# Impedance Matching. Using transformers Using matching networks

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1 Impedance Matching The plasma industry uses process power over a wide range of frequencies: from DC to several gigahertz. A variety of methods are used to couple the process power into the plasma load, that is, to transform the impedance of the plasma chamber to meet the requirements of the power supply. A plasma can be electrically represented as a diode, a resistor, and a capacitor in parallel, as shown in Figure 1. Most AC generators are designed to operate into a 5 Ω load because that is the standard the industry has settled on for measuring and transferring high-frequency electrical power. The function of an impedance matching network, then, is to transform the resistive and capacitive characteristics of the plasma to 5 Ω, thus matching the load impedance to the AC generator s impedance. Two of the most common methods for matching generator and load impedances are: Using transformers Using matching networks Figure 1. Simplified electrical model of plasma Although this is a very simple model, it represents the basic characteristics of a plasma. The diode effects arise from the fact that the electrons can move much faster than the ions (because the electrons are much lighter). The diode effects can cause a lot of harmonics (multiples of the input frequency) to be generated. These effects are dependent on the process and the chamber, and are of secondary concern when designing a matching network. This paper discusses both methods and also introduces you to the Smith chart, a tool that can be used to calculate the value and type of components needed to construct an impedance matching network.

2 a d v a n c e d e n e r g y When we compare the two impedance matching methods mentioned above, we see that they have different applications and strengths: Transformer Simplest type of impedancematching network Easy installation Simple design Low weight Inexpensive Best used in applications below 1 MHz Transformers Transformers convert source power from one voltage and current level to another voltage and current level. To achieve maximum power transfer from source to load, the load s impedance must match the characteristic impedance of the generator. The load impedance is transformed as a square of the voltage-transformation ratio. The ratio of the voltage transformation is referred to as N. N comes from the number of turns on the input winding divided by the number of turns on the output winding. A transformer with 2 turns on the input and 1 turns on the output is referred to as a 2:1 step-down transformer. If the load is a pure 1 Ω resistive load and a transformer with a 2:1 step-down ratio is used (N=2), the impedance that the generator sees is: 2 2 x1=4 Ω Transformers are good for nearly matching the resistive portion of the load. The impedances might not match exactly, so the generators must be able to absorb some reflected power (which usually results in heat). 4 Ω 1 Ω Matching Network Used in applications above 1 MHz Enables impedances to be matched more exactly than with a transformer Generally larger and more difficult to install than a transformer More difficult to design than a transformer Because matching networks do a good job of matching impedances, heating due to reflected power is not typically a problem. Figure 2. Schematic of a 2:1 step-down transformer Step Up or Step Down? The terms step up and step down refer to the ability of the transformer to change ( transform ) the voltage or the current that passes through it. The amount of power (VxI) that goes into a transformer is always equal to the amount of power that comes out (discounting negligible losses). This means that while a stepdown transformer is changing the input voltage to a lower output voltage, it is also changing the input current to a higher output current. Generally, it is the voltage we are referring to when we talk about stepping up or stepping down. 2

4 a d v a n c e d e n e r g y A B Resistance C Figure 4. The Smith chart Figure 3. Illustration of shunt and series elements in a circuit D Conversion Between Elements The following equations show one way to convert component values between series and shunt circuits: Rs=Rp/1+(Rp/Xp) 2 Xs=RsRp/Xp Rp=(Rs 2 +Xs 2 )/Rs Xp=(Rs 2 +Xs 2 )/Xs Rs=series resistance Xs=series reactance Rp=parallel resistance Xp=parallel reactance Smith charts contain many coordinate grids used to calculate electrical characteristics for electronic circuits. In modern use, transparent plastic sheets with Smith charts printed on them allow the designer to lay one chart on another to calculate device values. Many computerized design programs also incorporate Smith charts. To understand how a Smith chart is constructed, look at Figure 5, which is a basic X-Y coordinate system. We can plot impedance values on this chart in two dimensions. Smith Char ts Calculating the values of the devices needed to build an impedance-matching network requires an understanding of the characteristic impedance of the generator and the load impedance. The plasma can be modeled as having an electrical impedance. This facilitates the use of standard electronic matching network design techniques. Reactance to + Resistance to + In actual design, other factors, such as the power limitations of the components and the effects of the components packaging, must also be understood and considered. However, we will focus on impedance during this discussion because it is the fundamental parameter to understand. One way to determine the device values is to do a numerical analysis of the circuit, which can be time consuming and complex. Another way is to do a graphical analysis using a Smith chart, which is shown in Figure 4. to Figure 5. Basic X-Y coordinate system 4

5 w h i t e p a p e r Figure 6 is similar to Figure 5, but with logarithmic scaling on the X and Y axes. Reactance to + to - Resistance Figure 6. Logarithmic X-Y coordinate system to + Now, if we bend around the endpoints (infinity symbols) of the X-Y coordinate system as shown in Figure 7, we see that it forms a Smith chart (Figure 8). Reactance Resistance to + to - to + Bending the endpoints () around Figure 7. Bending the endpoints of the X-Y coordinate system so that it begins to resemble a Smith chart Reactance Resistance Forms a Smith Chart Figure 8. Voila! The X-Y coordinate system becomes a Smith chart A close examination of Figure 8 reveals the purpose of each line of the chart. The straight line running horizontally represents the real resistance of the circuit. This line has remained unchanged from the original X-Y coordinate system; only the to + is shown. This means that the circles (which are sometimes referred to as constant resistance circles) that cross the center line represent the constant, real resistance of the circuit. These are the vertical reference lines on the original graph. The arcs running adjacent to the center line represent the imaginary value of the reactance (capacitance and inductance) in the circuit. These were the horizontal reference lines on the original graph. Notice that the circle that runs through the center of the chart has a 1 label. This is referred to as the unit circle. When you work with a particular characteristic impedance, for example 5 Ω, all reference points on the chart are multiplied by the value of that impedance, and calculations are made based on the new reference point values. Inductors= Positive Values Capacitors= Negative Values Figure 9. Specific locations on the Smith chart determine device types and values. By examining Figure 9, we see that: Because capacitors have negative impedance values on the impedance chart, they are represented below the center line. Because inductors have positive impedance values, they are represented above the center line. Essentially, there are two forms of the Smith chart: the impedance chart and the admittance chart, shown in Figure 1. Impedance Chart -jb +jb Admittance Chart Figure 1. The two forms of the Smith chart 5

10 a d v a n c e d e n e r g y -j4 ms B This completes the design of a simple, single-point impedance matching network. AE engineers use Smith charts to design impedance-matching networks for various types of plasma systems. C For more information about Smith charts, refer to Electronic Applications of the Smith Chart by Phillip H. Smith (ISBN ). Z in =5 Ω, Y in =2 ms 4 ms (47 pf) +j5 Ω (587 nh) 1-j3 Ω Figure 21. Using an admittance chart to translate the susceptance along a constant conductance circle 1

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