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Tried and true, the Smith chart is still the basic tool for determining transmission line impedances.

When dealing with the practical implementation of RF applications there are always some tasks that appear nightmarish. One of these is the need to match the different impedances of the interconnected block. Typically these include the antenna to low-noise amplifier (LNA), RF ouput (RFOUT) to antenna, LNA output to mixer input, etc. The matching task is required for a proper transfer of signal and energy from a "source" to a "load".

At high radio frequencies, the spurious (wires inductances, interlayers capacitances, conductors resistances, etc) elements have a significant, yet unpredictable impact upon the matching network. Above a few tenths of MHz, 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.

There are many possible ways to do impedance matching. Some are:

- Computer simulations-Complex to use since such simulators are dedicated for differing design functions and not to impedance matching. The designer has to be familiar with the multiple data inputs that need to be entered and the correct formats. They also need the expertise to find the useful data among the tons of results coming out. In addition, circuit simulation software is not pre-installed on computers unless they are dedicated to such an application.

- Manual computations-Tedious due to the length ("kilometric") of the equations and the complex nature of the numbers to be manipulated.

- Instinct-This can be acquired only after one has devoted many years to the RF industry. In short, this is for the super-specialist!

- Smith Chart-Upon which this article concentrates.

The primary objective of this article is to refresh the Smith chart's construction and background, and to summarize practical ways to use it.

Topics addressed will include practical illustrations of parameters such as finding matching network componentS values. Of course, matching for maximum power transfer is not the only thing one can do with Smith charts. They can also help the designer optimize for the best noise figures, insure quality factor impact, asses stability analysis, etc.

A quick primer

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 100 MHz). This can be true for contingencies such as RS485 lines, between a PA and an antenna, between a LNA and down- converter/mixer, etc.

It is well known that to get the maximum power transfer from a source to a load, the source impedance must equal the complex conjugate of load impedance, or :

R subscript s + jX subscript s = R subscript L - jX subscript L (1)

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.

What it is

A Smith chart is a circular plot with a lot of interlaced circles on it. When correctly used, matching impedances, with apparent complicate structures, can be made without any computation. The only effort required is the reading and following of values along the circles.

The Smith chart is a polar plot of the complex reflection coefficient (also called gamma and symbolized by cap gamma). Or, mathematically defined as the 1- port scattering parameter s or s subscript 11.

A Smith chart is developed by examining the load where the impedance must be matched. Instead of considering its impedance directly, one expresses its reflection coefficient GL, which is used to characterize a load (such as admittances, gain, transconductances, etc). The cap gamma subscript L is more useful when dealing with RF frequencies.

We know the reflection coefficient is defined as the ratio between the reflected voltage wave and the incident voltage wave :

cap gamma = V subscript refl over V subscript inc

The amount of reflected signal from the load is dependent on the degree of mismatch between the source impedance and the load impedance. Its expression has been defined as follows:

cap gamma = V subscript refl over V subscript inc = Z subscript L - Z subscript O over Z subscript L + Z subscript O = cap gamma ??? + j cap gamma 1

(2.1)

Since the impedances are complex numbers, the reflection coefficient will be a complex number as well.

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 Z subscript o (the characteristic impedance) is often a constant and a real industry normalized value ie: 50 cap omega, 75 cap omega, 100 cap omega, 600 cap omega, etc. We can then define a normalized load impedance by:

z = Z subscript L/Z subscript o = (R + jX) / Z subscript o = r + jx (2.2)

With this simplification, we can rewrite the reflection coefficient formula as:

???

(2.3)

Here one can see the direct relationship between the load impedance and its reflection coefficient. Unfortunately the complex nature of the relation is not practically useful, so we can use the Smith chart is a type of graphical representation of the above equation.

To build the chart, the equation must be re-written to extract standard geometrical figures (likes circles or stray lines).

