Fluctuations in a quartz oscillator

FLUCTUATIONS IN A QUARTZ OSCILLATOR

A. N. Malakhov and N. N. Solin UDC 529.786.2

A detai led ana lys i s is given of ampli tude and f requency f luctuat ions in a quartz osc i l l a tor , due to f luctuations in different e l ements of the osc i l l a to r c i rcui t . The spec t r a l line width is found, and the contr ibut ions of f luctuations in va r ious p a r a m e t e r s to it a re invest igated.

1. I N T R O D U C T I O N

Fluctuat ions in quartz osc i l l a to r s have not been adequately inves t igated e i ther exper imenta l ly or theore t i ca l ly , in spite of the fact that they a r e of fundamental impor tance in p rac t i ca l appl icat ions. Exis t - ing expe r imen ta l data a re , fo r the mos t pa r t , m e r e l y descr ip t ive , w h e r e a s theore t ica l ana lyses a re mainly concerned with na tura l f luctuat ions [1-3]. The only exception is [4], where ment ion is made of quas is ta t ic f luctuat ions in the in terna l tube impedance.

In the p re sen t paper we r e p o r t a theore t ica l ana lys i s of f luctuations in a two-c i rcu i t (P ie rce- type) quar tz osc i l l a to r which a r e due to slow [in compar i son with cos (w0t) ] uneor re la ted f luctuations, for ex- ample , the quartz capaci tance , the capaci tance in the anode c i rcui t , the coupling capaci tance , the input tube capac i tance , and the t ransconductance of the tube.

This paper is, in fact , a continuation of [3], and we shall t he re fo re consider the same quartz osc i l - l a to r c i rcu i t and employ the same method of analyzing the fluctuation equations. We shall use the same notation for the c i rcu i t p a r a m e t e r s .

These assumpt ions will enable us to avoid a cons iderab le amount of detail and r ep roduce only the final exp re s s ions , leaving m o r e space for the ana lys i s of the r e su l t s .

It can be shown that the f luctuation equations, in our case , a re of the same f o r m as for na tura l noise [3]. The only di f ference is that the per turb ing fo rces f i (t) a re now due to f luctuations in the c i rcui t pa- r a m e t e r s and not to the shot and t h e r m a l noise . They a re given by:

where

f~(t) ----~-A s 2 [ ; C K - l - ; C + ( s l - h e ) g c ~ + S3 g~ gCs + n g ~ S ] , 2 . - s~ QK

2 g~

: f#) = ~ . g~ ~ c ~ . - ac + s~ , ,ac l + s~h~aC~ + ,tog~!, ~ . .-~.oS 2 . g x

f d t ) . . . . . /"-~[~O 4 -s~ s~ ~ d ] :2 L , . ~ ac +.--g~ (s, - h ~ ) ,~C, + s 3 h k ~ C , + n o s ~ % ' ~S .

(I)

~CK ~ - - Ac,( t ) ac - AC(t) ~C1-- AC~(t) C~ ' C ' Ct

~c, = Ace(t) ~s _'As_(t_~ C~ ' S

Gor 'k i i State Univers i ty . T rans la t ed f r o m Izves t iya VUZ. Radiofizika, Vol. 12, No. 4, pp. 529-537, Apri l , 1969. Original a r t i c l e submit ted October 16, 1967.

�9 1972C~ Bureau, a division of Plenum Publishing Corporation, 227 g/est I7th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

419

are the relat ive fluctuations in the quartz capacitance, the anode-c i rcui t capacitance, the coupling capaci- tance, the input tube capacitance Cs(t) , and the t ransconductance S(t) of the tube, respect ively .

According to [3], the coupling coefficients si = C~/(C + Cl), s 2 = C1/(C s + C~) and the relat ive differ- ence hk = ~'"21 . . . . ~k/2~r176 1-2 = Ck/(Ck + Cl + Cs) are of the order of/~a, where we are assuming #a = Q ~ and #k = Q~ �9 In pract ice, we always have #k -< #2a, where Qk and Qa are the Q fac tors of the quartz and anode circui ts , respect ively .

