)
= = £ S = = = = = = = = = = = = = = = = = = = = =
volume xxiv JAN U AR Y, 1945 number i
1 P- % S M S ,
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED T O THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COM M UNICATION
Intermittent Behavior in Oscillators . . W. A. Edson 1
Evaluating the Relative Bending Strength of Crossarms Richard C. Eggleston} 23
Mathematical Analysis of Random Noise (Concluded)
S. O. Rice 46
Abstracts of Technical Articles by Bell System Authors 157
Contributors to this Issue ... 159
AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK
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EDITORS
R. W. King J. O. Perrine
EDITORIAL BOARD M. R. Sullivan
O. B. Blackwell H. S. Osborne J. J. Pilliod
O. E. Buckley M. J. Kelly A. B. Clark S. Bracken
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Copyright, 1945
American Telephone and Telegraph Company
P R IN T E D IN U . S . A.
T h e Bell S y stem T e c h n ic a l J o u r n a l
Vol. X X I V J a n u a r y , 1 9 4 5 N o . /
Interm ittent Behavior in O scilla t
By W. A. EDSON
Oscillators of all sorts m ay, for certain values of the p a ram eters^
frequency disturbances. Usually the disturbance takes the form of a low-fre
quency interruption of the desired oscillation. B y the m ethod here presented it is possible to determ ine w hether or n ot such in term itten t behavior will occur in a given oscillator an d w hat circuit modifications are required to prom ote stability. T he intentional generation of a m odulated wave by control of the low frequency behavior of an oscillator is also considered. Oscillators of the nega
tive resistance type are n ot considered.
I . In t r o d u c t io n
T T H A S been know n for a long tim e th a t all kinds of oscillators are su b je ct to th e tro u b le v ario u sly referred to as in te rm itte n t oscillation, m o to r b o atin g , or squegging. I n conventional circuits such as th e H a rtle y th e phenom enon is m o st likely to be observed if th e grid leak a n d grid condenser are ab n o rm a lly large. I t is fo u n d t h a t th e tim e c o n sta n t of th is com bina
tio n m u st be reduced as th e frequency is raised an d as th e Q of th e resonant circ u it is decreased. A t frequencies above a few h u n d re d m egacycles th e p ro b lem of prod u cin g a p ra c tic a l circu it w ith suitab le m argin of sta b ility is q u ite difficult.
W ith th e a d v e n t of th e oscillator h av in g a u to m a tic o u tp u t control th e p ro b lem assum ed a new a s p e c t.1’ 3 B y ap p lica tio n of a n am plified control circ u it a high degree of co nstancy of o u tp u t to g e th e r w ith low harm onic o u tp u t is o b ta in ed . S a tisfa cto ry o p era tio n is secured, how ever, only w hen su itab le a tte n tio n is given to th e ch aracteristics of th e co n tro l circuit.
T h e in te n tio n a l generation of pulses b y m eans of in te rm itte n t oscillations of rela tiv e ly high frequency h as been stu d ied to some ex ten t, a n d circuits of th is k in d are em ployed in some television system s. U sually th e high- freq u en cy oscillation is lim ited to a sm all p o rtio n of th e low -frequency cycle,' th e charge sto red d u rin g th is period being allow ed to dissip ate itself relatively slow ly d u rin g th e rem a in d er of th e cycle.
I n all of these circuits sa tisfac to ry perform ance depends upon a p ro p er p ro p o rtio n in g of elem ents n o t d irec tly associated w ith th e op eratin g fre-
i L . B. Argimbau, “An Oscillator H aving a L inear Operating C haracteristic,” Proc.
I.R .E ., Vol. 21, p. 14, Jan . 1933.
2 J Groszkowski, “ Oscillators w ith A utom atic Control of the Threshold of Regenera
tio n ,” Proc. I.R .E ., Vol. 22, p. 145, Feb. 1934.
q uency. W hen co n tin u o u s oscillation is necessary it is d esirable to p ro v id e a d e q u a te m a rg in a g a in st in te r m itte n t o p era tio n . W hen in te r m itte n t o p e ra tio n is desired th e opposite is tru e . I n e ith e r case a n u n d e rsta n d in g of th e sam e g eneral p ro b lem is necessary.
T h e p re se n t analysis applies o n ly to o scillators of th e fee d b ack ty p e . N o m e th o d of ex ten d in g it to cover n e g a tiv e resistan c e oscillato rs such as th e D y n a tro n a n d th e T ra n s itro n h a s b een found. R e la x a tio n oscillato rs as such are n o t considered here inasm uch as th e y a re seldom affected b y in te r m itte n t o p era tio n . N o specific fre q u en c y lim its a p p ly b u t it is som etim es difficult a t v e ry h igh frequencies to achieve d esirab le v alu es of th e c o n s ta n ts.
A t v e ry low frequencies o scillators em ploying a u to m a tic o u tp u t c o n tro l are re la tiv e ly u n su ita b le because th e ir p erfo rm an ce te n d s to b e u n d u ly sluggish.
T h e te rm lin e ar oscillato r is used to in d ic a te a n o sc illa to r in w hich th e range of o p era tio n is con tro lled w ith in such lim its t h a t th e h a rm o n ic c o n te n t of th e o u tp u t is inapp reciab le.
T h e gen eral e q u a tio n d escribing a sim ple a m p litu d e -m o d u la te d w av e is V = Vo(l + m sin 2irft) sin 2-kFt
T h is m a y be ta k e n as defining th e m o d u la tio n fa c to r m , a com plex n u m b e r w hich is lim ite d to m a g n itu d e s b etw een zero a n d one.
I I . Ge n e r a l Th e o r y o f Os c i l l a t i o n
I t is fo und t h a t th re e se p a ra te fu n ctio n s a re n ec essary a n d sufficient for th e o p era tio n of a n oscillato r of th e fee d b ack t y p e.3 T h ese are in d ic a te d in th e block d ia g ra m of F ig. 1.
T h e am plifier m u s t b e p ro v id e d to overcom e th e losses of th e re s t of th e system . T h e pow er o u tp u t, if a n y , dep en d s u p o n th e fa c t th a t th e o u tp u t of a n am plifier is g re a te r th a n th e in p u t.
S electiv ity m u s t be p ro v id e d to in su re t h a t th e o u tp u t h a s a definite frequency. O rd in arily a tu n e d circ u it of re la tiv e ly h ig h Q is u se d a lth o u g h some excellent oscillators em ploy resista n c e -c a p a c ita n c e n etw o rk s. T h e te rm filter is em ployed as being sufficiently gen eral to in clu d e th e se extrem es.
A lim iter of som e form is n ecessary to estab lish th e level a t w hich su sta in e d
»oscillations occur. I n m a n y circu its th e fu n ctio n s of am plifier a n d lim ite r a re p erform ed sim u ltan eo u sly in th e v a c u u m tu b e . I n a n im p o r ta n t class of oscillators th e lim iter is a th e rm a l device such as a tu n g s te n la m p . I n the M e ac h am circu it th e fu n ctio n s of lim iter a n d filter are com bined in a bridge em ploying a tu n e d circ u it a n d a tu n g ste n lam p.
T o sim plify th e analysis it is co n v en ien t to assum e th a t th e am plifier of Fig. 1 is com pletely linear a n d o p era tes w ith e q u a l gain a t all frequencies
3 T his topic is discussed more fully in “ H yper an d U ltra-H igh Frequency E ngineering ” R. I. Sarbacher, and W. A. Edson, Jo h n W iley & Sons, Inc., 1943.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 3
from zero to infinity. S im ilarly th e filter is assum ed to consist of linear circu it elem ents a n d to h av e a definite curve of loss versus frequency. Asso
ciated w ith th is loss ch a racteristic is some specific p h ase c h a ra c te ristic.4 T h e lim iter is assum ed to h a v e a loss w hich is in d e p en d e n t of frequency b u t w hich is explicitly rela ted to th e in p u t (or o u tp u t) voltage.
