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<!DOCTYPE html>
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<title>Reactive Antenna Loads and Their NEC Models Index</title>
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<body>
<center>
<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models</h1>
<h3>L. B. Cebik, W4RNL</h3>
</center><br>
<br>
<br>
<center>
<img src="../images/colorbar.gif" align="middle" width="540" height="4">
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<br>
Although the basics of changing the electrical length of an antenna wire by adding a reactive load to it are seemingly well-known, the precise relationship between real loads provided to real antennas and modeled loads provided to modeled antennas is not so certain as one might think. Therefore, this series of notes will try to look at some of the ins and outs of the matter--all without pretending either to authoritativeness or completeness. It will be enough if we can get a better handhold on the subject.
<p>In the first note, I focus on the means for implementing loads in modes as complex impedances, that is, as resistance and reactances, and as series R-L-C loads. The effort will be to translate a reactance into an inductance as might be found in a solenoid inductor of standard single layer winding. Moreover, we shall also focus only on center or source- point loading, whether the math of the effects of loading is simple addition. Eventually, however, we shall have to return to R+/-jX and translate this into the other major means of implementation: the shorted transmission line stub of under 1/4 wl. That means of load implementation has another name: the linear load.</p>
<p>Center or source-point loads are not the only reactive loading we can use to electrically lengthen an antenna. With dipoles and similarly balanced antennas, we can place loading reactances almost anywhere along the element, with the proviso that they be symmetrical in placement and value with respect to the element center. Actually, such loads do not have to be symmetrical, but symmetry does simplify most design and analysis problems. So we shall remain symmetrical to a fault.</p>
<p>Next, we shall address both solenoid inductors and transmission line stubs placed well away from center on antenna elements. They will show some interesting patterns of which we should be aware before taking the next more difficult modeling step.</p>
<p>NEC offers us short-cut methods of modeling both the solenoid and the stub reactive load via purely mathematical models. Before we conclude this sequence of notes, we shall investigate modeling both types of reactive loads--linear or transmission line loads and solenoid loads--as physical wire assemblies. One reason for taking pains to develop the models used in the preliminary investigation of using the LD and TL functions of NEC for antenna loads is that physically modeling both transmission line loads and single layer solenoid (or helical) loads requires care lest one exceed one or another limitation of the NEC modeling calculation system. The early models are developed with the later physical models in mind.</p>
<p>The results of physically modeling linear or transmission line loads and of physically modeling coils (helices) may hold a few surprises relative to some commonly held beliefs about load models. For one example, not all apparent mid-element linear loads are in fact mid-element loads. For another example, loading coils turn out not to be simple inductances (or inductive reactances), but are instead a complex combination of inductance and radiating antenna wire.</p>
<p>Although these notes may be useful in demonstrating a few principles, their numerical results cannot be automatically transferred to some existent antenna making use of loads. The final note will suggest some of the possible transfer errors that might be made, as well as some of the modeling challenges yet to be met when working with existing antennas.</p>
<ul>
<li>
<a href="load1.html"><b>Part 1: Some Center Loading Basics</b></a>
</li>
<li>
<a href="load2.html"><b>Part 2: Some Mid-Element Loading Basics</b></a>
</li>
<li>
<a href="load3.html"><b>Part 3: Some Linear Loading Basics</b></a>
</li>
<li>
<a href="load4.html"><b>Part 4: Some Solenoid Loading Basics</b></a>
</li>
<li>
<a href="load5.html"><b>Part 5: Some Unfinished Business. . .</b></a>
</li>
</ul>
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<p><i>Updated 2-6-99. &COPY; 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.</i><br>
<br></p>
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<p class="topics-footer">Topics: <a href='../topics.html#loading'>Loading</a>, <a href='../topics.html#modeling'>Modeling</a></p><br>
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="keywords" content="Loading, Modeling">
<title>Reactive Antenna Loads and Their NEC Models Part 1: Some Center Loading Basics</title>
<link rel="stylesheet" href="../styles.css">
</head>
<body>
<center>
<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models<br>
Part 1: Some Center Loading Basics</h1>
<h3>L. B. Cebik, W4RNL</h3>
</center><br>
<br>
<center>
<img src="../images/colorbar.gif" align="middle" width="540" height="4">
</center><br>
<br>
Although the basics of changing the electrical length of an antenna wire by adding a reactive load to it are seemingly well-known, the precise relationship between real loads provided to real antennas and modeled loads provided to modeled antennas is not so certain as one might think. Therefore, this series of notes will try to look at some of the ins and outs of the matter--all without pretending either to authoritativeness or completeness. It will be enough if we can get a better handhold on the subject.
<p>For antennas up to (but not including) 1/2 wl long, the source impedance is capacitive. To bring the antenna to resonance, we electrically lengthen it by adding an inductive reactance. We can add the reactance at the element center, or we may split the load into two equal parts and place them farther out on the element. The require inductive reactance needed to bring an element to resonance increases the farther out along the element we place the reactances.</p>
<p>Since every method of implementing an inductive reactance has some resistive loss, the reactive device has a Q, a ratio of inductive reactance to resistance. The higher the Q, the lower the losses. The most common two means of providing an antenna in the HF region with an inductive reactance are the air-wound solenoid inductor and the shorted transmission line stub less than 1/4 wl long. Both are physical entities having both basic and functional loss mechanisms. For example, solenoid wire for any given RF frequency has an AC resistance that is a function of skin effect. In addition, the interaction of the individual turns of the coil may yield other losses. Whatever the source of the loss, the solenoids Q or figure of merit expresses them all within the resistive denominator of the ratio.</p>
<h3>Modeling Resistance-Reactance Loads</h3>NEC (either -2 or -4) provides several means of modeling inductively reactive loads. The most common type is the following:
<center>
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<p><b>Type 4: Complex loads consisting of a resistance and a reactance.</b> This is the native type of load for NEC. All loads are eventually translated into values of resistance and reactance. However, this type of load remains constant. That is to say, for the frequency sweep specified in the simple model in the figure, the values of resistance and of reactance remain the same for each frequency checked.</p>
<p>Type 4 loads are therefore not useful for frequency sweeps, since real solenoid components do not have the same reactance on every frequency. Reactance changes with frequency in relationship to a solenoid inductance. For most purposes in antenna work, the inductance and the resistance of the coil are treated as constants in simple inductive reactance loading problems.</p>
<p>Type 4 loads, however, are especially useful for initial design work. For example, if one has a short center-fed antenna, the reactive component of the source impedance will specify the absolute value of center reactive load necessary to eliminate it. The type of reactance introduced will, of course, be the opposite of the reactive component of the source impedance. since antenna wires shorter than 1/2 wl show a capacitive reactance at the source, a center reactive load must be inductive.</p>
<p>Type 4 loads are also very useful for setting up or exploring the effects of the load Q on antenna performance. The results will apply only at the frequency for which the values hold good. However, one can easily obtain a family of value curves for comparing various antenna parameters.</p>
<p>Let's set up a test dipole to see the type 4 load at work.</p>
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<p>The test dipole is exactly 1/4 wl long at 7.15 MHz, which makes it far short of resonant. We shall introduce a reactive load at the center, in the same position as the source or feedpoint. In NEC models, the source and the load are in series when the modeler specified the same position or segment for both the source and the load. The effect is comparable to splitting the center-loading solenoid at the center to insert the source (normally a transmission line for operational amateur radio stations).</p>
<p>The test model antenna uses lossless or perfect wire so that the only losses that appear will be from the load or loads we add. It uses 91 segments for reasons that will appear in later episodes. If this little test is replicated, be sure to use the same number of segments as in the example to obtain the same results. Antenna resonance in the modeling exercise is defined as a source impedance of less than +/-1.0 Ohm, and in most cases, the reactance will be less than +/-0.1 Ohm. Since there will be a slight drift in the source impedance as the segmentation of a given wire is changed, even for well-converged models, replication of results requires detailed replication of all of the model parameters.</p>
<p>We may determine the required load for the antenna simply by reading the source impedance before inserting a load. We shall use two diameters of wires for this example: #8 and #12 AWG (with diameters of 0.1285" and 0.0808", respectively). To cancel out the capacitive reactance, the #8 antenna requires a center inductive reactance of 833.3 Ohms, while the #12 wire antenna of identical length requires 889.4 Ohms. For any value of Q, the resistance is simply the reactance divided by the selected Q value. Thus, we can select a range of Q-values to check. In this case, let us check a perfect inductively reactive load (R = 0), along with finite Q-values of 1000, 750, 500, 250, and 100. For the two antennas--which differ only in wire diameter--we get two families of values, which the following graph artificially connects with an identifying line just to keep the two antennas differentiated from each other.</p>
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<p>First, let's remember that the two antennas required two different reactive load values solely as a function of the source impedance. Both antennas use lossless wire, so the source impedance difference is a function of wire diameter. Now, if we look at the left-most line, where Q is infinite because the load resistance is zero, the antenna gain is the same for both wires. The fact that the value (1.85 dBi) is less than the traditional values for a resonant dipole (2.14-2.15 dBi) results from the length of the wire. It is much shorter than a resonant dipole. The wire shows the same gain without a load in place. Part of the gain of an antenna is a direct function of its length. In fact, you wish to obtain more gain from a dipole, make it longer--up to about 1.25 wl long. You may not like the resulting source impedance, but the antenna will have more gain simply by being longer--until it reaches a length where the pattern begins to degenerate into multiple lobes.</p>
<p>Second, the divisions from left to right following no special pattern. Hence, the actual curve of the values is not apparent. The drop in gain from a perfect load to one with a high Q-value of 1000 is noticeable. The region between a Q of 1000 and a Q of 500 strongly suggests that the curve of gain does not track arithmetically. Indeed, the curve is much straighter if each selected Q-value is of a constant ratio with the immediately preceding value throughout the chart.</p>
<p>Third, the two families of values track each other, with lower gain values for the #12 wire antenna. If we remember that the necessary inductive reactance for the #12 wire antenna was higher than for the #8 antenna, then it follows that for any given Q, the resistance will also be higher. The higher the resistance of a center load, the lower the gain.</p>
<p>The type 4 load from which we derived the modeled data is a mathematical entity only. Whereas the wire is modeled as a physical entity, the load is only a mathematical modification of the results. Hence, any physical properties that the load may have will not play a role in its modeled performance. The length and diameter of the loading mechanism are never modeled in Type 4 loads, even if in reality they may play some kind of role in the performance of the antenna.</p>
<h3>Modeling Series Resistance-Inductance-Capacitance Loads</h3>Equally mathematical are loads of the following type:
<p><b>Type 0: Loads consisting of the following is series: resistance, inductance, and capacitance.</b> Missing elements are entered as values of zero (which the program automatically interprets as a missing value, not a zero value). to specify an inductor of 6 microH with a series resistance of 2 Ohms, one enters this: 2, 6E-6, 0. The final zero indicates that there is no capacitor.</p>
<center>
<img src="load1-4.gif" width="421" height="235" border="2" vspace="5" hspace="9">
</center>
<p>The LD line in the input file shows the load type and values for resistance and inductance, again with a missing capacitor. Another load type (Type 1) is available for parallel combinations of resistance, inductance, and capacitance. We shall not need it for this particular set of notes.</p>
<p>The advantage of using R-L-C loads is that NEC translates them into appropriate values of reactance for each frequency checked in a file request. Hence, with R-L-C loads, one may run a frequency sweep and have the correct reactance at each frequency is the sweep.</p>
<p>However, there is an accompanying limitation. The value of R remains constant. Therefore, the Q of the reactive device changes slightly with each frequency checked, since the ratio of reactance to resistance changes across the range of the frequency sweep. For limited-range frequency sweeps, such as across one of the HF amateur bands, the possible error is too small to be significant.</p>
<p>Of course, one needs to be able to translate reactance to inductance (or, in other cases, capacitance) and back again to set the proper values. If one begins with a value of inductance, one must find the value of reactance for the frequency in question in order to use a value of Q to set the resistance or to find the value of Q from an assigned or measured value of resistance. The standard relationship of reactance to inductance, where inductive reactance is 2 PI times frequency times inductance in basic units, is an essential utility tool for using R-L-C loads.</p>
<p>A set of frequency sweeps between 7.0 and 7.3 MHz in 0.05 MHz steps for the #8 perfect wire dipole, center-loaded at the design frequency of 7.15 MHz for resonance with an inductance of 18.549 microH will yield a family of SWR curves. For each curve, the source impedance against which the antenna impedance is compared is the resistive impedance of the antenna at resonance on the design frequency. Hence, each curve shows a 1:1 VSWR at design center.</p>
<center>
<img src="load1-1.gif" width="645" height="526" border="2" vspace="5" hspace="9">
</center>
<p>The graph shows a family of such curves for a perfect inductor (zero Ohms resistance) and for inductors with Qs of 1000, 750, 500, 250, and 100. The requisite values of series resistance for the inductor are 0.8333, 1.1111, 1.6666, 3.3332, and 8.333 Ohms from 1000 down to 100. Once more, as we suspected from the gain curves earlier, the results of altering Q are more geometric than arithmetic, varying more linearly as Q forms a constant ratio from one value to the next.</p>
<p>Apparent in the graph is the fact that as we lower the Q of the loading coil, the operating bandwidth--set by any arbitrary selection of a VSWR ratio--become wider in terms of frequency. Often, we simply summarize the effect by saying that the increased losses broaden the operating bandwidth, as if the summary were an explanation. It is not.</p>
<p>In fact, what occurs, as modeled, is interesting in itself--and somewhat obvious if we stop to think about it. Consider the following two tables of values of source impedance for the #8 and the #12 dipoles, each center- loaded. The #12 wire antenna requires a 19.798 microH inductor with suitable adjustments in the values of the series resistor for the Q-values indicated earlier.</p>
<pre>#8 Wire Dipole Source Impedance (R +/- jX Ohms)
Freq. Q=Inf. 1000 750 500 250 100
7.00 12.58-46.28 13.41-46.29 13.69-46.28 14.24-46.29 15.91-46.29 20.91-46.29
7.05 12.78-30.76 13.61-30.76 13.89-30.76 14.45-30.76 16.11-30.76 21.11-30.76
7.10 12.98-15.30 13.82-15.30 14.09-15.30 14.65-15.30 16.32-15.30 21.32-15.30
7.15 13.19-0.02 14.02-0.02 14.30-0.02 14.86-0.02 16.52-0.02 21.52-0.02
7.20 13.40+15.26 14.23+15.26 14.51+15.26 15.06+15.26 16.73+15.26 21.73+15.25
7.25 13.61+30.25 14.44+30.25 14.72+30.25 15.27+30.25 16.94+30.25 21.94+30.25
7.30 13.82+45.29 14.65+45.29 14.93+45.29 15.49+45.30 17.15+45.29 22.15+45.30
Delta R 1.24 1.24 1.24 1.25 1.24 1.24
Delta X 91.57 91.58 91.57 91.59 91.57 91.59
#12 Wire Dipole Source Impedance (R +/- jX Ohms)
Freq. Q=Inf. 1000 750 500 250 100
7.00 12.61-49.30 13.50-49.29 13.79-49.29 14.39-49.29 16.16-49.29 21.51-49.30
7.05 12.81-32.78 13.70-32.78 13.99-32.78 14.59-32.78 16.37-32.78 21.70-32.78
7.10 13.01-16.29 13.90-16.30 14.20-16.30 14.79-16.29 16.57-16.29 21.91-16.29
7.15 13.22-0.00 14.11-0.00 14.40-0.00 15.00-0.00 16.78-0.01 22.11-0.00
7.20 13.43+16.31 14.32+16.31 14.61+16.31 15.21+16.31 16.98+16.31 22.32+16.31
7.25 13.64+32.29 14.53+32.29 14.82+32.29 15.42+32.28 17.19+32.28 22.53+32.28
7.30 13.85+48.35 14.74+48.34 15.04+48.34 15.63+48.34 17.41+48.34 22.74+48.35
Delta R 1.24 1.24 1.25 1.24 1.25 1.24
Delta X 97.65 97.63 97.63 97.63 97.63 97.65</pre>
<p>The limits of accuracy are set by the last digit in each number, plus or minus 1, as a function of the sum of rounding conventions within the program. Within those limits, which vary only the value in the hundredths column, for each antenna, the range of variation of both resistance and reactance is invariant, regardless of the value of Q. Since Q is a function of changing R, the values for source reactance do not change at all. The values of source resistance is simply the values at infinite Q plus the series resistance of the inductive load.</p>
<p>In other words, the presence of a center load of finite Q does not alter the VSWR operating bandwidth curve by changing the source impedance other than adding the load resistance to the natural source resistance of the antenna without any load at all.</p>
<p>What the addition of the load series resistance does is to change the ratio of resistance to reactance at any of the frequencies swept, essentially raising the ratio of resistance to reactance. For the test antennas, the amount of change ranges from a fraction of an Ohm to under 10 Ohms. However, in each case, the result is a change in the impedance phase angle toward the resistive and away from the reactive.</p>
<center>
<img src="load1e1.gif" width="385" height="61">
</center>
<p>The basic SWR equation shows the importance of this simple adjustment in terms of a lowering of the SWR at any frequency within the sweep range, when the value of Zo is the same as the value of Rin at the design center frequency. With values of Xin constant, higher values for Rin and Zo will together lower the VSWR relative to a design center frequency value of 1:1.</p>
<h3>Transmission Line Stub Loads</h3>Before we turn away from the modeling of center loads with the mathematical models available in NEC, let's look at the alternative method of creating an inductively reactive load: the transmission line stub. These stubs present us with an immediate modeling problem. Modeled transmission lines appear in parallel with sources. So we must move them. Moreover, a single stub would be very long (over 20' long for a 600-Ohm line). Common practice is to split the stub into two equal parts, installing one on either side of the line at the main element feed point. Our simple modeling counterpart would be to place TL entries in NEC on each of the adjoining segments to the source segment.
<p>However, we cannot simply use the value of the center-loading reactance and split it into two pieces. If the reactance is split and moved away from the source point, the sum of the pieces will be greater than the value of the exactly centered single reactance. The center loading reactance required for the #8 perfect wire dipole that is exactly 1/4 wl long overall in free space was 833.3 Ohms. Two reactances placed about 9" apart center- to-center, one on either side of the source, require 426.3 Ohms each (or 852.6 Ohms total)</p>
<center>
<img src="load1-6.gif" width="444" height="413" border="2" vspace="5" hspace="9">
</center>
<p>With a type 4 reactive load, we can move in two loading directions. One of them is a Type 0 series R-L-C load, which requires an inductance at 7.15 MHz of 9.4892E-6 H on each side of the source. The other direction is a pair of transmission line stubs each side of center. Since the models of the stubs are mathematical only and a form of a NEC network, they will not show any interaction, loss, or other factors other than reactance.</p>
<p>The reactance of a shorted stub is equal to the characteristic impedance of the transmission line times the tangent of the electrical length of the stub in degrees or radians. Since we know that the stubs will each show 426.3 Ohms reactance, we can work backwards through the relationship to obtain the electrical length for a variety of common lines. Then we can obtain the physical length, since we know that a wavelength at the design frequency is 137.5624' long. For this exercise, we shall look at several common transmission lines with characteristic impedances of 50, 75, 300, 450, and 600 Ohms. We shall let the velocity factor be 1.00, since many of the lines offer options and variability. We have cut off the characteristic impedance at an upper limit of 600 Ohms, since with #8 wire, even 700-Ohm line requires over 22" of spacing without significantly altering the 600-Ohm values across the band.</p>
<p>Let's first determine the VSWR for a pair of perfect solenoid coils and then for lines of 300, 450, and 600 Ohms across the 40 meter band. Each lossless coil will have a 9.4892 microH inductance. The line lengths required by each of the lines (including the ungraphed 50 and 75 Ohm lines) will be as follows:</p>
<pre>Line Zo 50 75 300 450 600
L (degrees) 83.31 80.02 54.86 43.45 35.39
L (feet) 31.83 30.58 20.96 16.60 13.52
</pre>
<p>We create on the segments on each side of the source a TL or transmission line entry. Depending on the program we use, we might have to create a distant wire to terminate the stub or the program might do this for us and simply let us specify that we desire a short at end 2. Do not enter the length in electrical degrees, or a frequency sweep will not be accurate. That same electrical length, with a different physical length, will appear at each frequency checked by the sweep. Instead, enter the design center physical length of the stub, along with the characteristic impedance of the line. We shall explain in a moment why the 50- and 75-Ohm lines are not included in the graph below</p>
<center>
<img src="load1-7.gif" width="648" height="525" border="2" vspace="5" hspace="9">
</center>
<p>With all practical characteristic impedance lines, the SWR bandwidth is significantly narrower for a transmission line stub load than for a coil of indefinitely high Q. Note that the SWR bandwidth decreases as the characteristic impedance of the line decreases. The low impedance lines were omitted because their end values of SWR exceeded 100 and would have obscured the detail of the graph.</p>
<p>The reason for the narrower bandwidth is not Q as ordinarily understood in terms of resistive losses within the reactive loading device. Neither loading means has any resistive losses in the models. Rather, the stubs have reduced bandwidth because the value of reactance is a tangential function of their electrical length as the frequency changes, which yields a curve quite unlike the curve of solenoid coil reactance vs. frequency. The difference shows up vividly in these examples, since the required reactance is so high. For small-value line-vs-coil applications, such as the values associated with beta matches, the curves across a ham band will not be significantly different if the stub uses a high characteristic impedance.</p>
<p>To reinforce the differences between single-layer solenoids and transmission lines stubs as sources of reactance in antenna loading, let's look at tables of values for the perfect coil and the bank of transmission lines used.</p>
<pre> Source Impedance (R +/- jX Ohms) (Values rounded)
Frequency Coil Line Zo
MHz 50 75 300 450 600
7.00 13.1-47.3 13.0-207 13.0-154 13.1-64.8 13.1-56.3 13.1-52.7
7.05 13.3-31.5 13.2-147 13.3-107 13.3-43.3 13.3-37.5 13.3-35.0
7.10 13.5-15.6 13.5-78.5 13.5-55.9 13.5-21.7 13.5-18.7 13.5-17.5
7.15 13.8-0.0 13.8-0.2 13.8-0.1 13.8+0.0 13.9-0.0 13.8+0.0
7.20 14.0+15.7 14.0+91.5 14.0+61.2 14.0+22.0 14.0+18.8 14.0+17.6
7.25 14.2+31.1 14.3+201 14.3+122 14.2+43.9 14.2+37.4 14.2+33.9
7.30 14.4+46.6 14.7+334 14.6+205 14.5+66.1 14.5+56.2 14.4+52.3</pre>
<p>The table shows several features of using transmission line stubs as loading elements at or near the center of the antenna element. First, the range of reactance variation is higher in all cases than for the perfect inductor. Second, although the phenomenon is smaller by far, the range of source resistance values is also greater for transmission line stubs than for perfect (and, by reference to earlier data, for finite-Q) inductors.</p>
<p>Since all of the loading devices used here have no resistive losses, their Qs as determined by the usual ratio of reactance to resistance are identical. However, their operating bandwidths are not at all equal due to the manner in which the reactance changes with frequency for the two types of reactive loads. Although there are ways around the problem for special cases, there is no direct way to introduce losses into the transmission line model used by NEC. Hence, one cannot by this means explore the effects of accounting for losses in the stubs.</p>
<p>Despite this shortcoming, TL models of transmission lines stubs do reveal some properties of their use as reactive loads. The fundamental differences in behavior relative to inductors across a band of frequencies should be clear. Second, for single frequency use, stubs are equivalent in all ways to inductors. Finally, we may note in passing that inductively reactive stub loads have an alternative name: linear loads. That name has, unfortunately, led to a number of persistent misunderstandings about their basic nature. We shall have to return to that name later.</p>
<p>However, lest a small matter go unnoticed, let me set it out in relief. In splitting and moving the loading reactance only about 9" each side of the source, we found an increase in the required value of loading reactance. At the same time, the 7.15 MHz resonant source impedance for the antenna increase in the perfect reactance models from 13.19 Ohms to 13.76 Ohms. Although two cases alone do not constitute a trend, I shall bet that one is afoot. And with this, we have a prelude to looking at inductively reactive loads installed at a distance from the centered source of our very short dipole.<br>
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<p><i>Updated 12-11-98. &COPY; 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.</i><br>
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<title>Reactive Antenna Loads and Their NEC Models Part 2: Some Mid-Element Loading Basics</title>
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<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models<br>
Part 2: Some Mid-Element Loading Basics</h1>
<h3>L. B. Cebik, W4RNL</h3>
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In the first installment of these notes, we looked almost solely at short, center-loaded dipoles. Our test dipole is exactly 1/4 wl long (34.39') at 7.15 MHz and uses perfect or lossless wire of various sizes between AWG #8 and AWG #12. For reasons that will become apparent in still later episodes, the test model has 91 segments, each about 4.53" long. For Type 4 (complex R +/- jX) loads or for Type 0 (series R-L-C) loads, we placed the load on the center segment (#46).
