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SLAC KLYSTRON LECTURES

Lecture 2

January 21, 2004

Kinematic Theory of Velocity Modulation

George Caryotakis

Stanford Linear Accelerator Center

[email protected]


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K!"MATC T#"$%& $' V"L$CT& M$()LAT$!

In this sectin an! in the ne"t, #e $resent the thery %ehin! the $rinci$a& 'r(u&ae use! in the

!esi)n ' a($&i'ier *&ystrns+ The intent is t $ri!e the stu!ent r en)ineer #ith the assu($tins use! in their

!eriatins s that he r she can use the( crrect&y+ These assu($tins resu&t in the a$$r"i(atins necessaryt !erie ana&ytica& e"$ressins 'r the )ain an! %an!#i!th ' *&ystrns at &# r' si)na& &ee&s -s(a&&.si)na&/+

The athca! c!e, #hich is !iscusse! in a &ater &ecture, cntains such 'r(u&ae+ The thery %e&# is inc&u!e!

%ecause it is inc($&ete&y cere! in te"t%*s, an! %ecause there is #i!es$rea! cn'usin n h# t $r$er&y

treat cu$&in) ce''icients an! %ea( &a!in) in c($&e" caities+ This is a $r%&e(, $articu&ar&y at (i&&i(eter

#ae 're3uencies, #here e"ten!e! interactin caities are &i*e&y t %e use!+

Other c!es, such as AJ.!is* an! AIC, si(u&ate *&ystrn $er'r(ance usin) 'irst $rinci$&es

-e&ectrn !yna(ics an! a"#e&&5s e3uatins/+ These c!es are a$$r"i(ate n&y t the e"tent that they areeither ne.!i(ensina& -r t# !i(ensina&/, r e($&y t carse a )ri!, r cntain ther a$$r"i(atins+

Later &ectures !escri%e these c!es, as #e&& as t# #r*in) *&ystrn e"a($&es, in ne case c($arin) actua&

$er'r(ance t resu&ts 'r( three !i''erent c!es+ This is essentia& in'r(atin, since *&ystrn !esi)n in the

21st century is %ein) carrie! ut a&(st entire&y n c($uters -i+e+ #ith &itt&e, r n c&!.testin) r %ea(

testin)/+

The *ine(atic -n s$ace char)e/ ana&ysis %e&# '&&#s seera& authrs #h $u%&ishe! %*s n

*&ystrns shrt&y a'ter 662+ The e&city (!u&atin -r %unchin)/ 2.caity thery that '&&#s i((e!iate&y

is the n&y &ar)e si)na& ana&ytica& treat(ent ' *&ystrns in this cha$ter+ It &ea!s t the 7esse& 'unctince''icients ' the current har(nics an! a ca&cu&atin ' t#. caity a($&i'ier e''iciency+ The su%se3uent

!eriatin ' the cu$&in) ce''icient an! the state(ent ' the %ea(.&a!in) 'r(u&ae are %ase! n s(a&& si)na&

a$$r"i(atins, #hich are a&i! 'r a&& %ut the 'ina& caities ' a (u&ticaity *&ystrn+ Neither treat(ent ta*es

s$ace char)e int accunt+ This is !ne in the thir! &ecture ' the series+

A&& ca&cu&atins in this an! the ne"t &ecture are nn.re&atiistic+ The 'r(u&ae use! in si(u&atins in

&ater &ectures are hi)h&i)hte! in ye&&#+


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Cnsi!er a *&ystrn cnsistin) ' t# caities, a

%uncher an! a catcher, %th )ri!!e!+ -8i)+1/+ Let a

%ea( ' e&ectrns, #hich has %een acce&erate! %y a

$tentia& 9 t a e&city u, traerse the 'irst $air '

)ri!s, #here it is acte! u$n %y an r' &ta)e 91sinωt,

re!uce! %y a cu$&in) ce''icient + The &atter(!i'ies the &ta)e acrss the )ri!s t $r!uce the

e''ectie &ta)e (!u&atin) the e&ectrn %ea(+

E"$ressins 'r the cu$&in) ce''icient -aays &ess

than 1/ #i&& %e !erie! &ater+

2

0 0

1

2mu eV =

#here the e&ectrn char)e e !es nt carry its #n ne)atie si)n+ The e&ectrn ener)y is (!i'ie! %y the r'

'ie&! at the )a$ an! the '&&#in) re&atinshi$ can %e #ritten 'r the e"it e&city u:

2 2

0 1

1 1sin

2 2mu mu eMV t ω − =

-1/

-2/

'r( -1/ an! -2/, it '&&#s,

10

0

1 sin MV

u u t V

ω = + -;/

Fig. 1

The e&ectrns in the %ea( enter the )ri!!e! )a$ #ith ener)y,


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I' #e assu(e that 91/

10

0

1 sin2

t MV

u uV

ω ÷

≅ +

-

?/

@ is the %unchin) $ara(eter, an! θ0=ω&=u0+O%ius&y, #hen @ 1, ωt2 is (u&tia&ue! an! there ise&ectrn erta*in)


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1 2o t I dt I dt = 6e hae, !i''erentiatin) ->/

2

11 1 cs

dt

X t dt ω = −

8r( -B/ an! -/, can n# #rite

2 1=- = /t o I I dt dt =

An!, re$&acin) !t2=!t

1 %y its a&ue in E3+ -D/

-B/

-C/

-D/

1-1 cs /

ot

I I

X t ω =

−-10/

The 3uantity ' char)e &eain) the %uncher in the ti(e intera& t 1 t t 1 +dt 1 is I odt 1, #here I o is the

%ea( C current enterin) the %uncher+ This char)e, a'ter !ri'tin), enters the catcher in the intera& t 2 t

t 2+dt 2+ I' It -tta& current, !c an! r'/ is the current trans$rte! %y the %ea( t the entrance t the catcher,

then thru)h cnseratin ' char)e,


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8r @ F 1, the current at the catcher %ec(es in'inite, since %y ins$ectin ' 8i)+ 2, the 'inite char)e

trans$rte! 'r( the %uncher at t1 F 0 arries at the catcher in a Ger ti(e intera& -!t 2=!t1 F 0 at t1 F 0/

T ca&cu&ate It, ne (ust then su( the a%s&ute a&ues ' a&& current cntri%utins t I

t 'r( ti(e

se)(ents t11

, t12

, etc, at the %uncher as '&&#s,

0

11 12

1 1++++

1 cs 1 cst I I

X t X t ω ω

= + +

− − -11/

Fig 3(5)

The current #ae'r(s at the %uncher are sh#n in 8i)+ ; %e&#


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N#, since I t is c&ear&y a $eri!ic 'unctin ' Ht, it can %e e"$an!e! in a 8urier series, as '&&#s,

2 0 2 01

I cs - / sin - /Jt o n n

I I a n t b n t ω θ ω θ ∞

= + − + −∑the ce''icients are )ien %y,0

0

2 0 21= cs - / - /n t a I n t d t

θ π

θ π

π ω θ ω

+

−

= −∫ an!

0

0

2 0 21= sin - / - /n t b I n t d t

θ π

θ π

π ω θ ω

+

−

= −∫

-12/

-1;/

-14/

Usin) an! -B/ a%e, #e can n# #rite

01 1 1cs - sin / - /n

I a n t X t d t

π

π

ω ω ω π −= −∫

an!

01 1 1

sin - sin / - /n

I b n t X t d t

π

π

ω ω ω π −

= −∫

-1?/


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There're, the catcher r' current I t can %e #ritten as the '&&#in) series

0 0 1 0

1

2 - /cs - /t n I I I J nX n t ω θ ∞= + −∑

The n F 1 har(nic -the 'un!a(enta&/ is si($&y,

0

1 0 1 0

- /

0 1

2 - / cs- /

Re 2 - / j t

I I J X t

I J X e ω θ

ω θ

−

= −

=

Fig 4(5)

-1B/

-1/

%n is i!entica&&y e3ua& t Ger, since the inte)ran! a%e is an !! 'unctin ' t

1.

It turns ut that the e"$ressin -1?/ 'r the an ce''icients is a&s a re$resentatin ' the 7esse&

'unctins ' the 'irst *in! an! nth r!er -8i)+ 4/+

-1>/


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6hen @


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is $r%a%&y the (st i($rtant $ara(eter use! in the !esi)n ' a *&ystrn %ecause the *&ystrn

)ain is a 'unctin ' 2n -#here n is the nu(%er ' caities/, an! %ecause it is ery sensitie t the %ea(

!ia(eter, #hich is neer *n#n $recise&y+ ence !iscre$ancies in the ca&cu&ate! r si(u&ate! )ain ' a