A lthough am plifiers h av in g th e ideal perform ance in d ic ated are n o t p h y si
cally realizable th e re are no new or un fam iliar concepts involved. S im ilarly th e perfo rm an ce of p assive netw orks, such as c o n s titu te th e filter, has been extensively stu d ie d an d is well understo o d . I t is therefore ap p ro p ria te to d ev o te th e following section to th e th ird function.
F IL T E R
L IM IT E R
Fig. 1— F unctional block diagram of an oscillator.
I I I . Ty p e s o p Li m i t e r s
T h e lim iters w hich are now in com m on use m a y be se p arate d into four re la tiv e ly d istin c t groups.
1. V acuum tu b e s in w hich th e gain is decreased b y sim ple overload as the level of oscillation rises. T h is is th e m ost com m on form of lim iter.
2. V aristo rs in w hich th e im pedance d epends upon th e in stan ta n eo u s v alue of cu rre n t. C opper oxide, th y rite , an d electronic diodes are exam ples.
3. T h erm isto rs in w hich th e resistance depends u p o n th e rm s v alu e of c u rre n t b u t does n o t v a r y ap p reciab ly d u rin g a n y one cycle. C arbon an d tu n g ste n filam ent lam ps are th e m o st com m on exam ples.
4. V acuum tu b e s in w hich th e gain is reduced b y ap p lica tio n of a bias w hich depends upon th e level of oscillation. U sually th e bias is developed b y rectifying a p o rtio n of th e o u tp u t.
T h e lim iters of th e first tw o groups depend for th e ir o p era tio n u p o n th e g en e ratio n of harm o n ic v o ltag es a n d cu rren ts. T h e lim iters of th e second
4 H . W. Bode, “ Relations Between A ttenuation and Phase in Feedback Amplifier D e
sign,” Bell Sys. Tech. Jour., Vol. 19, pp. 421-457, Ju ly 1940.
tw o groups o p e ra te w ith v e ry little h arm o n ic d isto rtio n . T h e o u tp u t of o scillators em ploying such lim iters m ay , therefore, b e m ade q u ite free from h arm o n ic vo ltag es. O scillators of th is so rt are referred to as lin e ar b ecause th e tu b e or tu b e s serve as sim ple Class A lin ear am plifiers.
IV . Cr i t e r i o n o f Se l f Mo d u l a t i o n
T h e block d ia g ra m of Fig. 1 is c h a ra c te riz e d b y th e fa c t t h a t th e se p a ra te elem ents are co nnected to each o th e r in th e form of a n endless ring. T h e o u tp u t m a y be assum ed to come from a n y of th e th re e ju n c tio n s. I t is th is fa c t of closure w hich com plicates th e p ro b lem of o scillato r s tu d y . F o r p urposes of analysis it is co n v en ien t to open th e loop as show n in F ig. 2.
F o r th is exam ple it m akes no difference w here we choose to m a k e th e c u t, b u t in a c tu a l circuits some c a u tio n m u s t be exercised. T h is m a tte r is dis-
Fig. 2—T est for self-modulation in an oscillator.
cussed m ore fully la te r. I t is also necessary to choose th e im p ed an ces of th e te s t g e n e rato r a n d te s t d e te c to r so th a t th e o p e ra tio n of th e co m p o n en ts of th e .o rig in a l sy stem is n o t d istu rb e d .
If a continuous w ave of su itab le v o lta g e a n d fre q u en c y is su p p lied b y th e te s t g en e rato r it will be found th a t th e te rm in a l v o lta g e of th e te s t d e te c to r is id e n tic al in m a g n itu d e a n d p h ase w ith th a t of th e g e n e ra to r. I n th is co ndition th e req u irem en ts w hich are fu n d a m e n ta l to osc illa to rs a re s a tis fied. T h a t is, th e frequency a n d level a t w hich oscillation sho u ld occu r if th e circ u it w ere closed as in Fig. 1 h a v e b een estab lish ed . T h e n e t p h ase sh ift of th e system is zero a n d th e n e t gain is zero.
W h e th e r th e oscillations so p ro d u ced w ould b e sta b le or in te r ru p te d is now determ in e d b y ad d in g am p litu d e m o d u la tio n of rela tiv e ly low fre q u en c y a n d v e ry sm all m a g n itu d e to th e te s t g en e rato r. I t is clear th a t th is m o d u la tio n will be tra n s m itte d th ro u g h th e am plifier, filter, a n d lim ite r to th e te s t d e te c to r a n d th a t th e p h ase an d p erc en ta g e of th e m o d u la tio n m a y b o th be
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 5
m odified. B y exam ining th e transm ission of a lig h tly m o d u la te d w ave for vario u s frequencies of m o d u la tio n it is possible to d eterm in e w h eth er or n o t th e n o rm al oscillation will be self m o d u la te d w hen th e loop is closed as in F ig. 1.
T h e c a rrie r is held c o n s ta n t a t th e frequency F a n d am p litu d e V for w hich th e in p u t an d o u tp u t are id e n tic al, a n d th e frequency / of th e m o dulation is v a rie d from zero to infinity. I n th e following tre a tm e n t it is assum ed th a t th e significant p o rtio n of th e c h a racteristic is observed for m o dulation fre
quencies sm all com pared to F. T h e th e o ry is sim plified in th is w ay w ith o u t being seriously re stric te d in usefulness. T h e percentage of m od u latio n m u st be h eld v e ry low so as n o t to exceed th e n o rm al o p era tin g range of th e lim iter.
T h e criterio n is m o st co n veniently s ta te d in te rm s of th e transm ission of the m od u latio n envelope w hich m a y be considered as a v ec to r q u a n tity .
Fig. 3—N y q u ist diagram showing m agnitude and phase of loop transmission.
Legend: U is unstable
C is conditionally stable S is absolutely stable
A p lo t of th e v e c to r ra tio of o u tp u t to in p u t m od u latio n for v arious fre
quencies is p re p a re d as in Fig. 3. T h e sy stem characterized b y curve U is u n sta b le an d will g en e rate a self m o d u la te d ra th e r th a n a continuous w ave.
T h e system characterized b y curve S is uncond itio n ally stab le an d will be free from self m odulatio n . T h e sy stem ch aracterized b y curve C is condi
tio n a lly sta b le a n d m a y g enerate eith er a continuous or an in te rru p te d w ave d epending up o n th e m a n n er in w hich th e oscillation is s ta rte d a n d o th e r factors.
V . An a l o g y o p t h e Os c i l l a t o r t o t h e Fe e d b a c k Am p l i f i e r
T h e b eh a v io r of oscillators of th e ty p e here considered is en tirely d e
p e n d e n t u p o n feedback. I t is therefore ap p ro p ria te to review th e fu n d a m e n ta l principles w hich ap p ly to feedback in general.
I n th e feedback am plifier, n eg ativ e feedback is applied to im prove th e lin e arity , sta b ility , im pedance, or frequency ch aracteristics. C onsiderable im p ro v em en ts in som e or all of th e p ro p ertie s m a y be secured if a consider
able a m o u n t of n eg a tiv e feedback is ap p lied a n d p ro p e rly controlled. P o si
tiv e feedback is som etim es used to increase gain o r se lec tiv ity , b u t s ta b ility u n d e r such circum stances is poor. A n y considerable a m o u n t of p o sitiv e feedback resu lts in oscillation.
