The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

RF signal Optimization for Beamloaded Accelerator Control
M.A. Chernogubovsky
Department of Reactor Engineering, Japan Atomic Energy Research Institute,Tokai-mura, Naka-gun, Ibaraki 319-11, JAPANYo

Abstract

fx·(t + xcos(2'P-2B -2e(w)))dw = 2A, where x =-yo

Transient beam excitation analysis for resonant accelerator allows to determine the control signal for minimization of required RF source power with simplification of the signal implementation and with RF system
bandwidth minimization at prescribed errors in the fieldmagnitude. Initial results on the accelerating channel
optimization are obtained also.
1 Introduction

Coe(w) + iCo 0 (w), (*)-complex conjugate, and by applying Cauchy-Schwarz inequality the amplitude modulated signal optimum is proved for any B with global
minimum of required energy(2)
Since the second factor in (1) is w0 periodic, the form

D= M(t) cos(wot+'P) cos(wot+B)+L(t) cos(wot+'P), (3)Effective RF control [1) in the intense beam resonant
accelerator for single mode operation is realized at directional selective coupling, therewith RF source signal S,(t) = S_(t)+S(t), where S_(t) sets up therequired single harmonic Acos(w 0 t + 'P) field and S(t)compensates the beam excitation from t = 0 instant in
w E [wo- Yo, wo +Yo] frequency band, [2). RF systemtuning characteristics and S signal are defined by theparticlestotal energy derivative F(t) (Eq.(6) in [2]).
Assuming the required signal Ss(t) is synthesized
and the control system operates ideal, the particles energy variation will be T = 27r fwo periodic, I: Ea(t) =a

N

where M(w) is not wider than wE(-w 0 ,+w 0 ), L(w) isnot wider than (Yo- w 0 , w 0 - y 0 ) and also M(t) = L(t) =0 at t rf:. (0, KT + (), presents all possible solutions of
the optimization conditions (14) in [2). Thus, the beam
excitation signal in the optimized accelerator isN-1

I: £ 0 (t- (n- l)T), where NT is the beam duration_ -wrD(w). sin(:~ N) -i:~ (N -1),. ("w) eWF(w)1

where D(t)Wr,YoSln Wo

Eout = D(O) = M(O) cos('P- B)/2.

(5)

2 RF signal determination
The first term in exp.(4) presents the (w 0 ) band excitation that must be compensated by S signal. As far as
N-1

(1)

= d£a?) and also D(t)!t~t(O,KT+O = 0,

W1 - resonant frequency and norm, and A =J !F(w)l2dw/A 2 value, preassigned at S energy Eo min--Yo(4)

the output energy is defined by M ( w) function only,
since F {L(t) cos(w 0 t+'P)} does not contain a constantcomponent:n=1(N ~ 1), £ 0 (t) is the summary energy variation of theparticles that enter the resonator at t E [0, T], the lastparticle from these leaves the resonator at t = KT + ((K ~ N, 0 :::; ( < T), and £o(t)!t~KT+( = const(t) =Eout. Thus,

N-1cos(w 0 t+B) I: M(t-kT)+ I: L(t-kT)
k=O
1
k=O
V(t)=-AW1wr

imization in [2), with any finite precision for y 0 < wo/2
is A= s;utNwow;A- 2 W 1- 2 . Basic specifications of theaccelerated beam for prescribed resonator are included
indeed, but [2) optimization is restricted by the phase B
fixation of the signal carrier ex cos(w 0 t + B) from theEo minimum minimorum estimation. Complete solution is obtained at S = (Coe(w) + Co 0 (w))exp(i(B +6( w) + e( w)) consideration in the general form of theeven Coe(w),e(w) and the odd Coo(w),o(w) parts (fordefiniteness e(O) = 0), Eq.(ll) in [2) takes the form

M(t)!til(O,KT+() = 0, the envelope Mr,(t) =I: M(t-kT)
k=O
is periodic: Mr,(t) = Mr,(t+T) for t E [rv, (N -l)T],Tv = (K -1 )T +(. For sufficiently large t this term timedependence could be obtained from the infinite periodic
excitation in (w 0 ) band, that leaves behind a single harmonic from the line spectrum and the envelope is constant; without loss of generality the formM(t) = f(t)- f(t- T),(6)f(t)!t~O = 0, f(t)!t~r. = const = /., J'(rv) = 0,(7)satisfies this condition,

Mr,(t) = f(t)- f(t- NT),(8)exp.(6) integration for Eq.(5) yields the stationary level-221-fs = 2£outf(Tcos('P- B)).

