Afl-AI07 476 FOREIGN TECHNOL.OGY Oly WRIHT-PATTERSON AF5 ON F/S 9/1CALCULATING INPUJT RESISTANCE OF AN INS4JLATE0 VIBRATORt IN A SEMI-ETC(U)OCT 61 T N SOKOLOV

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANDARDS 1961 A

'71

FTD-ID(RS)T-1133-81

FOREIGN TECHNOLOGY DIVISION

MCC

CALCULATING INPUT RESISTANCE OF AN INSULATED VIBRATOR

IN A SEMICONDUCTING MEDIUM

by

Yu. N. Sokolov

* DTIO

N NOVI1 9 1981

LAA

~Approved for public release;distribution unlimited.

4 1LU ! ,1

FTD-ID(Rs)T-1133-81

EDITED TRANSLATION

MICROFICHE NR: FrD-81wC-000942

C ALCULATING ;NPUT IESISTANCE OF AN INSULATED'"VIBRATOR IN A SEMAONDUCTING MEDIUM;

B34IYu. N./Sokolov-.

02 --? kh.1 V ( .

Country of origiaU_SRTranslated by: ,"Victor Mesenzeff .>--"--- .

Requester: USAIf ....... -'..

Approved for public release; distribution 4unlimited.

THIS TRANSLATION IS A RENDITION OF THE ORIGI.NAL FOREIGN TEXT WITHOUT ANY ANALYTI-G6L OREDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY:ADVOCATED OR IMPLIED ARE THOSE OF THE SOURCEANDDO NOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISIONOR OPINION OF THE FOREIGN TECHNOLOGY DI- FOREIGN TECHNOLOGY DIVISIONVISION. WP.AFB; 05'O.

FTD-ID(RS,)T-113-81 Date 15 oat r9 81

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration Block Italic Transliterati..-

A a A a A, a P p P p R, r

6 6 5 6 B, b C c C S S, s

8 B B V, v T T Tm T, t

Fr r G, g Y y y U, u

A 9 D, d to $ 0 F, f

E e E a Ye, ye; E, e* X x X x Kh, kh

i m x Zh, zh Q4 Q V Ts, ts

3 3 3 s Z, z q 4 V V Ch, ch

H M W u I, i W W XU A Sh, sh

1 R i Y, y 14 of q Shch, shch

Ka K X K, k b 2 a

i a L, 1 M a Y, y

M At M, m bb b 6

HH mH, N, n 3 E, e

0 o 0 0 0, o h m 0 Yu, yu

nn /7 v P, p R a a Ya, ya

*ye initially, after vowels, and after -, b; t elsewhere.When written as & in Russian, transliterate as y9 or e.

RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian English Russian English Russian Englit--h

sin sin sh sinh arc sh sirih -

cos cos ch cosh arc ch coshtg tan th tanh arc th tanh .ctg cot cth coth arc cth coth-sec sec sch sech arc sch sech'cosec csc csch csch arc csch csch

Russian English

rot curlig log

;* - - .- - -" '/Z - --% ' . -,J " -- . ... . . . . rv ',"- - -

CALCULATING INPUT RESISTANCE OF AN

INSULATED VIBRATOR IN A SEMICONDUCTING

MEDIUM

Yu. N. Sokolov

This paper presents a generalization of the Bechmann formula forthe case of a designated arbitrary current distribution. The obtainedexpressions make it possible to calculate the radiation resistance ofthe vibrator, taking into account the effect of a dielectric coating.

Preliminary remarks and initial relations

When placing an electric vibrator in a semiconducting medium, most

often, it becomes advantageous to use a dielectric coating, which iso-

lates the vibrator from the ambience. When the conductivity of the

ambience is sufficient, the presence of such a coating leads to a sig-

nificant increase in the field excited by the vibrator.

We will not devote too much attention to this question in this work

but deal only with the field of the vibrator in its immediate vicinity

to determine its input resistance, taking into account the effect of

the isolating dielectric. The presents of the latter leads to the fact

that the wave number of electromagnetic vibrations along the vibrator

(y) is different from the wave number of the ambience (k) and, in con-

ducting media, this difference can be very significant.

The methods used to determine the value of y for a wire with a

dielectric insulation are known and are described, in particular, in

work [31 for any number of concentric sheaths surrounding the wire.

Thus, hereafter, we will assume that this value is known and that the

,, .. f 3 ,ii I i l I ..

current distribution along the vibrator follows the law

I(;)~i=4siny(1_ . (1)

The value of z, the electric-field component of the vibrator, we

determine with the aid of the expression

E"s0 CI . 'n, (2)

where

f few" (z -;J d

k is the wave nummber of the ambience.