*&ystrn can %e usua&&y trace! t inaccuracies in +

6e %e)in #ith the e3uatin ' (tin ' the e&ectrn in a )ri!!e! )a$ 'ie&! EG-G,t/, !e'ine! as

#here E( is the (a"i(u( a&ue ' the e&ectric 'ie&! in a caity interactin )a$ that e"ten!s 'r( G

F 0 t G F !+ The 'unctin '-G/ is a sha$e 'actr+ The (a"i(u( r' &ta)e acrss the )a$ is

A $ara(eter -'r s(a&&.si)na& a


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I' the 'ie&! in the )a$ #ere Ger, an! the e&ectrn e&city u0, the e&ectrn $sitin a'ter ti(e t #u&! %e

G F u0t+ Since the r' (!u&atin is s(a&&, #e can #rite, a$$r"i(ate&y

0 - /u t z h z α = +

6e n# intr!uce βe F H =u0 , the %ea( $r$a)atin 'actr+ The 'unctin h-G/ is in!eter(inate 'rur $ur$ses an! !es nt enter in the ca&cu&atins that '&&#+ Then -24/ %ec(es,

u&ti$&y n# %th si!es ' -2;/ %y 2!G an! %tain the !eriatie ' the s3uare ' the e&city, as '&&#s

22

1

22 2 - / j t

V dz d z d dz edt dt f z e dz

dt dt dt dt d m

ω = = ÷

Inte)ratin) 'r( G F 0 t a !istance G #ithin the )a$

2

210

0

2 - /

z

j t V dz eu f z e dz dt d m

ω

− = ÷ ∫ Su%stitutin) -2?/ int -2B/, inte)ratin) t the en! ' the )a$ -G F !/ an! !entin) the e"it e&city %y u,

#e hae

2 2 10

0

1- / - /

2

e

d

j z V m u u e f z e dz

d

β − = ∫

et z ω β ≅

-24/

-2?/

-2>/

-2B/

-2/


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N# reca&& E3+ -2/, re$r!uce! %e&#,

2 2

0 1

1 1

2 2 eff mu mu eMV eV − = =

#hich e"$resse! the chan)e in *inetic ener)y ' an e&ectrn a'ter )in) thru)h a narr# )a$,

#ith a &ta)e 91 acrss it -ti(e !e$en!ence (itte!/+ 9

1, #hen (!i'ie! %y the cu$&in)

ce''icient , yie&!s the e''ectie &ta)e actin) n the e&ectrns+ 6e n# hae an accurate

e"$ressin 'r + In a ne.!i(ensina& syste( it is, %y -21/, -2/ an! -2D/, The e''ectie

&ta)e is then,

0- / e

d

j z

eff z V E z e dz β

= ∫ an!,

1 0

1- / - / e

d

j z

e z M E z e dz V

β β = ∫

0

0

- /

- /

- /

e

d

j z

z

e d

z

E z e dz

M

E z dz

β

β =∫

∫

-2D/

-;0/

-;1/

-;2/


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-;;)

-;4/

E3+ -;2/ can %e )enera&iGe! %y e"ten!in) the inte)ratin er the entire G.a"is+ It can then %e

a$$&ie! t 2.!i(ensina& -)ri!&ess )a$, as in 8i)+ ? %e&#/

The cu$&in) ce''icient can then %e #ritten,

1

- /

e j z

z

eff

e

z

E e z V

M V

E dz

β

β

∞

−∞∞

−∞

= =∫

∫ In #r!s, is e3ua& t the cn&utin ' the a"ia& e&ectric 'ie&! E-G/ #ith the e"$nentia& e MβeG

-the e''ectie &ta)e/, !ii!e! %y the &ta)e -inte)ra& ' the 'ie&!/ acrss the )a$+

Nte a&s, that i' the 'ie&! EG-G/ is a $iece#ise cntinuus 'unctin ' G, it can %e #ritten in

ter(s ' the inerse 8urier inte)ra&,

1- / - /

2e j z

e z g E z e dz β β

π

∞

−∞

= ∫

#hich (eans that can %e a&s #ritten in ter(s ' the 8urier trans'r( ' the e&ectric 'ie&!