T he criterio n b y w hich sta b le feed b ack sy stem s are d istin g u ish e d from u n sta b le ones h as been p re se n te d b y N y q u is t a n d verified b y o th e rs.6’ 6 A closed feedback sy stem h a v in g in p u t a n d o u tp u t te rm in a ls is illu s tra te d in Fig. 4. In Fig. 5 th e loop is opened a t som e a r b itra ry p o in t a n d a te s t
oscillator a n d d e te c to r are connected. H ere as in F ig. 2 c e rta in p re c a u tio n s a s to im pedance are observed. T h e te s t g e n e ra to r m u s t p ro d u ce a p u re sinusoidal w ave of such sm all m a g n itu d e th a t no p a r t of th e te s te d sy stem overloads a n d th e v ec to r ra tio of th e d e te c to r v o lta g e to th e g e n e ra to r v o lta g e is observed for a large n u m b e r of frequencies. T h e p o la r p lo t of F ig. 3 applies d ire c tly to th e feed b ack am plifier ex cep t t h a t th e ra d iu s v e c to r rep resen ts th e tran sm issio n of a sim ple w ave r a th e r th a n of a n envelope.
5H . N yquist, “ R egeneration T h eo ry ,” Bell Sys. Tech. Jour., Vol. 11, pp. 126-147 Ja n ., 1932.
6 E. Peterson, J. G. K reer, & L. A. W are, “ R egeneration T heory an d E x p erim en t,”
Proc. I.R .E ., Vol. 22, pp. 1191-1210, O ct., 1934.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 7
T h e conditions of ab so lu te an d conditional s ta b ility an d in sta b ility are ex a ctly th e sam e as those alre ad y given.
I t m u st be a p p re cia ted th a t N y q u is t’s criterio n supplies no inform ation as to th e ty p e or frequency of o scillations w hich will be g en erated b y a n u n sta b le sy stem . T h is is tru e because th e analysis is lim ited to linear system s. T he only in fo rm atio n im p a rte d is th a t a v e ry sm all oscillation of some frequency will increase ex p o n en tially w ith tim e u n til th e a m p litu d e is lim ited b y the ac tio n of som e non-lin ear device. A sm all or rela tiv e ly large sh ift of fre
q uency m a y occur a n d th e oscillation m a y be reg u lar or in te rm itte n t. T he p re se n t w ork extends th e usefulness of N y q u is t’s criterion b y using it in m odified form to d eterm in e w h eth er or n o t a p a rtic u la r u n sta b le system (oscillator) h as or lacks s ta b ility as to self-m odulation. T h ere is no a p p a re n t reason w hy a system lacking in b o th fu n d a m e n ta l an d envelope s ta b ility m ig h t n o t be analyzed a th ir d tim e for th e sta b ility of th e self-m odulation.
V I. A n a l y s i s o f a n O s c i l l a t o r h a v i n g A u t o m a t i c O u t p u t C o n t r o l F ig u re 6 p rese n ts a sim ple form of feedback oscillator h av in g a se p arate rectifier as lim iter. F o r sm all am p litu d es of oscillation th e tu b e operates in a lin ear fashion w ith cathode self-bias. N o bias is produced b y th e diode rectifier u n til th e p e a k v o lta g e in th e coil L3 exceeds th a t of th e bias b a tte ry B . A ll v o lta g e in excess of th is v alu e is rectified, sm oothed b y th e condenser C, a n d applied to th e resisto r r as bias. I t is seen t h a t a sm all percentage change in th e o u tp u t level m a y re su lt in a large change in th e bias. A ccord
ingly a n o u tp u t w hich is q u ite sta b le w ith resp ect to th e tu b e condition an d ap p lied v oltages, except th a t of B , is to be expected.
T h e s ta b ility of th is circ u it w ith resp ect to self m o d u latio n is m osi con
v e n ie n tly te ste d b y opening th e oscillatory loop a t th e p la te of th e tube.
I n so fa r as th e p la te resistan ce of th e tu b e is high w ith resp ect to th a t of the associated circu it it is n o t necessary to co n tro l th e im pedances of th e te st g en e rato r an d d e te c to r ex trem ely ac cu rately . A block d ia g ra m equiv alen t to Fig. 6 is p rese n ted in Fig. 7. T h e conditions w hich m u st exist for the te s t of s ta b ility a re show n in F ig. 8. I n b o th th o se figures it should be noted t h a t th e gain co n tro l is a c tu a te d b y th e in p u t, n o t th e o u tp u t, of th e am p li
fier. I t is th erefo re possible for a m a rk e d decrease of o u tp u t v o ltag e to re su lt from a sm all increase of in p u t voltage. T h is beh av io r is v ery different from t h a t of th e conventional, b ack -actin g , autom atic-v o lu m e-co n tro l am plifier in w hich th e o u tp u t change is in th e sam e d irec tio n as th e in p u t change b u t of reduced m a g n itu d e . I t is th is difference w hich is th e basis of m o st difficulty w ith a m p litu d e controlled oscillators.
Fig. 8—T est for m odulation stability of autom atic o u tp u t control oscillator.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 9
Filter
T h e filter of Fig. 8 consists of only a single tu n e d circuit. I ts transm ission is rea d ily rep rese n ted in te rm s of th e circu it Q b y th e fam iliar universal resonance curve. T h e tran sm issio n of a m o d u la te d w ave th ro u g h such a passive n etw o rk is conveniently d eterm in e d b y se p aratin g th e w ave in to its ca rrie r a n d tw o sidebands. T h e carrier will be th e frequency F corre
sponding to zero p h ase sh ift w hich, in th is case, is also th e frequency of m axim um transm ission. T h e sidebands will be shifted in phase by equal
Fig. 9— Envelope transm ission of a m odulated wave through a single tu n ed circuit of selectivity Q.
Fig. 10— D a ta of Fig. 9 p lotted in polar form.
a n d opposite am o u n ts an d a tte n u a te d according to th e frequency / b y which th e y differ from th e carrier. T h is beh av io r is in te rp re te d in Fig. 9 as tra n s m ission a n d p h ase sh ift of th e envelope. I t is seen t h a t th e transm ission app ro ach es zero a n d th e p hase sh ift approaches 90° as th e m odulation frequency is indefinitely increased. T h e sam e d a ta is presen ted in p o lar form in Fig. 10. Specifically Fig. 10 shows th e v e c to r ra tio of th e m odula
tio n fa c to r m of th e o u tp u t w ave to th a t of th e in p u t w ave for all frequencies.
I n Fig. 9 th e m a g n itu d e an d p h ase angle of th e ra tio are show n separately.
Lim iter
T h e lim iting ac tio n of th e tu b e a n d diode com bination is determ ined b y d ire c t circu it analysis. F o r v e ry low m o d u latin g frequencies th e condenser
C of Fig. 6 serves only as a h igh-frequency b y -p ass; th e d ire c t v o lta g e across r being th e in sta n ta n e o u s difference b etw een th e p e a k v o lta g e in d u ced in Z3 a n d th a t of th e stabilizing b a tte r y B . F o r v e ry h igh m o d u la tin g frequencies th e m o d u la tio n as well as th e ca rrie r is b y-passed b y C a n d no m o d u la tio n vo lta g e ap p e ars across r. T h u s th e bias is c o n s ta n t a n d th e o u tp u t w ave is id e n tic al w ith th e in p u t w a v e . T h is corresponds to a n envelope tran sm issio n of (1, 0). F o r in te rm e d ia te valu es of th e m o d u la tin g fre q u en c y th e v o lta g e developed across r v aries in m a g n itu d e a n d p h ase a p p ro x im a te ly as if a c o n s ta n t c u rre n t of th e m o d u la tin g fre q u en c y / w ere ap p lied to r a n d C in parallel.
T h e o u tp u t of th e am plifier depends n o t o n ly u p o n th e b ia s developed across r b u t also up o n th e in p u t. F o r sy stem s h a v in g a large a m o u n t of co n tro l th e ac tio n of th e bias is p re d o m in a n t. T h u s for a low m o d u la tin g frequency th e v a ria tio n of th e b ia s overpow ers th e in itia l m o d u la tio n , th e phase of th e m o d u la tio n is reversed, a n d th e p erc en ta g e m agnified b y th e
Fig. 11—Envelope transm ission of a m odulated wave through controlled amplifier.
ac tio n of th e lim iter. I n Fig. 11 th e envelope tran sm issio n is p lo tte d in p o la r form for conditions of re la tiv e ly large a n d re la tiv e ly sm all a m o u n ts of control.
