dn Kp n Y 
= aad : a 161 x avi S (4 
dt Ar (ov) ove + 
Equation 3 is a power-balance equation 
that represents the rate of change of 
internal energy in the plasma, and Eq. 4 
represents a particle-number balance. 
Since all that follows is concerned en- 
tirely with obtaining and interpreting 
the mathematical solution of these two 
equations, an attempt will first be made 
to describe, and hopefully to justify, 
the terms of the two equations and the 
many assumptions that have been 
made. 
The assumption is made that some 
suitable configuration has been found 
for a fusion reactor, and this reactor 
will be first considered as operating on 
the DD reaction. None of the general- 
izations to be made is affected when 
the DT reaction is used. The deu- 
terium ions in the ‘“‘box’”’ have a den- 
sity n, are at an elevated temperature 
T, and are presumed to be confined in a 
nonspecified manner by means of a 
fixed constant magnetic field B. The 
process is assumed to be occurring on a 
per-unit-volume basis, and the over-all 
volume and geometry are ignored. As 
in the case of the fission-reactor equa- 
tions, the variables can also represent 
either total or average values for the 
entire box. 
dE/dt is the rate of change of energy 
within the “box” and can be represented 
Definitions of Auxiliary Constants 
Definition 
Auziliary 
constant DD reaction DT reaction 
Ki % 39k 
Ky lew law 
Kp,k 
Kp = (Bohm) Same as DD 
1B 
A K2Kane? Same as DD 
= Wg Kpne?T "2 
ane 2K pnoT’o 
B 2noaoKe Same as DD 
bem 2K gnoT'o”? 
— KT? 
C 2 Kp 2 Kp 
i pa ft es 
Ce ais Bias 
— 14Kano = VAI pie 
2K PAVE 
D —=-—7, Ese i 
30k 3k 
— Noao = 1onoao 
E Kino Same as DD 
F KiT Same as DD 
42 
by the rate of change of the thermal 
energy possessed by the system; hence, 
dE /dt = d(3gnkT)/dt, where k = Boltz- 
mann’s constant. The power balance 
consists of two parts: first, a production 
term representing thermonuclear energy 
release, and, secondly, various losses of 
power by specific processes. The pro- 
duction term is !gn%(av)aygW, where the 
constant W is the energy released to the 
charged particles per reaction. The 
energy possessed by the neutrons gen- 
erated in the DD reaction is presumed 
to be lost from the system, gainfully or 
otherwise, and therefore is not available 
to keep the reaction going. The (ov) avg 
term, the average of the product of 
cross section and relative velocity as 
given in Fig. 4, is dependent on tem- 
perature and is designated hereafter as 
a(T). 
The first loss term of Eq. 3 is given 
by 0.54 X 10-%n27", This is the 
Bremsstrahlung radiation loss from the 
system. Here the electrons radiating 
are assumed to be in temperature equi- 
librium with the ions (the electron par- 
ticle density, equals the ion particle 
density). 
The second loss term K p,nkT/7(T) 
is the power loss from the system by 
particle diffusion, where 7(7') repre- 
sents the average time taken by parti- 
cles beginning at the source to drift 
through the magnetic field to the walls 
of the box. This ‘‘confinement time” 
must be comparable to the average 
reaction time for a successful fusion 
reactor. 
Other loss terms can also be imagined, 
such as the radiative losses caused by 
impurities and the possible need to pro- 
vide internal power to heat up new par- 
ticles inserted into the system. These 
other losses only add additional nega- 
tive terms to Eq. 3 and are a re- 
finement that does not alter substan- 
tially the form or the approach to the 
control-stability problem. 
In Eq. 4 the term Kp,n/r(T) can be 
thought of in fission-reactor terms as 
the particle loss caused by leakage. 
Similarly, the second term, 1477(av) ave, 
is analogous to fuel depletion or burnup. 
The third term is a source or injection 
term which refers to the rate of addition 
of new particles to the system to com- 
pensate for the burnup and leakage. 
It will be noted that the roles of the 
source term in the fission and fusion re- 
actors are somewhat reversed. In a 
fission reactor as the power level is in- 
creased the effect of the source becomes 
smaller and smaller. In the fusion re- 
actor at higher power levels more and 
more particles are needed to keep the 
reaction going. Injection is probably 
a better term for use with fusion reac- 
tors in that the introduction of new 
fuel is actually implied. 
Mathematical Solution 
We must first consider the form of 
the confinement time 7(7’) with respect 
to the temperature and the magnetic 
field. The confinement time is in- 
versely proportional to an average 
(decibels) 
o 
zu 
&) 
ss 
a 
[= 
a 
Phase Shift 
(degrees) 
0.1 Ke) 
Frequency (cycles/sec) 
FIG. 3. Frequency dependence of 
transfer function An/AS relating time 
variation of ion density n and injection 
rate S for a fusion reactor operating 
on the DT reaction 
drift velocity. Post gives two forms 
for this velocity—the so-called “‘classi- 
cal’’ diffusion theory in which 
1(T) ~ B?T”/n and Bohm (4) diffu- 
sion theory where r(7) ~ B/T. For 
our purposes either form is suitable, 
and we first select Bohm diffusion. Let 
r B 
(TL). = 7 T 
where nisanappropriate constant. An 
indication will be given later as to the 
effect upon stability of changing the 
form of the diffusion equation. 
We can now rewrite Eqs. 3 and 4 in 
terms of the auxiliary constants de- 
fined in the table as follows: 
ee ‘ > apy 
Ky nt = Kin’a(T) — Kon? 
— KpnT? (8a) 
Kp J 
ae nT — \g6n’a(T) + S 
(4a) 
These equations are nonlinear, but 
as in the case of the fission reactor use- 
ful solutions are obtained by linearizing 
them. The object here is to derive a 
transfer function expressing the re- 
