Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
Rise time analysis of pulsed klystron-modulator for e$ciency
improvement of linear colliders
J.S. Oh!,*, M.H. Cho!, W. Namkung!, K.H. Chung", T. Shintake#, H. Matsumoto#
!Pohang Accelerator Laboratory, POSTECH, San 31 Hyoja-dong, Nam-ku, Pohang, South Korea
"Department of Nuclear Engineering, Seoul National University, Seoul, South Korea
#National Laboratory for High Energy Physics, Tsukuba, Japan
Received 20 April 1999; received in revised form 21 August 1999; accepted 18 October 1999
Abstract
In linear accelerators, the periods during the rise and fall of a klystron-modulator pulse cannot be used to generate RF
power. Thus, these periods need to be minimized to get high e$ciency, especially in large-scale machines. In this paper,
we present a simpli"ed and generalized voltage rise time function of a pulsed modulator with a high-power klystron load
using the equivalent circuit analysis method. The optimum pulse waveform is generated when this pulsed power system is
tuned with a damping factor of &0.85. The normalized rise time chart presented in this paper allows one to predict the
rise time and pulse shape of the pulsed power system in general. The results can be summarized as follows: The large
distributed capacitance in the pulse tank and operating parameters, < ]¹ , where < is load voltage and ¹ is the pulse
4
1
4
1
width, are the main factors determining the pulse rise time in the high-power RF system. With an RF pulse compression
scheme, up to $3% ripple of the modulator voltage is allowed without serious loss of compressor e$ciency, which
allows the modulator e$ciency to be improved as well. The wiring inductance should be minimized to get the fastest rise
time. ( 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
An e#e-linear collider at 500 GeV (center of
mass energy) has been proposed as a future accelerator. It is a large-scale machine. For instance, the
C-band scheme requires more than 4000 units of
klystrons and matching modulators [1]. The initial
design value of the wall plug power for the whole
RF system in the C-band scheme was 150 MW.
However, the previous e$ciency analysis of the "rst
C-band system shows that the present system re-
* Corresponding author. Tel.: #82-562-279-1143; fax: #82562-279-1199.
E-mail address: jsoh@postech.ac.kr (J.S. Oh).
quires 270 MW AC power due to the conservative design [2]. Therefore, the e$ciency improvement of the pulsed power system is a key concern
to reduce operational cost in future large-scale
accelerators.
The pulse tail is a result of stored energy decay in
the distributed elements of the pulse generator. In
a short pulse generator, most of this energy comes
during the rising period. Therefore, the rise time
of the modulator pulse has to be minimized to
achieve high e$ciency. In this paper, we derive
a simple and generalized form of the pulse rise time
characteristics of a modulator with a high-power
klystron load. As a result, the practical limits on
the rise time and reasonably achievable values are
described.
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 1 7 1 - 7
224
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
An RF pulse compressor such as SLED increases
the peak RF power for a "xed number of joules per
pulse [3}5]. The RF phase ripple and the energy
storage e$ciency of the pulse compressor are analyzed in detail, and the voltage ripple requirement is
revaluated. The rise time analysis of the C-band
system is included as an example case.
2. Optimum pulse waveform
In a conventional modulator, a line-type PFN
(pulse forming network) generates a high-voltage
pulse through a step-up pulse transformer. This
pulsed power system can be modeled by the circuit
shown in Fig. 1. The parameters in the "gure are
referenced to the secondary side of the pulse transformer. We can neglect the e!ect of the shunt resistance R and the shunt inductance ¸ during the
%
P
short rise time in the leading edge analysis.
The analytical solution of the load voltage < (t)
L
is described elsewhere [6}8]. The normalized load
voltage y(t) de"ned as
A
< (t) 1#m
y(t)" L
<
m
G
is given by
B
(1)
CA B
y(t)"1!exp(!at)
D
a
sinh(kt)#cosh(kt)
k
Fig. 2. Normalized rising pulse waveform as a function of
damping factor, p.
(p'1)
(2)
CA B
y(t)"1!exp(!at)
D
a
sinh(ut)#cosh(ut)
u
(p(1)
(3)
where
2pp
a"
,
q
Fig. 1. Equivalent circuit of a step-up pulse system. < is the
G
PFN voltage. S is the main switch. R is the PFN impedance.
