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Q U A R T E R L Y OF A P P L I E D M A T H E M A T I C S
V ol. I l l O C T O B E R , 1945 N o . 3
G R A P H IC A L A N A L Y S E S O F N O N L IN E A R C IR C U IT S *
BY
A L B E R T P R E IS M A N Capitol R adio Engineering In stitu te
1. In tro d u c tio n . As a general rule, a problem in physics is n o t considered solved unless th e solution can be expressed in an a ly tica l form . T h e sam e u sually holds tru e in th e case of engineering problem s, alth o u g h th ere th e a r t often progresses fa ste r th an th e th eo ry u n d er th e im p act of econom ic forces, an d th e engineer is often forced to seek a solution b y m ean s of an experim ental setu p , o r possibly b y m eans of som e num erical or g raphical process.
T h e d isad v an ta g e of a num erical or g rap hical m etho d is its lack of g enerality , its ten d en c y to w ard s inaccuracy, p a rtic u la rly ow ing to cu m u lativ e errors, an d its in
ab ility to exhibit o p tim u m values for th e p a ra m e te rs involved, p a rtic u la rly if these have to be in num erical ra th e r th a n in sym bolic form . On th e o th e r h an d , these m e th ods o ften yield answ ers to problem s t h a t th e an a ly tic a l m etho d c a n n o t handle, and fu rth e rm o re are often v ery effective as teach in g aids. T h is is p a rtic u la rly tru e of the graphical m ethods.
I t is th e p u rpose of th is artic le to illu stra te th e ap p licatio n of graph ical co n stru c
tions to problem s involving n o nlin ear circuits, p a rtic u la rly tho se con tain in g vacuum tubes. I t is th e w rite r’s hope t h a t som e m ath em atician will be sufficiently a ttra c te d to th is m eth o d to a tte m p t to establish it on a m ore general basis, possibly som ething akin to th e collection of theorem s of o rd in ary E uclidean o r of P ro jectiv e G eom etry.
2. D efinition of graphical m eth o d . Before proceeding w ith a d escrip tion of the m eth o d it will be desirable to define it. B y grap hical co n stru ctio n s are m e a n t those geom etrical manipulations by w hich a solution to a problem is o b tain ed . I t m ay be necessary to slide a curve represen tin g a relation ship betw een tw o variab les along th e axis of th e in d ep en d en t variable, an d to find (geom etrically) w here it intersects a n o th e r curve represen tin g a second relationship betw een th e tw o variab les. T h e m a n ip u la tio n s m ay be m ore involved th a n those of sim ple tra n sla tio n along th e axis, and it is to be stressed th a t th e restrictio n of ru ler an d com pass co nstru ctio n s is no t invoked in these m anipulations.
I t is a p p a re n t th a t th e m eth o d is n o t th a t usually u ndersto od by th e average engi
neer, nam ely, th e p lo ttin g of a com plicated generalized an a ly tica l expression to p er
m it values to be tak en off th e g ra p h in ord er to o b v iate th e need for co m pu tin g th e v alu e of th e expression every tim e th e problem arises.
3. Sim ple se rie s n o n lin e a r circuit. As an e lem e n ta ry exam ple of a grap hical con
stru c tio n , let us consider th e circ u it shown in Fig. 1, t h a t of a diode (tw o-elem ent
* R eceived M arch 13, 1945.
186 A L B E R T P R E IS M A N [Vo!. I l l , N o. 3
vacuum tu b e) in series w ith a resistance R and a source of d.c. p o ten tial E. I t is d e
sired to find th e c u rre n t flow in th is circuit.
I t is necessary to know th e v o ltag e-cu rre n t e —i relationship for th e diode, and for th e resisto r R. W e assum e for sim plicity t h a t th e la tte r is a linear resistance.
T h en th e e — i relationship is th a t shown in Fig. 2. T h e curve is a s tra ig h t line m aking an angle 9 w ith th e vo ltag e axis, such th a t
cot 0 = R, (1)
th e resistance of th e device. T h is slope is c o n sta n t an d hence R has a fixed v a lu e : so m an y v o lts per am pere, or ohm s.
On th e o th e r han d , th e diode has th e c h a rac te ristic shown in Fig. 3. H ere, for
* i
n eg ativ e values of voltage ( p ia te negative to cathode) no c u rre n t can flow; while for positive values of v oltage c u rre n t flows in such m an n er as to g enerate th e curve shown,.
T h e ideal diode w ould have the following eq u atio n for positive p la te voltages
i = A e3/2, (2)
b u t ac tu a l diodes d e p a rt to some e x te n t from th e abo ve eq u a tio n owing to such fac
to rs as initial velocity of emission of electrons from th e cathode, th e effect of th e su p p o rtin g m em bers for the cath o d e and plate, etc.
T h e diode is a nonlin ear device; first because of th e b re ak in th e curve a t th e origin an d second because even for positive p la te voltages th e e —i relatio nsh ip is usually n o t a s tra ig h t line. One can define th e resistance as
1) th e reciprocal slope of th e secant line to an y p o in t of th e curve (th is is th e so-called d.c. resistance) or
2) th e reciprocal slope of th e ta n g e n t line to an y p o in t of th e curve (this is usually called th e a.c., increm ental, or v a ria tio n a l resistance of th e device).
S uch concepts h av e lim ited u tility how ever, since th e resistan ce in eith er case is no longer a co n stan t, b u t a function of th e applied voltag e o r c u rre n t th ro u g h th e device.
1945] G R A P H IC A L A N A L Y S E S OF N O N L IN E A R C IR C U IT S 187
T h e grap h ical m ethod to be described tak es th e fu n d a m e n tal e —i relationship, or term in al c h a ra c te ristic as it has been called b y K irschstein , an d o p erates d irectly w ith it. F u rth e rm o re , th e curve does n o t h ave to be an aly tic, nor even expressible in th e form of an eq u a tio n ; it can sim ply be a p lo t of ex perim en tal d a ta , alth ou gh th is in volves in terp o latio n betw een exp erim entally determ in ed points.
T h e process of finding th e c u rre n t th ro u g h th e tw o in series for th e im pressed vo ltag e E is essentially th a t of solving th e tw o eq u atio n s for th e term in al c h a ra c te r
istics sim u ltan eo u sly u n d er th e condition t h a t th e sum of th e v oltag e d rop s across th e tw o elem ents m u st equal th e im pressed vo ltag e E. T h u s, if th e relation sh ip for th e one elem ent is i = f i ( e ) , th e n th a t for th e o th e r is i = f 2(E — e), an d it is desired to find a com m on v alu e of i th a t satisfies b o th relationships.
Since one or b o th of th e above eq u atio n s m ay be of degree higher th a n u n ity , th e an a ly tica l solution c a n n o t be effected by th e m etho d of d eterm in a n ts, b u t ra th e r by th e m eth o d of su b stitu tio n , an d finally resu lts in th e necessity for solving an eq u ation of degree higher th a n u n ity .
