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THEXRX OF OSCIuIlToR DESIGN

Ekich Hafner

I, IETRODUCTION:The approach t o the design of harmonic osci l ls to rs which w i l l bereported on i n t h i s paper has been developed in the course of a programaimed principaUy at establishing the fundamental limitations on the frequen-

cy stabi l i ty at tain able by means of mechanical resomtors. A major portionof the paper therefore, deals with the cr i ter ia which have t o be met i norder t o assure that th e frequency stabiliz ing chara cteris tics of a crystaluni t are fully trtiliz ed and that the influence of changes in the variouscircuit parameters on the output -frequency are kept t o a minimum.Harever, it i s the fonmrlation of the condition s for oscil lation i n anapparently new manner w h i c h i s of pivotal importance for t h e developentsi n t h i s paper. This new form, though only slightly different from previous

usage, opens the way t o achieving a very thorough qualitative understandingof oscillator performance whichwbe found usefW. i n a wide variety ofapplications.Many dif fe re nt methodsand techniques for the des gn of harmonicosci l la to rs have een reported on in th e l i terature l; and under Signa lCorps Contracts5- , Sane ma;y beconsidered more important han others,

but each one has added signif icant ly to the prese nt state -of-the -art. Adetailed discussion of their re la t ive merits however, would undoubtedly leadtoo far afield.

13 t

The success of any analysis of os c il la to r performance depends in nosmall measure on an adequate app reciat ion of the phenomena which determinethe amplitude of th e steady state os ci ll at io ns . Within the concept of theequiv alent lines rizat ion which, incidentally, will be used throwout t h i spaper, amplitude limit ing occu rs because the equivalent linear parametersof an ac tual ly nonl ine ar element dependupon the amplitude of the signal,Dasher and Wit@have investigated some aspects of t h i s phenomenon i nconnection with osci l la to r design considerations, while Reich lo asemphasized i t s major sfgnificance for t h e understanding of osci l la to rbehadour an a broader basis.11. ISCUSSIONS:

An idealized osciUiator circuit i s shown in the upper p a r t ofFig. 1, The general impedances Zl, Z2, and Z form the feedback networkand the elec tron tube suppl ies the energy necessary t o sustain osci l lat ions.If the concept of the equivalent linearization i s used, the vacuum tubecan be treated a s though it were a linear element whose a.c. output current

3

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il i s proportional t o t h e &.c. input vol tage vg and 180' out of phase withX L

il = -Q vgSince the tube , however, is act- a non-linear element, th e value of ~mdepends upon the amplitude of the signal v . A thorough discussion oft h i s subject, though most in terest ing i n i f s e w , i s outside the scope of t h i spaper and we w i l l assume here only that gm as a function of A, the ampliturleof the fuadamental component of the oscilLator signal, gives a steadilydecreasing curve such as shown in the lower pari., of Fig. 1. This assmptionmeets with the most camonly encountered behavior and does not impose aserious restr ict ion on the developnents of t h i s paper. The value of b,which represents t h e transconductance of the tube for inf ini tely smallsignals, as w e l l a s the exact shape of the ~m vs amplitude curve dependsupon t he type of tube and upon the d.c, po te nt ia ls on a l l th e tube electrodesHowever, t h i s curve i s independent f r o m the impedances Z l , 22, and 23 i n t h efeedback network.

The voltage vg i n Fig. 1 s related t o t h e input current il by therelat ion

v = il z1 z2Q ZSwith

zs - Z l + z, 23' (2 .4The re l s t ion (2), because of (l), educes t o the condit ions for osci l lat ionsi n the form

If we se tZi = Zi ePi (i = 1, 2, 3, S)

and separate the r e a l fram the imaginary part i n ( 3 ) w e obtain the twoconditionsel + e2 - es - T

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w h i c h have t o be sat isf ied sjmil-taneouslyfor steady st at e os ci ll at io ns tooccur. The condition (6) requires that the ratio of the impedance magni-tudes be smaller than The value of t h i s a t i o determines t h e ~mwhich must be exhibite& he active devlce dur ing steady state o sci l l at ionsandnence ii ae temr les ine amplitude of the oscillations by of theamplitude dependence of shown i n the lower p& of Fig. 1. If largeamplitude Osci Uat ions are required, therefore, it lq2ecessary t o make thedifference ~l lho as Large as possible. To obtain small amplitudes theopposite i s true, of course.It might be w e l l t o point out in t h i s connection, that an incidentalchange in any one of the impedance m i t u d e s w i l l change the value of theimpedance ratio in (6 ) and ence cause t o change. Since, however, avariat ion in ~m can take place only if t% amplitude of osci l lat ionschanges, the incidental variation in one of the imgedcwce magnitudes w i l lcause a change in mp llt ude which can be quite appreciable i f t h e slopeof the h vs amplitude curve i s smaJJ.. In order to steepen he slope oft h i s curve, par tic ula rly for low amplitude operation, one or more of thedoc. bias potent ials are freque nt ly made a fltnction of amplitude throughthe use of AGC circui ts .While equation (6) deternzines then the amplitude of os ci lla tio n, th eequation ( 5 ) determines the frequency a t which the osci l lat ions take place.Since all three impedances 21, Q,nd 23 in Fig. 1 nd therefore Zs asveU are, in general , fbct ions of frequency, one or more of t he phaseangles in (5) w i l l also be functions of frequency and t h e equation can be

satisf ied on l y a t one, o r more often a t severaldiscretepoints. Ifthere are several, stable oscillations w i l l occur a t that root of (5) a twhich (6) requires t h e lowest value of %.Any an al yt ic method t o solve ( 5 ) i s impractical for all but th e mostidea li ze d ci rc ui ts , however, it i s possible and indeed most i n s t ruc t ive to

solve t h i s equation without any additional approximations graphically in theimpedance plane.While t h i s technique i s by no means res t r ic ted to quar tz crys ta losc ill ato rs, we w i l l use in the following the Pierce and Mi lle r cir cui tst o i l l u s t r a te th e approach.Before we go in to it, however, we believe it quite useful t o reviewi n some d e t a i l the impedance diagram of a crys ta l unit because of i t s

fundamental importance fo r cr ys ta l os ci lla to r op er at io n and performance.2. THE IMPEDANCE DIAGRAM OF A QUARTZ CRYSTAII UNIT

The familiar equivalent electr ical circui t of a crys t a l unit withload capacitor i s sham on th e upper ri&thand side of Fig. 2 and next t o it,it i s dram again t o iden t i fy the spbols whch will be used i n describingi t s properties. Only the narrow fl-equency rage of a part icular crystalresponse i s of interest and it w i l l be a s m d t ha t X the reactance of themotional m, s th e only quantity whch changes appreciably with frequencyin t h i s range, i.e., Xo, XL, and Rlare assumed t o be constants.