G
G
¸ is the wiring inductance. ¸ is the leakage inductance of the
W
L
pulse transformer. R is the shunt resistance of the pulse trans%
former. ¸ is the shunt inductance of the pulse transformer.
P
C is the distributed capacitance of the pulse transformer. C is
D
L
the distributed capacitance of the load including the klystron.
R is the load impedance.
L
2pJp2!1
2pJ1!p2
k"
, u"
q
q
A
S
B
m
1
1
p"
cm# , q"2p
J¸ C
T T
m#1
c
2Jm(m#1)
S
¸
R
Z
T,
m" L , c" T , Z "
T
C
R
R
T
G
L
C "C #C .
T
D
L
The expanded view of the leading edge of y(t)
near the #at top is shown in Fig. 2 as a function of
the damping factor p. In this "gure, the normalized
time S is de"ned by S"t/q. The pulse shape and
the rise time are sensitive to the damping factor
p near the #at top. For a given value of ¸ C , small
T T
p gives fast rise time but generates large overshoot.
From Eq. (3), the maximum overshoot is located at
ut"n, and the corresponding peak value is given
by
AB
y
L "¸ #¸
T
W
L
A
A
p
a
"1#exp ! p
u
u
B
B
p
"1#exp !
p .
J1!p2
(4)
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
225
In general, a pulse #at-top with less than $0.5%
ripple is required to produce high-power RF with
small phase variations in high-energy linear accelerators. The damping factor p has to be larger than
0.86 to limit the overshoot to less than 0.5% during
the #at-top. So the optimum pulse waveform is
generated when the pulsed power system is tuned
with a damping factor of 0.86.
3. Rise time of the pulse
Fig. 3 shows the normalized rise time S as
3
a function of p for three amplitude fractions:
10}90% (S ), 0 to 95% (S ), and 0 to 99% (S ).
31
32
33
The rise time S is the standard de"nition that
31
characterizes the leading edge. However, the rise
time ¹ given by S is more suitable to estimate
3
33
the rise time e$ciency for a given #at-top width
¹ :
3&
¹
3& .
rise time e$ciency"
¹ #¹
3
3&
Fig. 3. Normalized rise time as a function of p.
(5)
The PFN impedance is normally made equal to
the load resistance to get the maximum energy
transfer e$ciency [6]. For a matched PFN (m"1),
the rise time ¹ is given by
3
¹ "S (p)q"J2pS (p)J¸ C .
3
33
33
T T
(6)
The klystron load is a nonlinear device since the
klystron current I is given by I "k<1.5 where k is
k
k
k
the perveance and < is the klystron voltage. This
k
means that the klystron impedance is continuously
changing during the pulse rise time. Fig. 4 shows
the rising edge of the waveform for a speci"c case,
p"0.9. These results were obtained using the
PSPICEt circuit simulation code. The klystron
voltage is rising faster than the resistor voltage, but
it approaches the #at-top more slowly with less
overshoot. As a result, the rise time for the klystron
load is longer. With the klystron load, a damping
factor of 0.8 yields the fastest rise time for less than
0.5% overshoot. We denote this optimum value by
p .
015
Fig. 5 shows the correction factor g(p) of the rise
time with a klystron load, that is, the ratio of the
Fig. 4. Rising waveform for a resistor and a klystron load.
rise time with a klystron load to the rise time with
a resistor equal to the klystron impedance at full
voltage. These values of g(p) were found by simulating a range of p values with the PSPICE code. The
correction factor is rather slow varying and changes less than 20% over the practical range of p.
A third-order polynomial "t to the correction factor yields
g(p)"!1.918p3#1.057p2#3.370p!1.331. (7)
The rise time with a klystron load, ¹ , is thus
k
¹ "g(p)¹ "J2ng(p)S (p)J¸ C .
k
r
33
T T
(8)
226
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
Fig. 5. Rise time correction factor for a klystron load.
Fig. 7. Example of pulse rise time versus damping factor.
Fig. 6. Equal rise time curves and damping factors.
Equal rise time curves as a function of total shunt
capacitance C and total series inductance ¸ are
T
T
plotted in Fig. 6 using (8). There are two constant
lines for each p in Fig. 6 because these are two
possible values of c for each value of p:
A B
1
1
p"
c# .
c
2J2
(9)
Since the discrete PFN is not a perfect transmission
line, it generates ringing in the leading edge. Therefore, the relatively small series inductance in a low
c system does not help much to reduce the oscillation amplitude. Also, in a high-voltage pulser, it is
hard to acheive low c due to the large leakage and
wiring inductance associated with the high step-up
ratio. For these reasons, we ignore the lower c solutions in Fig. 6 which are the lines with smaller
slopes.
