Smith Charts are easy…(The basics of antenna matching.)
When I decided to learn about Smith Charts and how to use them, I had no idea what a giant rabbit hole that was going to be. I have learned so much more since writing that first blog post about them that I wanted to revisit it and share some information with my readers. I love to learn new things and for some reason I have never taken the time (till recently that is…) to learn how to even read a smith chart, much less how to use them to design ANYTHING!
The nanoVNA: A Vector Network Analyzer for everyone!
I started down this path with looking at antenna data from one of my POTA antennas on the nanoVNA. The nanoVNA is one of those wonders of modern technology that has become accessible to the masses recently. You see, before a few short years ago a VNA (Vector Network Analyzer) would easily run in the 5 figures and some of the nicer ones would break 6 figures. Rohde and Schwarz come to mind here… Even their used ones trade today for thousands of dollars…
This is the reality of VNAs from just a couple of years ago. 4 and 5 figure prices for used units were common.
The nanoVNA changed all of that. It is small, runs on Linux and is now a single board computer in a tiny battery powered device that will literally sit in the palm of your hand. I have used one for my antenna setups for a couple of years now. I am actually on my second unit as the first one developed a fault and it was so inexpensive that I just bought a new one and threw the old one in the trash. You won’t see that happening with an HP / Agilent VNA!
There is a very good reason I use one for my antenna tuning needs too. The nanoVNA does everything those larger (and more expensive as well) ham radio specific antenna analyzers do that are available. You just have to learn a little more on how to use the nanoVNA. It also will perform functions that the antenna analyzers will not and this is where I love to bring the nanoVNA into the light. It has a smith chart function as well as an S21 input option that most, if not all ham radio antenna analyzers lack. You simply can not sweep filters with a MFJ antenna analyzer, at least not that I am aware of. This is where I found smith charts as I started to want to know what it was showing me on those charts. All of these nanoVNAs have a smith chart function and it is normally “on” when you start them up by default.
nanoVNA showing the Smith chart upon power up. Notice the diminutive size of the device compared to my fingers!
Smith Charts are NOT Scary…
In the above photo, the light blue trace is the one for the smith chart view. If you look really close, you can just barely make out the circular plot of the smith chart graph and in slight less illumination it does show up better. The light blue curved line represents the plotted values from 1.8 mhz up to 31 mhz (the two values are at the bottom) and you can move a marker along this plot as you go through that frequency range. In this photo I have stopped at 21.072 mhz as I was tuning my telescoping vertical to the 15 meter band to operate CW and FT8. It shows two more pieces of critical information though, the real reactance and the imaginary reactance..yes, imaginary… On these little nanoVNAs it shows up as either inductance (+j value when converted) and capacitance (-j value when converted). This antenna is measuring 50.13 ohms 577pf capacitive. We would need to convert this capacitance to a j value before we can plot it on the smith chart. I will get into that a little later in this article, but for now, this is a great tool if you are wanting to learn how smith charts work and I recommend you get one of these little wonders of technology for yourself and learn how to use it, even more so if you choose to use my link here to get it! As buying it from the link will help me maintain the website and costs you nothing extra.
One of my early attempts at working with a smith chart, it is covered in errors, but this is how you learn…
This is a plain, run of the mill smith chart (pictured above). This one shows two graphs in one chart. The red is the impedance graph and the blue one is the admittance graph.
Smith Chart: Understanding some basic concepts
The simple explanation for these two colors are as follows: red lines are series components and the blue are parallel components when you are using the chart for matching impedances between stages of devices. Like between the coax and the antenna as in the example I have drawn above. That is an antenna I use for POTA that I wanted to see what it would take to get it to 50 ohm resistive. (which is the point right in the middle of the chart BTW)
Already, I have given you three tidbits of intel about these charts and we haven’t even drawn on one yet… Another one is that the horizontal line through the middle is pure resistance and you will notice that it doesn’t say 50 ohms at the center, but rather the number 1. The chart is what is called “Normalized”. All this means is that you can assign what ever value you want to the chart and this center point becomes THAT number and all the others are relative to that value. Maybe you had two stages in an amplifier circuit that are 200 ohm impedance and you want to use this chart to match them, then the center is now 200 and you do simple math to get the other numbers from that point. Like the number 2 (moving to the right, remember we are using the red lines right now) will become 400 ohms as a result. Basically all the numbers are multipliers of the center value you assign. It really is that simple.