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If the impedance = R + jX of a crystal unit i s measured point bypoint as a function of frequency, it w i l l be found to describe a circ le inthe R-X plane. Such an impedance diagram of a crystal unit i s shotm as theheavily drawn c i rc le in Fig. 2. It can be described by theequation2 2

x 2 2 2(R-G)(x - x - XL) = (.eThe vector = % eJ x representing the impedancebetween the two

terminals of th e cr yst a l network follows t h i s c i rc le i n aclockwisedirection a s the frequency of operation is changed upward through therange of th e crys ta l response. Each point of t h i s circle corresponds t oa di ff er en t frequency and a measure of the change i n with frequencycan be obtained by proxiding the circ le with a frequency calib ratio n.

This i s the purpose of the set of circles drawn in Fig. 2 with thinlines, Eachoneof th es e ci rc le s, which fol low the equationr x 2

belongs t o a different value fo r X 1 and hence t o a d m e r e n t frequency.Those dzawn into the diagram, Fig. 2, are separated f r o m one another by&22 equal increment in Xl, and hence by an equal nmber of cycles, overa certain range of Xl around Xl = 0. Since the inter cepts of these circl eswith the impedance cir cl e of th e cr ys ta l unit identify the frequenciesa t which Z, assmes the corresponding values, a diagram such as shown inFig, 2 provides a rather i l lus t ra t ive picture of the behavior of the crys ta limpedance throughout the response range.

As an example we will consider the effects of a vsriat ion in the loadimpedance XL, From the equations (7) and (8) it can beseen that anychange in XL effects a simple t ranslat ion of the entir e d along theimaginary axis. If XL and are both capacitive as has been assumed here,an increase in the maguitude of X.,., w i l l cause the ent i re set of c i r c les tomove damward. Consequently, si nce the frequency calibration on theimpedance circle i s not a9fected thereby, the frequency a t which G has agiven phase angle, say 0 = 0, w i l l increase, h i l l thereby move into arange of the impedance c%cle where th e frequency ca li br at io n becomes m o r eopen. In an osci l la tor which should be frequency modulated, therefore,a given amount of mdulation in XLwcause a correspondingly smallermodulationof the olxtplxt frequency. On the other hand, i f XL i s madeinductive, for example, by using a condenser and c o i l i n series i n placeof CL, the distance between the center of the impedance ctrcle and thereal axis can be nade quite mall.. Since now the resonancefrequency ofth e network, i.e., th e frequency at which ex = 0, i s in a range of thehpe dan ce cir cle where the frequency calib ratio n poin ts are c losertogether, the same anount of modulation in % will cause a much Largermodulation of the output frequency.

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To help avoid a trivial t rap i n th e use of the impedance diagramla te r on, it should be emphasized that the impedance vector r;, s tar ts ofcourse a t the origin of the R-X coordinate system, while the impedanceci rc le touches the imaginary axis a t X = X, + XL.

I n an oscillator such as shown %nFig. 1, th e cr Js ta l networkcm take the place of any oneof th e th re e impedances Z1,Z$If it i s Z and the impedances Z1 and % are capacitive, the resultmgconfigurat?on i s that of the w e l l known Pierce oscillator, shownschematically in Fig. 3.

and z3 '

For the moment, it w i l l be assumed that the impedances Z 1 and +have been chosen i n i t i a l l y and that it shall be determined i f t h i sconfiguration w i l l osci l la te with a pafcticular crystal unit and. a t what pointon th e impedance ci rc le th e cr ys ta l w i l l be operated. According t o thecondition (5), osc illa tio ns catl take place only i f e the phase angle ofthe sum vector Zs = Z1 + $ +Z3, i s given by

e, = el + e2 +W can plot, there fore , the impedances Z and % in the impedance planeand f i n d the angle 81 + &z as well as e:= 81 + e2 + lao,uch as shownin t h e lef'thand diagram of rYg. 3. Rence, f o r t h e phase relat ion -bo besat isf ied, the sum vector Zs must f a l l along the broken line i n t h e f i r s tquadrant 0

To f i n d the magnitude of ZS, it i s on ly necessary t o f i r s t obtainZ + Iz/L and t o use t h i s impedance as the or ig in for the impedance diagramo h 'ne cry s ta l network. This i s shown in the righthand diagram of Fig. 3.The broken l ine circle denotes all possible values for Z while Z canonly assume v a l u e s along t he l i ne 3const, The in te rs ep t of tge setwo lin es her efo re determines Zs and Z . Apparently Z the impedance ofth e crystaL network,must a l w s have d nductive reac?&e which i slarger than the sun of the capacitive reactances of % .and % unless theresistive components of the l a t t e r are zero, i.e.unless and e2 areboth-goo.

If the diagram has been drawn t o scale, the length of 2 can bemeasured and the r a t i o gm

determined. If is Fnnnlr than osci l lat ions w i l l take place and,from the &tude curve, the%$litude of these osci l lat ions canbe found. e i r frequency folluws from the value of Z together withthe equation (8) The la t ter , incidentally, will gene$& be found mucheasier t o use, e i ther in the form shown or solved f o r X1, than most otherrelat ions f o r this purpose v

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From t h e diagram in Fig. 3, it c a n be readily appreciated that,f o r example, a decrease i n + XL will cause the impedance circle tointercept the line 8, = const a t a lower point. The magnitude of Zs,therefore , decreases and with it the ratio G, cal l ing for higher amplitudeoscil lat ion. This eff ec t of course becomes m r e prcnounced as the curvatureof the c r y s t a l impedance c ir cl e in cr ea se s ei th er due t o a larger value ofR or a SmaUer X . Drawing a se t of constant requency ci rc le s (8) intothe diagrau of Fig. 3 provides furthermore a very direct impression of theinterdependence of frequency of oscil lat ion, crystal res is tance R and loadreactance XL. It i s v e l l t o take note hereby of the fac t that the constantfrequency c i r c l e s (8) are independent of R1.4. THE MILLEB OSCILLATOR

If th e crystal. network takes the place of in Fig. 1, we obtainth e basic diagram of the Miller oscil lator as shown in Fig. 4. I n orderto explain the qualitative features of t h i s oscil lator, w e w i l l assume nowtha t Z1 and Z3 have been chosen in i t i a l l y and t ha t w e wish t o determine t h eImpedance, Q i n t h i s case, which the crystal network is required to exhibi tduring steady-state oscillations - provided oscillations are possible.

The phase relation (5 ) re qu ir es nowe - e2 = r + el.

