New method of analyzing wave processes in pulse generators based on lines with distributed parameters
A new method of theoretical analysis of wave processes in high-current pulse generators through the relations between integral values reflecting regularities of energy transfer in ideal lines with distributed parameters is described. The use of the method developed considerably simplifies the proced...
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irk-123456789-784102015-03-17T03:02:03Z New method of analyzing wave processes in pulse generators based on lines with distributed parameters Gordeev, V.S. A new method of theoretical analysis of wave processes in high-current pulse generators through the relations between integral values reflecting regularities of energy transfer in ideal lines with distributed parameters is described. The use of the method developed considerably simplifies the procedure of searching for an optimal - from the point of view of getting maximal efficiency – relation of impedances for pulse facilities on stepped lines including those with arbitrary number of cascades. High efficiency of the method is demonstrated by several examples. 2001 Article New method of analyzing wave processes in pulse generators based on lines with distributed parameters / V.S. Gordeev // Вопросы атомной науки и техники. — 2001. — № 5. — С. 39-42. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS numbers: 84.30.Ng; 84.70.+p http://dspace.nbuv.gov.ua/handle/123456789/78410 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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A new method of theoretical analysis of wave processes in high-current pulse generators through the relations between integral values reflecting regularities of energy transfer in ideal lines with distributed parameters is described. The use of the method developed considerably simplifies the procedure of searching for an optimal - from the point of view of getting maximal efficiency – relation of impedances for pulse facilities on stepped lines including those with arbitrary number of cascades. High efficiency of the method is demonstrated by several examples. |
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Gordeev, V.S. |
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Gordeev, V.S. New method of analyzing wave processes in pulse generators based on lines with distributed parameters Вопросы атомной науки и техники |
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Gordeev, V.S. |
| author_sort |
Gordeev, V.S. |
| title |
New method of analyzing wave processes in pulse generators based on lines with distributed parameters |
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New method of analyzing wave processes in pulse generators based on lines with distributed parameters |
| title_full |
New method of analyzing wave processes in pulse generators based on lines with distributed parameters |
| title_fullStr |
New method of analyzing wave processes in pulse generators based on lines with distributed parameters |
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New method of analyzing wave processes in pulse generators based on lines with distributed parameters |
| title_sort |
new method of analyzing wave processes in pulse generators based on lines with distributed parameters |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/78410 |
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New method of analyzing wave processes in pulse generators based on lines with distributed parameters / V.S. Gordeev // Вопросы атомной науки и техники. — 2001. — № 5. — С. 39-42. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT gordeevvs newmethodofanalyzingwaveprocessesinpulsegeneratorsbasedonlineswithdistributedparameters |
| first_indexed |
2025-07-06T02:31:03Z |
| last_indexed |
2025-07-06T02:31:03Z |
| _version_ |
1836862997592539136 |
| fulltext |
NEW METHOD OF ANALYZING WAVE PROCESSES
IN PULSE GENERATORS BASED ON LINES
WITH DISTRIBUTED PARAMETERS
V.S. Gordeev
Russian Federal Nuclear Center – All-Russia Scientific Research
Institute of Experimental Physics (RFNC-VNIIEF)
607188, Sarov, Nizhni Novgorod region, Mira Prospekt 37, Russia
E-mail: gordeev@expd.vniief.ru
A new method of theoretical analysis of wave processes in high-current pulse generators through the relations be-
tween integral values reflecting regularities of energy transfer in ideal lines with distributed parameters is described.
The use of the method developed considerably simplifies the procedure of searching for an optimal - from the point
of view of getting maximal efficiency – relation of impedances for pulse facilities on stepped lines including those
with arbitrary number of cascades. High efficiency of the method is demonstrated by several examples.
PACS numbers: 84.30.Ng; 84.70.+p
1 INTRODUCTION
A search for new circuits for multi-cascade pulse
generators is performed, as a rule, by two stages. Firstly,
a generator circuit is selected. Then transient processes
are analyzed basing on the subsequent consideration of
voltage or current waves propagating in lines. As a re-
sult, a mathematical expression for the voltage and load
current, involving impedances of all cascades can be ob-
tained. Impedances are optimized to get a maximum ef-
ficiency, voltage or current. A complexity of traditional
approach is conditioned by the necessity of accounting a
large number of wave limits in the circuits before such
an analysis. As a rule, these are circuits with a number
of cascades not greater than 2 or 3.
In the course of fundamental development of genera-
tors based on stepped lines [1-4] the author developed a
new method of search for an optimal relation of
impedances to reach 100% efficiency. Below, there are
contemplated some regularities of energy transfer in
generators with transmission lines the account of which
in some cases significantly simplifies a search for opti-
mal impedances. The method application is illustrated
on the example of generators with different techniques
of energy storage. All generators are formed by the ho-
mogeneous lines (cascades) of the equal electrical
length 0T . For each case there are defined conditions
when for idealized models the generator has 100% effi-
ciency at formation of a squared pulse of 02T duration
on the matched load.
