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Mode scalability in bent optical fibers

Ross T. Schermer

U.S. Naval Research Laboratory, Division of Optical Sciences, 4555 Overlook Ave. SW, Washington, D.C. 20375

schermer@https://www.sodocs.net/doc/3d13376208.html,

Abstract: This paper introduces a simple, analytical method for

generalizing the behavior of bent, weakly-guided fibers and waveguides. It

begins with a comprehensive study of the modes of the bent step-index

fiber, which is later extended to encompass a wide range of more

complicated waveguide geometries. The analysis is based on the

introduction of a scaling parameter, analogous to the V-number for straight

step-index fibers, for the bend radius. When this parameter remains

constant, waveguides of different bend radii, numerical apertures and

wavelengths will all propagate identical mode field distributions, except

scaled in size. This allows the behavior of individual waveguides to be

broadly extended, and is especially useful for generalizing the results of

numerical simulations. The technique is applied to the bent step-index fiber

in this paper to arrive at simple analytical formulae for the propagation

constant and mode area, which are valid well beyond the transition to

whispering-gallery modes. Animations illustrating mode deformation with

respect to bending and curves describing polarization decoupling are also

presented, which encompass the entire family of weakly-guided, step-index

fibers.

OCIS codes: (060.2310) Fiber Optics; (230.7370) Waveguides; (140.3510) Fiber Lasers;

(060.2320) Fiber Optic Amplifiers and Oscillators; (190.4370) Nonlinear Optics, Fibers;

(060.2280) Fiber Design and Fabrication.

References and links

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Expres s 14, 69-81 (2006).

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experiment,” IEEE. J. Quantum Electron.43, 899-909 (2007).

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Area Fibers,” Opt. Lett.32, 748-750 (2007).

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(2007).

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External Forces,” Appl. Opt.17, 2080-2085 (1978).

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24.R. T. Schermer is preparing a manuscript to be titled “Bend Loss in Weakly-Guided Fibers.”

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(1967).

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(1982).

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Amplifier,” Opt. Lett.25, 442-444 (2000).

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1. Introduction

Fiber-optic waveguides have long been a critical component of a wide range of photonic systems, utilized for communications [1,2], sensing [3,4], power delivery [5], and more recently optical sources and amplifiers [6,7]. Extensive information describing the behavior of optical fibers exists in the literature, based on both analytical models and numerical simulations. However, the vast majority of theoretical work is based on fibers that are assumed to be straight. To account for differences in fiber behavior caused by bending, perturbation methods are most often used [8,9], based on the properties of the straight fiber. These can work well for standard single-mode fibers, because the fields propagating along the fiber are only weakly distorted by bending. But for fibers supporting even a small number of modes, bending can greatly deform the mode fields [10-14], and render perturbation theory ineffective. In such cases the bent fiber must instead be studied through numerical simulation, for which a variety of techniques have been developed [15, 16]. Unfortunately both approaches have the drawback compared to simple analytical methods in that the calculated results apply to only a single particular fiber. Deducing the general behavior of bent fibers in relation to their many variable properties can therefore be time-consuming, and non-trivial.

This paper overcomes this limitation by introducing a simple, analytical method that allows the behavior of an individual bent waveguide to be extended to an entire family of similar designs. This is done by introducing conditions for which the mode fields propagating along one waveguide are essentially identical to those of innumerable others, except scaled spatially. Such an approach was taken by Gloge [17] years ago to generalize the behavior of straight, weakly-guided step-index fibers. This paper extends this powerful approach, however, to weakly-guided fibers that are circularly-bent and of arbitrary cross-section. The #86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007 (C) 2007 OSA26 November 2007 / Vol. 15, No. 24 / OPTICS EXPRESS 15675

method also applies to both single-mode and multi-mode fibers. As such, it provides a general and relatively simple framework for understanding the behavior of bent, weakly-guided waveguides.

In order to illustrate this approach, the paper also presents a comprehensive study of the modes of the bent step-index fiber. Included are animations of bend-induced mode deformation that, unlike previous studies [10-11, 13, 16, 18-20], are generally applicable to all weakly-guided, step-index fibers. Universal curves and simple formulae for mode area and propagation constant are also developed, which differ from those in the literature [13, 18-21] in that they remain valid well beyond the transition to whispering gallery modes, and are self-consistent. A detailed analysis of polarization decoupling as a result of bending is also provided, in the form of universal curves. These results are then extended to a wide range of more complicated, though commonly used, fiber geometries. The end result is a wide-ranging study of the modes of many common bent fibers, facilitated by the framework introduced in this paper.

Section 2 begins by introducing general conditions for mode scalability in bent step-index fibers. These are derived analytically in Appendices A through C, and shown to be in excellent agreement with bent fiber simulations. Section 3 follows with a detailed study of the mode fields themselves. It begins with animations of bend-induced mode deformation, followed by a simple expression that predicts when the propagating fields transition to whispering gallery modes. Mode areas and propagation constants are considered next, leading to universal curves and simple empirical formulae. Section 4 follows by generalizing these results to more complicated waveguide geometries, and Section 5 concludes with a summary of results. Polarization coupling, which is strongly inhibited by bending, is discussed in Appendix D. A list of symbols used is presented in Appendix G.

2. Mode scalability in bent step-index fibers

It is well known that mode fields of the straight step-index fiber may be scaled in size by proper adjustment of the fiber properties [1, 17]. However, no such conditions have yet been shown to exist for bent optical fibers. This section overcomes this limitation by introducing general conditions for mode scalability in the bent step-index fiber. The predicted behavior is then confirmed through bent fiber simulations. The conditions for mode scalability presented here will later be extended to fibers of arbitrary cross-section in Section 4.

2.1 Mode scalability in straight fiber

In the straight step-index fiber, the primary factor determining the form of the mode field distributions is the V-number [17],

NA ak V o =, (1)

where a is the fiber core radius, NA is the index-based numerical aperture,

22clad core n n NA ?=, (2)

n core and n clad are the core and cladding refractive indices, and k o is the vacuum wavenumber, related to the vacuum wavelength λo by k o = 2π/λo . For a given V-number, the transverse fields and the propagation characteristics of each fiber mode remain essentially the same, except in spatial extent, as core size, numerical aperture and wavelength are varied [1,17]. The modes of a given fiber therefore scale in size, without otherwise affecting their field distributions, while holding the V-number constant. Mode scalability is often expressed through use of the normalized propagation constant,

2222)(clad core clad

s eff s n n n n b ??=, (3)

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

where n eff(s) = βs /k o , βs is the modal propagation constant, and the subscript “s” denotes values specific to the straight fiber. This quantity is directly related to both the mode field distribution and its propagation characteristics [17]. Thus, it is significant that for a given V-number and particular mode, b s remains essentially invariant. Although a weak dependence on NA exists [17], this is relatively inconsequential for weakly-guided fibers, for which NA 2 << 2n clad 2.

2.2 Mode scalability in bent fiber

In the bent fiber, however, the situation is more complicated. Fiber curvature and bend-induced variations in the refractive index both tend to distort the mode field distributions

[10,14], and ultimately push them away from the center of curvature, as illustrated in Figs. 5 and 6 of the following section. As a result, both the mode field distributions and their propagation characteristics depend not only on V-number, but also on the bend radius.

Since a complete discussion of mode scalability in bent fibers is rather lengthy, for brevity much of the analysis of this paper has been included as appendices. Appendices A and B derive general requirements for mode scaling in arbitrary, weakly-guided dielectric waveguides, for the straight and curved cases, respectively. These results are then used in Appendix C to show that for the step-index fiber, mode scaling will occur if both the V-number and the dimensionless ?-number, a normalized bend radius defined as

????????≡?clad clad eff n NA k R , (4) are held constant. In this expression, the effective bend radius R eff is used to account for bend-induced stress in the fiber [14, 22], and is related to the actual bend radius R in silica glass by ()R R silica eff 27.1≈. (5) Weak-guidance (NA 2 << 2n clad 2) has also been assumed. In effect, just as the modes of the straight fiber depend solely on the V-number, in the bent fiber they are determined by the two quantities V and ?. This point is proven mathematically in Appendix C, which shows that each mode’s normalized propagation constant remains invariant for a given combination of V and ? (within the accuracy of the weak-guiding approximation). Further justification for these claims is provided through bent fiber simulations in the following subsection. Note that for a fixed wavelength, mode scaling requires the core size to vary inversely with numerical aperture, while the bend radius changes much more rapidly, as NA -3.

2.3 Agreement with BPM simulation In order to illustrate these points, a series of simulations of bent step-index fibers were performed for this paper using the beam propagation method (BPM) with conformal mapping

[15,16]. This approach is common for mode solving, and has been shown to accurately predict bend loss in both single-mode and multimode fibers [14]. A detailed description of the simulation procedure is provided in [14]. Schematic diagrams of the bent fiber and its conformal mapped equivalent are given in Figs. 1 and 2.

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Fig 1. Schematic diagram of the bent fiber (a), showing the bend radius R , core diameter 2a ,

and the cylindrical coordinates (ρ, φ, y). Also shown in (b) is the equivalent, straight fiber

obtained by conformal mapping to the coordinate system (x, z, y) as indicated. Refractive

index profiles are as indicated in Fig. 2. From [14].

Fig. 2. Bent fiber refractive index profiles, corresponding to the two coordinate systems in Fig. 1. The fiber’s physical refractive index is shown in (a), neglecting stress. In this case, n eff

decreases with distance from the center of curvature, in order to maintain a mode with constant

angular velocity. The index profile of the equivalent, straight fiber is also shown in (b), tilted

with respect to (a) as a result of the coordinate transformation. From [14].

Before considering simulation results in detail, it should first be noted that whereas in the straight fiber a mode’s linear velocity is spatially uniform, in the bent fiber it is the angular velocity which is conserved. This point is illustrated schematically in Fig. 2(a), which shows that the phase velocity c/n eff increases while moving away from the center of curvature, such that the angular velocity remains constant (the behavior of the conformal mapped fiber in Fig. 2(b) is analogous, provided the coordinate transformation). It is therefore necessary to reference the linear propagation constant β to radial position in the bent fiber. This is typically done by defining β at the center of the fiber,

R eff n =????????

=ρλπβ02, (6)

which ensures that the physical path length (along ρ = R ) does not vary as the fiber is bent, neglecting any applied tension. For this paper, however, it is advantageous to discuss propagation in terms of a normalized angular propagation constant, defined here as

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

()a

R clad core clad

real eff n n n n b +=??=ρ2222. (7) The added restriction in Eq. (7) compared to (3) accounts for the fact that in the bent fiber b would otherwise vary with position, as is evident from Fig. 2(a). This choice of reference also assures that all guided modes must fall within the range 0

it must reduce to the straight value b s at sufficiently large bend radii.