Since, however, 0 and 0 are of course not independent f r o m one another,the problems of shv ing r 5 ) is now sliatly more complicated, though notessential ly different from th e previous case. In Section 3 we had todetermine at f i r s t the loc i of all values Zs f o r which (5 ) is satisfied.This was done i n the lef'thand diagram in Fig. 3 and the curve happened tobe the straight l ine 8, = const. In the righthand diagram of t h i s f igurewe then found the actual operating point as the intercept of t h i s curvewith th e one describing those v a l u e s of Z which th e cr yst al network i scapable ofassuming. 3An identical procedure can be followed t o solve (5) i f % i s theunknown iurpedance. Disregarding f o r the mment thephysicalsignificanceof t h e 0' S, solving the equation ( 9 ) becanes a purely geometric problem.The solution describes a c i rc le i n th e impedance plane. Part of t h i s circ lei s dram i n t o t h e lefthand diagram i n Fig . 4. It goes through the or ig inof the impedance plane and through t h e end point of the vector Z1 + ZA t h i r d point on t h i s c i rc le i s found mos t conveniently by assuming 3*= goo. According to (5) , 8, i s then given by 8, = Q1 -~+hichi s readily obtained M m he graph. The c i rc le is then drawn through theseth re e p in ts , however, only the solid portion of it corresponds to physicallyrealizable values of Z and hence Z . In therighthandpicture of Fig. 4the impedance diagram 8f the crystaf unit is sham superimposed on the graphjust obtained, with i t s origin a t t h e inte rcep ts of thesetwo curves identifyhose values which satisfy the phaserelation and at the same time are orherystal network. O fthe two intercepts, only the upper one is of interest because it correspondsto a sub sta nti all y lower value of h. If t h e a t t e r i s smaller than ,the circuit w i l l osci l la te a t the frequency at which th e cr ys ta l exhib%the requ ired impedance Z2.

5 1 3

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The diagram in Fig. 4 s h ow s quite clearly that even though theimpedance $ i s inductive, the crystal unit i s s t i l l operated i n i t s l o wres ist ive region. By no means, therefore, should th e fa ct tha t the Millerosci l la to r i s knm as a parallel resonance os ci ll at or be taken t o imply ha tthe crystal uni t i s operated near i t s anti-resonance point, i.e . , in thehigh resistance range of i t s impedance circle. If the c r y s t a l i s operatedcorrectly, that is , according t o specifications, the Pierce oscillatoras well as the Miller osci l la to r will have the same output frequency whichi s only B different w a ~ f stat ing that the crystal i s operated on thesane point of i t s impedance circle in both cases.

Considerations such as were made f o r th e Pie rce oscil lat orconcerning the effects of a change in series load reactance % can ofcourse eas ily be trraslated t o app ly for the Miller osci l lator . Withs l igh t ly mre effort, the graphical construction in Fig. 4 together withthe equations (7) and (8) can be used to i l lu s t ra te the effects of achange in G, such as introduced by a load element in pa ra l l e l with thest at ic capacitance of the crystal uni t . Xr, thereby may or may not be l e f tto go to zero.Pri ncip all y fo r th e purpose of demonstrating some of the featuresof crystal operation in an osci l lator w e have assumed i n Sections 3 and 4

that the operating point of the c r y s t d unit i s what has t o be determinedwh ile he oth er impedances i n the ci rcu i t are known. However, oneproblemin designing an osci l la to r i s frequently to assure, by proper choice of theot he r ci rc ui t components, that the crys ta l un i t i s operated a t i t s specifiedoperating point in order t o obtain frequency correlation. This means thatthe impedances Z1 and 2 i n the case of th e Pierce oscillater o r 2 and 23for the Mill er osci llatgr have t o be determined, together with thehropervalues of Xr, and Xo, t o obtain the desired operation.

Since the conditions fo r os ci lla tio n can obviously be met with avery wide range of c i rcu i t impedances, the deci sive ques tion which has tobe answered i s what are the optinmu values.For frequency cont rol appli cations, the most pertinent criterionfor the performance i s undoubtedly the degree t o which the desiredfrequency of osci l lat ion i s maintained. It can easily be appreciated fromeither Fig. 3 or Fig. 4 that any change in, for example, Z1: will requireei ther + or Z3 o r both t o change if the conditions for oscillations

should again be sat i s f ied f o r the new value of Zl, .e., fo r Z1 +AZ1The rela tions hip betveen the various dZ S can be found quitegenerally by diffe rentia ting the condition for osci lla tio n ( 3 ). W find

Because the Z's are generally functions of a nuuiber of psrameters such asresistors, condensers, and inductors as well a s of frequency, i.e.,

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the differentials arenz ,=g

with like expressions for ZpEachoneof theAZ's

a phase angle

23 and AQ, I- 3 respectively.has, as a complex quantity a magnitude and

If (4 ) and (13) are substituted into (10) he equation can be separatedinto real and imaginary parts. The real pa.rt of (10) determines th e changesin ~m and hence in the amplitude of oscillation. The imaginary part,however, contains the eqxation forb, the change i n frequency. WithdZ, =AZ, +AZ2 +RZ which follows f r om (2a), the inaginary part of (10) becames3'

With (12) nd (13) it i s possible to cmpube from (14) th e frequency sh i f td& caused by a given set of parameter changes ( d d ). While the eqmtionlooks formidable i n i t s general form, it freq uent ly reduces to quite man-ageable expressions when a particular circu2-t i s being corsidered.Using the d i a g r a m of Fig 3 or Fig 4 it will be realized thateach one of the enns i n (14) represents a phase angle change. Forexample, the second term i s the ra t i o of that component of Z which i snormdl t o 2 t o t h e ma w t u d e of Z and hence rep res ent s the bane i neS due t o &e action of A Z . Simiflarily, the t h i r d term i s th e change i n8 due t o A Z a. s.0. The &ormation contained i n (14) could have beend h i v e d therhfore in principle by appropriately differentiating (S) ,harever, the resulting equation would be of very limited use zpd wouldagain have t o be transformed i nt o th e fo m (14) for practical application.Nevertheless the mere possibil i ty i s of great value conceptually in inter-preting t h e relat ion (14) * For example it demonstrates qu it e cl ea rl y thatth e sum of all phase angle changes caused by A Z 1 and A+ must be compensatedby an equal and opposite change i n 8, due to the act io n of ab Z3 ofappropriate magnitude. The change in e3, the phase angle of thehpedance23 , does not enter the s tab i l i ty relation directly.An important aspect of (14) i s the general type of design infor-mation whichcan be extracted fram it. We can use again the Pierc eosc i l l a to r to i l lus t ra te t h i s point.In the case of the Pierce osci l la tor Z3 repesen t s the crystalnetwork and dZ,/Q& wordinarily be so large compared ogZp/gw and QZ~/&Jt h a t the l a t t e r two can be neglected compased t o th e f i r s t . 4 Z1 and A 5

there fore represent changes in Z1 and Z2 which are caused by variations inany one o r more of the parameters d and o( of (ll) Since in t h i s typeosciUator Z and Z2 are nearly a,ld$s the &dances of networks consistingof sev era l e2ements i n p a r d l e l , it i s more convenient t o replace51 5

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ldZll by 1Zll ]dyr andld%I by I5 [AY2 . If i n addition it i s assumedt h a t the parameters of th e cr ys ta l network, h remain constant, th e equation(14) becomes