T h is, how ever, assum es t h a t term in al ch a rac te ristic s can be rep resen ted by pow er series. T h e g raph ical m ethod requ ires no such con dition ; it o p erates on th e graphical p lo ts d irectly. T h u s, suppose th e term in al c h a rac te ristic of th e diode is represented b y A O B , Fig. 4. L e t OC rep resen t th e m ag n itu d e of th e im pressed vo ltag e E. T h ro u g h C draw D C a t an angle 6, as show n, such t h a t co t 8 = R. T h e n th e in tersectio n of C D and AOB in D rep resen ts th e required solution, in t h a t D F is th e com m on c u rre n t in th is series c irc u it; O F is th e vo ltag e drop in th e diode; F C is th e vo ltag e d rop in th e resisto r R ; an d clearly O F + F C equals th e im pressed vo ltag e E . If E varies w ith tim e, D C can be sh ifted b ack an d fo rth along th e vo ltag e axis a t positions correspond
ing to th e in sta n ta n e o u s values of E , a n d th e in tersectio n s will furn ish th e corre
sponding in sta n ta n e o u s values of th e cu rren t.
T h e ab o v e solution rep resen ts a w ell-know n m ethod for solving tw o equations sim u ltan e o u sly w hen th e eq u a tio n s are of degree higher th a n th e first or even of tra n sc e n d e n ta l n a tu re . I t will be of in tere st, how ever, to see how th is m ethod is a p plied to a m ore com plicated circuit.
4. T rio d e tu b e a n d re sista n c e in serie s. T h e n ex t exam ple will be th a t of a three- elem en t o r trio d e tu b e in se
ries w ith a resistance an d a source of d.c. vo ltag e Ebb- T h e electrical connections are show n in F ig. 5. T h e a d d itio n a l com plication is th a t in th e trio d e th e p la te c u r
re n t is a function of tw o v aria b les; th e grid voltage an d th e p la te voltage. T h e term in al ch a rac te ristic m u st therefo re be represented b y a three-d im ensio nal p lo t in
volving th e p la te c u rre n t i p, th e grid v o ltag e eg (which is th e sum of th e in sta n ta n e o u s v alu e of th e a lte rn a tin g signal vo ltag e es an d th e c o n sta n t, d.c. bias v o ltag e E c), an d th e p la te v o ltag e ep.
T r io d e
S i g n a l V o U a g e <$>
HI
B ia s Vo lia g e +
L o a d r e s i s t a n c e
P l a t e S u p p l y Voltage
* 4
Fig. 5.
188 A L B E R T P R E IS M A N [Vol. I l l , N o . 3
T h e resulting p lo t is a curved surface in space. I t can be represented in tw o d i
m ensions b y a fam ily of curves w hich re p resen t discrete p ro jection s of th is surface upon a n y one of th e th re e coord in ate planes. F o r th e problem a t han d th e m o st useful s e t of pro jections is th a t upon th e ep — i p coo rd in ate plane, in th e form of a fam ily of ep — i p curves w ith eg as th e p aram eter. T h is is shown in Fig. 6 (solid lines). C urves for w hich eQ is p ositive h av e been o m itted for sim plicity.
A ssum e fu rth e r for sim plicity th a t Rl, th e load resistance, is linear. T h e c u rre n t th ro u g h it is a fu nction of b u t one voltage, th a t w hich m u st be app lied across its te rm in a ls to produce th e above c u rre n t flow. T o re p resen t its term in al ch a ra c te ristic
in th ree dim ensions, it is p lo tte d as a plane w hose in tersectio n w ith th e ep — ea co o rd in ate plane is a stra ig h t line p a ra l
lel to th e ea axis. In th is w ay th e c u rren t in it is in d ep en d en t of th e ea coordinate, an d is a (linear) fun ctio n of b u t one voltage, th a t corresponding to th e p late v o ltag e ep of th e tu b e. All po in ts of th is plane represen tin g Rl p ro je c t over to th e i p — ep co o rd in ate plane as a s tra ig h t FlG' 6‘ line t h a t is also th e in tersectio n of th e
above R l plane w ith th e i v — ep plane.
T h e stra ig h t line m akes an angle 6 w ith th e epaxis such t h a t cot 6 = R l , i.e., th e R l p lane is inclined a t th e angle dto th e e„ — e„co o rd in ate plane.
T h e graphical solution consists in d raw ing th e line of in tersection E A a t th e angle 6 to th e ep axis. T h e intersection of EA w ith th e tu b e fam ily of curves gives th e com m on vglue of c u rre n t flowing th ro u g h th e p la te circu it of th e trio d e an d R l in series, for an y given value of grid vo ltage e„. F o r exam ple, a t a m o m en t w hen th e signal voltage es is passing thro u g h zero, th e in sta n ta n e o u s value of th e grid voltag e e0 is sim ply th a t of th e bias b a tte ry , E c. T h e in stan tan eo u s value of th e p la te c u rre n t is BC, w here B is th e intersection of AC w ith th a t curve of th e p la te fam ily for which ea= E c. I t is fu rth e r to be n o ted th a t th e in stan tan eo u s p la te vo ltag e ep is OC, and th e in stan tan eo u s v alu e of th e voltage drop across R L is E C .
F o r o th e r in sta n ta n e o u s values of eg, o th e r curves of th e p la te fam ily are involved, an d th e process of d eterm in in g th e in stan tan eo u s values of p la te c u rre n t, p la te v o lt
age, and load vo ltag e (across R L) is identical to th a t described above. T h u s, for a signal voltage im pressed upon th e in p u t or grid circu it, th e o u tp u t signal v oltage be
tw een th e p la te and ground can be found. Such m a tte rs as th e am plification of th e stage, d isto rtio n in th e o u tp u t, etc., can th en be determ in ed .
In passing, w e m ay n o te here th a t th e locus of th e p la te c u rre n t for v ario u s values of e0 is th e intersectio n of th e tu b e surface and th e R l p lan e in space. T h is intersection is a curve in space, b u t fo rtu n a te ly its p rojection on th e ep—ip plane is a s tra ig h t line, nam ely th e intersection of th e R L plane itself w ith th e ev — i p plane. I t is for th a t re a
son t h a t th e ep — i p fam ily of th e tu b e curves is em ployed; th e graphical co nstru ctio n is sim ply th e p o in ts of intersection of a straight line representin g R l w ith th e above p la te fam ily.
The above problem can becom e m uch m ore com plicated u n d er certain conditions.
