THE IMPROVEMENT OF THE AUTO-TUNING SYSTEM IN KEK-PS MAIN RING RF CAVITY
Shigeshi Ninomiya
National Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ken, 305, Japan
ABSTRACT

When Ib is decreased by 100 A, S is negative,i.e., there is an excessive increment in ~. In thiscase, the permeability ~ is varied by Ib with noretardation.
Though the above equation is a result for 8Ib = +100 A, we assume that the retardation, caused by theferrite, is represented by the equation describing thecase where the bias current is increased. Taking thedifferentiation of the above equation with respect toThe performance of the auto-tuning system in KEKPS main ring was analyzed, and the system has been
improved. The error in resonance frequency is reduced
to one tenth of that of the old system.
INTRODUCTION

time,Ferrite loaded cavities are commonly used in theproton synchrotron.
The resonant frequency of thecavity is varied by the bias current which changes thepermeability of the ferrite.
The structure of thecavity used in KEK-PS is illustrated in Fig. 1, where
the flow of the bias current is shown by the arrows on
the lines. The dotted lines with arrows show the flowof the RF current.
The operating point on the B-H
curve of the ferrite is determined by the bias current.
The cavity resonates at the frequency satisfying
the relation to/2
(wave length)/4, and the light
velocity in the ferrite is given by 1//EV.
If we
assume that the cavity is filled with the ferrite, theresonant frequency of the cavity is written asd 8 ~(t) = - (K8Ib + 8~(t))/T
dt

Consisder the case where 8Ib has the time dependenceexp(jwt), then 8~(t) must also have the same dependenceon time. Therefore
8~(w) = -K8Ib(w)/(1+jwT)

The equation shows that the phase difference between
8~(w) and 8Ib(w) is 90° for the frequencies higher than
1/T(1 ~2Hz). We assume that~ is given by the sum of
its mean value and the incremental part cused by
8Ib (w),

flo =-11-

to/EV~ = ~I

where £ is the dielectric constant of the ferrite, and~ its permeability.
The effect of the bias current .on the resonant
frequency of the cavity is not linear. But in thisanalysis, we use a linear relation between them forsmall amplitude variation in bias current. In order toexplain the relation, we use the experimental results.
When the bias current Ib is increased by 8Ib at t
= 0, the variation in ~ is approximated with thefollowing equation (Fig. 2);8~(t)wherewhere ~I is the value of ~ at bias current Ib.
Hence the resonant frequency of the cavity iswritten as- 11-to~forS ~ 0.15,

where f!I = 11/(1 0 ~) is the resonant frequency at bias
current Ib; andT ~ 0.09 sec.

T'8Ib = 100 A.

Accelerating Gap-10 12--(1-(K/~I)8Ib(w)/(1+jwT)) - 1 / 2f!I(l-(K/~I)8Ib(w)) - 1 / 2 ·((l+jwT)/{l+jwT'~ 112 ,= - K8Ib(1 - S exp(- t/T))

K ~ 1.6 ~0 1/A,

- K8Ib(w)/(1+jwT)(T' > T)

Accelerating Gap/-iol2-I

c~~~l~ C8J
C8lJ:~~JJ1Fig. 1 The structure of the RF
cavity at KEK-PS main ring.

rzBe amFig. 2 The variation in~r.

-------------------- ----Ib CAJ>< ~ C8J ~ ':t I I
il I'I' ,;---+Ferrite

I r ---·--,.,_ ___•J)

H.T.

___ ....,.

- -+___ ...,
l~--('--II-III.. t-+---lt

BiasCurrent

100

0. 5 sec-t

When w << 1/T' (i.e., less than 2Hz), the imaginary parts in the equation of ~u can be neglected, andGbwfi
Fb (j w) = (j w) 2 + 2nwoj w + w6~u = ~ 1 (1 + Kblb(w)/(2~ 1 ~b~ 0 (w)or=where~ 1 Kblb(w)/(2~ 1 )b~o(w) is the incremental part of ~0·
In the case of blb(w) << ~ 1 /K, T' ~ T and the equation of b~o(w) also valid for the frequencies higher
than 1/T.
Hence there is no phase difference between b~o(w)and blb(w) for small amplitude variation in lb. Though
we used the rough approximations in the discussion, theresult is correct for the frequency range up to 1 kHz
For large amplitude, the relation be(see Fig. 4).
tween them is not so simple as the above discussion.

whereThe relation between ~0 and Ib
The relation between the resonant frequency ~0 of
If wethe cavity and bias current Ib is not clear.
limit our analysis within a small amplitude oscillationin Ib, the relation between ~0 and Ib can be approximated by the linear function with no retardation as
mentioned in the previous section. Since the variationin Ib from 100 A to 400 A causes the change in resonant
frequency from 6 MHz to 8 MHz,In tt.'s analysis, a simple model, which isdiagrammatically shown in Fig. 3, is used. If the bias
current is ideally programmed, the signals shown in thediagram are zero. The symbol b~ is the error in frequency between applied RF and resonance of the cavity.
The estimation of the reduction of b~ is the purpose of this analysis.