In Eq. (1) we have introduced the additional symbols defined by n = / ( 1 - 3/30R~/4), n o --/(1--floR~/4), which a re of the order of ~a, and the fur ther coefficient s 3 = Cs/(C 1 + Cs) , which is of the o rde r of p~ l = S/w1(C + C1) ~ ~a-

The fact that the fo rm of the fluctuation equations is the same as before leads to the following ob- vious consequence. If the spect ra of the pa ramete r fluctuations are constant within the band [0, Ha], where H a is the bandwidth of the anode circuit , such pa rame te r fluctuations may be rega rded as white, in which case the spect ra of the amplitude and frequency fluctuations of the self -osci l la t ions are of the same fo rm as in the case of natural noise. The spectra l distribution discussed in [3] will then also be valid in the present case. The only difference will be connected with the strong corre la t ion of the f i (t) [see Eq. (1)], which is absent in the case of natural noise.

On the other hand, the nonnniformity of the spect ra of pa rame te r fluctuations within the band [0, Ha], which occurs , for example, for f l icker- type pa ramete r fluctuations, gives r i se to a considerable modifica- tion of the amplitude and frequency fluctuation spectra of the quartz osci l lator , and affect substantially the shape and width of the self-osci l la t ion spectral line. Thus, the technical fluctuations in a quartz osci l la tor depend only on the proper t ies of the pa rame te r fluctuation spectra, their correla t ion, and their difference f rom the white noise spectrum.

2. A M P L I T U D E F L U C T U A T I O N S

Let us consider the expression for the perturbing force f i (t) which governs the direct effect of the fluctuations in the above circui t pa r ame te r s on the relat ive amplitude fluctuations a s (t) of the qua r t z - c i r - cuit oscil lat ions. It is readi ly found that the mean contribution of this force is due to the fluctuations in the quartz capacitance and the anode-c i rcui t capacitance. The fluctuations in the remaining pa rame te r s introduce a small fac tor which is of minor importance.

The same situation occurs for the relat ive amplitude fluctuations /3(t) in the anode circuit~whieh are due to the direct effect of the fluctuation fo rces f2 (t) and f3(t) . The only difference, as compared with the amplitude fluctuations of the quartz circuit , is that here the fluctuations in the coupling capacitance C 1 (t) will have an effect which is smal le r by a fur ther factor of #a 1 as compared with the f i r s t case, because the frequency of the generated osci l lat ions o~ 0 is close to the frequency of the quartz c i rcui t (hk is of the o rder of ~a).

Since s 2 ~ ~a, and gl and g2 are of the order of unity, it follows that f t (t) is smal le r by an o rder of magnitude than f2 (t) and f3(t) . This means that the intensity of quartz amplitude fluctuations due to tech- nical fluctuations in the c i rcui t pa rame te r s is lower by an order of magnitude than the intensity of ampli- tude fluctuations in the anode circui t . Hence, it follows that, as in the case of natural fluctuations, the small coupling between the quartz circui t and the anode circui t and, consequently, the tube (we are assum- ing zero grid currents) , which is determined by the coefficient s2, in some way "sc reens" the quartz ele- ment f rom fluctuations in other circui t elements.

If we use the method descr ibed in [3], we can readi ly obtain the following express ions for the relat ion between the spectra l densit ies of the quartz amplitude fluctuations Sai(~2) and the anode oscil lat ion Sfl (~2) with the spectral densit ies of fluctuations in the c i rcui t pa rame te r s S~k(~) (X~ = 6C k (t), X 2 = 5C (t), X 3 = 6Ci(t), X 4 = 6Cs(t),X ~ = 5S(t)) :

(D2 5

s ~ , ( ~ ) = ~ o ~ y , Ak(~) s~, k (~),

k=1 (2)

So(~) -- 4f~,] ,~=, Bk(S~) sxk(~)"

The frequency ~0 is of the o rder of the anode-c i rcui t bandwidth H a (see [3]).

420

TABLE 1

Ak(eo)

B,~ (ao)

Do~

D1 k

d N ~ q 0 ~-2 ~a d

d d

d

The f requency c h a r a c t e r i s t i c s Ak (~2) and B k (~) a re , in fact , the t r a n s f e r coeff icients between fluctu- a t ions in the p a r a m e t e r s Xk and the ampli tude f luctuations a 1 and /?. The s t ruc tu re of the exp res s ions for A k ( a ) and Bk(f~) is the same as that of A(~2) and B(g) found in [3].