F o r exam ple, if th e in p u t signal voltage is g re a t enough, th e grid can be driven posi
1945] G R A P H IC A L A N A L Y S E S OF N O N L IN E A R C IR C U IT S 189
tiv e w ith respect to th e cathode, w hereupon it draw s c u rre n t d u rin g th e positive peak of th e a.c. cycle. If th e signal source has appreciab le in tern al im pedance, th en a v o lt
age dro p will occur in th e source d u rin g th e above p o rtio n of th e cycle, an d th e ac tu a l vo ltag e applied to th e grid will differ from th e gen erated voltage e„
I t is therefore necessary to d eterm in e th e a c tu a l grid v oltag e before th e p late c u rre n t can be found. A n o th er com plication arises how ever, in th a t th e grid c u rre n t (and hence th e a c tu a l grid voltage) is a fun ction n o t only of th e po sitiv e grid voltage, b u t of th e p la te v o ltag e as well. T h is is because th e space c u rre n t divides betw een th e tw o electrodes in a m an n er depending upon th e tw o electrode voltages. A t th e sam e tim e th e p la te vo ltag e is a function of Rl an d th e grid voltage. T h u s th e above sim ple graphical co n stru c tio n can becom e q u ite involved if m erely th e in p u t signal is increased to a p o in t w here th e grid is driven positive.
5. T h e b ala n c e d am plifier. In ste a d of in v estig atin g such details, im p o rta n t th o u g h th e y m ay be, it will be of in te re st to exam ine a n o th e r ty p e of circu it v ery im p o rta n t in th e com m unication in d u stry . R eference is m ade to th e push-pull o r balanced am plifier. T h e circu it is shown in Fig. 7.
In (A) is shown th e a c tu a l circuit, w hereas in (B) is shown an idealization or eq u iv ale n t form b e tte r suited for th e p urp ose of analysis. In th e ac tu a l circu it (A), tw o tu b es are em ployed, in d u ctiv ely
coupled to each o th e r an d th e o u tp u t load resistan ce r L b y an o u tp u t tra n s form er. T h e signal on one-grid is 180 d e
grees o u t of p h ase w ith t h a t on th e o th er grid, as is suggested b y th e sym bols + e , and — e,. T h e bias v o ltag e E c, on th e o th er h a n d , is ap plied to b o th grids in th e sam e p o la rity ; an d th e p la te supply vo ltag e is applied to th e tw o tu b e s in th e sam e p olar
ity too, as show n.
T h e a c tu a l load resistance r L an d th e o u tp u t tra n sfo rm e r can be replaced b y th e c e n te r-ta p p e d in d u ctan c e an d reflected load resistance Rl as far as th e tu b es are concerned. T h e sim plified circu it is shown in (B ), Fig. 7. In using th is eq u iv ale n t cir
cuit, it is ta c itly assum ed t h a t th e a c tu a l o u tp u t tra n sfo rm e r is an ideal tran sfo rm er h av in g infinite p rim a ry and secondary o pen -circu it in d u ctan ce, no d istrib u te d ca
p a c ity , u n ity coefficient of coupling be
tw een w indings, etc. In th e eq u iv ale n t
circ u it th e c e n te r-ta p p e d in d u ctan ce is assum ed to be infinite in v alu e an d to have u n ity coupling betw een th e tw o halves of th e com plete w inding. O rdin arily this is a reasonable assum ption.
As a result, th e c u rre n t in one-half of th e w inding c a n n o t a t an y m o m ent exceed th a t in th e o th e r half for otherw ise an infinite co u nter-electro m otive force w ould be induced in th e w indings th a t w ould ten d to p re v e n t such an u n eq u a lity from ta k in g
O utput irons fo r m e r
B
Fi g. 7.
190 A L B E R T P R E IS M A N [Vol. I l l , N o. 3
place. T h e c u rre n ts in th e Avindings can v ary , how ever, provided th ey rem ain equal to one a n o th e r a t all tim es. F inally, tw o fu rth e r assu m p tio n s are m ade, nam ely, th a t th e signal voltages es an d — es applied to th e tw o grids are a t all tim es equal and op
posite to one a n o th er, an d th a t th e tw o tu b es h av e id entical term in al characteristics.
T hese tw o assu m p tio n s seem also reasonable.
C onsider first th a t ea equals zero (no signal is applied). T h e bias on each grid is £ c, an d th e p la te v o ltag e for eith er tu b e is £«,, hence th e tw o p la te c u rren ts h and J2 are equal to one an o th er. Since th e y flow in opposite d irection s from th e ends of th e w in d ing to th e ce n te r ta p , th ey balance each o th e r m ag netically in th e o u tp u t in d u ctan ce an d produce no voltag e across th e ends. C onsequently no c u rre n t flows in th e load resistance Rl-
Now suppose t h a t a signal vo ltag e is im pressed such t h a t th e to p grid is driven p ositive b y an a m o u n t e„i from its n orm al d.c. neg ativ e bias v alu e of E c, an d t h a t th e b o tto m grid is driven m ore neg ativ e by an equal a m o u n t, i.e., —e,i. T h e tw o p la te c u rre n ts will now v a ry in o ppo site directions, nam ely, I \ will increase an d J2 will d e
crease. H ow ever, th e sum of these tw o c u rren ts flows th ro u g h th e p la te pow er supply, and owing to th e infinite in d u ctan c e of th e ce n te r-tap p e d w inding, (Ii + I / ) / 2 flows dow n th ro u g h th e to p half of th e w inding, and an equal a m o u n t flows up th ro u g h th e b o tto m half, to com bine a t th e ce n te r ta p to furnish th e sum ( i i + J 2) flowing th ro u g h th e pow er supply.
Since ( / i+ / 2) / 2 is th e average betw een I \ and J 2, it is equal to n either, an d from th e principle of c o n tin u ity of c u rre n t flow, th e difference
h - K h + h ) = 1 ( 1 1 + h ) - I 2 = i ( h - h ) (3)
m u st flow th ro u g h R L. A quick check will in d icate t h a t K irchh off’s c u rre n t law is satisfied a t each ju n ctio n .
T h e c u rre n t (I\ — I i ) / 2 is th e o u tp u t cu rren t. In flowing th ro u g h Rl, it sets up a v o ltag e drop
E l = i ( h - I2) R l . (4)
H a lf of th is or El/ 2 ap p ears across each half of th e o u tp u t w inding of such p olarity th a t th e in sta n ta n e o u s p la te vo ltag e of th e to p tu b e is Ebb —( El/ 2 ) an d th a t of th e b o tto m tu b e is £¡,¡, + ( £ ¿72).
T h u s th e following facts h av e been b ro u g h t to lig h t:
1) T h e grid v oltages change b y equal b u t o pposite increm en ts from th eir com m on bias value £ c ow ing to th e cen ter ta p on th e in p u t tran sfo rm er secondary.
2) T h e p la te v o ltag es change by equal b u t opposite in crem ents from th eir com m on su p p ly v alu e £«, ow ing to th e ce n te r ta p on th e o u tp u t ind uctance. M o re
over, th e p la te vo ltag e increm ents are opposite in sign to th e corresponding grid voltag e increm ents.
3) T h e p la te c u rre n ts change in opposite d irectio ns in th e sam e sense as th e correspond ing grid voltages, b u t n o t necessarily to an equal degree. If th e tubes are n onlinear, as is usually th e case, th en th e increase in p la te c u rre n t of eith er tu b e for a positive in crem en t in grid voltage is n o t necessarily th e sam e as th e d e
crease in p la te c u rre n t for an equal neg ativ e in crem en t in grid voltage.
