XIX Sympozjum Srodowiskowe PTZE - materialy.pdf

XIX Sympozjum PTZE, Worliny 2009
analytical form of the known expression for magnetic field strength on a circular turn axis [2]:
2 2 ( x + r )
−1.
5
2
H A = 0.
5irk
0 k . Besides, these parameters are interconnected by the regularities of
direct current flow (Ohm’s law and resistance of the turn as a cylindrical conductor of the
cross-section of
2
. 25π
d and the length of 2 πrk
): i = U / R1
, R = ρr
/
2
.
0 c
178
1
8 k dc
Thus, for seven parameters of the system there are three relations which can be regarded as
three coupling equations connecting these parameters. Admission of the condition of thermal
equilibrium for the turn gives another additional coupling equation for the parameters. And
admission of the condition of extremum for a certain criterion also gives one equation, which
makes it possible to determine all the parameters of the considered system unambiguously
(the number of the equations is equal to the number of the unknown parameters). The analysis
of corresponding solutions for various criteria was the problem of this paper.
III. Thermal equilibrium equation for a one-turn coil
The condition of thermal equilibrium means that the quantity of the released heat (heat release
2
power Pi = i R1
) is equal to the quantity of the heat taken off the turn surface during cooling
(heat dissipation power P θ ): Pi = Pθ
. If specific power q of heat dissipation from a cooling
surface unit is included into the consideration, the mentioned condition of thermal equilibrium
can be written down in the form of the following expression for the turn current
1.
5
= 0 . 5πd
q ρ .
i c
IV. Coupling equations system for one-turn coil parameters
It is demonstrated in the paper that all the coupling equations of the coil parameters can be
brought to two equations of the following form:
1.
5
2 2 2 −1.
5
U = 4 πr
ρq
d , H ( π 4)
d ( q ρ)
r ( x + r ) .
k
c
A = c
k 0 k
Thus, if parameters A H and x 0 are known, as well as the condition of extremum for a certain
criterion of the optimization problem, it is possible to determine all the other turn parameters.
V. Minimization of the consumed power
As an example, the results of the solution of the problem of minimization of power consumed
by the turn are shown here, when the function of purpose can be presented as the function of
3 4 2 2 2 2
one variable r k (Fig. 1.): P 3
i = 2 16π
q ρH
( x0
+ rk
) / rk
. To be exact, the solution of the
mentioned problem resulted in getting the following relations:
0. 447x
2 2
rk = 0 , d 12. 0573
c
H Ax0
2
ρ
2 x
= , 3
0
U = 0.
7234 ρq
, i = 13. 145H
Ax0
.
q
H
A