or What Happens Along a Length of Feedline

We have looked (and will continue to look) at antennas that present complex feedpoint impedances, that is, impedances composed of both resistive and reactive components. In the shorthand that we use for calculations, Z = R ± jX.

We have previously noted that a transmission line is (among other things) an impedance transformer. An impedance presented to the line as a load will appear as a different impedance at every point along the line. We can calculate precisely the value of impedance for any point along the line, as illustrated in **Figure 1**.

Knowing the values of impedance along a transmission is very handy. For a monoband antenna, we can select the point along a line where R is 50 Ohms and insert a compensating reactance so that we can then run coax the rest of the way to the shack. For multiband antennas fed with parallel line, we can estimate how much line to add or subtract in order to let the values fall within the range of adjustment of our ATU.

To calculate the value of impedance anywhere along a transmission line, we need only know the following: the load impedance, the characteristic impedance and velocity factor of our feedline, the line length in any units, and the frequency of operation. For ultraprecision, we can improve the result by knowing the line loss figures for the feedline type and the frequency of operation. However, for many purposes, assuming lossless lines will provide all the accuracy needed.

Now some good news and some bad news. The bad news is that, even though the necessary equation looks straightforward, the use of complex numbers (a ± jb) results in a tedious job for hand calculation. However, we have good news: we can resort either to Smith charts or to computer programs to do the heavy work for us. HAMCALC has a program called "Transmission Line Performance" that will calculate both single values and tables of values of voltage, current, and impedance (and their phases), as well as resistance and reactance along a lossless transmission line.

Actually, for impedance, we only need to look at ½ wavelength of line, since in lossless lines, the values repeat themselves at that interval. In real lines with some loss, the values will change slightly as the line length is increased in multiples of ½ wl.

Note, however, that values of voltage and current only change once per full wavelength. Although this fact will not affect the work we shall do in this installment, failure to appreciate it has led to some interesting errors in analyzing antennas. For example, most accounts of the ZL Special, a phased 2- element array, have viewed it as a 135° phased impedance antenna. However, antenna phasing is a product of antenna currents, not impedance. When this fact is appreciated, the antenna is more properly analyzed as a -45° phased current array.

For most low-HF wire antenna matching problems, we can focus on the impedance. The question that faces users of multiband antennas is whether the length of line in use will present to the ATU a set of values for resistance and reactance that fall within the tuning range of the ATU. The actual range of permissible values varies with the ATU network type and components. But let's arbitrarily adopt these limits: R should range from about 20-25 Ohms to several hundred Ohms while jX should run no higher than ± a few hundred Ohms. These limits would place almost all the common ATU types within their high-efficiency matching zones.

**Line Length and Degrees**: The first step in analyzing our situation is finding the line length we have. Actually, we need only know where along the last ½ wl the line ends. We can calculate this length by plugging the following equations into a hand calculator:

1. Divide the total length of line by the length of ½ wl of line, adjusted for the velocity factor (VF) of the line in use. Common 450-Ohm line has a VF of 0.95, while common 300-Ohm line has a VF of 0.80. The tables below list line lengths for various increments and frequencies. Use the figure for 180° for the nearest frequency to the one in which you are interested.

2. Throw away the integer (the part left of the decimal, if any) and save the decimal part. Multiply this number times 180 to get the number of degrees along a half wavelength of transmission line that represents the termination point of your line.

Virtually all transmission line work is done in degrees relative to a full 360° circle. 180° represents the half-circle in which we are interested. The tables below list the lengths of transmission line for various degrees along the line for selected HF frequencies that are useful for estimating your line length. Line lengths are in feet for 450-Ohm and for 300-Ohm lines.

80-meters 3.6 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 21.63 18.21 60 43.26 36.43 90 64.89 57.68 120 86.52 72.86 150 108.15 85.00 180 129.78 109.29 40-meters 7.15 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 10.89 9.17 60 21.78 18.34 90 32.67 27.51 120 43.56 36.68 150 56.27 45.85 180 65.34 55.02 30-meters 10.125 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 7.69 6.48 60 5.38 12.95 90 20.51 19.43 120 30.76 25.90 150 38.45 32.38 180 46.14 38.86 20-meters 14.15 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 5.50 4.63 60 11.01 9.27 90 16.51 13.90 120 22.01 18.54 150 27.51 23.17 180 33.02 27.80 17-meters 18.1 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 4.30 3.62 60 8.60 7.25 90 12.91 10.87 120 17.21 14.49 150 21.51 18.11 180 25.81 21.74 15-meters 21.15 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 3.68 3.10 60 7.36 6.20 90 11.04 9.30 120 14.73 12.40 150 18.41 15.50 180 22.09 18.60 12-meters 24.95 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 3.12 2.63 60 6.24 5.26 90 9.36 7.88 120 12.48 10.51 150 15.60 13.14 180 18.73 15.77 10-meters 28.5 MHz L(°) L(ft): 450-Ohm 300-Ohm 30 2.73 2.30 60 5.46 4.60 90 8.20 6.90 120 10.93 9.20 150 13.66 11.50 180 16.39 13.80

**Sample Cases**: Since impedance transformations are a function of the number of degrees along a line we take a reading, regardless of frequency, we can look at some transformations that are interesting and list them in terms of degrees. You can then translate them into actual line lengths for your situation, depending on the line you are using and the frequency at which you encounter something similar to the example.

1. Mismatch with no load reactance:

Load Z = 150 ± j0 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 193 + j223 185 + j120 60 450 + j520 343 + j223 90 1350 + j0 600 + j0 120 450 - j520 343 - j223 150 193 - j223 185 - j120 180 150 + j0 150 + j0

Note the symmetry of the variations of values for this line with no reactance. Still, none of the values seems to exceed what a good ATU might handle. Line length is thus not at all critical.