<p>The one exception to this procedure arose when we substituted shorted transmission line stubs for the series R-L-C load, using the TL function of NEC. Since transmission lines appear in parallel with loads, we were forced to split the load and move the stubs to the segments immediately adjacent to the center segment (Segments #45 and #47). Two phenomena appeared:</p>
<p>1. Even with such a small spacing between loads, the source impedance of the dipole increased by a measurable amount. Using the #8 AWG wire model, the resonant source resistance climbed from 13.2 Ohms to 13.8 Ohms.</p>
<p>2. Likewise, the required reactance for loading the wire to resonance also increased. For the #8 wire model, the required reactance each side of center climbed from 416.65 Ohms to 426.3 Ohms just by moving the loads to a spacing of about 9" apart.</p>
<p>These two phenomena form part of a set of trends with which all designers of loaded elements must be familiar. We shall look at them initially by using the same load-creation facilities of NEC that we used in looking at centered loads. Before we are finished with these basics, however, we shall formulate some questions about the reliability of models to tell us the entire story using only the "regular" means of load creation: Type 4, Type 0, and transmission line (TL) shorted stubs.</p>
<h3>Load Position Trends</h3>In theory, it is possible to split a center inductively reactive load and to place the halves symmetrically anywhere along the wire of our short dipole. As we move the load halves outward, we must increase the value of reactance of each half in order to bring the dipole to resonance. The following exercise is designed to generate an appreciation of how much reactance must be pressed into service for the purpose of re-resonating the dipole as we push the loads further outward.
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<p>Let us take the #8 model and place the loads successively 10% further outward from center. In terms of total antenna length, this amounts to placing the load at the 45%-55% positions, the 40%-60% positions, etc. For a 91-segment dipole, we can only approximate these positions, but the results are certainly close enough to reveal the trends involved. The exercise can use simple R +/- jX loads. At each position, we increase the value of inductive reactance to bring the dipole to within +/- 1 Ohm reactance of resonance at 7.15 MHz.</p>
<p>Initially, we can let the Q of the load be indefinitely high (or infinite, meaning resistively lossless) by setting the value of the load resistance to zero. We obtain the following table of results.</p>
<pre>Position Position Reactance in Source Impedance Free Space
Approx % Actual % Ohms per Side R +/- jX Ohms Gain dBi
5/95 4.95/95.05 3948 37.7 + j 0.3 1.91
10/90 10.44/89.56 1939 34.4 - j 0.5 1.90
15/85 14.84/85.16 1369 32.0 + j 0.4 1.89
20/80 20/33/79.67 996 28.9 + j 0.8 1.88
25/75 24.73/75.27 817 26.5 + j 0.8 1.87
30/70 30.22/69.78 668 23.6 - j 0.6 1.87
35/65 44.62/65.38 586 21.3 + j 0.7 1.86
40/60 40.11/59.89 510 18.4 - j 0.5 1.86
45/55 44.51/55.49 465 16.2 + j 0.7 1.86
50/50 50/50 416.65 13.2 - j 0.0 1.85</pre>
<p>The most significant trend to note is the steady rise in the required reactance to bring the dipole to resonance as the load is moved progressively outward from center. The following graph shows the trend with some clarity.</p>
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<p>The source resistance and gain figures are predicated on the unlikely event of obtaining a perfect (lossless) inductively reactive load. A more realistic value would be a Q of about 300. This requires that for each load, we divide the reactance by 300 to obtain a series resistance for the total load. If we insert the required values of resistance into our R +/- jX loads, we obtain the following values at 7.15 MHz with our #8 lossless wire model. (Reactance values do not change and are not shown below.)</p>
<pre>Position Series R Source Impedance Free Space Inductance
Approx % Ohms R +/- jX Ohms Gain dBi microH
5/95 13.16 57.4 - j 0.9 0.08 87.9
10/90 6.46 44.7 - j 0.9 0.77 43.2
15/85 4.56 39.6 + j 0.2 0.96 30.5
20/80 3.32 34.9 + j 0.6 1.07 22.2
25/75 2.72 31.6 + j 0.7 1.11 18.2
30/70 2.23 27.9 - j 0.7 1.13 14.9
35/65 1.95 25.2 + j 0.6 1.13 13.0
40/60 1.70 21.9 - j 0.5 1.12 11.4
45/55 1.55 19.6 + j 0.7 1.02 10.4
50/50 1.39 16.0 - j 0.2 1.02 9.3</pre>
<p>The inductance that yields the required reactance is shown in the right- most column for convenience, and we shall return to it in a moment. First, however, let us look at the values from left to right. The series resistance values can be compared to the reactance values in the preceding table to confirm a Q of 300. The other values may benefit from a bit of graphing.</p>
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<p>The source resistance increases approximately, but not exactly, as the loss resistance increases for each pair of loads. The small differential between the reported source resistance and the sum of the source resistance with an infinite Q and the total of resistive losses results from placing the loads ever farther from the center or source point of the dipole. Nonetheless, the trend is abundantly clear.</p>
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<p>The graph that compares the free space gain of the dipole without load losses and with losses of a Q=300 load is perhaps more interesting. The lossless loads show a wholly insignificant variation of gain, regardless of load position. However, with a Q=300 load, the gain of the dipole shows a rise of 0.1 dB when the loads are place 20% to 60% of the total element length each side of center outward from the center--relative to the gain with the load centered. Beyond the 60% mark (20/80%, in terms of total dipole length), the gain decreases severely and rapidly as a result of the equal rapid increase in resistive loss that goes along with the steep rise in required loading reactance.</p>
<p>The 0.1 dB gain differential between center and mid-element loading is of questionable operational benefit. In a horizontal dipole, the use of mid-element loading in preference to center-loading is for other reasons, most often the higher source resistance of the dipole with mid-element loading. There remains a persistent myth that mid-element loading shows significantly higher gain than center loading. This myth derives from short mobile antenna experience using loaded monopoles. Field strength differences resulting from the environment within which such antennas work and from the very short lengths they use have become unwarrantedly generalized to cover all antennas making use of inductively reactive loads.</p>
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<p>A second source of the myth is an uncritical examination of current magnitude curves along loaded elements. The graph above displays the current magnitude for the R-L-C center-loaded element, the TL near-center-loaded element, and the mid-element R-L-C loaded element. Seemingly, the current magnitude (relative to a source value of 1.0) indicates a stronger far field based on the assumption that the field is roughly proportional to the current. However, that assumption needs considerable modification with loaded elements.</p>
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<p>One modification is occasioned by the current phase change along the element. For the mid-element loaded model, the rate of phase change along the element is much higher than for either center-loaded model. Moreover, the current magnitude curve is not a direct measure of the undissipated power in the element.</p>
<p>Using the strictly mathematical models of loads of all sorts, we may replace the Type 4 loads with either series R-L-C loads or with transmission lines (TL networks). The right-most column of the table shows the equivalent inductance values for the reactance loads used in this exercise. Although the weight and gain drop associated with such large inductance values is reason enough not to place loading inductors at the far ends of antenna elements, there is another reason as well. In at least the most extreme case (87.9 microH), for any practical single-layer solenoid coil-winding technique, we would not reach the required inductance before the antenna had passed resonance and had begun to show an inductive reactance at the source. We shall return to this point in a later episode.</p>
<h3>Alternative Means of Placing Loads</h3>Although certainly precise enough to show the important trends associated with mid-element placement of loads, the technique of choosing segments on which to place loads was only approximate. Except at the very center of the dipole, the loads were a bit off their desired marks.
<p>We could so segment the dipole that the loads would appear (as centered on their segments) in the desired places--but only for this exercise. If we chose other percentage values for load placement, further revisions of segmentation would be required. There is an alternative.</p>
<p>Consider the following problem (which, for the moment is academic only): suppose we wish to find the places along the dipole element where each of two mid-element loads are identical in required reactance to a single center load. For our 1/4 wl #8 perfect wire dipole, the required reactance of a center load is 833.3 Ohms to achieve resonance at 7.15 MHz. The question then is where along the wire to place two 833.3-Ohm loads also to achieve resonance.</p>
<p>The model has 91 segments. If we place the loads on segments 23 and 69, the source impedance reports as 27.11 + j 31.7 Ohms, which indicates the loads are too far inboard. If we place the loads one segment further outward--on segments 22 and 70--the source impedance reports as 26.34 - j 39.04 Ohms, which indicates that the loads are too far outboard. Within the limits of the model, neither placement satisfies the requirement of achieving resonance within +/-1 Ohm.</p>
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<p>These unsatisfactory situations are indicated in parts A. and B of the figure above. The figure also indicates an alternative modeling method that permits precise load location without sacrificing model accuracy. The model in C. is a 5-wire model, all composed of the same type of wire. Two outer sections and a center section are long wires having many segments. Two special wires have 3 segments each, with the load placed on the center segment.</p>
<p>The use of 3 segments on the load wires serves to ensure that the segments immediately adjacent to the load segment are equal in length to the load segment. The center wire and the outer wires are segmented so that each segment length is as close as possible--within the limits of the total segmentation of the antenna--to the length of the load wire segments.</p>
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<p>The .NEC model above shows the dimensions of the final model in terms of the wire geometry for the dipole with Type 4 reactive loads. For the #8 wire model, a center wire 199" long (+/- 99.5") used 45 segments, each 4.42" long. The outer wires are each 94.24" long and use 20 segments, each 4.71" long. The 3-segment load wires are 13.6" long, with segments 4.53" long. The total number of segments for the antenna remains 91.</p>
<p>The loads are centered 106.3" each side of the source. With this configuration, the reported source impedance is 26.78 + 0.53 Ohms, well within the limits set for resonance. The figures are for a pure reactance, with no series resistance.</p>
<p>The technique used here is subject to many variations, including re-segmentation to achieve even closer segment length equality along the antenna. Even more precise placement is possible to achieve resonance within narrower limits (for example, +/- 0.1 Ohm). However, for the purposes of this demonstration, the techniques used are sufficiently precise for both placement and resonance.</p>
<h3>Center vs. Mid-Element Loading</h3>Having placed the 833.3-Ohm load correctly at mid-element positions, it useful to take a preliminary look at the performance curves that result from applying the equivalent Type 0 (series R-L-C) load at both center and mid-element positions. The inductive equivalent of 833.3 Ohms at 7.15 MHz is 1.8549E-5 H. We can look at the curves both for infinite Q and for some representative Q, say, 300. The series resistance required for a Q of 300 at the design frequency is 2.778 Ohms.
<p>Once more, we shall use our #8 lossless wire 34.39' dipole.</p>
<pre>#8 Wire Dipole Source Impedance (R +/- jX Ohms)
Freq. Center Load Center load Mid-El. Load Mid-El. Load
MHz Q=inf. Q=300 Q=inf. Q=300
7.00 12.58-46.28 15.35-46.29 24.33-82.17 29.07-82.25
7.05 12.78-30.76 15.56-30.76 25.11-55.15 29.98-55.23
7.10 12.98-15.30 15.76-15.31 25.93-27.54 30.94-27.64
7.15 13.19-0.02 15.97-0.02 26.78+0.53 31.93+0.40
7.20 13.40+15.26 16.18+15.26 27.67+29.19 32.98+29.08
7.25 13.61+30.25 16.39+30.25 28.60+58.35 34.07+58.54
7.30 13.82+45.29 16.60+45.29 29.58+87.98 35.21+87.84
Delta R 1.24 1.25 5.25 6.14
Delta X 91.57 91.58 170.15 170.09</pre>
<p>Mid-element loading results in a larger swing of both source resistance and source reactance across the span of frequencies swept in this exercise. The result for resistance is larger not only in terms of Ohms, but also when taking the swing as a percentage of the design center frequency value. The result for reactance, however, is the opposite: as a percentage of the design center frequency resistance, the reactance swing is smaller for the mid-element load case than for the center loading case. Moreover, for the mid-element loads, there is not the virtual identity of swings for either resistance or reactance that holds over the frequency span swept by the models with center loads.</p>
<p>The smaller change of reactance relative to the resonant source resistance also suggests that the SWR curves for the mid-element loading cases might be broader than for their center-load counterparts. Of course, SWR is plotted relative to the design center frequency resistance value, where the antenna is resonant within the prescribed limits for this exercise.</p>
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<p>The graph confirms our suspicions. The upper two curves are for the test runs at an indefinitely high Q, while the lower curves are for the value Q=300. In both pairs, the curve for the mid-element loaded dipole is broader than its center-load counterpart. Two cautions attend these curves. First, the effect of wire losses are not accounted within the overall curve, since we are using perfect wire in order to isolate, as best we can within the model, the phenomena that may ascribed directly to the load itself. Second, the conditions of modeling must be remembered: the model is a free space model that, of course, does not take any ground effects into account. The modeled loads assign both the series resistance and reactance and do not take into account any further effects that might accompany a real single-layer solenoid inductor placed in center or mid-element loading service. Although specified in a Type 0 load in terms of a series combination of resistance and reactance, the loads are strictly mathematical entities whose physical properties are not calculated in the overall antenna evaluation.</p>
<p>The antenna free space gain for the various cases corresponds well to the curves earlier presented. The infinite-Q antenna shows a gain of 1.85 dBi for the center-loaded case and 1.87 dBi when mid-element loaded. When each loading inductor is set at a Q of 300, the gains are 1.02 dBi and 1.11 dBi, respectively. Again, these gain figures do not include any wire loss associated with the main element.</p>
<p>While we are exploring these modeling curves, we might as well compare transmission line loading when placed at the same mid-element point with the case developed in Part 1. In that instance, we placed a split load in the segments immediately adjacent to the source segment, because placement exactly on the source segment would not have yielded valid series load results. Even that small move of the load position required that we use two 426.3-Ohm loads, rather than a pair of 416.65-Ohm loads (833.3/2). The resultant 600-Ohm (velocity factor = 1.0) transmission line stubs were each 13.52' long. To replace the mid-element 833.3-Ohm loads with transmission line stubs with the same characteristic impedance requires lengths of 20.73' each. Once more, remember that transmission lines used in NEC models are lossless and mathematical. Within those constraints, the results are as follows:</p>
<pre>#8 Wire Source Impedance (R +/- jX Ohms)
Freq. Near-Center Mid-Element
MHz Placement Placement
7.00 13.09-52.66 23.84-110.0
7.05 13.31-35.04 24.76-74.67
7.10 13.53-17.46 25.74-37.85
7.15 13.76-0.03 26.78+0.42
7.20 13.98+17.57 27.90+40.47
7.25 14.21+34.88 29.09+82.25
7.30 14.44+52.30 30.38+126.0
Delta R 1.35 6.54
Delta X 104.96 236.0</pre>
<p>The mid-element transmission line load source impedance value verifies that the transmission line stub is the correct replacement for the reactance load of 833.3 Ohms. However, even with a 600-Ohm transmission line, the variance of reactance across the band is very large. As we have noted, with transmission lines of lower impedance, the variance will be larger still.</p>
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<p>Comparing the SWR curves (relative to an SWR of 1:1 at resonance) for the near-center and the mid-element transmission line loads shows an interesting result that is opposite to results from using series R-L-C loads. The mid-element loading stub system yields a narrower SWR curve than the near-center loading system. This consequence follows from the much high values of reactance required of the stub. Since the reactance of the stub is a tangent function of the electrical length of the stub, stubs over 45 degrees long change their reactance values rapidly. The required stub length in this case is 54.2 degrees.</p>
<h3>Concerns About Modeled Loads</h3>Commonly, series R-L-C loads are considered to be adequate models of single-layer solenoid inductor performance in a loaded dipole or similar element. Despite their strictly mathematical nature, that is, the fact that their physical structure does not play a role in the calculation of predicted antenna performance, the resulting antenna performance figures are considered accurate enough for most practical design and analysis purposes. The key loss factor in a solenoid inductor is the series resistive loss, and other losses and factors that might cause variations from the predicted outcomes are considered too small to be of significance. Consequently, only in the most critical cases are solenoid loading coils physically modeled--and usually only where such coils are very simple. An example of such a case is the solenoid in the middle of many automobile UHF antennas for cellular telephone service.
<p>Transmission line loads, on the other hand, are widely distrusted as models of linear loads used in HF antennas. Most obviously, they do not permit the accurate introduction of resistive losses in the transmission line. Nor do they permit, except as an external calculation, the introduction of the line's velocity factor. (Some programs permit velocity factor information to be introduced into the interface between the user and the core calculation, subsequently converting it to the values needed by the core's network calculations.)</p>
<p>Most antenna's employing linear loads use load wire having a diameter different from that of the main element. Although MININEC can directly handle these types of models, the required model often presses the segmentation limit of the program. (Some recent MININEC developments, such as NEC4WIN, have broken the 256-segment barrier, and one version permits virtually unlimited numbers of segments, but at the cost of slow calculation speeds.) MININEC has additional requirements in order to reduce errors at sharp corners, NEC (in either version 2 or version 4) is inaccurate when faced with angular junctions of wires having dissimilar diameters. Consequently, physically modeling existent antennas with linear loads is not normally attempted.</p>
<p>However, it is possible to physically model linear loads using a uniform wire diameter throughout the model. While the technique does not yield an accurate model of a particular antenna, it does produce quite accurate results that can reveal some interesting properties of linear loads. The technique may also permit some preliminary sorting of different functions performed by a linear load and the conditions under which it performs them. Physically modeling some linear load configurations will therefore be our next task.<br>
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<p><i>Updated 12-29-98. &COPY; 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.</i><br>
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<title>Reactive Antenna Loads and Their NEC Models Part 3: Some Linear Loading Basics</title>
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<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models<br>
Part 3: Some Linear Loading Basics</h1>
<h3>L. B. Cebik, W4RNL</h3>
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Linear loads do not model accurately using the TL facility of NEC. There are two major factors (and a number of minor ones) that contribute to the inaccuracy. First, the currents on the wires of the linear load are rarely, if ever, exactly equal in magnitude and opposite in phase. Consequently, the linear load does not as a pure transmission line. Second, most applications of linear loading snug the parallel load wires close to and parallel to the main radiating element. This placement creates complex interactions between the linear load wires and the antenna wire. often to the point that the contributions of each wire to the total far field and to the total element length cannot easily be distinguished.
<p>Nevertheless, NEC is able to physically model linear loads with a good level of accuracy, where "good" might be provisionally defined as providing reasonable guidance for building or providing reasonable calculations for analysis. Good accuracy, however, is subject to certain restrictions. For example, for best accuracy, the wires comprising the antenna element and the wires comprising the linear load must be the same diameter to avoid both closely spaced wires of differing diameters and wires of different diameters meeting at an angular junction. In both cases, NEC results can be quite inaccurate. One consequence of this restriction is that NEC cannot directly model most of the existing commercial antennas employing linear loading, since these antennas generally use element and load wires of radically different diameters.</p>
<p>MININEC is subject to neither of these restrictions. With due attention to the need for very short segment lengths at angular junctions and to the overall segment limitation of many versions of MININEC, the program is capable of directly modeling virtually any linear loaded element. MININEC was the program of choice in my 10-meter study of linear loaded dipoles and Yagis in "Modeling and Understanding Small Beams: <a href="../mu/mu4a.html">Part 4: Linear-Loaded Yagis</a>." <i>Communications Quarterly</i>, (Summer, 1996), pages 85-106. These antennas used 0.75" aluminum elements and #12 copper wire linear loads, and MININEC provided excellent building guidance. Despite the different materials used, no construction correctives could be directly attributed to the program limitation of specifying a single material for the entire modeled structure. For conservative predicted results, the higher loss material (aluminum) was used throughout the modeling.</p>
<p>Nonetheless, NEC (either -2 or -4) can model linear loaded elements where all wires are of the same diameter. This much capability is sufficient to reveal a number of linear load properties that are fundamental to this loading technique. Consequently, we may for this exercise employ some of the models already developed for other parts of this series and simply add to them various linear load configurations. Central to the work in this episode will be the 34.39' (412.68") short dipole that we have loaded in several ways. We shall principally use the #8 wire version. Throughout, we shall retain the use of 91 segments for the overall antenna main element, with an average segment length of about 4.53". This segment length will determine several other parameters in the course of physically modeling transmission lines or linear loads.</p>
<p>In previous exercises, we have used perfect or zero-loss wire for the dipole element. So long as we could specify load losses in Type 0 or Type 4 loads (series R-L-C and series R-X loads, respectively), we could isolate losses directly attributable to the load. In this exercise, we shall physically model linear loads as wires in the antenna structure. In order to leave the "main" element perfect and register the material losses of the load wires, we shall have to use either NEC-Win Pro or GNEC (NEC-2 and NEC- 4 programs from Nittany Scientific), as these are the only commercial versions of NEC commonly available that permit specification of different materials for each wire in the antenna.</p>
<p>Also see the <a href="../model/nec.html">Antenna Modeling Programs</a> page for more information.</p>
<h3>Physically Modeling Transmission Lines</h3>Modeled transmission lines must be parallel wire assemblies, since there is no practical way to model coaxial cables. (It is possible to simulate the cable braid with several wires of the same diameter as the center wires, but the cost in wire segment total makes the technique impractical here.) However, since a parallel wire line with significant spacing between wires represents the most common form of linear loading, this restriction will impose no problems.