2

The f i r s t term in (15) can be transformed =her by using anexpre ssion for the quali ty facto r of th e crys ta l network which i s validover th e en ti re range of th e cr ys ta l response. Such a formula f o r thecrys ta l Q has been found t o be given by

With (16) and (6) the equation (15) can be w rit te n in the following form:

wherebyReal Z3Qeff = Q sin(@ - es)1 Z s l 23

Evidently, a given set of Id fI I and Idbiiw i l l cause the smal les t change i nfrequency i f Qeff and ~m are as large as possible and Zlland l% ( aresmall.Qefe is the effect ive qual i ty factor of th e cr ys ta l network i n theosci l la to r c l rcu i t . It aJways i s smaller than Q*, the qual i ty factor ofthe cry st al network alone, with the factors contributing to the degradationapparent frm (18) Evidently, t h e maximum value fo r Qe i s and t oattain it, it would be necessary that I2 I = Real 2 and f6dz - 0,) = 90From Fig. 3 it i s seen that, approximatefy, lZsl = a e a l Z 1 + aeal % + Real 23Hence, the condition i Zs/ Real Z would requ ire the impedances Z1 and $t o be purely imaginary. This f a d has of course long been recognized andcan frequently be approximated fa i r ly w e l l in vacuum tube circuits. Fig 3also shows that the phase angledifference ( 0 - gs) w i l l be 90 only if

Zs in tersects the impedance circle of th e crys t a l a t right angle - asituation which i s not attainable in a Pierc e oscil lator, even i f t h e f i r s tcondition i s met, unless CO cancellation i s used. Since the sin function,however, i s quite f l a t around go0, t h i s source of Q degradation, by i tselP,i s frequently small, part icular ly at lower frequencies. A more extensive

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discussion of the effect ive qual i ty factor w i l l be found i n Section 7.Since Z1, Z2 and Z are re la ted to one another and the effect ivetransconductavlce of the ?&e, , y ( 6 ) , the lower lim it fo r 1 Z and j Z2 Ii s set by the available of%e ac tiv e device. To obtain helhighestkrequency sta bil ity , heore, eqgires that i s as close t o ~ m o s

possible. This means by impli ca tion hat he itude of theoscxllationswill be sxdl unless hevs amplitude curve i s extremely f l a t . Thel a t te r of course i s tmdes' 43b le because of poor amplitude stability aspointedout in Section 1. As th e cr ys ta l frequency i s se ns it iv e to amplitudevariations here i s an optimum point of oper ation on th e vsamplitudecurve which w i l l give the highest frequency stability and$1 s applies alsoi f the shape of t h i s curve i s modified by AGC action. Particularly inconnection with &C i t i s important t o note t ha t the frequency deviationdb/. according t o (17) depends ac tua l ly upn the impedance magnitudesand only becauseof (6) on t he effe ctive ~m of the active device. AGCshould ther efo re be appl ied sparingly so as not t o reduce th e operatingpoint of t h e active device and hence the available ~m more than necessa ry.

It has been assumed so far that th e parameters of the c r y s t a l networkd remain constantwhile the terminating impedances ZL and 5 me subjecttpc hange . According t o (17) it i s principally th e effect ive q u a l i t y factorof the crys tal unit which determines the sensi t iv i ty of the osci l latorfrequency t o the variations in Z and Z2. An essentially different ypeof condition, however, exists i$ 21 = A Z2 = 0 i s assumed and theparameters of the crystal network are varied. I n th is case the equation(14) reduces t o th e equation8 - e s = oRZ3

which s ta tes that Z can only change dong the l ine 8, - const. Thisl a t t e r f ac t can eas9ly be verified by using t h e diagram i n Fig. 3. SinceZ 1 and 2 re assumed constant, 9, i s not aFfected by a variation i n anyone of t e pa ram en ter sR 3 and hence remains constant. A specific exampleof the si tuat ion exist ing now has aheady been considered in Section 3where theffects of a discussed. It can be shownthat the change i n frequency of i n response t o a variat ion ofthe parameters i s controlled primari ly by t h e capacitance ratio Co/Clof the cqystal uhit and i s independent t o a large measure of the qualityfactor $.

All the consideratio ns made so fa r for the Pier ce osci llator apply,with some modifications, t o t h e Miller osci l la to r as well, however, i npractical applications it w i l l be found that the l a t te r i s more difficultt o design properly and i n addition i t s impedance le ve l appears t o be re-s t r i c ted t o a ra th er unfavorable range.

6. Generalization of the Os cil lat or Equations.When dealing with vacuum tube oscillators, particularly thoseemploying pentodes, the approach taken in Section 1might be found

acceptable f o r mst engineering applications without much hesi ta tion .The n a t u r e of presently available traslsistcirs however leaves no choicebut t o examine th e problem more c ar ef ul ly in order t o determine how mch517

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of the foregoing analysis needs t o be modified and anended fo r it t o applyto t rans is to r osci l la to rs as well.A perfectly general representation of a harmonic oscillator i ss h a m in Fig. 5. It cons ists of two four terminal networks in a cyclicasrangement. TheNetwork I shall cont ain he activ e element, o r elements

i f m r e than one i s used, while th e Network I1 contains all the elementsof th e biasing network and the a-c feedback Loop. The load i s natur-a part of Network 11.Within the concept of equivalent linearization it i s possible,using matrix theory, t o completely c hara cteri ze the sma l l signal performanceof the Metwork I by a se t of four constants, the most appro priate for theconfigurationbeing he a parameters. The inpubo utpu t re la tions f o rthe Network I are thereby given by

V 1 = 1 1 V 2 - a & 2$1= 9 1 2 - 922 12 (20)

It i s important t o emphasize tha t the parameters ( a ) areen ti re ly independent from the properties of the Network 11. %heir valuesdo of course dependupon the biasi ng cond itions and upon the signal. l evelin a manner s i m i b t o that explained in Sec tion 1 o r h.I n input-ohput re la t ions for t h e Network I1 areV = ah V4 - i433 = a& v4 - 14 (21)

Because of the cy cl ic arrangement of the networks shown in Fig. 5.= V2, i = -i4 nd V = Vl must hold during steadystate~ ~ ~ ~ % ~ ~t can% verified kuite readily with (20) and (21) hat forthese relat ions t o be possible, it

1 - a 2 1 4 2 - a1241w i t h

Equation (22) thereforei n general form, stibject only toof equivalent inearization. Itosci l lators al ike.

i s required that

+ A 4 Od- a 2 2 4 2 - la1

expresses the conditions for oscillationthe lim it at ions imposed by the conceptapplies t o vacuum tlibe and t rans is to r

If a feedback loop i n the fo? of a Tnetwork is assumed, assuggested i n Fig. 5, the parameters (aik) are given by518

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U\ Z G -Since every -earform, the above a s s m i o nfollowing considerations.

