Smith_Chart - Smith Chart Commentary

Smith Chart Notes

Keith Greiner

A Smith chart is a unique graphical tool for high frequency and microwave analysis that is used to design antenna and feed line systems. The chart is included in Visual Network Analyzer (VNA) displays that can tell users what inductance and capacitance values are needed to match the impedance between two components. The match might be from a transmitter to a feed line or from a feed line to an antenna. Because they are so specific and helpful for tuning, VNAs have become a standard tool for adjusting antennas and tuners.

A typical VNA screen has a number of charts as shown in Figure 1, below. Here we see an SWR sweep (ie; the white line that has the dip in the center around 14.040 Mhz.) and the circles of impedance, resistance, and admittance that are indicative of the Smith Chart. Text in white across the middle of the screen, shows what capacitance and inductance values, wired as an L network, would match the measured R and jX values to a load.

Figure 1. NanoVNA-F V2

Figure 1 shows a typical VNA screen. There is a multitude of information on the screen including resonance, SWR, Frequency of minimum SWR, and the distinctive circular Smith Chart.

Figure 2. NAB Smith Chart -- 1960

Figure 2 shows a paper Smith Chart that was published in the 1960 edition of the National Association of Broadcasters (NAB) Handbook. The chart is specific to an L network having a parallel (shunt) inductance and a series capacitance. It’s not surprising that engineers and amateur radio operators are aware of Smith Charts but do not fully understand nor regularly use them beyond the frequency and SWR adjustments. The NAB handbook included seven configurations and minimal instructions that limit usage to the most persistent of readers. Even then, calculations based on the chart still lack adequate precision.

The chart tool was invented by Philip Hagar Smith in 1939 as a pencil and paper nomograph and is one of the many innovative developments that came from Bell Telephone Laboratories (Bell Labs). A Japanese engineer named Mizuhiko Mishima created a graphical tool, but apparently that tool was not well known outside of Japan. Accounts vary, but apparently, the open-source software for the NanoVNA was developed with contributions by Yury Kuchura, Takahashi Tomohiro (edy555), Glen Hu (aka Hugen), and Dmitry (DiSlord), and Rune B Broberg (5Q5R). The software may be found on GitHub at: https://github.com/ttrftech/NanoVNA, https://github.com/hugen79/NanoVNA-H/blob/master/vna_math.c, https://github.com/hugen79, and https://github.com/NanoVNA-Saver/nanovna-saver

I assembled the tool at this link as a way to learn more about the Smith Chart applications and calculations. I wanted to know specifically how the background calculations work. I implemented the calculations in Mac Swift language which facilitates a display. The Swift program was translated to html/Javascript for use on this page.

A feature of this online tool allows you to left-click-and-hold the red dot and drag it around the screen and watch how the solution for real and imaginary (R and jX) measurements change. Output is in terms of the size of inductor and capacitor plus the L network configuration needed to match the dot location to a 50-ohm resistive load. The tool does not resize to a small telephone screen so it is best viewed on a laptop, desktop, or iPad. If you display the tool in a full web page view, you can left-click-and-hold as you drag the position of the red dot to locations on the chart.

Now, if you use the VNA tool or this online tool, and then go find inductors and capacitors of the exact sizes recommended here, do not expect that your assembly of parts will form a perfect match. It is possible to have a correct calculation, and still have variations. While my tests with an isolated function provided the same output as the VNA screen, there are many variables that could result in different outcomes in the context of the display and the feature that allows users to drag the red dot to a location. It is difficult to get the dot in the precise value shown on the VNA screen. If you are making comparisons, be sure that you note whether the Smith Chart measurement has the correct values for R and jX. If it shows the imaginary values as microHenrys or picoFarads, the values will need to be converted.

Figure 3. L-Networks

Figure 3a (on the left) shows an LTSpice model of a typical L network that can match the impedance of a source (ie: a transmitter) to a load (ie: an antenna or feed line) and having the inductor in series and the capacitor in parallel (shunt). Figure 3b (on the right, shows the same for an L network that has a capacitor in series and an inductor in parallel. Generally, speaking, one would first consider the 3a network if the antenna is too long, and the 3b network if the antenna is too short. The Smith Chart when used with a NanoVNA tool can help find the values of inductor and capacitor.

You can do many things with this online chart. You can use the default R, jX values and frequency or you can enter values you obtain from your VNA. To do that, set the VNA display to return the complex number of R and jX. Be careful, a VNA may display the complex and may also display the R value with an inductance/capacitance value. Use the complex number display option.

Drag the red dot around in a circle and watch how the solutions change. You may find that at some times, there is one solution, and at other times, there are two ore maybe even three solutions.

The NanoVNA finds exact values of the inductance and capacitance of an L network by using the quadratic formula to find an exact solution. Here is the basic procedure. The NanoVNA follows the following steps:

1. Take the measured values of resistance and reactance, R, and jX, find the initial value of impedance, where Z0 = (r^2 + jx^2)^0.5,

2. Sets the target total impedance of ZL, which in this tool, and the NanoVNA, is 50 ohms, and

3. Solves for values of r and x that make the load impedance 50 ohms.

The solution is found via the quadratic formula:

Where the values going into the formula are:

1. In the Hi-Z case where RL ≥ R₀ use the calcHiZ function

a = R0 - RL

b = 2.0 * XL * R0

c = R0 * ((XL * XL) + (RL * RL))

2. In the Lo-Z case where RL < R₀) use the calcLoZ function

a = 1.0

b = 2.0 * XL

c = (RL * RL) + (XL * XL) - (R0 * RL)

The calcHiZ and calcLoZ functions call the quadratic formula function and return values for;

xps = source-side shunt reactance,

xs = series reactance, and

xpl = load-side shunt reactance.

For an L network, the returned value xs and either xps or xpl can be non-zero.

Depending on the sign of the returned values, the program determines whether the values are inductive or capacitive.