In this work capacity of tokamak plasma is calculated using modeling of tokamak configuration as toroidal and coaxial capacitor. This value is very important and plays an important role in time- varying regimes in tokamak. For exact simulation of plasma behavior, this amount will be added to circuit equations and transport codes. Since capacitive properties of tokamak cause production of a radial electric field, it deserves our special attention.
Tokamak is a torodial shape magnetic confinement fusion device that is the best candidate for nuclear fusion reactors [
Estimation the value of capacitance of the equal circuit of tokamak can be done by a torodial coaxial capacitor (
The Laplace equation [
With substituting the above values in Equation (1) and with respect to that in this system
tion (1) becomes as follows.
Because of the uniqueness of the solutions of Laplace equation, there will be only one answer that will satisfy the above conditions.
It is obvious that in a constant
period is
scribed as a cosine Fourier series of
Now, it is necessary to obtain expressions for
the simplest solution, we first put the
the coefficients of
On the other hand, the solution to Laplace equation for such coordinate system would be in the form of
This solution provides the boundary conditions
With respect to that the boundaries are considered as co-potential surface, on them
electrical charge on each surface of torus is equal to
In this relation
By putting the calculated values for
Since
In limit of
the same formula for capacitance in concentric cylinders. It should be noted that
as
magnetic field, mass density and velocity of light, respectively. Because the values of
The electrical equivalent circuit for tokamak after taking into account the capacitor property is shown in
calculated by
equations,
In Equation (8),
The general solution of Equation (9) for the plasma current can be written as:
In Equation (10)
damping constant, respectively.
rent, respectively.
important factor that can play an important role in various regimes of the tokamak plasma. The values of natural frequencies for different tokamaks are in order of
In fact, according to the properties of electrical elements, in dc regimes (following the end of transient regimes),
The first effects of capacitance are weak damping oscillations with natural frequency in plasma current, radial electric fields and so on, detection of them is usually difficult. Its second effects are to cause some types of fluctuations in density and other plasma parameters at the edge of plasma. Such fluctuations have been observed in Damavand tokamak [
The experimental data of the Damavand tokamak are [
On the basis of dimension and condition of Damavand tokamak we would have:
After some calculating it would be:
And, therefore, it would be:
As previously noted, the first effect of the capacitance is weak damping oscillations with the frequency
In the experiments with Damavand tokamak that a sample of their results is demonstrated in
the negative spike of loop voltage is observed, the disruption instability happened and
damping oscillations. From the calculations using the presented model, it is seen that the frequency of these oscillations is approximately 100 kHz with damping in form of
them is observed [
When a power source (such as induced current, radio frequency wave, neutral beam) is injected into the plasma, there will be two working regimes:
1) The first one is a transient regime in which L and C along with R play role in equation of circuit and to simulate the behaviour of the plasma, using RLC circuit (
2) In the second regime that is a steady state regime, a simple resistive circuit with circuit equation of
Therefore, the capacitance of the plasma like its inductance, plays important role in some working regimes of tokamak. The importance of capacitance properties is not limited to regimes with time-varying loop voltages. In
each transient regime and regimes with time varying power source (such as radio frequency waves and neutral beam heating) the effect of capacitive property can be seen. In
The radial electric field in study of the H mode of tokamak plasmas is a very important parameter and several models have been proposed to explain the origin of them. In this paper, the effects of capacitive property on radial electric field in tokamak are briefly explained. It is proposed that a radial electric field is produced by a ra-
dial current in the form of
Then, a new model for equivalent circuit of tokamak plasmas on the base of capacitive property is explained. The plasma has the capacitive property of
in the form of
value for capacitance in Equation (8) and
In this article, the analysis on the base of this model is first accomplished for Damavand tokamak and a good agreement between results of this model and the experimental results is observed. This model may be extended in future for analysis of the performance of big tokamaks such as ITER.