However , the coeff ic ients in the expansions of Ak(~2) and Bk(f~ ) (we will not r eproduce these e x p r e s - sions he re because of the i r complexity) will be different both in appearance and in order of magnitude be- cause of the dependence of the f luctuation f o r c e s f l (t), f2 (t), and f3(t) on the f luctuations in the c i rcui t pa- r a m e t e r s . Graphs of Ak(~2) and Bk(a) will r e m a i n the same a s those of A(f~) and B(f~), shown in [3].

The o rde r of magnitude of the coeff ic ients A k and Bk is shown in Table 1. This table can be used to e s t ima t e the contr ibut ion of f luctuat ions in the va r ious p a r a m e t e r s to the ampli tude f luctuat ions of the anode and quar tz osc i l la t ions .

In the case of white p a r a m e t e r f luctuations, the spec t r a of the ampli tude f luctuations will , of course , coincide with the f requency c h a r a c t e r i s t i c s Ak(~2) and Bk(~2) and, as a l ready noted, they will be s im i l a r to the spec t r a of the natural ampli tude f luctuations.

Let us now cons ider the case when p a r a m e t e r f luctuat ions a re of the f l icker type. Suppose that there a r e p a r a m e t e r f luctuat ions whose spec t r a l densi ty is given by

Kk~ s~(a) = 2r f e I}" (3)

where Kkk = const , 0 < T < 3.

The spec t r a l dens i t ies of the ampli tude f luctuat ions for this case a re typical ly of the f o r m shown in Fig. t .

The values in the range 0 < 7 < 1 co r re spond to s ta t ionary p a r a m e t e r f luctuations (see [5], Section 1.6) and, in th is case , one can r ead i ly find the in tens i t ies of the ampli tude f luctuations, which turn out to be

(4)

where Aik = eonst, B i k = const, and A Ik < Bik"

In the ana lys i s leading to the above exp re s s ions we a s s u m e that c h (t) and fi(t) a re smal l enough. is r ead i ly seen that the condition for this is

K~,~ << 4~to ~ 1 s l n [ 2 (1 _ .r)] ~o~ Alk

It

(5)

This condition is thus essen t i a l ly the condition fo r the val idi ty of the pe r tu rba t ion theory . It follows f r o m Eq. (57 that , as 7 approaches unity, KX k mus t d e c r e a s e . This is connected with the fact that a ~ and

421

Fig. 1

S ffL)

(fi 2) as 7"*1. This in turn is connected with the fact that the spec t rum of pa ramete r fluctuations becomes nonintegrable near zero for 7 - 1, the corre la t ion functions for the p roces se s Xk(t), ~1 (t), and fl(t) do not exist, and the f l icker p roces se s themselves become nonstat ionary.

In this case , the conditions for (a~) and (f12) to be small reduce to a res t r i c t ion on the duration of the observat ions (for details see Section 4.6 in [5]).

3. FREQUENCY FLUCTATIONS

Let us now consider fluctuations in the quartz frequency v (t) = d~(t)/dt. By considering the fluctua- tion force fd t ) which direct ly affects fluctuations in the frequency v(t) we can establish that the main con- tr ibution of this force is due to fluctuations in the quartz capacitance C k. Fluctuations in C and C s have an effect which is lower by a factor of #a 1 , f i rs t ly , because of the weak coupling between the anode and quartz c i rcui ts (coupling coefficient s 2 ~ #a) and, secondly, because hk is small and of the o rder of Pa. Fluctuations in the capacitance C I have an effect which is smal le r by a fur ther factor of # ~ . The coupling coefficient s 1 ~ Pa is small.

It also follows f rom the express ion for f4 (t) that the frequency fluctuations of a quartz osci l la tor are due not only to fluctuations in the var ious circui t capaci tances (this is natural because w 1 and the frequency cor rec t ion A~ 0 depend on them), but also on fluctuations in the t ransconductance of the tube which is t r i - vial because w 0 does not depend on S. It is p rec ise ly for this r eason that frequency fluctuations depend on the derivative of t ransconductance fluctuations.

Therefore , although slow transconduetanee fluctuations do not affect the osci l la tor frequency, rapid t ransconductance fluctuations (and these are always present) do have a definite effect, although it is r e - duced by a factor of # ~ as compared with the effect of quartz capacitance fluctuations.