Loop Transm ission
T h e se p a ra te diagram s of Figs. 10 a n d 11 are com bined in F ig. 12 to d e
term in e th e sta b ility of th e system . F o r a n y chosen fre q u en c y / th e v e c to r of Fig. 10 is m ultip lied b y th e v e c to r of Fig. 11 co rresponding to th e sam e frequency to locate one p o in t of Fig. 12. T h e r e s u lta n t v e c to r h a s a n angle w hich is th e sum of th e tw o com ponent angles a n d a m a g n itu d e w hich is th e p ro d u c t of th e tw o co m ponent m ag n itu d es.
I t is seen t h a t th e loop m a y be m ade to cross th e axis co n sid erab ly to th e le ft of th e p o in t (1 ,0 ) if th e p o in ts A a n d A ' of th e p rev io u s figures co r
respond to th e sam e frequency. S im ilarly th e loop m a y b e m a d e to come v e ry close to th e p o in t ( 1 ,0 ) b y increasing th e size of C or low ering th e Q of th e tu n e d circ u it so th a t th e p o in ts B a n d B ' correspond to th e sam e frequency. W ith th e circu it elem ents d raw n in Fig. 6 th e s ta b ility m arg in
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 11
m a y b e reduced to zero, b u t a c tu a l looping of th e p o in t (1, 0) is n o t in d i
c a te d . P a ra sitic elem ents, n o t here considered, can readily affect th e perfo rm an ce enough to pro d u ce in stab ility .
V II. An a l y s i s o f t h e Ha r t l e y Os c i l l a t o r
T h e fam iliar H a rtle y O scillator circu it is show n in Fig. 13. In th is a rra n g e m e n t th e tu b e serves as am plifier a n d lim iter b y th e actio n of o v er
loading. H arm o n ic voltages a n d cu rren ts are p roduced b u t if th e selectivity of th e tu n e d circ u it is high th e v o lta g e re tu rn e d to th e grid of th e tu b e is n e a rly sinusoidal.
T h e sta b ility of th is circ u it is te ste d in exactly th e sam e w ay as was th a t of th e p rev io u s circuit. T h e loop is opened a t th e p la te of th e tu b e to d eterm in e th e tran sm issio n of a m o d u la te d signal. If, as is usu ally th e case, th e coupling of th e coil is close, th e filter reduces to a single tu n e d circuit.
T h e lim iting ac tio n resu lts from bias pro d u ced b y rectification a t th e grid.
A ccordingly th e block d ia g ra m of Fig. 7 is d irec tly applicable, an d the b eh a v io r of th e filter is correctly given b y Fig. 9.
G e n e r a l l y t h e c i r c u i t o p e r a t e s i n c l a s s “ C ” w i t h h i g h b i a s a n d l a r g e g r i d
vo lta g e swings. If th e tim e c o n s ta n t of th e grid-leak-condenser co m b in atio n is long in com parison to th e p erio d of a m o d u la tio n cycle th e b ia s w ill n o t be able to follow th e ap p lied v o lta g e a n d th e m o d u la tio n of th e o u tp u t w ill be larger th a n th a t of th e in p u t. M o reo v e r it is in p h ase w ith t h a t of th e in p u t. W hen th e m o d u la tin g fre q u en c y is low th e bias is ab le to follow th e level of m o d u latio n a n d th e o u tp u t m o d u la tio n is v e ry sm all. T h u s th e transm ission of a m o d u la te d signal is g re a te st a t high m o d u la tin g fre q u e n cies, an d th e m o d u la tio n o u tp u t is in p h ase w ith th e in p u t. B ecause of th e
Fig. 14— Envelope transmission of a m odulated wave through a grid-leak-biased Class C amplifier.
Fig. 15—N y q u ist diagram applying to Fig. 13.
ac tio n of th e grid-leak-condenser n etw o rk a p h ase sh ift a t in te rm e d ia te m o d u la tin g frequencies occurs. T h is b eh a v io r is rep rese n ted in p o la r form in Fig. 14.
T he sta b ility of th e system is d eterm in e d b y com bining in F ig. 15 th e se p arate diagram s of Figs. 14 a n d 10. As in th e p rev io u s sy stem a th o ro u g h ly stab le system resu lts if th e elem en t v alu es a re such t h a t th e p o in ts A a n d A ’ of Figs. 10 a n d 14 correspond to th e sam e freq u en cy . I f on th e o th e r h a n d th e elem ents are such th a t B a n d B ' correspond to th e sam e fre
quen cy th e curve loops (1, 0) in d ic atin g in sta b ility . I n g eneral s ta b ility is
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 13
p ro m o te d b y increase of th e Q of th e tu n e d circu it a n d b y decrease of th e tim e c o n s ta n t of th e grid-leak-condenser com bination.
V I I I . Th e La m p St a b i l i z e d Os c i l l a t o r
T h e circ u it of Fig. 16 is of p a rtic u la r in te re st because th e functions of am plifier, lim iter, an d filter are p erform ed se p arate ly b y u n its w hich are re a d ily identified w ith th e ir functions. T h e p rese n t m eth o d of analysis w as developed in connection w ith th is p a rtic u la r circuit. T h e o u tp u t freq u en cy a n d am p litu d e are b o th q u ite stab le a n d th e harm onic c o n ten t of th e o u tp u t is low.
U n d er o p era tin g conditions th e gain of th e tu n e d am plifier, which is o rd in arily in th e order of 40 db, is equalled b y th e loss of th e lam p bridge.
T h e lam ps o p era te a t such a te m p e ra tu re t h a t th e ir resistance is slightly less th a n th a t of th e associated linear resistors. If th e gain of th e am plifiers is for a n y reason som ew hat reduced, th e c u rren t th ro u g h th e lam ps decreases, th e te m p e ra tu re a n d resistance of th e lam ps is reduced, an d th e loss th rough th e brid g e is red u ced to th e new v alu e of am plifier gain.
T h e d-c ch a racteristic of a lam p bridge is show n in Fig. 17. A curve id e n tic a l w ith Fig. 17 is observed if th e m easu rem en t is m ade w ith an a lte r
n a tin g c u rre n t whose period is v ery sh o rt in com parison to th e th e rm al tim e-c o n sta n t of th e filam ents. U p to L th e o p eratio n is n early linear. In th e region of M th e o u tp u t is essentially in d ep en d en t of th e in p u t. A t N th e bridge is n e a rly b alan ced an d a sm all p erc en ta g e change in th e in p u t v o lta g e resu lts in a large a n d opposite percen tag e change in th e o u tp u t.
I t is th u s seen t h a t a n a lte rn a tin g c u rre n t hav in g a sm all superim posed m o d u la tio n of low frequency will resu lt in a n o u tp u t h av in g a considerably
larger p erc en ta g e m o d u la tio n in th e opposite ph ase. W hen th e m o d u la tio n freq u en cy exceeds a few h u n d re d cycles th e lam ps are u n ab le to follow th e in d iv id u al cycles a n d th e o u tp u t w ave is id e n tic al in form to th e in p u t. A t in te rm e d ia te m o d u la tin g frequencies th e tran sm issio n of a m o d u la te d w ave
Fig. 18— Envelope transm ission of a m odulated wave through a lam p bridge.
Fig. 19—Envelope transmission of a m odulated wave th rough two sim ilar tu n ed circuits of selectivity Q.
involves a p h ase shift. T h e b eh a v io r of a ty p ic a l lam p brid g e is p re se n te d in Fig. 18.