<p>The linear loads we shall use will have a spacing of 4.53" center-to-center between wires. The diameter of a #8 AWG wire is about 0.1285". We can calculate the anticipated characteristic impedance of the resulting parallel line from this standard equation:</p>
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<p>where Zo is the characteristic impedance in Ohms, S is the center-to-center spacing of the wires, and d is the wire diameter, with both S and d in the same units. For the #8 line, the Zo is 510.1 Ohms.</p>
<p>The standard equation presumes an indefinitely long line without connection wires to either a source or load. Moreover, it is invariant for any wire type and thus does not account for variations in the materials one might use for an open line. (In addition, it does not account for spacers, insulation, and other line variables that are not a part of this exercise.) In fact, wires physically arranged as parallel transmission lines do not model with precisely the same characteristic impedance as those yielded by the standard equation.</p>
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<p>We can model transmission lines to determine their characteristic impedance using the figure above as guidance. The "top" and "bottom" wires are single segments, each 4.53" long. The parallel wires will be an odd multiple of a quarter wavelength for the frequency tested. In our case, the frequency is 7.15 MHz, and we shall check the line using lengths of 1/4, 3/4, and 5/4 wl.</p>
<p>By adjusting the line to a length that yields resonance at the source for a given load, we can find the characteristic impedance of the line from the standard equation,</p>
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<p>where Zo is the characteristic impedance of the line, ZL is the load impedance, and ZS is the source impedance. If we use a purely resistive load and bring the line length to resonance (defined as usual as less than +/- 1 Ohm reactance), the calculation is simplified. Let us initially specify a load of 5,000 Ohms resistance with a zero-loss (perfect) #8 wire line and see what we get.</p>
<pre>Test Length Resonant ZS Zo Segs/Length Delta L"
in WL Length " R +/- jX Ohms
1/4 409.9 59.2 - j 0.1 544.1 91 / 4.50" 2.75"
3/4 1236.6 59.3 - j 0.0 544.6 271 / 4.56" 1.45"
5/4 2063.0 59.5 + j 0.0 545.3 451 / 4.57" 0.40"</pre>
<p>The "Delta L" figure is the departure of the resonant length from a perfect odd multiple of a quarter wavelength. The model has two wires at top and bottom which contribute to the overall line length and become error sources with respect to having a perfect parallel transmission line. However, the error they introduce decreases as the test line length increases.</p>
<p>The selection of 5000 Ohms as the resistive load for this construction was arbitrary. For any given test length for the line, selection of a different load value will give slightly different results. Let's look at a 3/4 wl (1236.6") line and examine the characteristic impedance yielded by various load values.</p>
<pre>ZL ZS Zo
Ohms R +/- jX Ohms
50 5920.0 - j 40 544.1
100 2961.0 - j 10 544.2
250 1185.0 - j 2.0 544.3
1,000 296.3 - j 0.0 544.3
2,500 118.6 - j 0.0 544.5
5,000 59.3 - j 0.0 544.6
10,000 29.7 - j 0.1 544.6</pre>
<p>With perfect wire, the range of variation in this second test of the characteristic impedance derived from the test model is not especially significant, with a maximum variance of about 0.5 Ohm for the range tested. The key variance in this exercise is between the modeled-derived Zo and the value calculated by the standard equation: about 33 Ohms--or a 6% variance, with the modeled impedance higher than the calculated value.</p>
<p>If we specify the #8 wire to be copper (Resistivity = 1.72E-08), we end up with slightly different tables relative to those for perfect wire:</p>
<pre>Test Length Resonant ZS Zo Segs/Length Delta L"
in WL Length " R +/- jX Ohms
1/4 409.7 60.1 + j 0.1 548.1 91 / 4.50" 2.98"
3/4 1235.6 61.7 - j 0.0 555.4 271 / 4.56" 2.44"
5/4 2061.0 63.4 - j 0.1 563.1 451 / 4.57" 2.40"</pre>
<p>The range of variation in derived Zo for the different test lengths, using the arbitrary 5000-Ohm load, is almost 15 Ohms. This represents a range of change of around 3%, which is well within the range of variation in real lines, especially those with insulation and a consequential velocity factor. However, the question remains as to whether the test yields accurate results. The second test provides the clue we need.</p>
<pre>ZL ZS Zo
Ohms R +/- jX Ohms
50 5669.0 - j 80 532.4
100 2901.0 - j 22 538.6
250 1179.0 - j 5.3 542.9
1,000 298.5 - j 0.5 546.3
2,500 121.0 - j 0.1 550.0
5,000 61.7 - j 0.0 555.4
10,000 32.0 - j 0.0 565.9</pre>
<p>Note: when using low values of load impedance, it may not be possible to remove all of the reactance to the arbitrary standard of resonance that we have been using (+/- 1 Ohm reactance). However, if the reactance value is less than about 2% of the resistive value and slight line length changes to achieve resonance do not change the resistive value significantly, then the resulting resistive value will be usable.</p>
<p>The test using different load values shows a wide range of values both above and below the values yielded by the perfect wire test. Since the approximate value of the characteristic impedance is also approximately the geometric mean between the adjacent load values (250 and 1000 Ohms), we can zero in on the actual line impedance by using that mean as a new load. If the first try does not yield a source impedance equal to the load we chose, we can adjust the load slightly until the results are equal within about 0.1 Ohms (to allow for rounding conventions in the software).</p>
<p>Using this method, the final value for the perfect wire Zo is a little over 544.3 Ohms, while the final value for the copper wire is a little over 544.8 Ohms: very close indeed. In fact, we have extended the number of significant digits with respect to the characteristic impedance of the line too far for practical purposes in order to ensure clarity of the mathematical progressions. A value of 544 or 545 Ohms would suffice for all practical enterprises. This conclusion means that running the characteristic impedance modeling test with perfect wire with any reasonable load would have produced an acceptable result with a single test run.</p>
<h3>Modeling Transmission-Line Loads</h3>We have lingered over the transmission line modeling test not only to show that transmission line characteristic impedance may differ in modeling software from values derived from standard equations, but as well to ensure that when modeling transmission line loads, we do not draw the wrong conclusions from the results we encounter. In order to get a handle on drawing the right conclusions, the next step is to model some transmission line loads using our #8 parallel transmission line. Because the principles do not change whether or not we account for wire losses and because the models are for demonstration purposes and do not represent structures anyone would actually build, we may use perfect wire transmission lines for these tests.
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<p>The models we shall construct use the basic 34.39' #8 perfect wire dipole. We shall insert transmission line loads at the segments adjacent to the feedpoint or source segment in the center and later at the points in the lines calling for 833.3-Ohm mid-element loads. From previous episodes of this exercise, we determined that loads placed immediately adjacent to the source for the 91-segment model required a reactance of 426.3 Ohms. We also used the 5-wire model to arrive at the positions of loads requiring 833.3 Ohms (the value of a single center load for this antenna).</p>
<p>For our model, we may place the transmission lines parallel to each other or at 180 degrees apart. I have chosen the latter type of model to minimize interactions between the transmission lines, especially in the close-spaced model. Whichever model we choose, there will be a significant radiation component at right angles to the radiation from the main element.</p>
<p>Let us take the Zo of the transmission line as 544 Ohms. Using the equations shown in earlier episodes, the 426.3-Ohm load requires a transmission line length of 174.6" while the 833.3-Ohm load is 260.7" long. Modeling these lines via the TL facility, in which the lines are mathematically but not physically modeled, would require these calculated values.</p>
<p>Physically modeling the lines tells a quite different story, one that reveals why using the TL facility of NEC yields inaccurate results. To achieve resonance with the closely-spaced transmission line loads required a length of 155.3" for each line, almost 20" shorter than called for by the equation. The mid-element model required loads that were 157" long, more than 100" shorter than called for by the standard equations. (For alternate models using lines that parallel each other, the values were 151.2" and 156.5" respectively.) Each loading transmission line replaced one segment of the main element and the spacing was set at 4.53" between transmission line wires.</p>
<p>A transmission line acts as a pure transmission line if and only if the currents everywhere along the line are equal in magnitude and opposite in phase on the two wires. An examination of the current tables produced by the NEC core reveals the degree to which this condition is not met by physical lines used as loads. In constructing models to perform this test, it is essential that the segments on each wire align perfectly.</p>
<p>The current phase differential along the transmission line wires does not exceed 0.4 degrees in either case, so current magnitude comparisons are sufficiently accurate for the level of analysis required here.</p>
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<p>The graph shows the current magnitude differential between the transmission line wires for both cases tested. Current differential is greatest where the transmission line joins the main element. It does not reach zero at the far end of the line. The level of current differentials is far less along the closely-spaced or "center" load lines than along the mid-element load lines. Note, however, that there is nothing like a linear relationship between the degree of current magnitude differential and the degree of shortening of the lines.</p>
<p>The current differential is an indicator of--but not a measure of--the fact that the lines have significant radiation. As radiating lines, they contribute linearly to the length of the antenna. To the degree that the wires in the load contribute to the linear length of the antenna wire, less reactive loading is required, and the lines required will be shorter than called for in standard equations. The higher the current magnitude differential, the greater the contribution to the antennas linear length and the higher the level of load shortening relative to equation-derived values. By mid-element, the contribution to the wires to the antennas linear length is so great that the load lines may be over a third shorter than their calculated value.</p>
<p>In designing this overall exercise, I specifically chose a very short dipole (about 1/2 normal length) in order to set the effects we have noted in bold relief. Antennas only mildly shortened will show lesser current differentials in each of the models, and the degree of load shortening will be correspondingly less. Nonetheless, I know of no way of eliminating the fact that transmission line loads act as both reactive loads and as part of the linear antenna length--short of eliminating transmission lines as loads.</p>
<p>The placement of the loads in this drill is far from optimal for maximum performance for the antenna. The gain of the close-spaced or center load model is 1.67 dBi, while the gain of the mid-element model is 1.88 dBi. The latter value is comparable to the values for zero-loss center and mid- element R-L-C loads (1.85 and 1.87 dBi, respectively), but there is considerable cross-polarized radiation that reduces the normal side nulls of a free space dipole model.</p>
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<p>Before we leave these informative but impractical models, we should briefly examine the SWR curves (relative to the source impedance at resonance for each model). The close-spaced or center loaded model shows a curve consistent with those we examined earlier in connection with transmission line loads generated with the TL facility. The mid-element line, in contrast, shows a very shallow curve, indicating a much wider operating bandwidth. This curve is in stark contrast to curves associated with TL- generated loads.</p>
<p>The source of the broader SWR curve for the mid-element transmission line load lies mostly in the fact that the load lines act to a much higher degree as part of the linear or radiating length of the antenna. Consequently, the SWR curve approaches that of a standard full-size dipole. The major difference is that the shortened dipole has a resonant impedance of about 20 Ohms. (The resonant impedance of the close-spaced model is under 10 Ohms. The impedance of a perfect center-loaded dipole in this series of models is about 13 Ohms, while a perfect mid-element loaded version shows a resonant impedance of about 27 Ohms.)</p>
<h3>Linear Loads</h3>The transmission line loads that we have just explored place the load lines at right angles to the main element in order to minimize interactions. Conventional linear loads tend to place the loading lines parallel to and close by the main element.
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<p>The sketch shows three typical linear load line assemblies. Actual antennas tend to fall into one of the three categories, although details may vary. The triangular load at the top feeds the center junction point of the linear load shorted lines, which then attach to the main element. Although we cannot accurately model actual structures in this exercise, the main element of most antennas of this type will be much larger in diameter than the load lines. For many purposes, the load lines may be placed in a plane with the main element, as shown in the middle sketch. Once more, the main or upper element will be larger in diameter than the load lines.</p>
<p>A third popular assembly is shown at the bottom as a mid-element linear load. In this case, a "main" or larger diameter element is the feedpoint. The apparent load lines begin at mid-element and return toward the center, terminating and returning to the completion of the main element. Ordinarily, the apparent load lines are placed symmetrically about the main element, forming a triangle or a single plane. Why the sketch refers to this schema as a "pseudo-mid-element linear load" will become apparent as we proceed.</p>
<p>The models for the tests were carefully constructed to achieve several goals. The segments throughout are all close to 4.53" long, and the spacing between lines is exactly that figure. Hence, the triangular models form equilateral triangles so that the influence of the main element is as equal upon both load lines as possible. The center separation between load ends (shorting segments) is also 4.53" or one segment's length. This procedure permitted alignment of all segments both between the load wires and on the main segment as well. All of the models are #8 AWG wire.</p>
<p>In this sequence of test, however, we shall look both at load lines that use perfect wire and lines that use other materials. The figure below shows an in-plane model in which only the load lines (wires 2 through 8) are copper. In this way, we may isolate the losses due to the load from the losses due to the main element wire, which remains a perfect or zero- loss wire like the ones used in earlier R-X and R-L-C load models.</p>
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<p>The table below presents the load line lengths, measured from the source point for each of the three types of loads modeled. The "triangular mid- element" load line is measured from the source point both to the start and to the finish of the line outward. For the center triangular and in-plane models, line lengths for perfect, copper, and aluminum load lines are recorded, while the mid-element line is recorded only for the perfect wire version.</p>
<pre>Antenna Load Material Load-Line Length (")
Triangular Center Perfect 175.2"
Copper 174.5"
Aluminum 174.35"
In-Plane Center Perfect 188.0"
Copper 187.3"
Aluminum 187.15"
Mid-Element Perfect 176.43" (outer)
171.90" (inner)</pre>
<p>The decrement in length for the loads with the use of materials having a reduced conductivity is identical for both the triangular and the in-plane models. The load length for the in-plane model is longer than for the triangular or the mid-element model. However, the mid-element model load structure is virtually identical in length to the equally triangular center loaded model. Note that the triangular model load lengths correspond very closely to lengths for a split center load reactance when calculated from standard equations using the 544-Ohm Zo of the load line. For the required 416.7-Ohm load, the required load line calculates to 171.7" or within 2% of the actual modeled line lengths.</p>
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<p>Although the triangular models are physically set to minimize unequal influence from the main element, the current differential on the load wires is not zero. The graph shows the two values of current differential to be almost precisely the same for the zero-loss wire models, with closely corresponding phase differences as well. With the exception of a 4.53" change of wire alignment, the two models are the same model in every operational respect. In the mid-element model, the so-called fed main element is actually one leg of the loading line, and the apparent second leg of the loading line corresponds to the main element in the triangular center-loaded model.</p>
<p>In contrast, the current differential between the load wires of the in- plane model is much higher. The higher differential is due to the differential interaction between the main element and the individual wires which are at different distances from the main element.</p>
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<p>A further insight into the difference between the triangular and the in- plane models can be gather from examining the currents on the main element outward from center. Note once more the close coincidence between the triangular models. In contrast, the current magnitudes along the outbound wire for the in-plane model are lower everywhere. Part of this reduction is due to the increased length of the in-plane linear load, which results in an electrical shift in the position of the beginning of the main element at the antenna's center.</p>
<p>As might be expected, the current distribution on the in-plane model results in a lower gain than for either triangular model. With perfect wire, both triangular models report an anticipated gain of 1.86 dBi, while the in-plane model gain is 1.79 dBi--not a big reduction, but noticeable. With copper loads, the triangular model gain is 1.35 dBi, while the in- plane gain is 1.30. An aluminum (6061-T6) loading wire set yields a gain of 1.11 dBi for the triangular models and 1.06 dBi for the in-plana model. The values for aluminum wire load lines (with perfect wire main elements) correspond closely to the values for center and mid-element load dipoles using R-L-C loads with a Q of 300 (1.02 and 1.11 dBi, respectively).</p>
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<p>Equally revealing about the behavior of linear loads are SWR curves for the three models, each predicated on the source impedance at resonance at the design center frequency of 7.15 MHz. The resonant source impedances are 12.0, 12.1, and 12.2 Ohms for the triangular center, in-plane center, and triangular mid-element models. The coincidence of the mid-element linear load model source impedance with the center loaded models and not with the mid-element transmission line load model examined earlier is further confirmation that the mid-element model is simply a minor variation of the triangular center loaded model. The SWR curves confirm this further by overlaying each other, while the in-plane curve is slightly broader, as one might expect of a very slightly lossier structure. The mid-element linear load model no where approaches the broadness of the mid-element transmission line load we explored earlier.</p>
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<p>The SWR curves for the triangular model for the three load line materials show the expected broadening of the curve with the use of materials with a lower conductivity. Similar results accrue to the in-plane model.</p>
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<h3>Conclusion</h3>The linear load is essentially a shorted transmission line stub placed in line with the main element of the antenna it loads. As an antenna load, the stub cannot act as a pure transmission line; the current can never be equal in magnitude and exactly opposite in phase on the two wires making up the load line, since the current magnitude and phase on the antenna element into which it is inserted is not the same at any two points along a path from the antenna center outward. However, the condition is most closely approximated when loads are as exactly at center as possible and symmetrically placed about the main element to equalize its influence on each load wire.
<p>Since linear loads are transmission lines and answer in part to the equations for calculating length from required loading reactance-- especially when placed at the center of the antenna structure--their operating bandwidth curves, as reflected in the SWR curves shown, are narrower than for corresponding R-L-C loads.</p>
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<p>The graph above plots curves for center and mid-element loads using perfect loads and loads with a Q of 300. Additionally, the graph records the SWR curves for the triangular linear load model using perfect and copper wire. note that the curve for the copper wire linear load is sharper than any of the R-L-C loads except the center load with infinite Q. The broadest lines in the curve are for R-L-C curves with a Q of 300.</p>
<p>Because the operating bandwidth of a linear load rests in large part on transmission line stub behavior and not on solenoid inductor behavior, losses cannot be directly correlated with operating bandwidth. Within each type of load, increases in bandwidth might point to increasing losses in the loads. However, cross-load type correlations are far less certain.</p>
<p>To the degree that the currents on the linear load transmission line stubs are unequal, the wires also contribute to the linear length of the antenna. If placed at right angles to the main element wire, the radiation appears as a cross polarized field relative to the radiation from the main element. Moreover, the larger the current magnitude differential between load line wires, the more the wires act as linear contributions to the overall antenna length and the shorter the required stub.</p>
<p>As with any complex electrical structure inserted into an antenna and modeled physically, the modeling calculations cannot directly account for all phenomena. For example, some theory exists to suggest that transmission line wires have greater losses for a given wire size and frequency of RF current than that same wires in an electrical circuit, due to the effects of the intense field between the wires. However, these effects turn out to be small, even for the long linear loads used in the very short modeled dipole. MININEC tests and other marginally valid tests within NEC suggests additional total field antenna gain losses between 0.1 and 0.2 dB. These losses, if present, have no noticeable effect on the operating bandwidth of the linear-loaded antenna models.</p>
<p>While the bulk of the transmission-line view of linear loads has been previously established, presenting it here is useful if--for no other reason--than because it raises an interesting question. Just as linear loads have unequal current magnitudes at their junction with the main antenna element as a function of the current distribution along an antenna element, it would appear that solenoid inductors used as loads would share the same behavior. If this turns out to be the case, then the mathematical specification of loads within NEC would be equally incomplete, despite the high trust antenna designers and analyzers place in R-L-C loads of Type 0 and Type 1.</p>
<p>It may be worth one more episode just to test the hypothesis and confirm (or disconfirm) the trust we place in Type 0 and Type 1 R-L-C loads.<br>
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<p><i>Updated 1-2-99. &COPY; 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.</i><br>
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<title>Reactive Antenna Loads and Their NEC Models Part 4: Some Solenoid Loading Basics</title>
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<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models<br>
Part 4: Some Solenoid Loading Basics</h1>
<h3>L. B. Cebik, W4RNL</h3>
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<p>The final question in our inquiry is whether Type 0, 1, or 4 mathematical loads (each within their application limitations) fully or adequately specify inductively reactive loads used in a shortened antenna. Answering this question via antenna construction would seem natural, were it not for our customary building practices and presumptions. When we install a loading coil and it proves too large relative to the wire length, we simply shorten the wire to match the coil. Then we fail to interrogate the situation further to discover why the coil was too large. Most of the time, we assume that we miscalculated or that the usual coil equations are not sufficiently accurate to yield a coil that requires no subsequent wire pruning.</p>
<p>As a consequence, few modelers have explored the behavior of helically wound single layer solenoid inductors in their models--simply because winding a coil is too laborious a task relative to using the loading facilities built into NEC. We have presumed that the loads--especially Type 0 series R-L-C loads--adequate specify the loads we need.</p>
<p>Yet nagging suspicions remain about such loads. The current differential that is present at each end of the load suggests that--like their linear load counterparts--these loads do not act like pure inductances, but may also function as simple current bearing wires. Models of helical dipole elements show very usable gain, which parallels the experience of those who have constructed such antennas. I guess we shall not know whether mathematical models of loads accurately reflect the operation of single layer solenoid inductances unless we actually wind some model coils and replace the Type 0 loads with them.</p>
<h3>Winding a Helix and Making a Model</h3>The basic model for the half-size (1/4 wl) 7.15 MHz dipole uses #8 (0.125" diameter) zero-loss wire that is 412.68" (34.39') long and divided into 91 segments. Each segment is about 4.53" long. For least error, the segments within the helix will be as close to this value as helix construction permits.
<p>The required center loading reactance for the antenna is 833.3 Ohms or 18.549 microH. We may let the coil be 3 segments long or 13.6" overall. Using the standard Wheeler equations (either the common 1928 version or the 1982 version), a circular single layer solenoid inductor of this length will require 10 turns and an 11.852" diameter.</p>
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<p>The most reasonable approximation of a circular solenoid is an octagonal structure. <b>Fig. 1</b> shows the resultant structure and the basic relationships that determine the coordinates of the points around each loop. For this exercise, let's adopt the convention of extending the turns along the X-axis and using the Y-axis and the Z-axis for the radial points of each loop. Under this convention, the point references in the lower part of the drawing refer to Y and Z coordinates. Beginning, perhaps, with half the diameter as the value of R (5.926"), we might place the first entry at 0, 5.296, 0 (assuming a free space modeling environment). Intermediate points between the horizontal and the vertical will require values of 0.707 R (3.744"). (Note: rounding to one less decimal place will make no significant difference in the long run.)</p>
<p>Each point progresses along the X-axis by the total length (13.6") divided by the number of turns (10) divided by the number of points in each turn (8), for a value of 0.17" in this case. Hence, the second point of the turn will be located at 0.17, 3.744, 3.744. We can then proceed to complete the first turn. Then, using whatever copy function may be present, we add progressively more turns, increasing the value for the X- coordinate by 1.36 with each turn.</p>
<p>The resulting solenoid will be about 2.5% short in overall wire length used compared to a truly circular solenoid, suggesting a slightly lower inductance than provided by the Wheeler calculation. Since the wire is small compared to the turns spacing, a slight additional variance may also be encountered. Of greater significance for the model, each wire in the helix is about 4.56" long, a close match to the length of the segments in the remaining portions of the dipole wires.</p>
<p>Once the coil is complete, it should be saved as a file, since we may use it later. For the initial test of center-loaded short dipole, we may replace the center 3 segments of the basic wire with the helix. The best coordinates for the coil may be -6.8 to + 6.8 on the X-axis to obtain a useful symmetry in the now enlarged geometry sheet--useful for visual error trapping. The formerly single wire is now two wires extending to + and to - 206.343" respectively, and each with 44 segments.</p>
<p>Part of the extended exercise in looking at NEC loads, both mathematical and physical, has involved sorting out the losses attributable to the load from those which may be associated with the linear wire portion of the antenna. For this purpose, we shall need to make the linear wire lossless and the coil wire of a suitable material, namely, copper. At present, only GNEC (Nittany-Scientific) permits a wire-by-wire material selection. <b>Fig. 2</b> shows a partial .NEC file from GNEC, with the dimensions in meters.</p>
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<p>The portion of the model shown shows the last few turns of the helix, wire by wire. The last wire (82) is the linear dipole wire in the +X direction. Skipping down to the Type 5 load lines (material loading), we see that wire 1 (and also 82) is skipped, being lossless, while the conductivity value for copper appears for each wire in the helix.</p>
<p>Finally, because the turns of the helix have an equal number of wires per turn, a split feed is used at the wires forming the very center of the inductor.</p>
<h3>The Center-Loaded Half-Length Dipole</h3>The model just constructed in a free space environment yielded an inductive reactance of over 86 Ohms, indicating not that our octagonal coil was slightly small, but rather that it was considerably too large. Achieving resonance required a significantly smaller helix. In order to preserve the length of the coil that was equal to 3 segments of the linear wire, the coil was reduced in diameter. With a diameter of 11.12" the loaded antenna achieved resonance (as defined in this study as less than +/- 1 Ohm source reactance). Wheeler calculations for a circular coil of this diameter, length, and number of turns yielded an inductance value of about 16.6 microH, roughly 10% less than the value called for by the mathematical load systems provided in NEC. The length of the coil wires reduces to about 4.25" each, which is unlikely to induce any significant error in the model.