1 (23)+ 3 zs * z1 + z2 + z3

z1passive network can be represented in t h i sdoes not res t r ic t th e generality of theof a tra nsi sto r are no t widely used and itwill be more convenient t o express the equat ion s i n terms of th e commonemitter "h" parameters. The l a t t e r are less abstract in t h e i r meaningand are usually obtained by direct measurement. Again, th e f a c t tha t thecommon emitter parameters are chosen here i s only a matter of conventionand does not impose any l imitat ions on the validity of the equations.The two fou r termina l networks i n Fig. 5 can always be defined such as t o

conform t o t h i s convention.The well known relations between the "arc nd "h" parameters are@v= by h12

$1( =l;] a a - - (241bh $1 h22 - h12 $1- - --If (23) and (24) are inserted into (22) itwill be found possible,after some re-arrangement of terms but without approximations, t o writethe generaland thus t osect ions f

condition for oscillations in the form

reduce it to the basic equation discussed i n the foregoingt h i s paper. The -1s in (25) are defined as follows:"m

h= 21 h12 t hah2 Z3hl 1 h21 1- -z = z1 + z* + 23

- - 1 A hEl z1 h l l- - + -

Clearly, i f th e elements of the pas sive feedback network in Fig.5are suitably redefined t o incl ude the effe ctive input and output impedancesof the active network dl he cons iderations based before on an idealizedos ci ll at or ci rc ui t remain valid, save fo r the mdification s required bythe fac t that t h e equivalent transconductance g m cannot ordin ar il y be assumedt o be a real quantity. 519

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The correc tion erm nvolv ing z i n (26) i s IikeLy t o prove neg-3l i g i b l e i n all but extreme cases. If it i s not suff ic ient ly small, ani ter at io n process has t o be used in solvin g (25) .It i s observed, t h a t according t o (26) , (28) and (29 ) t he t r ans i s t o ri n an osci l la tor i s equivalent t o a current generator of internal hpedanc e

h /Ah and n input impedance h I t s stren@;thil i s re la ted t o the nputv03 age by il = G vt,. Since% t ransistor thereby i s represented bya black box, t h i s statement applies equal& w el l to any other type ofamplifier, no matterhar simple o r complicated i t s structure. To apply,.there fore, the methods of os ci ll a to r analysis developed for an ideal currentgenerator t o any actual c i rcui t conf igurat ion it i s only necessazy t oco nsi der the inte rna l impedance as w e l l as the input impedanceof theequivalent generator as part of the feedback network.The parameters of t he ac ti ve device are s t i l l separated f romthose of the feedback network in the expressions (28) and (29). The basicadvantage of div idin g the osc illa tor int o an active and a passive networkas shmm i n Figure 5 i s s t i l l retain ed there fore, namely, th at th e para-meter of each one can be measured independently from the ot he r.The amplitude and phasefrom ( 2 5 ) by sepa rating the realwere found f r o m (3). They axe

whereby

re la t ions in t h i s general case are obtainedand imaginary parts jus t as ( 5 ) and (6)

Though the existence of a f i n i t e phase angle 8 g does require some minor modifi-'cation, the graphical method q l a i n e d previously i s readily applied t othesolution of (30). The effects of a f i n i t e 8 are most clearlydemonstrated q ua lit ati ve ly by us- th e co rr es po hi ng diagrams. For examplesince 0 i s usually negative, .e., of th e same sign as and l2 in aP ie rc e & c i D t o r , it i s noted tha t a 8 of pro r value can cause eSt o became zero even i f f 3 i s l es s han l8Oq Under certaincondlt ionstherefore, it i s possible t hag a f i n i t e 8 w i l l r e su l t i n improved oscillatorperfonnance. To identify heseconditio&, however,becomes increasinglymore difficult as the nwnber of variables increases and th ei r re la t i vetendency t o change i n value w i t h time and environmental conditiom i sconsidered.7. The Generalized StabilityRelations:

The f a c t t ha t ~m in (25) s no longer a rea l quant i ty i n thegeneral case also requires some modifications of t he s t ab i l i t y r e l a t i onsderived previously i n Section 5. To simplif'y w rit in g we w i l l , throughout

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Because of the large differences i n Q, the change in Z and Zwith frequency can be neglected campared t o and theref ore,%he segondand the third terms in (S)o not conta in AH . The f i r s t tern, hawever,can under these conditions be written with (16) aswhereby

Whatever the second an d t h i r d terms in ( 3 )might be, therefore,that is , whatever the change in the terminating impedances ZI and %,the resul t ing change i n fYequency w i l l be lowest i f the effect ive qual i tyfactor, Qeff, i s as Large as possible.The t r u t h of t h i s statement has of course long been recognizedeven i f the expression ( 3 6 ) has apparently not been avai lable before.The f i r s t general objective in designing an osci l la to r f o r maximumfrequency s tab ilit y the ref ore i s t o reduce t o a minimum all those factorswhich tend t o degrade the Q of the c r y s t a l unit once it i s incorporatedin to the c i r cu i t .The second cr ite rio n de rive s most d ir ec tl y m m considerationof the frequency changes caused by a change i n any one of the components

$:&ling parameter here i s the c ~ / ~ lat io of th e cr ys ta l network. Alarge r a t i o i s desired if the resulting frequency change should be small.For optimum stabi l i ty herefore, the c / ratio should be keptas argeas pract ical . To some extent t h i s cr i t e rc* on applies t o the proper choiceof the crys ta l un i t to be used in the circuit, since generally, f o r a giventh e c r y s t a l v i t h the higher resonance resistance w i l l have the higherr a t i o. However, i n some cases it might be desirable t o include animpedance matching network into 23 t o transform the cry sta l impedance t oa more favorable level.

It can be shown, as mentioned already i n Section 5, that the

Because of the mda me nta l iurportance of the effective qualityfactor fo r the proper design of an osciUator it might be w e l l t o b r ing( 3 6 ) in to a mre convenient form. W wfll use thereby the folluwingdefini t ions and relations.

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and with it obtain, except fo r terms which are sma l l of second order, t h efollowing expression for sff CO&, sin(@dZ? os- oA3+ e,)Qeff Qo 1 + - [ (40)I zm lFur the r t rans fomt ions do not appear advisable i n the general case. IfG1, G R and gim axe given quantities, the opthum ra t io I ll I + can

be de$e&ed from (40), however, the process i s quite cmbersame sinceit i s usually not possible t o ignore the dependence of the numerator onel and according t o (30). The form (40) appears t o be quite suitablefor the use of an it er at io n process, carr ied owt analytically, orgraphically i n the impedance plane.A substantial ly mre convenient expression for the effec t ivequality factor can be found, harever, if, as is very often the case, itcan be pov en justifi ed t o assme

0A z3 2 900 .f 8Ag = e, = 3 (4.1 )The numerator i n (40) then reduces t oa,, + g,) = COS 8,O&, - 8,- i ) 2 (42)

which can be approximated f o r reasonably small angles by

Because of (31)

so that the expression (40) can be rewritten approximately in the hi@;hlyuseful form:

Except for terms which are small of higher order, the relat ion (45) i sv a l i d i f the assumptions (41) are me and the phase angle 8, i s fimR17enough f o r (43) t o hold.Most frequently the values of GI, G2, % and R 3 are alreadydetermined once 821 active device and a cr ys ta l un it have been selected,

Leavbg the ra t io ( Z l ( / f% l i n (45) t o be adjusted fo r minimum depadation.Evidently t h i s requires

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G1 1 + 2R3G1The minimum value fo r ( 45 ) is then given by

Among the in teres t ing fea tures of (47) i s i t s dependence on R-the resistive component of thecrystal network. A s R i s decreased fr&some large val-ce, the Q degradat ion becomes event=& independent of Ii2nd the use of a c rys t a l un i t with a s t i l l lover R3 w i l l not result h 3

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For exeqple,with he same valuesfor Gl, , nd % assumed beforef o r a 2Nw t ransis tor one obta ins XII = 208sL , , 9= E n i f t h e c r y s t a lres is tance R 3 i s 1004. With X1 determined from 48) and (49) it onlyremains t o choose: XL i n Z such that XI + X2 + X L equals the reactance ofthe load capacitance spec%eied f o r t h e c r y s t a l unit t o be used and thedesign of the oscil lator i s essentially completed.One fac tor , however, ha s no t yet beenconsidered explicitly,namely the power dissipation in th e va ri ou s components.

8. DriveLevel and 0utp.Crt Pare r:During steady state osc i l l a t ions the vo l tages and cur re n t s in mo s c i l l a t o r c i r c u i t have a l l t o be re l a te d to one another in a veryde fi ni te manner and hence th e ir r e l a ti v e magnitudes are firmly established.

I n f a c t it i s prec i se ly th i s re l a t ion sh ip which leads t o t h e c o n d i t i o n sfor osc i l l a t ion s , no matter how theyareformulated. "he absolute magnitudeof t hev olt age san d cur ren ts remainundetermined unle ss he amp litud e depend-ence of thenonlinear elements is known. The l e f t s ide of equation (22) forexample i s a f 'unction of the transistor parameters (aik), a l l of whichdepend upon the amplitude of t h e signal. I1Jomally there will only be oneset of values ( a ), asstmed at a def ini te value for he signal amplitude,f o r which t he rigt side of (22) i s zero, for any other amplitude it willbe l a rger o r smaller than zero and the conditions for steady oscil lat ionsare not sa t is f ied . An accurate calculation of t h i s c r i t i c a l am plitude i spossible in principle i f t h e parameters (a! ) of the pas siv e network ar eknmm and theparameters (a ) of t h e a c t6 e device are avai lable asfunctions of signal am pl it ug . Such cal cul ati ons would obviously bequitecomp lica ted and, because of the nhe ren tsources of error ,a reveryl ike ly no t jus t i f i ed even i n extreme cases.

A more p r a c t i c a l approach can be taken if th e phase angles of th eimpedances Z and % as defined by (28) and (29) are i n the order of o rless than 10 . It 1 s then possible t o approximately consider Z1 and 3as independent of amplitude,using nominal valuesfor h u and hor values measured i n an impedance bridge at approximately the desiredamplitude i f increasedaccuracy i s required. This leavesrespectively, as t h e on ly amplitude dependent quantity i ni n ( 3 1 ) .If th e term with Z i s dele ted in (26 ) , the remainingexpressionfor % represents the effeczive transconduc tance of the active device with

i t s output short circuited and a sui tab le mdi fic ati on of any one of th emethods customarily used on vacuum tub es can be employed t o eva lua te i t smagnitudeand phase angle.Using reasonable care, it i s general ly poss ible to adjus t theparameters of the feedback networksuch th at th e impedance r a t i o IZsl/lZdt%i

f a l l s v i t h i n lO$?of a predeterminedvalue. If therefore a curve such asindicated i n Fig. 1has been ound expe rime ntal ly and thecorrespondingt o t h e desired amplitude of oscil lat ions determined,th e amp9 tude actual lyobtained canbe in er ro r by an amount which clearly depends on t h e slopeof t h i s curve. Par t icular i ly at low amplitudes, a 1% e r r o r i n t h eordin ate of this curve v i l l generally be found i nto ler abl e and the des ire d

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amplitude has t o be obtained by manual o r automatic fine adjustments of thebiascondi t ions of theactive device.With the otrtpuk ef fec t ive ly shor t c i rcu i ted during the @in measure-ment, th e amplitude of t he input voltage becomes t he cri t ic al var ia bl e f o rt h e % vs. amplitudecurves,with he bias conditions as parameters.Sincethe amplitudes of t he cur ren t s an d voltages a t the various points i n theo s c i l l a t o r are of course quite d i f f e ren t it was l e f t open so f a r whichmpl i tude i s best chosen as a reference and it i s now es ta bl is he d thatthe amplitude of t h e input voltage to the ac t i ve device i s t he log ica lchoice. The amplitudes of a l l other voltages and currents during ste.;,dys t a t e o s c i l h t i o n s can be derived from i t , using the condit ions forosc i l l a t ion .For the fol lowing considerat ion i n t h i s s ec t i on we vi11 again useth e ba si c ci rc ui t diagram shown i n Fig. 1, keeping in mind, however, that

%he mpedances 2, and % now include espect ively the effect iveoutputandinput impedance of theact ivedevice. If' the rms value of the i npu tvoltage Pg i s known, the parer, dissipated i n the impedmxe$ (k = 1, 2, 3) canbe

Normally t he load w i l l be part of Z and the pater del ivered into the loadcan be computed from Pl f o r any &icular case. For the prese nt purposeit e11 su f f i ce t o cons i de r P1 as synoqnmus wi th PL, t h e power into t h eload. If th e cr ys ta l network i s Z3 and the only re s is ti v e component i nt h i s network i s due t o t h e c r y s t a l unit, P3 w i l l be the power dissipatedin t he c ry s t a l .Since the r a t i o of t h e power del ivered to the load to the powerdi ss ipa ted i n t he c ry s t a l un i t i s frequent ly of considerable concern, wew i l l derive now an expression for PdP3 which v er y cleml y i l lus t ra tes t h ethe inf luence of the various ci rcui t parameters on th is ra t i o .Fr0m (50)

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In the gen era l case, the magnitude of the vector % f 23 i s best obtainedfmn iagrams such as used for the graphic al analysis of the phase relations(3O), which of course w i l l be very similar t o the one sham i n Fig. 3 . Ifhowever, th e phase angle of Q i s close to 900 and the phase angle of thesum vector Z, clo se t o zero, we c m set approximately

since, under theseconditions X1 + X2 + X 3 e O . With ( 5 3 ) and ( k g ) ,which also holds under these conditio ns, the power ra ti o ( 5 2 ) can bem i t t e n as