F ro m th e above facts several graphical co n stru c tio n s are av ailab le to d e te rm in e - th e p la te c u rre n t an d p la te vo ltag e v a ria tio n s in th e tu b es, th e o u tp u t c u rre n t and
1945] G R A P H IC A L A N A L Y S E S OF N O N L IN E A R C IR C U IT S 191
voltage, th e pow er o u tp u t, an d th e d.c. pow er in p u t. T h e following grap hical m eth od is preferred by th e a u th o r. In Fig. 8 is show n th e p la te fam ily of curves for eith er tu b e. If th ere is no signal in p u t, th e only voltages p re sen t are th e d.c. p o ten tials E bb applied to th e tw o p lates and E c applied to th e tw o grids. T h e c u rre n t th ro u g h eith er tu b e is th en /¡, = EbbB, a d irec t cu rren t.
N ow suppose t h a t equal an d opposite signal vo ltag es e, an d — e, are applied to th e grids in ad d itio n to E c. T h en th e c u rre n t in th e one tu b e will increase from BEbb to D G , an d th a t in th e o th e r tu b e will dro p to F H , as shown. T h e p la te vo ltage of th e first tu b e will d ro p from OEbb to OG = (E bb—A ep), and th a t in th e o th e r tu b e will rise by an equal amount to O H = (Ebb+ A e p).
I t is also clear from Fig. 8 th a t D J rep resen ts th e difference betw een th e tw o cu r
re n ts or (Ji —J 2), an d J F rep resen ts 2Aep, th e vo ltag e across th e o u tp u t in d u ctan ce and previously d enoted b y E L in Fig. 7. From E q. (4), it is ev id en t th a t
J F / D J = El/ ( I i - h ) = Rl/ 2 . (5) T h u s D F m akes th e angle 6 w ith th e ep axis such th a t
cot 0 = Rl/ 2 . (6)
I t is also ev id en t from th e geom etry of th e figure th a t D C = C F , i.e., t h a t th e o rd in ate th ro u g h Ebb bisects line D F in C.
T h e abov e facts suggest th e following m eth o d of grap hical co n stru c tio n . W e hold a rule a t th e angle 6 and slide it up or down until th e segm ent betw een th e desired ep — i p curves (corresponding to equal an d o ppo site grid vo ltag e excursions from th e bias value E c) is bisected by th e o rd in a te th ro u g h Ebb- T h e in tersection s of th e rule w ith th e tw o Cp—ip curves gives th e tw o in sta n ta n e o u s v alu es of th e tw o tu b e c u rren ts Ji an d h , corresponding to th e signal voltag es e, an d — e, and to th e p la te load re
sistance R l , or ra th e r to R l / 2 .
T h en a n o th e r p air of equal an d opposite grid signal voltag es are chosen, and th e process re p eated . T h is is contin ued until as m an y pairs of in sta n ta n e o u s grid signal voltag es h av e been used as is desired. F o r a sym m etrical signal v oltage, such as a sine w ave, in stan tan eo u s values for only o n e-q u arter of a cycle are required.
W hen th e abov e graphical co n stru ctio n is perform ed, th e re is o b tain ed a curve
192 A L B E R T P R E IS M A N [Vol. I l l , N o. 3
on th e p la te fam ily of curves such as th a t show n in broken lines A B C D E in Fig. 9.
T h is rep resen ts th e locus of th e c u rre n t for either tu b e over a cycle of grid signal v o lt
age. I t also represents th e term inal c h a rac te ristic for Rl as it appears to either tube in the presence of the other tube.
T h e significance of th e la st sta te m e n t is as fo llo w s: th e tw o tu b e s m ay be regarded as tw o gen erators connected to a com m on load R L. Owing to th e ir n on lin ear ch a rac
teristics, th e tu b es do n o t share th e load equally th ro u g h o u t th e signal cycle; th a t tu b e w hose a p p a re n t internal resistance is lower tak es a g re a te r share of the load, i.e., furnishes m ore th a n half of th e load c u rre n t ( h — I 2) / 2 flowing th ro u g h R L. As a result, Rl ap p e ars as a v ariab le or n onlinear resistance to e ith e r tu b e even tho ug h it is a c tu a lly a linear resistance, and its term in al ch a rac te ristic on eith er tu b e ’s ep — i p fam ily of curves is in itself a curved ra th e r th a n a s tra ig h t line.
L ack of space precludes a detailed discussion of th is in tere stin g circu it. H ow ever, several im p o rta n t fe atu res will be presented. As ind icated in Fig. 9, th e tw o ep — i p curves passing thro u g h B an d D, respectively, re p resen t equal and op po site grid swings. T h e corresponding c u rre n ts I x and I2 for th e tw o tu b es are B F an d zero; in sh o rt, th e tu b e experiencing th e n eg ative-grid swing has ju s t reached p la te c u rre n t cutoff.
F o r ep — i p curves passing th ro u g h A and E, corresponding to a still g re a te r grid swing for eith er tu b e, h is AG, and I2 still rem ains zero. T h is m eans th a t th e second tu b e is in o p erativ e over this p a r t of th e cycle and a c ts therefore as if it w ere discon
nected. U n d er these conditions R L ap p e ars to th e o p erativ e tu b e as i?z,/4, w hich can be expected since th e 2 to 1 tu rn s ra tio of th e o u tp u t in d u ctan c e will produce th is 4 to 1 im pedance tran sfo rm atio n if it is u nham pered b y th e o th er tube.
P o rtio n BA is therefore a s tra ig h t line w hose reciprocal slope corresponds to R l /4.
I t is easy to show t h a t if it w ere prolonged, it would pass th ro u g h Ebb- N o rm ally th e tu b es are o p erated so t h a t m axim um grid signal v o ltag e drives each tu b e a lte r
n a te ly to cutoff or beyond. M axim um o u tp u t occurs if i ? i / 4 equals e ith e r tu b e ’s a p p a re n t in tern a l p la te resistance a t th e p eak of th e cycle. T h e p la te resistance of e ith e r tu b e is given by th e reciprocal slope of th e ep — i p curve a t p o in t A. H ence a quick d eterm in a tio n for th e o p tim u m value of R l , or ra th e r R l /4, is to d raw a line thro u g h Ebb a t an angle equal to th a t of th e ep — i p curve a t p o in t A, and calculate from th e reciprocal slope of th is line th e v alu e of R l /4 an d hence of R l - T h e com plete
1945] G R A P H IC A L A N A L Y S E S OF N O N L IN E A R C IR C U IT S 193
ch a rac te ristic can th en be d eterm ined b y m eans of th e sliding rule as described p re
viously.
