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IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 5, SEPTEMBER 1996
Numerical Modeling of an Induction Plasma Installation Including the Generator State Space Model J.P. Ploteau, J. Fouladgar, F. Auger, A. Chentouf, G. Develey GE44 - LRTI, CRTT, BP 406,44602 Saint-Nazairecedex. France
Abstract-A complete inductively coupled radio-frequency plasma installation is modelized This installation includes an electrical generator coupled with a cylindrical inductor containing the plasma. The mathematical modeling of the generator with its triode uses a state space approach. A RungeKutta method is used to solve these differential equations. The electromagnetic equation of the inductor is solved by the moment method, whereas the electromagnetic equation of the plasma is solved by the finite difference method. The thermal equation of the argon plasma is solved by the confrol volume method. The simulations allows to study the influence of the installation components on voltages, currents, dissipated powers and generator performances.
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GENERATOR
LOAD
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Quartztorch
Fig. 1. Plasma installation
I. INTRODUCTION
11. PLASMA SIMULATION
In an induction plasma installation, a high frequency generator is coupled with a circular inductor. The inductor creates an electromagnetic field and ionizes a gas, usually argon, inside a cylindrical quartz tube (Fig 1). To determine the operating point of the generator, one should compute the impedance of the plasma and then solve the state equations of the generator. Since the impedance of the plasma depends on the temperature of the gas, and the temperature of the gas depends on the electromagnetic field, thus on the voltage and frequency, one should solve simultaneously the equations of the generator and the electromagnetic and heat transfer equations of the plasma and inductor [l]. In our previous work [2] we have developed a voltage driven formulation to compute the impedance of an induction radio frequency plasma using a combination of the moment method, finite difference method and control volume method. In this paper we introduce the generator state equations in the numerical simulation of the plasma so as to compute the operating parameters of the complete installation. The simultaneous resolution of these equations is very difficult. So we have developed a hierarchical algorithm which separates the different subsystems of the installation and their different physical equations. This algorithm allows also to separate the linear and non linear equations. The algorithm is developed under the PC Matlab environment.
The inductor is divided into N turns, each one with n elementary loops (31 and the plasma is divided into N, cylindrical volume elements. The Biot and Savart law is first used to compute the vector potential A in both the inductor and the plasma. This vector potential is then used to derive the current density J, from the voltage of each cylindrical loop **i”by means of the equation :
Manuscript received March 3, 1996. e-mail
[email protected], fax (33) 40-17-26-18; phone (33) 40-17-
(r,z) are the polar coordinates, po and o (T) are respectively the plasma permeability (i.e. vacuum) and the plasma electrical conductivity.
2640. This work was supported by “Electricit6 De France” and lhc “Centre F m p i s
+ jo
n
N
A,,, + jo EAk,, k=I
whereri is the radius, oi is the electrical conductivity of inductor element &