Though strictly speaking curved waveguides cannot support guided modes, since the fields must radiate to some extent [23], the term “guided” is used loosely here to apply to all modes for which 0 < b < 1. This criterion encompasses all propagating modes concentrated about the fiber core, and is thus analogous to that of the straight fiber. That each mode radiates as it propagates along the fiber also implies that its propagation constant β must be slightly complex. This has negligible impact on the results of this paper, however, provided that the loss remains reasonable [14].

The first step in the simulation process was to calculate the lowest-order modes of various step-index fibers, each with the same V-number, but a range of core sizes, numerical apertures, cladding refractive indices, wavelengths and bend radii. In each case the normalized angular propagation constants were then calculated, as defined for the bent fiber by Eq. (7). Figures 3(a) and 3(b) plot the normalized angular propagation constants calculated for the fundamental mode (LP 01) of various bent fibers, each with the same V-number, 7.375, reasonable for high-power fiber amplifiers. As shown, the different curves overlapped when plotted versus ?, indicating that the normalized angular propagation constants were the same for each given ?-number. Similar behavior was also observed as wavelength and cladding refractive index were varied. Together, these results demonstrate the validity of the normalization presented in Eq. (4). Furthermore, each mode’s field distribution did not vary perceptibly, other than in size, for a given combination of V and ?. Examples of this are presented in Fig. 4. Together, these results illustrate that for the step-index fiber, the modes scale for a given combination of V and ?.

00.20.40.60.8

0.010.11101001000R eff (cm)b

00.20.4

0.6

0.81110100100010000100000

Fig. 3. Normalized angular propagation constants of bent step-index fibers with the same V-

number, 7.375, but different core sizes and numerical apertures. When plotted versus R eff in

(a), each curve was distinct. However, when plotted versus ? in (b), the curves overlapped.

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

Fig 4. Mode field distributions for two different fibers, each with the same V = 7.375 and

? = 35.69. In each case the mode field distributions were identical, other than being scaled in

size. The circular outline marks the core-cladding interface. The center of curvature was to the

left of the figure. Subscripts “e” and “o” are added to the usual mode notation to differentiate

between mode orientations which are even and odd, respectively, in the vertical direction

normal to the plane of the bend. Simulated regions were much larger than shown.

3. Modes of the bent step-index fiber

The theory introduced in the previous section greatly simplifies the analysis of the bent step-index fiber, effectively reducing it to a problem of only two variables. In essence, it allows the behavior of any one fiber to be taken as representative of all fibers for which V and ? are the same, and vice versa. This section utilizes these results to discuss the guided modes of the bent step-index fiber in general terms, and in particular, how they vary in response to bending. This provides not only a detailed analysis of the bent step-index fiber, but serves as an example for understanding the more complicated fiber geometries discussed later in Section 4.

In order to determine the guided modes of the bent fiber, a semi-vector version of the imaginary-distance BPM was used [15], as described in detail in [14]. This approach included polarization dependence in the mode calculations, but it did not account for polarization coupling. As a result, the analysis was limited to the linearly polarized (LP) fiber modes, rather than the exact hybrid (EH and HE) modes [25]. We proceed however with the knowledge that hybrid modes may be approximated for weakly-guided waveguides by combinations of two oppositely polarized, degenerate LP modes [25]. Furthermore, as detailed in Appendix D, significant bending tends to inhibit such polarization coupling, in which case the modes of the bent fiber reduce to essentially linearly polarized (LP) states.

Polarization dependence in the bent fiber was also simplified by two key points. The first was that shape birefringence in step-index fiber is orders of magnitude less than the bend-induced stress birefringence [8], which allows it to be neglected. The second was that birefringence does not appreciably affect the distribution of the LP mode fields, except possibly in cases of extreme birefringence approaching the index step itself. The remainder of the paper therefore presents modes and propagation constants calculated assuming zero material birefringence. These are indicative of both polarization states. It is quite

straightforward to later account for the bend-induced birefringence, by adjusting the

calculated propagation constants accordingly as discussed in Appendix D.

3.1 Mode field deformation

In order to illustrate the impact of bending on the fiber modes, a set of simulated mode

profiles are presented in the animation in Fig. 5. Each frame plots the electric field magnitudes for the six lowest-order LP modes, for a particular value of ?, and the V-number 7.375. In each case the center of fiber curvature is located to the left of the figure, and the

core-cladding boundary was indicated by the circular outline. The modal notation LP mne and LP mno was also adopted to differentiate between mode orientations which were even or odd, respectively, in the (vertical) direction normal to the plane of the bend, while conforming to the usual LP mn notation of the straight fiber [17]. It is critical to note that Fig. 5 indicates how all weakly-guided fibers with the V-number 7.375 vary in response to bending. As will be shown momentarily, the mode progression is also representative of bent step-index fibers with different V-numbers as well.

Fig. 5. Variation of the lowest-order fiber modes with bending, for V = 7.375. Circular outlines

mark the core-cladding interface, and the center of curvature was to the left of each plot. The

subscripts “e” and “o” added to the names of the various modes denote whether each mode was

even or odd, respectively, in the (vertical) direction normal to the plane of the bend. As the

modes reach cutoff they disappear from the figure. (1940 kb).

To illustrate how the V-number influences the modes of the bent step-index fiber, Fig. 6 plots a similar animation to Fig. 5 for the case V = 29.5. As shown, in the initial stages of transformation the modes shown were similar to those in Fig. 5. However, this occurred at much greater values of ?(for reference, ?= 1000 corresponds to a bend radius of approximately 1.5m when NA=0.06 and λ0 =1μm). With continued bending, all the modes transitioned to whispering gallery modes well before reaching cutoff, and thus disappearing from the figure. In this context, the term “whispering gallery mode” refers to a mode confined at the inside of the bend by the fiber curvature, rather than the core-cladding interface. This definition is based on the familiar disc resonator, in which whispering gallery

Fig. 6. Variation in the mode fields with bending, for V = 29.5. The initial stages of mode

deformation were similar to those shown in Fig. 5, although at larger values of ?. Each mode

transitioned to a whispering gallery mode upon adequate bending, filling only a small fraction

of the fiber core. (2360 kb).

modes propagate along the innermost edge of a cylindrical boundary, confined entirely by the

surface curvature [26].

Together, the animations in Figs. 5 and 6 provide a representative picture of the variation

in the lowest-order modes in the bent step-index fiber. That the initial stage of mode transformation was similar in the two figures is a key point, because it implies that similar behavior should also be expected for different V-numbers, although at different values of ?. Furthermore, since the modes in Fig. 6 transitioned fully to whispering gallery modes, which

are relatively simply distributed, it is not difficult to envision their continued variation for larger values of V.

The similarity between Figs. 5 and 6 at different values of ? may be clarified by noting that in the following sections, fibers with different V-numbers will be shown to behave alike when maintaining the same value of ?(1-b s)/V. To illustrate this point, Fig. 7 displays modes of two fibers with different V-numbers, but the same value ?(1-b s)/V = 1. In both cases the mode field distributions were similar, other than in their degree of confinement to the core. Comparable behavior was also observed for other values of ?(1-b s)/V. Thus, although the mode field scale exactly for a given pair V and ? (under the assumption of weak guiding), they are also remarkably similar for a given value of ?(1-b s)/V. Figures 5 and 6 are therefore reasonably representative of the step-index fiber in general. Extending them to other V-numbers simply requires the ?-number in each frame to be adjusted, such that ?(1-b s)/V is conserved.

Fig 7. Mode field distributions for fibers with different ? and V , but the same value

?(1-b s )/V = 1. When this quantity was constant, the mode fields were similar, other than in

their level of confinement to the core. Confinement improved with increasing V-number, as is

typical of straight fibers. The value ?(1-b s )/V = 1 corresponds to ? = ?trans /2, where ?trans is

defined in Section 3.2.

3.2 Transition to whispering gallery modes

Inspection of the animations in Figs. 5 and 6 also indicates that the modes of the bent fiber may be grouped into two classes: perturbed modes at large bend radii, resembling those of the straight fiber, and whispering gallery modes at smaller radii, which fill only a fraction of the fiber core. It is clear that the most drastic field deformation occurs in the latter case. Establishing where the transition between the two cases takes place is therefore essential to understanding the bent step-index fiber.

This is accomplished by noting that in the conformal mapped coordinate system [16], bending causes the waveguide’s refractive index distribution to tilt according to [16,27]

???????

?+?eff straight bent R x n n 2122, (9) where n straight is the refractive index distribution of the straight fiber. This effect is shown schematically in Figs. 1 and 2. For adequately small bend radii, the sloping refractive index n bent falls below the modal effective index n eff within the fiber core, as illustrated in Fig. 8. Since the mode fields must decay in regions where n bent < n eff , this effectively confines the fields to a limited region of the core, where n bent > n eff . The end result is a whispering mode, as shown in Fig. 8.

The width of the guided region of the core (where n bent > n eff ) in the x-direction may also be shown with the aid of Eqs. (7) and (9) to be the smaller of 2a or

()2221core eff effx n b NA R W ?=

, (10)

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Guided (n bent > n eff )Evanescent (n bent < n eff )

Evanescent

(n bent < n eff )n bent |U|X

Fig. 8. Refractive index distribution and corresponding fundamental mode field profile |U| for a

bent fiber. Shown versus x , through the center of the fiber (y = 0). The mode fields are guided

(oscillatory) where n > n eff , and evanescent (decaying) where n < n eff . With sufficient bending,

the width of the guided region, and thus the mode, is reduced.

when measured through the center of the fiber (the latter condition for whispering gallery modes, and the former for perturbed). It follows that the transition between the whispering gallery and perturbed regimes occurs approximately when the widths W effx and 2a are equal. Thus, using Eqs. (1) and (4), this transition may be shown to occur at the ?-number

b V n n b V clad core trans ?????????????????≈?14142. (11)

In addition, it will be shown in Section 3.4 that the normalized angular propagation constant at this transition point, denoted b trans , is given approximately by

12?≈s trans b b , (12)

for all fiber modes. In light of this, the transition bend radius ?trans is instead defined by combining Eqs. (11) and (12), leading to

???

??????≡?s trans b V 12. (13) This definition is quite powerful, because it depends only on well-known properties of the straight fiber .

For reference in the above expressions, the normalized angular propagation constants of various straight fiber modes are plotted as Fig. 19 in Appendix F. They may alternatively be found from the characteristic equation [1], which in terms of normalized quantities, for n clad ? n core , is given by ()()()()

()s s s m s m s s m s m s b Vb m b V K b V K b b V J b V

J b ?+=′+?′??++++1111111

Here J and K are Bessel and modified Bessel functions, the prime represents differentiation with respect to the argument, and m is the azimuthal mode number, corresponding to the subscript in LP mn . This equation may easily be solved numerically.