Fig. 8 also reveals an in tere stin g point. CEbb is th e average betw een D G an d F H , i.e., it rep resen ts { I\-\-Ii)/2. T h is is th e m id-branch c u rre n t th a t flows th ro u g h th e p la te su pply, as ind icated in Fig. 7(B). F o r variou s p airs of values of I \ and J2 as d eterm ined by th e sliding rule, th e average, o r (/x-f-Jj) / 2 m oves up an d dow n along EbbA. T h is is a v ertical line or ord in ate, and ind icates t h a t th e resistance to th e m id b ran ch c u rre n t is zero. T h is has been ta c itly a s su m e d ; th e o u tp u t in d uctan ce an d the p la te sup ply have been assum ed to be free of resistance.
If th is is n o t th e case, th en a line m u st be d raw n thro u g h Ebb w hose reciprocal slope in d icates one-half th e value of th e m id-branch resistance t h a t is p resent, an d th e sliding rule m u st be bisected b y th is line ra th e r th a n th e o rd in a te E bbA, as is th e case in Fig. 8. F rom th is follows several fu rth e r in terestin g ch a rac te ristic s.1
A n o th er p o in t is th a t n o t only is th e locus of th e m id-branch c u rre n t along th e o rd in ate EbbA in Fig. 8, b u t th a t this c u rre n t executes tw o a lte rn a tio n s p er cycle of th e grid signal voltage. T h is m eans th a t th e m id-branch c u rre n t is a t least double the frequency of th e incom ing signal; actu ally , for p erfect sy m m etry , all th e even h a r
m onics g en erated by th e tu b es flow in parallel th ro u g h th e m id-b ran ch p ortio ns of th e circuit, w hile th e odd harm onics, including of course th e fu n d a m e n tal, flow th ro u g h th e o u tp u t resistance Rl■ T h u s, if th e tu b e ch a rac te ristic s are such t h a t th e second h arm onic is q u ite pro m in en t, b u t th e th ird (and higher) harm onics áre of small am p litu d e, th e n th e o u tp u t w ave will be a fairly faith fu l copy of the in p u t grid signal voltage an d th e stag e will exhibit little d isto rtio n . Such a tu b e ch a rac te ristic is pos
sessed, for exam ple, by th e 6L6 and 807 beam pow er tubes.
As in th e case of th e previous co n stru ctio n s for th e single-ended tu b e, vario u s de
grees of com plication can arise. F o r exam ple, if th e grids are driven positive so th a t grid c u rre n t flows, th e signal vo ltag e a t th e grids will be d isto rted , an d th is d isto rtio n m u st be determ in ed sep a rately before th e abo ve co n stru ctio n can be concluded. A n o th e r case is th a t w here th e m id-branch p la te su p p ly h as an in tern a l resistance th a t is a d e q u a te ly by-passed for th e even harm onics, all except th e d.c. com ponent. T h is rep resen ts a p a rtic u la rly difficult problem t h a t can be solved only b y a series of a p proxim ations.
6. R eactiv e circuits. T h e previous circu its co n tain ed o n ly resistances, linear of nonlinear. If reactances w ere p resen t, such as th e ce n te r-tap p e d o u tp u t in du ctance, th e y w ere assum ed infinite in value and so situ a te d in th e circu it as n o t to hav e an y appreciab le a.c. com ponents flowing in them . H ow ever, m a n y no nlinear circu its con
ta in reactances of finite value th a t influence th e beh av io r of th e circu it d irectly , and hence m u st be ta k e n d irec tly in to account.
Owing to lack of space, only the case of an ind uctan ce in series w ith a nonlinear resistance an d an a.c. source will be discussed here. C onsider th e circu it shown in Fig. 10. H ere a source of a.c. voltag e e is in series w ith a n o nlinear resistance r and in d u ctan c e L . T h e vo ltag e e is a know n function of tim e, and th e term inal c h a ra c te r
istic for r an d th e value of L is given. I t is desired to find th e c u rre n t flow in th is cir
cuit.
1 See, for exam ple, A . Preism an, Graphical constructions fo r vacuum tube circuits, M cG raw -H ill P ub
lishing Co., N ew York, 1943.
194 A L B E R T P R E IS M A N [Vol. III, N o. 3
W e hav e th e fu n d a m e n tal relatio n
e(t) = ir + L — •di
dl (7)
E xpressing E q. (7) in term s of finite increm ents, we o b tain
(8)
In E q. (8), it is assum ed th a t a t th e s ta rt, th e v o ltag e e h as a certain value e\, th e c u rre n t i has a certain value ii, an d ¿ = 0. T hese are th e in itial conditions. D u rin g a
a m o u n t A i i and to rem ain a t th e v alu e i i + A i i d u rin g th e tim e At. T h is is of course an app ro x im atio n , sufficiently close if At is ta k e n sufficiently sm all. U n d er these condi
tions E q. (8) holds.
T h e q u a n tity L / A t h as a finite value if A t is finite. I t can rep resen t th e co tan g e n t ances in series: th a t of r a t th e value A, an d th a t of L /A t. T h e graphical co nstru ction th en tak es th e form shown in Fig. 11, w here OA rep resen ts th e in itial v alue e\, and AB th e initial c u rre n t ii. W e now suppose th a t th e vo ltag e changes from e\ to ei+Aei in a sm all chosen tim e in terv a l At, an d let O D re p resen t ei+ A ei so th a t A D re p re
sen ts Aci.
T h e voltage across L is due to th e change of c u rre n t AA an d n o t d u e to i\ itself, w hich h as alread y been established in L . T h is is in dicated by th e fa c t th a t OA = ei rep resen ts th e dro p across th e nonlin ear resistance r; th ere is no v o ltag e dro p across L for ii a t th e tim e t — 0. H ence, in view of th e above, a p o in t C is located in line w ith B a n d d irec tly over D , an d th ro u g h C line E C is d raw n to re p resen t L / A t such th a t
T h e line E C h as been d esig n ate d b y th e a u th o r as a finite o p e ra to r because it re sem bles th e H eav iside o p e ra to r L p . T h e intersectio n of th is finite o p e ra to r w ith th e term in al c h a ra c te ristic of r in E gives th e v a lu e of Aii, n am ely , E J . H ere B J re p re
sen ts th e a d d itio n a l v o ltag e d ro p across r (in ad d itio n to th e original vo ltag e d ro p OA owing t o i l ) , an d J C re p resen ts th e v o ltag e d ro p a c ro s s L . In sh o rt, OA + B J rep resen ts (ii+ A ii)r; J C re p resen ts L ( A i i / A t ) ; an d OA + BC th erefo re re p resen ts ei+ Aei, an d hence satisfies E q . (8).
— [► Ł + A Ł
I-c.
