Proceedings o f the 22nd Linear Accelerator Meeting in Japan

pa-20

Transient Beamloading Analysis for Linac’s RF Control
M. A. Chernogubovsky
Japan Atomic Energy Research Institute (JAERI)
Tokai-mura, Naka-gun, Ibaraki 319-11, JapanABSTRACT

The beamexcitation properties at transient beamloading in linac are analyzed. Effective RF control design andthe control signal synthesis are considered at the directional selective coupling principle application. The resultson RF system characteristics determination for the accelerating structure precise tuning, for RF matching, andthe initial results on the control signal optimization are presented.

1 Introduction
The high-duty-factor linac’s main specifications leadto a need for excellent control of the accelerating
fields to obtain the high-quality beam characteris­tics and to minimize the beam loss possibility andthe resulting activation [1]. The well-developed [2,3]approximate method, assuming a slow variation of
amplitude and phase for the single harmonic field,
is inadequate for the rigorous analysis. Beamloaded
resonator equivalent circuit contains nonlinear ele­ments [4] even at [2,3] approximation and compete
analysis of the external properties is required also.

2 R F general characteristics
The resonant accelerator design usually employs sin­gle mode principle with the single-harmonic time de­pendence of the electric field,ei⑴ ニ A cos(u；c^+ p).

(1)

Particles dynamics analysis determines tolerable de­viations At4，Au；
o，Ap at this approximation withaddition oi the gradient (self-Coulomb) field. Thus,
RF control systemis to be designedfor this operating
condition implementation.
At eigenfunctions expansion the resonator vmode (spatial distribution {EU(R)1eigenfrequency
selfquality factor QV) norm Wu) timedependencies eu(t), hv{t) are defined byG+忐)％+夢=，(J乾地坦，fiv xノI n B (t) = f (^Eu( R ), j B( R, t)j d v ,I n L( t )=( eu ( R) J l ( R^) )d V ,、ら = 去 （1+ 士 X + 士 ん ；（2)where j s and jh are the beam and the coupling ele­ments current densities. The use of grid-control de­vice for RF source provides the relatively wide bandcontrol signal, and directional selective coupling [5]puts away the undesirable modes problem. The lin­
ear RF system (m identical loops) is presented byequivalent current source I s with output impedance
p at each loop reference plane, set (2) self-consistentsolution for simplified coupling device is/i’/ +=5^⑴ + F ⑴ ， ei = — ; (3)loaded quality factor of the operating mode
Ql =，Kl，兩 ;is⑷Si,Lor ,Qr —loop space, resonant frequency and un­
loaded quality factor; the beamexcitation and source
signals V(t) — -cor f {E^ R) , j B {R,t))dv/Wx, S s(t) =Is(t)Lo^fimKi/Wi ， Consider the beamwith as muchas desired but finite duration r*nj which leading
front enters the resonator at t =0, the source signal
S s(t) = S S - (t) was started before ぞ=0
and sets up the required field (1)，whereas S(t) starts
at /= 0. The form (3)linearity yields the condition
for S(t) synthesis: the solution h of
hn + -7z—h ; + UJ^h rr-f(t)(5)= hf(0) = h(0) = 0) must be zero1for Z>0 andthe control systemmust hold up V ( t ) |t>0variations in required limits. To define Fourier F \ iuj}
spectrum V{u) consider the energy variation £a of
1T rivial solu tio n 5 ⑴ = —V (t) m ay b e unrealizable sinceth e lim ite d b a n d w id th of th e source, in a d d itio n it is fa r fromth e b e st in th e m ean in g of th e re q u ire d R F energy.

the beam particle a with charge e, which moves
on f*a(t) trajectory with ⑴ velocity in a mode
field E = ev{t)^Ev{R), [6]: ^ ニ e ( v a{t),E ) =eu(t)^e (圮 (り，瓦 (r:(り)).The (2) integral I n B (t) =Ee ^(t), E l/(rl(t))^

0that ev{i) ^a in the resonatorsootherwise
where the extension ofaSaif) definition on the constant value eliminates thebivariant condition2. Thus, V{t) satisfies the equa­tion
A cos(o；o^+ ^ ^ ( t )dSa^~ dTm.

