APPLIED PLASMA RIESEARCH
VI.
Active Plasma Systems
A.
Academic Research Staff
Prof. L.
Prof. A.
Prof. iR. iR. Parker
Prof. K. I. Thomassen
D. Smullin
Bers
Graduate Students
G. HI. Neilson
J. J. Schuss
NI. D. Simonutti
N. J. Fisch
S. P. Hirshman
C. F. F. Karney
J. L. Kulp
1.
EFFICIENCY CONSIDERATIONS
N1. S. Tekula
A.
I.
Throop
D.
P.
C.
R.
Watson
\Widing
FOR A PLASMA GUN SYSTEM\
National Science Foundation (Grant (G -28282X1)
S.
Hirshman, L.
P.
D. Smullin
As reported in Quarterly
Progress
IReport No.
105 (pp. 89-93),
we
have
been
studying the efficiency of electrical energy conversion to plasma energy in a plasma gun
system. This report gives a summary of our latest results.1
General Efficiency Criteria
Several measures of efficiency have been proposed by Jahn" and
1en dl. 3
The
dynamic efficiency is defined by
id
1
2
2
s
W
p
dT is the total work done on the sheet gas by the magnetic forces
(including shock heating). In this measure of efficiency, only the final mass nmotion of
the plasma is considered as a figure of merit. An alternative definition is the electrowhere
V
p
my
= ft
o
s
mechanical efficiency, a measure of the ability to couple electric energy into total
plasma energy
,
o
where E o is the initial energy stored in the external power supply. This efficiency takes
into account the fact that gas particles undergoing inelastic collisions with the snowplow
piston are heated and thus represent a useful conversion of electric energy into plasma
QP{ No.
109
(VI.
APPLIED PLASMIA IRESEARCHII)
Finally, the kinetic efficiency, a measure of the ability to couple electrical
energy.
energy to energy of notion of the sheet, is defined by
1
2
E
Note that -q1
=
o
; therefore,
cc
1i\ is always less than either -yd
or jelll
The choice
of p to be used in any efficiency analysis depends on the particular task for which the
gun is to be used.
For ion thrusters,
for plasma injection applications,
p (t
LR
Ie
Et)V
EXTERNAL
CIRCUIT
L it
my is the
both p i and
relevant efficiency to optimize, while
n (n are useful efficiency measures.
,
GUN
(a)
Fig. VI-1.
(a) General gun circuit.
bank with crowbar.
I
(b) Capacitor
TO GUN
-.--
CROWBAR APPLIED HERE
(b)
To obtain physical insight into the dependence of
parameters,
E(t) (Fig.
it is useful to express all
\-I-la)
variables
mK
and
e1m on circuit and gun
in terms of the external circuit energy
and the gun inductance I,(t), which is proportional to the instantaneous
sheet position and hence representative of the "state" of the sheet in the accelerator.
rThe result
1
is
L
E
1+
ofL
+Lif
and
1
S=em
QPR{ No.
109
2 , mf
ftf
"
0
d
\m
er)
2
,
(2)
APPIED PIASMXA RESEARCH)
(VI.
where subscript f refers to the final accelerator state.
It is
clear that to optimize
>> le is desirable; furthermore, the externally stored energy should be supplied
One way to
to the gun as quickly as possible and remain out of the external circuit.
rlem' Lf
prevent energy
E from returning to the power circuit is to crowbar the power supply
when E is at minimum.
A typical circuit of this kind, which will be analyzed in greater
detail, is shown in Fig. VI-1b.
The second term in Eq.
2 represents losses from shock heating, and the effect on
efficiency is not as readily interpreted as rln1 was before.
Independent Parameters of a Gun System
In any real gun system, there are numerous performance requirements that impose
For injection (or thruster)
constraints on the otherwise independent gun parameters.
applications, the type of working gas,
the final velocity,
and the kinetic energy of the
accelerated plasma are to be considered prescribed by the requirements imposed by
fields,
heating capabilities, plasma penetration into magnetic
and so forth.
