Tutorial on RF impedance matching using the Smith chart. Examples are shown plotting reflection coefficients, impedances and admittances. A sample matching network of the MAX2472 is designed at 900MHz using graphical methods.
When dealing with the practical implementation of RF applications, there are always some nightmarish tasks. One is the need to match the different impedances of the interconnected blocks. Typically these include the antenna to the low-noise amplifier (LNA), power-amplifier output (RFOUT) to the antenna, and LNA/VCO output to mixer inputs. The matching task is required for a proper transfer of signal and energy from a "source" to a "load."
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At high radio frequencies, the spurious elements (like wire inductances, interlayer capacitances, and conductor resistances) have a significant yet unpredictable impact on the matching network. Above a few tens of megahertz, theoretical calculations and simulations are often insufficient. In-situ RF lab measurements, along with tuning work, have to be considered for determining the proper final values. The computational values are required to set up the type of structure and target component values.
The primary objectives of this article are to review the Smith chart's construction and background, and to summarize the practical ways it is used. Topics addressed include practical illustrations of parameters, such as finding matching network component values. Of course, matching for maximum power transfer is not the only thing we can do with Smith charts. They can also help the designer with such tasks as optimizing for the best noise figures, ensuring quality factor impact, and assessing stability analysis.
Before introducing the Smith chart utilities, it would be prudent to present a short refresher on wave propagation phenomenon for IC wiring under RF conditions (above 100MHz). This can be valid for contingencies such as RS-485 lines, between a PA and an antenna, between an LNA and a downconverter/mixer, and so forth.
For this condition, the energy transferred from the source to the load is maximized. In addition, for efficient power transfer, this condition is required to avoid the reflection of energy from the load back to the source. This is particularly true for high-frequency environments like video lines and RF and microwave networks.
A Smith chart is a circular plot with a lot of interlaced circles on it. When used correctly, matching impedances, with apparent complicated structures, can be made without any computation. The only effort required is the reading and following of values along the circles.
A Smith chart is developed by examining the load where the impedance must be matched. Instead of considering its impedance directly, you express its reflection coefficient L, which is used to characterize a load (such as admittance, gain, and transconductance). The L is more useful when dealing with RF frequencies.
In order to reduce the number of unknown parameters, it is useful to freeze the ones that appear often and are common in the application. Here Z0 (the characteristic impedance) is often a constant and a real industry normalized value, such as 50, 75, 100, and 600. We can then define a normalized load impedance by:
Here we can see the direct relationship between the load impedance and its reflection coefficient. Unfortunately, the complex nature of the relation is not useful practically, so we can use the Smith chart as a type of graphical representation of the above equation.
Figure 4a. The points situated on a circle are all the impedances characterized by a same real impedance part value. For example, the circle, r = 1, is centered at the coordinates (0.5, 0) and has a radius of 0.5. It includes the point (0, 0), which is the reflection zero point (the load is matched with the characteristic impedance). A short circuit, as a load, presents a circle centered at the coordinate (0, 0) and has a radius of 1. For an open-circuit load, the circle degenerates to a single point (centered at 1, 0 and with a radius of 0). This corresponds to a maximum reflection coefficient of 1, at which the entire incident wave is reflected totally.
Figure 4b. The points situated on a circle are all the impedances characterized by a same imaginary impedance part value x. For example, the circle = 1 is centered at coordinate (1, 1) and has a radius of 1. All circles (constant x) include the point (1, 0). Differing with the real part circles, can be positive or negative. This explains the duplicate mirrored circles at the bottom side of the complex plane. All the circle centers are placed on the vertical axis, intersecting the point 1.
To complete our Smith chart, we superimpose the two circles' families. It can then be seen that all of the circles of one family will intersect all of the circles of the other family. Knowing the impedance, in the form of r + jx, the corresponding reflection coefficient can be determined. It is only necessary to find the intersection point of the two circles corresponding to the values r and x.
