Mathematical model of electronic optical moire pattern of magnetic field on electronic equipment element defects Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Математика. Физика

УДК 621.3.4:537.533.35

MATHEMATICAL MODEL OF ELECTRONIC OPTICAL MOIRE PATTERN OF MAGNETIC FIELD ON ELECTRONIC EQUIPMENT ELEMENT DEFECTS V.M. Ivanov1, E.B. Vinokurov1, E.A. Pechagin 1, M.I. Potapochkina2, A.V. Lanovaya3

Departments: «Electrical Equipment and Automation» (1),

«AppliedMathematics» (2), «Higher Mathematics» (3), TSTU

Represented by a Member of the Editorial Board Professor V.F. Kalinin

Key words and phrases: electromagnetic field; field distortion; moire.

Abstract: Problems of interconnection between electronic equipment element performance and generated electromagnetic fields are viewed in the article. It is pointed out that concentration of electromagnetic energy near defects may result in electronic equipment failure. Techniques of observation and measuring of magnetic fields near defects on flat conductors are discussed and methods of their calculation with the help of electron optical moire are offered in the article.

Performance of electrical equipment elements is closely connected with generated electromagnetic fields. There are usually some kinds of defects on thin flat conductors which appear in the process of production and operation and further define their electromagnetic compatibility. Concentration of electromagnetic energy near these defects may result in electronic equipment failure. That is why monitoring and measuring of magnetic fields near defects appears to be an actual task. Let us view some investigation results of these fields observation near defects on flat conductors and present methods of their calculation with the help of electron optical moire.

As model sample there was used cuprum plate with central aperture (80 x 80 x 1) mm, placed in the column of electronograph EG-100A so as to enable the cathode beam to slide down the surface maximally close to the defect (Fig. 1).

Magnetic field excitation in the plate was initiated by constant voltage U commutated by contacts of polarized relay. Power supply of relay winding was carried out from audio-frequency oscillator with frequency of 32 Gertz and controlled by an oscillograph. Moire pattern was obtained when two views were put together; that of undistorted grid and grid distorted due to Lorentz power appearing in magnetic field (Fig. 2).

For quantity analysis of magnetic field near defect by amount of cathode beam displacement on electronograph screen let us use the equation of charged particle motion on axis x

d 2 x e

dt2 m

= — W0 Hx (1)

e T

vx = H"0V0 I Hx (z)dz :

m J

where vx - velocity of electron escape from the active field region; v0 - particle acceleration velocity; Hz - magnetic field near aperture; e, m - electron charge and mass; |a 0 - magnetic constant.

Finally we obtain

e L

— V0 —

m v0

x = — V-0— I Hx(y)dy,

m v* J

(3)

where v0 =

2eU

m

dz = v{)dt; x = vx

L_

v0

; U - accelerating voltage of 40 kV.

Visible on the screen beam deflection in a magnetic field can be calculated by the equation

L e L

v0

— e — (*

— = - Ц0— I Hx (y)dy , (4)

7U m 7)^ J

v0

where L - distance between the object and the screen equal to 0,4 m.

Let us specify magnetic field distribution on axis x which approximates rather precisely integration. Function des-crybing bell distribution is used more often when flat magnetic lens are calculated [1]

Hx (y) = Hx\ 2 , (5)

1+u

{ 5d

where d - aperture diameter.

After data substitution the calculated ratio between deflection x' and Hx(y) is

x = 3,5 -10-6 Hxmax. (6)

Fig. 1. Experimental model (sample with central aperture):

SE - source of electrons; G - grid; object (bent plate with aperture); E - electronograph; e - cathode beam; U - source potential

Fig. 2. Electron-optical moire patterns of magnetic fields:

a - plates without aperture; b - plates with aperture, I = 5 A

Beam deflection can be easily measured by moire patter, e.g.

x' = km = 6 • 0,5 -10-3 m ,

where k - equipotential number; m = 0,5-10 m - the size of amplified grid cell picture. Magnetic field strength is found from the expression (6)

rr _________

n xm ~ *

a—6

3,5 40'

Obtained data are compared with theoretical ones get by the formula

Hx (x y ) = — nR

1 + i|2 +f x'2

R

■к (ф)

1 -

2

2

.-I2 +fx 2 E <ф>

R J

R

(8)

(9)

Hx (y)

Distance from aperture center y, mm

a)

Hy (x)

Distance from aperture center x, mm b)

Fig. 2. Distribution of axial (a) and radial (b) components of magnetic field in direct current:

1 - experimental; 2 - theoretical

1

+

where I - electrical current on the plate; R - aperture radius; к(ф), Е(ф) - elliptic integrals of first and second type [2].

