Setup & Principle

The setup used for measuring the power actually consumed in radio-frequency excited, capacitively coupled plasmas is presented in the figure below:

It consists of an RF oscillator providing purely sinusoidal waves in the range of 25 KHz to 250 MHz, a linear RF amplifier, a conventional wattmeter and, depending on the excitation frequency,  a π- or L- type matching network.  The Faraday cub hosts a shunt circuit, a passive (typically 1:100 or 1:1000 voltage probe) and an inductive current probe. The digital two channel oscilloscope and the personal computer that are also shown in the figure are necessary for power and impedance calculation.

The method is based on recording voltage and current waveforms near the RF electrode but outside the vacuum chamber and the use of an equivalent electrical circuit representing the elements from the point of measurement up to the electrode surface.

This equivalent electrical circuit accounts for the stray impedance of the cell and the shunt and is used to convert the voltage and current waveforms measured to the equivalent waveforms at the RF electrode surface.

The shunt circuit consists of a high voltage variable air capacitor and a high voltage inductor and it is used to null the parasitic current that flows from the RF line to the ground through the RF electrode insulator at the plasma off condition.

Fast Fourier Transform of the voltage and current waveforms is also essential especially for the accuracy of the calculation of the phase difference between voltage and current in the discharge off and on conditions and consequently for the accuracy of the calculation of the discharge impedance. A more precise description of the electrical circuit and of the method in general is presented below. 

RF Power measurement and control

The equivalent electrical circuit is presented in the figure above.

V_n and I_n are the voltage and current waveforms measured using the voltage and current probe, while V_en, I_en are the equivalent voltage and current waveforms at the RF electrode.

The electrical analysis of the circuit shows that that the n^th harmonic of the complex electrode voltage V_en and the complex electrode current I_en are related to the n^th harmonic component of the measured complex voltage V_n and the measured complex current I_n by the relations (absence of shunt circuit):

For the actual cell parameters, the parasitic currents are large and the equivalent magnitudes are very sensitive even to small errors in the measured values. In fact, the electrode current I_en is a very small quantity compared the measured current I_n. This is because the actual electrode current results as the difference between two large numbers: the measured current minus the displacement current  jω_nCV_n due to the capacitance between the RF electrode and the grounded shield. To avoid this difficulty the shunt circuit consisting of the capacitor C_s and the inductor L_s is connected to the electrical circuit. The shunt is tuned by exciting the empty cell with a sinusoidal voltage and adjusting the capacitor C_s until the measured current is nulled. In this ideal situation the net reactance jωL_s+1/jωC_s of the shunt cancels the net reactance of jωL+1/jωC of the cell equivalent circuit.

The shunt and the cell however are not pure reactances. They have also stray resistive components which cause additional losses upon them. Therefore the shunt does not exactly nulls the measured current. It is observed that the current takes a minimum value in the tuned situation. The remaining residual current is ~3.5% of the current without the shunt present. The attachment of the shunt circuit results to additional and considerable ohmic parasitics.

Measurements of this residual current have lead to the inclusion of this stray resistive component of the cell - shunt in the equivalent circuit. Namely, with the shunt properly tuned, the input impedance V₁/I₁ of the empty cell and shunt circuit is independent of the excitation voltage. Moreover, this residual current is in phase with voltage at the point of measurement when the cell is driven with a sinusoidal voltage. This behavior is as if the residual current flows, in parallel to the shunt circuit, from the RF power lead to the ground. This is taken into account by introducing a resistance R_L in the equivalent circuit parallel to the shunt. The additional capacitor C_L is introduced to account for the existence of the dc bias of the powered electrode which is due to the capacitive coupling of the power generator to the cell. This capacitance is considered to be infinite and is not involved in the calculations.

Finally, taking into account the shunt elements and the resistance R_L the equations used for converting the measured values to the equivalent values at the electrode are given by the relations:

Experimental determination of the circuit elements

The determination of the exact values of the electrical components that are involved in the equivalent circuit combined to the measured values V_n, In can lead to the calculation of the voltage V_en and the current I_en at the surface of the RF electrode. The values of the cell capacitance C and cell inductance L are determined by exciting the empty cell with the shunt circuit disconnected at different driving frequencies. The results of the cell impedance that is calculated from the measured values of voltage and current, are further used to calculate the inductive L, capacitive C and resistive R components of the cell.

The experimental measurements and the fitting with the RLC model used for the calculation of voltage and current waveforms on the RF electrode are shown in the figure above. The excellent agreement between the experimental and the RLC model results, confirms the applicability of the equivalent circuit.

These results are applied to derive the following electrical characteristics for the chamber used in this example: inductance L=40nH, capacitance C=285pF, resistance R=0.8Ω and a resonance reactor frequency of 43MHz.

Furthermore, the resistance R_L that has been introduced to the equivalent circuit to counterbalance the remaining residual current can be calculated by using the following procedure. The shunt circuit is connected to the circuit, is tuned and the cell is excited at different driving voltages. The values of voltage and current are recorded and plotted in the figure presented above. The slope of the curve result to the calculation of the resistance R_L=1700 Ω.

In addition the values of the shunt circuit elements Cs and Ls are known as the capacitor is scaled and the inductor has fixed values. As the values of all the elements that are involved to the equivalent circuit can be determined, the values of the voltage V_en and current I_en at the surface of the RF electrode can be calculated. These values can be further used for the calculation of:

A complete description of the method as well as the extension to frequencies higher than 13.56 MHz are given in the following PTL_UP publications:

1. "Power dissipation and impedance measurements in radio frequency discharges"
   N. Spiliopoulos, D. Mataras, D. E. Rapakoulias
   J. Vac. Sci. Technol. A 14, 2757 (1996) ^©PDF

2. "Frequency variation under constant power conditions in hydrogen radio frequency discharges"
   E. Amanatides and D. Mataras
   J. Appl. Phys. 89, 1556 (2001) ^©PDF

Applications of the method are presented in a number of publications as well (see publications list).

Current and voltage waveforms - harmonics

The usefulness of the method is shown in the above figure.

The black curves in figures a, b represent the voltage and current waveforms as measured. The measurements are in an Argon discharge at a gas pressure of 100 mTorr, frequency of 13.56 MHz and excitation peak to peak voltage of 100 Volts.

The red curves represent the voltage and current waveforms transformed by applying the equivalent circuit for the RF electrode.

The measured and the calculated waveforms of the voltage differs only in phase, the amplitude being almost the same. In addition the contribution of higher harmonics is almost negligible being less than 2% of the fundamental frequency. On the other hand, there is a significant difference between the current waveform at the point of measurement and at the RF electrode. The current waveform at the RF electrode that represents the real discharge current differs in phase, amplitude and distribution of harmonics compared to the measured current. The difference in phase is the result of the delay that is introduced by the current probe. The difference in amplitude is the result of the current flow though the shunt circuit and the flow from the RF electrode to the ground. The strong hanarmonicity (about 20 % of the fundamental frequency, figure d) is the result of the non-linear response (non-linear impedance) of the discharge.