## Matching Antennas to Coax

This article will show how to use coaxial cable and open wire line to transform the impedance of certain non-resonant antennas to 50 ohms. Twenty-meter dipoles will be used as examples but once the technique is understood it may be used on other bands. In fact, this method could be applied to other types of antennas, but for simplicity only dipoles will be discussed.

We know that a half wave resonant dipole one-quarter wavelength, or some multiple thereof, above ground has an impedance of around 72 ohms (all dipoles discussed here will be center fed). If the dipole is connected to 50-ohm coax an SWR of about 1.44:1 will exist on the coax. As we move away from the resonant frequency of the antenna the SWR will increase. One problem is that most modem transmitters work best when feeding a load that is close to 50 ohms. We could lower the dipole to less than one-quarter wavelength above ground and get a feedpoint impedance of 50 ohms but that would result in increased ground losses and most of the radiated energy would go straight up. That is usually not what we want. Ideally the impedance of the dipole would be matched to the coax at the feedpoint of the antenna. While that is usually impractical, the method explained here will accomplish exactly that. Fortunately, we do not have to use a resonant dipole. If we make the dipole longer than one half wavelength both the real and the imaginary parts of the feedpoint impedance will increase. If we make the dipole shorter the impedance will decrease. We can use this to our advantage. We will come back to this point in a minute but now let’s look at a transmission line.

To keep things simple, I will refer to coaxial cable as 50-ohm line. RG-58, for example, is really 53.5 ohms. Coaxes of other impedances could be used, but as stated in the first paragraph the purpose here is to explain the technique.

#### Now Lets Start

Let’s start by taking two equal length pieces of coax, placing them side-by-side, and connecting only their shields (or outer conductors) together at each end. The shields will not be connected to anything else. By connecting to the center conductors we now have 100-ohm line. If we place a 50-ohm load on one end of this line we will have a 2:1 SWR on the line. Here is how I define coax as 50 ohms: two pieces of RG-58 connected this way gives us 107 ohms, and connected to a 50-ohm load we would have a 2.14:1 SWR. Some of the impedances to be discussed would have to be normalized to 107 ohms. By normalizing to 100 ohms we only need to move the decimal point over two places.

**Figure 1** shows a Smith Chart normalized to 100 ohms with a 2:1 SWR circle. We know that by choosing the proper length of line we can match any impedance on the circle to 50 ohms. The trick will be to find a dipole length that has impedance that falls on the circle and determine what length of line will be needed to match that impedance to 50 ohms.

Thus the procedure is to:

**1) choose an impedance for the impedance transforming transmission line2) determine the SWR that will be on that line with a 50-ohm load connected to one end3) draw that SWR circle on a Smith Chart normalized to the impedance selected in step 14) find a length of dipole, at the desired height, that has a feedpoint impedance that falls on that SWR circle5) determine the length of the impedance transforming transmission line that will be needed.**

If you are using open wire line I suggest you model the system and tweak it before you build it. If you do not have access to modeling software you can use the lengths given in this article and scale them to your frequency of interest. As you will see the absolute length of the dipole or matching line is not critical, but the length of the other may need to be tuned for best results. You will probably want to start out by making the dipole or matching line a little longer than calculated so that you can “prune and tune”. Now back to the dipole.

#### Optimising Impedance

I used Brian Beasley’s Antenna Optimizer (AO) program to determine the impedance of many lengths of dipoles. Only the lengths of interest will be discussed here. All of the dipoles discussed here are made of #12 AWG and are 33 feet high unless otherwise stated (this is one half wavelength high). All dipole lengths mentioned here refer to the end-to-end length. For reference, this dipole was resonant at 34 feet 2 inches.

A length of 33 feet yielded a feedpoint impedance of about 69 – j49 ohms, which falls on or close to the 2:1 SWR circle (see Figure 1 point A). Note that the numbers in Figure 1 should be multiplied by 100 to get the numbers shown in the text. By drawing a line from the center of the chart through the 69 – j49 ohm point and continuing to the outer edge of the chart we see that an electrical length of about 0.105 wavelength will give us 50 ohms. The wavelength of 14.175 MHz is about 21.2 meters. The last number times 0.105 gives us about 2.22 meters. That number times the velocity factor of coax (about 0.66 for non foam RG-58 or RG-8) gives us about 1.5 meters or about 5 feet (see Figure 2). The SWR of this antenna and matching line should be less than 1.5:1 across the entire 20-meter band.