<p>The following table provides some basis for making comparisons between the physically modeled load and the Type 0 loads used in earlier parts of this exercise. In all cases, the linear portion of the antenna is composed of lossless wire.</p>
<pre>Load Free Space Source Impedance
Gain dBi R +/- jX Ohms
Lossless Type 0 1.85 13.19 - j 0.02
Q=300 Type 0 1.02 15.97 - j 0.01
Copper Helix 1.35 10.61 + j 0.64</pre>
<p>The higher gain provided by the physically modeled center load should not be surprising, considering that the helix contains wire that radiates. See <b>Fig. 3</b>.</p>
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<p>The small but definite vertically polarized radiation field results from radiation from the portions of the helix that are vertically oriented. This field along would not suffice to provide the added gain over the Q=300 center loaded model, but the entire 13.6" length of the inductor would. More precisely, that portion of the overall inductor wire beyond the 13.6" it replaces in the overall antenna length would suffice to provide additional gain.</p>
<p>The actual Q of the inductor cannot be ascertained with accuracy. Some calculations estimate the Q at over 400, but only under the conditions that the wires be closer to half the spacing between turns. The modeled helix uses much smaller diameter wire, resulting in a decrease in Q, perhaps to the 200-300 range at best.</p>
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<p>Despite the lower Q estimate, the source impedance of the physically modeled helical loaded dipole is lower than even the Type 0 model at zero load loss. As <b>Fig. 4</b> shows, the SWR curve is slightly steeper than even the R-L-C lossless load model--and a lossless helix yields a steeper curve yet.</p>
<p>The actions of the physically modeled helix load are in part a function of the current magnitude and phase profile along the coil. <b>Fig. 5</b> shows the two curves from the center point outward toward the end.</p>
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<p>The current magnitude differential between the source and the coil end is approximately 20% of the maximum value. Since the inductance of a helix is in theory predicated upon equal current magnitudes at both ends of the coil, the differential currents dictate that in part the indictor does not act solely as an inductor. The physical model of the center load provides indications that the helix also radiates in the manner of any length of wire included within the total length of the dipole--in terms of gain over the Type 0 load, the vertical radiation field, and the reduction in the required inductance to effect loading to resonance.</p>
<p>There are potential error sources within the model. First, the octagonal inductor with less then optimal wire diameter relative to runs spacing casts some doubt on the precision of the calculations relating inductance to turns, diameter, and length. However, most of these factors suggest that the octagonal coil's inductance would be low, when in fact, the model showed it to be very much higher required for resonance.</p>
<p>Second, the helix design, although apt to the modeling task, is very large and open compared to coils one might ordinarily apply as center loads in a shortened dipole. However, as an error source, the helix configuration would be a matter of degree and not of fundamental principle of operation. What the dimensionless load of a Type 0 load does not show is the relatively large early drop of current magnitude along the total length of the dipole. Fig. 5 shows the more dramatic drop occasioned by the physically modeled coil. Although different coil configuration may alter the amount of drop somewhat, the current magnitude curve is unlikely to resemble that for the Type 0 load.</p>
<p>The plausibility of the physical model might be put to still another test. The original short dipole model was designed to permit one to place load values equal to the center load value at specific mid-element points. If the center load helix as modified to achieve resonance acts like a Type 0 load, then placing the same coils at the specified points should also result in resonance.</p>
<h3>The Mid-Element-Loaded Half-Length Dipole</h3>The original model was modified into a 5-wire model having the same operational characteristics as the original. The inserted 3-segment wires were placed at points between 99.5" and 113.1" from the center so that the Type 0 load was positioned 106.3" from the center. At these positions either side of center, a load of 833.3 Ohms (18.549 microH) resonated the antenna. The value of these two loads was the same as the value of the single center loading reactance.
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<p>As shown in <b>Fig. 7</b>, the helix developed for the center load may directly replace the inserted 3-segment wires in the 5-wire model. By copying the original helix and changing the values for all of the X coordinates by the same amount, each coil may be placed in the correct position. The connection points to the coils are very slightly displaced relative to the axis line, but not by an amount to show up in a movement of the antennas resonant frequency. The 11.12" diameter (R = 5.56") helix was copied and transferred to these points and the model run.</p>
<p>The result was a source impedance of about 29 + j 229 Ohms, indicating that the inductor size (16.6 microH) was much too large for resonance. This result was consistent with but considerably larger than expected. To create helices that allowed the overall antenna to achieve resonance required a reduction in the coil diameter to 10.2" (R = 5.10"). By Wheeler calculations, the inductance was reduced to about 14.3 microH for a truly circular coil. The wire length from coil point to coil point diminished to 3.90" compared to linear wire segment lengths of 4.53" each.</p>
<p>The amount of reduction in coil size was about 14% relative to the final center loading coil and almost 23% relative to the inductance required by a Type 0 load. This reduction is consistent with several factors involved in inductive loading, including the fact that the further out along an element one places the inductor, the less it acts as an inductor and the more it acts as a compact form of wire length to increase the overall wire length to resonance. By the outer ends of a shortened dipole, helices and others forms of element extension cannot be viewed as inductances almost at all.</p>
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<p>The current magnitude and phase profile within the helix may be of some surprise to some. Along the mid-element loading helix, after size adjustment to resonance, the current does not show a steady progression lower in magnitude. Instead, there is a current peak about a fourth of the way outward from the inboard end of the coil. From that point onward, the current magnitude descends rapidly. The phase transition, although not linear, generally follows the progression along the entire antenna element.</p>
<p>Compared to the center loading coil, the current magnitude excursion within the inductor is about 34% of the maximum value, and the end to end difference is about 30% of maximum current magnitude. This compares to the 20% differential from center to end for the center loading coil. The greater differential of current magnitude along the mid-element inductor wire is another indicator that it is functioning to a greater extent than the center loading coil as part of the element length and hence requires less inductance and inductive reactance to effect loading to resonance.</p>
<p>Like the physically modeled center load, the physically modeled mid-element load shows a much larger drop in current at its outer end than does a corresponding Type 0 load. The dramatic decrease in current immediately outside the loading helix appears vividly in Fig. 9, with the Type 0 load curve shown for comparison.</p>
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<p>Despite the close correspondence of current up to the load position and the dramatically lower current beyond that point for the physically modeled helix, the model shows a significantly higher gain than either the corresponding Type 0 load model or even the physically modeled center load. The following table summarizes the gain situation for various models.</p>
<pre>Antenna Free Space Gain Relative to a
Gain dBi Full Length Dipole
Full-Length 2.14 ---
Half-Length Lossless Wire 1.85 -0.29 dB
Type 0 Center Load, Q=300 1.02 -1.12 dB
Helical Center Load, Copper 1.35 -0.79 dB
Type 0 Mid-Element loads, Q=300 1.11 -1.03 dB
Helical mid-Element Loads, copper 1.71 -0.43 dB</pre>
<p>The gain figures strongly suggest that the current within the turns of the helix is not going wholly to waste within a tightly coupled inductive field. That part of the current not devoted to the field, as indicated by the current differential at the coil ends and among the turns is contributing to the total field of the antenna.</p>
<p>The use of Type 0 loads to characterize both center and mid-element loading strongly suggests that there is little difference between the gain potential of two antennas of equal length loaded in each manner. however, physically modeled loading helixes tell a quite different story. Not only do both types of loaded antennas potentially perform better than mathematically inserted loads would indicate, but as well the empirical experience giving the nod in gain to mid-element loading gains some support from the physically modeled loading coils.</p>
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<p>Although the physically modeled center loaded antenna showed a steeper SWR curve than its Type 0 counterpart, the mid-element loaded antennas show a good correspondence between mathematical and physical loads. In fact, as shown in Fig. 10, the mid-element physical helices correspond nicely to mathematical loads with a Q of 300. This curve corresponds reasonable well with estimates for the Q of the coil at or just below 300, based upon reductions from the calculated Q of over 400 that presumed an ideal wire diameter to spacing ratio. The resonant source impedance of about 24.6 Ohms is a bit lower than the Type 0 model at Q=300, where the source impedance was about 31.9 Ohms.</p>
<p>It should be noted for accuracy and completeness that the actual resonant frequency for the model using physically modeled mid-element coils with a 10.2" diameter was 7.16 MHz, and that the curves for this antenna actually run from 7.01 to 7.31 MHz. This 10 kHz displacement did not seem great enough to warrant reconstruction of the coils through the iterations necessary to move the resonant frequency. The estimated diameter would have been something of the order of 10.205" +/- 0.002" for resonance at 7.15".</p>
<h3>Potential Errors and Conclusions</h3>The potential error sources for the mid-element load model remain the same as for the center loading coil. The octagonal helices may in fact show a lower inductance than the values calculated by the Wheeler equations. Moreover, the widely spaced open helix may prove a better radiator than some more tightly wound loading coils. Unique to the modified mid-element loading coils is the shortening of the wires and a somewhat greater differential to the lengths of the segments in the linear wire portions of the antenna.
<p>Some of these possible error sources work in the wrong direction relative to the results, for example, the reduced inductance of octagonal helices relative to truly circular ones. The sum of the remaining error sources appears to be far short of what would be required to account for the radical reduction in required inductance for the loading coils. Indeed, the most telling part of the demonstration is the further reduction in coil size required in moving the center-loading inductor out to the position where the same value should have served as a mid-element loading inductor. Although absolute values of the inductors involved may be off the mark by some small amount, the trends are in al probability accurate reflections of anticipated antenna performance.</p>
<p>As a consequence of these data and considerations, the bottom line seems inevitable: Type 0 (and by extension, Type 1 and Type 4) loads do not fully characterize the performance of inductively reactive loads in the form of single layer solenoids. For many purposes, that characterization may be adequate to a design task. For example, where design criteria permit adjustment of the antenna outer section length, designing to the Type 0 load value may cause not harm.</p>
<p>However, using mathematical loads alone can give a potentially misleading impression of the gain of the loaded antenna. The gain for physically modeled loads is considerably higher than the gain for mathematical loads. Where multi-element arrays--such as Yagis--may employ center or mid-element loads in all elements, Type 0 load models may seriously underestimate the overall array gain.</p>
<p>This exercise, along with the previous portions of this study, should be viewed as only a beginning to the investigations into modeling loaded shortened antennas. Further work might yield additional insights into the behavior of linear loads examined in the last episode. Likewise, for helical inductive loads, much remains to be done with respect to the use of more compact forms of inductance. Even using these models, additional data can be gathered for the use of linear wires with non-zero losses and for the use of other helix materials, such as aluminum.</p>
<p>Nonetheless, if the present demonstration holds up as reasonably valid, then the suspicion raised at the outset has been established. In several important respects, the mathematical loading features provided within NEC do not fully characterize the action of these loads when physically modeled. Nevertheless, modeling actual loading inductors that form junctions with elements of radically different diameter from the coil wire and modeling small-diameter coils without violating adjacent segment length restrictions may prove to set major challenges for the amateur and the professional modeler. This study was carefully designed to use models that stayed well within all NEC restrictions. Most real antennas that use inductive reactance to electrically lengthen a physically short element will press those restrictions or result in quite large models.</p>
<p>In other words, what we can establish in principle in a straightforward manner might prove to be an exceptional modeling challenge when real antennas and their parts are involved. Loads are not quite as simple as casual modeling might lead us to believe.</p>
<p>Also see the <a href="../model/nec.html">Antenna Modeling Programs</a> page for more information.</p>
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<p><i>Updated 2-4-99. &COPY; 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.</i></p>
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<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Notes on Reactive Antenna Loads and Their NEC Models<br>
Part 5: Some Unfinished Business</h1>
<h3>L. B. Cebik, W4RNL</h3>
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<p>In the first four sections of this series on inductively reactive loads used to electrically lengthen physically shortened antennas, we uncovered some interesting disparities between the standard mathematical models of loads and loads which are physically modeled. If the demonstration models are valid, then the following summary points are true in principle.</p>
<p>1. Because the current magnitude and phase in a transmission line used to load a short antenna do not show a true transmission line relationship in a physical antenna, the use of the TL facility in NEC is not a reliable model of the real line and should be confirmed with a physically modeled transmission line load.</p>
<p>2. Linear loads are a variety of transmission line stub load, but their behavior should be physically modeled for greatest reliability, especially when folded back from a mid-point along the so-called main element. Within the length of the linear load, which wire comprises the "main" element is ambiguous, since the radiation from that portion of the antenna is a function of the closely coupled fields from the element and the load wires in the stub.</p>
<p>3. To a lesser but still significant degree, physical models of single layer solenoids, when modeled as helices, depart from the Type 0, 1, and 4 NEC mathematical models of them when placed as loads along an antenna element. In general, such loads show composite properties as inductances and as added wire length, the latter of which properties contributes to the radiation field of the antenna. The effect increases as the inductors are moved outward from the center of a dipole, with less inductance being required for the load than for a center loading coil. Even a center loading coil requires less inductance in physical models than called for by the NEC mathematical load.</p>
<p>Although the exercises have demonstrated the effects, they fall far short of establishing a set of universal principles, mostly due to the limitations of NEC. Some of the potential error sources in NEC-4.1 have been reviewed in "NEC-4.1: Limitations of Importance to Hams," <i>QEX</i> (May/June, 1998), pages 3-16.</p>
<p>Throughout the exercises, NEC-4.1 was used in the EZNEC Pro and the Nittany-Scientific GNEC implementations. In all models, unless otherwise specified, the linear wire was modeled as lossless in order to sort out losses and performance curves due to the load from those attributable to the linear wire. Hence, all gain figures in previous exercises will be higher than with any real antenna using copper or aluminum elements. All models were constructed in free space.</p>
<p>The model short dipole for 7.15 MHz was 1/2 full size at 34.39' (412.7") long and used #8 AWG (0.1285" diameter) wire. All physically modeled load structures used the same wire in order to avoid possible errors resulting from angular junctions of wires having dissimilar diameters. The original antenna was assigned 91 segments, each about 4.53" long. The selection of segment length served a number of purposes. First, it set the spacing of linear load wires from each other at a reasonable distance for accurate modeling. Second, it permitted the construction of octagonal single layer solenoid inductors that were exactly 3 segments long and whose wire lengths for each portion of each turn were close in length to the adjacent segments of the linear wire.</p>
<p>The original model was revised for mid-element loading exercises into a 5- wire model. 3-segment wires were placed on each side of the center wire so that the center segment of the insert was positioned exactly where a mid- element load was equal in size (reactance and inductance) to a corresponding center load such that either system brought the antenna to resonance with no change in the overall length of the dipole. The overall segmentation was not changed, so that segment length among models remained close to equal. A 3-segment long (13.6") physical inductor model replaced the center 3 segments of the model for center loading and replaced the two inserts for mid-element loading tests.</p>
<p>Subject to continuing review, it is believed that these models fall well within the limitations of NEC-4.1's ability to provide accurate results. It should be unnecessary to add that the relationship of the NEC-4.1 modeling system to real antennas is not like the relationship of a child's plastic toy to a real automobile. Rather, NEC-4.1 (and other version of NEC and MININEC) are complex mathematical systems based upon both fundamental antenna theory and fundamental mathematical principles for calculating the various performance parameters of many types antennas. Those who dismiss antenna modeling in general largely reveal only a general lack appreciation of the role of mathematics in both antenna theory and antenna practice.</p>
<p>Those who develop, implement, and work with any version of NEC are also conscious of its limitations. All version of NEC have been extensively tested against experimental results for many fundamental antenna types. However, it is possible to create antenna geometries that exceed the limitations of the calculation system. In such cases, NEC yields results that do not tally with either theory or practice. Since no program developer or development team can fully predict all of the antenna geometries (or environmental circumstances) that users will model, they can only enumerate some of the program limitations. Users will discover other limitations in the course of their work, and these limitations set the challenges for further development.</p>
<p>This note is about some of the limitations that apply to the extension of the models used in this exercise set to various real antenna designs.</p>
<h3>Linear Load Limitations</h3>The difficulty of extending the models used for linear loads in these notes to real antennas can best be illustrated by taking up a few concrete examples of extant antennas.