~ = 4.2 x 10 4 (rad./secYA

Correction amplifierThis amplifier is the compensation network for theauto-tuning system.
The transfer function of this amplifier is written

The relation between ~E and b~
The symbol <PE means the phase difference between
the applied RF voltage of the cavity and its current.
The characteristics of the cavity are equivalent tothat of the R-L-C parallel network in the vicinity ofWhen the cavity is tuned tothe resonant frequency.
resonate at the applied RF frequency, <PE = 0. If there±
is an error in resonant frequency given by b~
11/4 rad., where Q(=~o/(2Q), <PE takes the value ofHence inR/(~ 0 L)) is the quality factor of the cavity.
linear approximation,as;jw + w2
Gcwl------Fc (j w) = jw + WJ
w2andWj

211 X 10 rad./sec,

w2

211 X 8 X 10 2 rad./sec,

Gc80 V/V.

+.

~.E = - !2 Q b~~0

The value of WJ determines the loop gain at lowerfrequency. Since loop gain at lower frequency is sufficiently large, we limit the increase in loop gain to
the value at 10 Hz. And w2 is introduced to compensate
the phase lag at higher frequency.
The former system had complicated compensation
networks at high frequencies, and individual adjustments of the parameters were not possible.

where b~ = ~ -~0·Phase detector
The phase detector detects <PE' and gives the errorvoltage bVP with sensitivity Gp;
bVBias-current power supply
Readjustment of the bias-current power supply wasnecessary to make the system most suitable condition.
The transfer function of the power supply can be approximated byIsolation
Amplifier

0.5,

n211 x 2.1 x 10 4 rad./sec.

Before the adjustment, the parameters of thispower supply were n ~ 0.3 and wo ~ 211 x 4k rad./sec.
The frequency response had a maximum at 4 kHz anddecreased at the frequencies higher than 4 kHz. Since
the phase characteristics changed abruptly near thepeak frequency, the phase lag of the former power
supply was difficult to be compensated with the othercircuits.

THE ANALYSIS OF THE AUTO-TUNING SYSTEM

whereGb = 100 A/V,

andp

where G = 3 V/rad.
pThe other elements
The isolation amplifier is made to have a frequency
response of 100 kHz with unity gain. Our analysis islimited within the frequency range of 20 kHz, hence its
effect can be neglected from this analysis.
The variable gain amplifier has also good frequency response and its effect is simply described by aIn normal operation, it is adjusted
consta~t GT(< 1).
as GT - O.li.
The linear-gate circuit has the gain of 0.5.

Correction Bias Current Bias Current
P. S.
ProgramAmplifier

OPEN LOOP GAIN AND ERROR REJECTION(l) The open loop gain of the system F.(jw) is given by
the multiplications of the transfer functions of theelements connected in the loop.

1----1· Cavity

j w + w2
F (. ) = - G
0 (j w+w1) ( (j w) 2+2nwoj w+wt)
JwcpE
Variable Gain
AmplifierPhase
Detector

f:::.!J,

whereFrequency Error4G0 = 1IQ
W2 Gc Gb Gp GT x 1 • OS . x 10~u ~( ra d/ sec ) 2Fig. 3 The block diagram of the auto-tuning system.
Q = 40, and ~ 0 = 211 x 6 ~ 211 x 8 Mrad/sec~
101,We compared the above equation with the measured
values in Fig. 4. The good agreement in gain curve
between measured and calculated values is obtained.
The difference in phase at the frequencies higher than
1 kHz is caused by the characteristics of the ferrite.
The FFT processor is used in t~e measurement withapplying white noise into the loop 3 •quency range of the feedback is 400kHz).

WI(i.e., "' 4 ·CONCLUSION
The improvements are shown in Figs. 5 and 6.
The phase errors ~E of the three cavities innormal operation are reduced to nearly 2° in RF phaseangle which is , one tenth of the error existed in theold system.
The analysis is elementary, but it gives a guidein diagnosis of the auto-tuning systems.

(2) If we express the error in resonant frequency without feedback by 6Q(off) and the error with feedback by
6Q(on), then their ratio is given by
6Q(on)
6Q(off)xACKNOWLEDGEMENTS

1 + F(jw)

For the frequency range whereequation can be approximated by

The author wishes to express his appreciation toAssist. Professor K. Muto, for his help and useful
discussions.
The author also expresses his appreciation toProfessors T. Kamei and M. Kondo for their interest on
this work.

ljwl < wL, the above.!4~
6Q(on) _6Q(off) - Gowz

REFERENCES
The ratio has the minimum value of 1/400 at thefrequencies less than WI• It approaches to 1 at thefrequency near 400 x WI• Therefore the effective fre-1)
2)
3)

Gain [dB]

Q.A. Kerns et al., IEEE NS-12, 185, 1965.
T. Kamei et al., SJC-A-68-4.
YHP 3582A manual and application note 245-J3.

Phase [deg]r-----------------------------------------------,-40"
6040
Fig. 4 The open loop gain of thesystem #3 at 6MHz.

20•0measured

x measured

gainphase')(Fig. 5 AVp signals of the 4 cavities. The system marked #3 isalready improved.

-20~~~~--~--~~~~~~~~--~--~~~~

10100

lk
Frequency [Hz]

IOk

lOOk

CavityNumber
#0

#1

#2

#3

102

Fig. 6 AVp signals. The system
marked #0 is not improved.
(Note the difference inthe vertical scales.)