First, equation 2.3 is reversed to give:

z = r + jx = 1 + cap gamma subscript L over 1 - cap gamma subscript L = 1 + cap gamma subscript gamma + j cap gamma subscript i over 1 - cap gamma subscript gamma - j cap gamma subscript i

(2.4)

and,

r = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2 over 1 + cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + cap gamma subscript i superscript 2

(2.5)

By setting the real parts and the imaginary parts of (equation 2.5) equal, we obtain two independent new relationships:

r = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2 over 1 + cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + cap gamma subscript i superscript 2

(2.6)

x = 2 cap gamma subscript i over 1 + cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + cap gamma subscript i superscript 2

(2.7)

Equation (2.6) is then manipulated, by developing equations (2.8) through (2.13), into to the final equation (2.14). This equation is a relationship in the form of a parametric equation (x-a) superscript 2 + (y-b) superscript 2 = R superscript 2),in the complex plane (cap gamma r, cap gamma i), of a circle centered at the coordinates (r/r+1, 0), and having a radius of 1/1+r.

r + r cap gamma subscript gamma superscript 2 - 2r cap gamma subscript gamma + r cap gamma subscript i superscript 2 = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2

(2.8)

cap gamma subscript i superscript 2 + r cap gamma subscript gamma superscript 2 - 2r cap gamma subscript gamma + r cap gamma subscript i superscript 2 + cap gamma subscript i superscript 2 = 1 - r

(2.9)

(1 + r)cap gamma subscript i superscript 2 - 2r cap gamma subscript gamma + (r + 1)cap gamma subscript i superscript 2 = 1 - r

(2.10

cap gamma subscript gamma superscript 2 - 2r over r + 1 cap gamma subscript gamma + cap gamma subscript i superscript 2 = 1 - r over 1 + r

(2.11)

cap gamma subscript gamma superscript 2 - 2r over r + 1 cap gamma subscript gamma + r superscript 2 over (r + 1) superscript 2 + cap gamma subscript i superscript 2

- r superscript 2 over (r + 1) superscript 2 = 1 - r over 1 + r

(2.12)

(cap gamma subscript gamma - r over r + 1)superscript 2 + cap gamma subscript i superscript 2 = 1 - r over 1 + r

+ r superscript 2 over (1 + r) superscript 2 = 1 over (1 + r) superscript 2

2.13)

(cap gamma subscript gamma - r over r + 1)superscript 2 + cap gamma subscript i superscript 2 = (1 over 1 + r)superscript 2

2.14)

(See Figure 3 for further details)

When developing the Smith chart, there are certain precautions that should be noted. Among the more important are:

- All the circles have one same, unique intersecting point at the coordinate (1, 0).

- The zero cap omega circle where there is no resistance (r = 0) is the largest one.

- The infinite resistor circle is reduced to one point at (1, 0).

- There should be no negative resistance. If one (or more) should occur, we will be faced with the possiblity of oscillatory conditions.

- Another resistance value can be chosen by simply selecting another circle, corresponding to the new value.

Back to the drawing board

Moving on, we use equations (2.15) through (2.18) to further develop equation (2.7) into another parametric equation. This results in equation (2.19).

x + x cap gamma subscript gamma superscript 2 - 2 x cap gamma subscript gamma + x cap gamma subscript i superscript 2 = 2 cap gamma subscript i

(2.15)

1 + cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + cap gamma subscript i superscript 2 = 2 cap gamma subscript i over x

(2.16)

cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + 1 + cap gamma subscript i superscript 2 - (2 over x) cap gamma subscript i = 0

(2.17)

cap gamma subscript gamma superscript 2 - 2 cap gamma subscript gamma + 1 cap gamma subscript i superscript 2 - (2 over x) cap gamma subscript i

+ 1 over x superscript 2 - 1 over x superscript 2 = 0

(2.18)

(cap gamma subscript ??? - 1) superscript ??? + (n - 1 over x) superscript 2 = 1 over x superscript 2

(2.19)

(See Figure 3a for further details)

Again, 2.19 is a parametric equation of the type (x-a)superscript 2 + (y-b) superscript 2 = R superscript 2, in the complex plane (cap gamma r, cap gamma i), of a circle centered at the coordinates (1, 1/x) and having aradius of 1/x.