The foregoing discussion of pa rame te r fluctuations as a factor affecting fluctuations in v(t) r e f e r s also to the slow component of fluctuations in the frequency of the anode c i rcui t which are va(t) = v(t) + d~bl(t)/dt, where r is the phase difference between the quartz and anode oscil lations, since in this case we may suppose that va(t) -~ v(t).

There is an essent ia l difference between the fast fluctuations Va(t ) and v(t) since, in this case , v(t) do l 2_ da~ ,

is smal le r by a fac tor of ~a ~ than dr = m 0 dt dt whereas , according to Eq. (1), v(t) is smal le r than

d~ 2 da 3 dt and ~ by a factor of pE 1, owing to the weak coupling between the anode and quartz c i rcui ts (s 2 N Pa

and m 0 is of the order of p~

422

If we use the method desc r ibed in [3], we can der ive the following re la t ion between the spec t ra l den ' s i t ies of the quartz f requency fluctuations, and the anode f requency fluctuations, on the one hand, and the spec t r a l dens i t ies of the c i rcu i t p a r a m e t e r f luctuations, on the other :

5

&(f~) = o~ ~, N"k(Q) Sx~(f~), k = l

5

s,(a) = u ~ s s~{a) k = i

(6)

The f o r m of the functions Nvk(f~) and Nvak(f~) in the case of capaci tance f luctuations (k = 1, 2, 3, 4) will be the s ame as for the analogous functions Nv (S2) and Nva (f~), given in [3].

The f o r m of the functions Nv5 (~2) and Nva 5 (~2) cor responding to t ransconductance f luctuat ions is somewhat m o r e compl ica ted . Because v(t) depends on the der iva t ive dS/dt, the quantity Nv5 (f2) will contain t e r m s of the f o r m af~ 2 and b~22 + c~ r which, however , i nc rea se with f requency v e r y slowly within the f r e - quency band [0, Ha].

As in the case of na tura l f luctuations, the spec t r a l dens i t ies S V (f~) and Sva(~2 ) a re equal for ~2 <<wl/Qk (because the spec t r a l dens i t ies of the de r iva t ives da2/dt and d~s/dt a re ze ro for ~ = 0), and: then diverge s t rongly as ~2 i n c r e a s e s . For sufficiently high f requency Sva(~2) is g r e a t e r than Sv (~2) by a fac tor of #~2. This occu r s because for f requenc ies ~2 ~ ~20 the c i r cu i t s have no effect on each other owing to iner t ia , and since the i r Q f a c t o r s a re v e r y different , the spec t r a l dens i t ies Sv (~) and Sva(~) mus t differ sharp ly f r o m each other .

Despi te the fact that the functions Nvk(f~) have a compl ica ted f o r m and have a number of ex t r ema , they can, neve r the l e s s , be r ep l aced by the approx imate values

g~k(~2) = Dok = const,

since all the changes in Nvk (~) lie within one o rde r of magnitude. The o rde r of magnitude of the coeff i- c ients D0k is shown in Table 1.

Fo r f requenc ies ~2 ~ ~21 (f~l is of the o rde r of the bandwidth of the quartz c i rcui t ) , the functions Nvak(~) a r e of the s a m e o rde r as the cor responding functions Nvk(~). At f requenc ies c lose to f~0, when the influence of the quartz c i rcu i t is reduced, the Nvak ( f t ) increase by a fac tor of #a -2 , whe rea s at f requen- c ies f~ > f~0, they tend to the constant value Dlk. There fo re , in the ent i re f requency r ange which we are consider ing, we can r ep lace the exact va lues of Nvak (~) by the approx imate express ion

N~ak(Q ) = DokQ~ + Dlkfl ~

The c h a r a c t e r i s t i c f ea tu re of N v ak(f~) is the i r substant ia l reduct ion when ~2 << ~2~, as compa red with the s i tuat ion for ~2 >> f~0- f f w e suppose that SXk(~2) = const , the function Sva(~2 ) will be s im i l a r to Nvak(~2). This f o r m of the curve ( rapid fal l as ~ -* 0) is analogous to the f requency fluctuation spec t rum of a noisy osc i l l a to r locked to another noisy osc i l l a to r [6]. This analogy enables us to cons ider a two-c i r cu i t quar tz osc i l l a to r as a s y s t e m of two mutual ly synchronized osc i l l a to r s (using a single tube), one of which is the s ing l e -c i r cu i t quar tz osc i l l a to r and the other the anode c i rcu i t osc i l l a to r . In this case , the "good" osc i l - l a tor (having the higher Q circuit) begins to influence the "poore r " osc i l l a to r at sufficiently low fluctuation f requenc ies cor responding to the band width of the high-Q c i rcui t .