3.3 Effective mode area

The effective mode area, defined as the area where the fields surpassed the 1/e 2 power level, is plotted for the fundamental mode of various bent fibers in Fig. 9. In each case the mode area was normalized to that of the straight fiber to demonstrate a clear trend. In the perturbation region, where ? > ?trans , the mode areas remained relatively constant as the fibers were bent. However, in the whispering gallery region (? < ?trans ), the mode areas decreased steadily with bending, following essentially the same path. For reference, the area of the guided portion of the core (where n core was greater than n eff ) is also plotted in the figure. This illustrates that the reduction in mode area with bending followed the same trend as the area of the guided region, as expected.

0.01

0.11

0.0010.010.1110100

?/?trans

A b e n t /A s t r a i g h t

Fig. 9. Effective mode area A bent of the fundamental (LP 01) mode of various bent fibers. Mode

area is normalized to that of the straight fiber A straight for comparison. For all ? > ?trans the

mode areas were essentially the same as those of the straight fiber. For ? < ?trans , they

decreased steadily, following a path which was independent of fiber V-number. This variation

in mode area followed the same trend as the variation in area of the guided region, indicated by

the dashed line. Each curve was truncated where simulated radiation loss became excessive

(over 10-4 dB/λ for typical fiber NA ).

The effective mode areas of the lowest-order modes of the bent fiber were also plotted in Fig. 10, for the V-number 29.5. These follow the same trend as the fundamental mode, although with some ripples caused by reorientation of the fields as the fiber was bent. From these curves it is clear that the quantity ?trans

marks the approximate boundary of the whispering gallery mode region, where the effective mode areas begin to fall off. This was true of all modes and fibers simulated. Figures 9 and 10 demonstrate not only when each mode’s area begins to fall off (at ?trans ), but also how it varies with bending. In each case mode area tracks the size of the guided region, as shown. A mathematical expression for the size of the guided region is given

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

0.01

0.1

110

0.0010.010.1110100

?/?trans

A b e n t /A s t r a i g h t

Fig. 10. Effective mode areas of the lowest-order bent fiber modes, for V = 29.5. Mode areas

were normalized to those of the straight fiber for comparison. For all ? > 2?trans the effective

areas were essentially the same as those of the straight fiber. For ? < ?trans , they decreased

steadily, following a similar trend. Ripples in the curves for higher-order modes were related

to significant reorientation of the mode fields in the whispering gallery region.

in Appendix E. Such results are of particular importance for devices that rely on large mode areas for optimum performance, such as current high-power fiber lasers and amplifiers

[28,29].

It is next useful to return to Eq. (13) and consider the relationship between ?trans and core size in detail. Note that the right side of Eq. (13) depends only on V-number for each given mode. It is thus a simple matter to determine ?trans uniquely for each value of V , with the aid of the characteristic Eq. (14). The whispering gallery transition may then be related to the area of the core, through the expression

220224NA V a A core πλπ==, (15)

as has been done in Fig. 11 for various LP 0n modes of the step-index fiber. Each curve indicates that ?trans increases roughly as A core 3/2, while holding NA/λ0 constant. Other modes not shown in the figure also follow the same trend. Thus, as core size increases for a given mode and value of NA/λ0, the whispering gallery transition takes place at a correspondingly larger bend radius.

This limitation is considerably relaxed, however, with increasing mode order. Higher-order modes therefore allow much tighter bending without significant reductions in mode area. This is an added benefit compared to the usual motivation for using higher-order modes in large mode area (LMA) fibers: their reduced probability of inter-modal scattering [30, 31]. Such an approach has limitations, however, imposed by the restriction that ?trans must remain

greater than 4V . This constraint is indicated in Fig. 11 by the dashed line, which marks the limit as b trans goes to zero (and thus b s goes to ?, through Eq. (13)). Along this dashed line, ?trans increases much more gradually than for each individual mode, as A core 1/2.

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

Fig. 11. Relationship between core area and the whispering gallery transition bend radius,

?trans , in step-index fiber. For each particular mode, the bend radius where the whispering

gallery transition occurs must increase as A core 3/2. However, with increasing mode order, ?trans

is reduced significantly. The fundamental minimum value of ?trans for a given core area is

marked by the dashed line, which corresponds b s = ?, and thus b trans = 0. Along this boundary,

?trans increases as A core 1/2. The shaded region marks where bend loss becomes prohibitive, for

typical fiber NA (the dotted line indicates where loss was of the order 10-7 dB/λ0).

It is also worthwhile to consider how the numerical aperture enters into these results. This is accomplished by recasting Eq. (13) in terms of non-normalized quantities, which leads to an expression for the bend radius at the whispering gallery transition

2)(12???????????????=NA n b a R core s trans eff . (16)

This radius may be reduced, for a given core size, both by reducing b s (propagating higher-order modes), and by increasing the fiber NA . The latter conclusion is contrary to the conventional approach for obtaining large mode area, that of reducing the numerical aperture

[32]. However, it has the advantages that with increasing NA the modes become more strongly guided [17], less prone to inter-modal scattering [30-31], and scaleable to larger areas for a given bend radius. Its primary disadvantage is that that more modes are guided as the numerical aperture increases, so mode selection [28,31,33-35] becomes more challenging.

3.4 Propagation constants

The variation in the propagation constant of the bent step-index fiber is plotted in Fig. 12, where Δβ2 = Re(β)2 - βs 2, for the lowest-order modes, and the V-numbers 2.36, 7.375, and 29.5 (the normalization on the vertical axis will be discussed in a moment). In each case, when ? > ?trans the propagation constant increased to second order or greater in R -1, as predicted by perturbation theory [8]. However, in the whispering gallery region the propagation constant increased more gradually. This indicates that in addition to marking where mode area begins to fall off, ?trans is indicative of where simple perturbation theory breaks down.

#86279 - $15.00 USD

Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

0.010.1110

?/?trans

(Δβ2/k 02N A 2)(b s /(1-b s )) Fig. 12. Variation in propagation constant β with bending, for a variety of modes and V-

numbers. Where ? > ?trans , the propagation constant increased roughly to second order in R -1.

For the LP 11e mode, the change was significantly less than other simulated modes. By

normalizing the vertical axis by (k 0NA)2(1-b s )/b s , the curves for similar modes were made to

overlap rather well, regardless of V-number. The curves all converge in the whispering gallery

region.

Figure 12 also shows that when Δβ2 was normalized according to (k 0NA)2(1-b s )/b s , the curves were relatively independent of the fiber V-number for each particular mode. Such a result is in keeping with the previous observation that modes are similar at a given value of ?(1-b s )/V , which is easily shown to be equal to 2?/?trans . Thus, in addition to the mode fields being similar for a given value of ?/?trans , their propagation characteristics were as well. This leads to the conclusion that the curves in Fig. 12 are reasonably general, at least in the range tested where b s > ?.

That the curves in Fig. 12 all converge together in the whispering gallery region, regardless of mode or V-number, is remarkable. In order to clarify this trend, Figure 13(a) plots the same data from the whispering gallery region of Fig. 12, but on different axes. The dashed line in the figure marks where the two axes were equal, and matched the data quite well for the full range ? < ?trans . This implies that an approximate expression for the real part of β in the whispering gallery region is, for all modes,

()()???

??????

???????????????????????????+≈3/22022111Re trans trans s s s b b NA k ββ (? < ?trans ) (17) Although this expression represents an empirical fit, it was remarkably accurate for all modes and fibers simulated.

That such a simple expression exists for the propagation constant, despite wide variations in the field distributions, is related to the fact that the modes shift away from center of the fiber with bending. As the modes shift outward, they must travel further around the curve than if centered about the core, and the resulting path delay causes their propagation constants to increase accordingly. In the perturbation region this effect is relatively minor because the outward motion is constrained by the core-cladding interface. However, in the whispering gallery region it becomes the dominant effect.

#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

101001000

1101001000

(?trans /?)-(?trans /?-1)2/3[(Δβ2/(k 02N A 2))(b s /(1-b s ))]+1

0.00010.0010.010.110.00010.001

0.010.11

2V/?b s -b

Fig. 13. (a). Simulated data from the whispering gallery region of Fig. 12, plotted along different axes. The dashed line marks where the horizontal and vertical axes are equal, and

matched the data well for all mode and fibers simulated. (b) Variation in the normalized

angular propagation constant with bending, for a variety of modes and V-numbers. All data

corresponds to the perturbation region, ? > ?trans . The dashed line indicates where the

horizontal and vertical axes are equal, and matched the data extremely well for all fibers and

modes simulated.

The variation in the normalized angular propagation constant with bending is plotted in Fig. 13(b), for a variety of modes and V-numbers. Each point in the figure corresponds to the perturbation region (? > ?trans ), and falls along the line

?V b b s 2 (? > ?trans ) (18)

with considerable accuracy. Such a result is consistent with the fact that β changes very little with bending for ? > ?trans , which causes n eff to remain relatively constant at the center of the fiber. Thus, as the refractive index distribution tilts with bending, it is easily shown that to first order b must decrease according to Eq. (18).

This is an important result because when combined with Eq. (11), it leads to the simple expressions for b trans in Eq. (12), and ?trans in (13). Furthermore, Eq. (12) implies that the only modes capable of transitioning to whispering gallery modes before reaching cutoff (b = 0) are those for which b s is greater than ?. This explains why analysis of the bent single-mode fiber is relatively simple: it is incapable of transitioning to a whispering gallery mode since b s is always less than 0.53 (see Fig. 19). Single-mode fibers are therefore reasonably well-described by perturbation theory (b s can be slightly larger than ?, but this is relatively inconsequential). Multimode fibers, on the other hand, and in particular those modes for which (1-b s ) is small, exhibit more complicated behavior due to the existence of the whispering gallery transition.

An empirical relation for the normalized angular propagation constant in the whispering gallery region may be derived from Eq. (17), with the aid of Eqs. (1-7) and (9), ()???

????????????????????????+??????????≈s trans trans s s s b b b b b 11113/2. (? <

?trans ) (19) This expression is useful for predicting bend loss in step-index fibers, as will be discussed in a future publication [24]. It is also valuable for predicting mode areas since it is directly related to the size of the guided region (where n eff < n core ), as discussed in Section 3.3.

4. Extension to other fiber geometries

Although the discussion thus far has focused on simple step-index fiber, much of the preceding analysis may also be applied to more complicated fiber geometries. The following #86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

subsections discuss how the results of Sections 2 and 3 may be extended to weakly-guided fibers of more complex geometry.

4.1 Mode scalability in arbitrary weakly-guided fibers

Section 2 introduced two conditions for mode scaling in bent step-index fibers: one corresponding to the bend radius (?), and the other to the fiber cross-section (V ). With more complicated fiber geometries analogous conditions also exist, such that the modes of most fibers are also generally scalable. These are derived in Appendices A and B, and summarized below. The only restriction inherent in the following conditions is that the fiber be weakly-guided (Δn << n ), from which it follows that it must also be slowly bent (W effx << R ). Since this condition is easily satisfied, the following results cover a vast range of fiber geometries.