2 SOME REGULARITIES OF ENERGY
TRANSMISSION IN IDEAL LINES
From Maxwell equations one can get two integral
relationships. The first one determines a relation be-
tween time integral of voltage for arbitrary closed cir-
cuit L and a change of magnetic induction flux through
an arbitrary surface S , which is supported by this cir-
cuit: )()0(
0
tФФUdt msms
t
−=∫ , where ∫=
L
ldЕU
. The sec-
ond relationship defines a relation of time integral of
conduction current flowing through a closed surface ,S
with a charge change in the volume V limited by this
surface: ( ) ( )tqqIdt
t
−=∫
0
0 .
Provided the energy is extracted out from the vol-
ume V by the time moment 0t , the formulas have a sim-
pler form:
=
=
∫
∫
0
0
0
0
).0(
),0(
t
m
t
qIdt
ФUdt (1)
(2)
Following equations (1), (2) is a necessary and suffi-
cient condition for a complete extraction of energy. It is
convenient to select the time moment as a beginning of
the integration interval 0=t , when, as a result of com-
mutation, in the generator electromagnetic waves ap-
pear. In this case for generators with a capacitive energy
storage the right part of equation (1) is equal to zero,
and for generators with an inductive energy storage - the
right part of equation (2) equals to zero.
The formulas obtained do not find a wide applica-
tion when solving electrotechnical problems, as )(tU
and )(tI in most cases change by unknown law that ma-
kes impossible their integration. The situation is differ-
ent for stepped lines, when pulses are in the form of
squared steps, and time integration amounts to summa-
tion of constant magnitude products. The time integral
of voltage equaling to zero, as a necessary condition for
a complete energy extraction from capacitive genera-
tors, was mentioned earlier in paper [5].
3 CAPACITIVE GENERATOR
Let us consider as an example a circuit of a genera-
tor (Fig. 1) formed from 2≥n cascades with
impedances nZZZ ,...,, 21 . Cascades with impedances
nn ZZ ,1− are charged from the external source up to the
voltage 0U . After charging is finished ( 0=t ), a switch
1S is closed. The load 1ZZ L = is connected to the gen-
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 39-42.
39
erator by a switch 2S ( 0nTt = ) with delay by 02T with
regard to arrival of the first wave from switch 1S to it.
As the voltage along the circuit L (Fig. 1a) differs
from zero only in the cross-section AB , equation (1)
takes the following form:
∫ =
0
0
0
t
ABdtU . (3)
As cascades with numbers )2(1 −÷ n are not initial-
ly charged, for the surface S (Fig. 1a) intersecting a
grounding electrode near the load and at the juncture of
i and 1+i cascades, equation (2) is converted into the
form:
∫ =
0
0
0
t
Idt . (4)
a
b
Fig. 1. Capacitive generator circuit.
Let the wave arrives to the cross-section AB at the
time moment it . The voltage and current in this cross-
section should differ from zero in the time interval
)4( 0Ttt ii +÷ . If one designates the amplitude of voltage
and current in the time intervals )2( 0Ttt ii +÷ and
)4()2( 00 TtTt ii +÷+ as 11 , ii IU and 22, ii IU , equations
(3), (4) takes the form 021 =+ ii UU , =+ 21 ii II
1/ ZUI LL = . Energy transmitted through the cross-sec-
tion AB should be equal to the energy absorbed in the
load: 01
2
02211 2)/(2)( ТZUТIUIU Liiii ⋅=⋅+ . When solv-
ing jointly equations with the use of relations
iii ZIU =11 / and )/(2 11111,1 iiii ZZZUU += −−− we ob-
tain:
1
2
11
2
1 ///2 ZUZUUZU LLiii =+ . (5)
After subtraction from (5) of the similar equation for
the voltage 2,1−iU with regard to 112UU L −= let us find
the final expression for relation of impedances of cas-
cades with numbers 2−≤ ni :
)]1(/[2 1 +⋅= iiZZ i (6)
In this case the voltage on the load is
)1()]/(2[2 1,2
3
1
11,2 −−=+−= −
−
=
+− ∏ nUZZZUU n
n
j
jjjnL . (7)
Now let us line a circuit (Fig. 1b) passing along the
grounding and high-voltage electrodes of a line with
impedance nZ through switch 1S and junction of lines
with impedances 1, −nn ZZ . When closing the switch 1S
the voltage along the circuit differs from zero only at the
junction of cascades with impedances nZ , 1−nZ . One
can demonstrate that in the time interval 00 T÷ the volt-
age is 0U and in the interval 00 3TT ÷ -
{ ⋅+++−−= −−−−−− )/(2[)(/)( 2121110 nnnnnnnn ZZZZZZZZUU
)]}( 1 nn ZZ +− . Further the voltage in this cross-section
should be equal to zero, for, otherwise, the residual en-
ergy will not have a chance to arrive at the load by the
time moment 0)2( Tnt += . From the condition of equa-
tion to zero of time integral of voltage in this cross-sec-
tion (1) we obtain:
+++− −−−− 1111 /(2[)/()( nnnnnnn ZZZZZZZ
2/1)])( 12 =++ −− nnn ZZZ . (8)
Through the left bound into the volume limited by
the surface S (Fig. 1b) in the time interval 00 3TT ÷
there flows out current )2/(0 nZU , and through the left
bound in the time interval 00 )2( TnnT +÷ there flows
out current 1/ ZUL . The voltage on the load can be de-
termined with regard to (7):
)/()1( 2120 −−− +−= nnnL ZZZnUU . As for the case under
study =)0(q 100 / −nZТU , equation (4) is transformed
as:
)](/[)1(2/1/1 21121 −−−− +⋅⋅−−= nnnnn ZZZZnZZ . (9)
Solving equations (8), (9) with regard to (6) we find:
)]1(/[2 1 += nnZZn , ])1/[(2 11 nnZZn −=− .