These
three constraints are sufficient to allow an optimum determuination of all other system
per unit length Y. is
(In a practical system, the inductance
parameters.
severely
restricted by geometrical considerations and should be chosen as large as possible;
cf. Eq.
1.)
Our procedure consists in solving a normalized set of equations1 which completely
describes the gun system of
found that il
(i)
and -e
'ig. VI-1b.
For a uniform distribution of filling gas, we
depend only on two coupling parameters:
Energy paramneter, e
/1 (o1. Physically,
E
E is a mieasulre of the energy
"match" between the gun and the external circuit.
(ii)
and t
c
Time parameter,
= E
O
V I .
00
Here
t
j4
/t
, where t
=
/u
with u0 the
w,
is a measure of the matching of circuit
(t ) gun time scales.
snowplow velocity,
(I
c
) and intrinsic
L
In the particular case considered here,
L0 =
1
2
CV
0
and
0
= CV
/l., so that C =
1
-
e
L
0
and
= 2to/(LC)1/
Therefore,
for any gun system driven by a capacitor bank and obeying the snowplow
dynanics, the efficiency of enetrv conversion depends only on the two parameters
lurthe rmore, %with l,
Fu.
and
in., and uf=
u
E
sp(cified, the remaining system
parameters can be expressed solely in terms of c and p.,provided we make an optimum choice of system length.
There is an efficiency,
m,
associated with any pair of coordinates (C, K).
eral, there will be level curves of constant
Tj
In gen-
in c, K space, and any point in this space
for which 1j is a maximnum represents an optimum solution to the efficiency problem.
There are, however, certain practical constraints that limit the region of accessibility
QPIR No.
109
(b)
Fig. VI-2.
(a) Kinetic efficiency level curves.
(b) Electromechanical efficiency level curves.
10
Fig. VI-3.
QPR No.
109
Final velocity fraction (K) curves.
100
10
0.1
1
10
41
4
Fig. VI-4.
Typical constraints V
,
C,
Le
2
15
Fig. VI-5.
Intersections of regions of accessibility (net region shaded). O (E= 0. 25,
= 3.5) is the point of optimum attainable efficiency.
80
10
0.01
i
o
L
0.1
QPR No.
109
101
0.75
ELECTROMECHANICAL
0.50
THERMAL
0.25
1.0
0.5
0.5
2.
2.0
1.5
KINETIC
5 .
1.0
1.5
2.0
2.5
TIME
CAPACITIVE
0.50
INDUCTIVE
GUN
1.0
0.5
2.0
1.5
2.5
1.0
0.5
TIME
Fig. VI-6.
0
1.5
2.0
2.5
TIME
(t = 0. 25,
F = 3. 5):
Optimum realizable parameters.
1 .0
1.0
r-ELECTROMECHANICAL
0.75
Z 0.75
THERMAL
Ou
Z
-
0.50
0.50
-
a
/
0.25
KINETIC
R/
3
4
1
5
2
TIME
3
4
5
TIME
1.0
CAPACITIVE
0.75
z
0.50
INDUCTIVE
0.25
GUN
1
2
3
4
1
5
TIME
Fig.
QPR No.
109
VI-7.
0
2
3
TIME
(E = 0. 04,
S= 5):
102
Optimum parameters.
(VI.
in c,
APPLIED PLAS\IA RESEARCH)
4 space, and hence the optimum value of jj.
The contours of constant kinetic and electromechanical efficiencies were numerically computed and are shown in Fig. VI-2 for the c, . space of practical interest.
Figure VI-3 shows curves of constant final velocity fraction K(E, k).
Note the great sen-
sitivity of the efficiency curves on c ~ Le/Lo, as expected from Eq.
1.
Once these curves have been obtained, they can be used to design a plasma gun that
The typical constraints 5 kV < Vo < 30 kV,
will operate in the most efficient manner.
