Characterization of Coupling Coefficient for Transformer Model Application in Non-Invasive Plasma Monitoring

Coupling coefficient ($k$) in transformer model, which is widely used in the circuit analysis of inductively coupled plasma(ICP) discharges, is a global value that indicates the degree of coupling by the change of magnetic flux between the primary antenna current and the secondary plasma current. Although the determinants of $k$ include the skin depth as a function of electron density and effective electron collision frequency, those plasma parameters can be calculated inversely from the relation between $k$ and electrical parameters at the antenna of ICP without additional diagnostic tools at a given pressure. In real-time non-invasive plasma diagnosis method using the determined $k$ based on the transformer model, the magnetic inductance due to the discharge current path ($L_{p}$) is assumed to be equal to the inductance of antenna ($L_{a}$) to simplify the circuit analysis model1. This study clarifies the applicable discharge condition of $L_{p}=L_{a}$ assumption in transformer model analysis by measuring voltage and current(V-I) of the antenna and plasma parameters. Difference between calculated values of $k$ from the measured antenna V-I and plasma properties in various conditions explains the possible range of $L_{p}=L_{a}$ assumption. Experiment was conducted on H-mode ICP discharge and the results are in good agreement with the predictive trends based on numerical approach of electromagnetic model. Verification of coupling coefficient in transformer model with electric signals and plasma parameters shows that monitoring of coupling coefficient can be adopted for real-time monitoring of volume averaged plasma parameter in non-invasive ICP plasma diagnostics.