In ideal case such a generator possesses 100% effi-
ciency and forms a rectangular voltage pulse of 02T du-
ration and amplitude 2/0nUU L = on the matched load.
Addition of each supplementary cascade to the genera-
tor raises the voltage by 2/0U .
4 INDUCTIVE GENERATOR
Let us consider a stepped forming line (SFL, Fig. 2)
constituted by n subsequently connected cascades. In
the initially closed circuit composed by electrodes of
SFL and current opening switches 1S and 2S under the
action of the external source the current is created 0I
and magnetic energy is stored in the SFL. At the time
moment 0=t the opening switch 1S disconnects the cur-
rent source, and the load is connected to the stepped
forming line at 2S operation at the time moment 0nTt = .
Fig. 2. Inductive generator circuit.
For the closed surface S covering a part of one of
electrodes of SFL from the output (point A ) to the
juncture of i and 1+i cascades (point B ), according to
(2) for circuits with an inductive energy storage
∫ =
0
0
0
t
Idt . As at 0≥t 1S is opened, the value I repre-
sents the current at the juncture of i and 1+i cascades.
Before the current arrival from 1S the current BI equals
0I . At the time moment 0iTt = the current becomes
40
equal ⋅− iI 21{0 })]/([
1
2
11∏
+
=
−− +
i
i
iii ZZZ and remains con-
stant in the time interval 00 )2( TiiT +÷ . At 0)2( Tit +>
the current 0=BI . Equating the BI current time inte-
gral to zero we obtain 0)2( Tint −−= or
( )[ ]∏
+
=
+−
−− ⋅+=+
1
2
)1(
11 2)2(/
i
j
i
jjj iZZZ .
The given equation is reduced to the form:
)]1(/[2/ 1 += iiZZi
. (10)
Provided that (10) is adhered, on the matched load
there is formed a rectangular current pulse of the ampli-
tude 2/0nI and duration 02T , during this pulse the en-
ergy is fully delivered to the load.
5 INDUCTIVE-CAPACITIVE GENERATOR
The generator (Fig. 3) contains 2≥n cascades with
impedances nZZZ ,....,, 21 . The magnetic energy 0LW is
stored in all cascades under the action of the 0I current,
formed by the external source. Simultaneously, the elec-
tric energy 0CW is stored in 1−n and n cascades
charged up to the voltage 0U from another source. At
the time moment 0=t the switch 1S is closed and
opening switch 2S connects the load at the time mo-
ment 0)2( Tnt −= . Switch 3S must close the grounding
electrode before the first wave arrives from 1S .
Fig. 3. Inductive-capacitive generator circuit.
Applying the integral equation (1) for the circuit
passing through a short-circuited electrode of stepped
lines, supposing that the voltage on the load is LU we
obtain:
0
1
000 2/)(2/)( ТZIТLIU
n
i
iL ∑
=
⋅== , (11)
where L is a total inductance of stepped line. The ener-
gy transmitted to the load with regard to (11) is
∑
=
==
n
i
iLLLL ZTIITIUW
1
000 5.02 . (12)
As 00 CLL WWW += we get:
( ) ( )λ+=+= 15.0/15.0/ 000 LCL WWII , (13)
where the factor 00 / LC WW=λ .
