1. Smith Charts Explained
2. Smith Chart Basics

The Smith chart is a polar plot of the complex reflection coefficient (also called gamma and symbolized by Γ). Or, it is defined mathematically as the 1-port scattering parameter s or s11. A Smith chart is developed by examining the load where the impedance must be matched. 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.

• Now, you have learned all basics of Smith chart and you know the chart is consisted of 3 very basic parameters, (Γ, z, y), and they can be converted among each other based on a.
• What is the Smith Chart? The Smith Chart is the graphical representation of a complex mathematical equation. It is the circular plot of the characteristics of microwave components. The Smith Chart is the most used tool for microwave engineers to visualize complex-valued quantities and calculate the mapping between them.
• A Smith chart, shown in Figure E9-3 above, is a chart designed to solve transmission line problems graphically. While a complete discussion of the theory behind the Smith Chart is outside the scope of this study guide, a good discussion of the Smith Chart can be found on the ARRL website. The Smith chart coordinate.

This page describes the free Smith Chart impedance matching program from Iowa Hills Software.

See this page for a set of useful matching circuit equations.

Smith Charts allow the user to design impedance matching circuits. This is done by entering a load impedance on the chart, then altering that impedance with series and shunt components. This program allows for the following components to be used.

You may choose to work with ideal components (no parasitics or loss), or you may specify the Q and parasitic reactance associated with your components. It is quite interesting to see the effects of the parasitics, even for the simplest of circuits. It is also nice to be able to adjust a component value after it has been placed. This can be done by selecting the text cell below the schematic symbol and using the arrow keys to modify the value.

The program was written for practical use. As such, the user must specify the tolerance for the capacitors and inductors which in turn specifies the discrete values used by the program. For example, if specifying 10% inductors, then the allowed inductor values go as 1.8, 2.2, 2.7, etc. You cannot place a 2.45673 nH inductor for example. The effect of using these discrete values is shown with the short changing color segments. For example, a segment drawn as green, dark green, green, dark green, etc. would represent 1.8, 2.2, 2.7, 3.3 etc. This gives the user a good sense of circuit's sensitivity.

The program can also plot S parameter data, along with their stability circles and the stability factor k. The user may match to S11 or S22 directly, or choose to match to the optimal bilateral matching impedance (if k >= 1).

An FAQ
A frequently asked question when getting started with the Smith Chart is: To match Z to 50 Ohms, do I start on the Smith Chart at Z, or Z conjugate?

We start at Z, but a simple example shows an important aspect about these matches. Here we match a load of 75 - j100 to 50 Ohms at 1 GHz. Then on the right, we work through the problem in reverse to end up at the conjugate of our load.

This simple example shows a fundamental fact about circuits. If a circuit is designed for maximum power transfer, then you can break the circuit at any point and looking to the left you will see Z, and to the right you will see Z conjugate. If this isn't the case, maximum power transfer isn't occurring.

Q Circles

The program allows for Constant Q and VSWR circles to be plotted on the chart. Q circles are used as a way to control the bandwidth of the match. The following example should make the point.

In the plots below we show two ways to match a load of 118 - j99.5 to 50 Ohms. In the first example, we draw a Q=1 circle on the Smith Chart. We then choose a circuit topology and component values that keeps the impedance traces within the Q Circle. Then we use a Q=6 circle, and do the same thing. Notice the effect on the bandwidth of the match.

It should ne noted that in general, these matching circuits are seldom this easy. It is more typically the case that the load being matched to is much further out on the Smith Chart. In other words, the Q of the load is high. In those situations, the bandwidth of the match is controlled by the load, not the matching circuit. The only way to improve the Q of the match (widen its bandwidth) is to add either shunt or series resistance to the match, which of course introduces losses.

It is worth noting that a transformer traces out the same path on the Smith Chart as a Q Circle. VSWR circles can also be plotted on the chart, but they have little practical value in the design of a matching network.

Smith Charts Explained

Q=1 Match

Q=6 Match This match uses values that push the impedance traces out to Q= 6. Note the narrower bandwidth of this match as shown by the return loss plot.

Q circles show a fundamental fact about matching. It is usually the case that the engineer needs a broadband match, but to have a broadband match, we must stay within a low Q circle. Notice that if the load you are matching to is purely resistive, then it is possible to stay within an arbitrarily low Q circle, but you may need to use a large number of components to get back to 50 Ohms.

