The causality of the complex permittivity for a plasma model in which both loss and restoring terms are included is shown. Discussions as to what occurs in this causality property when the loss and restoring terms vanish are presented. Results are shown to agree with earlier work

Cockrell, C. R.

English

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NASA
Technical Memorandum _01671
Causality of Plasma Permittivity
with Loss and Restoring Terms
C. R. Cockrell
(NASA-TM-I01671) CAUSALITY OF PLASMA
PERMITTIVITY WITH LOSS AND RESTORING TERMS
(NASA) 15 p CSCL 201
G3/32
N90-19453
Unclas
0264840
December 1989
fU/ A
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23665-5225
https://ntrs.nasa.gov/search.jsp?R=19900010137 2020-03-19T23:03:24+00:00Z
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CAUSALITY OF PLASMA PERMITTIVITY WITH LOSS AND RESTORING TERMS
ABSTRACT
The causality of the complex permittlvity for a plasma model,
In which both loss and restoring terms are included, is shown.
Discussions as to what occurs to this causality property when the
loss and restoring terms vanish are presented. Results are shown
to agree with earlier work.
INTRODUCTION
The expression representing the complex permittlvlty of a
plasma material is derived through the equation of motion of
electrons in the presence of an applied electrlc field |1-2]. In
many plasma models, the material is assumed Inhomogeneous in the
direction normal to the plane layer and homogeneous in the
unbounded lateral directions. The inhomogeneity of the material
is expressed in the complex permittivity representation through
the plasma and collision frequencies. An electron density which
is identified with the plasma frequency and a loss term which is
identified with co111slon frequency vary as one moves into the
material. It is this spatial dependency of the plasma and
colllslon frequencies which require numerical evaluation of
Maxwell's equations in the inhomogeneous plasma material as a
function of the applied electrlc fleld frequency.
Since the complex permittivlty is dispersive, its fre_ehcy _
behavior for a fixed plasma profile is also of interest. In
particular, the complex permlttlvlty should be causal; that is,
the time response for the complex permlttivity must exist only for
times greater than zero. The expressions given in the literature
representing complex permittlvlty [1-3], which are derived with
only losses (colllslon frequency) assumed in the model, are shown
to be noncausal [4]. It was shown in reference 4 that the system
can be made causal by the addition of a term proportional to the
impulse function 6(_).
The purpose of this paper is to show that the complex
permittivity representation, in which both loss and restoring
terms are Included in the plasma model, obeys the prlnclple of
causallty. By a proper llmltlng procedure as the restoring term
vanishes, the same impulsive term that appears in the solution
given in reference 4 is recovered.
2
PROOF OF CAUSALITy
The dlfferential equation which represents the dynamics of
electron motion in a plasma or dielectric material in the presence
of an applied electric is given as [5]
d2x(t) + m v _ + b x(t) = - q 6(t) (I)
m dt 2
where m is the mass of an electron, q is the charge of an
electron, u is the collislon frequency (corresponds to loss) of
the plasma, b Is the coefficient of the restoring force term, x(t)
is the dlsplacement of the electrons, and _(t) iS the applied
electrlc field. The derivation of the complex permlttlvity based
on equation (1) Is given in reference 5. Therefore, without
proof, the complex permlttlvlty for this model is given as
o
2 _2 2(2_ ) _ _ v
i - P - j P
( 2__2)2+ 2p2 ( 2__2)2+ 2p2
(2)
where _p is the plasma frequency, u is the collision frequency of
the plasma, _ = b/m, c Is the free space permittivlty, and _ is
O
the frequency of the applled electric fleld.
Rewrite equation (2) as
2
C
c(_) = c p o (3)
o 2__2_j_v
The inverse Fourier transform is defined as
(t) = _-_ c (_) eJ_td_
--CO
Substituting equation (3) into equation (4),
_(t) - 2n c - eJ_td(_
o ( 2__2_j(j v
--00
(4)
(5)
3
or
2 00
c I 1 eJ_td_c(t) = Co6(t ) - _ 2_p2_j_jv (6)
--CO
The integral in equation (6) can be readily evaluated by contour
integration, once the poles of its integrand are determined.
The poles are determined by setting the denominator of the
Integrand in equation (6) to zero. The dlscrlminant obtained by
using the quadratic formula permits, in general, three solution
sets. The solutlon set corresponding to the condition v2<4_ 2 is
considered Invalid because as _ is ultlmately allowed to approach
zero, v must be real. For the condition P2>4B 2, the poles are
given as
and
o2: ]
iv)
(8)
which are shown in figure 1.
ej_t
The complex function f(_) - is meromorphlc in the
_2-p2-j_ v
upper half plane of the complex-_ plane and analytic in the lower
half plane of the complex-_ plane. From Cauchy-Goursat theorem,
the closed contour integral of f(_) for t < 0 is zerb [6]; that
is,
f(_) d_ = 0 (9)
where the closed contour is indicated by the dashed lines in
figure 1. Rewriting equation (9) over the Indlvldual contours
shown in the figure:
I_ eJ_t I eJ_t
_2__2_j_ v d_ + 2__2_j_ v
-_ Cr<
It is easily shown that the contribution
zero [6]. Equation (10) now becomes
I eJ_t
d_ = 0
_2-_2-J_v
d_ = 0 (10)
from the Or<
, t < 0
Integral is
(II)
4
which says that c(t) in equation (6) is zero for t < O.
