sponse of the dependent variables 7 
and n to sinusoidal variations of the 
injection term S. The characteristic 
equation so determined will then indi- 
cate whether the system is stable and 
will show the effect of the various 
parameters on stability. The process 
consists in assuming that each of the 
time variables, n, 7, and S, is made up 
of a steady-state value and a small 
sinusoidal variation about the steady- 
state value: 
n = no + An(t) 
T=T+AT(t) 
S = So + AS(t) 
The variable coefficient a(7') also is 
linearized as 
a(T) = a+ Ka AT 
We neglect second-order differentials 
and note that a new term K, is intro- 
duced which is analogous to the tem- 
perature coefficient of reactivity in a 
fission reactor. These new variables 
can now be introduced into Eqs. 3a and 
4a. With new constants, as defined in 
the table, we have finally 
5 ACL iy 
ai ar a +B An (5) 
d An 
“a ~ CAT + DAn+ AS (6) 
These equations are now linear with 
constant coefficients; so we can take the 
Laplace transforms. The zero condi- 
tions are ignored as we have already 
subtracted the steady-state terms. 
Es AT(s) + Fs An(s) = A AT(s) 
+ BAn(s) (7) 
sAn(s) =C AT(s)+D An(s)+AS(s) (8) 
The transfer functions for An and AT 
can now be obtained directly using 
AS(s) as a forcing function. 
AT(s) 
AS(s) 
_ B — Fs 
 Es?+ (FC —ED—A)s+AD— BC 
(9) 
An(s) 
AS(s) 
Es—A 
 Es?+ (FC —ED—A)s+AD—BC 
(10) 
These transfer functions, which are 
of the form 
ste 
(s + a)(s + b) 
start off with a finite gain at zero fre- 
quency. They have three break points, 
assuming a and 6 are real, and the 
amplitude response ultimately falls off 
at arate of 6db per octave. The roots 
can very easily be obtained once specific 
numerical values are used. 
Stability Criteria 
We now are in a position to examine 
the stability of the system. The sys- 
tem is stable if the roots of the charac- 
teristic equation (the denominator of 
Eqs. 9 and 10) have negative real 
parts. A necessary and sufficient con- 
dition for the roots of a quadratic equa- 
tion to have negative real parts is that 
all of the coefficients have the same 
sign. As we know that the energy EL 
oO 
o 
“ 
> 
E 
oO 
= 
a 
2 
a 
{= 
i] 
c 
S 
a 
@ 
+ 
° 
a 
c 
fe) 
= 
o 
is) 
o 
[4 
20 40 60 80 100 120 
Kinetic Temperature (kev) 
FIG. 4. Temperature dependence of 
reaction-rate parameter for DT and DD 
reactions assuming Maxwellian particle 
distribution. This figure from article by 
R. F. Post (7) 
must be positive, to have a stable sys- 
tem it is necessary that 
FC —-ED-—A>0 
AD — BC >0 
(11) 
(12) 
We can now see what these criteria 
mean in terms of the original constants. 
First using inequality 11 and substitut- 
ing the expressions from the table, we 
find that, for stability, 
3gkao — 34kKalo > 14KaW 
2K nT 
no 
— WKsTo% — (13) 
This inequality reveals several interest- 
ing bits of information. It will be 
noted that the terms to the right of the 
inequality are related directly to the 
gain and loss terms of Eq. 3 with the 
last two terms representing the losses. 
From the negative signs it appears as 
though the greater the losses the higher 
the possibility of stability. Physically 
this seems reasonable, as a system with 
large losses is heavily damped and not 
capable of continuous oscillations. 
Other inferences can be drawn from 
this inequality by examination of Ka, 
the temperature coefficient. If Ka is 
negative, meaning a decreased reaction 
rate with an increase in temperature, 
the system is clearly stable, since the 
other pertinent constants are all posi- 
tive. (From Fig. 4, (ov) for the DT 
reaction shows a negative slope over a 
portion of its range. The DD reaction 
does not exhibit this characteristic in 
the indicated range.) If AK, is zero or 
slightly positive, the system also ap- 
pears to be stable. The maximum 
positive value permissible for Ka de- 
pends on 7'o, No, and also K p, which is a 
function of the confinement process and 
field strength. Without considerable 
refinement and good numerical data the 
above statements can be interpreted 
only as trends either toward or away 
* from stability. 
The remaining stability criterion can 
be obtained by substituting the original 
constants into inequality 12 
KK; 
2K Te ao - : ) 
3k 
+ 16K paonoT' % 
> KpTo% (2 ae Kno) 
2 IK 
— 4K.— ao (14) 
This inequality does not appear to have 
the obvious physical significance that 
inequality 13 does. However, if we 
assume that 13 imposes a lower bound 
on the temperature for stability, at a 
high temperature there seems to bea 
form of density limit from 14. It can 
be seen that at high temperatures and 
high densities the KeT'o*Kano term 
might ultimately predominate if Ka re- 
mains positive and finite. Stable oper- 
ation then appears to have a low-temp- 
erature limit and a high-density limit 
at high temperature. 
To indicate that the trend remains 
the same regardless of the form of the 
diffusion, we can again solve Eqs. 3 and 
4 using the classical type of diffusion 
rather than the Bohm type. The ine- 
43 