A fiber of arbitrary cross-section, supporting the transverse mode field distributions U T (r T ), is first considered. Appendices A and

B show that an infinite number of other fibers also exist that will support identical mode field distributions, except scaled in size by the factor M such that

()()M

r U r U T T T T =′′, (20) where T r ′ represents the scaled coordinate system M r r T T ≡′. (21)

These fibers differ from the original only in that their refractive index profiles are also scaled, prior to bending , according to the expression ()()()2201

22,M

k r g M n M r n T ref T straight ′+=′ . (22) Here the function g 1 describes the spatial variation in the fiber’s index profile, and n ref is a spatially invariant reference index, though not necessarily the background. Both are known at M = 1, as determined by the original fiber’s refractive index profile. The spatial variation in Eq. (22) is thus defined from the start. The form of n ref 2(M) may be chosen arbitrarily, however, provided that the fibers remain weakly guided. For bent fibers the effective bend radius must also scale such that

()()M n M k M R ref o eff 232∝,

(23) where again the proportionality depends on the original fiber at M = 1. These relations demonstrate how wavelength, refractive index, and bend radius are all interrelated, such that the mode field distributions may be scaled exactly for any weakly-guided fiber. Equations

(22) and (23) are therefore analogous to those presented earlier for V and ? in the case of step-index fiber. However, they apply to all fiber geometries, under the assumptions of weak-guiding and slow bending.

Furthermore, when Eqs. (22) and (23) are satisfied, each mode’s effective index will also scale according to

()()()201

22M k c M n M n ref eff +=, (24)

where the constant c 1 depends upon the fiber geometry and the particular mode, and is presumably known for the original fiber. This expression is in turn analogous to the result from Section 2 that the normalized angular propagation constant will remain invariant for a given pair V and ?. Such is evident from Eqs. (7), (22) and (24). However, Eq. (24) represents the general form.

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Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

4.2 General condition for similar modes

In addition to the previous relations that describe how the modes will scale exactly in arbitrary bent fibers, there also exists an analogous expression to that of Section 3.1 for which the modes of the bent step-index fiber will be similar, but not exactly the same, for constant ?(1-b s )/V . In fibers of arbitrary cross-section, this condition generalizes to

()()[]()T eff T straight eff

r g M n r n M R ′=?′ 222

, (25)

where the function g 2 is determined by the refractive index profile n straight at M = 1. This relation describes a family of bent fibers for which modes will be similar to each other from fiber to fiber, such as those illustrated in Fig. 7 at the end of Section 3.1. It is important to note, however, that Eq. (25) is in no way related to Eqs. (23) and (24), which pertain to mode scaling in its exact form. It is also worth noting that unlike Eq. (22), the magnitude of n straight remains constant in Eq. (25).

4.3 Comparison to step-index fiber

That mode scaling is possible in all weakly-guided fibers suggests that certain behavior discussed in Section 3 may also be extended to more complex fiber geometries. With more complicated refractive index profiles, however, the fiber NA given by Eq. (2) is no longer valid (here the NA is based on the index step, rather than far field output, which is an important distinction [14]). Consequently, the quantities V , ? and b s are no longer defined. Nevertheless, much of the previous analysis may still be applied to an important class of fibers such as those in Fig. 14, provided some clarification.

Fig. 14. Refractive index profiles of similar bent fibers, neglecting stress, assuming cylindrical

symmetry. Also shown is the variation in the modal effective index in the ρ-direction when

bent to the same bend radius. All modes indicated are confined to the region from –a to a , and

reach the whispering gallery transition at the same bend radius as shown. Variation in n eff from

fiber to fiber due to differences in the refractive index profiles have been omitted for the

purpose of illustration.

Although the fiber cross-sections in Fig. 14 differ substantially, each has the same index step at the same position, |ρ-R| = a . For the moment we also consider modes with similar

values of n eff

, as shown, such that their fields are guided in the region from #86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

–a to a. Since perturbation theory indicates that β should vary to second order or greater in R-1 for all fibers symmetric in the ρ-direction [8], and in this regard the bent fiber has been shown to follow perturbation theory until reaching the whispering gallery transition, it follows that n eff will remain relatively constant at the center of each fiber prior to this transition. Thus, as the fibers in Fig. 14 are bent, their effective indices will pivot about the fiber center as shown. The result is that each mode in the figure transitions to a whispering gallery mode at essentially the same bend radius. The only disparity stems from the fact that n eff should vary somewhat from fiber to fiber due to differences in the refractive index profiles.

These examples demonstrate that the behavior of more complicated fiber geometries is often closely related to that of the simple step-index fiber. The primary difference, as far as the whispering gallery transition is concerned, is that the refractive index step at the boundary that confines the mode must be used when computing V, ? and b s, in this case |ρ-R| = a. Equation (13) will then express the bend radius where the size of the guided region begins to vary. This point corresponds to the whispering gallery transition in all examples of Fig. 14, at which the modes are no longer entirely confined by the index step. However, it is not necessarily true of all fiber geometries.

In general, beyond ?trans the modes and propagation constants will vary differently from fiber to fiber due to differences in their refractive index profiles. In light of this, the simple step-index fiber merely serves as a guide for the other fibers in Fig. 14. Nonetheless, behavior should be quite comparable for the fibers in Figs. 14(a) and 14(b), and to a lesser extent that in 14(c). This is due to the fact that the guided region remains identical same in each case.

5. Summary

This paper has introduced a general and relatively simple framework for understanding the behavior of bent, weakly-guided fibers and waveguides. Much of the analysis has dealt specifically with the bent step-index fiber, resulting in a rather comprehensive study of its modal behavior. Some of the more important points are summarized as follows:

1)Just as the modes of the straight step-index fiber depend solely on the V-number, in the

bent fiber they are determined by the two quantities V and ?. The mode field distributions all scale exactly as V and ? are held constant, and their normalized angular propagation constants remain the same.

2)The modes of fibers with different values of V and/or ? are similar, but not exactly the

same, when bent such that ?/?trans is conserved.

3)The transition to a whispering gallery mode occurs approximately at the ?-number ?trans.

4)Prior to the whispering gallery transition, mode areas and angular propagation constants

are both relatively unaffected by bending. Thereafter they depend primarily on ?/?trans, regardless of the particular field distribution.

5)Modes for which b s < ? reach cutoff (b = 0) before reaching the whispering gallery mode

transition.

6)The bend radius of whispering gallery transition may be reduced, for a given core area,

both by reducing b s (propagating higher-order modes), and by increasing the fiber NA.

7)The lowest-order fiber modes transition to linearly polarized states when bent to the

whispering gallery transition, and beyond (see Appendix D). Although exceptions are possible, they are unlikely under the assumption of weak-guiding.

8)All quantities necessary for computing V, ? and ?trans are those of the straight fiber, and

are thus well known.

In addition, much of the preceding analysis for the bent step-index fiber may be extended to the specific group of common fiber designs indicated in Fig. 14.

More generally, this paper has shown that the modes of any bent, weakly-guided waveguide may be considered as representative of those of an entire family of waveguides, for which two scaling conditions apply: one for the waveguide cross section, and the other for the bend radius. This allows the behavior of a single waveguide to be extended to the entire

group of similar designs, and as such, greatly simplifies the analysis of bent, weakly-guided fibers and waveguides.

Acknowledgment

The author would like to thank James H. Cole, Carl A. Villarruel, and Frank Bucholtz for many illuminating discussions, and Jeff Salzano for assisting with the multimedia. This research was performed while the author held a National Research Council Research Associateship Award at the U.S. Naval Research Laboratory, Washington, DC.

Appendix A: General conditions for mode scaling

It is well-known that the modes of the straight, weakly-guided step-index fiber may be scaled in size, without otherwise affecting the transverse fields, while maintaining the same V-number from fiber to fiber. The purpose of this appendix is to introduce analogous conditions for straight, weakly-guided waveguides in general. Such conditions, when satisfied, allow the modes of one particular waveguide to be extended to an entire family of other waveguides, with different mode sizes, refractive indices and wavelengths.

The electromagnetic wave equation may be expressed in terms of electric field in a general dielectric medium as

()()()()E E E ×?×????????????=+?μμεεμεω1122. (A1)

The left side of this expression is the homogenous wave equation, while the right side accounts for variations in the dielectric with position, and reduces to the usual boundary conditions at abrupt interfaces. An identical expression also holds for the magnetic field, but with ε and μ reversed. If it is assumed that the dielectric does not vary in the z -direction, and supports at least one guided mode, then each of the guided modes may be expressed in the form ()()()z j r U E r E T o β?=exp , (A2) where T r is the coordinate transverse to the z -direction. By combining Eqs. (A1-A2), it follows that the guided modes must satisfy the relations ()()()()T T T T T T T T U U U ×?×????????????=?+?μμεεβμεω11222 (A3) ()()()()[]

T z T T T T z T U U U j U βμμ

εεββμεω+????????????=?+?11222 (A4) for their transverse and longitudinal vector components, respectively.

The boundary conditions on the right in (A3-A4) determine the propagation constant β of each guided mode. They also cause the normal components of the fields to become discontinuous across discrete boundaries, and lead to coupling between polarizations. The latter two conditions make it impossible for a second waveguide with different material properties to support exactly the same modes, or even spatially scaled versions, except for the trivial cases in which √ε and √μ are either exactly the same as the original, or scale inversely with waveguide size. In any other situation, the field discontinuity must necessarily be altered. However, for weakly-guided waveguides [17], in which the relative variation in ε and μ is small (Δε << 2ε, Δμ << 2μ), the field discontinuity at the boundaries is negligible. This allows the scalar form of the wave equation, ()

0222

??+?i T U βμεω, (A5) to instead be used, subject to the conditions that U and ?×U must remain continuous. In this weakly-guided regime, it is possible for different waveguides to support otherwise identical mode field distributions, except scaled in size. For the remainder of the appendices, weakly-#86279 - $15.00 USD Received 9 Aug 2007; revised 10 Oct 2007; accepted 12 Oct 2007; published 12 Nov 2007