The reverse operation is also possible. Knowing the reflection coefficient, find the two circles intersecting at that point and read the corresponding values r and on the circles. The procedure for this is as follows:
Because the Smith chart resolution technique is basically a graphical method, the precision of the solutions depends directly on the graph definitions. Here is an example that can be represented by the Smith chart for RF applications:
It is now possible to directly extract the reflection coefficient on the Smith chart of Figure 5. Once the impedance point is plotted (the intersection point of a constant resistance circle and of a constant reactance circle), simply read the rectangular coordinates projection on the horizontal and vertical axis. This will give r, the real part of the reflection coefficient, and i, the imaginary part of the reflection coefficient (see Figure 6).
The Smith chart is built by considering impedance (resistor and reactance). Once the Smith chart is built, it can be used to analyze these parameters in both the series and parallel worlds. Adding elements in a series is straightforward. New elements can be added and their effects determined by simply moving along the circle to their respective values. However, summing elements in parallel is another matter. This requires considering additional parameters. Often it is easier to work with parallel elements in the admittance world.
If we know z, we can invert the signs of and find a point situated at the same distance from (0, 0), but in the opposite direction. This same result can be obtained by rotating an angle 180 around the center point (see Figure 7).
Of course, while Z and 1/Y do represent the same component, the new point appears as a different impedance (the new value has a different point in the Smith chart and a different reflection value, and so forth). This occurs because the plot is an impedance plot. But the new point is, in fact, an admittance. Therefore, the value read on the chart has to be read as siemens.
Let's consider the network of Figure 8 (the elements are normalized with Z0 = 50). The series reactance (x) is positive for inductance and negative for capacitance. The susceptance (b) is positive for capacitance and negative for inductance.
The circuit needs to be simplified (see Figure 9). Starting at the right side, where there is a resistor and an inductor with a value of 1, we plot a series point where the r circle = 1 and the l circle = 1. This becomes point A. As the next element is an element in shunt (parallel), we switch to the admittance Smith chart (by rotating the whole plane 180). To do this, however, we need to convert the previous point into admittance. This becomes A'. We then rotate the plane by 180. We are now in the admittance mode. The shunt element can be added by going along the conductance circle by a distance corresponding to 0.3. This must be done in a counterclockwise direction (negative value) and gives point B. Then we have another series element. We again switch back to the impedance Smith chart.
Before doing this, it is again necessary to reconvert the previous point into impedance (it was an admittance). After the conversion, we can determine B'. Using the previously established routine, the chart is again rotated 180 to get back to the impedance mode. The series element is added by following along the resistance circle by a distance corresponding to 1.4 and marking point C. This needs to be done counterclockwise (negative value). For the next element, the same operation is performed (conversion into admittance and plane rotation). Then move the prescribed distance (1.1), in a clockwise direction (because the value is positive), along the constant conductance circle. We mark this as D. Finally, we reconvert back to impedance mode and add the last element (the series inductor). We then determine the required value, z, located at the intersection of resistor circle 0.2 and reactance circle 0.5. Thus, z is determined to be 0.2 + j0.5. If the system characteristic impedance is 50, then Z = 10 + j25 (see Figure 10).
Another function of the Smith chart is the ability to determine impedance matching. This is the reverse operation of finding the equivalent impedance of a given network. Here, the impedances are fixed at the two access ends (often the source and the load), as shown in Figure 11. The objective is to design a network to insert between them so that proper impedance matching occurs.
At first glance, it appears that it is no more difficult than finding equivalent impedance. But the problem is that an infinite number of matching network component combinations can exist that create similar results. And other inputs may need to be considered as well (such as filter type structure, quality factor, and limited choice of components).
The approach chosen to accomplish this calls for adding series and shunt elements on the Smith chart until the desired impedance is achieved. Graphically, it appears as finding a way to link the points on the Smith chart. Again, the best method to illustrate the approach is to address the requirement as an example. 152ee80cbc
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