BSTJ 30: 3. July 1951: Waves in Electron Streams and Circuits. (Pierce, J.R.)

10.8 MB·English

Waves in Electron Streams and Circuits

hy Je, Pomme
ssn

Ismeancanine

OF recent yeu & aud deal of work to ayanal sone small
linear perturbations of unin clouds of electrons and fon" A rater
of qustons can Te Fed coneeiing Ue plied nbeapresaos a sick
rater iors

Fir of all, for thereto be a vey diet physical intepeetston, the
pertera stale mute est at sae Li or place ae then beanie
fn the maoner decribed ny the perurbtion, This condition este, for
instante rs or lati seam reat current shot into n
Tong metal te aed tively hui mgr el, seer,
thesrmet is mule large enous, the aeons oy uprated
tne seth of parurbation can ba used aly to etch such stabi
alt deters wat frm the ay will eine, Tel same ving
ut deaing lysed acepotaions In petariatne of aaorm
Dass and hntely eed clouds of lange sale thee ener
ator can be sbowa to et physi, or wos te els ean he sw

Pe Saree apa if ictasene wa Sum of Heim?
ane) fae. ps hens tye en ae, Sern
{ANE Hide Sues Ching Bie 8 Mkts Bars? Poe. 8, Vo 2, pm
TER Ns, Lin Sate Cart a Este rae he Perce as
svne ne munconon SrRPANS AND cance or

lo le ayprotirintons to thos which goal Le bina for mane reise
‘but mathemabealy more reiractory stustiaw

Dil misnverretatons fave art heasgh combing non-ativi
teatone of moten ith Masel’ cps-inne an she astoeaing san
ance to terms ef te ender a *

Finely, raring the al le ie wall, iki ingate to draw eonchsons
lean che extminstion of pascal sluts of serena epations. 8
‘ery simple example, it is spose to determine the gain of an amplier

ube hick wean eleriun stream simpy By exseining vars waves”
hich eam carl an ester, Th ong @ psalms
Sot only salve the Wfetentialequntin iovlvsl ft e mant say the
"ppoprite lnnsnary coming a el

Tn all suc onion as Core as been eanerning waves in chu a
éeciors and iors seems to have arser not ttl ack ef nama
fmbitousess ul raver temigh spl eros in physi iaterpeitio,

"The folowing, uel ronreos Heel wth same patie types of
‘waves and sith Ue importance an consequences of BtGng only
onditions. The wack eats a very my ts, simpled and sbetacted
from «physically realise, Te se yas mace so simple in onder
to rvoi? nainful mathematics wich right anecure tae actual ple tae
mace. The parnae sn eapore thc simp thorcughls, aiding bie
imisanderstneings It ib obiece? ta nates sre Shan 0" be
{uel at ses rng, brense no-one could misunderstand them soy
Tran only mpl thie ¥ id misunderstand same of the male vero
erin

1. Woy Ane Wares Isineavean?

We will censidr he case of nating or thin ez of electrons acass
which we ear aseune tha Lae eat Ceili cometart Tn ae elas
tee ase then ll egeone 4 8en vy smal eegon have “hr same
Seley. thos redecting che thenzal velocity aseibiton | We acme
Una the ow is sacotaedout ‘lly of change,7 wk the change per uit
dass charateraie uf nertrone thot, we neglect individual atetactions
‘etwecn elec al comer nly a sort af twee fect

‘We sll rice the quate involved inthe following fae

selvity = 9b ae

"7ke at sania ot mbce eae wil Be
628 rm nrc gysrioe TremucaL JOURNAL, UL 1951
lve ty a constant compe and ¢ fs sll Msctaating ora, om
erent
ergs density = @= po

‘hore avin cs the erage or €-% ernonent, wbch wl of coaree be
eget end pfs the a, coupon

ceawostine ramon density
inn

Here Je ie the averge ora 6 current Geis and, ss the electrons are
‘esa to move in the 2 dieetio, the earrens density inthe be dies
tion i asen as —/. Ke oer work Fave used fat ye caveat eater
‘han as current dey; Thope tht thi wll rause no contain,

Sin eesmed that has isn average ld. Tes assum tha there =n
se, Geld inthe x dizesian ols ant hi scaled

‘We have to equations to wcce with, One

Hae e/1m che ciage-ecvass ans af the elector, kes as 2 postion
(quity The tne eerie 5 C-at moving wh an eects, We ean
stead take dexivatives at: fel pnt

ao bd)

ae bd

Fetw)

an

‘he ls ca the second Ha are ren Ineause Oxi = 0 as ~ 0
Torthe, ket as conser 42ers of eatin of (11) for Geli in which 2
tras the same fore fs rel pace Eat vere a rng, Av Bs made
tena an sale» sil Become seller ad asa, are“ wm e993,
‘Which fa protet of vo ane units, wl beet eltively scaler
Saves De ROSCRON Stmugae AD eoMEE co