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<p><b>Fig. 1</b> shows a sketch that outlines the basic design features of one commercially manufactured antenna element using a linear load. To this point, I have been unable to construct a model of this element that falls within NEC limitations.</p>
<p>The element uses a series of tubes of decreasing diameter as one moves from the center point outward. The NEC-2 limitation with tapered diameter elements is largely, but not completely, overcome in the NEC-4.1 algorithms. Modeling this element in NEC-2 with Leeson corrections (or, more properly, conversion of the element into its equivalent uniform diameter equivalent) is not possible, since the linear load junctions would block correction implementation (and the implementation of the correction would be inaccurate, since the element is not continuous).</p>
<p>In this particular design, the linear load wires are very thin compared to the center element, and they are very closely spaced to the center element. All versions of NEC yield erroneous results when wires of different diameter are closely spaced. Moreover, there is a right-angle junction of wires of different diameter, another case in which NEC yields erroneous results.</p>
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<p>A second design of commercial origin appears in <b>Fig. 2</b>. In this case, the previously noted limitation involving angular junctions of wires having dissimilar diameters also applies to this design, since the load wires are very much thinner than the center element tubing. The tubing also uses a diameter tapering schedule, adding a definite limitation to modeling with NEC-2 and a minor limitation for NEC-4.1.</p>
<p>The wires of the load are not parallel to each other. This feature would make the correlation of the load to transmission line stubs more complex, although for general purposes, the averaged characteristic impedance of the line might be used with fair accuracy. However, to the degree that physical models of parallel transmission lines yield results at variance form standard calculations, determining the characteristic impedance of the physically modeled line will prove more difficult.</p>
<p>A third feature of this design is the placement of the long shorting bar near the center of the element. Physical models of shorted transmission line stubs take into account the current magnitude and phase of the shorting bar. In this instance of a long bar, the bar itself becomes an integral part of the line, one having a rapidly decreasing spacing between opposing segments. The design also may place the shorting bar at a distance from the element center point than used in the exercise models. Hence, the correlation drawn in the exercise between mid-element loads folded back to the center and loads properly called center linear loads may be less certain for this case.</p>
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<p>We can illustrate the difficulties of modeling a real linear load in NEC by looking once more, in <b>Fig. 3</b>, at the model of the in-plane center linear load, discussed in part 3. In that model, using #8 wire uniformly throughout the model, several facets of real antenna construction were simplified. The spacing between the three wires was set at 4.53" center-to-center. This spacing produced identical surface-to-surface spacings that were 0.1285" less than the center-to-center spacing.</p>
<p>If we increase the "main" element to a 1" diameter, center-to-center spacings no longer correlate to surface-to-surface spacings, changing the relationship among the elements. In addition, we encounter the potential problem in NEC of closely spaced wires with different diameters.</p>
<p>Fortunately, this model just barely fits within the MININEC segmentation limit and can be evaluation on that system. MININEC does not show a limitation with respect to closely spaced wires with different diameters. However, MININEC models tend to clip corners with angular connections, and we should expect (without invoking length tapering) some required change in the length of the loading wires if we hold the overall length constant.</p>
<p>The following table shows the results of modeling the in-plane antenna in both NEC-4.1 and in MININEC. Load line lengths (from center to short) are given to indicate the require modifications to achieve a value close to resonance (in this demonstration, loosely defined as under +/- 5 Ohms). In addition, values are shown for models using lossless, copper, and aluminum wire throughout.</p>
<pre> MININEC NEC-4.1
Main Free Space Feed Z Free Space Feed Z
Element Gain dBi R+/-jX Gain dBi R+/-jX
#8 AWG (Load = 194.5") (Load = 188")
Lossless wire 1.85 11.9-j1.2 1.79 12.1-j0.0
Copper 1.30 13.5-j1.2 1.25 13.7+j1.5
Aluminum 1.04 14.3-j1.2 0.99 14.5+j2.4
1" (Load = 158") (Load = 165.3")
Lossless wire 1.85 11.9+j4.5 1.25 13.7-j0.5
Copper 1.39 13.2+j4.5 0.83 15.1+j0.9
Aluminum 1.17 13.9+j4.5 0.62 15.9+j1.6</pre>
<p>In the table, gain figures are shown to 2 decimal places in order to show numerical trends that would have been erased by excessive rounding. (For operational purposes, figures to a single decimal place would suffice for many comparative cases, and the decimals might be dropped from certain generalized discussions. However, those who believe that showing gain to multiple decimal places should be dropped in all cases simply lack an appreciation for all of the purposes for which the antenna modeling programs can be put to effective use. Each degree of precision in reportage has its proper context. My own preference would be to see modeling reports overly precisely reported and then to do my own rounding than to see such figures prefiltered.)</p>
<p>The gain and impedance figures for the #8 model show excellent correlation between the modeling systems, with a maximum gain variance of 0.05 dB and a maximum resistive variance of 0.2 Ohms. If we look at the MININEC column and read downward, then the reported performance of the 1" main element version of the modeled antenna shows very sensible results. The lossless version of the model shows minuscule differences from the #8 version (actually none in the truncated decimals used here). The copper and aluminum versions of the 1" version show increases in gain over the #8 version as the material becomes more lossy. This is a function of the increased surface area of the larger element, and a similar trend shows up in full size dipoles. Likewise, the feedpoint impedance of the 1" antenna decreases more rapidly from the #8 version as the material loss increases, again for the same reason of lower overall losses. Once more, this trend also shows itself in unloaded antennas.</p>
<p>Regardless of the absolute accuracy of the MININEC figures, the trends in numbers are not only intuitively sensible, they correlate with the behavior of models of unloaded antennas. In contrast, the NEC-4.1 figures for the 1" diameter main element model report decreases in gain and increases in feedpoint impedance for each material assignment. These numbers indicate either that the wire spacing for the element sizes involved and the frequency of test (7.15 MHz) has crossed the threshold at which NEC-4.1 no longer delivers accurate results or that the angular junction of the #8 wire with the 1" wire is yielding less than accurate results--or both. It is common in either error mode for NEC to report a gain that is too high/low while reporting an source impedance that is too low/high.</p>
<p>Allowing for corner junction errors, MININEC remains the program of choice for modeling complex geometries that involve junctions of wires having dissimilar diameters and/or closely spaced wires of differing diameters. Recent developments in MININEC to break the segment limit, such as in NEC4WIN, will go far to make more complex MININEC models possible. However, not until the other MININEC limits are overcome--namely, raising the speed of matrix execution and grafting on the Sommerfeld-Norton ground calculation system--will the program be fully competent for all purposes.</p>
<h3>Solenoid Coil Limitations</h3>
<p>In the physical models of single-layer solenoid inductances used as loads, the constraints imposed on the exercise dictated the use of helices with a relatively large diameter (11.85") and a long length (13.6"). The wire size (0.1285") was relatively thin compared to the turns spacing (1.36"), which yielded a coil Q somewhat less than the theoretical maximum for the overall configuration.</p>
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<p><b>Fig. 4</b> shows some of the key parameters of coil construction that play a role in physical determination of coil inductance. Length, diameter and turns spacing are parts of the classic 1928 Wheeler approximation for single-layer solenoids (as found in ARRL <i>Handbooks</i> since almost time immemorial):</p>
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<p>where L is the inductance in microHenries, d is the diameter, l is the length, and n is the number of turns, and where d and l are in the same units. The number of turns can be determined from the length and turns spacing (or vice versa). For coils used at upper HF and higher, the lead length becomes a significant factor in the coil's inductance. In antenna modeling, the lead length is generally absorbed by the linear element length.</p>
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<p>In <b>Fig. 5</b>, we find some variations on the coils used in the physical models. If we let A represent the coil configuration used, we might obtains the same inductance (and reactance) using a coil like that in B, which uses the same spacing, but a smaller diameter and more turns. C spreads the turns over a longer length, while D uses a shorter length, closer spacing, and a larger diameter, perhaps to effect the highest Q configuration that might be practical. E employs fatter wire to achieve a high Q by optimizing the wire-diameter-to-turn-spacing ratio. Each of the preceding variants on the original coil was based on theoretical factors. However, in practical antenna design, one might also accept a somewhat lower Q in order to achieve a design that slipped wind by virtue of its small diameter, as suggested in F.</p>
<p>Each coil configuration is likely to affect antenna performance slightly, due to different radiation field additions that depend on diameter and turn spacing. Some of the variants are more easily modeled than others. With proper attention to segment lengths, coils that use the same size wire as the main or linear element can be modeled with fair ease. However, length tapering of the linear element as it approaches the helix may be necessary for coils with small diameters. Where coils use wire diameters that differ from that of the main element, NEC (in any version) will exhibit a tendency toward erroneous results.</p>
<p>The consequence is that modeling solenoids as physical loading elements in an antenna may not in cases be practical, although the task itself may become design-specific, especially in view of all of the variations possible. Thus, it may be some time before there is a more general profile of the affects of coil design on the performance of shortened antenna elements that use them. However, whatever differential there is between a physical coil and a mathematical load will apply equally to both NEC and MININEC, since both use essentially the same mathematical loading schemes, allowing for the fact that NEC places the load on a segment and MININEC places a load on a pulse or junction of segments.</p>
<p>It is perhaps a curiosity (since multiple types of coils have not been checked) that the loading coil used as the center load in part 4 of this series--and also used as the mid-element load--can be trebled to yield a 30-turn helical dipole only 40.8" long by less than 12" diameter. The resonant frequency of this experimental design was 7.105 MHz, only slightly lower than the standard design frequency used throughout the demonstration. If the exercise has any utility at all, it lies in the confirming the soundness of the advice to model solenoids using the material that will actually form the coil.</p>
<pre>Material Free Space Feedpoint Impedance
Gain dBi R +/- jX Ohms
Lossless wire 1.76 0.3 + j 0.5
Copper -5.46 1.3 + j 1.6
Aluminum -6.99 1.9 + j 2.2</pre>
<p>The impracticality of the design, as indicated by the source impedance, is less important than a comparison of the anticipated gain of the antenna from the lossless wire to normal materials. Lossless wire makes the antenna look like a promising performer, while the gain figures for real materials tell the true story. There is no magic in this system. Indeed, it packs over 93' of wire into the helix, more than required for a standard 40-meter dipole. Yet, at 7-9 dB below the performance of a full size dipole, signals would be down by only about 1.5 S-units. Such performance would be usable under certain kinds of operating conditions, although the losses associated with feeding the antenna might attenuate signals even more than the basic design itself.</p>
<p>Obviously, those interested in helical antennas will always benefit from making them as long and open as possible. an adequate model of a helical monopole or dipole will always take the form of a physical model.</p>
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<p>One final note on solenoids seems necessary, based on <b>Fig. 6</b> above. A common practice, especially in commercial antenna manufacture is placing a solenoid within a shield, and the shield may be connected (or not) to either of the linear wires extending from it. At present, I am unaware of any technique of modeling this situation which has been correlated to experimental results. On can surround the solenoid with a series of wires or even a wire grid--with one end connected to an antenna wire if desired. To what degree antenna fields arising from the solenoid will interact with the shield and what the consequences for the overall antenna field will be remains an area yet to be fully explored. It, too, is likely to remain a project-specific task for antenna designers. If we add in the variables of coil construction that might be present within the shielding structure, the task is no small one. Note that the term "shield" is here used by convention. To whatever degree that casing plays a role in radiation field production, it shielding effects may be only mechanical. Its electrical role may turn out to be either simple or complex.</p>
<h3>Conclusion</h3>These notes are intended to provide a reasonably fair view of the limitations of the modeling studies undertaken in the first 4 parts of this series. They provide some cautions against taking the results of the work as numerically general principles, which they are not. They also caution against hasty modeling that might unwittingly cross the boundaries of NEC capabilities. Finally, they also outline a host of work that might be done to extend the study and to make its conclusions both more general and more precise.
<p>Some of the further work can be done with some modeling ingenuity. Other parts of the work might be undertaken in alternative programs, such as MININEC. However, there may be a residue of the effort that may have to await the next generation of modeling cores.</p>
<p>Also see the <a href="../model/nec.html">Antenna Modeling Programs</a> page for more information.</p>
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<p><i>Updated 2-7-99. &COPY; 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.</i></p>
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<h1><img src="../images/pattern.gif" align="left" width="85" height="85" border="2" vspace="5" hspace="9">Modeling Loads and Transmission Lines<img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10"></h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Questions often arise about how properly to model loads and transmission lines within various versions of NEC and its commercial implementations. So I thought a brief primer might be useful. The utility of a primer is unquestionable, but that it can be brief quickly proved false.
<p>Although the information is covered in both the NEC manuals and the instruction manuals that accompany commercial implementations, something more systematic seems needed--something with examples carried from one level to another so that users can see the transitions and get a sense of what is and is not possible--and how to get around what is not possible. I also discovered that we cannot begin in the middle, assuming that readers know the basics. When it comes to antenna modeling, modelers come at the enterprise from many different backgrounds, enter it with different interests, and get into modeling loads and transmission lines with different goals and interests. Hence, everybody knows the basics in a slightly different way.</p>
<p>So we shall begin at the beginning--with simple antennas and simple loads-- and work our way to more interesting examples. Along the way, we shall have occasion to mention some useful adjunct calculation programs and some equations for use on calculators. Modeling programs just do not do everything we need to do to make good models. But there are some utility programs that can take most of the drudge work out of modeling.</p>
<p>In the end, we shall quit far short of exhausting the subject, but a little ways beyond exhausting the writer. However, incomplete, I hope the information is useful to you.</p>
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<li><b><a href="loadtl1.html">Part 1: Getting Oriented: Basic Loads and Simple Antennas</a></b></li>
<li><b><a href="loadtl2.html">Part 2: From Place to Laplace: Converting X and R to C, L, and R</a></b></li>
<li><b><a href="loadtl3.html">Part 3: Transmission Lines as Lines</a></b></li>
<li><b><a href="loadtl4.html">Part 4: Transmission Lines as Lumped Constants</a></b></li>
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<p><i>Updated 12-29-97. &COPY; 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.</i><br>
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<h1><img src="../images/pattern.gif" align="left" width="85" height="85" border="2" vspace="5" hspace="9"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Modeling Loads and Transmission Lines<br>
Part 1: Getting Oriented: Basic Loads and Simple Antennas</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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NEC models antennas as collections of straight wire segments. Within NEC programs, we specify the antenna structure as one or more wires and then subdivide each wire into one or more segments. One way in which we can modify one or many segments of an antenna is to load them.
<p>A load is simply a combination of resistance or reactance assigned to a segment. NEC is capable of assigning load as series R-L-C circuits (load type 0), parallel R-L-C circuits (load type 1), or as complex series R +/- jX impedance loads (load type 4). (NEC may also assign loads as series or parallel R-L-C values per unit of length (load types 2 and 3), but we shall not have occasion to deal with this option in this series.) Figure 1 surveys some of the options.</p>
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<p>In this part of the series, we shall deal only with series R +/- jX impedance loads. They are among the simplest to implement and manipulate, although they do have some limitations. We shall also mention the load that many users do not think of as a load: wire conductivity (load type 5). (NEC-4 is also able to handle wire relative permeability.)</p>
<p>For many modelers, this first part is a review of the most basic sort. However, within the review there might be a detail or two that will make a more advanced procedure a little bit clearer.</p>
<h3>A. The Load Entries: Material Loading</h3>NEC input consists of a series of lines of entry in a model description. Each line corresponds to a "card" in older FORTRAN input systems. It has a label and a sequence of entries that the program interprets as values and instructions. CM entries are comments; GW entries are straight wire specifications; etc. Load entries carry the designation LD.
<p>Unless we wish to use lossless wire in our model, we ordinarily specify a material or materials for the antenna elements. With commercial implementations of NEC, this specification may occur as a special entry apart from program means of introducing other types of loads. Nevertheless, the specification results in LD entries of type 5. Here is a typical entry:</p>
<pre> LD 5 1 1 101 5.8001E7
Type Tag 1st-Seg Last-Seg Conductivity</pre>
<p>The entry spacing has been widened to permit labeling the parts. A Type 5 load is a round wire internal impedance specification. The Tag indicates the wire number, here the only wire of a simple dipole antenna. The first and last segment designators set the limits of application of the value. Finally, the conductivity figure is in Siemens(mhos)/meter. The 1-wire antenna has 101 segments, all with the conductivity of copper.</p>
<p>Various implementations of NEC may handle the conductivity figure differently. Some provide named entries (for example, "copper"). Others provide conductivity figures or provisions for conductivity entry so that users can specify exact material properties. Still others provide custom entry in terms of resistance per unit of length, the inverse of conductivity. However entered, the entry is translated to an input of the type shown or its programmatic equivalent.</p>
<p>Unless an antenna uses materials of very specialized sorts, performance will not be altered to a significant degree by specifying either copper of aluminum (in its own various types). EZNEC permits only a single wire material specification, and users may use with confidence in the results either the conductivity of the "worst" material or the conductivity of the most dominant material in the antenna. However, NEC itself permit all manner of combinations of materials to be specified. NEC-Win Pro permits all the element load entries permitted by NEC itself, if modeling changes of material is an important consideration. With access to individual line entries, one may also within NEC-Win Pro load all segments or all wires to the same value in a single line by specifying the tag and the first and last segment entries as 0.</p>
<h3>B. The Load Entries: Spot Loads</h3>All spot or specifically placed loads within NEC consist of mathematical objects that do not themselves radiate, whatever the size of their physical implementations. However, the segment(s) on which they are placed do enter into the calculations for mutual impedance and current and are thus a part of the calculations for antenna pattern and other properties. Consequently, the error introduced by treating the loads themselves as non- radiating is normally insignificant. Spot loads do affect the current magnitude and phase along the antenna wire in mathematically correct ways to replicate reality very closely. Hence, they are normally a very accurate way to model loading coils and capacitors, traps, and resistances.
<p>Entering spot loads differs according to the implementation of NEC. Since we shall restrict ourselves in this part of the series to complex impedances composed of resistance and reactance only, we shall speak only in terms of type 4 loads for the moment. A basic NEC entry cord for such a load might look like the following:</p>
<pre> LD 4 1 77 77 0 870.4
Type Tag 1st-Seg Last-Seg Resistance Reactance</pre>
<p>At wire (tag) 1, on segment number 77 (first and last segment), we introduced a load of type R +/- jX with a resistance of 0 ohms and an inductive reactance of 870.4 ohms. (R and jX are always entered in ohms.) Although a 0-ohm resistance is does not coincide with reality, introducing pure reactances as a step in the modeling process can be useful, as we shall see down the line.</p>
<p>Specifying spot loads within commercial implementations of NEC may take a number of forms. In EZNEC, there is a separate load screen, a sample of which is replicated in Figure 2.</p>
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<p>Making an entry consists of adding the desired number of loads and then specifying their positions and values. In EZNEC, the user specifies a position in terms of a percentage along the wire. The program places the load on the segment closest to the specified percentage and lists the segment number and the actual percentage along the wire. R and X entries are self-explanatory.</p>
<p>NEC-Win Pro permits three modes of load entry. One is to create a load line of the type shown by typing it into the antenna description. One may also access a "Place Load" box from the description and enter the requisite data, as shown in Figure 3 (which replicates load #1 in the EZNEC figure).</p>
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<p>An alternative exists within the NEC-Win Basic module, which is provided for rapid entry of basic antenna information. The user may "drag and drop" loads (and sources) and then access a data box for numerical entries and load-type specification. Although the first two methods refer only to tags, this system also provides a percentage along the wire, which is useful for calculating the precise linear position of loads. Figure 4 shows an example.</p>
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<p>Virtually all other implementations of load placement in commercial implementations of NEC will be a variant on these systems.</p>
<h3>C. Simple Loads and Simple Antennas</h3>The Type-4 loads we place on an antenna consist of series R-jX circuits. These circuits are also in series with other features of the wire, including the source. Figure 5 presents a simple dipole, modeled in 11 segments, with possible loads. One position is at the source, simulating a series loading inductor (or capacitor). Other loading positions are indicated by the outer 2 loads, usually spaced equidistant from the center on symmetrical antennas like the dipole.
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<h4>1. Loading Short Dipoles: X Only</h4>To illustrate the utility of modeling loads, let us consider a 20-meter wire dipole of #14 copper wire. (I shall not show the LD line for the type-5 element loading information.) A full (1/2 wavelength) resonant dipole in free space would be just about 33.736' long (404.84" or 10.282 meters) at 14.15 MHz. The free space gain for reference is 2.08 dBi. The numerical precision outstrips real building conditions, but is here used as also a measure of resonance. Resonance in an antenna is the condition of having a feedpoint impedance that is purely resistive, that is, with 0 reactance. For modeling purposes, depending upon one's goal, resonance may be approximated for any design frequency with greater or lesser limits. In most pure modeling exercises, I tend to use +/- j1 ohm as the limits of resonance. In this example, the feedpoint impedance happens to calculate to 73.06 - j0.01 ohms. This particular model was done in NEC-4 using 101 segments. (If not pressing the limits of either program, NEC-2 results might still vary from those of NEC-4 by an amount that has no practical significance. Moreover, results from different implementations of NEC may also vary, according to input and output processing--but, again, only by amounts that have no practical significance.)
<p>Like loads, sources are distributed along a segment, although it is often convenient to think of them as appearing in the center of a segment. Hence, for a single-wire element, we need an odd number of segments in order to truly center a single source. (We shall take up split sources in just a little bit.)</p>
<p>This 20-meter full-size dipole will be our reference point. Let us now cut the wire in half, for a length of 16.868' (2.571 m). We shall retain the 101 segments, since each segment turns out to be just about 2" long (about 5.1 cm). At 14.15 MHz, gain is 1.73 dBi, and the feedpoint impedance registers as 12.67 - j870.4 ohms.</p>
<p>Let us introduce a complex or R +/- jX load at the center of the antenna (the 50% mark or segment 51). Let it be a pure inductive reactance of 870.4 ohms (R=0; jX=870.4). Loads are considered by NEC to be in series with sources. Therefore, it should be no surprise that rerunning the model produces feed point impedance of 12.67 - j0.04 ohms. The effect is equivalent to introducing a perfect center loading inductor having the require reactance. If the inductor could be perfect, the gain of the antenna would not change: 1.73 dBi.</p>
<p>As an interesting exercise, let us replace the center load with two other loads spaced away from the center. In fact, let us place them at the points where equal inductive reactances of 870.4 ohms (with no resistance) also produce resonance. These positions happen to be at segments 25 and 77 of our 101 segment model, a little over 24% inward from the ends. In other words, the positions are just beyond the midpoints of each half of the dipole. Interestingly, the antenna gain is 1.76 dBi, an insignificant amount greater than with center loading. (Mid-element loading of dipoles and dipole-based antennas is done not for added gain, but for other antenna properties, some related to the feedpoint impedance on the prime and other frequencies, some related to mechanical considerations).</p>
<p>The feedpoint impedance climbs to 26.08 + j1.65 ohms. (This value is as close to resonance as one may come in the model without changing the value of the loads.) The higher feedpoint impedance is one of those good reasons for using mid-element loading in preference to center-loading.</p>
<p>As an initial exercise, try moving these mid-element loads, without changing the reactance, one segment closer to center and then one segment farther from center. Notice that a 2" change in placement per load results in over 60-ohms reactance at the feedpoint. You may also replicate this entire sequence using other element wire diameters. Expect to find a different resonant length for the basic reference antenna, as well as changes in the requisite loading to restore resonance using either a center or mid-element loading technique. The patterns will familiarize you with trends that may be useful in the design of more advanced antennas.</p>
<p>As a further exercise, the modeler may also wish to check all of the points along the antenna length between center loading and the point at which mid- element loads reach the value of the center load. One will discover a curve of ever-more rapidly increasing reactance values. In fact, if one continues the exercise beyond the point used in this exercise, there will come a point where adding reactance to achieve resonance is no longer effective in terms of practical inductors. Yet helical structures placed at the ends of short inductors can resonate the antenna. Are they not inductors, that is vehicles of inductive reactance?</p>
<p>In fact, although the structures exhibit inductive reactance, at the ends of antennas they function primarily as lengths of wire necessary to achieve the current distribution along the entire wire to produce a feedpoint impedance that is wholly resistive. Many non-helical structures, including so-called capacitive hats, may achieve similar results. The inductive reactance of end coils is largely incidental to most amateur designs. However, it may be used for other purposes, such as in a trap. (Models suggest that the trap is not an appendage to the antenna length that it terminates, but is indeed a part of that length.)</p>
<h4>2. Split Sources and Loads (and Alternatives)</h4>The procedures noted so far for applying loads to antennas, especially at sources, are applicable only to linear elements. When elements meet at angles and the apex of the angle is the source, we must use another procedure. The procedure is also applicable to linear elements, although hardly ever used except in exercises like this one.
<p>Our original dipole consisted of a single wire, divided into 101 segments. Let us replace that wire with two wires, each half the length of the original. We shall assign 50 segments to each and let them join in the middle. Unless we use a different feed system, the source will not be centered.</p>
<p>NEC is able to use a split source. If we specify either a split voltage or a split current source, we should place the source on the last (right-most) segment of the first wire--the segment joining the second wire. It pays to use any graphical antenna viewing system to ensure that the source is properly place. On some systems, the source will show up as two sources, one on the last segment of the first wire and the other on the first segment of the next wire.</p>
<p>NEC treats these two sources as being in series, and the total impedance is simply the sum of the individual impedances of the source. Normally, only the composite values is shown in the outputs of commercial implementations. For our reconfigured dipole (with no change in total length, the split feed system returns a gain of 1.73 dBi (identical to that of the center, single source model) and a feedpoint impedance of 12.93 - j879.4 ohms. The source impedance is within a percent or so of the earlier model.</p>
<p>Loading the antenna mid-element requires no change except for the precise value of the load assigned to each of the two med-element loads. Moreover, because the segmentation varies very slightly from the single-source model, moving the loads inward or outward by one segment may surround true resonance, rather than hitting it with precision. (Obviously, other models may reverse the effect.)</p>
<p>Loading at the antenna center, however, calls for a change in procedure. Instead of assigning a single load to a single segment, we create two loads, each half the total load. We assign these loads to the segments containing the sources. Hence, the last segment of wire 1 (Seg 50) receives a load of R=0, X=j439.7, and the first segment of the next wire (Seg 1) receives an identical load. In our example, the source impedance returned a value of 12.93 + j0.04 ohms.</p>
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<p>The figure shows a schematic representation of the principle of split sources and loads applied to an inverted Vee. For simplicity, I simply rotated the dipole elements downward (still in free space) by 45 degrees. This actually made the elements a bit shorter than half size. But since we are going to load the elements, the difference make no big difference.</p>
<p>Using a split source, the model returned an impedance of 6.84 - j900.9 ohms. Adding purely reactive loads of -j450.45 ohms to each segment containing a source produced a source impedance of 6.843 + j0.05 ohms, showing that the technique indeed works.</p>
<p>There is an alternative to using split sources and loads. At the junction of the wires where we desire to place either a source or a load (or both), we may do a little substitute modeling. At the center point of the antenna element, create a short wire that is horizontal. To this very short wire, connect the angled wires at their normal lengths or their lengths minus half the length of the new wire.</p>
<p>Recommendations concerning the minimum length of this wire vary. However, if the wire has the same diameter (or radius, in pure NEC terms) as the connecting wires and if the segments are about the same length as the segments in the joining wires, it will make little difference whether the new center wire has 1 or 3 segments or what its precise length is. Place a single source on the center segment of the new wire.</p>
<p>For the inverted Vee with 100 segments, I created first a 3 segment wire 6" long. This preserved the approximate 2" length of all segments. I made no direct changes in the angled-wire lengths, but simply connected them to the new wire. This did alter the overall length of the antenna very slightly. The source was placed at the center of this new wire (Wire 2, Seg 2). The source impedance returned a value of 6.72 - j855.6 ohms. A single purely inductive load of the same absolute value placed on the same segment (in series with the source) eliminated the reactance.</p>
<p>I also use a single 2" long horizontal segment as the center wire. This changed the antenna length slightly less than the 6" center wire. The source was placed on the only segment of this Wire (Wire 2). The source impedance returned was 6.687 - j875.7, which a suitable inductive load placed on the same segment also eliminated.</p>
<p>With angled wires, expect minor variations in the values returned between the split source/load technique and the center section technique. In these examples, the values are well within the range of adjustments one normally expects to make with real antennas on site. However, the models with the longer horizontal center sections consistently showed a slightly higher gain than the split source model, and this effect showed up over real ground as well. The effect results from a portion of the antenna at the high current region being immune from the partial canceling of fields that results from the Vee configuration. Expect up to a 0.1 dB difference in gain, depending on the length of the center section.</p>
<p>These notes apply to heavily segmented models, such as the ones used in these examples. Ordinarily, NEC yields accurate results with linear elements with as few as 11 segments per half wavelength. Moreover, the fewer the segments, the faster the run-time for a model. However, where geometries grow complex even to the slightest degree--as in the case of the Vee--attention to segment length is advisable for the most accurate results. For a given configuration, the modeler should always achieve convergence to the degree required by the practical application. Convergence is simply raising the number of segments per wire (usually as a percentage) until further increases yield output figures that do not significantly vary from those produced by the next lower number of segments. These exercises, of course, are unduly segment and precision laden in order to familiarize you with the numbers and movements of numbers you may expect from the program.</p>
<h3>D. Reactance, Resistance, and Q</h3>Unfortunately, I know of no way of implementing an inductive (or capacitive) reactance as an antenna load without also having a series resistance. The ratio of the reactance to the series resistance defines the Q of the component used as the loading device.