In order to generalize the rise time curves, shunt
capacitance and series inductance are normalized
using the load resistance R . The range of the
L
horizontal and vertical axes correspond to the
practical range of series inductances (300}1200 lH),
shunt capacitances (100}300 pF), and load resistances (700}1,400 )). For given values of R , C ,
L T
and ¸ , one can easily evaluate the rise time of
T
a pulsed power system and the magnitude of overshoot. For example, system &A' in Fig. 6 has a rise
time of 1.5 ls with p"0.9. From this point, one
could reduce the rise time to 1.1 ls by decreasing
the series inductance by about 30% (&B'). However,
making the inductance even smaller to shorten the
rise time (&C') yields too large of an overshoot
(greater than 0.5%).
Fig. 7 shows typical pulse waveforms for the
C-band modulator. The waveform with the slower
rise time (1.9 ls) was measured initially. The rise
time was reduced to 1.2 ls by reducing the wiring
inductance. This also yielded a faster fall time.
4. Minimum rise time
The equivalent circuit parameters of the pulse
transformer are related to geometrical and material
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
parameters. The constant gradient or tapered
bi"lar secondary winding technique is widely
used in high-voltage pulse transformers [9,10].
The leakage inductance, distributed capacitance,
and shunt inductance of the secondary side are
given by
pN2*;
#
¸ " S
L
l
#
[nH]
A B
n!1 2; l
##
C "0.0885 e
D
3
n
*
N2A
¸ "4pk S .
P
3 l
.
(10)
[pF]
(11)
[nH]
(12)
where N is the number of turns in the secondary
4
coil, * is the insulation distance between layers in
cm, ; is the average circumference of the layers
#
in cm, l is the winding length in cm, e is the
#
3
relative dielectric constant of the insulating material between the layers, n is the step-up ratio, k is
3
the relative pulse permeability of the core, A
.
is the cross-sectional area of the core in cm2,
and l is the mean magnetic path length of the
.
core in cm. The pulse magnetization of the core is
given by
*BN A "108< ¹ ,
4 .
4 1
(13)
where *B is the average increment of the magnetic
#ux density of the core in gauss, < is the load
4
voltage in volts, and ¹ is the pulse width in sec1
onds. Using Eqs. (10), (11) and (13), the transformer
time constant g is given by
A BA B
;
#
g"J¸ C "1.67]10~3Je
L D
3 A
.
A B
1
](< ¹ )
.
4 P *B
n!1
n
(14)
The time constant is proportional to operating
parameters, < ]¹ , geometric parameters, ; ]A~1,
4
1
#
.
and material parameters, e0.5]*B~1. For a given
r
c, total series inductance is given by
¸ "¸ #¸ "(C #C ) (cR )2.
T
W
L
D
L
L
(15)
227
Using Eqs. (14) and (15), ¸ and C are given by
L
D
1
¸ "
[(a2C !¸ )
L
L
W
2
AB
#J(a2C !¸ )2#4a2g2]
L
W
1
C "
[!(a2C !¸ )
D
L
W
2a2
(16)
A B
(17)
#J(a2C !¸ )2#4a2g2]
L
W
where a"cR . Using (15), (8) can be written as
L
¹ "J2n g(p)S (p)cR (C #C ).
33
L L
D
k
Substituting Eq. (17) yields
(18)
C
p
¸
¹ "
g(p)S (p) aC # W
k J2
33
L
a
SA
#
B
D
2
¸
aC ! W #4g2 .
L
a
(18a)
With optimum parameters, p ("0.8) and
015
c ("1.66), Eq. (18) can be simpli"ed as follows:
015
¹ "4.77R (C #C ).
(19)
k
L L
D
With "xed C and a, C "0 yields minimum rise
L
D
time
¹ "4.77R C .
(20)
.*/
L L
The actual rise time is always larger than the value
given by Eq. (20). This equation gives a useful
measure of the minimum rise time.
In general, to get a fast rising pulse for given
values of a and C , ¸ and g should be minimized.