Equipotentials obtained by lines of vertical grid coincidence and characterizing radial magnetic field component are treated in similar way. In this case deflection on axis y can be calculated from the formula

У = Vy — = e ц0 — I Hy (x)dx . (10)

V0 m V0J

After substituting of numerical data and bell distribution when d = 0,5-10-6 m in equation (10) and integrating it we obtain the calculated ratio

Hym =^-6. (11)

^ 5-10-6

Theoretical and experimental dependence of axial magnetic field strength H x ( y) of a flat conductor with aperture when voltage is constant are shown on Fig. 2, a.

Data obtained on radial magnetic field component Hy (x) are treated in a similar

way.

It follows from the comparison of theoretical and experimental data that divergency is not more than 5 % of the whole range, which proves the moire pattern adequacy to real heterogeneous magnetic field of a flat conductor.

It must be stressed that in all experiments the aperture in the plate was characterized by maximal moire pattern distortion.

Suggested technique of electron-optical moire may be used not only for magnetic field topography visualization but also for defects detection due to maximal moire pattern deflection.

References

1. Кельман, B.M. Электронная оптика / В.М. Кельман, С.Я. Явор. - М.-Л. : Изд-во АН СССР, 1963. - 362 с.

2. Иванов, В.М. Исследование концентрации энергии в области непроводящих включений плоского проводника с током средствами электронной микроскопии / В.М. Иванов, Е.А. Печагин, А.Н. Уваров // Электрика. - 2003. -№ 7. - С. 34-37.

Математическая модель электронно-оптической муаровой картины магнитного поля на дефектах элементов электрооборудования

В.М. Иванов1, Е.Б. Винокуров1, Е.А. Печагин1,

М.И Потапочкина2, А.В. Лановая3

Кафедры: «Электрооборудование и автоматизация» (1),

«Прикладная математика» (2), «Высшая математика» (3), ГОУ ВПО «ТГТУ»

Ключевые слова и фразы: искажение полей; муар; электромагнитное

поле.

Аннотация: Работа элементов электрооборудования тесно связана с генерируемыми электромагнитными полями. В тонких полосковых проводниках всегда имеются монтажные, технические, геометрические и структурные

дефекты, которые рождаются в процессе их производства и эксплуатации и определяют их электромагнитную совместимость. Концентрация электромагнитной энергии вокруг этих дефектов может привести к отказу электрооборудования, поэтому наблюдение и измерение магнитных полей вокруг дефектов, которые локализуются в малых объемах, является актуальной задачей. Приведены некоторые исследования по наблюдению этих полей вокруг дефектов на плоских проводниках и представлен механизм расчета с помощью электронно-оптического муара.

Matematisches Modell des elektronenoptischen Moire-Bildes des magnetischen Feldes auf den Defekten der Elemente der elektrischen Ausrastung

Zusammenfassung: Die Arbeit der Elemente der elektrischen Ausrustung ist mit den erzeugten elektromagnetischen Feldern eng verbunden. In den feinen Streifenleiter gibt es immer die montagen, technischen, geometrischen und strukturellen Defekte, die im Laufe ihrer Produktion und der Ausbeutung entstehen. Die Konzentration der elektromagnetischen Energie um diese Defekte kann zum Versagen der elektrischen Ausrustung bringen. Die Beobachtung und die Messung der magnetischen Felder um Defekte, die in den kleinen Umfangen lokalisiert werden, ist eine aktuelle Aufgabe. Wir werden einige Forschungen nach der Beobachtung dieser Felder um die Defekte auf den flachen Leitern anfuhren und wir werden den Mechanismus der Berechnung mit Hilfe des elektronenoptischen Moires darstellen.

Modele matematique de l’image de moire electronique et optique du champ magnetique sur les defauts des elements de l’equipement electrique

Resume: Le fonctionnement des elements de l’equipement electrique est etroitement lie aux champs magnetiques generes. Dans les conducteurs plats il y a toujours les defauts de montage, techniques, geometriques et structurels qui apparaissent au processus de leur production et exploitation et qui determinent leur compatibilite electromagnetique. La concentration de l’energie electromagnetique autour de ces defauts peut ammener l’equipement electrique a bloc. C’est pourquoi l’observation et la mesure des champs magnetiques autour des defauts qui se localisent dans de petits vulumes est un probleme actuel. Citons quelques etudes de l’observation de ces champs autours des defauts sur les conducteurs plats et presentons le mecanisme du calcul a l’aide de moire electronique optique.