some kind of clock wave regenerator circuit for synchronous timing recovery, in sampled modem receivers the timing is recovered asynchronously to the incoming data stream, with no adjustment being made to the input sampling rate. All timing corrections are accomplished by digital operations on the sampled data stream, and timing recovery is asynchronous with the uncontrolled, input A/D system. A good timing error measurement algorithm is a zero crossing tracker proposed by Gardner. Digital, speech rate (2400 - 4800 bps) M-PSK modem receivers employing Gardner's zero crossing tracker were implemented and tested and found to achieve BER performance very close to theoretical values on the AWGN channel. Nyguist pulse shaped modem systems with excess bandwidth factors ranging from 100 to 60 percent were considered. We can show that for any symmetric M-PSK signal set Gardner's NDA algorithm is free of pattern jitter for any carrier phase offset for rectangular pulses and for Nyquist pulses having 100 percent excess bandwidth. Also, the Nyquist pulse shaped system is studied on the mobile satellite channel, where Doppler shifts and multipath fading degrade the pi/4-DQPSK signal. Two simple modifications to Gardner's zero crossing tracker enable it to remain useful in the presence of multipath fading.

Koblents, B.↗

NASA NTRSDate not supplied

Isolator Shock Train Response to Dual-Mode Scramjet Throttling-Like Forcing

Typical dual-mode scramjet designs employ an isolator component situated between the inlet and combustor. The isolator fluidically buffers the inlet from the combustor via the formation of a shock train, an interconnected series of shock-boundary layer interactions. The isolator is required during ramjet-engine operation and at the lower edges of the scramjet operational envelope. In an operational engine, transient behavior imposes fluidic dynamics due to attitude adjustments, combustor throttling, aberrant resonances within the combustor, etc. To accommodate these transients, the shock train adjusts on both temporal and spatial scales. Sufficiently large downstream pressure increases in the combustor play a critical role during unstart, whereby strong compression waves propagate upstream in the isolator to ultimately force the shock train out the inlet, if it cannot stabilize to the pressure increase. This critically reduces the engine mass flow rate, potentially causing a catastrophic reduction in thrust and loss of the vehicle. A nearly instantaneous unstart can, in principle, be predicted by a ratio of the normal shock pressure jump in an inviscid isolator. However, lower-magnitude perturbations, below this bound, also induce unstarts. Understanding the transient behavior inherent in unsteady isolator operation is critical to reducing safety margin in isolator length and refining engine control algorithms to improve engine performance. The Isolator Dynamics Research Lab (IDRL) is a cold-flow, direct-connect isolator facility ideal for assessing the fundamental fluid dynamics of shock trains. The internal flow of a scramjet isolator is modeled using a 2D, symmetric Mach 2.5 nozzle followed by a 695 mm long, 25.4x50.8 mm rectangular isolator test section. A blockage cone on a high-speed ball screw actuator downstream of the isolator is pulsed to simulate the throttling behavior in an operational combustor. Wall flush high-speed static pressure transducers detect the streamwise shock train within the isolator. Time delays between the onset of the throttling and the response motion of the shock train are analyzed to assess the controlling scales and parameters. From this analysis, the response time of the isolator shock train system is found to depend primarily on the isolator back pressure time-rate-of-change but quickly saturates to a maximum propagation speed along the isolator. These results inform dual-mode scramjet engine designers of the critical parameters and limits of an unstart resistant engine.

scramjet↗

NASA NTRS1993

Instability modes excited by natural screech tones in a supersonic rectangular jet

The evolution of hydrodynamic instability modes self-excited by harmonically related natural screech tones was experimentally investigated. A convergent rectangular nozzle with an aspect ratio of 9.63 was used to produce a supersonic shock containing jet. Measurements in the flow-field were made using standard hot-film probes positioned only in the subsonic (outer) portions of the flow. The hydrodynamic instability mode observed in the shear layer at the screech frequency was observed to be antisymmetric (sinuous) about the smaller dimension of the jet, whereas its harmonic was observed to be symmetric (varicose). In addition, the near-field noise measurements indicated that the radiated screech tone noise was out of phase on either side of the small jet dimension whereas its harmonic was in phase over the same region. To our knowledge such an observation on the nature of the harmonic has thus far gone unreported and therefore is the focus of the present work. The hydrodynamic instability modes occurring at the screech frequency and its harmonic satisfied the conditions for resonance. Detailed measurements of the coherent wave evolution in the streamwise and spanwise directions indicated that strong spanwise variations were present beyond x/h = 8. Details of the screech noise radiated by the coherent instability modes are also presented in this paper.

Raman, Ganesh↗

NASA NTRS2000

Effects of Geometric Azimuthal Asymmetries of the PPM Stack on Electron Beam Characteristics

The effects of geometric azimuthally asymmetric properties of a periodic permanent magnet (PPM) focusing stack on electron beam characteristics obtained using a fully three dimensional (3D) particle-in-cell (PIC) code will be presented. The simulation model, using MAFIA (Solution of MAxwell's equations by the Finite-Integration-Algorithm), incorporates 3D behavior of the beam immersed in static fields calculated directly from the exact geometry and material properties of the 3D magnetic focusing structure. The Hughes 8916H, 18-40 GHz helical TWT for the millimeter-wave power module (MMPM) was used as a prototype. Firstly, the effects of C-magnets used at the input and output of the TWT to allow for coupling of the RF signal into and out of the tube are considered. The 8916H input and output C-magnets differ because coaxial couplers are used at the input and waveguide couplers are used at the output The repositioning of the beam from its central axis due to the inclusion of the output C-magnet was found to be most significant. The modeled output C-magnet and its orientation in the Cartesian coordinate system is shown, and a two-dimensional beam profile including the output C-magnet is also shown. A table presents the shift of the beam center off the central axis relative to the average radius of the beam at the longitudinal points A, B and C designated on an enclosed figure. Secondly, the addition of shunts, or rectangular iron pieces applied manually by a skilled technician in order to improve beam transmission, is considered. The shunts are applied to the top of the tube; thus, azimuthal symmetry of the focusing stack is interrupted. Although shunts are typically added during RF focusing, they are also typically added at the input section of the tube where RF forces are minimal, making an electron optics analysis meaningful. Because several shunts are usually applied to one pole piece, the simulations have been simplified by modeling a half washer with the same radius and longitudinal length as a shunt over the entire x, positive-y half of the transverse plane. A modeled pole piece and shunt as described are shown. Lastly, in order to study the effects of magnet misalignments, a magnet in the PPM stack was arbitrarily chosen and adjusted so that its central axis was shifted both 0.7 percent and 1.0 percent of the magnet outer diameter in the positive-y direction. In practice, positioning the magnets so that their central axis is accurately aligned with the central axis of the tube is challenging. Thus, it is a strong possibility that one or more magnets will be misaligned relative to the tube central axis.

Kory, Carol L.↗