Now let’s examine what happens if we vary the height and length of the dipole. We will only examine this dipole, but what we learn here can be applied to the rest of the dipoles described in this article. We will start by changing the height. Let’s say the dipole is only 32 feet high. Its feedpoint impedance would be about 72 -j49 ohms (see Figure 1 point B). This point is slightly closer to the center of the chart and would require a little bit more rotation to bring it to a pure resistance. When rotated to the real axis its impedance would be slightly higher than 50 ohms. If we move the dipole up to 34 feet high its impedance will be about 66 – j49 ohms (see Figure 1 point C). This point is slightly farther from the center of the chart and would require slightly less rotation to bring it to a pure resistance. When rotated to the real axis its impedance would be slightly less than 50 ohms. Once you have about the correct length of dipole and impedance matching section changing the height affects mostly the real part of the impedance and changing the length of the matching section affects mostly the imaginary part of the impedance. Note that if the impedance falls on the real axis between 45.5 and 55 ohms the SWR is less than 1.1:1.

#### Varying Length

Next let’s see what happens when we vary the length of the dipole. If the dipole is made 33 feet 3 inches long its feedpoint impedance will be about 71 -j37 ohms (this is point D on Figure 1). We see that it will need less rotation than the original dipole. When it is rotated to a pure resistance that resistance will be about 55 ohms. This is not bad, but if we raised the dipole the real part of its impedance would be lowered, as explained in the paragraph above, and the SWR would be even closer to 1:1. Even if you left its impedance at 1.1:1 at mid band, the SWR would still probably be less than 1.5:1 across the 20-meter band. If the dipole is shortened to 32 feet 9 inches the feedpoint impedance would be about 67 -j58 ohms and is shown as point E on Figure 1. This shorter dipole would require more rotation than the original dipole, and would rotate around to about 45 ohms. Again, not bad at all, but you could lower the height a bit to get closer to 50 ohms.

We know from experience that working with the woven outer conductor of coax makes coax a pain in the hands. Based on that and what we have seen in the proceeding two paragraphs, I recommend that after you have gone through the procedure above and calculated the length of 100-ohm line that you will need, use that length of line and tune the system by shortening the length of the dipole, and varying the height if need be, to obtain the best SWR. Obviously you would start with the dipole a little on the long side.

Another interesting length is 35 feet 8 inches, which has a feedpoint impedance of about 88 +j65 ohms. This is point F on Figure 1. This is not quite on the 2:1 circle but close enough for this discussion. By drawing another line we see that we will need about a .363 wavelength to bring this impedance around to 50 ohms. Using a little math, 21.2 times 0.363 times 0.66 gives us about 5.2 meters or about 17 feet. Again refer to **Figure 2**.

The 33-foot dipole uses the least amount of coax in the matching section and means the least amount of weight hanging from the center of the dipole. This is a factor to consider if your dipole is hung from its ends. The longer dipole will give slightly more gain: about 0.1 dB according to AO. This small increase in gain will probably be lost in the longer matching section since the matching section has a 2:1 SWR on it and therefore more loss than the coax with the 50-ohm load. You could get about 0.2 dB more gain from the shorter dipole by raising it one foot higher. So the second dipole is probably not worth building but it helps illustrate the technique.

I found two other lengths of dipole that had impedances that fall on or close to the 2:1 SWR circle. The first is 101 feet 6 inches that yielded a feedpoint impedance of about 118 – j76 ohms and the second 104 feet 6 inches to yield a feedpoint impedance of about 145 +j70 ohms. These are points G and H respectively on **Figure 1**. These dipoles are about three half-wavelengths long and have a radiation pattern that has more lobes than a half-wave dipole, but for amateur radio that is usually not a problem. The same comments about the length of the matching section and tuning by varying the height and/or length of the dipole apply here too.