If th e Q of th e grid an d p la te circu its are b o th re la tiv e ly high th e filter circ u it m a y be ta k e n as e q u iv a le n t to tw o se p a ra te tu n e d circu its. T h e transm ission of each is given b y Fig. 9. T h e com bined tran sm issio n of th e p a ir is given in p o la r form in Fig. 19. B ecause tw o tu n e d circ u its are
em ployed, th e d ia g ra m of Fig. 19 differs m ark e d ly from th a t of Fig. 10.
Specifically th e p h ase sh ift corresponding to a given v alu e of a tte n u a tio n is g re a tly increased. As in prev io u s cases th e curve of over-all loop tra n sm is
sion m a y or m a y n o t loop th e p o in t (1,0) d epending up o n th e relative fre q u en c y scales. T h u s if th e p o in ts A a n d A ' of Figs. 18 an d 19 correspond to th e sam e freq u en cy th e N y q u ist d iag ram passes n e a r th e p o in t (2, 0) in d ic atin g in sta b ility . I f th e p o in ts B an d B ' correspond to th e sam e fre q u en c y th e loop passes v e ry n e a r to th e p o in t (1,0) a n d in sta b ility is likely.
B y m a k in g th e tu n e d circuits v e ry selective or b y reducing th e th erm al tim e c o n s ta n t of th e lam p circ u it th e p o in ts C an d C' m a y be m ade to cor
respond to th e sam e frequency. I n th is case th e loop passes to th e left of th e p o in t (1, 0) a n d th e sy stem is abso lu tely stable. T h e sam e resu lt m ay b e secured m ore easily b y m ak in g one of th e tu n e d circuits m uch more selective th a n th e o th er. T h is is o rd in arily accom plished b y increasing the Q a n d im pedance level of th e grid circuit while keeping th e Q a n d im pedance level of th e p la te circu it m uch low er so as to p rovide a su itab le pow er o u tp u t to o p era te th e lam p bridge.
I X . Th e Va r i s t o r St a b i l i z e d Os c i l l a t o r
A circu it w hich differs from t h a t of Fig. 16 only in th a t th e lam ps are replaced b y v a risto rs is show n in Fig. 20. A t low levels of oscillation the im pedance of th e v a risto rs is relativ ely high, th e loss of th e lim iter is low a n d th e am p litu d e of oscillation rises. A t some higher level th e v a risto r im pedance is reduced, th e bridge approaches balance to th e fu n d am en tal frequency, a n d a sta b le condition is reached. B ecause th e in itial u n b alan ce of th e bridge is opposite to th a t of Fig. 16 a reversal of phase is necessary to estab lish oscillation.
T h e sta b le co ndition reached differs from th a t of th e lam p stabilized o scillator in t h a t th e v a risto r goes th ro u g h its e n tire range d u rin g each high- fre q u en c y cycle. T h e lam p resistance changes b y only a sm all am o u n t d u rin g a n y one cycle, its resistan ce d epending on an in te g ratio n of m a n y p rev io u s cycles. T w o im p o rta n t facts arise from th is difference. H a r
m onics are p ro d u ce d in th e bridge an d , in so far as th e v aristo rs face re a c t
ances of th e se harm o n ic frequencies, in te rm o d u latio n m a y produce cu rren ts of fu n d a m e n ta l freq u en cy b u t shifted in p h ase w ith respect to th e original.
T h u s th e brid g e m a y p ro d u ce a p h ase sh ift w hich is a fu nction of level of the oscillation frequency. A d eg ra d atio n of frequency sta b ility results from such a condition. M o re im p o rta n t to th e p rese n t p roblem is th e fac t th a t all m o d u la tio n frequencies are tra n sm itte d alike. A sm all m odulation is reversed in p h ase an d m agnified b y a n am o u n t d epending upon th e bridge balan ce b u t n o t up o n th e m o d u la tio n frequency.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 15
B ecause th e lim iter in tro d u ces no p h ase sh ift it follows t h a t th e envelope loop tran sm issio n is m erely a n enlarged a n d reversed copy of t h a t for th e filter. T h is can loop th e (1,0) p o in t only if th e re are a t le a st th re e s h u n t elem ents in th e filter section. T h a t is, in sta b ility can re su lt only if th e p h ase sh ift of th e filter sy stem exceeds 180° for frequencies re la tiv e ly n e a r th e o p era tin g frequency. T h is circu it is therefore m uch less likely to p ro d u ce in te r m itte n t op eratio n th a n a n y o th e r circu it h ere considered.
X . Ne g a t i v e Fe e d b a c k i n Os c i l l a t o r s
B ecause po sitiv e feedback is th e necessary co n d itio n for th e o p e ra tio n of an o scillator it is n o t obvious th a t th e ap p lic a tio n of n e g a tiv e fee d b ack is ever desirable. A ctu a lly it is fre q u e n tly possible to in tro d u c e n e g a tiv e feedback in to an oscillator w ith no loss of perfo rm an ce a n d u n d e r c e rta in circum stances ad v a n ta g e s are gained.
T h e circ u it of Fig. 16 serves as a co n v e n ie n t exam ple. R e m o v a l of th e ca th o d e by-pass condenser is likely to reduce th e am plifier gain b y a b o u t 6 db a n d to increase th e sta b ility of th e gain w ith resp ect to ap p lied v o lta g es b y a corresponding a m o u n t. C oincident w ith rem o v al of th e b y -p ass condenser th e op eratin g level drops a sm all a m o u n t, th e brid g e loss decreases 6 db to reestablish equilibrium , a n d th e stabilizing effect of th e b rid g e is c u t in half. A ccordingly th e over-all sta b ility of th e o u tp u t w ith resp e ct to applied voltages is unchanged. T h e a d v a n ta g e s gained are t h a t th e loss w hich m u st be held in th e bridge is reduced so th a t s tr a y re a cta n ce s are less likely to d istu rb th e o p eratio n , a n d t h a t th e h arm o n ic c o n te n t of th e o u tp u t is reduced.
S ta te d in a different w ay, th e o u tp u t sta b ility of an o scillator u sing a n o n feedback am plifier is lim ited in p ra c tic e b y th e brid g e b alan ce w hich m a y be m a in ta in ed . A fter th is value of gain h as b een reach ed a d d itio n a l s ta b ility
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 17
m a y be secured b y su pplying increased in h e re n t gain w hich is offset b y d ire c t n eg a tiv e feedback.
T o clarify th e m a te ria l alre ad y p rese n ted an d to convey some ad d itio n a l concepts a n o scillator h av in g a large am o u n t of control w ill be designed.
T h e block d ia g ra m is to be t h a t of Fig. 7 an d th e circu it is to be sim ilar to t h a t of Fig. 6.
I t m a y readily be seen th a t th e gain co n tro l m u st satisfy tw o fu n d am e n tal req u irem en ts. I t m u st deliver a d-c bias w hich increases rap id ly w ith increase of th e level of oscillation an d it m u st n o t re tu rn a n y appreciable v o lta g e of oscillation frequency. O therw ise th e frequency will be affected b y th e elem ents in th e co n tro l circuit as well as those in th e filter, a n d th e p erfo rm an ce will be generally poor. B ecause of its balance a push-pull rectifier is helpful in m eeting th e la tte r requ irem en t. T he prin cip al req u ire
m e n t is achieved b y am plification an d b y th e use of a co n sta n t co u n ter emf or b ac k bias. N o bias is pro d u ced u n til th e level of oscillation exceeds some th resh o ld value. A bove th is thresh o ld th e bias increases ap p ro x im ately v o lt for v o lt w ith th e p e a k v alu e of th e signal. T h e sam e am plifier w hich is used to increase th e co n tro l m a y be used ad v an tag eo u sly as a buffer so th a t ap p reciab le pow er o u tp u ts m a y be produced w ith o u t degrading th e frequency o r a m p litu d e sta b ility .