2. Mismatch with small inductive reactance.

Load Z = 150 + j30 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 208 + j260 206 + j153 60 538 + j564 419 + j226 90 1298 - j259 577 - j115 120 380 - j475 282 - j209 150 179 - j188 166 - j90 180 150 + j30 150 + j30

Note the disruption to the symmetry due to the presence of reactance. Although the values appear reasonable, I would avoid a 90° length, if possible.

3. Mismatch with small capacitive reactance.

Load Z = 150 - j30 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 179 + j188 166 + j90 60 380 + j475 282 + j209 90 1298 + j259 577 + j115 120 538 - j564 419 - j226 150 208 - j260 206 - j153 180 150 - j30 150 - j30

Notice the pattern of values and signs in the comparison of this case and the preceding one, where the only change is the type of reactance in the load.

4. Mismatch with larger inductive reactance.

Load Z = 150 + j150 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 290 + j438 339 + j316 60 1172 + j598 781 - j52 90 675 - j675 300 - j300 120 213 - j321 142 - j132 150 137 - j69 115 - j8 180 150 + j150 150 + j150

This plot reveals that as the reactance goes up, part of the curve of values along the line grows quite steep. In this case, it is the first 90° of each half wavelength. The second 90° provides the best region for connection to an ATU.

5. Mismatch with larger resistance and inductive reactance.

Load Z = 600 + j150 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 643 - j105 435 - j252 60 435 -j180 200 - j165 90 318 - j79 141 - j35 120 307 + j50 155 + j90 150 397 + j164 267 + j221 180 600 + j150 600 + j150

As the resistive component of the antenna feedpoint impedance goes considerably higher than the characteristic impedance of the feedline, (also making the ratio of resistance to reactance greater), the best matching region becomes the mid-region of the half wavelength line section, where the rate of change of resistance and reactance values is also the lowest. However, none of the values shown should present most ATUs with any problems.

6. Mismatch with the larger resistance and a high capacitive reactance.

Load Z = 600 - j1000 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 138 - j371 81 - j314 60 83 -j85 41 - j92 90 89 + j149 40 + j66 120 89 + j149 69 + j268 150 1189 +j1216 365 + j812 180 600 - j1000 600 - j1000

As the reactance grows larger than a resistance which is already larger than the characteristic impedance of the line, the region of easiest match to an ATU shifts. For this capacitive reactance, 30° to 120° becomes the best region; for an equivalent inductive reactance, 60° to 150° would be the best region.

7. Mismatch with very large resistance and inductive reactance.

Load Z = 2000 + j2000 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 295 - j960 116 - j606 60 77 - j327 33 - j203 90 51 - j51 23 - j23 120 59 + j192 28 + j143 150 138 + j587 70 + j432 180 2000 +j2000 2000 + j2000

As the resistance and inductive reactance reach the 2K level, as is common when 80-meter and 40-meter antennas are used on WARC bands, line lengths near 90° become more favorable for easy matches. Note that 300-Ohm line becomes less favorable for use, since at 90° it present a very low impedance to the ATU.

8. Mismatch with very large resistance and capacitive reactance.

Load Z = 2000 - j2000 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 138 - j587 70 - j432 60 59 - j193 28 - j143 90 51 + j51 23 + j23 120 77 + j327 33 + j203 150 295 + j960 116 + j606 180 2000 - 2000 2000 - j2000

Notice the reversal of the progression of values, but not signs attached to the reactance, compared to the previous case. Also compare cases 7. and 8. to cases 2. and 3.

Cases 7. and 8. also show why the use of the 4:1 balun built into many ATUs should be avoided. With higher feedpoint impedances, the region of best match along the line encounters low impedance values. Assuming a perfect 4:1 ratio and no loss (actually, a rather poor assumption when there is significant reactance), the values seen by the ATU would be very low, either beyond the range of the ATU or subject to further losses in the network.

9. Mismatch with very large resistance and extremely large inductive reactance.

Load Z = 2000 + j10000 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 18 - j863 8 - j556 60 5 - j286 2 - j185 90 4 - j19 2 - j8 120 5 + j234 2 + j162 150 13 + j707 6 + j487 180 2000 +j 10K 2000 + j10K

Note that the transformer effect of the feedlines, as their characteristic impedances become very small compared to the magnitude of the feedpoint impedance (especially the reactance), confines the high impedance values to a small region near 0 and 180°. The remainder of the line section shows very low values of resistance.

10. Mismatch with very large resistance and extremely large capacitive reactance.

Load Z = 2000 - j10000 Ohms Line Length R ± jX (degrees) 450-Ohms 300-Ohms 30 13 - j707 6 - j486 60 5 - j234 2 - j162 90 4 + j19 2 + j9 120 5 + j286 2 + j185 150 18 + j863 8 + j556 180 2000 - 10K 2000 - j10K

These final two cases demonstrate why half wavelength loops and similar antennas are unusable with standard lines and ATUs. Indeed, with the end- fed Zepp, if the feedlines were not at least slightly out of balance, effecting a match with common forms of ATUs would not be possible, since the impedances presented to the feedline would be even higher.

Exploring the behavior of impedances along a feedline can be very instructive in estimating the most favorable line lengths for presenting the ATU with not only a load it can match, but one which it can match efficiently. The samples presented here only scratch the surface of the subject. However, they do show some of the emergent patterns of how the values change with different ranges of antenna feedpoint impedances. I heartily recommend that you get a copy of HAMCALC and run the values presented with some of the antennas we have already explored and will yet explore in future episodes. What you learn just might increase your antenna system efficiency a notch or two.

*Updated 11-11-98. © L. B. Cebik, W4RNL. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.*