I
A . e,
D G
Ą+Ae,
e
Fi g. 10. Fi g. 11.
sm all tim e in terv a l At, ei is assum ed to change in sta n tly to ei+A e! and rem ain a t this value d u rin g the in terv al At, and sim ilarly ii is assum ed to change in s ta n tly b y an
of som e angle 6. T h e n — as far as A ii is concerned— th e circu it consists of tw o resist-
cot <f. EC B = L/At. (9)
1945] G R A P H IC A L A N A L Y S E S OF N O N L IN E A R C IR C U IT S 195
T h e p o in t E is p rojected over to F d irec tly ab ov e C and D, an d F D rep resen ts th en th e new v alu e of c u rre n t ¿i+A i'i, a t th e end of th e tim e in terv a l At. A n o th er sm all tim e in terv a l can now be chosen, p referably equal to th e previous one, so th a t L / A t rem ains a t th e sam e angle to th e e-axis as before. W e suppose t h a t in th is new tim e in terv a l, e changes from ei+ A ei to e i+ A e i+ A e 2. L e ttin g OG re p resen t th e new v alue of voltage, we p ro jec t F o ver to H d irec tly above G. T h ro u g h H we draw H K p arallel to C E , in tersectin g th e term in al ch a ra c te ristic for r in K . T h en K L rep resen ts th e new in crem en t of c u rre n t A i it E L th e ad d itio n al vo ltag e d rop across r, an d L H th e new v oltag e d rop across L . I t is ev id en t t h a t E q. (8) is once again satisfied. I t is also ev id e n t t h a t IG rep resen ts (ii+ A i’i-l-Aii), th e new v alu e of c u rre n t a t th e end of th e second tim e in terv al.
P o in ts B, F , an d I re p resen t th ree p o in ts on th e overall terminal characteristic for L an d r in series for th e given function e{t). If e{t) is a periodic voltage, th e overall term in al ch a ra c te ristic will spiral aro u n d counter-clockw ise an d u ltim a te ly form a closed curve, th e s te a d y -sta te solution for th e given circu it an d given fun ction e(t).
T h e initial open bran ch es of th is spiral' re p resen t th e tra n s ie n t solution. If r is a linear resistance so th a t its term in al ch a ra c te ristic is a s tra ig h t line in stead of th e curve show n in F ig. 11, th e closed loop will be an ellipse inclined to b o th axes; if on th e o th e r h and r is nonlinear, th e closed loop will be som e form of d isto rted ellipse d epending upon th e n o n lin earity of r. I t can be shown from th e g rap hical co n stru ctio n th a t th e ta n g e n ts to th e closed loop a t th e p o in ts w here it in tersects th e term in al c h a r
a c te ristic for r are parallel to th e e axis and hence p erp en d icu lar to th e i axis.
7. R elax a tio n oscillator. S im ilar m eth o d s can be developed for r in series w ith a condenser C, and for L C r circuits, an d for p arallel as well as series arran g em en ts.
Owing to lack of space these will n o t be tre a te d h e re.2 An in tere stin g case is th a t of a nonlin ear resistance h av in g a su itab le n eg a tiv e b ra n ch , in series w ith a p u re in d u c t
ance. F o r g rap h ical purposes th e sim plest form for th e term in al ch a ra c te ristic of r is possibly t h a t of th re e in tersec tin g s tra ig h t lines, as show n in Fig. 12. Such a ch a r
a c te ristic m ay be ap p ro x im a te d b y a tu b e h av in g p ositive feedback, by a d y n a tro n , etc. U sually a d.c. p olarizing v o ltag e is required, b u t th is m erely rep resen ts a tra n s la tion of th e axes an d does n o t m a te ria lly change th e co n stru c tio n or resu lts as o btain ed in Fig. 12, in w hich th e im pressed voltage is assum ed to be zero.
s Cf. Preism an, loc. cit., p. 109.
196 A L B E R T P R E IS M A N [Vol. I l l , N o . 3
S uppose t h a t th e initial conditions a t i = 0 are th a t e = 0, an d t h a t f = AB, th e p eak c u rre n t for th e left-hand p o rtio n of r. T h en C will be th e sta rtin g po in t, w here CO = AB. T h ro u g h C th e finite o p e ra to r L / A t is d raw n corresponding to a tim e in
terv a l At. If At is sufficiently sm all, L / A t will be p ra ctically a horizontal line th ro u g h C.
In Fig. 12, L / A t has been d raw n w ith a finite tilt to clarify th e co nstru ctio n , an d is represented b y C D . T h is finite o p e ra to r curve in tersec ts th e term in al c h a rac te ristic for r in D , as shown.
T h e c u rre n t therefore decreases from CO to D G . P o in t D is projected over to the i axis as p o in t E. F rom E, E F is draw n parallel to C D u n d er th e assu m p tio n t h a t th e second tim e in terv a l is equal to th e first. T h e c u rre n t now decreases from D G to F H . P o in t F can now be p rojected over to th e i axis an d th e process rep eated . I t is clear from th e figure th a t th e intersections will proceed dow n th e rig h t-h an d b ran ch of r to I, hop over from I to J, d irectly opposite I, th en proceed from J up to A, hop over to D, an d re p e a t th e first set of intersections. As At app roach es zero, th e finite o p era
to r curve appro ach es a horizontal position, D G = = C O = A B , an d th e p o in ts of in te r
section becom e m ore an d m ore closely spaced so t h a t th e y form essentially all th e po in ts of ID an d JA .
T h e overall term in al ch a rac te ristic is by definition all th e p o in ts betw een C and K in th a t th e overall im pressed voltage has been assum ed zero, so th a t th e po in ts m u st lie along th e i axis, and th e c u rre n t range is from C to K . H ow ever, a m ore sig
nificant term in al ch a rac te ristic in this case is th e relatio nsh ip betw een th e c u rre n t and th e voltag e across eith er circuit elem ent. T h e v oltag e across th e in d u ctan ce, for exam ple, is equal an d opposite to th a t across r w hen ta k e n in a circu ital direction, since th e algebraic sum of th e tw o m u st equal th e im pressed voltage, w hich is zero.
A ccording to th is definition, th e term in al c h a rac te ristic is represen ted b y such p o in ts as D, F , etc.; in th is case, it is lines D I, I J , JA , and A D , trav ersed in th e order given. T h is m eans t h a t for th e circ u it given, th e term in al ch a ra c te ristic is v ery sim ply given b y a q u a d rila te ra l involving th e tw o p ositive resistance p o rtio n s of th e term in al c h a ra c te ristic for r contained betw een th e ir peak values A an d I.
T h e tim e required to trav e rse these p o rtio n s d epen ds upon th e relaxation tim e for L in series w ith th e increm ental resistance of r for each p ortio n, u n d e r th e p rop er initial conditions. T h e tim e required to trav e rse th e horizontal p ortio n s A D and IJ is infinitesim al, an d is in d ep en d e n t of th e shape of th e n eg ativ e resistance p o rtio n A I provided i t has no m axim a or m inim a exceeding or less th a n A an d I, respectively.
T h e device o p erates continuously as an oscillator w ith a period of oscillation d e te r
m ined b y th e tw o relaxation tim es.
S im ilar conclusions can be draw n for shapes of r o th e r th a n th ree s tra ig h t lines.