I t will be assum ed th a t a Q of 100 is available in th e coil an d th a t a fre
quen cy of one m egacycle is to be gen erated . T h e transm ission of a m o d u la te d w ave in te rm s of th e sideband displacem ent th ro u g h such a one-circuit filter is show n in Fig. 21. B ecause th e cutoff occurs v e ry slow ly it will be co n v en ien t to in c o rp o rate a ra p id cutoff in th e au x iliary filter of th e gain control, th u s avoiding a n excessive p h ase sh ift a t a n y one frequency.
T h e circ u it fea tu re s a lre a d y discussed are show n in Fig. 22. A basic oscillator w ith a single tu n e d coil, a buffer am plifier h av in g little selectivity a n d th erefo re co n trib u tin g v e ry little to th e eq u iv alen t filter section, a source of biasing v o lta g e, a b alan ced rectifier, a n d a n au x iliary low -pass filter are show n. T h e condenser C is only large enough to allow th e rectifier to be d riv en w ith o u t serious loss a t one m egacycle. I t h as rela tiv e ly little effect up o n th e m o d u la tio n perform ance.
I t is assum ed t h a t th e buffer-am plifier, rectifier, etc. are so chosen th a t a m o d u la tio n of v e ry low frequency of one p a r t p er m illion applied a t th e p la te te rm in a l of th e oscillator will resu lt in a m od u latio n of one p a r t in a th o u sa n d re tu rn e d to t h a t p o in t. T his is eq u iv alen t to saying th a t th e envelope gain is 60 d b a t low frequencies, an d corresponds to 60 db of
n e g a t i v e feedback in a conventional am plifier.
T h e auxiliary filter will be designed to ap p ro x im ate th e a tte n u a tio n an d X I . De s i g n o f a Co n t r o l l e d Os c i l l a t o r
Fig. 21— Envelope transm ission through tu n ed circuit.
f - ' v / s E C .
Fig. 23— C haracteristics of auxiliary filter.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 19
p h ase c h a racteristics show n in Fig. 23. T h e choice of th is p a rtic u la r shape is b est explained b y reference to Fig. 2-4 w hich p rese n ts th e over-all envelope loop tran sm issio n of th e system . I t is seen th a t th e p hase sh ift is relativ ely c o n s ta n t a t 90° over a wide b a n d of frequencies a n d t h a t th e gain falls off a p p ro x im ate ly lin early over th e sam e b an d . In p a rtic u la r th e gain becomes zero a ro u n d 5000 cycles w hereas th e phase does n o t reach zero below 500,000
Fig. 25— Configuration of auxiliary filter.
cycles. I n te rm s of N y q u is t’s criterio n th is represents a v ery stab le system w hich is little d istu rb e d b y tra n s ie n t effects. A system hav in g even g reater s ta b ility could be achieved b y beginning th e cut-off a t lower frequencies.
I t w ould th e n be fo u n d th a t th e o u tp u t w as som ew hat sluggish in reaching a new equ ilib riu m a fte r being d istu rb e d . Such a beh av io r is n o t uncom m on b u t is g enerally undesirable.
E le m en ts w hich give a p p ro x im ate ly th e characteristics called for in Fig.
23 are show n in Fig. 25. T h e p ea k of loss a t one m egacycle is co n trib u ted b y th e series reso n a n t tra p . T h e re st of th e beh av io r is due to th e 0.5 yuf condenser in com bination w ith th e associated resistors.
X I I . Au x i l i a r y Co n t r o l o r Th e r m a l l y Li m i t e d Os c i l l a t o r s
I n th e M e ac h am a n d ce rta in o th e r oscillator circ u its a th e rm is to r is asso ciated w ith re a c tiv e elem ents in a brid g e c irc u it w hich fu n ctio n s as b o th lim ite r a n d filter. I n these circu its a large increase in th e fre q u en c y s ta b ility is observed. T h is m a y som etim es be co n v en ien tly expressed as a m agnifica
tio n of th e effective Q of th e filter.
T h e ad v a n ta g e s of g re a t freq u en cy s ta b ility a n d good a m p litu d e s ta b ility of these system s are accom panied b y a n u n d esira b le te n d e n c y to w a rd in te rm itte n t operatio n . T h e th e rm a l c o n s ta n ts of th e th e rm is to r a re n o t rea d ily a d ju sta b le. M o reover a d ju s tm e n t of th e re a c ta n c e s to secure
su itab le envelope s ta b ility is likely to im p a ir th e fre q u en c y o r a m p litu d e s ta b ility for w hich th e circ u it is chosen.
T h is dilem m a m a y be resolved b y th e a d d itio n of a n a u x ilia ry n e tw o rk w hich does n o t affect th e envelope tran sm issio n to v e ry low frequencies b u t does m odify th e b eh a v io r a t higher frequencies in such a w a y as to p ro m o te th e sta b ility of th e system .
A sim ple circu it illu stra tin g th e p rinciple a p p e a rs in Fig. 26. I t w ill b e no ticed th a t th e circ u it is so arra n g e d t h a t th e av e rag e bias ap p lied to th e tu b e is only th a t due to th e ca th o d e resistor. T h e ste a d y v o lta g e d eveloped across C i b y th e rectifier is u n ab le to affect th e b ia s because of th e b locking condenser C 2. A ccordingly th e rectifier circ u it does n o t affect th e n o rm a l o p era tin g condition, w hich is c h a racterize d b y a b rid g e loss e q u a l to th e am plifier gain. T h e ad d e d elem ents com e in to p la y o n ly if th e re is a te n d ency to w a rd self-m odulation. T h e n d isp lac em e n t c u rre n ts of m o d u la tio n freq u en cy flow th ro u g h C2 in such a m a g n itu d e a n d p h a se as to m odify th e tu b e gain an d com p en sate th e m o d u la tio n re tu rn e d from th e bridge.
T h e exact n a tu re of th e co n tro l w hich m u st be ad d ed is b est ascertained b y opening th e circ u it a t th e p la te of th e tu b e. T h e loop transm ission of a m o d u la tio n envelope m a y th e n b e determ ined, e ith er experim entally or an a ly tica lly . If in sta b ility is found an auxiliary circu it m u st be designed to p ro d u ce a n over-all sy stem w hich is stable. I n general th e elem ents of th e a u x ilia ry circ u it are to be chosen so t h a t th e loop transm ission is con
siderably less th a n u n ity in th e region of zero phase. T h is is ordinarily accom plished b y increasing th e final cutoff frequency a t w hich th e over-all loop envelope transm ission is negligible.
I N T E R M I T T E N T B E H A V I O R I N O S C I L L A T O R S 21
X I I I . A Se l f Mo d u l a t e d Os c i l l a t o r
T h e p revious sections h a v e been d evoted p rim a rily to th e problem of p rev e n tin g self-m odulation in oscillators. L e t us now consider an oscillator hav in g envelope in sta b ility . T h e N y q u ist d iag ram indicates t h a t self
m o d u la tio n will occur an d tells th e appro x im ate frequency of th e envelope w ave. M ore d etailed analysis of th e circuit is necessary to determ ine the w ave form of th e envelope a n d th e m a n n er in w hich its am p litu d e is lim ited.
I f a circ u it is to fu n ctio n well as a n oscillator th e N y q u ist d ia g ra m for th e o p era tin g freq u en cy m u s t loop th e (1, 0) p o in t w ith considerable m argin.
T h is is n ecessary so th a t a sm all loss of gain will n o t sto p oscillation. A t th e o p era tin g level th e lim iter reduces th e loop transm ission to u n ity . I n th e region of (1, 0) am p litu d e s ta b ility is favored if th e ra te of change of gain
w ith resp ect to level is high. S im ilarly th e fre q u en c y s ta b ility is fav o red if th e r a te of change of p h ase w ith resp e ct to fre q u en c y is hig h .