7
10e
mass 10
L
in Fig. VI-4 for hydrogen gas of total filling
are shown
5
kg and final velocity 105 m/s. In Figure VI-5, the intersection of all three
H, and C
<
100 iF
The point O (E = 0.25,
constraints is plotted, with the net region of accessibility shaded.
4= 3. 5) appears to be the point of optimum attainable efficiency for this design, and corresponds to
TK
=
current, voltage,
2 5%o
and
nem = 60%.
The (normalized) system variables (velocity,
efficiencies) are plotted in Fig. VI-6 for point () as functions of time.
Note how the initial capacitive energy is entirely converted into inductive and plasma
energy.
In Fig. VI-7 the point 0
(E= 0. 04,
5) demonstrates the favorable effect obtained
i=
by lowering the circuit inductance L e approximately an order of magnitude.
Note the
25% gain in plasma efficiency over point 0.
0.50
0.50
0.25
0.25
ELECTROMECHANICAL
KINETIC
S_
KINETIC
i------T
-
8
4
ELECTROMECHANICAL
KINETIC
12
16
0.4
20
0.8
1.6
2.0
1.6
2 0
1.0
1.0
0.75
1.2
TIME
TIME
-
0.75
CAPACITIVE
2
2
z 0.50
0.50z
CAPACITIVE
u
u
0.25
0.25
TOTAL INDUCTIVE
GUN INDUCTIVE
4
8
12
16
0.4
20
(a)
(b)
Inefficient mode of operation.
(a) P 1: E = 0 025, i = 0. 2.
(b) P2: e = 4,
No.
109
1.2
TIME
Fig. VI-8.
QPRP
0.8
TIME
103
L = 10.
(VI.
APPLIED PLASMA RESEARCH)
Furthermore, we may conclude that there are only two basically different ways in
which the plasma gun system can operate inefficiently.
These two cases are represented
by P1 and P2 in Fig. VI-5, and correspond to small i and large c (and kL), respectively. When i is small, the gun is electrically "too short," and the acceleration
process is
over before the external circuit can transfer its energy to the plasma
(Fig. VI-8a).
When E (and
i4)
are large,
energy is transferred only from the capacitor
to the external inductor and never gets into the gun circuit.
This inefficiency is illus-
trated in Fig. VI-8b.
Conclusion
We have found that with reasonable circuit and gun parameters,
it is
possible to
design a plasma gun that meets typical operating specifications with kinetic efficiency
of 25% and overall plasma efficiency (including shock heating) of 60%.
estimates were based on a final plasma velocity of -105
below the thermonuclear threshold.
It can be shown
final velocity by a factor of 10 is to decrease the
Therefore,
I
The efficiency
m/s, which is (for hydrogen)
that the effect of increasing the
system efficiencies to
< 5-10%.
there is a severe reduction in system efficiency as the velocities required
for thermonuclear applications are approached,
a situation that casts doubt on the appli-
cability of a plasma accelerator for thermonuclear heating.
References
1.
Detailed results are presented in S. P. Hirshnian, S. M. Thesis,
Electrical Engineering, M. I. T. , January 1973 (unpublished).
2.
R. G. Jahn, Physics of Electric Propulsion (MIcGraw-Hill Book Company,
New York, 1968).
3.
C. W.
QPR No.
Mendel, J.
109
Appl. Phys. 42,
5483 (1971).
104
Department of
Inc.,
APPLIED PLASMA RESEARCH
VI.
Laser-Plasma Interactions
D.
Academic and Research Staff
Prof. II. A. IHaus
Dr. P. A. Politzer
Dr. A. H. M. Ross
Prof. E. V. George
Prof. A. Bers
J. J. McCarthy
W. J. Mulligan
Graduate Students
1.
C. W. Werner
D. Wildman
D. Prosnitz
Y. Manichaikul
J. L. Miller
SHORT PULSE PROPAGATION
National Science Foundation (Grant GK-33843)
Ross
A. H. M.
A computer code has been developed to solve the wave equation for electromagnetic
propagation in resonant media.
