Antenna -- Feed line Matching Simplified
By
Donald Sanders W4BWS

This article is aimed at the amateurs just beginning to learn more than the basics of antennas and radio theory. Many amateurs who, having passed the General or Extra license exam, have the desire to learn and understand the how and why of the function of antennas and their equipment.

You probably have wondered why some Amateurs have tremendous signals and get terrific signal reports. You probably think: He must be running more than a kilowatt. Those outstanding signals most often are primarily due to a perfectly matched and resonated Dipole or beam. Often home brewed.

Outstanding signals result from a combination of things such as antenna, location, height and matching, as well as power, good grounding and other factors. Don't let anyone convince you antenna theory does not work. If you are willing and able to squeeze the few extra dBs out of your dipole or beam, you will have an outstanding signal also. Many well-known DXers use only 100 watts. It's all about the antenna.

Much has been written and published, over the last 60 years I have been a ham, on the various methods of matching feed lines to antennas and radios. Many vendors, as well as authors of homemade equipment, expound on their many bridges, antenna test equipment, and antenna tuners. But low SWR is not always a good indication of a good match between antenna and feed line. While it is important, SWR can often be 3:1 or higher and the antenna will work well. What is more important is the efficient transfer of power from transmitter to feed line and feed line to antenna. Maxwell has made this evident in his books "Reflections". In this paper I will talk about two of the most important factors relating to antennas, Impedance and matching.

Most antenna articles give an approximation of the antenna impedance (load) and amateurs build their systems based on these approximations. However, many factors will influence the load impedance, also called antenna radiation resistance. Height above RF ground, wire size and type, nearby metal objects, house wiring and ducts are just a few of these influences.

Impedance

There are several basic principles one must keep in mind. In reading the following information, take it slow, read it several times, and you will understand and remember much of what has been known about antennas for over 80 years, but is often forgotten.

First is the question of impedance (Z) vs. height of an antenna. One of the first questions a newcomer asks is, what feed line should I use to feed my antenna?, or What is the impedance of my antenna?

The impedance of the dipole or beam changes as the antenna is raised or lowered to different heights above ground. The ARRL Antenna Handbook (1949 ed. Pg 48 - yes I keep my old handbook as theory doesn't change much) states that the radiation from the antenna reflects back from the ground below the antenna to create a secondary current into the antenna. This current will be in phase or out of phase with the primary current depending on the height. This second current is the one that affects the impedance (Z) of the antenna. Rotating a beam often will show changes in the SWR indicating changing impedances caused by objects below the antenna which affect the reflected secondary signal. Also with a beam, the low Z is caused by the secondary currents from the reflector and director. This is covered by Bill Orr in his Beam Antenna Handbook and in the ARRL Antenna Handbook.

If the antenna is raised or lowered, a higher current in the antenna feed point, at the same value of power, means the effective resistance (Z) of the antenna has changed. If power is constant, and the antenna resistance (Z) changes, the current at the feed point must also change to meet the requirement of the formula: P=I (squared) R.

For example, an antenna resonant at 3900 kHz with a 2: 1 SWR and operated at 3800 kHz will have a higher SWR and less gain. The higher SWR means more losses, inefficient operation of low pass filters and band pass filters, and modern transmitters. They are usually designed for low SWR and 50 ohm impedance. This is why impedance matching and SWR of less that 2:1 are important.

The theoretical input impedance (Z) of a dipole antenna in free space is 73 ohms. This is only under ideal conditions with the "perfect dipole" and is not seen in actual antenna installations. "Free Space" applied to wavelengths at lower HF frequencies is an illusion. Actual installations will depend on location and other factors that are not easily determined or calculated, and rarely anticipated.

If the amateur could reach the input terminals of his dipole, vertical or beam in its actual position for use and make direct connection with an impedance bridge, without influencing the readings due to his body being near the antenna, he would be able to determine the value of the impedance to be matched.

Then the rest of the task of matching would be easy. There is a method to solve this dilemma without Sky Hooks, or Cherry Picker vehicles. There are several tools available to the amateur to use to determine the actual antenna impedance from the ground. These are 1/4 or 1/2 wave coaxial or parallel line sections and simple manual RF bridges or automatic/scanning antenna analyzers. Many are shown in the advertisers in QST and CQ magazines. But there are several basic requirements to be met first.

