Studies on characteristics of resistive power calculated with discrete Fourier transform in a pulse-modulated radio frequency discharge

In a pulse-modulated (PM) radio-frequency (RF) capacitively coupled plasma, the DFT (Discrete Fourier Transform)-calculated RF power and the corresponding phase shift between voltage and current measured with calibrated voltage and current probes present oscillations in the pulse rising and falling edges. The oscillating phase shift between voltage and current obtained in the falling edge is outside the expected value for a resistive-capacitive RF discharge. Numerical simulation and analytical analysis are made to interpret these abnormal characteristics and seek an approach to obtaining the reliable resistive (active) RF power. The oscillation is proved to be originated from the oscillating non-zero reactive RF power of the capacitor(s) in the load. At the time instant when the reactive RF power within an integer RF period is not zero, the reactive RF power is mistakenly regarded as the active RF power in the DFT analysis, as a result, the corresponding phase is thus incorrect and even outside the expected value for a resistive-capacitive load. The resistive RF power and the phase can be only correctly calculated at the time instant when the reactive RF power is zero. For a series (or parallel) RC (resistor-capacitor) load and a combined RC load with the dominated series (or parallel) RC impedance, the time instant of the zero reactive RF power is calculated with one of the two proposed empirical formulae. In practice, the DFT-calculated resistive RF power is obtained according to the following procedures: (1) applying DFT to the measured RF voltage and current signals to obtain the power and time instants for minimal phase shifts between voltage and current; (2) selecting the empirical formula to calculate time instants of the zero reactive RF power; (3) getting resistive powers at time instants of the zero reactive RF power. In real PM RF capacitively coupled plasmas, the empirical formula for the series RC load is selected to calculate the resistive RF power. The accuracy of DFT-calculated resistive RF power is proved to be related to two kinds of errors. The first is the error of the time instant of the zero reactive RF power calculated using the empirical formula. This error is relatively lower when the requirement that the dominated parallel or series RC impedance is met and is almost independent of the impedance phase angle of a combined RC load. The second is the error of the DFT-calculated resistive RF power compared with the corresponding time i