If we r e s t r i c t our at tention to quas t i s ta t ic f luctuat ions (i .e. , f r equenc ies 2 << ~21),we can show that

N , , k ( f l ) = N , ak ( f l ) = Do k = const,

and this is now an exact r e su l t . In this way, instead of Eq. (6) we now obtain :5

o~ ~ Do*Sxk(2)" (7) k = l

423

It follows that for quasis ta t ic p a r a m e t e r f luctuat ions the spec t r a of the anode and quartz f requency f luctu- at ions a re wholly de te rmined by the spec t r a of the p a r a m e t e r f luctuations.

4 . W I D T H A N D S H A P E O F T H E S P E C T R A L L I N E

It follows f r o m the foregoing d iscuss ion that the spec t r a l dens i t ies of the f requency fluctuations, S~ (f~) and Sv a (~ ) , do not vanish for ~ = 0 if the spec t r a of the p a r a m e t e r f luctuations extend to zero . It is well known that, in this case , the spec t r a l line of the osci l la t ion b roadens and the line width is of ma jo r in te res t .

Examples of f requency fluctuation spec t r a for f l i cke r - type p a r a m e t e r f luctuations a re shown in Fig. 1 (see Fig. 3 in [2]).

The approx imate express ion for the spec t r a l dens i t ies of the quartz and anode f requency f luctuations for ~ < f~l is seen f r o m Eq. (6) to be

5

S,('2) == S , ( s = -~- ]~]Do k 2=1s k = l

(8)

Using the r e s u l t s of Section 4.5 in [5], these spec t r a l dens i t ies co r r e spond to the following expres s ions for the re la t ive width of the osci l la t ion line, which is due to c i r c u i t - p a r a m e t e r f luctuations:

5

~f = 2,2[ .~ Do kK~k] ''2 ('f ---- 0,9),

5

~f= XDo,K~k-~blnl~2br) (7 = 1),

k ~ l

5 1[2

(9)

The shape of the spec t ra l l ines of the anode and quar tz c i rcu i t osc i l la t ions cor responding to the spec t ra l densi t ies given by Eq. (6) a re difficult to es tabl i sh because of computat ional compl ica t ions . Nev- e r the l e s s , it may be concluded that the shape of the spec t ra l line is in te rmedia te between the resonance shape and the Doppler shape for 0 < Y < 1, and as Y approaches unity, it a s s u m e s the pure Doppler shape (within the band containing mos t of the power) . When i -<- 7 < 3, the f requency f luctuations become non- s ta t ionary and, the re fo re , the width and shape of the spec t ra l l ines become functions of the t ime of m e a - su rement . The spec t r a l - l ine shape a s s u m e s the Doppler f o r m only a f t e r a ce r t a in definite t ime has e lapsed (see Section 4.5 in [5]).

5 . C O N C L U S I O N S A N D N U M E R I C A L E S T I M A T E S

We may conclude f r o m the foregoing ana lys i s that, in the case of technical f luctuations in a quartz osc i l l a tor , the posi t ion of the spec t ra l line and its width for the anode osci l la t ion will be p rac t i ca l ly the same as for the quartz osci l la t ion (this was also the case for the natural f luctuations). The bas ic di f ference between the quar tz and anode osci l la t ion s pe c t r a is , t he re fo re , only in the "pedes ta l s . " We may, the re fo re , suppose that the substantial i nc rease in Sva(~2) at f requenc ies ~ > ~1 leads to a higher spec t r a l line ped- es ta l for the anode osci l la t ions and to a m o r e va r i ed line shape as compa red with the quartz osci l la t ion pedes ta l . It follows that whenever it is n e c e s s a r y to have a lower pedesta l , it is be t t e r to take the osc i l la - t ions f r o m the quartz c i rcui t . If, on the other hand, we a re in te res ted only in the peak value of the spec- t r a l line of the quar tz osc i l l a tor , it is i m m a t e r i a l whether the output osci l la t ion is taken f r o m the quartz or the anode c i rcui t .