L W
This follows from Eq. (18a). The primary wiring
inductance is practically limited to &1.0 lH due
to the dimensions and layout of the high-voltage
components. From Eq. (14), g can be reduced by
increasing core cross-sectional area because ; is
#
proportional to A0.5. Using Eqs. (12) and (13), the
.
pulse droop D is given by
A BA B
R ¹
R
10~7 *B2
L (A l ).
D" L P "
(21)
..
2¸
k
<2¹
8p
P
S P
3
A large cross-sectional area produces a large pulse
droop so it has to be limited by the droop requirement.
228
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
Table 1
Rise time parameters of the C-band modulator
Load parameters
Matching parameters
Rise time factors
Insulator parameter
Transformer parameters
Minimum rise time
Fig. 8. Distributed capacitance and potential lines in the pulse
tank with E3746 C-band klystron.
To estimate the correct rise time, it is very important to have a realistic measure of the load
capacitance C . Fig. 8 shows the results of the
L
distributed capacitance obtained using the DENKAI2 code [11] for the pulse tank and klystron
assembly of the C-band system [12]. The right half
of the "gure shows electrostatic potential lines in
R "1104 ), C "120 pF,
L
L
< "350 kV, ¹ "3.5 ls
S
P
m"1, p "0.8, c "1.66
.*/
.*/
S (p )"0.61, g(p )"1.06
r3 .*/
.*/
e "2.2
r
n"15, *B"1.8 T
¹ "0.63 ls
.*/
the region of the klystron tube and between the
coils of the pulse transformer in the tank. The
calculated capacitance of each section for the
C-band system is shown in the left half of the "gure.
The total distributed load capacitance C is
L
115 pF. If one adds the capacitance between the
corona ring and other structures, which are not
included in this estimation, it becomes approximately 120 pF.
Table 1 lists the rise time parameters for the
C-band klystron-modulator. It is assumed that the
transformer is immersed in the insulation oil
(e "2.2). Due to the distributed load capacitance
3
of 120 pF, the rise time is larger than
¹
("0.63 ls) given by Eq. (20). Table 2 shows
.*/
the calculated result of the rise time and corresponding transformer parameters. It is possible to
get the rise time less than 1 ls with a primary
wiring inductance of 2 lH. A rise time e$ciency of
about 75% with a #at-top of 2.5 ls is achievable
with 1 lH wiring inductance. If the wiring inductance is relatively large, smaller leakage inductance
and larger distributed capacitance are required (see
Table 2
Rise time evaluation of the C-band modulator
Core dimension
g [ls]
¸ [lH]
W
¸ [lH]
L
C [pF]
D
¹ [ls]
k
4 cores (5]5 cm2)
A "100 cm2
.
; "72 cm
C
4 cores (7]7 cm2)
A "196 cm2
.
; "92 cm
C
0.11
1.0]n2
314
41
0.85
0.07
2.0]n2
185
69
1.00
1.0]n2
251
22
0.75
2.0]n2
114
48
0.89
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
Eqs. (16) and (17)). From Eqs. (10) and (11), this can
be done by increasing the winding length although
this increases C . It means that the transformer
D
size has to be increased to meet the requirement.
However, the rise time cannot be improved signi"cantly even with a core twice as large. Therefore,
the rise time is determined by the distributed load
capacitance and the transformer time constant
which is proportional to the operating parameters,
< ]¹ .
4
1
5. Relaxation of ripple requirement
The tight ripple speci"cation of the #at-top can
be relaxed if a pulse compressor is used. This is
a common technique in electron linear accelerators
to increase peak power. For purpose here, it can be
thought of as a narrow band RF storage device.
The klystron output RF can be described by
v(t)"A(t)sin(2pf t#*/(t)),
(22)
#
where f is the operating frequency, */(t) is the
#
phase ripple due to the voltage ripple, and A(t)2 is
proportional to the instantaneous RF power. The
maximum RF phase ripple $*/ of the klystron
.
tube is given by
A B
A BA B
*/
¸
e<
*<
. "! 0 (c2!1)~1.5
4
. , (23)
%
2p
j
m c2
<
0
%
4
where c "1#eV /(m c2), ¸ is the tube length
%
4 %
0
from input cavity to output cavity of the klystron,
j is the free space wavelength of the klystron
0
operating wave, m is the electron rest mass, and
%
$*< is the maximum allowable #at-top ripple.