Get the picture?

To complete our Smith chart, we superimpose the two circle's 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.

It's reciprocating too

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 x on the circles. The procedure for this is as follows:

- Determine the impedance as a spot on the Smith Chart.

- Find the reflection coefficient (cap gamma) for the impedance.

- Having the characteristic impedance and cap gamma, find the impedance.

- Convert the impedance to admittance.

- Find the equivalent impedance.

- Find the components values for the wanted reflection coefficient (in particular the elements of a matching network see Figure 6).

To extrapolate

Since the Smith chart resolution technique is basically a graphical method, the precision of the solutions depends directly on the graph definitions. Here are some examples that can be represented by the Smith chart for RF applications:

- Example 1: Consider the characteristic impedance of a 50 W termination and the following impedances:

Z subscript 1 = 100 + j50 cap omega Z subscript 2 = 75 -j100 cap omega, Z subscript 3 = j200 cap gamma, Z subscript 4 = 150 cap omega, Z subscript 5 = x (an open-circuit) Z subscript 6 = 0 (a short circuit), Z subscript 7 = 50 cap omega, Z subscript 8 = 184 -j900 cap omega.

Then, normalize and plot (see Figure 5.) The points are plotted as follows:

z subscript 1 = 2 + j, z subscript 2 = 1.5 -j2, z subscript 3 = j4, z subscript 4 = 3, z subscript 5 = 8, z subscript 6 = 0, z subscript 7 = 1, z subscript 8 = 3.68 -j18S.

- It is now possible to directly extract the reflection coefficient cap gamma 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 cap gamma r, the real part of the reflection coefficient) and cap gamma i, the imaginary part of the reflection coefficient (see Figure 6).

- It is also possible to take the eight cases presented in Example 1 and extract their corresponding cap gamma directly from the Smith chart of Figure 5. The numbers are:

cap gamma subscript 1 = 0.4 + 0.2j, cap gamma 2 = 0.51 - 0.4j, cap gamma 3 = 0.875 + 0.48j cap gamma 4 = 0.5, cap gamma 5 = 1, cap gamma 6 = -1, cap gamma 7 = 0, cap gamma 8 = 0.96 - 0.1j.

Working with admittance

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. It requires considering additional parameters. Often, it is easier to work with parallel elements in the admittance world.

We know that, by definition, Y = 1/Z and Z = 1/Y. The admittance is expressed in mhos or cap omega superscript -1 (in earlier times it was expressed as Siemens or S). And, since Z is complex, Y must also be complex.

Therefore, Y = G + jB, (2.20) where G is called "conductance" and B the "susceptance" of the element. One must exercise caution, though. By following the logical assumption, one may conclude that G = 1/R and B = 1/X. This, however, is not the case. If this assumption is used, the results will be incorrect.

When working with admittance, the first thing that one must do is normalize y = Y/Y subscript o. This results in y = g + jb. So, what happens to the reflection coefficient? By working through thefollowing :

cap gamma = Z subscript L - Z subscript ??? over Z subscript L + Z subscript ??? = 1 over Y subscript ??? - 1 over Y subscript ??? ocer 1 over ??? + 1 over Y ??? = ??? - Y subscript L over Y subscript ??? + Y subscript 1 = 1 - y over 1 + y

(2.21)

It turns out that the expression for G is the opposite, in sign, of z, and cap gamma(y) = -cap gamma(z).

If we know z, we can invert the signs of G 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 degrees around the center point (see Figure 7).

Of course, while Z and 1/Z 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, etc.). 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 mhos.

While this method is sufficient for making conversions, it does not work for determining circuit resolution when dealing with elements in parallel.