The quantity r will represent the external radius of the dielectric

insulation.

Making use of the widely known method of induced electromotive for-

ces, we determine the radiation resistance of the vibrator by the rela-t ion 2rtion -- 2 E, P* (z) dz,

(4)

where

is(z) si * (I-z), (5)

y* is the value of conjugation with y.

The solution of this problem, i.e., determination of expression (4),

can be obtained using the Bechmann method [2], which, however, requires

special consideration in this case.

Generalization of the Bechmann formula for the case of an arbitrary

distribution of current along the vibrator

The quantity Zz is determined easily with the aid of (2) and (4)and leads, as is known, to the Bechmann's formula if the distribution

VI(G)of current along the vibrator obeys the relation -0---=--I(:) . We

will show that the expressions analogous to the Bechmann formula have

a place also in the case when

S(:- ) -_ - - .( 6 )

a:2

*1

H2

]I

In this section, no additional conditions are imposed on the quanti-

ty y, generally speaking.

Having substituted in (4) the value of Ez from expression (2). weobtained the following expression after a double integration by parts:

I

Z, 1-(-_ " $2"(z): (7)

Then, having substituted expressions (2) and (3) in (4) and using

the same procedure of double integration by parts, in this case per-

formed with respect to C, we obtaint

- 2 P (z) (. dz -

Is (z 'a/ ( ) J dz + (K'-,) * (z) 17, dzJ. (8)

in which the following is introduced:

In deriving expression (8), we made use of the fact that

After introducing the designation &-f(r=Of, r)

tiC P4, (z) f (r) dz (9)

and excluding the last integral from expressions (7) and (8), we find

--[ ' -x' r/7~ 2 i _____]__"-- ____()1 (0

Z1=__1[iIzXI 17z (10)

Expression (10) is obtained with the consideration of the conditions:

0 with z=-=O, I(t)=I(z)=0 with z=C=.

For the generalization of these results, we note that in determin-

ing the value of mutual energy of two vibrators located at the axes

and 2 with different constants of distribution of the electromag-

netic waves along them, which are designated by yj and y7, respectively,

we have the relation which is analogous to expression (A):

3

~ ~ ~ ~ ~~~~~~---- -- . .... l=aImr.. . I =i I... . .II. .

-2 VI V2 J,

where

17, - I ()T d t; n7. if (" ) T z;RR

R is the-distance from the C axis to the surface of the vibrator at the

z axis.

Expression (11) can be used, for example, for calulating asymmet-

rical vibrators whose arms are under different conditions.

These expressions can be easily generalized for the case of any

designated current distribution which permits expansion in:harmonic

components.

Derivation of calculation relations for radiation resistance

Having expanded the first term in expression (10), we obtain

[7ir,( ei id+( -.[n/'J -" _o____ "'"+"----

, - -='. sin y (1 d_. (12)_4+ = (I -- '(Si -- ') R

"00

Let- us consider the integrals entering this expression. We will

conduct the analysis for the case of relatively long vibrators, when

' the condition I/r is satisfied.

After transforming the first integral to the form

0 -,- f- -Vie.. ',- - I

-" and then after multiplying and dividing each term by e , and also

after introducing the substitution x=I-)+,+(I--)-. we obtain

2_ -K -, 4V 4 f! e 2 dx -e e dx .

21 (J X

4- - -.*- ---- *1~*

In the last expression the change in the factors e 2 A for the

case f~r can be disregarded, then, with an accuracy to the terms of the

order of kr, we obtain- I *-Ei!, - Ei-- l( -- yJ-Ei-- +l V+ -I n- V (15)

where Ei(z) is the integral exponential function.

On the basis of (14), when necessary, it easy to obtain an expres-

sion for I, with the consideration of the following expansion terms of

the factors e 2 a

After presenting the third integral in expression (12) in the form

e"'- -"1"; d e-v -Ik I'i. d)- (16)

and after multiplying and dividing each term by elv , after the substi-

tution x=,+Vr+ 2:under the integrals, we obtain expressions which are

totally analogous to the integrands in (14). Calculation of the integ-

ral yields

'i' eJ [Ei (- i(ic+ j) + Ini(2+)EI

2 (17)-e-'[Eil(-,(K-y)J+In i(w_-, E

In the second integral entering expression (12), we can disregard

the value of r immediately, which leads to the expression

I. = [{ [Ei - i 21 (,c +.y)I-Ei (-il(,c ± )1] -2i (18)

The second terms entering expression (10) is calculated exactly

*the same way, and the final formulas differ only in the substitution

of y* for y.