Another impor tant conclusion follows f r o m a c o m p a r i s o n between the values of the coeff ic ients D0k (Table 1). The o r d e r of magnitude of the coeff icients D0k enables us to m e a s u r e the contr ibut ion of the

424

pa rame te r fluctuations to the technical spectra l line width of the generated oscillation. It tu rns out that the l a rges t contribution to this line width is due to the quar tz -capaci tance fluctuations (if they are strong

0 The fluctuations in the remaining p a r a m e t e r s are highly attenuated by the enough) for which D O ~ ~ Pa �9 circui t . This enables us to obtain an approximate es t imate for the quar tz-capaci tance fluctuations by using the technical spectra l line width of the quartz osci l lat ions obtained experimental ly. If we take this line width relat ive to the spectra l line obtained in [7] (i.e., 6f = 3.3" 10 7 and 5 f= 5.5 �9 10-9), we have(forT = 0.9)

K a c K " ~ 10-14

K ~ c , , ~ 10 -1~

(3f = 3.3" 10-7),

(V = 6 5 . io-~).

Consequently, one would expect that the spectral density of the quar tz-capaci tance oscillations is of the order of

10-14-- lO-m (10) &c,,(-q) - 2~ [ ~ IT

We note that, as far as we know, there are no published measurements of quar tz-capaci tance fluctuations or any es t imates of their o rder of magnitude.

Fluctuations in the capaci tances C and Cs, for which the coefficients D02 and D04 are lower by a fac tor of tZa-2 than D o ~, are of the next order of small quantities as far as the contributions to the frequency fluctuations of the quartz osci l la tor are concerned. Consequently, for these fluctuations we must have K6C and K6C k ~ t~a 2 to ensure that the technical line width is the same. When ~ = Qa ~ 102, we must have K/~ C and K6C k N 10-t0for 6f = 3.3"10 -7 , and K6C, K6C k ~ 10-I4 for 6f = 5.5- 10 -8.

According to es t imates given in Section 3.4 of [5], in the case of fluctuations in the tube capaci tances , one may expect that K6C and K6C k are of the o rder of "~ t0-~4-10 -I6. Thus, a technical spect ra l line width of 5f = 5.5 �9 10 -9 may be completely due to fluctuations in C and Cs, whereas the technical fluctuations in a se l f -osc i l l a to r with a spec t ra l line width of the o rder of 10 -7 should be unaffected by them.

For fluctuations in the coupling capacitance C 1 and in the t ransconductance S of the tube (using Table 1 to est imate the corresponding D o 3 and Dos ) we should have K6C ~ and K6S "~ pa 4 i.e., KSC 1 and K6S ~ 10-~ for 6f ~ 10-Tand KSC 1 and K6S N 10-10 for 6f -~ 10 -$ (when p a ~ = 102). Such ah igh fluctuation intensity for the coupling capacitance is unexpected, but for the tube transeonductance fluctuations this o rder of K6S can be obtained if the circui t is supplied by "poor" supply sources {in the sense of f l icker fluctuations; see Table 1 in [5], Section 3.4), or when a tube has a "poor" cathode.

It follows that the main contribution to the technical line width of the quartz osci l la tor is due to fluctuations in the tube t ransconductance and quartz capacitance.

The authors are indebted to I. L. Bershtein and M. E. Gertsenshtein for useful c r i t i c i sms .

L I T E R A T U R E C I T E D

1. M. E. Zhabotinskii and P. E. Zi l 'berman, DAN SSSR, 119, 918, 1958. 2. A. N. Malakhov, Izvest iya VUZ. radiofizika, 9, no. 3, 622, 1966. 3. A. N. Malakhov, Izvest iya VUZ. radiofizika, 1_~1, no. 6, 850, 1968. 4. Yu. 1~. Aptek and D. P. Filatov, Radiotekhnika i elektronika, 1_~1, 759, 1966. 5. A. N. Malakhov~ Fluctuations in Self-Oscillating Systems [in Russian], izd. Nauka, Moscow, 1968. 6. A. N. Malkahov, Izvest iya VUZ. radiofizika, _8, no. 6, 1160, 1965. 7. D . A . Dmitrenko and A. I. Chikin, Izvest iya VUZ. radiofizika, _6, no. 6, 1271, 1963.

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