Fun fact: All the values below the resistive center line are capacitive and all the numbers above the line are inductive. So when you see one of those weird numbers like 117-j68 This is the “complex” impedance…duh, it even looks complex…haha. But to put it on the chart, it has to be what is called “normalized” and this simply means that you divide the first AND the second number by whatever you assigned to the center of the chart. For most ham radio and especially ham radio antenna stuff, this is 50 ohms. So this complex impedance normalized will be 117/50=2.34 AND 68/50=1.36. BUT since it is a negative “j” value it will now read 2.34-j1.36. This can be plotted on the smith chart directly now. The first number is ALWAYS found on the center “resistive” line first.Then after this point is found you find the second number on the perimeter ring as you see in the photo below. The reason I started at the bottom and not the top (which has the same exact numbers) is because it has a negative symbol in front of the letter “j”. If it is negative, it always goes in the bottom half and if it is positive, then it goes in the top half.
Smith chart showing the initial plot for the value 2.34-j1.36 and how to find it.
Then you simply follow the circular lines from each number out to where these two cross and this is your start point on the chart. Now this particular smith chart only shows the impedance curves so you need the second half (which is a literal mirror image of this one laid on top of it and in a different color) to be able to do a parallel → series type solution to solve for this. If we didn’t have the other half of the chart then we would never be able to put a component in parallel with the load as that is what the other half of the chart is for. Right now we can only add series components to get us up to the resistive line, but it will only give us the native resistance we started with and only eliminate the j portion of the value if we did that. It still doesn’t solve for the movement we need to get to 50 ohms resistive. I guess we could add a parallel resistor to lower the value, but that is really lossy option and we don’t want to put a resistor in parallel with our antenna, that is just burning RF and not putting it into the antenna. Remember a 50 ohm dummy load presents a perfect match to the transmitter output, but with radiate very little of that energy out to the world, instead it is turning this energy into heat…
Knowing this, the solution would be to add a “shunt” inductor to move the plot point up on the chart till it intersects the 50 ohm curve on the TOP HALF OF THE CHART. Once it intersects this line, we will add a series capacitor to bring the value back down to the center point, thereby matching the two stages perfectly. If this is clear as mud, then watch the video below for a visual explanation of what I just typed as well.
Add to all the stuff thus far the following tip as well, anytime you want to move the plot point up on the chart, towards the center line or above it, then you will use an inductor. By the same token, when you want to move the plot point down somewhere lower on the chart from where you are, you will use a capacitor. If you move along the red lines of the chart then it will be series with the load for either device (inductor or capacitor), if it follows a blue line then the device will be in parallel with the load. The below video by W2AEW does a really great job showing this visually so if you are a visual learner, this video is for you.
Matching Impedance with a Smith Chart…the easy way…
A good point someone made in one of the tutorials that I either read about or watched a video on said to remember you might need a specific solution for other reasons as well as to match impedance. Most of the time there is at least two ways to solve for these problems. In the example to follow below, I opted for a parallel capacitor and a series inductor, but let’s say you have two amplifier sections and you want to keep the DC voltage separated in the two stages, will then you would use a shunt inductor first then a series capacitor to finish the solution, this would match our impedance as well as block the DC voltage from the first stage as well. You see this is useful in more than one way.
Going back to the first image above of my hand drawn plot. If you will notice I drew a circle on that chart. This is called the Unity circle (for the reactance side) for some reason…. The point here is that no matter where you move the point along this circle the first number will always be 50. (If you used 50 ohms for that center point, from here on out I am going to assign 50 ohms to this value as that is what I use as a ham radio operator). Anywhere along this line, other than where it crosses the horizontal line in the center of the chart, there will be a “j” value added to it. If it is below the horizontal line, it will be negative and above will be positive “j” value. Now, at this point, if you want to use this chart to match a load, this is where the really easy stuff ends. Past this point you will need to use math…I know we have done some division earlier to get the normalized value so we could plot it above, but now we have to start calculating things like reactance and component values and such and as you know, this requires math… So buckle up as we match the 2.34-j1.36 mythical antenna from the second chart above to a 50 ohm transmitter output.
I have a two color chart below to make this happen as we have to use both colors to get it to work. The red dotted line is the arc on the impedance graph (red circles) for the 2.34 resistive value. You can see how I tracked it down into the negative region from the horizontal resistive line (this horizontal line is where you will always start these plots BTW) till it met the -1.36 blue dotted arc (denoted on the lower perimeter in red reactance arcs, you can see the numbers on the lower edge where I started). Yes, these charts are not precise to 5 decimal places, but are useful for ham radio ops and engineers use them for a myriad of other applications other than antenna matching like we are doing here. It will get you close enough to make the system usable and that is the whole point for us anyway. Anyway, these two initial lines meet at the start point of the process, the impedance of the antenna as measured…
two color smith chart showing two possible ways to solve for L networks to match a transmitter to an antenna.