1

2- / - /e e M g

V

π β β = -;?/


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-;>/

-;B/

8r (st cases, #here the 'ie&! can %e c($ute! e"act&y #ith c!es such as SUER8IS r A8IA

-see Cha$ter 10/, an! #here the 'ie&! is an een 'unctin ' G, a ery si($&e e"$ressin resu&ts+

cs

- /

z e

e

z

E z

M E dz

β

β

∞

−∞ ∞

−∞

=

∫

∫

A$$&yin) it t the case ' a )ri!!e! )a$ #ith a cnstant 'ie&! ' (a)nitu!e EG F9

1=! %et#een !=2

an! !=2 -an! shi'tin) the ri)ina& ri)in/, #e %tain the #e&&.*n#n e"$ressin

1

2

1

2

- / sin1 2- / cs

2

e j z d e z

e eed

V d E z e dz d

M zdz d V d

β β

β β β

∞

−∞

−

= = =∫

∫

This ca&cu&atin !e(nstrates the use ' a si($&i'ie! 'r( 'r the cu$&in) ce''icient #hen the

'ie&! ' interactin is an ana&ytic 'unctin ' G+ In c($uter si(u&atins, the 'ie&! ' an e"ten!e!

interactin caity can %e !eter(ine! %y si(u&atin, &iste! in E"ce& -'r e"a($&e/ #ith cs-% eG/ an!c($ute! usin) E3+-;4/ 'r sy((etrica& 'ie&!s, an! -;1/ 'r ar%itrary 'ie&! !istri%utins+ This is a (uch

%etter (eth! 'r ca&cu&atin) , i' the caity interactin 'ie&! is *n#n in !etai&+

K&ystrn en)ineers )enera&&y ! nt re&y n -;B/ t !esi)n their tu%es, %ut rather e($&y (re

accurate, t#.!i(ensina& (!e&s+ T !erie a 2.!i(ensina& e"$ressin 'r the cu$&in) ce''icient

in un)ri!!e! cy&in!rica& !ri't tu%es, ne (ust: a/ assu(e a certain e&ectric 'ie&! at the e!)e ' the )a$ at

r F a, c($ute! 'r( the !etai&e! nature ' the !ri't tu%e ti$s, r !eter(ine! %y si(u&atin, an!

ca&cu&ate at r F a, %/ 'r( that %un!ary cn!itin, c($ute as a 'unctin ' r thru)hut theinteractin re)in, an! c/ aera)e the cu$&in) ce''icient er the %ea(+


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The )a$ 'ie&! is !e'ine! as %e're, this ti(e #ith an r.

!e$en!ence:

- , , / - , / j t z m E r z t E f r z e ω =

an! at r F a,

- , , / - , / j t z m E a z t E f a z e ω =

9arius authrs hae a(use! the(se&es er the years c($utin) cu$&in) ce''icients 'r run!

ti$s, *ni'e.e!)e ti$s, s3uare ti$s, etc+ 6arnec*e P uenar!4 assu(e !ri't tu%es en!in) in *ni'e.e!)es t %tain the e"$ressin %e&# 'r the 'ie&! at the )a$, at the !ri't tu%e ra!ius a+

1

2

1- , /

1

z

V E a z

d z

d

π =

− ÷

8i) ?

-;/

-;D/

-40/


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Usin) the ear&ier 'r(u&a in E3+ -;1/

1

21 10 0

1 1 1- , / - , /

1

e e

d d

j z j z

e z

V M a E a z e dz e dz

V V d z

d

β β β π

= = − ÷

∫ ∫

6ith a )! ta%&e ' inte)ra&s, #e %tain the (st c((n&y use! cu$&in) ce''icient at r F a,

0- , /2

ee

d M a J

β β

= ÷

In the a%sence ' s$ace char)e an! 'r 'ie&! aryin) sinusi!a&&y #ith ti(e in a acuu(,

a"#e&&5s e3uatins re!uce t the #ae e3uatin,

22

, 2 2

1 z r z z

z

E E c E

∂∇ = ∂

#here c is the e&city ' &i)ht+ Su%stitutin) 'r( -4?/,

22

2

z z

z

E E

E ω

∂= −

∂#e hae,

2 2

z z E k E ∇ = −#here * F #=c+ I' the ariatin #ith G is as eMβG, then,

2 2 2- /r z z

E k E β ∇ = −

-41/

-42/

-4;/

-44/

-4?/

-4>/


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Since, un&i*e in a c&se! #ae)ui!e, the a"ia& e&ectric 'ie&! cannt %e Ger n a"is at the center ' the )a$,

the s&utin t -?0/ cannt %e a J0, %ut an I

0 7esse& 'unctin, #hich is nt Ger at r F 0 as J

0 is+ This (eans that

β* -r that #ae the $hase e&city u is &#er than c an! the a$$r$riate 'unctin is Ι0(γ r )e MβG, #ith,