拟Bent函数的构造

第27卷第5期2010年10月工程数学学报 CHINESE JOURNAL OF ENGINEERING MATHEMATICS Vol.27No.5 Oct.2010 文章编号:1005-3085(2010)05-0865-08 拟Bent函数的构造? 张习勇1,郭华2,滕吉红2 (1-解放军信息工程大学信息工程学院四系，郑州450002; 2-北京航空航天大学计算机学院，北京100083) 摘要:拟Bent函数在密码系统中可用作非线性组合函数和消息摘要函数，因而具有很好的密码学性质。本文通过计算布尔函数的Walsh谱，从映射的角度确定了变元个数不超过六的拟Bent函数的代数结构；提出了一般交换群上的扩展组合函数族的概念，研究了这类函数的性质，利用商群给出了扩展组合函数的下降构造，通过组合函数给出了提升构造法，从而得到了一种由扩展组合函数族构造拟Bent函数和Bent函数的递归构造法，由这种方法可以构造大量拟Bent函数和Bent函数。另外也构造了几种参数的布尔扩展组合函数。关键词:拟Bent函数；扩展组合函数族；组合函数族；递归构造分类号:AMS(2000)94C10;06E30中图分类号:TN918.1文献标识码:A 1引言和基本定义在密码学中，人们设计非线性组合函数时考虑的主要准则有：高的代数次数，高的非线性度，满足一定阶的相关免疫性和一定阶数的扩散准则等等。然而，这些准则相互之间有制约关系[1]。针对这种情况，人们相继提出了部分Bent函数[2]，半Bent函数[3]等概念，这些函数通过牺牲Bent函数的扩散性，来换取满足其它的一些准则。文献[4]等提出了k-阶拟Bent函数的概念，它是上述函数的更大类。由于拟Bent函数不具有极端的密码学性质，能很好地综合上述密码学准则，因而在这类函数中，较易找到满足上述准则的非线性组合函数。研究拟Bent函数的中心问题包括拟Bent函数的性质、构造与应用等。在构造方面，文献[4]利用矩阵，研究了拟Bent函数的密码学性质，文献[5]利用组合函数族给出了拟Bent函数的一种递归构造等等。在拟Bent函数的结构刻画方面，目前还没有相关的结论。本文从映射的角度，刻画了六元二阶拟Bent函数的代数结构，从而实际上确定了变元个数不超过六的拟Bent函数的代数结构。在构造方面，从线性空间和映射的角度，给出了拟Bent函数的四种不同构造法。文献[5]利用组合函数族(BF)给出了拟Bent函数的递归构造，但是不能构造Bent函数，本文进一步从不同的正交角度提出了扩展组合函数族(EBF)的概念，从而结合组合函数族(BF)从正交函数组的角度，给出了拟Bent函数和Bent函数的递归构造。本文中G表示有限Abel群，G?表示G的特征标群，C表示复数域，S={z∈C||z|= 1}，ξm表示m次本原单位根。对任意的f:G→C，f的傅立叶变换定义为 f(χ)= a∈G f(a)·χ(a),?χ∈G?. 收稿日期:2008-12-15.作者简介:张习勇(1975年6月生)，男，博士，副教授.研究方向：密码学. ?基金项目:国家自然科学基金(60803154).

台达伺服调试经验故障排除

Q1：伺服电机与普通电机有何区别？ A1：伺服电机与普通电机最大的区别在于电机转子和反馈装置。伺服电机转子表面贴有强力磁钢片，因此可以通过定子线圈产生的磁场精确控制转子的位置，并且加减速特性远高于普通电机。反馈装置可以精确反馈电机转子位置到伺服驱动器，伺服电机常用的反馈装置有光学编码器、旋转变压器等。 Q2：伺服驱动器输入电源是否可接单相220V ？ A2：台达伺服1.5KW（含）以下可接单相/三相220V电源，2.0KW（含）以上只能接三相220V电源。三相电源整流出来的直流波形质量更好，质量不好的直流电源会消耗母线上电容的能量，电机急加减速时电容会对母线充放电来保持母线电压稳定，因此三相电源输入比单相电源输入伺服的特性会好一些，三相电源输入提供的电流也更大。 Q3：伺服驱动器输出到电机的UVW三相是否可以互换？ A3：不可以，伺服驱动器到电机UVW的接法是唯一的。普通异步电机输入电源UVW两相互换时电机会反转，事实上伺服电机UVW任意两相互换电机也会反转，但是伺服电机是有反馈装置的，这样就出现正反馈会导致电机飞车。伺服驱动器会检测并防止飞车，因此在UVW

接错线后我们看到的现象是电机以很快的速度转过一个角度然后报警过负载ALE06。 Q4：伺服电机为何要Servo on之后才可以动作？ A4：伺服驱动器并不是在通电后就会输出电流到电机，因此电机是处于放松的状态（手可以转动电机轴）。伺服驱动器接收到Servo on信号后会输出电流到电机，让电机处于一种电气保持的状态，此时才可以接收指令去动作，没有收到指令时是不会动作的即使有外力介入（手转不动电机轴），这样伺服电机才能实现精确定位。

ASD伺服常见问题处理方式

ASD伺服常见问题处理方式 1,伺服驱动器输出到电机的UVW三相是否可以互换？不可以，伺服驱动器到电机UVW的接法是唯一的。普通异步电机输入电源UVW两相互换时电机会反转，事实上伺服电机UVW任意两相互换电机也会反转，但是伺服电机是有反馈装置的，这样就出现正反馈会导致电机飞车。伺服驱动器会检测并防止飞车，因此在UVW接错线后我们看到的现象是电机以很快的速度转过一个角度然后报警过负载ALE06。 2,伺服电机为何要Servo on之后才可以动作？伺服驱动器并不是在通电后就会输出电流到电机，因此电机是处于放松的状态（手可以转动电机轴）。伺服驱动器接收到Servo on信号后会输出电流到电机，让电机处于一种电气保持的状态，此时才可以接收指令去动作，没有收到指令时是不会动作的即使有外力介入（手转不动电机轴），这样伺服电机才能实现精确定位。 3,伺服驱动器报警ALE01如何处理？检查UVW线是否有短路。如果把UVW线与驱动器断开再通电仍然出现ALE01则是驱动器硬件故障。 4,ALE02过电压/ALE03低电压报警发生时如何处理？首先使用万用表测量输入电压是否在允许范围内；再次是通过驱动器或伺服软件示波器监视“主回路电压”，这是直流母线电压，电压伏数应该是输入交流电压的1.414倍，正常来讲应该不会有太大的偏差。如果偏差很大需返厂重新校准。ALE02/ALE03报警是以“主回路电压”来判断的。 5,在高速运行时机台在中途有很明显的一钝，观察发现是中途有ALE03报警产生，但是一闪就消失了，如何解决这个问题？在高速运行时会消耗很大能量，母线电压会下降，如果输入电压偏低此时就会出现ALE03报警。报警发生时伺服马上停止，母线电压恢复正常，报警自动消失，伺服会继续运行，因此看起来就是明显的一钝。这种情况多发生在使用单相电源供电时，建议主回路使用三相电源供电。参数P2-65 bit12置ON可使ALE03报警发生时，母线电压恢复后报警不会自动消失。 6,伺服驱动器报警ALE04如何处理？ AB系列伺服驱动器配ECMA马达时功率不匹配上电会报警ALE04，除这种情况外刚一上电就报警ALE04就是电机编码器故障。如果在使用过程中出现ALE04报警是因为编码器信号被干扰，请查看编码器线是否是屏蔽双绞、驱动器与电机间地线是否连接，或者在编码器线上套磁环。通过ALE04.EXE软件可以监测每次Z脉冲位置AB脉冲计数是否变化，有变化则会报

台达伺服调试

何謂伺服的低頻擺振？當發生低頻擺振時如何處理？若系統剛性不足，在定位命令結束後，即使馬達本身已經接近靜止，機械傳動端仍會出現持續擺動。低頻抑振功能可以用來減緩機械傳動端擺動的現象。低頻抑振的範圍為 1.0 ~ 100.0Hz。本功能提供手動設定與自動設定，但目前只有ASDA-A2系列機種支援此功能。低頻抑振方式分為自動及手動方式： (1) 自動設定若使用者難以直接知道頻率的發生點，可以開啟自動低頻抑振功能。此功能會自動尋找低頻擺動的頻率。若P1-29設定為1時，系統會先自動關閉低頻抑振濾波功能，並開始自動尋找低頻的擺動頻率。當自動偵測到的頻率維持固定後，P1-29會自動設回0，並會將第一擺動頻率設定在P1-25且P1-26設為1。第二擺動頻率設定在P1-27且將P1-28設為1。當P1-29自動設回零後，低頻擺動依然存在，請檢查低頻抑振P1-26或P1-28是否已被自動開啟。若P1-26與P1-28皆為零，代表沒有偵測到任何頻率，此時請減少低頻擺動檢測準位P1-30，並設定P1-29 = 1，重新尋找低頻的擺動頻率。 (2) 手動設定低頻抑振有兩組低頻抑振濾波器，第一組為參數P1-25 ~ P1-26，第二組為參數P1-27 ~ P1-28。可以利用這兩組濾波器來減緩兩個不同頻率的低頻擺動。參數P1-25與P1-27用來設定低頻擺動所發生的頻率，低頻抑振功能唯有在低頻抑振頻率參數設定與真實的擺動頻率接近時，才會抑制低頻的機械傳動端的擺動。參數P1-26與P1-28用來設定經濾波處理後的響應，當P1-26與P1-28設定越大響應越好，但設太大容易使得馬達行走不順。參數P1-26與P1-28出廠值預設值為零，代表兩組濾波器的功能皆被關閉。伺服煞車電阻使用時機為何？當伺服驅動器搭配馬達運轉時，若驅動器面板出現ALE05（回生能量異常）時，代表馬達回生產生的能量超過驅動器內建回生電阻所能消耗的能量，此時必須安裝回生電阻，提高驅動器回生能量消耗速度。 ASDA-A2系列內建回生電阻規格：

