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ARinn. _ axi xi in =_axi x1 i=I i=1. Powering both parts of expressions (4) in a square and taking into account absence of a correlation between parameters x , we ...
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Defense Technical Information Center Compilation Part Notice ADP013915 TITLE: Statistical Analysis of Internal Parameters of Radiating Systems with Reactance Elements DISTRIBUTION: Approved for public release, distribution unlimited

This paper is part of the following report: TITLE: 2002 International Conference on Mathematical Methods in Electromagnetic Theory [MMET 02]. Volume 2 To order the complete compilation report, use: ADA413455 The component part is provided here to allow users access to individually authored sections f proceedings, annals, symposia, etc. However, the component should be considered within

-he context of the overall compilation report and not as a stand-alone technical report. The following component part numbers comprise the compilation report: ADP013889 thru ADP013989

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MAIETP02 PiRoc II.N(;s

446

STATISTICAL ANALYSIS OF INTERNAL PARAMETERS OF RADIATING SYSTEMS WITH REACTANCE ELEMENTS V.V. Ovsyanikov Dniepropetrovsk National University., Ukraine 49050, Dniepropetrovsk, Naykovy lane 9, korp. 12, ph. + 380 (56) 776-90-92: e-mail: root(apl .net-rff.dsu.dp.ua ABSTRACT The mathematical model of the statistical analysis of a standing-wave ratio of a voltage on an input of the radiator is offered depending on small random fluctuations of values of reactive elements, included into the radiator and from coordinates of their inclusion. From the results of a statistical estimation it follows, that minor random deviations of parameters of reactive loads of the short vibrators firom design values result in essential changes of a standing-wave ratio, that must be taken into consideration at designing and using of similar radiating systems One of the important internal radio parameters of radiating systems (RS) with impedance elements is the standing-wave ratio of voltage ( Ks ) on input connectors RS, which as is known is expressed through active (Ril) input impedance at given frequency as follows [I]:

Ks

1+

1-4R !'

(I+RN

and reactive ( Xi, ) components of

-4R

+1)2+xi

" L(1+R ,) 2

x

2

(1) where Ri 1

X

are normalized on a wave impedance of a feeding channel (Wi I)

components of input resistance of a radiating system. In their turn components R!]' and A'] are functions of values (x ) of geometrical parameters RS (d,ci-,), included in it of impedances (Z) and coordinates of their inclusion (hz ), operational frequency (f ) of an exciting source 1U, wave impedance W.A! etc (see fig. 1). These relations can be presented as:

Ri,: R(xjl.,2... xA)=:R(xi). i=1.2..... ', Xli',=X(xl,x

2

,... XN)=X(xi).

i=1,2 ..... N;

(2)

Taking into account, that R!' , XN, , xi are random quantities and regarding systematic components of errors of values RN,!,

KlI:'ir UK

I,A INI ,17

X?, and parameters xi, which can be defined and

INPY N7iVAm A'. C().VI.-1RF.v "I: ON Alfl.II/L IT

"I11. Alk/11/I), 1 IN I1.:CV( " k.;, A'P ,,'("

TII:OIy

MMET*02 PROCEEDINGS

447

eliminated, we consider expressions (2) with the account of only random errors (Ax, the estimation of which is fulfilled below. Let's expansion (2) in a Taylor's series near to average values of xi or their mathematical expectations [2]: N n 1N 2 n Rin +A Rnn

D( aRlfý

R

l

~llN

Rin 2

N nN i"+A AXn =

+...,

2X-n

a~i+icxi i=1 Xyn Ax. + 2I2+....

2

xi

(3)

i=1 a Xi ~ i-Ix are small in comparison with values xi we

Provided that the random errors Axi

neglect addends containing hjwers of A x above the first. Further, subtracting (2) from (3) we receive values of random errors as: NRnn ARinn

_

AR inAx; = xi axi

AX in n

=_axi

i=I

XinA

(

(4)

x1

i=1

Powering both parts of expressions (4) in a square and taking into account absence of a correlation between parameters x , we determine dispersions of components Ri and Xin

R

N

2

~2(Rý= aR in

2(xi);

axi

InJ

C72(Xn)Z=j

cy2 (Xi

in s=j

At known dispersions (5) we determine a dispersion KS, considering similarly to the previous making of expressions (5) provided that the random errors A and AXn are small in comparison with values of the relevant components Ri and X i.

a 2(Ks)=

MKs + MKs in

22(Rý Y ). 2(Xi

(6)

in)

Derivatives from Ks for expression (6) is determined from (1) as: ý2-Xn2 -1I aKs

4(Rin2

Xin

I

(j1+UI

HX- inA 2

Wl-4N-l

i +

Kll;.Ir, UKAINFi,

1X-TIH INTFTINATIONAIL

CO)NHLRI-c'Nh"ON

in n

...

. I 4Rin

(7)

N

in

MATHEMATICAL. METHODS IN ELECTR.7'1OMAGNf;TIy(' TH.!'ORy

MAJEPh 02 PROCAEEDINGS

448

aK

=

ax1

(8)

+

in

22

1

IRn

n-1

4Ri +

4Rn

RXi) +

-

2

41R.

Y........

and dispersions of expressions Rýn and X " for the studied radiator we shall find from the formulas (5). Thus, with the help of expressions (4) - (8) it is possible to design a statistical estimation any RS with included reactive elements. Thus, it is necessary to know particular relations such as (2) for Ripn and X ý, and derivatives from them on the conforming parameters xi. Let's put the results of a statistical estimation Ks

on an input shortened twice

(d=0,12X) concerning resonant length of the symmetrical vibrator with included in radiating branches the inductive loads depending on random fluctuations of values of these loads and places of their connection (fig. 1). An estimation is designed by method with usage of the theory of an equivalent long line. The results of research are shown in Cr2K, a fig. 1. As follows from the charts L, hL = var L U L of a fig. I random changes of parameters of included reactances IL= var; d L L=consL result in essential oscillations KS. "•L= var, For small-sized radiators of such hL= onrs' type it is possible to explain this

"•A L/L

o

0,05 Ah LA17L Relative the variations L and Ii

the random errors Fig. 1. Dispersion K, versus L an hireactances. L and h1.

phenomenon by narrowing of their bandwidth because of linear shrinkage of length. Let's mark, that in the not shortened radiators of a fluctuation of icue forcoretion reatacs included for correction distribution of a current in them

with the purpose. for example, dilating of frequency range or the correction of the directional diagrams do not result in such sharp of variations the input characteristics. REFERENCES [1] V.V. Ovsyanikov. Wire Antennas with Reactive Loads.- Moscow: Radio i Svyaz, 1985.120 p. (in Russian). [2] Ya. S. Shifriiin. Statistical theory of the antennas // Chapt. IX in the boock: Reference book on the antenna teclnmics.-V.1 -Moscow: Radiotechnics,1997.-P.148-205.(in Russian).

KIri,ý UKRAIN'E,

IV-Yi1

IA'Tvu I

ONAr ..IL CONIImr.\',N(',:"

Al"

'.miAI T!'. ( Ai:1wt. )S IN Elu( ,IY

•IO .Ii(;vnh (' TIllOYoi