2 2 2k γ β = −

Since the a"ia& 'ie&! is c($&ete&y s$eci'ie! at the sur'ace r F a, it can %e !escri%e! as a

8urier inte)ra& er %, 'r any a&ue ' r insi!e that sur'ace

( )

( )0

0

- , , / - / j t j z z I r

E r z t e g e d I a

ω β γ

β β γ

∞−

−∞

= ∫ #here, as %e're, %ut n# as a 'unctin ' r as #e&&,

1- , / - , /

2

j z

z g r E r z e dz β β

π

∞

−∞

= ∫

( )

( )0 - /

0

- , / - , / e j z z I r

E r z g r e d I a

β β γ β β γ

∞−

−∞

= ∫

-4B/

-4/

-4D/

-?0/

Let us n# 'in! the e''ectie $tentia& -inte)ra& ' the 'ie&! e"$erience! %y an e&ectrn

trae&in) #ith the %ea( e&city u0at a ra!ius r a&n) a &ine $ara&&e& t the a"is+ Let the e&ectrn

enter the )a$ at t F 0+ Since ω t β e z the e&ectric 'ie&! at r, G is


18/23

18

Let us n# inte)rate er G , at an ar%itrary ra!ius r t %tain the e''ectie $tentia& 9e''

, as in E3+

-;;/+

( )

( )

0 - /

0

- , / e j z

eff

I r V dz g r e d

I a

β β γ β β

γ

∞ ∞−

−∞ −∞

=

∫ ∫ 6e #i&& n# use the '&&#in) i!entity:

- /2 - /e

j z

ee dz β β πδ β β

∞−

−∞

≡ −∫ The !e&ta 'unctin δ-β

e.β/ has the $r$erty that, #hen (u&ti$&ie! %y anther 'unctin ' β, an!

inte)rate! er β 'r( (inus t $&us in'inity, it returns that 'unctin ea&uate! at βe+ Cnse3uent&y,

( )

( )

( )

( )0 0- /

0 0

- , / 2 - , / - /e j z

eff e

I r I r V dz g r e d g r d

I a I a

β β γ γ β β π β δ β β β γ γ

∞ ∞ ∞−

−∞ −∞ −∞

= = −∫ ∫ ∫

an!,

( )

( )

( )

( )0 0

0 0

2 - , / - / 2 - , /e

e e

e

I r I r g r d g r

I a I a

γ γ π β δ β β β π β

γ γ

∞

−∞

− =∫

-?1/

-?2/

-?;/

-?4/


19/23

19

#here, )e2 F β

e2 * 2+ Then,

( )

( )0

0

- , / 2 - , / e

eff e e

e

I r V r g r

I a

γ β π β

γ =

an!

( )

( )0

1 1 0

- , / 1- , / 2 - , /

eff e e

e e

e

V r I r M r g r

V V I a

β γ β π β

γ = =

#h

ere, 2 2 2

e e k γ β = −6e #i&& n# !r$ the e su%scri$t 'r ) 'r( this $int n, #ith the un!erstan!in) that ) #i&& aays %e

re$resente! %y E3+ ?B an! #i&& aays %e rea&, #here *&ystrns -r T6Ts/ are cncerne!+ On the ther

han!, it is cnentina& in (st te"t%*s t *ee$ the su%scri$t e in βe F H=u

0+ N#, ea&uatin) 9

e''at r

F a, an! c($arin) #ith -;?/ an! -42/

( )( )

0

0

0

- , / - /2

ee

I r d M r J I a

γ β β γ

=

#hich is the )enera& e"$ressin 'r the cu$&in) ce''icient at ra!ius r insi!e a )ri!&ess cy&in!rica& )a$ '

ra!ius a an! #i!th !+

-??/

-?>/

-?B/

-?)