While the ultimate goal of this effort is to produce a
realistic numerical model of short-pulse amplifiers, including the effects of collisional
exchange processes and rotational state degeneracy,
preliminary testing with the ideal-
ized model of a two-level medium has produced new results concerning the so-called
07r pulses.
1
The one -dimensional wave equation in the slowly varying envelope approximation
is
1I a_
aJ
c at
at
a--+
az
a3
2y
e-
i
-o
2v
0
,
where 0- is a phenomenological conductivity included to account for nonresonant losses,
ii-s the admittance of free space, and
vo = \
of the electric field and polarization,
6' and Y are the complex amplitudes
respectively.
(1)
E(z, t) = LRe [6(z, t) exp{i(ot-ko
0 )}].
Schr6dinger's equation for a gaseous medium is most easily written in terms
sity matrix distribution function p(z, v, t).
i
dp
-dp)
~dt
1where
d
QPR No.
-
a
109
+ v
Th
a
collision'
is the convective derivative.
105
The constitutive relation is
2
of a den-
(VI.
APPLIED PLASMA
P = n
RESEARICH)
dv Tr (ip(z. V, t)),
The nornmalization of p is taken to be
where no is the mean particle density.
S
(p) = 1.
dtv Tr
For a system of two nondegenerate levels these relations become
apaa
at
at
-
a
= + - Im
at
(iabab
h
Im
(
p
)
ab
,!
;ab)
-bbb
ab
i
Y(z, t)= 2no
a
a
(2)
rbb
(3)
ap
+
a aa
+
aa-Pbb
dv Pa (z,v,t).(5
-cc
The collision ter'm has been represented by phenonmenological
relaxation
irates and
pumping terms.
In the limit of very short pulse lengths, where
electromagnetic ones can be neglected,
all terms in (2)-(5)
analytical solutions of (1)-(5)
Among them are the nr pulses, which leave the mnediun
passage as before their arrival.
other than the
can be found.
in the same state after their
1Particularly interesting are the pulses with bipolar
envelopes such that their area is zero(,
the Or, pulses.
XWhile most coherent propagation
effects are drastically modified by degeneracy- of the participating
has been shown, both theoreticallyI and experimentally,3
atomic levels, it
that On pulses can exist in
more complicated mendia.
Since the closed-form solutions have been found only fotr completely unbroadened
absorbing media, and because they cannot a(:commodate arbitrary inputs, it it necessary
to use numerical methods for mnore realistic situations.
Because of the interest in Or
pulses, we present solutions of (1)-(5) in homoneneously and inhonogeneously broadened
absorbing (pbb(t= 0)
1,
paa(t= 0) = 0) eases
for inputs that would otherwise
e
Obe
pulses.
As Lamb has shown,
trated in Fig. VI-9,
there are two simple types of Or pulses.
The first,
separates into twvo relatively inverted 2-, pulses.
illus-
The effect of
finite phase damping is shown in Fig. VI-10 where the homnogeneous linewidth has been
QPR{ No.
109
106
0=0
0
-20
0
-12
-12
-8
-4
4
0
8
12
16
20
-8
-4
-8
-4
-8
-4
8
i2
0
4
8
(e)
3
4
0
24
4
(d)
6
20
24
2
16
20
24
2
16
20
(0
t
e
n
-2 0
-8
-12
-4
4
0
8
12
16
20
12
24
0
0-
10
S= 2
2
-8
8
4
0
-4
12
0
20
16
-12
24
8
(f
(c )
T
Fig. VI-9. Evolution of a separating 07 pulse in E two-level medium of zero spectral
width.
Pulse widths:
1 =
1,
2
3.
ordinate the electric field envelope;
sionless.)
Abscissa represents retarded time;
6 depth in medium.
