COMSOL 6.4 - Impedance Matching

A radio frequency (RF) power supply is used to supply power to a capacitively coupled plasma (CCP). Commercially available RF power supplies are specified to deliver power into a 50 ohm load, and interconnecting coaxial cables and connectors have a 50 ohm characteristic impedance. On the other hand the CCP impedance is typically not equal to 50 ohm and has both resistive and reactive components. For instance in this example the plasma impedance is

at 13.56 MHz and 10 W. A matching network is introduced between the RF power supply and the CCP allowing maximum power transfer and a safe operating region for the power supply. In operation the match components may be fixed or variable and “tuned” either manually or automatically. Modern plasma reactors may be powered by multiple power supplies at different operating frequencies. This note shows how a match can be designed using dimensions of the chamber, properties of the feed gas and the specification of the power supply.

Model Definition

Figure 1 is a schematic of a RF power supply (Vs, Rs) connected to a L–type match network (Lm, Cp) with an additional DC blocking capacitor (Cb). The match network is connected to a plasma load with a parallel stray capacitance (Cs). The plasma voltage and current are V and I.

Figure 1: L–match including blocking and stray capacitance.

Figure 2 shows a simplified schematic, in this case the blocking capacitor (Cb) is removed (since the discharge is symmetric and there will be no DC self-bias). The stray capacitance (Cs) is ignored.

Figure 2: L–match used in this model.

In order to compute the optimum values for the parallel capacitor Cp and the inductor Lm, the plasma impedance is required, Zp. Using

and the generator source resistance, Rs, the following can be used to compute Cp and Lm. Defining:

(1)

and

(2)

then the match inductance is given by:

(3)

and the parallel capacitance is given by:

(4)

Results and Discussion

In order to evaluate the results, two meaningful measures are used, the maximum power transfer coefficient and the efficiency. The maximum power transfer is given by:

(5)

where Vs is the generator voltage, and the maximum power transfer coefficient is then given by:

(6)

The efficiency is then simply:

(7)

where Ps is the power lost in the generator.

Figure 3 shows that the maximum of power transfer coefficient is 1 when the plasma power dissipation is 10 W. This is expected because the values of Cp and Lm used to calculated the match components were determined at 10 W plasma power.

Figure 4 and Figure 5 show a plot of maximum power transfer coefficient as a function of frequency and pressure respectively. Matching occurs at 13.56 MHz and 1 torr corresponding to the conditions used to determine Cp and Lm. At low pressure the plasma is more resistive and a significant mismatch occurs. The effect is less pronounced at higher pressures.

Effect of Harmonics

Finally, the plasma impedance was measured at 50 W, a match was designed. Figure 6 shows the power transfer coefficient as a function of power. It is evident that the maximum value is around 45 W, 10% lower than expected.

The match parameters are calculated from the voltage and current amplitudes at the fundamental frequency. However the current waveform has a significant third harmonic contribution (4.5%) which accounts for the difference between the measured and calculated power at maximum power transfer coefficient.

These results show how a matching network for a CCP reactor can be designed using the dimensions of the reactor, the properties of the feed gas and the characteristics of the generator. Estimates of the power transfer coefficient as a function of power frequency and pressure can be made. The same procedure can be used for 2D reactor geometries.

Figure 3: Plot of the power transfer coefficient and efficiency versus power.

Figure 4: Plot of the power transfer coefficient and efficiency versus frequency.

Figure 5: Plot of the power transfer coefficient and efficiency versus pressure.

Figure 6: Plot of the power transfer coefficient and efficiency versus power at high power.

Reference

1. M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, 2005.

Plasma_Module/Capacitively_Coupled_Plasmas/impedance_matching

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, The model is set up in an identical way to the computing plasma impedance model, except an external circuit is used to drive the discharge.

click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Add parameters to compute the match inductance and capacitance.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L	0.025[m]	0.025 m	Discharge gap
de	0.3[m]	0.3 m	Electrode diameter
As	0.25pide^2	0.070686 m²	Electrode area
P0	10[W]	10 W	Input power
f0	13.56E6[Hz]	1.356E7 Hz	Frequency
p0	1[torr]	133.32 Pa	Pressure
T0	300[K]	300 K	Temperature
Rp	42.696[ohm]	42.696 Ω	Plasma impedance, real part
Xp	-156.62[ohm]	-156.62 Ω	Plasma impedance, imaginary part
Rs	50[ohm]	50 Ω	Generator impedance
fmatch	13.56E6[Hz]	1.356E7 Hz	Frequency at perfect match
Xmd	sqrt(Rp*Rs-Rp^2)-Xp	174.28 Ω	Match help variable
Bm	((1/(Rp*Rs))-(1/Rs^2))^0.5	0.0082721 S	Match help variable
Lm	Xmd/(2pifmatch)	2.0455E-6 H	Inductance
Cp	Bm/(2pifmatch)	9.7091E-11 F	Capacitance