On the other hand, if the load is highly reactive, 2 + j50 for example, then the load has a Q of 25, and it is impossible to do a lossless broadband match, no matter how many components you use.

Bilateral Matching

To start, let us make three points about this subject that need to be understood.

1. Bilateral matching is done with 2 port s parameter data. This is typically done when designing the input and output matching circuits for a high frequency transistor. The transistor manufacturer typically supplies the s parameter data.

2. Bilateral matching is only used for devices with |S12| > 0. This is generally the case. The only 2 port devices with S12 = 0 are ideal isolators.

3. If S12 = 0, then there is no path from port 2 to port 1, and the bilateral match reduces to a conjugate match. Or said a bit differently, if S12 for your device is very small, then bilateral matching isn't required and a simple conjugate match will suffice.

The basic idea for bilateral matching is fairly simple. If S12 ≠ 0, then there is a signal path from the output back to the input. The effect of this is that the device's input impedance is affected by the load, and the device's output impedance is affected by what is connected to it's input. Consequently, the only way to match such a device is to calculate the input and output matches simultaneously. We do this by using the device's s parameter data and the equations that follow.

The following is a screen shot from a MathCAD file. The equations were taken from High Frequency Amplifiers by Ralph S Carson, Wiley and Sons, 1982. The final results are the variables Z1opt and Z2opt. These two variables can be a bit confusing. i.e. Are we calculating the impedance of the device, or the impedance the device wants to see.

There is less confusion if one remembers that the outcome of these equations will yield a result fairly close to a simple conjugate match. So it helps to note their value relative the Z1 and Z2 as calculated from S11 and S22. The conjugate of Z1opt and Z2opt are relatively close Z1 and Z2. Thus, the conjugate of Z1opt and Z2opt represent the optimal impedance of the device itself.

Smith Chart Basics

Consequently, if using a Smith Chart to calculate a matching network, start at the conjugate of Z1opt and Z2opt (start at the device, as always), and work toward 50 Ohms.

The Iowa Hills Smith Chart program will do these calculations. The user simply needs to read in an s parameter file, and specify the match frequency.

Antenna Matching

Many people want to use a Smith Chart to design an input match to an antenna. Be warned however, this probably will not work.

If you use a network analyzer to measure an antenna's input impedance, you will notice that S11 can be made to move all over the place by moving an object around the antenna. This is a very good sign as it means that the antenna is well coupled to the atmosphere and any changes in the antennas surroundings, which also changes the amount of signal reflected back to the antenna, appear as a change in S11. Only very poor antennas are insensitive to their surroundings.

It has been my experience that good antennas are also very sensitive to their source impedance. Which is to say, a change in the source impedance also changes the impedance of the antenna. (Emphasis on the word 'experience' here, I have no theory to back this statement.) The result of this is; there is not a good way to calculate a matching circuit for antennas.

It is certainly the case that some low frequency antennas use matching circuits, because they don't have a choice. To the best of my knowledge however, these circuits are either hand tuned, or tuned programmatically in the field, to minimize the reflected power.

An antenna is supposed to be the impedance matching circuit between 50 Ohms and 377 Ohms. If the antenna isn't presenting 50 Ohms to you, the problem should be fixed by modifying the antenna. A way to think about this is; if an antenna looks reactive, it is by definition storing energy, and this is the last thing you want an antenna to do as it will inevitably lead to losses. Low loss antennas look purely resistive.

In general, using an impedance matching circuit to fix an antenna's input impedance is a poor approach to take. The final result will likely be a very lossy antenna. Of course, in some situations, you don't have a choice in the matter.

Miscellaneous Tools

A pad calculator, units converter, and free-space path loss calculator are built into the program, because it was easy to do and they are handy to have in the lab. Experienced engineers probably have the units and path loss conversions memorized, but the pad calculator is less trivial.

Calculating the resistor values for a pad isn't particularly difficult, but there are limitations the RF engineer must pay attention to if the load and source are to remain matched. The screen shot below illustrates this point.

In this example, we want a 3 dB pad between a 50 Ohm Source and a 75 Ohm load. As the program shows however, the minimum attenuation for this configuration is 4.0 dB if both the source and load are to remain matched. If we are willing to allow either the source or the load to be unmatched, then we can use an L pad as shown here.

This Smith Chart program is free, and available on the Download Page. Merge pdf.

RF = ReFined

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