In the video above, he used the path of going up (like my green dots) and then following the 50 ohm arc back down (black dots from the green intersection point). But he was using a simple circuit built on a SMA connector. For an antenna installation though, it really makes more sense to me to actually follow the black line down from the start point and then follow the yellow line up the 50 ohm arc instead. Let me explain…
If we use the black arc that goes down first and the yellow one back to the center then what that looks like in the real world is a small capacitor from the base of the antenna to ground (this is what the black dotted line is after all). You see the black like is moving DOWN which means it will be a capacitor. We move it down till it meets the “50 ohm” circle so we can add the next part to move the point to the center of the chart. That next part isa series inductor between the base of the antenna and the coax. What us ham radio ops would call a load coil… This makes way more sense to me in this application than the other path which would be a “shunt” inductor or what would be an inductor from the antenna base to ground and then a series capacitor between the base and the coax. This is what would be similar to an L type antenna tuner to be honest but we are planning on using individual components in this lab… You see the yellow line is going UP to the central 50 ohm point, so this means it is an inductor and it is moving along the red lines on the chart which means it will be in series with the antenna.
This is actually really simple to be honest:
Red lines and your “movement” is going down - Series Capacitor
Red lines and your “movement” is going up - Series Inductor
Blue lines and your “movement” is going down - Shunt (Parallel) Capacitor
Blue lines and your “movement” is going up - Shunt (Parallel) Inductor
Do you see the pattern? Tracking up the lines on the chart is always inductors and tracking down on the lines is always capacitive. Like wise, if you use a blue line for your movement the the device will be in parallel and if the color of the line is red then it will be a series part. Now the color codes of my dotted lines above is purely there to make it easier to see what is happening and mean nothing other than that. The colors I am really concerned with are the two colors of the chart itself.
Math with a Smith Chart to match an Antenna
So the math is actually really simple to be honest. If you look at the first arc (the black line moving down) it is read on the blue chart (admittance - which is the opposite of resistance…) as it is moving along the blue circles. it is moving from about .19 to .48 on the blue chart (this is called suseptance, but you really don’t need to know that for this application) as you can see the lines run out to the perimeter where it is marked. This is .29 distance units on the admittance half of the chart. This .29 is the opposite of resistance so we then have to invert it (1 divided by .29), so it is a reactance value that we can use to do the math with, which turns it into 3.449 or as it should read -j3.449 (remember, because it is below the center line of the chart) and this is then multiplied by 50 (the value we assigned to the center point to start with) to get the actual reactance. 3.449 × 50 = 172.45 ohm of capacitive reactance. We now know everything in the formula to turn this into capacitance… since it is the same formula, you just switch out the two values. Super simple to be honest. Xc= 1/(2 x Pi x f x C) is turned into C=1/(2 x Pi x f x Xc) as you can see, we just plug in the numbers and then we get the capacitance. BTW, we are making this for 10 mhz so we can listen to WWV in Ft Collins… haha, why not?
Mathing this first step involved counting on the chart and subtracting the smaller number from the larger, then inverting it since it is on the blue lines (because the blue lines represent the opposite of resistance and we need it to be a resistance value), then multiplying that number by the center assigned value (in this example it is 50 ohms) and we ended up with 172.45 ohms of reactance. Now we turn this into C= 1/(2×3.1415×10e6×172.45) which is C=92.3pf
Once this step is done, we simply run up the red unity circle to the 50 ohm point in the middle and do the same thing but for inductance (since it is moving up on the chart). It looks like it is on about 1.42 on the red lines at the bottom of the chart, just take a look and see what I am talking about… Since it goes all the way up to the horizontal line, we just use this number and multiply it by the center value again (are you noticing a trend here yet?) and we get XL=1.42×50 so XL=71 ohms of reactance The we flip the inductive reactance formula and this time the formula looks a little different since it is XL=2 x Pi x f x L so to get inductance from inductive reactance the formula looks like this L=XL/(2 x Pi x f) so now we know all the numbers for this one too. It looks like this now L= 71/(2×3.1415×10e6) and this equals L=1.13uH of series inductance since this movement happened on the red lines and went UP.
So adding a 92.3pf shunt capacitor from the base of the antenna radiating element to ground (honestly an appropriately sized trimmer cap like a 50-100pf trimmer would be optimal since the calculated size is such an odd value...sure as a bird flies, this is some sort of common size…lol) Next is to insert a base “load” coil between the feedline and the antenna that is 1.13uH in size. This will match our mythical vertical to be resonant with the 10 mhz WWV signal as close to perfect as humans can get it. It SHOULD (if the parts are the right value) move the Smith chart plot to the center point on the chart which the goal for impedance matching.
Clear as mud again, right? HaHa… You can see it seems like a lot, but once you do it a couple of times, it really does get a lot easier to understand. I recommend you watch the video a couple of times and print off a chart from the web to practice on like this one.