台达伺服器报警与处理

台达伺服器异警处理 RLE01：过电流：主回路电流值超越电机瞬间最大电流值1.5倍时动作 1．驱动器输出短路：检查电机与驱动器接线状态或导线本体是否短路，排除短路状态，并防止金属导体外露 2. 电机接线异常：检查电机连接至驱动器的接线顺序，根据说明书的配线顺序重新配线 3. IGBT 异常：散热片温度异常，送回经销商或原厂检修 4. 控制参数设定异常：设定值是否远大于出厂预设值，回复至原出厂预设值，再逐量修正 5. 控制命令设定异常：检查控制输入命令是否变动过于剧烈，修正输入命令变动率或开启滤波功能 RLE02：过电压：主回路电压值高于规格值时动作 1．主回路输入电压高于额定容许电压值：用电压计测定主回路输入电压是否在额定容许电压值以内（参照11-1），使用正确电压源或串接稳压器 2. 电源输入错误（非正确电源系统）：用电压计测定电源系统是否与规格定义相符，使用正确电压源或串接变压器 3. 驱动器硬件故障：当电压计测定主回路输入电压在额定容许电压值以内仍然发生此错误，送回经销商或原厂检修 RLE03：低电压：主回路电压值低于规格电压时动作 1．主回路输入电压低于额定容许电压值：检查主回路输入电压接线是否正常，重新确认电压接线 2. 主回路无输入电压源：用电压计测定是否主回路电压正常，重新确认电源开关 3. 电源输入错误（非正确电源系统）：用电压计测定电源系统是否与规格定义相符，使用正确电压源或串接变压器 RLE04：RLE04：Z 脉冲所对应磁场角度异常 1．编码器损坏：编码器异常，更换电机 2. 编码器松脱：检视编码器接头，重新安装 RLE05：回生错误：回生控制作动异常时动作 1．回生电阻未接或过小：确认回生电阻的连接状况，重新连接回生电阻或计算回生电阻值 2. .回生用切换晶体管失效：检查回生用切换晶体管是否短路，送回经销商或原厂检修 3. 参数设定错误：确认回生电阻参数（P1-52）设定值与回生电阻容量参数（P1-53）设定，重新正确设定 RLE06：过负载：电机及驱动器过负载时动作 1．超过驱动器额定负载连续使用：可由驱动器状态显示P0-02设定为11后，监视平均转矩[%]是否持续一直超过100%以上，提高电机容量或降低负载 2. 控制系统参数设定不当：机械系统是否摆振、加减速设定常数过快，调整控制回路增益值、加减速设定时间减慢 3. 电机、编码器接线错误：检查U、V、W 及编码器接线是否准确 4. 电机的编码器不良：送回经销商或原厂检修 RLE07：过速度：电机控制速度超过正常速度过大时动作 1．速度输入命令变动过剧：用信号检测计检测输入的模拟电压信号是否异常，调整输入变信号动率或开启滤波功能 2. 过速度判定参数设定不当：检查过速度设定参数P2-34（过速度警告条件）是否太小，检查过速度设定参数P2-34（过速度警告条件）是否太小 RLE08：异常脉冲控制命令：脉冲命令的输入频率超过硬件界面容许值时动作 1．脉冲命令频率高于额定输入频率：用脉冲频率检测计检测输入频率是否超过额定输入频率，正确设定输入脉冲频率 RLE09：位置控制误差过大：位置控制误差量大于设定容许值时动作 1．最大位置误差参数设定过小：确认最大位置误差参数P2-35（位置控制误差过大警告条件）设定值，加大P2-35 （位置控制误差过大警告条件）设定值 2. 增益值设定过小：确认设定值是否适当，正确调整增益值 3. 扭矩限制过低：确认扭矩限制值，正确调整扭矩限制值 4. 外部负载过大：检查外部负载，减低外部负载或重新评估电机容量。更换摇床电机。 RLE10：芯片执行超时：芯片异常时动作 1．芯片动作异常：电源复位检测，复位仍异常时，送回经销商或原厂检修 RLE11：编码器异常：编码器产生脉冲信号异常时动作

台达伺服电机驱动器的常见问题

三相機種的變頻器是否可以接單相入力電源？台達變頻器為單相及三相機種，其最大的差異在於電容的配置。單相機種會配置比較大的電容，因此若三相機種只接單相入力，可能導致輸出電流不足，且會發生欠相的異常。為確保系統正常運行，請搭配使用正確的電源系統。變頻器使用在硬體上需加裝PG卡，在PG卡上的開關設置編碼器為Open-Collector或是 Line-Driver型式，並設置正確的電壓大小。在參數上，設定編碼器每轉的脈波數及輸入脈波型式。以台達VFD-VE系列變頻器為例，選用EMV-PG01X的PG卡，且編碼器一圈有1024個脈波，為Open-Collector 12V型，此時，PG卡需設置(如下圖) 在參數設定方面，需設定參數10-00每轉脈波數為1024。另外，在設定10-01之前，需先確定該編碼器的脈波型式為AB相、脈波加方向或單一脈波，再加以設定。之後只要將參數00-04設為7，就可以在使用者顯示的內容看到馬達實際由編碼器回授的轉速。無感測向量控制 a.優異開迴路速度控制，不必滑差補償 b.在低度時有高轉矩，不必提供過多之轉矩增強 c.更低損耗,更高效率 d.更高動力響應- 尤其是階梯式負載 e.大馬達有穩定之運轉 f.在電流限制，改善滑差控制有較好之表現在台達交流馬達驅動器的輸入

電源輸入側電抗器用於變頻器/驅動器輸入端，電抗器保護著靈敏電子設備使其免受變頻器產生的電力雜訊干擾(如電壓凹陷、脈衝、失真、諧波等)，而藉由電抗器吸收電源上的突波，更能使變頻器受到良好的保護。變頻器/驅動器輸出側電抗器在長距離電纜接線應用中，使用IGBT保護型電抗器於馬達與變頻器之間，來減緩dv/dt值及降低馬達端的反射電壓。使用負載電抗器於輸出端，可抑制負載迅速變化所產生的突波電流，即使是負載短路亦可提供保護。何謂控速比可控速範圍是以馬達的額定轉速為基準，在定轉矩操作區中為維持額定轉矩，其額定轉速與最低轉速的比值，例如一典型交流伺服馬達的可控速範圍為1000:1，亦即若馬達的額定轉速為2000 rpm/min，其最低轉速為2 rpm/min；而且在此控速範圍內，由無載至額定負載時，其轉速誤差百分比值均能滿足所設定的控速精度，如+-0.01%。轉速誤差百分比值是由下式計算：(如下圖) 什麼是變頻器的失速防止功能？如果給定的加速時間過短，變頻器的輸出頻率變化遠遠超過轉速的變化，變頻器將因流過過電流而跳機，而自由運轉停止，這就是失速。為了防止失速使馬達繼續運轉，就要檢出電流的大小進行頻率控制。當加速電流過大時，適當放慢加速速率。減速時也是如此。兩者結合起來就是失速防止功能。變頻器的哪些模式可以調整馬達轉速？變頻器上的轉速控制主要有以下： 1. 直接從變頻器面版上的可變電阻調整 2. 外接類比電壓或電流信號來調整 3. 利用變頻器的多功能輸入端子可達成多段速控制 4. 台達變頻器支援Modbus通訊，可利用上位控制器以通訊的方式改變變頻器轉速。請問可以，只要韌體版本為4.08版，即可運轉到2000Hz。請問不可以，因為EF輸入端子是數位端子，只有開及關的狀態而已，所以不能作為PTC的輸入端。請問

台达伺服定位控制案例

X1 Y0脉冲输出Y1正转/反转Y 脉冲清除 4DOP-A 人机 ASDA 伺服驱动器【控制要求】 ● 由台达PLC 和台达伺服，台达人机组成一个简单的定位控制演示系统。通过PLC 发送脉冲控制伺服，实现原点回归、相对定位和绝对定位功能的演示。 ● 下面是台达DOP-A 人机监控画面：原点回归演示画面相对定位演示画面

绝对定位演示画面【元件说明】

【PLC 与伺服驱动器硬件接线图】台达伺服驱动器码器 DO_COM SRDY ZSPD TPOS ALAM HOME

【ASD-A伺服驱动器参数必要设置】当出现伺服因参数设置错乱而导致不能正常运行时，可先设置P2-08=10（回归出厂值），重新上电后再按照上表进行参数设置。【控制程序】

M1002 MOV K200 D1343 Y7 Y10 Y11 M20 M21 M22 M23 M24 M1334 Y12 M1346 M11 X0 X1 X3 X4 X5 X6 X7 M12 M13 设置加减速时间为 200ms Y6 M10 伺服启动伺服异常复位M0M1M2M3M4M1029 DZRN DDRVI DDRVI DDRVA DDRVA ZRST K10000 K100000K-100000K400000K-50000K5000 K20000 K20000 K200000 K200000 X2 Y0 Y0 Y0 Y0 Y0 Y1 Y1 Y1 Y1 M1M0M0M0M0M2M2M1M1M1M3M3M3M2M2M4 M4 M4 M4 M3 M0 M4 原点回归正转圈 10跑到绝对坐标，处400000跑到绝对坐标，处 -50000定位完成后自动关闭定位指令执行伺服计数寄存器清零使能反转圈10伺服电机正转禁止伺服电机反转禁止PLC 暂停输出脉冲伺服紧急停止伺服启动准备完毕伺服启动零速度检出伺服原点回归完成伺服定位完成伺服异常报警

台达A系列伺服电机调试步骤

台达A系列伺服电机调试步骤 The Standardization Office was revised on the afternoon of December 13, 2020

第七轴通过伺服电机运行的调试步骤一、概述此文档将介绍如何通过西门子PLC来控制伺服电机的正转、反转、以某一速度进行绝对位置的定位以及电机运行错误后如何复位，伺服驱动器如何设置参数等一些最基本的伺服电机的运行操作步骤。二、需准备的材料 1、西门子S7-1200系列PLC一台（我们准备的S7-1200 CPU1215C DC/DC/DC） 2、台达伺服电机ECMA-L110 20RS一台 3、台达伺服控制器ASD-A2-2023-M一台 4、威纶通触摸屏MT-8012IE一台 5、博途V15设计软件 6、威纶通设计软件三、调试步骤及简单说明调试之前首先将所有设备按照安装说明书上控制接线部分的介绍正确的接入电源，所有设备中需要特别注意的是伺服控制器的进线是三项220V 的电压。建议先让伺服电机在无负载的作用下正常运作，之后再将负载接上以免造成不必要的危险，伺服驱动器的控制用CN1信号端口来接线控制（CN1端口如何接线将提供接线图来接线）。

1、伺服驱动器的参数设置 1）、伺服驱动器面板介绍 2）、启动电源面板将显示以下几种报警画面，根据需要将参数调整到位。画面一：将参数P2-15、P2-16、P2-17三个参数设定为0

画面二：将参数P2-10~P2-17参数中没有一个设定为21 画面三：将参数P2-10~P2-17参数中没有一个设定为23

3）、以上步骤调整好之后可以利用JOG寸动方式来试转电机和驱动器，操作步骤如下图

比较PageRank算法和HITS算法的优缺点

题目：请比较PageRank算法和HITS算法的优缺点，除此之外，请再介绍2种用于搜索引擎检索结果的排序算法，并举例说明。答： 1998年，Sergey Brin和Lawrence Page[1]提出了PageRank算法。该算法基于“从许多优质的网页链接过来的网页，必定还是优质网页”的回归关系，来判定网页的重要性。该算法认为从网页A导向网页B的链接可以看作是页面A对页面B的支持投票，根据这个投票数来判断页面的重要性。当然，不仅仅只看投票数，还要对投票的页面进行重要性分析，越是重要的页面所投票的评价也就越高。根据这样的分析，得到了高评价的重要页面会被给予较高的PageRank值，在检索结果内的名次也会提高。PageRank是基于对“使用复杂的算法而得到的链接构造”的分析，从而得出的各网页本身的特性。 HITS 算法是由康奈尔大学( Cornell University ) 的JonKleinberg 博士于1998 年首先提出。Kleinberg认为既然搜索是开始于用户的检索提问，那么每个页面的重要性也就依赖于用户的检索提问。他将用户检索提问分为如下三种：特指主题检索提问（specific queries，也称窄主题检索提问）、泛指主题检索提问（Broad-topic queries，也称宽主题检索提问）和相似网页检索提问（Similar-page queries）。HITS 算法专注于改善泛指主题检索的结果。 Kleinberg将网页（或网站）分为两类，即hubs和authorities，而且每个页面也有两个级别，即hubs（中心级别）和authorities（权威级别）。Authorities 是具有较高价值的网页，依赖于指向它的页面；hubs为指向较多authorities的网页，依赖于它指向的页面。HITS算法的目标就是通过迭代计算得到针对某个检索提问的排名最高的authority的网页。通常HITS算法是作用在一定范围的，例如一个以程序开发为主题的网页，指向另一个以程序开发为主题的网页，则另一个网页的重要性就可能比较高，但是指向另一个购物类的网页则不一定。在限定范围之后根据网页的出度和入度建立一个矩阵，通过矩阵的迭代运算和定义收敛的阈值不断对两个向量authority 和hub值进行更新直至收敛。从上面的分析可见，PageRank算法和HITS算法都是基于链接分析的搜索引擎排序算法，并且在算法中两者都利用了特征向量作为理论基础和收敛性依据。