while the field contains single harmonic and the loop
voltage is nonzero at too only. So, at uj + ujo all
the incident power will be reflected, this propertyprovides support signals for the control system. The
value onojo is determined by selecting the component—wo) + e~lVo6{uj + u；o)] 3 in the beam
excitation signal, so the resonator must be tuned inaccordance withVq-(jjr A(8)

Vqcos 也 、:於 仍 如 )}:oo oo/

外 = vq — ip, forresonance condition;
therewith RF source matching is realized at⑻

As far as L0 o ^u j ri and the Eq.(5) left-hand side op­
erator has the narrow bandwidth, the desirable so­lution is obtained at V{uj) compensation by S(lo)only in [u)L^h] band, Fig. la. The ujl,
values are
determined for concrete control systemunder admis­
sible A A 1Acjo，.

sm 也

Au>r j爪m (corfiKi)

The same Yin(^o) value is obtained for the resonatorwithout the beam (Vo = 0 in Eq.(7)) but with newresonant frequency ujeq and new unloaded quality
factor Qeq:
weq •ん灯^^，Q e q :we qujrA

V q co s( 9 )Au>r

For real coupling devices nothing more than some in­consistency between the real and the second Eq.(4)coefficients, indeterminacy of the reference plane po­sition, the frequency shift from the device reactancetake place, since the real device properties are con­stant in vastly wider band thanFig. la. The
last expressions do not contain these factors and canbe used for precise tuning.

3 Control signal properties
Consideration of Eq.(6) for the functions1Sac( t ) =Sa(t) —£a(t —r c)，Tc > 0; Fc(t) = F(t) - F(t - Tc)
and also f Fc(t)dt ニ 0 yields

IFig.1 . Spectrums transformations according toEq.(6) and input conductance Yin(uj).
At the required solution of Eq.(5) with (1)form ofei⑴ the input conductance in the reference plane is(7)Yin(u;)m ( u r fiKi)^ a c { t ) jd tI

丨尸H I 2. Tc-w}

duj

27T

The second factor brings out |F(cu) |2 filtration in^ 1/TC bandwidth. The a particle is inside the res­onator at ^G [rai , Tao]; for max{rao-rai} 《 Tc < rinwith Tin increase the left-hand side integral approx­
imates to n.n.(X^a(Tc))2, for Tc > rin with Tc in­crease it approximates to TC'(Y] Sa{Tao)Y ，but r2-n,Tc

A7rujr—u；o) + e~l(p6(to + wq)],L0U)U)r -Qr . Finite values are only at cu

where ミ士o；o，Fig.la，since the Is source spectrum is finite,
2T h e re su lt is ex act for different p a rticle charges also.

3T h is form continuous e xtension onto t G ( —oo; + o o ) iscorrect since th e second te rm in Eq.(T) presen ts only th e ra tioof tim e functions.
4T h e どa ⑴ ex ten sio n delivers 广 °° |^Sa\dt > oo, so th a t

Fourier im age of th e E q.(6) in te g ral m ay be nonexistent.

must remain finite. Nevertheless, the definition of (6)spectrums
E K a ( r ao ) e - ^ T- + J / V a( t ) e - ^ ^ ^aVバ o;):r“Kat a; = 0 gives F( 0)>iニi^ E ^ ( rao).

(10)

Thus,af |F( cj) | dto over the low frequency band presents
the beam’s main characteristics and may be taken as
preassigned.
Consider the symmetrical band signal S o(lj) =S( uj) + Sad[^), Fig.l; in the general case=Co(uj—coo)eZO!^w~UJo^-\-Co(—Lo—ujo)e~ZOi^~UJ~LUo\ whereCo{y), a(y) are defined in y G [— y o ，+Z/o ] ， Vo == ^had —^o- Now the transformed spec­
trums (Fig. lb) coincide in the low frequency band,
wherein the squared modulus integration yields Ew =yo

Eq7t — JC q[u) \duj) where E is the S signal energy,VbVb = ^ h ~ ^ o , and the So energyCAq(lo ) Co( — <^) c o s ( j ) d u j

E0 :(11)7T

yowhere A = f |Ffo；)| dco/A^ value is prescribed.
—yo
C auchy-Schw arz in e q u a lity aD p licatio n d eterm in es
Eq, E g lo b a l m in im u m at
j=2 ( p—a ( — c j )—a ( u ; ) =：0 ，Cq{( j o ) =Cq{ — uj) .