<p>Knowing or being able to estimate the Q of a loading capacitive or inductive reactance allows us to calculate the resistance needed to complete a realistic load. Most commercial inductors I have encountered have had a Q between 150 and 300. As an exercise, let's go back to the shortened dipole with 101 segments and see what these particular Qs do to antenna performance. The following input line shows a typical entry:</p>
<pre> LD 4 1 77 77 2.9 870.4
Type Tag 1st-Seg Last-Seg Resistance Reactance</pre>
<p>With a center load of j870.4 and infinite Q (or zero resistance), the antenna gain was 1.73 dBi, with a source impedance of 12.67 - j0.04 ohms. If we let Q=300, then the resistance in series with the reactance is about 2.9 ohms. Under these conditions, the antenna model returns a gain of 0.83 dBi and a source impedance of 15.57 - j0.04 ohms. The source impedance is clearly the sum of the load loss resistance and the source impedance without a loss. If we let Q=150, the load resistance is 5.8 ohms, returning a gain value of 0.09 and a source impedance of 18.47 - j0.02.</p>
<p>Using the model with two mid-element loads of the same value as the center load and positioned to provide resonance, we come up with comparable figures. With no resistance, the model showed a gain of 1.76 dBi and a source impedance of 26.08 + j1.65 ohms. With a Q of 300, each load has a series resistance of 2.9 ohms. This model shows a gain of 0.93 dBi and a source impedance of 31.55 + j1.52 ohms. Lowering the Q to 150 requires a series resistance of 5.8 ohms, with a consequential gain of 0.24 dBi and a source impedance of 37.01 + j1.37 ohms.</p>
<p>Notice that resistive component of the source impedance is close to but not quite the sum of the load resistors and the source resistance of the unloaded antenna. This difference exists because the current magnitude at the load is not the source current, but a slightly less value of about 97% source current. The loads, of course, are positioned only a little over 1/16 of a wavelength away from the source.</p>
<p>NEC also returns data on the loads, which is accessible in the outputs of most implementations. Among the direct or calculable data is the power consumed in the loads. With a Q of 300, the center-loaded model consumes nearly 19% of the power in the load, while the mid-element loaded model consumes about 17% of the power between the two loading inductors. Hence, there is about a 0.1 dB difference in the gain of the two models.</p>
<p>At a Q of 150, the center-loaded model consumes over 37% of the power in the load, while the mid-element loads together consume a little under 30% of the power. Hence, there is a greater (0.15) difference in the gain levels of the two models. If mid-element loading begins to show a distinct advantage over center loading (apart from source impedance and mechanical considerations), it is with low Q loads.</p>
<p>As with models using pure reactances, the numbers yielded by your specific implementation of NEC may vary slightly from these. However, you should notice the same trends. As an exercise, you may wish to run load Qs ranging from 100 to 1000 for various load positions and values in order to familiarize yourself with the trends involved. Be sure to make use of all relevant outputs from NEC, including load data and current distribution along the antenna.</p>
<h3>E. Capacitive Reactance as a Load</h3>So far, we have dealt only with inductive reactance as a load. This feature of the exercise has been a product of our initial model: a heavily shortened dipole and some angular cousins. However, capacitive loading is also possible and often useful.
<p>Consider a 20-meter dipole of #14 copper wire, which for free space resonance was 33.7' long. The source impedance was just about 73 ohms. Suppose, as an exercise, we make the antenna 36' long. The source impedance now registers as 90.9 + j102.8 ohms: highly nonresonant.</p>
<p>However, using the same techniques as shown earlier, let us insert in the model a load in the same segment as the source. The load will be R=0, jX=- 102.8 ohms.</p>
<pre> LD 4 1 51 51 0 -102.8
Type Tag 1st-Seg Last-Seg Resistance Reactance</pre>
<p>Now the model will show a source impedance of 90 - j0.05 ohms. The lengthening of the element increased the gain by 0.06 dB, but we ignore that small amount.</p>
<p>Although we may divide an inductor into two parts, we must use separate capacitors to place equal amounts of capacitive reactance on each side of the source. Let's revise the load to place a reactance of -j51.4 ohms on each side of the load, by using the segments just before and after the source segment.</p>
<pre> LD 4 1 50 50 0 870.4
LD 4 1 52 52 0 870.4
Type Tag 1st-Seg Last-Seg Resistance Reactance</pre>
<p>Now the source impedance is 89.25 - j0.32 ohms.</p>
<p>Although the movement of the loading was small, the trend is evident. Just as placing inductive reactance farther outward on an element moved the source impedance closer to that of a standard dipole (relative to center loading), so too, placing a capacitive reactance farther outward on an element will also move the impedance closer to that of a standard dipole (again, relative to center loading). You can continue to move the capacitive reactance load outward to familiarize yourself with the trends in effects.</p>
<p>Here, the exercise is largely academic. However, the scheme of lengthening a driven element to raise the resistive component of its source impedance and then using capacitive loading to compensate for the resulting inductive reactance at the source has been used by at least one Yagi manufacturer in the past. The case involved a Yagi, whose driven element showed an impedance below 50 ohms. Lengthening and loading it provided the desired source impedance to match a 50-ohm coaxial cable feedline system.</p>
<p>Other applications of intentional capacitive loading include the controlled current distribution antenna, described by WB9RQR and W9WQ in the <i>ARRL Antenna Compendium</i>, Vol. 2 (pp. 132-135). A less load-laden antenna using capacitive loading to control antenna characteristics is the modified extended double Zepp (also called the double extended Zepp) described by N6LF in the <i>ARRL Antenna Compendium</i>, Vol. 4 (pp. 78-80).</p>
<p>Most capacitors whose ratings are not exceeded by an antenna application have Qs well in excess of 1000. Consequently, it is much safer to model capacitive loads without adding in a series R value than it is inductive loads. However, in some instances, capacitor leakage cannot be ignored. In such cases, estimates of Q will result in series resistance values calculated just as for inductive reactances. However, in most cases, the values will be much smaller, since the Qs will still be quite high.<br>
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These exercises take us just about to the limit of resistance-reactance (Type 4) loads. Although easy to use--and often the first step in placing loads into amateur antennas--they suffer an important limitation. Real capacitors and inductors show a change of reactance as we change frequency, even slightly. Specified reactances maintain the value assigned. Consequently, they do not accurately represent element loading across a span of frequencies, even as narrow a span as a single amateur band.
<p>In order to more accurately model the behavior of loads, we shall have to learn how to place series and parallel combinations of resistors, capacitors, and inductors as loads. In the process, we shall not only be able to model inductors and capacitors individually, but as well we shall be able to model traps and other tuned circuits placed within antennas. However, all that is for another segment of this series.<br>
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<p><i>Updated 12-18-97. &COPY; 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.</i><br>
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<h1><img src="../images/pattern.gif" align="left" width="85" height="85" border="2" vspace="5" hspace="9"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Modeling Loads and Transmission Lines<br>
Part 2: From Place to Laplace: Converting X and R to C, L, and R</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Although complex resistance-reactance loads are handy for rapid antenna development, they are limited in application. NEC does not change the value of reactance as the frequency changes. However, real-world components with which we implement reactances do change their reactance value with frequency. If we intend to evaluate an antenna over a range of frequencies, we shall have to employ different kinds of loads that use actual capacitance and inductance values. We have two choices: series R- L-C circuits (type 0) and parallel R-L-C circuits (type 1). Since the series circuit most closely coincides with the complex R +/- jX loads we have been using, let's begin there.
<h3>A. Series (Type 0) R-L-C loads: Finding and Entering the Numbers</h3>If we know a set of values for R, L, and C (or any one of them), we can go straight to the data entering process. However, if we have begun our design process with R +/- jX, then we must convert the reactance to a value of inductance or capacitance.
<p>Although the equations for conversion are common handbook material, let's repeat them here for reference (in case you have a calculator handy):</p>
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<p>where Xl is in Ohms, L is in Henries, and F is in Herz. Incidentally, NEC works almost exclusively in basic units for inductance and capacitance, so using the most basic equations is apt.</p>
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<p>where Xc is in Ohms, C is in Farads, and F is in Herz.</p>
<p>Now we may turn to a definite example. We shall resurrect the 1/2-size #14 copper wire dipole in free space, with its 2 mid-element loads. For the moment, we shall let the resistance be zero and worry only about the load reactance, which was 870.4 Ohms.</p>
<p>Using the appropriate equation, we discover that this reactance, at the target frequency of 14.15 MHz, yields a required inductance of 9.79 microH. However, for modeling purposes, we should get used to specifying this and similar inductances in the computer form of engineering notation, hence 9.79e-06 H. (Likewise, although most HF capacitance values are in pF, for example, 23 pF, we need also to express them in a similar way, that is, as 23e-12 F. Notice that we insert no spaces in any part of the number.)</p>
<p>The following figure shows the antenna description in NEC input file format for the 20-meter dipole with the loads expressed in type 0 format:</p>
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<p>Each of the load lines (ignoring the type 5 wire loading) follows the conventions just noted:</p>
<pre> LD 0 1 77 77 0 9.79e-6 0
LD 0 1 25 25 0 9.79e-6 0
Type Tag 1st-Seg Last-Seg Resistance Inductance Capacitance</pre>
<p>For type 0 loads, some of the entries have changed their meanings, specifically, the entries after resistance. Note that the order in which we enter the loads makes no difference to the calculation, although--except for examples like this one--I tend to enter all loads in some logical order. That practice makes modification easier, since I can more easily find the load I wish to modify and less easily select the wrong one. following the order of the elements in the antenna and working from left to right or from End 1 to End 2 for each element generally works in most cases.</p>
<p>The inductance entry is in the notation we described. In some cases, exponents with leading zeroes will be truncated to a single digit; in others, the leading zero will appear. Where values are missing (we had no resistance and no capacitance in this example, enter zero. NEC knows to treat these as missing values, whether we are dealing with series or parallel loading circuits. NEC does not create inappropriate short circuits for missing values. The user has to do that by specifying very small values other than zero.</p>
<p>Besides allowing the user to enter a load line in the antenna description, NEC-Win Pro provides two other means for entering load data. One is the load screen accessed from the load line.</p>
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<p>On this screen, we specified a Series RLC load and entered zeroes for the missing values and the inductance value. This screen produces the entry line for the load at segment 77.</p>
<p>Within the NEC-Win Basic model, calling up the drag-and-drop screen produces the following screens:</p>
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<p>Although the data is the same, this screen set permits us to specify the load position as a percentage of element wire length.</p>
<p>Within EZNEC, entering R-L-C loads is a 3-screen sequence, as shown in the following partial replication of EZNEC screens.</p>
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<p>Since EZNEC defaults to R+/-jX values, we changed the load type. If we had been working from a previous model that had R+/-jX loads, these loads would have been deleted. Therefore, we had to enter two new loads in the exact positions occupied by the old loads, as seen on Screen 1.</p>
<p>We begin the process of providing values for these loads on Screen 2, by selecting either a series or parallel type load.</p>
<p>On screen 3, we enter the values for the R, L, and C components of the load, using the engineering-computer notation, as shown.</p>
<p>Once we enter the values, they appear at the bottom of the screen as Laplace coefficients for the equations in the program that supply the NEC core with its input data. I have treated this example as if I were modifying existing values. Had there been no previous entry, all numerator (Num) values would have been 0.000E+00. However, note that the inductance value appears as the numerator for S^1.</p>
<p>For simple series loads, the Laplace coefficient for resistance will appear as the numerator for S^0 and the inductance will appear as the numerator for S^1. In some complex loads, some of the coefficients will not be the same as the values entered. Therefore, it is a good practice to keep a record of the actual R, L, and C values entered for each model.</p>
<p>Now that we have our model, we might as well run it. EZNEC Pro, using NEC- 4 returned a gain of 1.76 dBi and a source impedance of 26.08 + j1.65 Ohms. NEC-Win Pro, running NEC-2 return values of 1.76 dBi and 26.09 + j2.13 Ohms. In short, no difference.</p>
<h3>B. The Significance of R-L-C Loads</h3>Placing loads as series R-L-C circuits results within EZNEC in a number of cases that may initially mystify the user, since these loads are transformed and displayed in the load menu as Laplace coefficients. For the case of the "inductance only load" of 9.79e-6 H, the coefficient lines looked like this (to S^2 only, since everything beyond that point is zero):
<pre> Load 1: S^0 S^1 S^2
Num 0.000E+00 9.790E-06 0.000E+00
Den 1.000E+00 0.000E+00 0.000E+00</pre>
<p>If we add an arbitrary 2 Ohms resistance in series with the inductor, we see the following table:</p>
<pre> Load 1: S^0 S^1 S^2
Num 2.000E+00 9.790E-06 0.000E+00
Den 1.000E+00 0.000E+00 0.000E+00</pre>
<p>For inductors and resistors in series, we now know where to look for the values we inserted.</p>
<p>With capacitive loads, the table takes on a different look. Consider once more our slightly long dipole that we center loaded with a capacitor having a reactance of -j102.8 Ohms. If we convert the reactance to a capacitance at 14.15 MHz, the value required is 109.4 pF or 109.4e-12 F. If we now plug this value into the EZNEC load screen, we see a table like this one:</p>
<pre> Load 1: S^0 S^1 S^2
Num 1.000E+00 0.000E+00 0.000E+00
Den 0.000E+00 1.094E-10 0.000E+00</pre>
<p>The capacitive value shows itself in the denominator line at S^1. Complex loads having resistance and capacitance are a bit more complex in appearance. Let us place an arbitrary 2 Ohm resistor in series with the capacitor. The coefficients now look like this:</p>
<pre> Load 1: S^0 S^1 S^2
Num 2.000E+00 2.188E-10 0.000E+00
Den 0.000E+00 1.094E-10 0.000E+00</pre>
<p>The resistance is in the numerator line at S^0, while the capacitance value remains where it was. The new number in the numerator line at S^1 is the product of the two values. Incidentally, note that all values are transformed to the form x.xxx, with whatever exponent is needed to make the figure equal to the number entered.</p>
<p>The long dipole with capacitive loading is also interesting because it seems (and only seems) to suggest that it makes little difference whether we use a complex R +/- jX or an R-L-C load when evaluating an antenna across a ham band. A frequency sweep of the capacitively loaded dipole from 14.0 to 14.3 MHz produces the following table of source impedance values for each type of load:</p>
<pre> Frequency Load=R+/-jX Load=R-L-C
Source Impedance Source Impedance
14.00 86.88-j17.46 Ohms 86.87-j18.58 Ohms
14.05 87.90-j11.64 87.90-j12.37
14.10 88.94-j 5.85 88.94-j 6.23
14.15 90.00+j 0.05 90.00+j 0.04
14.20 91.05+j 5.78 91.05+j 6.13
14.25 92.13+j11.70 92.13+j12.42
14.30 93.22+j17.48 93.22+j18.55</pre>
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<p>The SWR curves are hardly distinguishable, as one might expect from the table of source impedance values. However notice the steeper rise in reactance with the series R-L-C load, which provides the correct reactance of the loading capacitor for each frequency checked.</p>
<p>When the characteristics of an antenna vary more extremely across a band of frequencies, the trend becomes more noticeable and important. Consider our short, mid-element-loaded dipole once more. Just as an example, we shall use the model that does not introduce the series resistance that represent losses due to having a finite Q. The table below provides the source impedance values reported by each type of load, while the graph supplies comparable SWR data.</p>
<pre> Frequency Load=R+/-jX Load=R-L-C
Source Impedance Source Impedance
14.00 25.13-j25.45 Ohms 24.84-j42.26 Ohms
14.05 25.44-j16.46 25.24-j27.82
14.10 25.76-j 7.44 25.66-j13.14
14.15 26.08+j 1.65 26.08+j 1.65
14.20 26.41+j10.62 26.52+j16.47
14.25 26.74+j19.93 26.96+j31.77
14.30 27.08+j29.07 27.42+j47.06</pre>
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<p>Clearly, the R+/-jX loaded model provides an excessively optimistic report of the antenna operating bandwidth--by nearly a factor of 2. Once more, just to be repetitive, accurate modeling of a loaded antenna across a band of frequencies requires the use of R-L-C loads.</p>
<h3>C. Parallel (Type 1) R-L-C Loads and Traps</h3>Any series load having a series resistance as well as reactance can be converted into a corresponding parallel load (type 1). If we begin with series values of R and C or L, we must convert the C or L into a value of +/-jX. For the record, the conversion equations are the following. For series to parallel conversions:
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<p>where Rs and Xs are the series values of resistance and reactance and Rp and Xp are the corresponding parallel values of resistance and reactance.</p>
<p>For parallel to series conversions:</p>
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<p>with the same designations.</p>
<p>Parallel R-L or R-C circuits do not have much utility, since they are only the converted counterparts of series R-L and R-C circuits. However, for the exercise, you can convert the mid-element loads with a Q of 300 into their parallel counterparts, arriving at values of Rp=261243 and Xp=870.41 Ohms from the series values of Rs=2.9 and Xs=870.4 Ohms. The reactance, of course, yields a required inductance of 9.79e-6 H</p>
<p>For NEC-Win Pro, we would simply select a parallel RLC circuit and enter the calculated values of R and L, with zero for the missing C. In EZNEC, we would similarly select a parallel RLC load and enter "261243,9.79e-6,0." In the coefficients table, we would see</p>
<pre> Load 1: S^0 S^1 S^2
Num 0.000E+00 2.558E+00 0.000E+00
Den 2.612E+05 9.790E-06 0.000E+00
</pre>
<p>We recognize the values of R and L in the S^0 and S^1 positions of the denominator line, while the value in the S^1 position of the numerator is simply the product of the two.</p>
<p>To further establish the equivalence of appropriately converted series and parallel loads, here are sample results for the mid-element loaded short dipole run with both types of loads in NEC-4 within EZNEC Pro:</p>
<pre>Frequency Series RLC Model Parallel RLC Model
Gain (dBi) Source R+/-jX Gain (dBi) Source R+/-jX
14.00 MHz 0.92 30.07-j42.36 0.94 29.96-j42.38
14.15 0.93 31.55+j 1.52 0.93 31.55+j 1.52
14.30 0.94 33.13+j46.95 0.92 33.26+j46.95</pre>
<p>The most useful application of parallel loads is with traps and similar RLC circuits. A trap is a resonant L-C circuit used to provide multiband service from a single element. There are several studies devoted to traps listed in the main index to these articles. Here we shall only cover certain essentials related to traps as computer modeling loads.</p>
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<p>The figure shows a typical dual-band trap dipole for 20 and 10 meters, resonant on 28.5 and 14.175 MHz. The element material is 0.5" diameter aluminum. The traps are resonant on 27.75 MHz to equalize performance on the two bands.</p>
<p>The models for this consist of three wires. The first is 6.41' (76.9" or 1.95 m) long with 12 segments. The last wire is identical to the first. Between them is a wire 16.36' (196.3" or 4.99 m) long, with 31 segments. The first and last segments are loaded by parallel R-L-C circuits representing the traps.</p>
<p>The figure also shows trap circuit-equivalents. The upper schematic provides the basis for modeling a trap. The main losses in a trap are normally in the coil, given its finite Q. For this model, a Q of 200 was selected as being slightly more conservative than some commercial traps available. However, Qs from 100-400 have been successfully constructed.</p>
<p>The trap values are 1.2 microH and 27.4 pf to achieve resonance at 27.75 MHz. The reactance of each component is about 210 Ohms at resonance. With a Q of 200, the coil series resistance is about 1.05 Ohms. To create a true parallel circuit, we must convert the series inductive leg to its parallel equivalent. The inductance remains almost identical to its series value, but the equivalent parallel resistor becomes quite large.</p>
<p>However, we must do the conversions at other than the trap resonant frequency. At 28.5 MHz, the net reactance of the circuit is capacitive, and the required parallel resistance is nearly 43 kOhms. At 14.175 MHz, the net reactance is inductive and the required parallel resistance is about 21.3 kOhms. There is a program within <a href="../radio.html#HAMCALC">HAMCALC</a> and also within another trap article in this collection that will perform the requisite calculations, as well as explain the procedure in detail.</p>
<p>In effect, we have two sets of parallel R-L-C loads, one for 10 meters, on for 20 meters. (The amount that the parallel resistance changes within a band the size of 20 meters is too small to be significant, especially in view of the rounded figures used in most cases for coil Q, although one may calculate the resistor's value for each frequency of interest.)</p>
<p>Entering the parallel R-L-C circuit for which ever band interests us is done in the same way as for series loads, except that we select the parallel option and enter figures for all three types of components. The following figure shows a typical NEC-Win Pro entry screen for one of the loads for the 10-meter model.</p>
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<p>With EZNEC, entry procedures are also the same as with a series load. The Laplace coefficient lines that result may seem mysterious at first sight:</p>
<pre> Load 1: S^0 S^1 S^2
Num 0.000E+00 5.155E-02 0.000E+00
Den 4.296E+04 1.200E-06 1.413E-12</pre>
<p>S^0 in the denominator line is the resistance, while S^1 in the same line is the inductance. S^1 in the numerator line is the product of the two numbers. It would appear that S^2 in the denominator line must have something to do with the capacitance, but the value seems to have no relationship to the 27.4 pF entered. However, if we divide S^2(d) by S^1(n), we arrive at our original capacitance. This is handy to keep in mind if you forget to record the capacitance you entered.</p>
<p>At 28.5 MHz, the model (in NEC-4) returns a gain of 2.08 dBi, with a source impedance of 85.69 + j0.02 Ohms. The same model using NEC-2 returns the same gain and a source impedance of 85.57 - j0.33 Ohms, establishing equal reliability in both programs for handling parallel R-L-C loads. At 14.175 MHz, the model with the appropriate parallel resistor yields a gain of 1.97 dBi with a source impedance of 63.16 - j0.63 Ohms. (Once more, these figures are excessively precise for practice, but are given to illustrate the numbers provided by the programs.)</p>
<p>As with all loads, the parallel R-L-C load is placed in series with sources and transmission lines. There is no direct provision for placing a load in parallel with sources or transmission lines within NEC.</p>
<h3>D. R-X and R-L-C Loads in Parallel with the Source</h3>Not all loads that we might use at the same segment as the source go in series with the source. One good example is the beta or hairpin match. Ultimately, this matching system places an inductance or inductive reactance in parallel with the source. The question is this: can we effectively model a beta match as part of the antenna? The answer is a qualified "yes."