F o r exam ple, r can have th e form of a cubic p arab o la. T h is case h as been tre a te d an a ly tica lly b y V an d e r P o l.3 H ow ever, he s ta rte d w ith an L C r parallel circu it o r double-energy condition. F o r such a circu it th e term in al c h a rac te ristic is a closed curve or loop th a t exceeds th e above q u a d rila te ra l in size. As C approaches zero, th e loop shrinks an d ap p e ars to have as its lim it th e ab ov e q u ad rila te ral. H ow ever, th e an a ly tica l m etho d required th a t som e c a p a c ity be p re sen t even in th is lim it, relax a
tion case, and it h as been suggested th a t in a p ra ctical circu it th e re w ould alw ays be som e residual s tra y ca p acitan ce present.
5 B . V an der Pol, The nonlinear theory o f electric oscillations, Proc. I.R .E ., 22, 1051-1086 (1934).
1945] G R A P H IC A L A N A L Y S E S OF N O N L I N E A R C IR C U IT S 197
T h e re are o th e r g raphical m eth o d s for h andling th e double-energy case, n o tab ly th a t b y L ién a rd4 an d a n o th e r by K irsc h ste in.5 U n fo rtu n ately , th ese co n stru c tio n s be
com e in d e te rm in a te in n a tu re as C ap proaches zero, so th a t alth o u g h th e relaxation condition is suggested by them , it c a n n o t be conclusively show n to be th e lim it form.
T h e co n stru ctio n given here s ta r ts o u t w ith m erely L an d r, an d requires no C for its arg u m e n t. I t ap p e a rs to give th e lim it case d irectly an d p resen ts no in d eterm i
n a te considerations. I t has seem ed to th e a u th o r th a t th e necessity for req u irin g a ca p acity to be p resen t, no m a tte r how sm all, w as an unnecessary restrictio n , an d th a t th e a rg u m e n t ad v an ced th a t a n y p ra ctical circu it would h ave som e cap acity , a p peared to be ra th e r irre le v an t, since th e notion of a circu it is in itself an idealization of w h a t is really a field problem . In tre a tin g an electrical problem as a circu it problem one assum es t h a t th e circ u it elem ents are ideal in d u ctan ces or cap acitan ces or resist
ances an d develops th e vario u s theorem s on th is basis.
S im ilar resu lts can be o b tain ed for a ca p acitan ce in series w ith a n o nlinear resist
ance h av in g an S -shaped term in al c h a rac te ristic p rov id ed th a t it is tu rn e d th ro u g h a rig h t angle from t h a t show n in F ig. 12, i.e., provided t h a t it is a single-valued function of th e c u rre n t ra th e r th a n of th e voltage. A fam iliar exam ple is th e neon tu b e relaxa
tio n oscillator em ployed to g en e rate a saw -to o th vo ltage. I t is also possible to develop a g raphical co nstruction em ploying th e finite o p e ra to r m eth od for an L C r circuit, and in th is case L o r C m ay be p erm itted to app ro ach zero, dep end ing upon th e position of th e S-shaped c h a ra c te ristic for r, w ith o u t th e co n stru c tio n becom ing in d eterm in ate . F o r exam ple, th e constru ctio n reduces to th e form given in connection w ith Fig. 12 if C is m ade to ap p ro ach zero and r has th e term in al c h a rac te ristic show n in th e figure.
8. C onclusions. T h is concludes th e discussion on som e g rap hical m eth o d s for solv
ing n o n lin ear electrical circuits. Sim ple series circu its involving resistan ce elem ents only, are v ery sim ply solved by finding th e intersection s of th e ir term inal ch a rac te ris
tics. T h is can th en be extended to m ore com plicated resistances in which th e c u rre n t is a function of tw o voltages, as in th e case of a triod e tube.
T h e n ex t circ u it considered is t h a t of th e ideal balanced am plifier h av in g p erfectly m atch e d tu b es an d feeding th e load resistance th ro u g h an ideal tran sfo rm er. H ere th e coupling of th e tw o tu b es th ro u g h th is ideal tran sfo rm er req uires a special co n stru c
tion involving th e sliding of a rule a t a fixed angle along th e tu b e characteristics. T h e w ave sh ap e of th e o u tp u t an d of th e m id-branch c u rren ts is th en discussed, an d it is show n t h a t owing to th e sym m etry' of th e circu it th e form er can co n tain only odd harm onics; an d th e la tte r, even harm onics.
F inally , a sim ple case of a re activ e circu it involv ing a n o n linear resistance in series w ith an in d u ctan c e is tre a te d . H ere th e concept of a finite o p e ra to r curve corresp on d
ing to L / A t is developed and th is curve is em ployed to solve th e circuit. S im ilar m e th ods are av ailable for cap acitiv e circu its an d for double-energy circu its involving b o th L an d C. T h e m ethod is applied to a su itab le n eg ativ e resistance in series w ith an inductan ce, and it is show n in a d irec t m an n er th a t th is circ u it can prod uce relaxa
tion oscillations.
4 A. Liénard, Élude des oscillations entretenues, R ev . G en. Élec. 23, 9 0 1 -9 4 6 (1928). See also P . LeCorbeiller, The non-linear theory of the m aintenance of oscillations, Journal IE E (London) 79, 3 6 1 -3 7 8 (1936).
5 F . K irschstein, Uber ein Verfahren zu r graphischen Beliandlung eleklrischer Schwingungsvorgange, Arch. Elek. 24, 731 (1930).
198
P R E S S U R E F L O W O F A T U R B U L E N T F L U ID B E T W E E N T W O IN F IN IT E P A R A L L E L P L A N E S *
BY
P. Y . C H O U
California In stitu te o f Technology
1. In tro d u c tio n . T h e solution of th e N a v ier-S to k es d ifferential eq u a tio n s for th e ste a d y lam in ar flow th ro u g h a channel o r a circu lar pipe is well know n for its m a th e m atical sim plicity. T h e reason for th is sim plicity is t h a t for such flows P ra n d tl's b o u n d a ry lay er eq u atio n s hold rigorously for th e e n tire region of th e fluid. In o th e r w ords th e b o u n d a ry lay er extends up to th e c e n te r of th e channel, w hereas in th e case of th e flow aro u n d a solid ob stacle th e re is only a th in lay er of viscous fluid a t ta c h e d to th e surface of th e obstacle.