(9)

The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Since the envelope energy in [-yo, +Yo] band is determined for any M~(t) by Eq.(2),JYo!

4-yo

2IF {f(t) _:_ f(t- NT)}I 2 dw = &outNwo , (10)cos 2 (~- B)

the most complete compensation is obtained for envelopewith the minimal spectrum width\ the closely approximating acceptable form 2 isf(t) = fs ·[e-~a2-eIF {f(t)}l is used instead of the general IM~(w)l, because the constant component is excluded and the maximal elun(t) value is attained at the commensurable withKT instant. This type error dependence is presented onFig. 1 at concrete parameters (Qr - selfquality factor,
the frequency deviation with the loaded quality factor
are determined by Eq.(8,9) in [2] for RF matching realization), but the similar dependence takes place in thegeneral case also.

_u2

a2

t E [O,rv]_!L
1- e a 2(11)t E [rv, oo);

1'where x = tjT, u = rv/T; parameter a remains free,because the low-frequency filtering in Eq.(10) at N'5PK
eliminates a dependence on a, so that the equation yields
only N '5P 2K +w 0 j(21ry 0 ) additional estimation for N
lower bound.
To minimize the required bandwidth y 0 consider theerrors of RF control system. The beam excitation spectrum is infinite in the any case (the finite time duration
signal), and uncompensated outside of the 2y0 bai1d signal (4) excites elun(t) field (Eq.(3) in[2]), which maximal
normalized amplitude estimation isaw2 -w20.006

Fig. 1 Reciprocal off-band overshoot (normalized); b =1.1/1.68, u=367,Qr=10 4 values are taken from
[5,6], cos(~-0)=0.5.
The second type error is caused by not ideal compensation inside 2y0 band. For normal error difference between V and S, which power density spectrum is w(w ), the mean square field error is e1a =1 wo+Yo1fw}yo w(w)where ~L(w) =WWr r QL, QL - loaded quality factor; the integrand function resonance filtration allows
to bound the upper limit gr and to neglect the L functions sum contribution from exp.( 4); b is the beam power
to the cavity power losses ratio. Here IMm(w)l =140

Iw;( +i~L(w))
QL
12.
.
.
dw.Smcethlserrorm1

creases with the band extension while (12) error decreases, the optimal Yom value in the meaning of theminimal total error can be presupposed, however, thenormalized e1a value

AA

1
A(13)

=1 The width can be defined by 0 2
J:'oo IME(w)l1w 2 dw
value similarly to moment method (3,4] as far as the sharply defined ME(O) component is prescribed; and the function G(w)=F{[l(t+Tv)-l(t-Tv)](i(Tv -t)+ j(Tv +t))- fs }, which time
duration is evaluated by the accelerating periods number, gives thesame value as ME(w) for any f(t) that satisfies Eq.(7,8). CauchySchwarz inequality consideration for=J

J:'oo t2 G( t)2 dt ·jfii withat d~~t) ex tG(t) that is satisfied at Tv -+ oo for Gauss timedependence G 00 ( t).
2 Real f(t) can not present G 00 (t) exactly, but under the use ofthis form the loss2( '"u.
1 uerf(h-)----ea >2a,;'2; aAa =1(1radditional condition f'(O)0 yields 0?:
forv f(t) 2 dt) 2 /Ta,
where Ta -equivalent duration of G(t). The last integral is limited (to be more precise, this energy in [-yo,+Yo] band is prescribed, but it can not be determined under N :> K condition),
Ta ::; Tv ' and 0 minimum is achieved in the equality case, i.e.,

0/0min =must be prescribed as far as this error exists during all
the accelerating time interval; therewith the autocontrol
system precision Aa is specified by the normalized power
error at the determined (Eq.(2)) S power,I (1-e-a:('" >2) 'erf- error function, is not very substantial for uja > 2.