.
Then, Eq. (22) can be written as
v(t)
u(t)"
"sin(2pf t#*/ sin(2pf t))
#
.
3
A(t)
(24)
where f is the fundamental ripple frequency. Using
3
the following relations,
sin(2pf t#*/ sin(2pf t))
#
.
3
"sin(2pf t)cos(*/ sin(2pf t))
c
.
3
#cos(2pf t)sin(*/ sin(2pf t))
#
.
3
229
cos(*/ sin(2pf t))
.
3
"J (*/ )#2J (*/ )sin(4pf t)
0
.
2
.
3
#2J (*/ )sin(6pf t)#2
4
.
3
sin(*/ sin(2pf t))
.
3
"2J (*/ )sin(2pf t)
1
.
3
#2J (*/ )sin(6pf t)#2.
3
.
3
Eq. (24) can be written as
u(t)"J (*/ )sin(2pf t)
0
.
#
#J (*/ )[sin(2p(f #f )t)
1
.
#
3
#sin(2p(f !f )t)]
c
3
#J (*/ )[sin(2p(f #2f )t)
2
.
#
3
#sin(2p(f !2f )t)]#2
(25)
#
3
where J is the Bessel function of the "rst kind with
n
order n. The #at-top oscillation is caused by the
wave re#ections between the primary and the secondary transformer coils [6]. The fundamental frequency of this oscillation is usually larger than
1 MHz. The pulse compressor has a high unloaded
Q of &105, a loaded Q of &104, and a narrow
o
L
bandwidth of less than a few 100 kHz, which means
that the energy in the side band terms in Eq. (25)
cannot be stored in the pulse compressor. Therefore the reduction of the storage e$ciency of the
pulse compressor is simply given by J (*/ )2.
0
.
Typical parameters of the C-band system are summarized in Table 3. As shown in Table 3, even
though there is $3% voltage ripple and $8.63
phase ripple, an energy storage e$ciency of 99% is
achievable. Of course, the ripple after the phase #ip
that is used to discharge the pulse compressor
should be less than 1% because the beam acceleration is done during this period. However, it is not
di$cult to adjust the #at-top near the pulse tail
using tunable inductors in the PFN.
In the matched PFN, the maximum possible
overshoot is 4.3% which occurs for p"0.71 (i.e.,
c"1). Thus one can reduce the rise time without
serious degradation of the energy e$ciency by using a pulse compressor and having the pulsed
power system tuned to as small a damping factor as
possible. Table 4 shows the rise time assuming
a maximum ripple less than $3%, and using
230
J.S. Oh et al. / Nuclear Instruments and Methods in Physics Research A 443 (2000) 223}230
Table 3
RF phase ripple and pulse compressor e$ciency
Operating frequency, f
#
Cavity distance, ¸
0
Operating voltage, <
4
Voltage ripple, $*<
Phase ripple, $*/
Pulse compressor e$ciency, J (*/ )2
0
.
5,712 MHz
30 cm
350 kV
$3%
$8.63
0.99
< ]¹ , are the main factors determining the pulse
4
1
rise time in the high-power RF system. With the
pulse compression scheme, up to $3% voltage
ripple is allowable without serious loss in e$ciency,
which allows the modulator e$ciency to be
improved as well.
References
Table 4
Rise time with a pulse compressor (p "0.75)
.*/
Core dimension
g [ls]
¸ [lH]
W
¸ [lH]
L
C [pF]
D
¹ [ls]
k
4 cores (5]5 cm2)
A "100 cm2, ; "72 cm
.
C
0.11
1.0]n2
208
61
0.65
2.0]n2
111
145
0.84
S and the parameters, p "0.75, c "1.4,
32
.*/
.*/
S (p )"0.53, g(p )"0.98. With the looser
32 .*/
.*/
ripple constraints, the rise time decreases from 1.0
to 0.84 ls for 2.0 lH wiring inductance. Thus, for
a #at-top of 2.5 ls, a rise time e$ciency of about
80% is achievable with 1 lH wiring inductance.
6. Conclusion
The generalized rise time chart presented in this
paper is useful for predicting the rise time and pulse
shape of a pulsed power system. The minimum rise
time and corresponding circuit parameters can be
evaluated using the simpli"ed equations with reasonable accuracy. The large distributed capacitance
in the pulse tank and the operating parameters,
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