T h e ste a d y tu rb u le n t flow th ro u g h a channel or a circu lar p ip e is m ore com pli
ca te d in th e sense t h a t all th e eq u atio n s of m ean m o tio n an d th e eq u a tio n s of d ouble and trip le correlatio n previously d ev e lo p e d1-2 h av e to be utilized to ac c o u n t for th e m ean velo city d istrib u tio n in th e en tire region of th e ch annel, an d t h a t th e y ca n n o t be fu rth e r sim plified b y physical a rg u m e n ts as proposed, for exam ple, b y th e b o u n d a ry layer th eo ry . H ow ever, if we exam ine th e alg ebraic e q u a tio n t h a t re p resen ts th e m ean velo city d istrib u tio n across th e channel, we n o tice t h a t it has fu n c tio n a l be
h av io u r sim ilar to t h a t of th e form ula for th e m ean velo city d is trib u tio n w ith in a tu rb u le n t b o u n d ary la y e r.3 In o th e r w ords, th e tu rb u le n t flow in a chan nel bears som e resem blance to th e corresponding lam in ar flow on th e whole, th o u g h its d etailed s tru c tu re is m uch m ore com plicated as will be seen soon.
In w h a t follows we shall first d eterm in e th e m ean v elo city d istrib u tio n based upon th e eq u a tio n of m ean m otion an d th e eq u atio n s of d ou ble co rrelatio n , b y giving th e trip le correlatio n s th e ir values in th e m iddle of th e channel. T h is p ro ced ure leads to good re su lts in th e th e o ry of th e spread of tu rb u le n t je ts and w akes (references a t th e end of II ), b u t in th e p re se n t case it only agrees w ith th e ex perim en t in th e ce n tral p o rtio n of th e channel, w hile it fails w hen th e side is ap p ro ached. W e shall also see t h a t th e m ean sq u ares of th e th re e com p o n en ts of th e v elo city flu ctu atio n agree q u ali
ta tiv e ly w ith o b serv atio n in th e correspo nding region.
T h e second d e te rm in a tio n given below for th e m ean velo city d istrib u tio n utilizes eq u a tio n s of m ean m otion an d b o th th e equ atio n s of do ub le an d trip le correlation by neglecting term s involving q u ad ru p le co rrelation s. I t will be show n t h a t th e trip le co rrelatio n s w hich re p re se n t th e tra n s p o r t of tu rb u le n t energy p lay a p a rtic u la rly im p o rta n t role in th e v ic in ity of th e wall of th e ch annel, and th erefo re can n o t be dispensed w ith for a b e tte r re p re se n ta tio n of th e m ean v elo city d istrib u tio n .
F ro m th is second d e te rm in a tio n w e shall find t h a t n eglect of te rm s involving q u a d ru p le co rrelatio n s is ju stifia b le as a first a p p ro x im a tio n . In o th e r w ords th e eq u atio n s
* R eceived D ec. 19, 1944.
1 P. Y . C hou, Chin. Journ. o f P h ys. 4, 1-33 (1940). T h is paper w ill be referred to hereafter a s I.
3 P. Y . Chou, On velocity correlations an d the solutions o f the equations o f turbulent fluctuation, Quart, of A p pl. M ath . 3, 3 8 -5 4 (1945). T h is paper will be referred to as II.
3 N . H u, The turbulent flo w along a sem i-infinite plate (unpublished).
P R E S S U R E FL O W B E T W E E N P A R A L L E L P L A N E S 199
of m ean m otion an d of th e double and trip le co rrelatio n s are sufficient in tre a tin g tu rb u le n t flow problem s even though th e re is a wall p resen t. H ence th e m ath em atica l p ro cedure is c o m p arativ ely sim ple in a n o th e r sense t h a t th e building of eq u atio n s satisfied b y hig h er ord er correlations can b e d ropp ed up to th e p re sen t degree of a p proxim ation.
T h e first d e te rm in a tio n reveals t h a t th e v a ria tio n of th e m ean sq u ares of th e t u r b u le n t fluctu atio n is slow er th a n th e corresponding v a ria tio n of th e m ean v elo city d istrib u tio n across th e channel, w hich agrees q u a lita tiv e ly w ith experim ent. In view of th e fa c t th a t m easu rem en ts of th e m ean squ ares of th e co m p on ents of tu rb u le n t flu ctu atio n h av e n o t been rep o rted sy stem atically in th e lite ra tu re for th e flow un d er consideration, we shall o m it th e q u a n tita tiv e com parison of th e th eo ry w ith th e ex
p erim e n ta l d a ta now available on th ese q u a n titie s.
In th e second d e te rm in a tio n th e m ean velo city d istrib u tio n will be calcu lated b y assum ing c o n s ta n t m ean sq u ares of tu rb u le n t velo city co m p o n en ts across th e channel.
T h is is ju stifia b le due to th e slow er v a ria tio n of these fu n ctions across th e channel, and fu rth e rm o re th e m ean velocity d istrib u tio n rem ains p ra c tic a lly unchanged in th e m ajo r p o rtio n of th e chan n el—w ith th e exception of th e im m ed iate neighborhood of th e w all— w hen th e c o n sta n t valu es assum ed for th ese fu n ctio n s a re differen t from each o th er. T h is p ro ced u re of assigning c o n s ta n t values to th e m ean sq uares of th e v elocity flu ctu atio n s an d th en ca lcu latin g th e m ean velo city d istrib u tio n can be con
sidered as th e initial step in a m ethod of ite ra tio n w hich will be explained in §3 below in g re a te r detail.
In th e final section w e shall in d icate th e u n c e rta in tie s connected w ith th e correla
tion in teg rals p ointed o u t before (II, §8). T h e y are p ro b a b ly n o t im p o rta n t for th e m ean v elocity d istrib u tio n , because th e y involve possibly th e m ean sq uares of th e tu r b u le n t flu ctu a tio n w hich are ta k e n to be c o n sta n t for th e p re sen t calcu lation T hese u n c e rta in tie s could be rem oved, if we h ad b e tte r ex perim en tal inform atio n on th e v a r
iatio n of th e tu rb u le n t level across th e channel an d on th e velocity co rrelation betw een tw o d istin c t p o in ts in flows such as th e one exam ined here. In o th e r w ords th e p re sen t th e o ry is p erh ap s sufficient so fa r as th e m ean velocity d istrib u tio n is concerned, and it p o in ts o u t th e possibilities for fu tu re in v estig atio n s in tu rb u len c e along b o th ex
p erim en tal an d th eo re tic al lines.
2. M ean velocity distribution based upon the solution of the equations of mean motion and of double correlation. As before (I, §4) we ta k e th e p ositive x-axis (x = x l) as th e direction of m ean m otion of th e fluid, th e y-axis (y = x 2) p erp en d icu lar to th e tw o parallel planes form ing th e channel, and th e 2-axis (z = x 3) p arallel to th ese planes.
T h e p lan e in m id-channel is chosen as th e xs-plane. F rom th e eq u a tio n s of m ean m otion we h av e
tu/p = — ul<r — vdU /dy, (2.1)
w here
— dp /p d x = u l /d , <r = y/d . (2.2) T h e q u a n tity 2d re p resen ts th e w id th of th e ch an nel an d Ur is th e so-called friction velocity.
E q u a tio n (2.1) defines th e shearin g stress ri2 in term s of y and d U / d y . E x ce p t in th e im m ed iate neighborhood of th e wall th e viscous stress is sm all, so t i2 is a linear