20/23

20

Nte that the tta& cu$&in) ce''icient is the $r!uct ' the cu$&in) ce''icient a at r F a,

an! aer

#hich re$resents an aera)in) ' the cu$&in) ce''icient er the %ea( crss.sectin+

I', instea! ' %tainin) -42/ 'r( an ana&ytic e"$ressin ' the a"ia& e&ectric 'ie&! at r F a, #e

%taine! that 'ie&! %y si(u&atin an! use! -;;/ t ca&cu&ate a cu$&in) ce''icient at r F a, #e#u&! (u&ti$&y the resu&t #ith a ca&cu&ate!

aer + re )enera&&y, then

2 2

0 1

00

- / - /- = 2/

- /a a!er e

I b I b M M M J d

I a

γ γ β

γ

−= =

S(e ty$ica& nu(%ers 'r the $ara(eters use! in ea&uatin) : I' %e! F1, )a F 1 an! %=a F 0+>, #e

%tain a F 0+B>?,

aer F 0+>D an! F 0+?;+

-?D/


21/23

21

2

0 4

b e

e

M ""

β β

∂= −∂

E3uatin ->0/ is 3uite )enera& an! a$$&ies t a&& )a$ sha$es, as #e&& as t the e"ten!e! r

sheet %ea( caities that #i&& %e !iscusse! &ater+ Its (eanin) is that the rati ' the %ea( &a!in)

cn!uctance % an! the %ea( cn!uctance

0 F -I

0=9

0/ is entire&y !eter(ine! %y the cu$&in)

ce''icient an! its ariatin #ith the C %ea( &ta)e+ In the ne"t &ecture, #e #i&& see that the

(a)nitu!e ' the cu$&in) ce''icient in e"ten!e! caities is $ti(iGe! %y synchrniGin) the

$hase e&city ' the e"ten!e! caity t the %ea( e&city, an! that the sta%i&ity ' these caities

!e$en!s n a $sitie %=

0

Usin) ->0/ #ith an! a &itt&e !i''erentia& ca&cu&us, #i&& eri'y the %ea( &a!in) 'r(u&a 'r

$&ane )ri!!e! )a$s,

0

sin = 2 sin = 21 cs = 22 = 2 = 2

b e ee

e e

" d d d " d d

β β β β β

= −

2 2

0 1

0

- / - /

- /a

I b I a M M

I a

γ γ

γ

−=

->1/

->0/

->2/

T cnc&u!e this &ecture, #e nee! t $ri!e a 'r(u&a 'r the %ea( &a!in) cn!uctance %+

Reca&& that in E3+ -2?/ a%e, ter(s in #ere nt inc&u!e! in the !eriatin ' + It turns ut that

ca&cu&atin 'r %ea( &a!in) re3uires these t %e ta*en int accunt, an! the !eriatin ' an

e"$ressin 'r % %ec(es ery cu(%ers(e+ 6e #i&& state the resu&t %e&# an! e"$&ain it

3uantitatie&y+


22/23

22

I' the e"$ressin -?/ 'r #ere (ani$u&ate! as in E3+ ->1/ t %tain %= 'r thecy&in!rica& nn.)ri!!e! )e(etry, an! the resu&t #ere then aera)e! er the %ea(, a ery

c($&icate! e"$ressin 'r %ea( &a!in) #u&! resu&t+ Such a 'r(u&a #u&! a&s %e rather

&i(ite! in its a$$&icatin, since the resu&t #u&! nt %e a$$&ica%&e t e"ten!e! )a$s+ A %etter

$rce!ure #i&& %e !escri%e! in Lecture 4+

A *&ystrn caity e3uia&ent circuit, inc&u!in) the %ea( &a!in) cn!uctance -an!

susce$tance/ is sh#n in 8i)+ > %e&#+ A a&ue 'r 7 % is $ri!e! in the 6arnec*e %*+ The

!etunin) e''ect is s(a&& an! ca$acitie+

Fig 6


23/23

23

[1] Feenberg, E., “Notes on Velocity Modulation,” Sperry Gyroscope Lab. Report 5521-1043,

Sperry Gyroscope o., !nc., Garden ity, N", Sept #$, 1%&$.

[#] Fre'lin, (. )., *. +. Gent, . -. . -etrie, -. (. +allis, and S. G. /o'lin, “-rinciples o0Velocity Modulation,” IEEE Journal %, -art !!! *, 1%&2, -g. 34$5%14.

[] 6ranc7, G. M. (r., “Electron 6ea' oupling in !nteraction Gaps o0 ylindrical Sy''etry,”

IRE Trans. on Elec. De ., May, 1%21, -g 1%5#82.

[&] +arnec9e, ., and -. Guenard, Les Tubes Electron!"ues a #o$$an%e par &o%ulat!on %e

'!tesse( -aris: Gaut7ier Villars, 1%$%.

[$] Ge;arto;s9i, (. +., and +atson ). *.,)r!nc!ples o* Electron Tubes( . Van Nostrando'pany, !nc. 1%2$.

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