For t > 0 the contour may be closed in the upper half plane
as indicated by the solld llne shown in figure 1. The closed
contour integral must again be zero; that is,
f(w) dw = 0
(12)
Expanding the total contour into the individual contours as shown
in figure i:
I f(w) dw + f(w) d_ + f(_) d_ + f(_) d_ = 0 (13)
-_ Pl P2 Cr>
The contribution from the contour Cr> is zero. The contributions
from second and third integrals are evaluated by residue theory.
It is easily shown that
_ _ /v 2
e
Pl
e jwt dw = - 2. (14)
w2-82-J_p
and
I eJ_t
d_ = 2n
P2
With these equations, equation (6)
v/v 2- 4_ 2
t 4p2 ]
-_[_-_/_ 2-
e
,//v 2- 482
(15)
becomes
(t) = c _(t)
0 v,Islnhl v _,02i}+ 2 2 ......P _oe 2 /v 2- 482 (16)
for 4_ 2 < v 2 and t z O. For 4B2< v 2 and t < O, _(t) equals zero.
The condition 4B 2 = v 2 and t > 0 can be evaluated by
considering the contour integration path as shown in figure 2. In
this case there exists a pole of order two at _ = iv/2. Evaluating
by residue theory,
I I-0
_(t) = Co_(t) - _-_ p w _ =Jr 2
and, since there are no poles in the lower half plane, e(t) is zero
for t < 0. Hence, equation (6) becomes
vt
2 2
co8(t) + _p c t e
o
0 , t < 0
• t-> 0
(18)
This result can also be obtained from equation (16) through a
proper limltlng procedure.
Letting _ approach zero• that is the tem corresponding to
the restoring force vanishes• equation (16) for 4B2< v2 becomes
2
(E) = I_
[ 0 t <0
, t-> 0
(19)
which, once normalized with respect to c , is the same as equation
o
(7) in reference 4. Letting B approach zero in equation (18),
which Implles v goes to zero because v2= 4_ 2, the expression for
c(t) becomes
c(t) = c 6(t) + (2 c t t > 0
o p o , - (20)
which, once normalized with respect to Co, is the same as equation
(19) in reference 4 ....
It Is emphasized at this point that _(t) obeys the principle
of causallty whenever both the loss and restoring terms are
included (see equations (16) and (18)). Allowlng B to approach
zero or both _ and v to approach zero in the general solutions
(equation (16) and equation (18)), causal solutions are obtained
(equation (19) and equation (20)• respectively). However, if B=0
in equation (1), the direct derivation of c(_) and its inverse
Fourier transform lead to a noncausal solution (see equatlons (4)
and (5) in reference 4). Also, if p = 0 and v = 0 in equation (1),
the direct derivation of c(_) and its inverse Fourier transform
lead to a noncausal solutlon. Reference 4 shows, however• that
each of these noncausal solutlons can be made causal by the
addition of an impulsive term. The next section shows• through a
limiting procedure as B approaches zero in equation (2), that the _
proper impulsive term given in equation (6) in reference 4 is "
recovered.
6
P_RMITTIVITY AS RESTORING TERM VANISHES
In this section equation (2) is examined as _ approaches
zero. Its inverse Fourier transform is written explicitly as
I{- [ ]}1 _ 2 2_(t) = e _(t)+llm _F_ 2 + eo_ P /3 eJ_td_ (21)
o /3-> 0 (_2-/32) 2+_2v (_2-B2) 2+_2v2
The evaluation of this integral as _ approaches zero is performed
by contour integration. The poles of the integrand are
_l=J_ +
_2 = J _
1 (-V/V2- + v l_3 = j 2 4_2
1 [-/v2- 4/32 v}_4 = J _
(22)
where the condition 4_ 2 < u2 is the only valid solution because u
is real and greater than zero. These poles are indicated in
figure 3.