<p>The beta match is a modified L-network for transforming a higher (feedline) impedance to a lower (antenna) impedance. An L-network consists, when transforming to a lower impedance, of a shunt or parallel reactance on the high impedance side and a series reactance of the opposite type on the low impedance side.</p>
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<p>As the figure shows, the modification consists in letting the antenna feedpoint impedance supply the series reactance and then installing only the shunt reactance. Normally, we make the antenna a bit shorter than resonant to provide a capacitive series reactance and then add a shunt inductive reactance. The inductive reactance may be in the form of an inductor or a shorted transmission line length (the hairpin). However, for special cases, we might also make the antenna long, providing a series inductive reactance and then install a shunt capacitive reactance.</p>
<p>Beta values are easily calculated, either from the following equations or from a program such as found in the HAMCALC collection. Let's sample the normal case, letting the series reactance be capacitive and the shunt reactance be inductive.</p>
<p>The common factor in L-networks is delta, which may be called the working Q or measure of efficiency of the network.</p>
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<p>From delta, we can calculate the requisite values of Xa (the series reactance at the antenna feedpoint) and Xl (the shunt inductor).</p>
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<p>Now let's return to one of the antennas in the first episode of this series. Consider the half-length 20-meter dipole with mid-element loading. This model used a single wire of 101 segments 16.868' long. The mid- element loads were on segments 25 and 77 and had values of 870.4 Ohms. The source impedance of this antenna was 26.08 + j1.65 Ohms.</p>
<p>The transformation from 26 Ohms to 50 Ohms for coax feed is a good job for the beta match. Using the equations above, we find a delta of about 0.95, resulting in an Xa of 24.97 and an Xl of 52.7. Now, let's model this antenna, working toward the beta match.</p>
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<p>The figure indicates several steps we need to take in order to ensure an adequate model of the antenna and its beta match. Step 1 we have already taken in modeling the original antenna. The gain, using lossless mid- element loads in this exercise, is 1.76 dBi.</p>
<p>Step 2 consist of preparing the model for a parallel load. Break the single antenna wire into 3 parts, the two outer parts having 50 segments each, with the loads installed where they physically had been in the original model. This would place them on segment 25 (49%) of the left wire and segment 26 (51%) of the right. At the center is a single segment wire the length of the segment that it replaced (about 2" in this example). When we run this model, we obtain a feedpoint impedance of 26.08 +j1.68 Ohms and a gain of 1.76 dBi, establishing that we have not made anything but a cosmetic change in the model.</p>
<p>Step 3 will establish the antenna shortening in order to provide the desired series source capacitive reactance. Here, we have two choices. We may physically shorten the outer wires, or we may simply reduce the mid- element loads a bit. Let's do the latter. Lowering the mid-element load reactances to 857 Ohms each produces a feedpoint impedance of 25.63 - j23.20 Ohms.</p>
<p>Step 4 requires that we add to the physical structure of the antenna. We add 3 1-segment wires to form a box around the center source wire. On the wire parallel to the source wire, we initially install a resistive load of 10 MOhms. This resistive load produces what is essentially an open circuit in the new structure so that we may check the effects, if any, of the structure on the antenna performance. Under these conditions, we obtain a feedpoint impedance of 25.97 -j23.55 Ohms, and an antenna gain of 1.70 dBi.</p>
<p>Although these changes are small, they do indicate that the structure is not insignificant. Had we used fewer segments, each would have been longer, and the box wires would have had greater effects on performance of the model. Reversing the positions of the source and the resistive load provide another indication that the structure is not insignificant: the feedpoint impedance and the gain change by very noticeable amounts. In practice, we tend to ignore small lumps and bumps in linear antenna element structure, but careful modeling suggests that their cumulative effects on performance may be more than trivial.</p>
<p>Step 5 consists in replacing the resistive load with the reactive load indicated by our work with the equations. Placing a purely inductive reactance of 52.7 at the parallel load point yields a source impedance of 53.28 - j0.10 Ohms. Using a very small structure to model the parallel load produced results very close to predictions from the equations.</p>
<p>Step 6 takes into account the Q of the beta coil. Let us assume a small inductor with a Q of 300 and hence a series resistance of 0.18 Ohms. This small move produces a source impedance of 53.13 + j0.12 Ohms, all the while making no change in the antenna gain of 1.70 dBi in free space.</p>
<p>If we wish to check the performance of the beta matched antenna away from the design frequency (14.15 MHz), we must make one further move: change the load type from a Type 4 (R +/- jX) load to a type 0 (series R-L-C) load. The resistance remains at 0.18 Ohm, while the requisite inductance is 0.59 microH. At the target frequency, the source impedance is reported by NEC as 52.91 + 0.13 Ohms.</p>
<p>The importance changing to a series R-L-C load becomes apparent the moment we frequency sweep the completed antenna model from 14.0 to 14.3 MHz.</p>
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<p>As the graph shows, assuming that the reactance of the coil will remain "close enough" to the value at the design frequency produces an overly optimistic curve of SWR. Using a series R-L-C load for which NEC provides the correct reactance at each frequency checked shows a much narrower (and more correct) operating bandwidth for the antenna. The difference is more than slight.</p>
<p>As the exercise demonstrates, it is possible to model loads in parallel to a source using the smallest feasible structures and paying close attention to details along the way. Among those details are deviations from the modeled performance without the added structure and an evaluation of the degree to which those deviations also apply to the real-world antenna. Equally important is using the right load type to model antenna performance across the relevant band of frequencies.<br>
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Although we have not fully covered all of the possibilities for using loads in NEC antenna models, perhaps we have gone far enough to let you continue on your own. If that is true, then we can turn our attention to a related subject, the entry and use of transmission lines in NEC models.<br>
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<p><i>Updated 12-28-97. &COPY; 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.</i><br>
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<p class="topics-footer">Topics: <a href='../topics.html#loading'>Loading</a>, <a href='../topics.html#modeling'>Modeling</a>, <a href='../topics.html#transmission-line'>Transmission Line</a></p><br>
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<title>Modeling Loads and Transmission Lines Part 3: Transmission Lines as Lines</title>
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<h1><img src="../images/pattern.gif" align="left" width="85" height="85" border="2" vspace="5" hspace="9"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Modeling Loads and Transmission Lines<br>
Part 3: Transmission Lines as Lines</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Modeling transmission lines between the source and the antenna can take two forms. We can physically model parallel transmission lines. And we can model them as TL inputs. Each method has its advantages and limitations.
<h3>A. Physical Models of Parallel Transmission Lines</h3>Consider a simple antenna fed at the center. The source is in series with the antenna wire. Any transmission line we use to feed the antenna thus forms a series circuit with the antenna wire. At the base of the transmission line, the actual source is now in series with the two wires making up the line.
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<p>These simple facts determine the basic rules for modeling a parallel transmission line as a parallel set of wires. First, we break the single antenna wire element into two segments, with the centermost ends spaced the same distance apart as the spacing of the wires in the transmission line.</p>
<p>In the figure are a set of arrows indicating ends 1 and 2 of each wire, using a left-to-right convention for this example. Note that we keep the series of wires continuous so that End 2 of Wire 1 connects to End 1 of Wire 2, etc. This is crucial to ensure that the transmission line has equal and opposite currents at every point. (For many placements of transmission lines, the "equal and opposite currents" situation will not fully materialize in reality. However, we always model the transmission line in the manner shown.)</p>
<p>The source wire may simply be a one-segment wire from one side of the transmission line to the other. This wire will only be as long as the spacing between transmission line wires, and its segment length sets some specifications for the transmission line segmentation.</p>
<p>Segments that meet at angular junctions should be about the same length and in any event within about a 2:1 length ratio. Closely spaced transmission lines require much shorter segments due to the close spacing of their wires. The similar junction at the top of the assembly, where transmission lines wires and antenna wires meet, dictates short antenna wire segments lengths. Even for simple antennas, the entire assembly can quickly grow past the overall segment limit (usually abut 500) for modestly priced commercial NEC packages.</p>
<p>One possible solution is to taper the segments lengths along the transmission line, especially at the end closest to the source. EZNEC provides for element length tapering. However, the shortest segment length selected should not be the standard value (0.0025 wavelength), but should match the element spacing. However, NEC-2 recommends that no segment be shorter than 0.001 wavelength long, which the spacing of a 300-ohm line will violate at many HF frequencies. Errors for lines in the 450-ohm to 600-ohm range run about 5 ohms difference from full segmentation (at the test frequency of 14.15 MHz and save from 30 to 130 segments per 1/4 wavelength of each side of the transmission line.</p>
<p>Physical transmission lines are usually calculated according to one of two equations:</p>
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<p>where Zo is the characteristic impedance of the resulting air-insulated parallel feedline, S is the center-to-center wire spacing, and d is the wire diameter in the same units as S. The first form is handiest for calculators, while the second form is somewhat more accurate for closely spaced lines.</p>
<p>Program suites such as <a href="../radio.html#HAMCALC">HAMCALC</a> have modules for calculating the Zo from the wire size and spacing of air-insulated lines--or for calculating any third factor from the other two. Note that these equations do not account for insulation other than air, and the velocity factor is either 1.0 or very close to it. Therefore, any models of physical transmission lines should be based on a velocity factor of 1.0 and the results later readjusted for the actual velocity factor of the line used. Likewise, any losses due to the dielectric must also be accounted for after modeling.</p>
<p>Interestingly, an extensive series of models using NEC-4 reveals that transmission lines between 300 and 600 ohms show a higher impedance than that calculated for them with the equations. The figure immediately below shows the test modeling arrangement. Instead of an antenna structure at the top of the transmission line, the lines were bridged with a resistive load.</p>
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<p>The following table shows the results of using the arrangement with a carefully adjusted 1/4 wavelength line of #14 AWG copper wire at 14.15 MHz (210"). Theoretically, the source impedance should be the same as the load resistance only when both are equal to the line impedance, according to the standard 1/4 wavelength line equation:</p>
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<p>where Zo is the characteristic impedance of the line, Rl is the load resistance, and Zs is the reported source impedance, all in Ohms.</p>
<p>First, there is a small and uneliminable reactance at the source from the test model. Second, the wires are very close together, approaching the limits of NEC-4 to handle, even when segments are carefully paralleled. In all cases, the modeled impedance is higher than the calculated impedance by from 5% to 7%.</p>
<pre>Line Zo Wire Spacing Modeled Zo New Wire
Calculated Dia. (inches) in Ohms Size for Zo
600 0.064" 4.78" 635 0.086"
450 0.064" 1.37" 483 0.084"
300 0.064" 0.39" 316 0.075"</pre>
<p>All results used segments lengths equal to the wire spacing. As the segment lengths were lengthened, the 600-ohm line was least affected, but the 300-ohm and 450-ohm lines changed values for each new segment length tried--without a definitive pattern. As the segment lengths increased, the far field from the test model increased from nearly -40 dBi to values in the low 30s.</p>
<p>Increasing the wire size brought the line impedance close to the calculated value. Equally, one might widen the spacing.</p>
<p>NEC-2 results with the same models correlated well with NEC-4 results with segment lengths greater than 0.005 wavelength. With shorter segments, the two programs apparently respond somewhat differently to closely spaced wires. Tapered-length transmission line models correlated well between the two programs, especially in the 450-ohm to 600-ohm Zo range.</p>
<p>For many applications, the exact line impedance may be unimportant. However, when physically modeling lines where their impedance does make a difference, running a test model to determine the line Zo is recommended. Physical models of transmission lines are at best somewhat cumbersome. However, they are important to any application where the lines may experience unequal currents and therefore radiate. Typical of such applications is the off-center-fed wire antenna. Comparing the far fields (including both vertical and horizontal components) of the antenna alone and the antenna with its parallel feedline will provide an estimate of such things as the field shape modification occasioned by the feedline, any change of elevation angle of maximum radiation, and any far-field gain changes. An examination of the currents along the transmission line may also be useful in determining the degree of unbalance occasioned by the off-center feed point. Similar exercises may prove useful with other antenna types.</p>
<p>Within their limitations, physically modeled parallel transmission lines can provide useful information, even if in some cases exactitude is sacrificed. To date, I have found no useful technique to model physically a coaxial cable such that the cable model fits within the segmentation limitations of versions of NEC aimed at radio amateurs.</p>
<h3>B. The NEC TL Card</h3>NEC-2 and up provide an alternative (TL input) means of modeling transmission lines of any type. However, the models are mathematical and not physical. By physical, I mean any portion of the antenna structure for which mutual impedances, currents, and fields are calculated. Although the terminals of transmission lines are real and interact with other parts of the model in regular ways, the line itself is not treated as a physical element. Hence, radiation from transmission lines that are handled by this alternative TL input means is not accounted for by the model.
<p>In exchange for this limitation, the modeler acquires the ability to place one or more transmission lines into the model. The lines may have any Zo and any velocity factor. Moreover, the transmissions lines do not add to the total wire and segment count. For a given antenna structure, core runs will be faster. Alternatively, where there is a limit to the number of segments allowed by a program, more segments are available for detailing the antenna structure itself.</p>
<p>Transmission lines inserted via the TL input will be treated as lossless. Consequently, if line losses are significant to the analysis, they will have to be separately calculated via a program such as Dean Straw's TL.</p>
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<p>The figure shows the NEC input file for a simple 20-meter dipole with a transmission line running from the antenna's normal feedpoint to another point used as the source. The terminating point of a transmission (as well as the starting point) must be a wire. When dealing with transmission lines used to feed an antenna, the "other wire" may be a very short single segment wire. Wire 2 in the sample is the terminating wire for the transmission line. It is only 0.003 wavelength long. The reason for keeping such wires very short is to ensure that they are minimally affected by the main antenna field and likewise minimally affect that field. For most amateur purposes, keeping the current on the secondary wire to 0.1% of the highest level of current on the antenna structure is sufficient to safeguard the accuracy of the reported antenna performance results.</p>
<p>The actual transmission line is specified on the line marked TL. The significance of the entries is as follows:</p>
<pre> TL 1 8 2 1 50 10.6 0 0 0 0
1st End Seg. 2nd End Seg. Zo Length Real Imag. Real Imag.
Wire # # Wire # # Ohms meters Adm. Adm. Adm. Adm.
W1 shunt W2 shunt</pre>
<p>The input entry tells us that the transmission line runs from segment 8 (the center) of wire 1 to segment 1 of wire 2. The Zo is 50 ohms. If the entry had been -50, the negative sign would indicate to the program to cross or reverse the terminals at the far end. The length is 10.6 meters, about 1/2 wavelength at the test frequency of 14.15 MHz. Note that there is no provision on the input line for a velocity factor. Any input system that permits entry of a velocity factor pre-calculates the length of an equivalent line with a velocity factor of 1.0 and enters that figure in the length place.</p>
<p>The last four places are for shunt admittances containing conductivity and susceptibility factors for ends 1 and 2 of the transmission line. We shall not use them in this exercise.</p>
<p>Note also that the distance between the two wires specified in the GW, or wires, entries is 69 feet. However, the length of the transmission line used in calculations will be the one specified in the TL card unless a zero is entered for length. In that case, the program will use the straight-line distance between the two terminating points of the transmission line as the line length. This latter entry capability is useful for placing straight line phasing lines between fixed elements and similar applications.</p>
<h3>C. Entering Transmission Line Data</h3>Commercial implementations of NEC use a variety of methods for entering transmission line data. Of course, the TL input may be entered in the editor by directly typing in the requisite information.
<p>NEC-Win Basic and the Basic module of NEC-Win Pro use a special window to simplify data entry:</p>
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<p>Although the label says, "No Transmission Lines Defined, this window shows one in the process of being defined. The terminating points are defined in the upper left, while the Zo and Length are immediately below. Note again that there is no provision for a velocity factor: the line length for a VF of 1.0 must be pre-calculated by the user. In the lower left is an entry to specify phase reversal or normal connections, with the unused shunt entries immediately above.</p>
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<p>NEC-Win Pro also accesses a similar window from the TL line of the antenna description. Although styled slightly differently, the data entry scheme is identical to the one just described.</p>
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<p>EZNEC enters transmission lines on a separate page, somewhat similar to the one used for loads. The partial replications in the figure take you through the transitions as you move from one item to the next in creating a transmission line. The top section shows the complete data entry, ready for revision, if one chooses to do so. The remaining partial screen section change as the user highlights one item after the other.</p>
<p>The second screen appears when either end of the transmission line is illuminated. The next screen highlights the length, which in EZNEC is entered in the same units as the wire dimensions. In this case, it is feet. If the user specified the length with a "d" suffix, the entry is interpreted as electrical degrees, which is then calculated into meters for the NEC core. Be careful not to use electrical degrees for a frequency sweep designed to show the properties of a fixed length of transmission line over an amateur band. An entry of "a" for the length will use the straight-line distance between the specified wire segments as the line length.</p>
<p>When either the Zo or the velocity factor are illuminated, the user has the option of selecting a transmission line from a table of both coaxial cables and parallel lines. Alternatively, the user can enter the required values. Finally, the user can construct an air-insulated parallel line by supplying the wire diameter (in measured units or as an AWG value) and the wire spacing.</p>
<p>The normal-reverse flag is activated in EZNEC by specifying N or R as the final line entry.</p>
<h3>D. Basic Transmission Line Applications</h3>The most basic transmission line application is as a feedline from a power source to the antenna, as shown in the figure.
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<p>Although the figure shows 11 segments, the antenna wire can have any (odd) number of segments (if a true center connection is desired). The second wire is very short and has one segment. The transmission line goes from the center of one wire to the next.</p>
<p>Consider a 20-meter dipole similar to the one in the figure, but with 101 segments of #14 copper wire. The total length is 33.696' (5.141 m). The second wire is 0.02' (0.006 m) long and has 1 segment. (This length at the test frequency of 14.15 MHz is about 0.0003 wavelength long, while the recommended NEC-4 minimum wire length for such applications is about 0.0001. For NEC-2, the wire should be at least 0.001 wavelength long.)</p>
<p>The NEC-4 source impedance of the antenna alone is 73.06 - j0.01 ohms. Now let's do a familiarization exercise based on the fact that a lossless feedline repeats the antenna source impedance every multiple of 1/2 wavelength. First, we move the source from the long antenna wire to the short second wire. Then, we specify a 50-ohm, VF=1.0 transmission line between the center of the antenna (the former feedpoint or segment 51) to the center of the second wire.</p>
<p>Since TL-type transmission lines are in parallel with sources, the effect of placing both the source and the transmission line on the second wire is to place a source in series with each wire of the transmission line, just as we would model the system physically. Except for the fact that the TL- type transmission line will not even show copper wire loss, the results will be identical for physical and TL transmission lines.</p>
<p>For example, let us assign to the transmission line a length that is exactly 180 electrical degrees. The resulting reported source impedance will be 73.06 - 0.08 ohms in NEC-4. (As with all models in the exercises in this series, minor adjustments will have to be made for NEC-2, adjustments that are only apt to yielding the excessively precise outputs used for numerical comparison, but which have little or no practical application to real-world antennas.)</p>
<p>EZNEC permits the specification of transmission line lengths in electrical degrees (and automatically specifies a velocity factor of 1.0). For other programs, we may have to calculate precisely the length of the transmission line that is exactly 180 electrical degrees long at the test frequency. At 14.15 MHz, this length is 34.7552' or 10.593 m. Using either of these values, the source impedance reported is the same as just noted.</p>
<p>Now let us substitute a truer 50-ohm coaxial cable, perhaps RG-213, with a velocity factor of 0.66. The requisite length of the cable is simply 0.66 times the value used with the assumed velocity factor of 1.0, or 22.9384' (6.9914 m). Using this transmission line, the reported source impedance is 73.06 - j0.00 ohms.</p>
<p>A real coaxial cable, even brand new, will not deliver lossless performance. If we use a program like TL, we shall discover that RG-213 at the specified length of nearly 23' (nearly 7 m) with an antenna impedance of 73.06 - j0.01 ohms will show that the source end an impedance of 71.91 - 0.03 ohms. This value represents the apparent SWR decrease (from 1.46 to 1.42) due to line loss with which most antenna users are familiar. The line also creates a 0.195 dB power loss, which amounts to 4.4%. The loss is a combination of basic line loss plus the additional loss due to the SWR level.</p>
<p>There is no convenient way to incorporate this loss into the antenna wire at either end of the model. Nor should we want to. Whatever the power delivered to the antenna, its gain and other characteristics remain constant (assuming that material junction losses can be neglected). If the losses and gains within an overall antenna-plus-transmission line-plus- source system are important, they can easily be calculated externally to modeling the antenna itself.</p>
<p>This exercise has little practical value other than familiarizing the modeler with the placement of TL transmission lines within a model. The model used here can be extended by changing the length of the transmission line in increments of 10 electrical degrees and observing the change in source impedance across a full half wavelength. One might also use 300-ohm and 450-ohm transmission lines in order to observe the greater excursions of resistance and reactance with each 10 degrees of electrical length. All of these are familiarization exercises, designed to develop a set of reasonable expectations for models employing TL transmission lines.</p>
<p>In each case, we must remember that the values given are for a lossless line. The TL transmission line will return the same value of source impedance for 0, 180, 360, and 540 electrical degrees, or multiples of 34.7552' (10.593 m) for velocity factors of 1.0. For comparison, run each length through a program such as TL to determine the losses for each length and the anticipated impedance presented at the source end of the line.</p>
<h3>E. Sample Practical Applications</h3>There are many practical combinations of antenna elements plus feedline(s) for which modeling is very apt. The figure below illustrates only three of them.