Let us now consider the surface S (Fig. 3). Current
flowing through the surface S at the point A in the
time 0)2(0 Tin −−÷ is equal to 0I . After the arrival of
the voltage wave of iU amplitude at the i cascade from
1S at the time moment 0)2( Tint −−= the current in-
creases up to ii ZUI /0 + and remains constant during
the time interval 02T . Beginning with the time moment
0)( Tint −= the current in this cross-section must be
equal to zero. The current flowing out through the sur-
face S at the point B remains equal to 0I until the
switch 2S is opened. Then in the time interval
÷− 0)2( Tn 0nT this current equals to the load current
1/ ZUL .
Taking into account that 00 =q the relation (2) will
be written in the form
+=+ 0000 2)/( iTITZUI ii 10 /2 ZTU L
or
1/2)2(/2 ZUiIZU Liii +−⋅= (14)
Taking into account that
∏
−
=
++⋅=
1
1
11 )]/(2[
i
j
jjji ZZZUU , (15)
2/101 ZIUUL ⋅+= , (16)
from (14) one can get a relation
( ) =− iL ZZIU /2/2 10
[ ] ∏
−
=
++⋅+−=
1
1
10 )]/(2[/2)2(
i
j
jjjiL ZZZZUiI .
Solving this equation together with a similar equa-
tion written for a cascade with a number greater by one
unit we find a recurrence formula +=+ 01 /2[ IIZZ Lii
)/2/()]2( 0 iIIi L +−+ and an expression for
impedances of cascades with numbers )2(1 −÷= ni :
)]1()/[()1(1 −+⋅++= iiZZi λλλλ (17)
After the switch 1S is closed the energy must be fully
extracted from the cascade with the impedance nZ in
the time interval 00 T÷ . In the mathematical form this
condition has the form:
00 IZU n ⋅= (18)
To avoid a recharge of this cascade one must avoid
appearance of the voltage wave reflected from the junc-
ture of n and 1−n cascades:
( ) ( ) /2/ 10110 −−− ++− nnnnnn ZZUZZZZU
( )( ) 0][ 211 =++ −−− nnnn ZZzZ (19)
When closing the switch 1S the voltage wave will
go into cascade 2−n
( )21202 / −−−− += nnnn ZZZUU . (20)
The amplitude of this wave can be also determined
from equation (15) taking into account (16), (17):
( )[ ] ( ) /25.0/2/ 0
3
1
112 IIZZZUU L
n
j
jjjn −⋅=+= ∏
−
=
+−
( ) ( )[ ] ( ) ( )2/15.01/ 01
3
1
−++=−++∏
−
=
nIZjkjk
n
j
λλλ . (21)
Equating the right sides of (20) and (21) we obtain:
( ) ( ) /15.0/ 012120 +=+ −−− λλIZZZZU nnn ( )2−+ nλ .
From (19) we find:
)/()( 12121 −−−−− −+= nnnnnn ZZZZZZ . (22)
Substituting the expression for nZ in (21) with re-
gard to (18) we find the impedance 1−nZ :
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 41-42.
41
)]1()2/[()1(11 −+⋅−++=− nnZZn λλλλ . (23)
The optimal impedance nZ is found after substitu-
tion of the expression (23) in (22):
)1/()1(1 −++= nZZn λλλ . (24)
The voltage on the load is equal to
λλ 2/)1(0 −+= nUU l . (25)
The optimal relation of generator impedances
(Fig. 3) is determined by equations (17), (23) and (24).
Besides, there should be matched the amplitudes of 0U
and 0I according to expression (18). Polarity of charged
voltage should be so that after the switch 1S is closed,
the current in the line n decreases.
6 CONCLUSION
There was developed a new method of calculation,
whose employment significantly simplifies a search for
optimal relation of impedances of multi-cascade genera-
tors on stepped lines possessing in the ideal case 100%
efficiency. A high efficiency of the method was demon-
strated on the example of generator circuits with differ-
ent methods of energy storage.
REFERENCES
1. V.S.Bosamykin, V.S.Gordeev, A.I.Pavlovskii. New
schemes for high-voltage pulsed generators based
on stepped transmission lines // Proc. of IX Intern.
Conf. on High Power Particle Beams “BEAMS
92”. Washington, DC, May 25-29, 1992. v. 1,
p. 511-516.
2. V.S.Gordeev, V.S.Bosamykin. Schemes of high-
power pulsed generators with inductive storages on
stepped lines // Proc. of XI Intern. Conf. on High
Power Particle Beams “BEAMS 96”. Prague, 1996.
v. 2, p. 938-941.
3. V.S.Bosamykin, A.I.Gerasimov, V.S.Gordeev.
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shapers on the basis of stepped transmission lines
for high-current accelerators // Problems of Atomic
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Research (35). 1999, No. 4, p. 68-70.
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pulse-line cavities with external pulse injection //
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