The lengths of 100-ohm line mentioned above could be made one-half wavelength longer (or some multiple thereof). Making this shielded line longer can provide a bonus. We know that if we feed an antenna with coax the RF power will go up the inside of the coax to the antenna where it will be radiated. Some of this radiated power will come into contact with the outside of the coax. Since the coax is connected to ground at the transmitter, some current will be induced on to the outside of the coax and flow to ground. This current flow can cause problems. By using enough shielded feedline to reach physical ground, and not connecting the shields to ground, we have a “sort of free choke balun” since no current will flow on the outside of the coaxes. Making this line longer also has drawbacks that will be explained below.

#### Open Wire Lines

Now let’s consider using open wire line instead of coax. Open wire line has several advantages over the dual coax method described above. One advantage is weight: open wire line is usually lighter for a given length than coax. It is cheaper too since you can make it yourself using a couple pieces of wire and a few insulators. If you make it yourself you can build it so that the spacing between the wires can be varied. That allows you to vary the impedance of the line, which gives you one more variable that will be discussed below. Also, open wire line has less loss than coax.

Let’s start with 300-ohm line because 300-ohm twinlead is widely available. You can make 300-ohm line by placing two pieces of #12 wire 1/2 inch apart center to center. A 50-ohm load on a 300-ohm line gives a 6:1 SWR. My modeling with AO showed that a dipole 31 feet 3 inches long would have a feedpoint impedance of about 59 -j 125 ohms. This is on or close to the 6:1 SWR circle. About 0.065 wavelengths will rotate this dipole’s impedance around to 50 ohms. That works out to about 4 feet 6 inches of line. See Figure 3 point A. Note that the numbers in Figure 3 should be multiplied by 300 to get the numbers shown in the text. I modeled this antenna with 300-ohm line and found that 31 feet for the dipole length and 4 feet 7 inches for the matching line were the optimal numbers for lowest SWR (see Figure 4). The SWR at the low impedance end of the 300-ohm line was below 1.4:1 across the 20-meter band.

#### More Modeling

Further modeling revealed that any dipole length between about 30 feet and 32 feet 6 inches when connected to the correct length of 300-ohm line would yield an SWR of less than 1.5:1 across the 20-meter band. This shows that the length of the dipole is not very critical as this is a variation of nearly 10 percent. But for each length of dipole there is a unique length of 300-ohm line that will result in minimum SWR and those lengths varied from 5 feet 11 inches for the 30-foot dipole to 2 feet 4 inches for the longest dipole.

The next interesting dipole was calculated to be 44 feet 6 inches long. It had a feedpoint impedance of about 200 +j484 ohms, which is on or close to the 6:1 SWR circle. Drawing another line shows that we need about 0.33 wavelengths of 300-ohm line. Going through the math I found that is about 23 feet of 300-ohm line (see Figure 3 point B). But when I put all of that into the model I found that the impedance on the other end of the 300-ohm line was 53 – j 10 ohms, which is an SWR of about 1.2:1. In fact the modeling results said to increase the length of the dipole to 45 feet 3 inches to get the impedance on the other end of the 300-ohm line down to about 50 ohms. The reason for this is that there is some loss in the 300-ohm line. This will be discussed in more detail later. The model also indicated that 22 feet 10 inches was the optimal length for the 300-ohm line (see Figure 4 again). This antenna and feed line had an SWR of less than 1.4:1 across the 20-meter band.

While I was modeling the above dipole and feed line I found that any length dipole from about 43 feet to about 50 feet long with about 24 feet 3 inches to about 21 feet of 300-ohm line respectively would also yield an SWR of less than 1.5:1 across the 20-meter band. This is a variation of more than 15 percent in the length of the dipole.

The 45 foot 3 inch dipole yields only about 0.5 dB more gain than the 31 foot 3 inch dipole. The longer dipole will require more 300-ohm line than the shorter dipole. Since open wire line usually weighs less and has less loss than coax, even with a high SWR, the longer dipole might be a better choice in this case.

Since open wire line is easier to work with than coax, it might be easier to shorten the impedance matching section instead of shortening the dipole to tune for best SWR.