In R vel3 designed o x i l l a t o r the Q degradation should be kept a t aminbm, t ha t is , t h e r a t i o 1 X11 / I X21 i s given by (48). Hence,

i s the expression for the pawer rat io i n a Pie rce type osc illa tor whosecomponents me ad ju st ed f o r a re la t ive maximum i n effect ive Q. G1 and G2ar e th e conductances of the term ina ting networks Z 1 and Q respectivelyand R i s th e re si st iv e component of th e cr ys ta l network.3U s i n g ag ai n the approximations which lead from (52) t o (55), oneobtains i n a similar manner the following expressions:

In the preceeding sec tion we had chosen the following paraMeters asrepresentative of a 2Nm t r ans i s to r c i r cu i t G1 = 10-4 mho, G2 = lom2mho~m = 4 x 10-2 mho, R 3 = lOOa. Inserting these values into the relations(55) t h o @ (58) we f i n d P&= 5.3 x loe2 P2/P3 = 1.4 x 10-2, P3 = .69 Tg271 = 19 Fig. Hence, if thedrive evel of thecrys t a l unit i s P3= wattsit follows t h a t P1 = 5.3 x watts, P2 = 1.4 x watts, 5 = 1.2 x 10-3V and T l = 2.3 x lom2V. The corresponding Q degradation has previouslybeen determined t o D = 8.5%. Q

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It i s noted that even i f the maJor p& of the conductance G 1i s the load, the out put power from a Pierce type osc i l l a to r i s only asmall f r ac t i on of the power dissipated i n t he c rys t a l un i t and any attemptt o i n c r ea s e th i s fr a c t io n WiU result in increased Q degradation, evenunder optimum conditions. A s i s t rel l holm, largeoutput power and highs tab i l i ty are contradictory equirements.Nevertheless, he elations(55) through (58) together with (47) will be found use fu l t o r each anacc ept able compromise fo r a given application.

It i s noted further that for low c rys t a l d r i ve t he voltages TIand Ti are a t an extremely low level. If thebiasingvol tagesareadjusfed very careful ly, steady state operation of t h e o s c i l l a t o r can beachieved a t the se ev els even without AGC. The ampli tude stabi l i ty,however i s generallynotadequ ate under the seconditions and some formof a r t i f i c i a l l e v e l c o n t r o l i s nearly always required.111. CONCLUSIONS:

I n the foreg oing discu ssions we have ill u s tr a te d an approach to %hedes ign of osc illat ors which circumvents to a large measure the extremecomplexity of theanalyt icalexpressionsusuallyencountered. By re-pr es en ti ng fo r th e most pa rt th e network impedances in polar form it vasalso pos sibl e o avo id approximations i n t he early stages of the develop-ment and th us to dem ons trat e he nter rela tion s between th e variouselements in general erms.It i s an axiom of any analyt ic t reatment that the content of anequation describing a ph ysi ca l phenomenon cannot be changed by any mountof transformation. The in ter pre tat io n and solut ionof heequation however,ver y ofte n dependson one 's abi l i ty and good fortune to bring i t i n to aform which in i t s e l f suggestsalready hesolut ion. This of course i st r u e i n the present analysis a s wel l and a case in poin t i s t h e phasere l a t ion(5) . me st ructure of t h i s r e l a t i on as sham i s such as t o a c t u d l ydemand a graphical method for it a solution. A measure of t h e d i f f i c u l t i e s

involved in ex t ra ct i ng Lhe sal i ent fea ture s of th is equat ion by any othermeans are the comp lexities encounte red i f one attempts t o transform (5 )in to the condi t ion f o r osc i l l a t i on as it resul ts from repres enting theimpedances Z1, +, and 23 i n rec tangular coord ina tes in i t i a l ly .After choosing t h e bas i c o sc i l l a t o r diagram as shown i n Fig. Iand writing the condit ions for osc i l l a t i on in the form ( 3 ) , t h e r e s t oft he analysis of t he i dea l i zed o sc i l h t o r fo l l ow s almost inevitably oncethe impedances are expressed i n polar form. In pa rt i cu lar , th e graph ica l

in terpre tat ion of the phase relat ion and furt her on the s t a b i l i t y r e l a t i o n sare direct consequencesof t h e i n i t i a l approach.The equation (25) , however, which st at es in ef fe ct th at anya rb i t r a ry o sc i l l a t o r can be treated rig or ou sly by t he method developedf o r an i d e a l i z e d o s c i U t o r , i s again an example of t h e +.pemendoussimplification which can be achieved i f su f f i c i en t effort is expended intot ransforming the ini t ia l ly obtained equat ions until they are most amenablet o i n t e rp re t a t i on and solut ion.

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Throughout the main body of t h i s paper we have res t r ic t ed the de-t a i l ed discuss ions pr incipal ly t o tho se osc il l ato rs which have th e cr ys ta lu n i t i n t h e t o g se ct io n of he"network.Although t h i s covers a fairlyLarge cla ss of ci rc ui ts , it i s admittedly a configuraticn which i s mosteasilyanalyze d by thepresent echniq ue. Themore important c i r c u i t s int h i s category are t h e P i e r c e o s c i l l a t o r w-d t h e C1 meter c i r c u i t , as wsllas the transform er coupled oscil lator and their various modifica tions.

Comon t o all of t hes e c i r cui ts are thr ee and only three e s s e n t i a lnodes in t h e feedback network.More complicatedfeedback ci rc u it s can, as line ar pass ive networks,always be t r ans fe r red into equivalent II networks, but the equivalent imped-ances are generally not independent from one another and are often physical lynot realizable. While the con ditio ns for osci l lst i on can ther efor e be

expressed formally through th e same equations which ap pli ed for the thr eenode c i rcui ts , he nterpreta t ion i s f requent ly so d i f f i c u l t h a t h e r ei s no pr ac ti ca l advantage t o be derived from it. Rather, such c ir cu it sare b e s t t r e a t e d by d e r i r h g t h e "at' parame ters of the feed back networki n terms of the inpedances zk as they actually appear between th e vari ousnodes. If the act ive element i s then assumed a t f i r s t t o bean idealcur ren t genera to r , the cond i t ion fo r osc i l l a t ion invo lves on ly the pma-meter i.e. the ransfer dmit tance of the networknd the esul t in@equation w i l l have the smallest number of terms possible with t h i s t yp ec s c i l l a t o r . By propergrouping of th e terms anexpression which resembles(25) cm of te n be obtaine d and th e developments of th e th r ee nodeosc i l l a to r s can be used as a guide in the ana lys i s .

For tra ns is to r os ci l l at or s, th e approximation of the active deviceby an ideal current generator i s rarely adequate and additional complexit ieshave t o be expected, even i f only h= and are onsidere d, .e., i fh== h22 = 0 i s assumed. Nevertheless, for exmple in the case o f a"bridged TI' osci .Uto r, very simple and sur pris ingl y accu rate relat ionshave been obta ine d by expanding t he ba si c approach.