Denoting the first integral by 11 and the second Integral by
12 , equation (21) is rewritten as
where
and
e(t) = eo6(t ) + 11 + 12 (23)
2 00
c _ [ (2 + j_v)
= - llm _ ..... eJ_td_ (24)
II _ 0 2. J ( 2__2)2+3V2
--CO
12
2 o3
= llm
p_ 0 2n 2+cj2p2
--00
e j _td_ (2 5 )
Equation (24) can be evaluated in a straightforward but lengthy
process using the contour shown in figure 3. For brevity, only
the results are indicated:
2
I1 _ o p 1 t > 0 (26)v _- - e , -
At this point it might appear that as p approaches zero 12 would
vanish. This is incorrect because the range of integration
includes _ = 0, and thus the square bracket in equation (25)
varies as _2/ 4 and is not zero. Contour integration, using the
contour shown in figure 3, yields -
2
c
- ° P (27)I2 2v
Substituting equations (26) and (27) into equation 123), the
expression _ becomes
2
c _ [ -vt]
_(t) = c alt) + o p z - e , t _ o 128)
o V
An interesting result can be obtained if equation (25) is
rewritten as
I2O I1io
J _ 0 (w2-p2)2+_2v 2
comparing
definition can be made:
thls equation wlth equation
eJ_td_ (29)
(27), the following
a(_)- lira 1 [ _2v ]
_ 0 Tr ( 2__2)2+B2u2
(30)
This is one of the many different definitions which can be made
for impulse functions. Definitions of others are given in
reference 7. It is observed that as B approaches zero, therefore,
equation (2) must become
2 _2 _p_P + _ -J_ a(_) - J _2 + vc(_) = co 1 - _2 + v2 u 2 (31)
which, once normalized with respect to c,
equation (6) in reference 4.
is Identlcal to
CONCLUDING REMARKS
The causality of the complex permittivlty for a plasma model,
in which both loss and restoring terms are included, has been
shown. Whenever the loss term or the restoring term or both were
allowed to vanish in the time representation for this plasma
model, the solutlon still obeyed the principle of causality.
However, direct solutlons of the differential equations without
both of these terms produced noncausal results. Through an
appropriate limltlng procedure for the restoring term vanishing,
the impulsive term given in an earlier paper was recovered-- the
term required for causallty.
I. Ramo,S.; Whlnnery,J.R.;and Van Duzer,T.: Fields and Waves in
Communication Electronics, John Wiley and Sons, 1967.
2. Jordan,E.C.; and Balmain,K.G.: Electromagnetic Waves and
Radiating Systems, Prentice-Hall, Inc., 1958.
3. Croswe11,W.F.; Taylor,W.C.; Swift,C.T.; and Cockrell,C.R.z
"The Input Admittance of a Rectangular Waveguide-Fed Aperture
Under an Inhomogeneous Plasma: Theory and Experiment,"
IEEE Trans. on Antennas and Propagation, Vol. AP-16,
pp.475-487. July 1968.
4. Cockrell,C.R.I " On the Causality of Plasma Permittivlty," NASA
TM 101551, Langley Research Center, April 1989.
5. Collin,R.E.z Foundations for Microwave Engineering, McGraw-Hill
Book Company, 1966.
6. Marsden,J,E.; and Hoffman,M.J.z Basic Complex Analysis, W.H.
Freeman and Company, 1987.
7. Bracewell,R.M.: The Fourier Transform and Its Applications,
McGraw-Hill Book Company, 1965.
10
Cr< "
• s
Complex -co
_"_ t>O
,. / Cr>
!
!
#
I
t
J
• t<O¢,
s
plane
Figure 1.-Contours for equation(6) with 4_ 2< _2
11
Complex-co plane
Figure 2.-Contours for equation(6) with 4132=t_2
12
(0 2
(0 4
Complex-co plane
Figure 3.-Contours for equation(21)
13
Report Documentation Page
1. Report No.
NASA TM-101671
4. Title and Subtitle
Causality of Plasma
Restoring Terms
2, Government Accession No
Permittivity with Loss and
7, Author(s)
C. R. Cockrell
9. Pedorming Organization Name and Address
NASA Langley Research Center
Hampton, Virginia 23665-5225
12. Sponsoring Agency Name and Address
National Aeronautics and Space
Washington, DC 20546-0001
Administration
L--
15. Supplementary Notes
3. Recipient's Catalog No.
5. Report Date
December 1989
6. Performing Organization Code
8. Performing Organization Report No.
10. Work Unit No.
583-01-11-12
11, contract oi Grant No,
-1-3_-Typ---eof Report and Period Covered
Technical Memorandum
14 Sponsoring Agency Code
16. Abstract
The causality of the complex permittivity for a plasma mode], in which both loss
and restoring terms are included, is shown. Discussion as to what occurs to this
causality property when the loss and restoring terms vanish is presented. Results
are shown to agree with earlier work.
17. Key Words (Suggested by Author(s))
causality
plasma
restoring term
19, Security Classif, (of this report)
unclassified
I
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unclassified
T 18 DistributionStatement
I unclassified - unlimited
Subject
2t Nol of pages
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22. Price
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