‘han the other «wo tems involving La our snl so Uheacy we net
‘the tem 05/4, and write

Suh Ln aa

Wenote then that this sypocimates the tue egunton for small values of
Ean sony.
‘Welhiwe another equting

bunt hated a

“his isthe equation af eontoty, or of conznetion of charge, I we
Inurgeate ie over m msl dscance a2 we obtain

— finn ~~~ Slee

be cuanticy p-t p)de'S the charge per uot cross ection ia the essance
‘e. Thus, the Haha side the rte wish ebaege ia the data
Greases. The quantity te hit is obviously the Tae at which charge
er unit nos set een ont of the pee 8 long.

se ery out the operations (2.3) 9 bal

oe ae ae
ss aude 0, dpsed = 0

ate 1
eo To a
‘We cl to al ht the convection cuaent is geen by
= To oF edie
as
FoI po oe ante et
“Tae erm pat cast term nl st be ieatiid with =
<h= pom an

“The tom pf prec of ee utils, Suppose we salve al ou: eq
Hoss wmgating e ‘Then, the er caused by this approxiatioe wil he
Tessa pu sale, tia al sgl lev. Tm ote

Fm pie ttes an
60) rm nua ssc mBemSTeML orm, gra 1951

We now have three anprosinate equations, wid are good approxima
Lina ual signal levels

-fk (2)
Boe as
oman an

ie ean einiotep a » or these equations al lain an equation 16
Teng aad, Lod ts we ele (1) fe +

Aieentn
ai aay
Bi at mal
we (14)

eta, wah
BT pea ae

erent (1.7) wth repect 105,

aG)t 6m

and sultat for dr, taining

asi

“This san cenation relating # wali devtivs with BT a nex equ
tion: thf aud Sts derivative, and 2 expone 16 the fis rae ony
Tis boczage we heve reget we Enea ters, yng ha a ets
‘hey ane sll earapred wit The Tse ten

"Non, the elon ow interest wit soroundings of some sr on, we
sally, witha ciel. et sm omni a aa egpmple of aie tase
feison ine witha disxibned espaianse C por unis engl ans 2 die
shuns! vetance 1, per wil leash, whist wll wall a how ase
Save thatthe rvs steam Home slong very Sige to Ce Hae. Tae
‘WAVES DF RIEETBON SEREALG AND cousTS ot

if the cute of ofthe cletrn scam, where = the ares af cto Do,
‘ane with distance current J ill oy ato the Line por ugh
shun i Fig 1 where

19)

10)

aay,

ca

i
Litt

We can eliminate Phy dierentnting

a
We van forherierty she fil acting onthe electrons ex
ov
ne feo)

3, let us seplace by cass of (1.18)! Bw erent ith
spect te and age with respon We in
692 ame mets wvonims mremaear yoURKAL, JU 1981

We can subsite for ain ram (1.12) and obtain

“Thus, we hove tained a linear pri eens eqution in x ad
So far, nothing Ins been seid out wave oz wavalle behaioe. We
nngat sive (25) for vay boundery cooiios os Yap js vases
that wechove iy emy mean, ae hy nga dfecensial nalyze er a dll
Computer ‘There & however, a rel statis tecaigee fer dealing wih
Unearpurtil feel equttions with eosstanteooicnts sc as (18)
fs Tenn that they have soli of tha form
Pe deem (1s),
As (1.1) isan emt sal eeuaton if (115) a solution, te ren par
Hf (115) isola mbtin, i,
Ro(aedies)

ina solution, Tener, we ay rgacd dhe ol yt of the comples ¥ xt the
eve elsial solution.
Tye eueitate(

15) into (L440 ne stale

1)

sip sul (Ee =
E (a6)

+ alcals — Cut = 1

‘Now (1.16) is an alae eqution nw aed 8. How ar we to inexpeet i?
Suppuee we ave intneasin dovires drvzn ftom sizascia) yeerairs,
such a sepli* ie ueans theta al, ad thas the rian f=
(Geen of the apple wigeal We my shen zeal (16) as am ect in
[Bad es it fourth cegroe eqvation, tate wl i geeral be time nis
‘We may sel tase aa pteinne to foce waves, wheee voltages vary 28

Te aes
Ven teed
vin agin

"
ih, he to and be nest

ee
eaves 19 RoReTEONT STRFANS AND eMCETS as

Bach ofthese fou compareats ea souton ofthe fifteutiatenstion. The
solutica ofan acual posica pote sil be the suum ef the fur compar
nents, of we like, thereat part of th sus, aud the apd Seton
ar~ 4a, whic are general comples, wil depend «particular
psi! probes whi is,