Resonance

First the antenna must be resonant at the desired frequency of measurement. While 468/F (Mhz) is used to determine the length of a dipole, many factors will cause the actual resonant frequency to vary for a given length of antenna. Therefore an RF bridge or GDO should be used to determine the actual resonant frequency of the antenna. Then it should be lengthened or shortened to move it to the frequency desired. This is the subject for another paper but many methods have been published. I suggest reading the ARRL Antenna Handbook and looking at several back QST issues for the methods to tune the antenna to a desired frequency.

For only under the condition of resonance will the antenna present a nearly pure resistance and no reactance. While this is the desired condition, this is almost impossible, since in practical use it will almost always have a small reactance.

The resistance will include the resistance of the wire plus the radiation resistance of the antenna. Usually below 30 MHz the resistance of the wire is so small it can be ignored. However at VHF the "skin effect" resistance cannot be ignored. This leaves just the radiation resistance of the HF antenna to be measured and it will be the impedance (Z) of the antenna.

Impedance

The use of a half wave, or multiple thereof, coax or parallel conductor feed line connected to the antenna will allow the antenna to be raised to its operating position for measuring impedance. An RF Bridge should be used to measure the exact length of the feed line. It will be close but not exactly equal to 984VF/F (MHz) for 1 wave length or 492VF/F (MHz) for 1/2 wave length due to the velocity factor of the line. The RF Bridge will take this factor into account.

An electrically equivalent 1/2 wave feed line or multiple thereof will reflect the impedance at one end to the other end at the exact frequency of 1/2 or 1 wave length. The RF Bridge will then measure the impedance of the antenna as if it was at the antenna.

The wave length of transmission line is derived from the formula:
L (wavelength in meters) = 300,000/KHz.
A quarter wave section is equal to 75000/KHz times VF. VF is the velocity factor of the line. 492VF/f = feet for 1/2 wave. 246VF/f = 1/4 wave.

Matching Antenna to Feed Line

Often I hear on the air and see in email forums questions by amateurs. Can I use 75 ohm coax from my antenna to my 50 ohm radio? My dipole is 73 ohms; can I use 75 ohm coax to feed it? My vertical is 36 ohms, can I use 50 or 75 ohm coax to feed it?

This shows a lack of understanding of some antenna basics. First of all, as previously stated, the dipole most often will not be 73 ohms impedance. Usually lower when closer than 1/2 wave to the ground. A full wave closed loop will be 80 to 150 ohms and a half wave open loop about 10 ohms impedance.

Using a 50 ohm coax with a 36 ohm vertical may present with a fairly low SWR and many hams do this without problems. However, when running low power, QRP-5 watts to 100 watts, every watt radiated matters. Therefore, it is important to have the best match of feed line to antenna possible.

Many articles in QST and QEX Quarterly have covered the subject of matching antennas to feed lines. A quick search will show 1/4 wave lines, matching stubs, inline sections of different impedances, Omega and Gamma matches, Paulsey stubs, baluns, ununs, and parallel 1/4 wave transformers to name a few. Quite confusing for new amateurs and usually requiring use of mathematics which may be unfamiliar.

What I want to present is a specific specialized use of 1/4 wave line sections. I will show a little basic math and then a graphical solution to make it easier for the newer amateurs.

492/f (KHz) = feet of 1/2 wave and 246VF/f (KHz) = 1/4 wave in feet.

The quarter wave line must be an electrical equivalent of a quarter wave, meaning the frequency and VF must be taken into account. Naturally, the frequency will be the same as the resonant frequency of the antenna to be matched. The quarter wave transformer , often referred to as a "Q" section, is connected in series between the main feed line and the mismatched non-reactive load, i.e., a resonant antenna.

A quarter wave line, when terminated in a non-inductive resistance or resonant antenna, will reflect back to the input end certain resistive impedance. This impedance will be equal to:

Z = v (Zr X Zo) or rewritten as Zo = Z(squared)/Zr or Zr = Z(squared)/Zo

Where: Z = the 1/4 wave coax line impedance,

Zr = terminating resistive impedance (antenna) ( input resistive impedance as read on the RF impedance bridge at the end of the 1/2 wave line).

Zo = the impedance of the feed line.

The above formula is another form of the familiar equation: Z(squared) = Zo X Zr. This is the form most amateurs use, with the terms being the same as the above formula. The bottom line is : the impedance of the matching section is equal to the square root of the antenna impedance times the feed line impedance.

The average amateur does not like to perform large amounts of calculating or graphing of data. Many newer hams are products of our education system and do not know how to do anything, except perhaps basic addition and subtraction of small numbers. Many younger hams may not have been exposed to Algebra and extraction of square roots. Many hams also do not have 1/4 wave lengths of many different impedance coax or parallel lines for experimentation and probably do not want to make special impedance value lines. Therefore I have done most of the math and included a graph of several standard impedance lines and how they can be used to match antenna impedances to coax or antenna system tuners.