台达伺服常见故障分析与解决

1.增量型伺服初次上电报警解决步骤：报警代码涉及参数设定值 AL013 P2-15 0 AL014 P2-16 0 AL015 P2-17 0 更改完参数，需重新上电。 2.绝对值伺服初次上电报警解决步骤：除了以上问题，还有绝对值伺服本身的设定参数。绝对值伺服上电会报AL060 绝对值伺服设定步骤： 2-08 先设 30 再设 28（断电上电） 2-69 1 2-08 271 2-71 1 0-49 1 看0-51 0-52 3.测试过程中，出现报警及解决方法：报警代码涉及参数（故障原因）设定值（或修改方法） AL006 启动短时间报警电机堵转， U V W 接错 AL011 位置检出器异常查看编码器线，或排除干扰 AL018 若是伴随 AL011 出现按 AL011 处理 AL009 P2-35 上限值检查负载或者电子齿轮比设定 AL018 确认以下条件是否产生：正确设定参数 P1-76 与 P1-76< 电机转速与P1-46 ： 1 46 4 19.8 10 P1-76> 电机转速与 6 1 46 4 19.8 10 60 6 电机转速P AL024 AL026

编码器初始磁场错误电机接地端是否正常接 (磁场位置 UVW 错误地 2. 编码器讯号线，是否有与电源或大电流的线路分开，避免干扰源的产生 3. 位置检出器的线材是否使用隔离线1. 电机接地端是否正常 1. 请将 UVW 接头的接接地地端(绿

2.编码器讯号线，是否有色 )与驱动器的散热部分与电源或大电流的线路分开，避免干扰源的产生 3. 位置检出器的线材连接 2. 请检查编码器讯号线，是否有与电源或大电流的线路确实的分隔开 3. 请使用含隔离网的线材 4.当运行过程中电机出现明显的抖动或震动：需手动调增益看看效果手动模式调增益：当P2-32 设定为 0 时，速度回路的比例增益（ P2-04), 积分增益（ P2-06）, 和前馈增益（ P2-07), 可自由设定。比例增益：增加增益会提高速度回路响应带宽积分增益：增加增益会提高速度回路低频刚度，并降低稳态误差。前馈增益：降低相位落后误差另外在排除干扰的过程中需要注意：信号线归结在一起，电源线归结在一起。两者之间至少保持 30 公分距离，以减少在运行过程中强电对弱电造成信号上的干扰！

pagerank算法实验报告

PageRank算法实验报告一、算法介绍 PageRank是Google专有的算法，用于衡量特定网页相对于搜索引擎索引中的其他网页而言的重要程度。它由Larry Page 和Sergey Brin在20世纪90年代后期发明。PageRank实现了将链接价值概念作为排名因素。 PageRank的核心思想有2点： 1.如果一个网页被很多其他网页链接到的话说明这个网页比较重要，也就是pagerank值会相对较高； 2.如果一个pagerank值很高的网页链接到一个其他的网页，那么被链接到的网页的pagerank值会相应地因此而提高。若页面表示有向图的顶点，有向边表示链接，w(i,j)=1表示页面i存在指向页面j的超链接，否则w(i,j)=0。如果页面A存在指向其他页面的超链接，就将A 的PageRank的份额平均地分给其所指向的所有页面，一次类推。虽然PageRank 会一直传递，但总的来说PageRank的计算是收敛的。实际应用中可以采用幂法来计算PageRank，假如总共有m个页面，计算如公式所示： r=A*x 其中A=d*P+（1-d)*(e*e'/m) r表示当前迭代后的PageRank，它是一个m行的列向量,x是所有页面的PageRank初始值。 P由有向图的邻接矩阵变化而来，P'为邻接矩阵的每个元素除以每行元素之和得到。 e是m行的元素都为1的列向量。二、算法代码实现

三、心得体会在完成算法的过程中，我有以下几点体会： 1、在动手实现的过程中，先将算法的思想和思路理解清楚，对于后续动手实现有很大帮助。 2、在实现之前，对于每步要做什么要有概念，然后对于不会实现的部分代码先查找相应的用法，在进行整体编写。 3、在实现算法后，在寻找数据验证算法的过程中比较困难。作为初学者，对于数据量大的数据的处理存在难度，但数据量的数据很难寻找，所以难以进行实例分析。

台达A2系列伺服电机调试步骤(2019.7.12)

第七轴通过伺服电机运行的调试步骤一、概述此文档将介绍如何通过西门子PLC来控制伺服电机的正转、反转、以某一速度进行绝对位置的定位以及电机运行错误后如何复位，伺服驱动器如何设置参数等一些最基本的伺服电机的运行操作步骤。二、需准备的材料 1、西门子S7-1200系列PLC一台（我们准备的S7-1200 CPU1215C DC/DC/DC） 2、台达伺服电机ECMA-L110 20RS一台 3、台达伺服控制器ASD-A2-2023-M一台 4、威纶通触摸屏MT-8012IE一台 5、博途V15设计软件 6、威纶通EBproV6.0设计软件三、调试步骤及简单说明调试之前首先将所有设备按照安装说明书上控制接线部分的介绍正确的接入电源，所有设备中需要特别注意的是伺服控制器的进线是三项220V 的电压。建议先让伺服电机在无负载的作用下正常运作，之后再将负载接上以免造成不必要的危险，伺服驱动器的控制用CN1信号端口来接线控制（CN1端口如何接线将提供接线图来接线）。

画面二：将参数P2-10~P2-17参数中没有一个设定为21 画面三：将参数P2-10~P2-17参数中没有一个设定为23

3）、以上步骤调整好之后可以利用JOG寸动方式来试转电机和驱动器，操作步骤如下图 4）、JOG模式调试正常后，在通过PLC控制伺服电机运转，需设定以下几个参数用来。 ①、P1-01设定成Pt模式 00000

PageRank算法的核心思想

如何理解网页和网页之间的关系，特别是怎么从这些关系中提取网页中除文字以外的其他特性。这部分的一些核心算法曾是提高搜索引擎质量的重要推进力量。另外，我们这周要分享的算法也适用于其他能够把信息用结点与结点关系来表达的信息网络。今天，我们先看一看用图来表达网页与网页之间的关系，并且计算网页重要性的经典算法：PageRank。 PageRank 的简要历史时至今日，谢尔盖·布林（Sergey Brin）和拉里·佩奇（Larry Page）作为Google 这一雄厚科技帝国的创始人，已经耳熟能详。但在1995 年，他们两人还都是在斯坦福大学计算机系苦读的博士生。那个年代，互联网方兴未艾。雅虎作为信息时代的第一代巨人诞生了，布林和佩奇都希望能够创立属于自己的搜索引擎。1998 年夏天，两个人都暂时离开斯坦福大学的博士生项目，转而全职投入到Google 的研发工作中。他们把整个项目的一个总结发表在了1998 年的万维网国际会议上（WWW7，the seventh international conference on World Wide Web）（见参考文献[1]）。这是PageRank 算法的第一次完整表述。 PageRank 一经提出就在学术界引起了很大反响，各类变形以及对PageRank 的各种解释和分析层出不穷。在这之后很长的一段时间里，PageRank 几乎成了网页链接分析的代名词。给你推荐一篇参考文献[2]，作为进一步深入了解的阅读资料。

PageRank 的基本原理我在这里先介绍一下PageRank 的最基本形式，这也是布林和佩奇最早发表PageRank 时的思路。首先，我们来看一下每一个网页的周边结构。每一个网页都有一个“输出链接”（Outlink）的集合。这里，输出链接指的是从当前网页出发所指向的其他页面。比如，从页面A 有一个链接到页面B。那么B 就是A 的输出链接。根据这个定义，可以同样定义“输入链接”（Inlink），指的就是指向当前页面的其他页面。比如，页面C 指向页面A，那么C 就是A 的输入链接。有了输入链接和输出链接的概念后，下面我们来定义一个页面的PageRank。我们假定每一个页面都有一个值，叫作PageRank，来衡量这个页面的重要程度。这个值是这么定义的，当前页面I 的PageRank 值，是I 的所有输入链接PageRank 值的加权和。那么，权重是多少呢？对于I 的某一个输入链接J，假设其有N 个输出链接，那么这个权重就是N 分之一。也就是说，J 把自己的PageRank 的N 分之一分给I。从这个意义上来看，I 的PageRank，就是其所有输入链接把他们自身的PageRank 按照他们各自输出链接的比例分配给I。谁的输出链接多，谁分配的就少一些；反之，谁的输出链接少，谁分配的就多一些。这是一个非常形象直观的定义。

台达伺服基本参数设置

台达伺服基本参数设置 1．新伺服驱动器一般会报警。如：ALE13（紧急停止）解除方法P2-15参数值设为122 ALE14（逆向极限异常）解除方法P2-16参数值设为0 ALE15（正向极限异常）解除方法P2-17参数值设为0 2．脉冲设置P1-00设为2 （伺服OFF时设置有效） 3．电子齿轮比设置。（1）台达伺服速比12.5 丝杆导程10mm P1-44分子＝编码器线数X减速比=2500X12.5 P1-45分母=每毫米脉冲数X螺距=1000X10 （2）山洋速比150 旋转轴P1-44分子＝编码器线数X减速比=131072X150 P1-45分母=每毫米脉冲数X360=1000X360 （3）台达伺服速比20 同步带314 m m /转P1-44分子＝编码器线数X减速比=2500X20 P1-45分母=每毫米脉冲数X314=1000X314 4.马达平滑度调节，主要调P2-00 （位置控制比例增益初值35）（速度控制增益初值500 ），使P2-00 P2-04值慢慢调大。（参考值P2-00 80-120 P2-04 800-1400）山洋RS2伺服基本参数设置 1.Group C 00设为01(00为绝对式，01为相对式) 2.Gr1 02设为60（位置环比例增益1，初值35，调整马达平滑度，慢慢调整） 3.Gr1 03设为600（位置环比积分时间常数1，初值1000，调整马达反应，慢慢调整） 4.Gr1 13设为100（速度环比例增益1，初值50，调整马达平滑度，慢慢调整） 5. Gr1 14设为30（速度环比积分时间常数1，初值20.0，调整马达反应，慢慢调整） 6.Gr8 00设为00（位置，速度，转矩指令输入极性） 7.Gr8 10设为02（位置指令脉冲选择） 8.Gr8 13设为电子齿轮比的分子 9.Gr8 14设为电子齿轮比的分子 10.Gr9 00设为0C（正转超程功能） 11.Gr9 01设为0A（逆转超程功能） 12.Gr9 05设为01（伺服ON功能）