The admittance Smith chart

In the previous discussion we saw that every point on the impedance Smith chart can be converted into its admittance counterpart by taking a 180 degrees rotation around the origin of the cap gamma complex plane. Thus, an admittance Smith chart can be obtained by rotating the whole impedance Smith chart by 180 degrees. This is extremely convenient since it eliminates the necessity to build another chart. The intersecting point of all the circles (constant conductances and constant susceptances) is at the point (-1, 0) automatically. With that plot, adding elements in parallel also becomes easier. Mathematically, the construction of the admittance Smith chart is created by:

cap gamma subscript L = cap gamma subscript gamma + j cap gamma subscript i = 1 - y over 1 + y = 1 - g - jb over 1 + g + jb

(3.1)

then, reversing the equation:

y = g + jb = 1 - cap gamma subscript L over 1 + cap gamma subscript L = 1 - cap gamma subscript gamma - subscript j cap gamma subscript i over 1 + cap gamma subscript gamma + subscript j cap gamma subscript i

(3.2)

g + jb = (1 - cap gamma subscript gamma - subscript j cap gamma subscript i)(1 + cap gamma subscript gamma - ??? over (1 + cap gamma subscript gamma + subscript j cap gamma subscript i)(1 - cap gamma subscript gamma + ??? = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2 - j2 cap gamma subscript i over 1 + cap gamma subscript gamma superscript 2 + 2 cap gamma subscript gamma + cap gamma superscript 2 subscript i

(3.3)

Next, by setting the real and the imaginary parts of Equation 3.3 equal, we obtain two new, independent relationships:

g = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2 over 1 + cap gamma subscript gamma superscript 2 + 2 cap gamma subscript gamma - cap gamma subscript i superscript 2

(3.4)

b = -2 cap gamma subscript i over 1 + cap gamma subscript gamma superscript 2 + 2 cap gamma subscript gamma + cap gamma subscript i superscript 2

(3.5)

By developing Equation 3.4 we get:

g+ = g cap gamma subscript gamma superscript 2 + 2g cap gamma subscript gamma + g cap gamma subscript i superscript 2 = 1 - cap gamma subscript gamma superscript 2 - cap gamma subscript i superscript 2

(3.6)

cap gamma subscript gamma superscript 2 + g cap gamma subscript gamma superscript 2 + 2 g cap gamma subscript gamma + g cap gamma subscript i superscript 2 + cap gamma subscript i superscript 2 = 1 - g

(3.7)

(1 + g) cap gamma ??? + (g + 1) cap gamma ??? = 1 - g

(3.8)

cap gamma subscript r superscript 2 + 2g over g + 1 cap gamma subscript gamma + cap gamma subscript i superscript 2 = 1 - g over 1 + g

(3.9)

cap gamma subscript gamma superscript 2 + 2g over g + 1 cap gamma subscript gamma + g superscript 2 over (g + 1) superscript 2 + cap gamma subscript i superscript 2 - g superscript 2 over (g + 1) superscript 2 = 1 - g over 1 + g

(3.10)

(cap gamma subscript gamma + g over g + 1) superscript 2 + cap gamma subscript i superscript 2 = 1 - g over 1 + g + g superscript 2 over (1 + g) superscript 2 = 1 over (1 + g) superscript 2

(3.11)

(cap gamma subscript gamma + g over g + 1) superscript 2 + cap gamma subscript i superscript 2 = 1 over (1 + g) superscript 2

(3.12)

Which again is a parametric equation of the type (x-a)superscript 2 + (y-b)superscript 2 = R superscript 2 (Equation 3.12), in the complex plane (cap gamma r, cap gamma i), of a circle with its coordinates centered at (-g/g+1 , 0) and having a radius of 1/(1+g). Furthermore, By developing (Equation 3.5), we show that :

b + b cap gamma subscript gamma superscript 2 + 2b cap gamma subscript gamma + b cap gamma subscript i superscript 2 = -2 cap gamma subscript i

(3.13)

1 + cap gamma subscript gamma superscript 2 + 2 cap gamma subscript gamma + cap gamma subscript i superscript 2 = -2 cap gamma subscript i over b

(3.14)

cap gamma subscript gamma superscript 2 + 2 cap gamma subscript gamma + 1 + cap gamma subscript i superscript 2 + 2 over b cap gamma subscript i = 0

(3.15)

cap gamma subscript gamma superscript 2 + 2 cap gamma ??? + 1 + cap gamma ??? + 2 over b cap gamma ??? + 1 over ??? - 1 over ??? = 0

(3.16)

(cap gamma subscript gamma + 1) superscript 2 + (cap gamma subscript i + 1 over b) superscript 2 = 1 over b superscript 2

(3.17)

which is again a parametric equation of the type (x-a) superscript 2 + (y-b) superscript 2 = R superscript 2 (Equation 3.17).