Figure 1 (see below) presents graphically a series of load lines covering several of the commercially available feed lines of various impedances. Use the RF bridge reading at the end of the 1/2 wave feed line as the antenna feed point impedance, Zr. This will be the load resistance and is shown on the bottom horizontal scale. This value will allow selecting the closest impedance cable for the 1/4 wave section to match to the 50 ohm output of the transmitter.

Of course, the graphs allow matching most any load impedance from 1 to 1000 ohms to any impedance desired from 10 to 1000 ohms. The graphs allow quick and reasonably accurate choice of matching section to provide the most efficient transfer of power from the transmitter to the antenna for radiation of the signal. After you become familiar with the graph, you will be able to use it for a variety of matching solutions.

Actual use examples

From the chart horizontal load of 20, go up vertically to the horizontal line from 50 on the left vertical scale. Where they cross read the value of the impedance of the slanted line that intersects them. This is the impedance of the 1/4 wave matching section. This graph shows approximately 30 ohms.

Since this is not a standard value, we have to make a 30 ohm section. Fortunately, a 1/4 wave each of 50 ohm and a 75 ohm paralleled will provide a 30 ohm cable. Here is how we calculate this.

Zo (section impedance) = Z1 (cable 1 impedance) X Z2 (cable 2 impedance) / Z1 + Z2 = 75 ohms X 50 ohms / 75 ohms + 50 ohms = 3750/125 = 30 ohms.

Remember, the velocity factor may be different for each cable. This means they will be different lengths. This is OK since using the RF Bridge to make the electrically equivalent 1/4 wave cable will factor the VF into the proper length at the desired frequency.

2. A 1/2 wave 40 meter dipole folded into a square loop of 16.5 feet each side, to fit a small garden, has an impedance of approximately 10 ohms.

From the graph, 10 ohms load to 50 ohms cable cross at the load line of approximately 20 ohms. Again this is not a standard value. However, two 1/4 wave lengths of 50 ohm line in parallel will give us a 25 ohm section. From the chart this section will transform the 10 ohms to 60 ohms. This will be about 1.2: 1 SWR and a good match for the transmitter.

3. A one wave 20 meter quad loop has approximately 120 ohms impedance. From the graph, a 75 ohm 1/4 wave section will transform these 120 ohms to about 40 ohms. Again about 1.2:1.0 SWR and a good match to the transmitter.

4. A 20 meter 2 element beam has approximately 25 ohms impedance. From the graph, a 35 ohm 1/4 wave section will transform the 25 ohms to 50 ohms. Two parallel 1/4 wave 75 ohm cables will make a 35 ohm matching section.

5. You need an emergency or quick 40 meter antenna. A 66 foot length of wire is available. It can be used as an end fed half wave. However the impedance of an EFHW is very high- 2K to 5 K ohms. From the graph it will be noted that from 2K on the left margin, a line horizontal to the 300 ohm slant line and then vertically will intersect with the horizontal at 30 to 50 ohms. Thus a 1/4 wave of 300 ohm line (about 24.5 feet at 7.1 mHz) will transform the EFHW to near 50 ohms. An ugly 1:1 balun made from 50 ohm coax and a short lead to the transmitter will allow operation without a transmatch antenna system tuner.

6. Rather than a EFHW, a folded dipole made from splitting 66 feet of lamp zip cord( or using it without splitting), shorting the ends and cutting one wire at the middle for a feed point. This antenna will have an impedance of approximately 300 ohms (usually 200 to 300 ohms depending on the height above ground). From the graph, a 100 ohm line 1/4 wave long will match it to present approximately 35 to 50 ohms. The lamp zip cord used as a balanced feed line has about 105 ohms impedance. Thus the whole antenna and feedline can be made from zip cord. An ugly balun and short line to the transmitter will allow operation on 40 meters. The 1/4 wave line would be 246X.71(VF)/7.1MHz. This is about 24.6 feet of zip cord.

I hope this graph will help to simplify matching antennas to feed lines for the beginning amateur and kindle a desire to experiment with different types of antennas. This is what amateur radio is really about, learning, experimenting and building to increase knowledge and understanding. Have Fun.

NOTE: If the graph below is unreadable, just click on it for a much larger view. You may save (download) it for future reference as needed when it loads in the larger image. In some browsers, you may have to click on it again to expand it. It's file size is over 1 Meg so dialup users allow time for download.