Whe hes boon the prone of thi argumnut? Fst ofall i is ateuded
‘ovindicnte how the waves get ino the plture, The diferent equions
or long beam of rorsnnt avenge ve oy ard charge dots ute
nears y negiecing ter ich te prota of oe quanti 3p
race Hy thie mae var part leentat equation with ennatant
Dances which wiste Cad cm oo, This wes combined with the
Uae patil diferent eration for x unior fansminsn ie cca,
aed an overall pt ier eatin fr ¥ ae sind, an
vith constare coofcierts, Such an equetion could be seved by sy means,

Bet i leroan to have mavestice anions rs te alton of og
pysiealprobiem emis be sam a ll ach soon
‘Ip geaeral, we wll not expect so simple a relation between {and V or H

ns (112), that for a srple srantisinn ie. barter for bro electron
‘reams thereon behave anno expreand go spy at as em
in (18). Nonetbeles, we wil int wiveottions in which al quantities acy
sith fine and dstanre ae

oem

asslongae
(Uy the de, bens propertiss the wads cet Ow) al the
Sri properties donot vary wide
(2) te signal amplitude is Jw enough go thee tem inveleng products
of ee. queattes cap be negeted
‘Whex this i 20, the slution of a phic! problem cam be expressed as
the au, or the veal ato them, ck weve ction ten wich
‘he proper ampli t

UL Thue Couronne Waves
(ae we are comvincod thatthe clition of our vlan ca be expres
athe eam ofa number of aves whi ae soations of near para dt
festa esuaton, iis singlet to use tie fet dircty in Ending certain
opertes of the wovee of wich the elution i to Be made up,
Tet a, for neta, a (18) ala she ctor

| Anadkos eveni contin et sto ae a ea aan,
68s nkat susie teemOAL JOURNAL, EHY 1951

Inothse wots, at Bin (18) be one of the wave comments ofa soaton
‘Then (18) corns

Cab ant alah — ie Eat

oy

ese g the cleric onstant of wes, has Leon intedue! For suns
‘lich ill become apparent ates, 1 fees of acnet to ntrvnes aer
supe peometes

tes en
her ox
fm ten

“The quantity i ell che glaema freeney (a radian frequency].
Fes ive esate p's ngative y weal he pace constant of a Sv
travelng with the elvisonweoey Wille would be te ose constant
hs weve tnveliag with » pha velacgy equ tothe eleien velodty,
End having » Eequency oy we tay messy rem fp a8 a convenient
parameter wis incentes asthe Beam curent siege. To teres of
fy an

pa Bo tiers ea
a — ta)
“this nay seem a strana: for ia which to ite the equation, I. wil
perhaps sec les stunge,sowever Herr Lit th carent eens 7
tra disterie ries is given by

1 jt

“Thus, we ses hat for real valu of 8 the eectron convection custo den
sly is that whieh wl crespond. co @ negative sree constant ot 3
Sevativecapnctane, Ita magrtle depends on 2. which ix propothnal
Tnvthe sie beam sume: density; an the magritede becomes vers las
tien the phase wlaily of the eve approacbce the velocity af the ec-
ious that ie wea 8 approaches fe
[SAUTS 1 RIRETRON STERAMS AK CIRCUS es

Spouse me cus atin of ante. We can wre te total eciron
envcetion coca. Tym he fo

re 06
nt °
Gs a en

We wil call Fhe elartonie admittance; 2 fa measured in miko meter

Later we il deal with waves fn whi the ell slam trols
owes Us the eno, aad i i ineesting to see under what camdiione
{ge cam take place Tel Uneampite of the wave ander consideration vary
‘wih distance se

‘We may take the cinder nalune of the prqegation ear inte aoa
by autettuing ln 2,

Bean ih as
Baia t& ”
“his ead
te gaia
ee wo)
_ wre ate — 8) — fe — a)

Hee = a)" # ut)
‘The eecom strum can tansler cuergy to the ctcuit only ifthe veal
ait af T, i negative @ negative conductance), For a rave bith ite
"renee in the ciertion of elev tow (ihe “+2 frelon), a be ponte
tnd the electrons conductance wil be neti #2, > that be
‘eetron velocity ie gar than the phase velocity of the wave?

or a weve whisk etree on the = dietio, the mace wll
‘oe neatve ifthe electron vel earl than the phe welcty of
the wave

Tirso soir te intention of war th lato tsa wi he
desi Bee here sone posit of codon. Ta (1.12) the del caused
Dy impreing a current ob a cit was ealulted. Thi may be Ukesed
te the volage along an impedance Z chused by an ipesed current f

re haalig Rey HS pact Wie rob alhekay

t