This conditions are satisfied if V(to) in [u)L^had\band is the spectrum of an amplitude modulated
signal. Usually deviation (8) is much smaller than
the required band, ^ yo and the second term in
Ett expression is negligible; further analysis assumes
symmetrical band S(u;). In the general caseS(t) = A 0(t) cos(u)0t +^) +B 0(t) sin(u；o^+(f) ,(12)the F(cu) compensation (Fig. lb) in the band-^o(^) — •^I(u;)|a；e[-y0，+y0] and ill the _F+2(U；) —^(^)lwe[2co0-yo,2w0+yo] with P-2レ ） bands5 proves
the realizability of Ao(t), Bo(t) and their spectrums
ル—

2i[F0(u;)-2F+2(u;+2u；o)e-^]^2F0(u;) Bj， 一ス

u —Thus, the signal S(t) = A mod(t) cos(Lu0t + p + 少 (t))
contains amplitude and phase modulations
A mod(t) = ^ ( ( F - ^ { F 0(u;)})2+^(F- 1{i(F0(c) - 2F+2(u; + 2co)e- ^ ) } ) 2) " ， ⑴ ）邺 ）=a—巴， (ぬ;:ド ， ’} );5F ( uj) c o m p en satio n only in th e low frequency b a n d is in ­consisten t because of th e ’’m irro r ch annel” effects.

where F{uj) is defined by Eq.(10). The signal energy
will be minimal at
2Fi .2(cj-h2cj0) e - i2^f^{Z} = 1(14)又^{Z} =const(u;)
F o ( c ^ ) —，in this case the phase modulation is absent.
The ujQ harmonic for (7) form is selected at zerocorrelation of the residual V{t) —Vbcos(a;of + Vo)
signal with cos(u；o^+ (f) Vp in the finite V(t) in­
terval, at any finite precision uJoTin 》 1 it yields= arg(V"(o;。)），F。= + |F(o；o)|；with the use ofEq.(12)Fo = 4j ^li；

Vocosル -2Fp(0)(15)^o = a rg (F +2(2cc；o)) •A

4 R esum e
1.Basic idea for the control and the condition Eq.(5)for state equation are obtained. 2. Accelerating res­onator external characteristics for RF feed systemdesign are defined, Eq.(9) determine beam’s equiva­
lent for the system precise tuning. 3. The reflectedwave properties: full reflection at u: ^ ujq with zeroreflection on cj = cjq form the possible support forthe control.4. Spectrum F(u) (Eq.(10)) completely
determines the control signal Eq.(13) and the tuningcharacteristics Eq.(15,9). 5. Optimal control signaldelivers minimum of RF source energy, the signal
is simple for realization since it is only the ampli­
tude modulated. The optimum conditions Eq.(14)are also expressed in terms of F( lu).
The (14) conditions allow to consider RF system
optimization.

R eferences[1]Lynch M. T. et al. :”Proc. of the 1994Inter­national Ldnac Conf. Tsukuba, Japan”，Na­tional Laboratory for Hign Energy Physics[2] Pedersen F.: IEEE Trans. Nucl. Sci” NS32, 2138 (1985).
[3] Wilson P. B. :”AIP Conference Proceedings87”，American Institute of Physics, 450(1982).
[4] Jones R. M. et al. :”1988 Linear Accelerator
Conf. Proc. Virginia”，CEBAF-Report-89001，82 (1989).
[5] Chernogubovsky M. A. et al.Portable
433MHz RFQ Linac RF System”，to be
published in Nucl.Instr.Meth.Phys.Res.A.
[6] Landau L. D., Lifshitz E. M. :”The Classi­
cal Theory of Fields” [in Russian], Nauka,
Moscow, ch. M (1988).