Equivalent impedance resolution

When solving problems where elements in series and in parallel are mixed together, one can use the same Smith chart and rotate it around any point where conversions from z to y or y to z exist.

Let's consider the network of Figure 8 (the elements are normalized with Z subscript o = 50 cap omega). The series reactance (x) is positive for inductance and negative for capacitors. The susceptance (b) is positive for capacitance and negative for inductance.

The circuit needs to to be simplified (see Figure 9). Starting at the right side, where there is a resistor and 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. Since the next element is an element in shunt (parallel), we switch to the admittance Smith chart (by rotating the whole plane 180 degrees). To do this, however, we need to convert the previous point into admittance. This becomes A'. We then rotate the plane by 180 degrees. 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 counter-clockwise direction (negative value) and gives point B. Next, 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 degrees 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 has to be done counter-clockwise (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 (since 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 cap omega, then Z = 10 + j25 cap omega (see Figure 10).

Matching impedances by steps

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 is 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 (filter type structure, quality factor, limited choice of components, etc.).

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.

The objective is to match a source impedance (Z subscript S) to a load (Z subscript L) at the working frequency of 60 MHz (see Figure 11). The network structure has been fixed as a lowpass, L type (an alternative approach is to view the problem as how to force the load to appear like an impedance of value = Z subscript S (a complex conjugate of Z subscript S). Here is how the solution is found.

The first thing to do is to normalize the different impedance values. If this is not given, choose a value that is in the same range as the load/source values. Assume Z subscript o to be 50 cap omega. Thus z subscript S = 0.5 -j0.3, z*subscript S = 0.5 + j0.3 and Z subscript L = 2 -j0.5.

Next, position the two points on the chart. Mark A for z subscript L and D for Z*subscript S.

Then identify the first element connected to the load (a capacitor in shunt) and convert to admittance. This gives us point A'.

Determine the arc portion where the next point will appear after the connection of the capacitor C. Since we don't know the value of C, we don't know where to stop. We do, however, know the direction. A C in shunt means move in the clock-wise direction on the admittance Smith chart until the value is found. This will be point B (an admittance). Since the next element is a series element, point B has to be converted to the impedance plane. Point B' can then be obtained. Point B' has to be located on the same resistor circle as D. Graphically, there is only one solution from A' to D, but the intermediate point B (and hence B') will need to be verified by a "test-and-try" setup. After having found points B and B', we can measure the lengths of arc A' - B and arc B' - D. The first gives the normalized susceptance value of C. The second gives the normalized reactance value of L. The arc A' - B measures b = 0.78 and thus B = 0.78 x Y subscript o = 0.0156 mhos. Since omega C = B, then C = B/omega = B/(2 pi f) = 0.0156/(2 pi 60 superscript 7) = 41.4 pF. The arc B - D measures x = 1.2, thus X = 1.2 x Z subscript o = 60 cap omega. Since omega L = X, then L = X/omega = X/(2 pi f) = 60/(2 pi 60 superscript 7) = 159 nH.

Conclusion

Given today's wealth of software and accessibility of high-speed, high-power computers, one may question the need for such a basic and fundamental method for determining circuit fundamentals.

In reality, what makes an engineer a real engineer is not only academic knowledge, but the ability to use resources of all types to solve a problem. It is easy to plug a few numbers into a program and have it spit out the solutions. And, when the solutions are complex and multifaceted, having a computer handy to do the grunt work is "back-savingly" handy.

However, knowing underlying theory and principles that have been ported to computer platforms, and where they came from, makes the engineer or designer a more well rounded and confident professional-and makes the results more reliable.

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