(All units dimen-
Fig. VI-10.
t 0
Separating 07T pulse emerging from
a homogeneously broadened attenab = 1. (Other parameters
uator:
e
-10999
-12
-8
-4
8
4
0
12
20
16
24
as in Fig. VI-9.)
Fig. VI-11.
0
Separating 07, pulse emerging from a
Doppler -broadened attenuator: AwD=
e -I 0
20
-8
-12
-4
4
0
8
12
20
16
24
1/ -2.
(Other parameters
Fig. VI-9.)
as
in
T--
Fig.
VI-12.
0
e
-1
Separating OT pulse emerging from
an attenuator with both broadening
5
0
mechanisms:
-12
QPR No.
-8
109
-4
0
4
8
12
16
20
24
107
ab = 1, AD = 1 /
2.
(Other parameters as in Fig. VI-9.)
0
e
t
e
0
-8
-12
-4
4
0
24
8
12
16
20
8
12
16
20
24
10
16
20
24
(d)
10
e
0
-I
-1 0
-12
-8
-4
0
4
8
2
6
20
2
-8
4
4
24
(b)
e
5
0
e
0
:2
SIJ
-1 .0
-8
-12
8
4
(C)
0
-4
12
-1
24
20
16
12
-8
-4
8
12
(f
T--
Fig. VI-13.
4
0
T-
Evolution of cohesive O7 pulse in a two-level medium of zero spectral
width.
1 =1,
Pulse widths:
T2
3.
VI-14.
0 40 -Fig.
4=5
Cohesive O7 pulse emerging from
a homogeneously broadened attenuator: yVab = 1. (Other parameters
in Fig. VI-13.)
e
-0
Sas
-12
-8
0
-4
8
4
12
16
24
20
as in Fig. VI-13.)
7----
5
e
Fig. VI-15.
emerging from a
Cohesive OrT pulse
Doppler-broadened attenuator: Aw D =
0
1.0o
-12
-8
-4
8
4
0
12
16
20
24
1 / -. (Other
Fig. VI-13.)
parameters
as in
T-
Fig. VI-16.
o40
e
Cohesive Om pulse emerging from an
attenuator
with both broadening
mechanisms: yab = 1, AWD = 1/ 2.
0
-0.40
-12
-8
-4
0
4
8
12
16
20
24
T-
QPR No.
109
108
(Other parameters as in Fig. VI-13.)
(VI.
APPLIED PLASMA RESEARCH)
chosen to be equal to the inverse pulse width of the shorter of the two pulses after separation. Figure VI-11 illustrates the same situation for inhomogeneous (Doppler)
broadening with no phase damping.
Figure VI-12 combines both broadening mecha-
nisms.
The second type of 07r pulse,
in the absence of broadening,
remains localized;
it
resembles a modulated hyperbolic secant in which the modulation varies with time.
Figures VI-13 through VI-16 illustrate the effects of the various broadening mechanisms
on this pulse.
broadening.
Note the relative insensitivity of this pulse to the pure inhomogeneous
In all cases,
however,
energy loss to the medium eventually reduces the
pulse to the linear regime.
Work continues on a more sophisticated model of a medium containing the effects
of a rotation spectrum. This will be essentially a two-vibrational-level rotator that
should provide a reasonably realistic model of many molecular lasers.
References
1.
G. L. Lamb, ,Jr., "Analytical Descriptions of Ultrashort Optical Pulse Propagation
in a Resonant Medium," Rev. Mod. Phys. 43, 99-124 (1971).
2.
A. II. M. Ross,
June 1972.
3.
H. P. Grieneisen, J. Goldhar, N. A. Kurnit, A. Javan, and H. 1t. Schlossberg,
"Observation of the Transparency of a Resonant Medium to Zero-degree Optical
Pulses," Appl. Phys. Ietters 21, 559-562 (1972).
QPR \o.
109
Ph. D.
Thesis,
1Department of Electrical Engineering,
o109
M. I. T.,