Thus, the final expression for the radiation resistance of this

system we obtain in the formza-- -0-InA +n'A'I, (19)

V's g

whereY'- ICS, , n' g2 V-"

L_

According to (12), (15), (17), and (19), after all the transfor-

mations, the expression for A has the form:

A = Ei (). 1) - Ei Qa 0 +.In A! + e'2 (Ei (. -,) -

-E O- E .2 1) - In + E}- i'" Ei (2kO- Ei ) l-

-Ei,)-I-n ----2 +E - 2- [Ei(,.., + I --- E]--(+-4t--1 (20)

+ e'- I Ei(4, / + In 2 _ E]

where E is the Euler constant,),, -- 1( -- ¥), )t= -i( +T),(21)

a and 8 are determined by the relation y=S-i.

The expression for A' is obtained from expression (20) by substi-

tuting -a for , i.e., y* for y.

The arguments of the functions entering the expression for A have

the form: ).=(-*) -;=-i(c+*).

For further analysis, it is convenient to transform formula (19)

to the form

z = 2LL [f--[(A +l A') i -L'(A-A]. (22)

where

Let us consider some of the particular cases of this solution.

/ The simplest relations are obtained when riI <. Then we find the fol-

lowing with an accuracy to the terms of the order of (Kil) and (yI)

A =-4(ip1+cxO(In 1). A'=-4(ip1-cl)(In 1 (23)

and according to (22)

Z2 -i 0 (In -- (24)

Expression (22) permits a limited transition o-0 when yk, which

becomes possible if we make use of the relations:

Ei0. _Eij 1) + i N, i)i e '(25)

El ( 1. i) _e Ei (A, 1 + i - .

6

and, in the expressions for A and A', proceed to the functions only of

the arguments X, and X2 (the remaining functions of A, and X2 can be ob-

tained from (25) by replacing 1 by 21 or r).

Finally, in the case where y-k, from (19) we have

Z A'°.

The arguments of functions Ei(z) assume the form: .;=-2a,;= i2,

after which, it is possible to proceed to the functions Si(x) and Ci(x)

of the actual arguments and, ultimately, evolve the actual and imaginary

parts. We will not write out the expressions obtained in this case,

since they are congruent, with respect to accuracy, with the expressions

obtained by Yu. K. Murav'yev, who used the method of induced electro-

motive forces for the case y=k.

In conclusion, it is necessary to note that this analysis does not

fully exhausts the solution of the problem of input resistance of the

isolated vibrator, since introduction into the calculation of the quan-

tity y. which is different from the wave number of the ambience, makes

it possible to determine with certainty only q - the magnetic-field

component. As regards the quantity z - the electric-field component,

which enters the expression of energy connected with the vibrator as

a second cofactor, then for a precise solution of the problem this value

must be determined not on the surface of the insulation but on the sur-

face of the wire, taking into account the reflections at the insulation-

medium boundary. In the absence of losses in the wire insulation, the

magnitude of the active component, however, does not vary as compared

with the actual part of expression (22). With regard to the active

component yielded by this expression, in the case of thick insulations

and strong reflections at the insulation-medium interface, this solution

needs further refinement, which can be obtained on the basis of this

solution by introducing a reflected wave when considering the electrical

field.

In the course of this work the author was fortunate to receive

valuable suggestions and constant consultation from the Dr. of Tech.

Sciences G. A. Lavrov to whom he is deeply indebted.

The author also expresses his appreciation to the Dr. of Tech. Scien-

ces Yu. K. Murav'yev, who made a number of valuable remarks while re-

7m

L. .s L ' ' ' " "" ... . ___-_.... ___........ ______r_........ __" ____--w" . *.. ___ _ •____ ......__________ ',m - % ..-

viewing this work.

Bibliography

s. jissapoB r. A.. KmKA3 e i A. c. flpH3eMrn.e H loi3emjmde WHerni. 4.314Cmercxoe PIAoD. M4.. 1965.

2. CP e TTO H .jx~ A. TeopmR I.eTO~rel3a roCueltll.a. 1949.J. A a u p o a r. A., C o v o0.1o3 10 H. K pacqier% DNt)oh,)ro Coipoyll 1.1 j'wnjj a1i-

TSIHU 2 MHOrOC.1iHOA lL4.lH~aptiqecxoA o6o.ioqxe. Tp ,IIJ HH-Ta UIMMIr~.Bwn. 74. tba-e, cHeapav. 1968.

8I