If a circ u it is to fu n ctio n well as a self-m o d u lated o scillator, th e a b o v e conditions m u st be m e t a n d in a d d itio n th e N y q u is t d ia g ra m for th e envelope m u s t m e et sim ilar req u irem en ts. T h a t is, th e re m u s t b e a lim ite r a n d filter in a d d itio n to th e effective am plifier in th e envelope system .
A circ u it w hich m eets these req u ire m e n ts is show n in Fig. 27. I t is seen to be sim ilar to t h a t of Fig. 6 b u t to h a v e a m ore co m p lic ate d low -frequency p a th . T h e o p era tio n is b est explained in te rm s of th e re la tiv e size of th e v ario u s elem ents. T h e b y -p ass condensers C i a n d Cz are c o m p a ra tiv e ly sm all. T h e blocking condensers C3 a n d C4 are q u ite large. T h e choke L \ is large. T h u s these elem ents serve as open o r s h o rt circ u its b u t do n o t e n te r in to th e se ttin g of e ith e r of th e frequencies.
T h e sta b ility te sts are carried o u t b y opening th e m esh a t th e p la te of th e tu b e. A t th e o p era tin g frequency, as defined b y th e p la te coil a n d condenser th e loop gain is high a t low levels. T h u s th e fu n d a m e n ta l co n d itio n s for oscillation exist.
T h e n e x t ste p in th e analysis is to su p p ly a signal of su ita b le m a g n itu d e an d frequency to reduce th e loop tran sm issio n to (1, 0). A sm a ll m o d u la tio n of v ery low fre q u en c y is re tu rn e d m agnified a n d rev e rse d in p h ase, as w ith previous system s. T h e p h ase of th e envelope tran sm issio n changes w ith increase of m o d u la tin g freq u en cy u n til i t is zero a t th e re so n a n t fre q u en c y of i s a n d C5. A t th is freq u en cy a considerable gain exists so t h a t th e N y q u ist d ia g ra m for th e envelope also loops th e p o in t ( 1 ,0 ) .
T h e tu n g ste n lam p in co n ju n ctio n w ith th e o th e r im pedances of th e brid g e serves to lim it th e degree of self-m odulation. T h e o p e ra tin g fre q u en c y m a y be se t b y m ean s of C6 in co n ju n ctio n w ith a su ita b le v a lu e of T 6. T h e op era tin g a m p litu d e m a y be controlled b y a d ju s tm e n t of th e b ia s b a t te r y B . T h e frequency of th e self-m odulation is se t b y m ean s of C5 in co n ju n ctio n w ith ¿5.
X IV . Co n c l u s i o n s
A m e th o d of ap p ly in g know n feedback th e o ry to th e p ro b lem of self
m o d u la tio n in oscillators h a s been p rese n ted . A lth o u g h th e discussion h a s been lim ited to electrical circuits it is clear t h a t th e an aly sis is ap p lica b le to o th e r system s, such as electrom echanical o r m ech an ical oscillators.
T h e analysis h a s been applied to several fam iliar o scillators to illu s tra te th e m e th o d a n d to clarify som e d etails of th e ir o p era tio n . A sam ple design of a bias controlled o scillator is p rese n ted to show ap p lic a tio n to new designs.
T h e ap p lica tio n of bias co n tro l to th e rm isto r sta b ilized oscillato rs is described. T h e design of a self-m odulated o scillator is u n d e rta k e n to show how in te n tio n a l m o d u la tio n m a y b e in tro d u c ed a n d controlled.
Evaluating the R elative B ending Strength of Crossarms
By RICHARD C. EGGLESTON
/^ A 'V E R a m illion crossarm s are produced an n u a lly in th e U nited S tates.
I n th e open wire lines of th e Bell S ystem alone th ere are now ab o u t 20 m illion arm s in use. I t is n a tu ra l, therefore, th a t public u tility engineers should h a v e a n in te re st in th e stre n g th of such an im p o rta n t item of outside p la n t m a te ria l; an d , co nsequently, an in te re st in a n y tool or m eans of evalu
atin g th e stre n g th of such m a teria l. I t is believed th a t th e m om ent diagram is a convenient a n d reaso n ab ly reliable tool for estim a tin g th e loads an arm will su p p o rt, for m easuring th e effect of k n o ts of v arious sizes an d of pinhole locations on a rm stre n g th , a n d for answ ering sim ilar questions rela tin g to th e b ending stre n g th of crossarm s u n d e r v e rtic a l loads.
Tw o m o m e n t d iag ram s are show n in Fig. 1 for Bell S ystem T y p e A cross- arm s; a n d in th e pages th a t follow are p rese n ted th e m eth o d used in con
stru c tin g th e d iag ram s a n d a discussion of th e ir use. W hile th e calculation resu lts ap p ly p a rtic u la rly to th e ty p e an d q u a lity of arm referred to, th e y w ould also b e of value as a tim e saving reference in fu tu re studies th a t m ay be p roposed rela tin g to th e stre n g th of th e sam e or o th e r ty p e s of arm s involving different k n o t allow ances.
T h e resistin g m o m e n t of a b ea m is th e p ro d u c t of its section m odulus b y th e u n it stress on th e rem o test fiber of th e beam . T h e section m odulus of a b eam of u n ifo rm cross-section is c o n s ta n t a n d readily determ inable. T h e section m odulus, how ever, of a beam of n onuniform cross-section, such as a crossarm , v aries because of th e different cross-sectional shapes a n d dim en
sions involved.
In th is s tu d y th e following five different shapes w ere recognized:
(1) R oofed section betw een pinholes (2) R oofed pinh o le section
(3) R oofed b rac e b o lt hole section (4) R e c ta n g u la r pole b o lt hole section (5) R e c ta n g u la r section w ith o u t b o lt holes
T h e dim ensions of th e sections in v e stig a ted were as follows:
Section of Arm
D im ensions
M inim um N om inal
(Inches) 3 * x 4 *
x 4 3~i6 x 4 ^ -
(Inches) 3 f x 4 * 3 J x 4 * 3 | x 4 J
23
Since th e re is little , if an y , engineering in te re st in th e s tre n g th of s tru c tu ra l m em bers of m axim um size, no in v e stig a tio n s w ere m a d e of sections of m axim um dim ensions.
Fig. 1— M om ent diagram for T ype A southern pine and D ouglas fir crossarm s per Specification AT-7075:
G raph 1—Resisting m om ents of arm s of nom inal dimensions, straig h t grained an d free from knots. (Fiber stress 5000 psi)
G raph 2— Resisting m om ents of arm s of minimum dimensions, having m axim um slant grain (1" in 8"), and containing knots of the m axim um sizes p e rm itte d (viz., sizes shown a t bottom of arm sketch). (Fiber stress 3250 psi)
G raph 3— Bending m om ents from a load of 50 pounds a t each pin position. j
S ection m odulus calcu latio n s w ere m a d e of each sh a p e of m in im u m a n d no m in al size, b o th w ith a n d w ith o u t k n o ts. T e sts h a v e show n t h a t, b e
R E L A T I V E B E N D I N G S T R E N G T H OF CR OS S A R M S 25
cause of th e d isto rtio n of th e grain t h a t occurs a ro u n d them , k n o ts are fully as injurious to th e stre n g th of s tru c tu ra l tim b ers as k n o t ho les.1 Therefore, m dealing w ith sections con tain in g k n o ts, it w as assum ed for th e purposes of th is s tu d y th a t th e k n o t exten d ed across th e section in th e sam e m anner as a hole h a v in g a d ia m e te r eq u a l to th e d ia m eter of th e k n o t. I t was also assum ed t h a t th e k n o t w as located in, or reasonably close to, th e m ost d am ag in g p o sitio n in th e a rm section.