NT woj+Yo~ w(w)dw.
Eo 'IT(14)

Assuming the autocontrol error to be white noise,Eq.(12),(13) immediately yield the equation for the extremal bandwidth values. Minimal bandwidths exist; overshoot dependence (Eq.(12) and Fig. 1) determines the range of a values that restricts the number of the extremals, and under the reasonable condition (elunM/A) ~ (A/A) the optimum (aopt, Ymopt)
can be determined. The minimum dependencies forelunM < 10- 2 -A at Yo/wo < 0.025 are presented in Fig. 2;-222-

The lith Symposium on Accelerator Science and TechnolO!JY· Harima Science Garden City, 1997

at the use of, e.g., the upper band (aopt ~ 184), the loss
is fl/flmin = 1.019, so that any other possible solutions
could give not more than 2% advantage in the bandwidthvalue.

EoA(t)-r;;;;0.5(.::la=2.8·10- 4 ; elunM/A=3.1·10- 6 )(8.8·10- 4 ; 1.5-10- 6 )100

0.0224

200300tjT

Fig. 3 Normalized energy for different a; u = 367.
0.02224 Conclusion

0.0220 r---.;.__ __0.01710.0169180190200210aFig. 2 Minimal bandwidths at A/A = w- 3 and required control precision Lla , off-band overshoote1~nM /A values in optimal points for nominal (-), doubled (-), and half(-) beam currents; u = 367, b = 1.1/1.68, Qr = 104 values aretaken from [5,6], cos(<p-0)=0.5.
Thus, the required signal in wE [wo- Yo, wo +Yo] isAbw;F {f(t)-f(t-NT) cos(wot +B)}

S(w) =(15)isQr cos(<p-B)

where f(t) is defined by Eq.(l1,9) for prescribed
Eout, N, K, B values; the optimized a 0 pt, bandwidth
Ymopt, and required autocontrol precision are determinedby Eq.(12,13) and Eq.(14) at prescribed random error
and ma~imal tolerable overshoot error in the field magnitude.
3 Accelerating channel characteristics

As far as M(t) is already defined, exp.(3) integration
at separating out the terms without high-frequency oscillations gives the accelerated particles energy variation,averaged over the period[OA(t) _-Eout-(:!!:..)2.

1-e a[erf(x-;:;-u)+erf0t}] -xe-(~) 2 ,xE[O, 1]a'{!
a'{! [erf(x-;:;-u)+erf(x-~- 1 )]-e-(~) 2 , xE [1, u](x-u)- e-(*) 2 - a'{!erf(x-~- 1 ),xE[u,u+1]

1- e-(~) 2xE[u+1,oo);

The developed optimizations under effective RF control provide the following. (1) The beam excitation does
not contain phase modulation. (2) The required RF
source power is minimal at prescribed resonator. (3)RF source bandwidth is minimal for prescribed random
error and maximal admissible overshoot error in the fieldmagnitude. The optimization gives only a few percents
disadvantage in reference to unrealizable minimal bandwidth limit. (4) RF signal implementation is the simple.
(5) The optimized characteristics have only a weak dependence on the beam current deviations.
Conc~rning the beam dynamics, the (1) and (5) results
allow to presuppose the minimal particle losses. The
extremums are global (in the same meaning as indicatedfor (3) result), so that the use of the averaged energy
variation law, which is maximally correlated with the
optimized one will provide the better results if it can
not be implemented exactly.
References[1] Lynch M. T. et al., "RF System Considerations forLarge High-Duty-Factor Linacs", Proc. of the 1994
Int. Linac Conf. Tsukuba, Japan, (1994) 454.
[2] Chernogubovsky M. A., "Transient Beamloading
Analysis for Linac's RF Control", to be presented
at The 22nd Linear Accelerator Meeting in Japan,Sendai, 1997.
[3] W. McC. Siebert, "Circuits, Signals, and Systems",
McGraw-Hill, N.Y., 1986.
[4] I. S. Gonorovsky, M. P. Demin, "Radioengineering
Circuits and Signals" [in Russian], Radio i Svyaz,
Moscow, 1994.
[5] M. Martone, ed., "IFMIF Final Conceptual Design
Activity Report", ENEA, 1997.
[6] K. Sawada, "Comments on RFQ dynamics", Minutes of the Second IFMIF-CDA Design Integration
Workshop (JAERI-Conf 96-012), 1996.

EoA(t) laws are. rather different for different a, Fig. 3.
However, under the optimized a value the optimal RF
field characteristics are obtained with actually insensitive to the beam current variations, Fig. 2, so that thechannel with a = aopt is the best.

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