The card ina l ru le he re , as in t r e g t i n g any o s c i l l a t o r , i s t orepresent the impedances between t h e network nodes i n general terms andto int rod uce the ir res ist ive and reactiv e components only i f it can nolonger be avoided and only af t e r a c lea r qua l i t i a t ive F ic tu re of th e i rrespec t ive ro les has beenobtained. There i s abso lute ly no need t o useth e R's, L's and C ' s of th e ac tu al network elements during the analysis.Within the frame work 02 the equivalent l inearization, these elements serveno other purpose i n t h e o s c i l l a t o r t h a n t o phys ica l ly rea l ize the res i s t iveand re ac ti ve components of th e re sp ec ti ve impedances and th e i r v a l u e s c mbedetermined accordingly at the ver y end of th e analysis.

This of course, i s a l so the prin cipl e which has been followedthro ugh out he developmentsof t h i s paper. The grap hica l solu tion of t h ephase rela tion prov ides a very c lea r in te rp re ta tion of the ro le of the threeimpedances involved. While it may be desirable or even necessary to usethese graphs for the quant i ta t ive analys is i n extreme cases, it i s pr inc ipa l lythe ir qu al i ta t ive fea tur es which havebeen important in the l a t e r sect ionsof the paper by providing guide lines t o acceptable approximations.

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The s ta bi l i ty re l a t io ns are s t i l l treated i n general terms andonly af ter the expression f o r the e f fec t ive qua l i ty fac to r was obtained i nSection 7 d i d we introduce the r e s i s t i v e and reactive components of thecircuit impedance - or thecorrespondingconductance and susceptance i ft h i s more convenient. Even if it vas necessary i n th e numericalexample t o choose l ikely va lues fo r the c i rcu i t losses , the re was no needa t any po in t in th e development t o s t i p u l a t e how the re ac ti ve componentsa r e t o be realized physically.

-'With a given set of c i rcui t losses , the analysis of an o s c i l l a t o ri n the general Pierce Configuration results in specific values f o r /X,-/andj$1 f o r which os ci ll at io ns oc cu r with a minimm on Q degradation. Thesiws o f these reactances are determined by th e phase re la ti on . Theymust ap-parently 8e ei ther both negat ive or both posi t ive if the ac t iveelement i s a 180 phase shift device.A nuniber of f a c to rs have t o be considered i n determining hov th eimpedances 21 and !+ thus specified are t o be real izedphysical ly .Firs t ,they have t o provide a d.c. pat h for bia sin g and one of them, generallyZ 1 has t o include the loading effect due to t h e amplifier stage followingth eos ci ll at or . Second, t h e i r frequency dependenfe must be such as t oprevent osci l la t ions a t any but the desired frequency.

The preferred way of bias ing a t r a n s i s t o r is through a l a rger e s i s t a n c e i n tp5,fptter path and very small res i s tances in the base Etnacol lectorpaths - Hence if Z1 and Z2 consist ach of a p a r a l l e lL-C combination, th e in du ctor s provide very low res is tance d . c . pathsfor biasi ng, besid es, ncide ntall y, improving the noi se performance ofthe o s c i l l a t o r . If overtonecrystals are used, one of these L-Ccombinations must be resonant above the freq uenc y of the next lower overtoneso tha t i t s reactance a t th e frequency of t h i s overtone i s already posit ivewhile the reactance of the other L-C cambination must s t i l l be negative.The other L-C combination must have i t s resonance frequency below th efundamental mode of t h e c r y s t a l t o p r e v e n t o s c i l l a t i o n s at any one of theovertones below the desired one.A t the frequency of the desired overtone both L-C combinationst r i l l then be capacitive. The actual values of the L ' s and C ' s necessaryt o meet all these requirements w i l l genera.Uy be found t o be i n a reason-able range and t o meet the addit ional requirement that no o s c i l l a t i o n s bepossible a t frequencies belov th e lower of the two resonances, where bo threactances axe posit ive.The condensor bypassing t h e emitter res is tance must be chosen aslarge as possible but small enough f o r th e corresponding R-C time constantt o be sma e r than the s ta r t up time of the osc i l l a to r t o he lp p reventsqueggin8 The Lat ter i s more l ikely t o occur with biasing throughinductors than through resistors, however, it can readi ly be avoided i nmost cases.Nevertheless, even if ' it does res-nd f r e q w n t l y t o simplecorrective measures, the phenomeron of squegging shows a number of featureswhich seem t o defy a simple explanation. A detailed study of i t s causesand the condi t ions for i t s occurrence may w e l l reveal eventually some very

f'undamental aspects of the behaviour of nonlinear active devices.

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IV. ACKNOIm-:It i s a pleasure to gratef 'u l ly acknowledge th een thu sia st ic ass is ta nceof M G. Davidson of USAELRDL, who has carr ied out a grea t many experi-ments and measurements on os ci l l at or s which have been indis pens able for

th e developnent of thepresent heory. MM. H. Thorn hasconstructedmost of t h e c i r c u i t s with painstaking care.v. REFERENCES:1. R. A. Heising: " Q u a r t z Crystals fo r E l ec t r i ca l C i r cu i t s "V a n Nostrand 1946.2. WA. Edson: "Vacuum W e Oscil lators" John Wiley 19533. W. Herzog: "Oszillatoren mit Schwingkristallen"Springer 19584. H. J. Reich:"FunctionalCircuits and Osc il lators " V a n Nostrand 19615. B. J. Dasher & S. N. Witt Jr: "Transis tor Osci l la tors of W e n d e dFrequency Range" Final Report,Si gn al Corps Contract DA36-039-sc-42'7l.2.Engineering Experiment Station, Georgia Insti tute of Technology, June 19556. H. E. Gruen & A. 0. Pl a i t : "A study of C r y s t a l O s c i u t o r C i r c u i t s " ,Fi na l Report , Signal Corps Contract DA36-039-sc-64609, Armour ResearchFoundation, Augu s t 19577.W. L. Smith, I' An Ultra-PreciseStandardof Frequency" Final Report,Signal Corps Contract~~36-039-sc-73078, B e l l Telephone Laboratories,December 1960.8. H. R. Meadows & D. Fi r th : "Quartz Crystal Osci l l a torCi rcu i t s" Fina l

Report, signal Corps Contract ~~36-039 -sc-8889 2, TheMa@;navox Company;December 1962.9. B. J. Dasher & S. N. W i t t op c i t

10. H. J. Reich op.cit.U . H. J. Reich op. c i t .12. WG. C a d y , "P ie zo el ec tr ic it y" McGraw H i l l 194613. R. F. Shea Edit . "!transis tor Circuit Engineering" John Wiley 195714. W. W. G h n e r : "Ftxmsistors:Principles, Design andApplications"Van Nostrand 1960.15. H. J. Reich "Amplitude Instability i n OscilLBtors", h o c IRE(Correspondence) v01 51 P 381 Fe3 1963.

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A

FIGURE 1

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LtlIIIIIIIIIIII

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IaU

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/

I

\

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m- + CT

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i 0x

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536