台达伺服报警查询

台达伺服驱动器异警处理 RLE01：过电流：主回路电流值超越电机瞬间最大电流值1.5倍时动作 1． 2．驱动器输出短路：检查电机与驱动器接线状态或导线本体是否短路，排除短路状态，并防止金属导体外露 2. 电机接线异常：检查电机连接至驱动器的接线顺序，根据说明书的配线顺序重新配线 3. IGBT 异常：散热片温度异常，送回经销商或原厂检修 4. 控制参数设定异常：设定值是否远大于出厂预设值，回复至原出厂预设值，再逐量修正 5. 控制命令设定异常：检查控制输入命令是否变动过于剧烈，修正输入命令变动率或开启滤波功能 RLE02：过电压：主回路电压值高于规格值时动作 1．主回路输入电压高于额定容许电压值：用电压计测定主回路输入电压是否在额定容许电压值以内（参照11-1），使用正确电压源或串接稳压器 2. 电源输入错误（非正确电源系统）：用电压计测定电源系统是否与规格定义相符，使用正确电压源或串接变压器 3. 驱动器硬件故障：当电压计测定主回路输入电压在额定容许电压值以内仍然发生此错误，送回经销商或原厂检修

RLE03：低电压：主回路电压值低于规格电压时动作 1．主回路输入电压低于额定容许电压值：检查主回路输入电压接线是否正常，重新确认电压接线 2. 主回路无输入电压源：用电压计测定是否主回路电压正常，重新确认电源开关 3. 电源输入错误（非正确电源系统）：用电压计测定电源系统是否与规格定义相符，使用正确电压源或串接变压器 RLE04：RLE04：Z 脉冲所对应磁场角度异常 1． 2．编码器损坏：编码器异常，更换电机 2. 编码器松脱：检视编码器接头，重新安装 RLE05：回生错误：回生控制作动异常时动作 1．回生电阻未接或过小：确认回生电阻的连接状况，重新连接回生电阻或计算回生电阻值 2. .回生用切换晶体管失效：检查回生用切换晶体管是否短路，送回经销商或原厂检修 3. 参数设定错误：确认回生电阻参数（P1-52）设定值与回生电阻容量参数（P1-53）设定，重新正确设定 RLE06：过负载：电机及驱动器过负载时动作

台达B2伺服驱动器

台达B2伺服驱动器台达伺服驱动器恢复出厂设置： A、通电后，P2-08设置为10，断电重启（如遇到无法设置成功请断开使能）。 B、这时出现AL-13，伺服报警（报警内容参照用户手册报警对照表） C、把P2-15设置为0。P2-16设置为0。P2-17设置为0. D、重新启动后报警消失注意：进行模式选择后需要重新上电才能生效。 L1C,L2C控制电源接AC220V 。R-S主回路电源接AC220V 注意：通电时尽量使控制电源先得电，主回路电源后得电。至少同时得电，不能主回路电源先得电。 U-V-W接电机U接电机的红V接电机的白W接电机的黑绿接驱动器接地处短接P-D。一、位置模式：例：螺杆螺距为5MM，要使每个脉冲当量为1微米。要求按下启动按钮滑台前进2mm，3s 后后退5mm，再7s后前进3mm，4s后如此循环。按下停止按钮停止。 A：电子齿轮比分子B：电子齿轮比分母脉冲数x A/B=编码器分辨率（当电子齿轮比为1时，编码器反馈回的脉冲个数，编码器的分辨率为16000） 5mm=5000微米5000 x 160/5=160000（由于电子齿轮比分子设置为16时伺服转动过慢，因此放大10倍，设置为160。）所以滑台要移动2MM那么上位机发送2000个脉冲。参数设置： P1-00，设置为2,。脉冲+方向 P1-01，设置为0 P1-44 设置160 电子齿轮比，分子 P1-45 设置为5 电子齿轮比，分母（需要修改分母时必须断开使能）接线： CN1共有44个接头，其中17,11,35短接 37接PLC的Y3 41接PLC的Y0 14为公共端COM，接PLC输出公共端 9为使能端，短接14 CN2接编码器接线端子号及端子功能如下两张图：

20160310_台达伺服位置控制的应用和调试

台达伺服位置控制的应用和调试 1 PLC和伺服驱动器的接线方式天银一般只用位置（PT）模式标准接线（脉冲与方向的），只用9，14，35，37和41四个端子，其中： 9号端子，伺服启动； 14号端子，COM-； 35号端子，指令脉冲的外部电源，COM+；（台达脉冲命令输入使用内部电源） 37号端子，伺服方向； 41号端子，伺服脉冲，外部输入脉冲的频率确定转动速度的大小，脉冲的个数来确定转动的角度。

2 伺服参数调试 2.1 脉冲个数确定 le 如果我们拿到一台伺服驱动器，不知道参数是否正确，需要把P2-8 设为10 即为恢复出厂设置。复位完成后既要开始设置参数，最先要搞清楚电机转一圈需要多少脉冲，计算公式如下：分辨率 / 1圈脉冲数 = P1-44/P1-45 式中：P1-44，电子齿轮比分子 P1-45，电子齿轮比分母（一般不动）再结合齿轮比，同步带周长或丝杆的间距，就可以确定我们达到要求要发多少脉冲了。 2.2 参数调试 2.2.1 基本参数（伺服能够运行的前提） P1-00 设为2，表示脉冲+方向控制方式； P1-01 设为00 ，表示位置控制模式； P1-32 设为0 ，表示停止方式为立即停止； P1-37 初始值10，表示负载惯量与电机本身惯量比，在调试时自动估算； P1-44，电子齿轮比分子； P1-45，电子齿轮比分母； P2-15，设为122； P2-16，设为123； P2-17，设为121。 2.2.2 扩展参数（伺服运行平稳必须的参数，可自动整定，也可手动设置） P2-00 位置控制比例增益（提升位置应答性，缩小位置控制误差，太大容易产生噪音）。 P2-04 速度控制增益（提升速度应答性，太大容易产生噪音）。

台达伺服问答

01、问台达交流伺服系统ASDA-M系列所提供DI/O功能与交流伺服系统ASDA-A2系列有何差异？答台达交流伺服系统ASDA-M系列各轴各提供6个DI,3个DO；共有18个DI,9个DO。交流伺服系统ASDA-A2则提供8个DI,5个DI。 ASDA-M系列硬件的DI与DO分别在三轴的50 PIN Connector上，透过韧体的转换，可以将各轴6个DI与3个DO整合之后分配给其他轴使用。为避免一些共享DI重复及节省DI脚位，可透过参数设定三轴共享DI，目前提供三轴共享DI： ,伺服启动:设定数值为0101(A接点)，0001(B接点) ,异常重置: 设定数值为0102(A接点)，0002(B接点) ,紧急停止: 设定数值为0121(A接点)，0021(B接点) 在指定各轴DI/O的参数设定上，DI(P2-10~P2-15)及DO(P2-18~P2-20)功能参数设定中增加位4作为各轴DI/O的指定。

02、问当连接绝对型伺服系统时，如何设定绝对型编码器？答设定步骤如下: 1.确认P2-69参数目前设定值(0x0èINC ；0x1èABS)，P2-69如果有修改设定必须重新上电功能才会生效，此参数特性与P1-01属同一类型。 2.接上电池盒(已经连接编码器端与驱动器端，电池也安装上)，首次上电会跳ALE60，此时需坐标初始化，ALE60才会消失。 3.坐标初始化有三个方法尚未作坐标初始化时驱动器会出现ALE60，可以透过以下初始化方式排除: (1)参数法：设定P2-08è271后，设定P2-71è0x1,，此时ALE60会消失，但是当电池电量低于会跳ALE61，否则正常情况面板看到会出现00000。 (2)DI法：设定ABSE(0x1D)与ABSC(0x1F)，当ABSE(ON)，ABSC设定由OFF变为ON，系统将进行坐标初始化，完成后编码器脉波将从重设为0且PUU将重设为P6-01数值。 (3)PR回原点法：若设定在PR控制模式时，可以执行PR回原点方式完成坐标初始化。

台达伺服基本参数设置

xx伺服基本参数设置 1．新伺服驱动器一般会报警。如： ALE13（紧急停止）解除方法P2-15参数值设为122ALE14（逆向极限异常）解除方法P2-16参数值设为0ALE15（正向极限异常）解除方法P2-17参数值设为 02．脉冲设置P1-00设为2（伺服OFF时设置有效） 3．电子齿轮比设置。（1）xx伺服速比 12.5丝杆导程10mmP1-44分子＝编码器线数X减速比=2500X 12.5P1-45分母=每毫米脉冲数X螺距=1000X10 （2）山洋速比150旋转轴P1-44分子＝编码器线数X减速比 =131072X150P1-45分母=每毫米脉冲数X360=1000X360 （3）台达伺服速比20同步带314 m m /转P1-44分子＝编码器线数X减速比=2500X20P1-45分母=每毫米脉冲数X314=1000X314 4.马达平滑度调节，主要调P2-00（位置控制比例增益初值35）（速度控制增益初值500），使P2-00P2-04值慢慢调大。（参考值P2-0080-120P2-04800-1400）山洋RS2伺服基本参数设置 1.Group C00设为01(00为绝对式，01为相对式) 2.Gr102设为60（位置环比例增益1，初值35，调整马达平滑度，慢慢调整） 3.Gr103设为600（位置环比积分时间常数1，初值1000，调整马达反应，慢慢调整） 4.Gr113设为100（速度环比例增益1，初值50，调整马达平滑度，慢慢调整）

5.Gr114设为30（速度环比积分时间常数1，初值20.0，调整xx反应，慢慢调整） 6.Gr800设为00（位置，速度，转矩指令输入极性） 7.Gr810设为02（位置指令脉冲选择） 8.Gr813设为电子齿轮比的分子 9.Gr814设为电子齿轮比的分子 10.Gr900设为0C（正转超程功能） 11.Gr901设为0A（逆转超程功能） 12.Gr905设为01（伺服ON功能）

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