#### No Transmatch Needed

An antenna system using a dipole and twinlead might be quite useful for campers or others for whom weight and/or space must be minimized. Campers might want to use 300-ohm twinlead since it is lighter than coax, has less loss than coax, is readily available and is less likely to get tangled than open wire line. A minimum weight scenario would be a slightly shorter than resonant dipole fed with a little over a half wavelength of 300-ohm twinlead going straight into the radio — no transmatch needed!

As mentioned above if you are making open wire line you can construct it so that you can vary the distance between conductors and in so doing vary the line’s impedance. Let’s now consider a dipole 60 feet long. This length was chosen because it has a feedpoint impedance of about 1620 +j1870, which falls on or close to the 9:1 SWR circle of a Smith Chart normalized to 450 ohms. This dipole’s impedance can be matched to 50 ohms using about 0.273 wavelengths of 450-ohm line, which is about 19 feet. For the sake of illustration, let’s normalize 1620 +j1870 to 300, 450 and 600 ohms. We get 5.4 +j6.24, 3.6 +j4.2 and 2.7 +j3.1 respectively. Figure 5 illustrates these points. Point A is for the 600-ohm line, point B is for the 450-ohm line and point C is for the 300-ohm line. The circle is the 9:1 SWR circle.

By inspection we see that the lower the impedance of the line, the shorter the line needs to be to rotate the dipole’s impedance to a pure resistance. Start by making the open wire line a little longer and a little wider than you initially think it needs to be. Then tune the line by shortening it until you get minimum SWR. If the SWR is not as low as you want, move the wires a little closer together, and if need be, shorten them again until you get an SWR of 1:1 or close to it. Of course, you could also vary the height of the dipole to help achieve a 1:1 SWR. You can make 450-ohm line by placing two pieces of #12 wire about 1.75 inches apart center to center. In this case AO showed that 1.5 inch spacing center to center was closer to optimum. This is due to line loss, which will be discussed in a minute.

#### Switching Lines

Finally, we could make a dipole 65 feet 6 inches long. That is pretty close to being two half waves in phase, and the feedpoint impedance will be about 4400 ohms! A section of open wire line consisting of two pieces of #12 wire 2-1/8 inches apart and about 17 feet 8 inches long will match that antenna to 50 ohms. The SWR should be 1.5:1 or less across the 20-meter band. Then, if you can figure out some way to switch out the open wire line and connect the coax directly to the center of the dipole you will have a “usable” antenna on 40 meters. The SWR will be about 2:1 at the ends of 40 meters and about 1.6:1 at mid band. Certainly there is a way to obtain a lower SWR on 40 meters and still get a good SWR on 20 meters, but someone else can figure that out.

As stated above, the sections of impedance matching transmission lines can be increased by one-half wavelength or some multiple thereof. In addition to reducing the bandwidth of the system, there is something else that needs to be considered: all transmission lines have loss. A result of this loss is that the impedance we are trying to match does not really rotate around the center of the Smith Chart, but rather it “spirals in” towards the center. Whatever the load is, as the transmission line gets longer, the impedance the generator sees becomes closer to the impedance of the line. If you are using one or more half wave sections of line, in addition to the matching section, it will be necessary to design the antenna to have a slightly higher feedpoint impedance in order to “spiral in” to 50 ohms. This effect will be more pronounced with the dual coax method than with open wire line since coax has more loss. In the case of the 60-foot dipole bringing the feed line wires closer together lowers the impedance of the line which has the same effect as raising the impedance of the antenna.

#### Summary

This article has shown how the proper length of non 50-ohm transmission lines can be used to match certain impedances to 50 ohms. It has shown how to determine what impedances can be matched this way and how to determine the proper length of the non 50-ohm line. Having the mechanism to match the impedance of the antenna to the 50-ohm transmission line immediately between the antenna and the 50-ohm line is advantageous because it minimizes the loss in the 50-ohm line. This method can also eliminate any currents that might flow to ground on the outside of the 50-ohm line. Antennas other than dipoles could be used if they have the proper feedpoint impedance.

Originally posted on the AntennaX Online Magazine by Jim Westbrook, K1FD

*Last Updated : 26th May 2024*