I n th e calcu latio n s of th e section m odulus of all roofed arm sections, it w as necessary first to co m p u te th e m om ents of in e rtia of th e whole or p a rts of th e to p segm ents of such sections (viz. n om inal a n d m inim um sections
betw een pinholes, a n d nom inal a n d m inim um pinhole sections). A ccord
ingly, fo u r such co m p u tatio n s were m ade an d th e resu lts used in calculating th e section m oduli of all th e roofed sections investigated. T h e details of th e fo u r co m p u tatio n s a re show n in th e A ppendix. T o insure u niform ity in th e resu lts, th e degree of precision used in these com putations was con
sid e ra b ly g re a te r th a n is o rdinarily em ployed in dealing w ith tim b er p ro d u cts. A ll of th e w ork, how ever, was done on a com puting m achine, an d it w as ju s t a b o u t as easy to ca rry th e operations to eight decim al places (which w as th e ca p a c ity of th e m achine used) as to a lesser num ber. As a m a tte r of in te re st in th is connection, it was found b y a c tu a l tria l in C o m p u tatio n I th a t a b su rd resu lts w ould occur if fewer th a n five decim al places were used.
F o r convenience, all of th e section m odulus calculations were m ade in ta b u la r form . I n such form th e procedure em ployed w ould n o t be readily
1 Pg 6 D ep t C ircular 295, U. S. D ept, of Agriculture, “Basic Grading Rules and W ork
ing Stresses for S tructural Tim bers,” by J. A. Newlin and R. P. A. Johnson.
a p p a re n t. T herefore, a sam ple calc u latio n follows show ing th e m e th o d of finding th e section m odulus of th e b race b o lt hole section co n ta in in g a f inch k n o t.
Sam pie Calculation
R eferring to Fig. 2, it will be noted th a t the k n o t an d bolt hole divide th e section into three p a rts: the top segm ent (T ) and two rectangular portions (R l and R 2). T he m om ent of inertia (/) of such a compound section about its n eu tral axis (at a distance c from M -M ) is equal to th e sum of the m om ents of inertia ( IT , IRX and IR 2 ) of the com ponent p a rts T, R l an d R2 ab o u t axes through their own centers of gravity, plus the areas of th e com
ponent p a rts m ultiplied by the squares of the distances of th eir own centers of g rav ity from the n eutral axis of th e compound section. T he section m odulus (S) of th is section is found, of course, by dividing its m om ent of inertia by the distance (y) from the n eu tral axis of the section to the m ost rem ote fiber.
Dimensions:
Areas:
Moments about M — M :
Moments of Inertia:
b 3.1875"
k - 0.7500"
d = 3.7625"
h i = 0.7000"
h2 = 1.9375"
g = 0.1330"
t = 3.8955"
r l = 2.6625"
r 2 = 0.96875"
D = 4.09375"
T
=
0.7099 sR l = 2.2313 R2 = 6.1758
(W idth of Section) (D iam eter of K not)
(See C om putation I in Appendix) (d - 2.125" - 0.1875" - k) (2.125" - 0.1875")
(See C om putation I) (<f + I)
( i h i + 2.3125") ( I h2)
09375" (D epth of Section) s. (See C om putation I)
(bhl) (bh2) 9.1170 sq. ins.
T t = 2.7654 R l r l = 5.9408 R2r2 = 5.9828
14.6890 = 9.1170 c; an d hence c = 1.6112
I T = 0.0053 (See C om putation I)
12)
12) IR 1 = 0.0911 (bhl I R 2 = 1.9319 (bh23 T (t - c) 2 = 3.7043 R l (rl - c)2 = 2.4661 R2(c - r2)2 = 2.5490
Section Modulus:
I = 10.7477 y = 2.48255 (D * c)
S = - = 4.3293 y
T h e sam e general p ro ced u re show n in th is sam ple c a lc u latio n w as fol
lowed in dealing w ith th e o th e r cross-sectional shapes. F o r th is reason, only th e final resu lts of th e several calcu latio n s are p re se n te d ; a lth o u g h , for
R E L A T I V E B E N D IN G S T R E N G T H OF CROSS A R M S Ta b l e 1 .— Section M odulus of Roofed Sections between Pinholes
27
Knot Diameter—Inches
N o
K not 1 a4 l li 2 21 3
Calcidation 1:
(K n o ts located at top of section)
Section Size*:
M in im u m ...
E nd. M in ...
N o m in al...
Calcidation 2:
(K nots located at b o tto m of section)
Section Size*:
6 . 8 6
7.37
5.08 4 .78 5.50
3.57 3.91
2.33 2.13 2.59
1.35 1.54
0 .64 0 .53 0.76
M in im u m ...
Calcidation 3:
(K nots located im m ediately below top segment)
Section Size*:
6 . 1 1 5 .24 4 .42 3.71 3.05 1.92 1.05 0.47
M in im u m ...
E nd. M in ...
N o m in al...
8.03 7.65 8 .60
5.45 4 .5 6 3.86 3.65 4 .16
3.34 2.95 2 .50 2.34 2.37
Ta b l e 2 .— Section M odulus of Roofed Pinhole Sections Knot Diameter—Inches
N o Knot i 1 14 l 1 1 2
Calcidation 4:
(K nots vertical) Section Size*:
M in im u m . . . . E nd. M in. . . . N o m in al...
4 .5 0 4 .2 9 5.11
3.84 3.21 2.63
2.50
2 . 8 8
2.25
Calcidation 5:
(K nots horizontal) Section Size*:
M in im u m . . . . E nd. M in. . . . N o m in al...
3.63 2.96 2.40
2.26 2 .76
1.97 1.41
1.33 1.64
1 . 1 1
* Section Sizes:
M inim um = 3 ^ " x 4 ^ "
E nd. M in. = 3t s " x 4" (viz. minimum a t end of arm) N om inal = 3 J" x 4y s"
convenience, reference is ma.de to th e calculations b y n u m b e r in th e pages t h a t follow. T hese resu lts are show n in T ables 1, 2, 3 a n d 4, a n d a brief discussion of th e scope a n d use m ade of th e m follows.
Ta b l e 3 .— Section M odulus of Bolt Hole Sections Knot D iam eter—Inches
KnotNo i I i 3
4 l 1 1 1 1
Calculation 6:
Brace bolt hole section
Section Size*:
M inim um . N om inal . .
7.97 8.55
6.47 5 .28 4 .3 3
4 .7 1
3 .58 2 .6 2
2 .7 8 Calculation 7:
Pole bolt hole section
Section Size*:
M inim um . N om inal . .
9.25 9 .7 4
7.42 5.63
6 .05
3 .2 4 3 .61
2 \" K not 1.51
1 . 6 6
3" K n o t .75 .85
* Section Sizes:
M inim um == 3 t s " x 4 -^ "
End. M in. = 3 ^ " x 4" (viz. m inim um a t end of arm ) N om inal = 3 |" x 4 ^ "
Ta b l e 4 .— Section M odulus of Rectangular Section without Bolt Holes (Calculation 8)
Section Size K not D iam eter Section Modulus
M inim um ( 3 ^ " x 4 ^ " ) (No K not) 9 .3 2
i4 8 .2 4
12 7.22
34- 6.2 8
1 5 .4 0
1 1 3.8 4
2 2.5 4
9 1 1.51
3 .75
N om inal (3 j" x 4§") (No K not) 9.7 8
i4 8.67
12 7 .62
34 6 .64
1 5 .72
I f 4 .1 0
2 2 .74
9 1 1 . 6 6
3 .85
Ro o f e d Se c t i o n s Be t w e e n Pi n h o l e s
As in d ic ated in T a b le 1, th re e ta b u la r calcu latio n s w ere m a d e for roofed sections betw een pinholes. I n C alcu latio n s 1, 2 an d 3 it w as assu m ed th a t th e k n o ts p re se n t were located (1) a t th e to p , (2) a t th e b o tto m , a n d (3) im m ed ia te ly below th e to p segm ent of th e section, resp ectiv ely . T h e re su lts rela tin g to th e 3 y s " x 4 ^ " section a re p lo tte d as C urves 1, 2 a n d 3, respectively, in Fig. 3.