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<p>At the upper right is a pair of horizontal antenna elements spaced some distance apart. The spacing might be anything from 1/8 to 1/2 wavelength. We may feed the antennas either in phase or out of phase. To explore the various situations, we need only create a third wire. Then we create transmission lines from wire 1 and from wire 2 to wire 3, where we also locate the source. In order the reverse the phase of feeding, we simply specify one of the two transmission lines as "reversed" or "180-degree phase reversal," which enters a minus sign in front of the Zo we specify for the two lines. We can even experiment with different Zos for the transmission lines and different lengths. The models, even though subject to slight inexactitudes due to the lossless nature of the transmission lines, will teach much about phasing lines with a common feed point.</p>
<p>At the bottom is the KB8I phasing system for feeding two 1/4 wavelength verticals that are spaced 1/8 wavelength apart. TL1 and TL2 are 50-ohm coaxial cables 157 degrees long, each terminating with a wire in the model. Between the wires is another cable, TL3, which is 29 degrees long. To create this model, we might place the verticals over perfect ground. If we wish to model over real ground, we would elevate the verticals and their ground planes for NEC-2. In either event, wires 3 and 4 should be elevated. There are methods for calculating the phasing lines to achieve a desired current phasing of -135 degrees for one of the verticals with the other at zero degrees, giving a broad, nearly cardioidal pattern to the array. Of course, we can replicate Al Christman's dual directional array switch simply by moving the source from wire 3 to wire 4 and back again. For this array, which permits a direct 50-ohm coaxial cable feed from the model source to the shack, it is wise to look at the current magnitude and phase at the based of each antenna as well as the overall current and source impedance. You might even wish to experiment by replacing the 50- ohm cable with 75-ohm cable, just to see what happens. Be sure to remember to calculate for the velocity factor when translating the cable lengths in electrical degrees into feet or meters.</p>
<p>At the upper left is an interesting system for improving the operating 2:1 SWR bandwidth of an antenna-feedline system for dipoles. This can be operationally important to amateurs using gear with sensitive SWR shut-down systems with certain antenna situations. Consider a half wavelength center-fed dipole for 40 meters, which we might model as #14 copper wire 67.2' long with 101 segments. Let us place this antenna at a height of 50' above medium earth. We would obtain the following values of source impedance and 50-ohm SWR from NEC-4:</p>
<pre>Freq. 7.0 7.05 7.10 7.15 7.2 7.25 7.3
R 83.28 84.88 86.44 88.07 89.72 91.38 93.06
X -32.5 -21.3 -10.1 - 0.9 12.0 23.0 34.0
SWR 2.03 1.85 1.76 1.76 1.84 1.99 2.19</pre>
<p>Even allowing for the lower SWR that appears at the shack end of the line due to line losses, the band edge values may activate power reduction circuitry in some gear. As a step toward resolving the problem, let us create a short wire spaced well away from the dipole. From the antenna, run a 50-ohm coaxial line exactly 1/2 wavelength long at the band center, 7.15 MHz. RG-213, with a velocity factor of 0.66, would be 45.40' long for this application. Be sure to move the source to the new wire (wire 2 in the diagram). Because the transmission line is less than 1/2 wavelength long at 7.0 MHz and longer than 1/2 wavelength long at 7.3 MHz, we obtain the original feedpoint impedance only at the band center, 7.15 MHz. At other frequencies across the band, we obtain the following values for the lossless line:</p>
<pre>Freq. 7.0 7.05 7.10 7.15 7.2 7.25 7.3
R 90.09 87.80 87.13 88.07 90.58 94.75 100.8
X -27.1 -17.3 - 8.0 0.9 9.5 17.9 26.0
SWR 2.03 1.85 1.76 1.76 1.84 1.99 2.19</pre>
<p>Note that the SWR has not changed, but the impedance values have, except for mid-band. The resistive component has gone up, and the reactive component has gone down. In fact, the values are sufficiently higher than 50-ohms to make a 1/4 wavelength 75-ohm matching section useful to transform them to lower values. So let's add a new short wire (wire 3) and move the source to it. Then let's create a 75-ohm, 0.66 velocity factor transmission line from wire 2 to wire 3. The line will be 22.70' long. From it--again allowing for the fact that the model uses lossless line--we obtain the follow source values:</p>
<pre>Freq. 7.0 7.05 7.10 7.15 7.2 7.25 7.3
R 56.45 61.26 63.92 63.86 61.31 56.96 51.72
X 16.0 11.5 5.6 - 0.7 - 6.2 -10.1 -12.2
SWR 1.38 1.34 1.30 1.28 1.26 1.26 1.27</pre>
<p>The result is a set of SWR values across 40-meters that should satisfy any piece of amateur equipment. The line losses at any point in the band may be calculated from TL and compared to the use of a single 50-ohm cable or to the use of a single 75-ohm cable with impedance transformation (and network losses) at the shack. There is not likely to be a significant difference.</p>
<p>For developing the general picture of what occurs all across the band with this system of matching, modeling both the antenna and the transmission lines provides one of the most efficient methods. I am indebted to Dave Leeson, N6NL, for bringing the matching system to my attention. Dave attributes it to Frank Witt, AI1H and other sources. All in all, it provides a good exercise in seeing the modeling capabilities of NEC in handling multiple lengths of transmission line on the way from the source to the antenna.</p>
<p>Not all transmission lines must have a short wire at one end. Consider the following sketch of a ZL Special. Pioneered, although apparently not invented, by George Prichard, ZL3MH (later ZL2OQ), the antenna consists of two elements spaced about 1/8 wavelength apart. The feedpoint is the junction of the forward element and the phase line (transmission line) to the rear element.</p>
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<p>Let's build one of these antennas as a model. For 28.5 MHz, we might choose 3/4" diameter aluminum tubing and make two identical elements, each 16' long with 51 segments. We shall space them 3.46' apart, which is just about 0.1 wavelength. We shall set one element as the forward element and place the source at its center. We create the phasing line by placing a transmission line between segments 26 of each of the two elements. Let us follow Prichard's lead and use 71- ohm parallel line, which had a reported velocity factor of 0.67. Let's be sure to place the traditional half twist in the line by reversing the phase. If we place the antenna at 35' above medium earth, NEC will return a gain of about 11.8 dBi, a front-to-back ratio of about 25.7 dB, and a source impedance of about 7.2 + j8.8 ohms. (The low source impedance did not bother Prichard, since he fed his arrays with parallel feeders and an antenna tuning unit.) These facts are perhaps the least interesting in connection with the ZL Special.</p>
<p>Now let's examine the current tables available as an output from NEC. If we used a current source of 1 for the antenna, we shall find the current at the center of the forward element to be about 0.455 at -16.69 degrees. At the center of the rear element, we find the current to be about 0.462 at 159.78 degrees. The rear element has a magnitude of 1.015 and phase of 143.09 degrees relative to the forward element. Yet, the phase line length was only 0.1 wavelength long physically and 0.15 wavelength long electrically. How is this possible?</p>
<p>To find out, let's make a slight change in the model. If the forward element was modeled from left to right (-8.0' to +8.0'), then let's make the rear element in the reverse direction (+8.0' to -8.0'). Next, let's change the transmission line back to its normal or non-reversed orientation.</p>
<p>When we run this model, we obtain identical performance figures to those cited above. However, the current tables show another story. The forward element remains as it was, but the rear element shows a center current of 0.462 at -20.22 degrees. The rear element is -36.91 degrees relative to the forward element, which is equivalent to a phase angle of 323.15 degrees. Of course, subtracting 180 degrees from this figure yields 143.15 degrees, which we obtained from the first version of the model.</p>
<p>The current phase change of (-)36.91 degrees along the 71-ohm, 0.67 VF phase line of 3.46' from the rear element to the forward element is precisely what we need to provide the two elements with the relative current magnitude and phase angles so that the array achieves a front-to-back performance far in excess of what a similar driven element-reflector Yagi might obtain with the same spacing. The details of the analysis of ZL Specials go way beyond our purpose here, but the ZL Special is an excellent example of modeling transmission lines between two elements.</p>
<p>There are numerous other feedline uses of feedlines, but these examples should get you started. We may also use feedlines for other things, for example, as substitutes for lumped components. However, that set of applications will require that we search for shorted and open lines, and they can only be found in the next part of this series.<br>
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<p><i>Updated 12-28-97. &COPY; 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.</i><br>
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<h1><img src="../images/pattern.gif" align="left" width="85" height="85" border="2" vspace="5" hspace="9"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10">Modeling Loads and Transmission Lines<br>
Part 4: Transmission Lines as Lumped Constants</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Lengths of transmission line can be used as "lumped constants," that is as inductive or capacitive reactances. In many cases, the use of a transmission line to provide a desired reactance is preferable to or at least as equally good as the use of a capacitor or inductor. The reasons are many and depend on the particular circumstances of the application. For some purposes, transmission line lengths have a higher Q; in others, they are more durable or better shaped. In antenna work, transmission line lengths--or stubs--are very useful, and therefore worth modeling in place.
<h3>A. Stub Principles</h3>The general principles of creating reactances from transmission line stubs is well known. The figure shows the general technique.
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<p>Shorted stubs less than 1/4 wavelength long provide inductive reactance. At exactly 1/4 wavelength long, the stub becomes the equivalent of a very high impedance parallel resonant circuit, creating a virtual open circuit between the terminals. Longer than 1/4 wavelength, the shorted stub becomes a capacitive reactance.</p>
<p>Open stubs follow an opposing path. Shorter than 1/4 wavelength, they provide capacitive reactance. At exactly 1/4 wavelength, they are the equivalent of a very low impedance series resonant circuit, creating a virtual short circuit between the terminals. Longer than 1/4 wavelength, the open stub becomes an inductive reactance.</p>
<p>We can easily calculate the reactance of a length of shorted transmission line if we know the characteristic impedance (Zo)--along with the velocity factor of the line. For shorted stubs, the equation is as follows:</p>
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<p>where Zo is the characteristic impedance of the line, X is the reactance, and "beta l" is simply the line length in electrical degrees or radians. Of course, you must pre-calculate how many electrical degrees or radians there are in the physical length of line with which you start. Likewise, if you turn the equation around to determine the length of line required for a certain reactance, you will have to translate the result from electrical degrees or radians in the units of measure of length you are using.</p>
<p>For open stubs, we use the equation</p>
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<p>where the terms have the same meaning. The value of X will be either positive or negative, indicating inductive or capacitive reactance. At exactly 90 degrees, the tangent of the angle climbs without limit, resulting in a value of reactance that is zero for open stubs and that is indefinitely high (some folks like to say infinite) for shorted stubs.</p>
<p>At zero degree lengths and at multiples of 180 degree lengths, the value of the tangent of the angle is zero. The stubs reverse their roles with their 1/4 wavelength counterparts.</p>
<p>There are a number of applications of shorted and open stubs in antennas, and we can model most of them. However, we must always remember that transmission lines (TL entries) in NEC represent lossless lines. For all but short lines, we should externally calculate line losses in the final evaluation of any antenna design where the line or stub is a part of the antenna.</p>
<h3>B. Entering Stub (TL) Data</h3>Every transmission line or TL entry in NEC requires two wire segments to act as terminations for the line. In effect, the TL entry opens the wire segment and inserts the transmission line segment in series with it at that point. Without wire segments at either end, the transmission line cannot be formed.
<p>This process is invisible with EZNEC.</p>
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<p>As the screen replica shows, we have inserted 2 stubs in Wire 1. The first is a shorted stub, which we specified simply by keying "s" for end 2 of the transmission line. The remainder of the entry of length, Zo, and velocity factor are normal. We use a similar procedure for the second open stub by specifying "o" for end 2 of the line.</p>
<p>What the entry does not show you is a second wire created by the program to act as the other termination of the line specified. The line is created about 100 wavelengths away from the antenna in all 3 axes, X, Y, and Z. Additionally, the line is about 0.002 wavelengths long and divided into two segments, one to act as the terminal point of each of the two loads. The wire has a radius of about 0.0001 wavelength, the smallest permissible under NEC. Each of these measures ensures that the wire will interact minimally with the other wire segments, thus not disturbing the accuracy of the impedance, current, and field calculations.</p>
<p>You can easily recreate this situation with programs, such as NEC-Win, where you may have to create your own stub conditions.</p>
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<p>The TL window in NEC-Win presumes that you have created the second wire. In this case, we are creating a shorted stub at wire 1, segment 25, and it goes to wire 2, segment 2. We are using 450-ohm transmission line with a length of 3.687 meters (12.09'). Note, however, one final necessary element of the stub. End 2 must specify the conditions for a short circuit across the terminals, which means an indefinitely high value of "shunt admittance" (the inverse of a shunt impedance). Since the boxes are small, I have move the "real" (conductance) component to the right to show the exponent value, while the "imaginary" (susceptance) component remains shifted left. Values of 1E+10 are so high as to be indistinguishable from a true short circuit.</p>
<p>The corresponding input line in the text editor would appear as follows:</p>
<pre> TL 1 25 2 2 450 3.687 0 0 1E+10 1E+10
1st End Seg. 2nd End Seg. Zo Length Real Imag. Real Imag.
Wire # # Wire # # Ohms meters Adm. Adm. Adm. Adm.
W1 shunt W2 shunt</pre>
<p>For an open circuit, we would use the same procedure, but place values of 1E-10 in each of the boxes. This low a value of admittance corresponds to a value of 1E+10 for impedance, indistinguishable from an open circuit. Although such values are manually placed in this NEC-Win example, EZNEC places them automatically and invisibly to the user.</p>
<p>Now that we can create transmission line stubs, we can litter our antennas with them. Unlike physically modeled transmission lines and stubs, these TL entries occupy no wire segments, speeding core calculation runs and allowing greater antenna detailing. They appear in series with loads on the same segments they occupy, but in parallel with sources on the same segment.</p>
<h3>C. A Basic Stub Use</h3>One basic use of transmission stubs is to replace inductors as loads along antenna elements. To understand their use within models, let us return to our running model: the 101 segment #14 copper wire model in free space of a dipole shortened to about half-normal length. In past exercises, we loaded this dipole at segments 25 and 77, first with a reactance of 870.4 ohms, and then with an inductor of 9.79e-6 H, at 14.15 MHz. For a lossless inductor, we obtained a gain of 1.76 dBi in free space, with a feedpoint (source) impedance of 26.08 + j1.65 ohms.
<p>Now let's replace the R-X and R-L-C type loads with a transmission line stub. In fact, let's try a couple of different stubs.</p>
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<p>First, we can replace the loads with TL entries at the same segments using 450-ohm, 0.95 VF transmission line. The standard equation tells us that the line needs to be 11.494' (3.503 m). This will show up in the TL line in the text editor as 3.687 m, the equivalent length of line with a VF of 1.0. (The second wire is assumed to have been placed either manually or automatically by the program.)</p>
<p>At our test frequency, the model returns (in NEC-4) a source impedance of 26.09 + j1.72 ohms, and a gain of 1.76 dBi in free space. The slight numerical differences from the R-x and R-L-C load models, of course, make no difference at all.</p>
<p>Now let's replace the 450-ohm line with 50-ohm coaxial cable with a VF of 0.66 (RG-213). The required line length will be 11.05' (3.369 m), which will translate to 5.104 m of 1.0 VF line in the TL entry.</p>
<p>Using this line in the model, it returns a source impedance value of 26.09 + j2.147 ohms and a free space gain of 1.76 dBi. Again, no difference.</p>
<p>Although we recognize that these lines are lossless in the model, losses in transmission line lengths--when used within their recommended limits--tend to be negligible for such short lengths. We would expect no changes in the real antenna that are attributable to line loss adjustments, in contrast to those occasioned by wire and line cutting and assembly precision. However, transmission lines have at least wire losses which given them finite Qs in applications such as these. Although the Qs may be higher than those of inductors, we may not expect true lossless performance.</p>
<p>An exercise such as this one can mislead as well as instruct. At first sight, it would appear that we might easily replace our loading inductor with transmission line stubs and have the same antenna. This would be true if and only if we operated the antenna only at 14.15 MHz.</p>
<h3>D. Frequency Sweeping Inductive and Stub Loads</h3>Let us take the trouble to run frequency sweeps of our half size dipole using all three loads: the inductor, the 450-ohm stub, and the 50-ohm stub. Across 20 meters from 14.0 to 14.3 MHz, we would obtain the following sets of values. The values are grouped according to source resistance, source reactance, and SWR relative to 26.08 Ohms (the value at resonance).
<pre>Source Resistance (in Ohms):
Type Load FQ: 14.0 14.05 14.1 14.15 14.2 14.25 14.3
Inductive 24.84 22.24 25.66 26.08 26.52 26.96 27.42
450-ohm Stub 24.38 24.92 25.49 26.09 26.70 27.36 28.04
50-ohm Stub 20.25 21.65 23.50 26.09 29.98 36.50 49.50
Source Reactance (in Ohms):
Type Load FQ: 14.0 14.05 14.1 14.15 14.2 14.25 14.3
Inductive -42.3 -27.8 -13.1 1.65 16.47 31.77 47.06
450-ohm Stub -68.9 -46.1 -22.5 1.72 26.59 52.64 79.38
50-ohm Stub -320. -240. -136. 2.15 198.8 502.9 1038.
SWR relative to 26.08 Ohms:
Type Load FQ: 14.0 14.05 14.1 14.15 14.2 14.25 14.3
Inductive 4.54 2.82 1.65 1.07 1.85 3.12 4.90
450-ohm Stub 9.35 5.07 2.34 1.07 2.65 5.71 10.50
50-ohm Stub &gt;100. &gt;100. 32.3 1.08 52.5 &gt;100. &gt;100.</pre>
<p>Clearly, something is different about the stubs relative to the inductor-- and between the stubs themselves. Although the rate of change in source resistance is moderate as we move from one load to the next, the rate of change in the source reactance is much higher for the stubs--and extremely high for the low-impedance transmission line. The 2:1 SWR operating bandwidth for the inductive load is less than 200 kHz, for the 450-ohm stub less than 100 kHz, and for the 50-ohm stub less than 20 kHz.</p>
<p>Stubs change reactance according to changes in the TAN function of the electrical length in degrees or radians and relative to the characteristic impedance of the transmission line. Inductors, however, change reactance according to changes in frequency. The differences in the way each changes reactance do not show up if the required inductive reactance of a shorted stub is low--in the 0 to 100 ohm range.</p>
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<p>The graph compares change in reactance at 14.0 and at 14.3 MHz for an inductor (inner lines) and 50-ohm transmission stubs (outer lines) when the load at 14.15 MHz is assigned values between 0 and 100 ohms. Even at the upper end of the graph, the differential between 50-ohm stubs and an inductor is a matter of a few ohms. However, when the required inductive reactance is greater--between 100 and 800 ohms--a more radical divergence appears.</p>
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<p>As the graph shows, the inductor changes its reactance by only small amounts all the way up to 800 ohms. However, the 50-ohm transmission line stub changes its value more rapidly as the frequency departs from the design center as the load value increases. The result is that for antennas requiring high values of inductive reactance as loads, the 50-ohm stub permits only a very narrow operating bandwidth.</p>
<p>The 50-ohm stub is a very extreme case, used here to dramatize the differences in the rate of change of value between transmission line stubs and inductors with frequency movement from the design center. Fortunately, the higher the impedance of the transmission line stub, the smaller the difference between its change of reactance and the change of reactance of an inductor over the same frequency deviation from design center.</p>
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<p>The graph above is developed on a different premise. Let us use the change of reactance of the inductor as a base line. If the design center is 14.15 MHz, then at a frequency of 14.3 MHz, the percent of change in reactance will be constant over the entire range of reactances. In contrast, the percent of change of transmission line stubs will very, increasing with the reactance required, and so too will the ratio of the change in reactance in stubs to the change in inductors.</p>
<p>The graph shows the ratios for 50, 100, 300, and 600 ohm lines, relative to the inductor's baseline. The lesson is clear: for the widest bandwidth when using transmission line stubs, use the highest feasible characteristic-impednace transmission line. Even so, a lossless inductor will always show a wider operating band width than an equally lossless transmission line stub.</p>
<p>For inductors, Qs of 150 to 300 are common in antenna loading service. These Qs generally further increase the operating bandwidth at the cost of slight losses in antenna gain due to the resistive losses. In contrast, transmission line stubs less than 1/4 wavelength long rarely have Qs less than 300, and often they can reach 1000 to 1500. Hence, stubs offer slightly higher gain at the expense of operating bandwidth for any given antenna design that is inductively loaded.</p>
<p>For the antenna modeler, this exercise should reinforce the importance of making frequency sweeps across every the range of frequencies for which the antenna is designed. The sweep must be run using the correctly modeled load for the application. Even if losses are not apparent from the modeling exercise itself, they can be separately calculated and factored into the design.</p>
<h3>E. Other Applications of Stubs</h3>Element loading is not the only possible application for modeled transmission line stubs. We can also combine stubs with normally terminated transmission lines to explore the properties of the combined array. A common application in this vein is the matching-section-and-stub system used to match an antenna's odd source impedance to a standard feedline.
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<img src="ltl4-4.gif" width="441" height="338" border="2" vspace="5" hspace="9">
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<p>Consider the following example, corresponding to the sketch. We have an extended double Zepp antenna with a design center frequency of 28.5 MHz which we wish to feed with 50-ohm coax. Can we design a match and stub system to do the job? If we can design the system, what will be the 2:1 SWR operating bandwidth of the resulting antenna and match system?</p>
<p>The antenna model consists of one wire, 44.2' (13.48 m) long, with 31 segments. We shall place this antenna at a height of 35' (1 wavelength) above medium ground conditions. The program returns a gain of 10.76 dBi and a source impedance of 135.3 -j680.1 ohms. The source impedance becomes the starting point for our match-and-stub system. We shall use 450-ohm, 0.95 VF parallel line for the transition to our 50-ohm coax.</p>
<p>Because transmission lines are impedance transformers, we can find 2 points along each half wavelength of the initial 450-ohm line at which a properly sized stub (either open or shorted) will yield an impedance of 50 +/- j0 ohms. (The limiting case is where the impedance along the line never reaches 50 ohms, in which case, we must select a different Zo line for the match and stub system.) We can use either a Smith Chart or a utility program, such as the one included in the <a href="../radio.html#HAMCALC">HAMCALC</a> collection, to simplify the calculation process.</p>
<p>For the case in hand, we find that lengths of A. 4.983' and B. 5.441' are able to support stubs for the transformation. At point A., a shorted stub 1.176' long or an open stub 9.373' long will do the job. At point B., a shorted stub 12.216' long or an open stub 7.020' long will also do the job.</p>
<p>Ordinarily, we select the combination of matching section and stub that is shortest. In this example, the 4.983' + 1.176' stub combination is the most compact.</p>
<p>To place these lines in the model, we create a second wire, 0.02' long at a height of about 30' up. The exact height is not critical, since we shall control the length of the match line with the TL entry. If we must manually create the distant wire for the termination of the shorted stub, we do so at this time.</p>
<p>Then we enter two transmission lines, both 450 ohms and 0.95 VF. The first goes between wire 1, segment 16 and wire 2, segment 1. The length is 4.983 feet. The second is a shorted stub from wire 2 segment 1, with a length of 1.176 feet. (Adjust lengths for a VF of 1.0 if directly entering values into the TL line: 1.599 m for the match line and 0.378 m for the stub.) Be certain to move the source from the center of wire 1 to wire 2, segment 1.</p>
<p>Running this revised model yields at 28.5 MHz a source impedance of 49.97 - j0.01 ohms, showing the coincidence between the external calculations and our modeling. The gain, of course, does not change. However, once we have established the fact that our external calculations and our modeling work coincide, is there any use to installing the transmission lines in our model?</p>
<p>The answer is affirmative, for we can use this installation to determine the anticipated 2:1 SWR operating bandwidth of the antenna. If we perform a frequency sweep across the first MHz of 10 meters, we obtain the following data:</p>
<pre> Frequency Source Impedance 50-Ohm SWR
in MHz R +/- jX Ohms
28.0 111.0 - j26.13 2.37
28.1 93.43 - j22.74 2.02
28.2 79.03 - j17.74 1.71
28.3 67.30 - j11.98 1.43
28.4 57.75 - j 5.97 1.20
28.5 49.97 - j 0.01 1.00
28.6 43.57 + j 5.74 1.20
28.7 38.29 + j11.21 1.45
28.8 33.89 + j16.36 1.73
28.9 30.22 + j21.18 2.07
29.0 27.12 + j25.70 2.47
</pre>
<p>The data tell us that we are likely to be able to operate the antenna without equipment indisposition over at least 0.8 MHz bandwidth. Moreover, using the curve as a guide, we are likely to be able to make field adjustments of the actual match and stub system to bring it into line with the portion of the band we most wish to use. Since we are using a mere 6' (2 m) of line for the match and stub system, line losses will be negligible within it. In any event, the losses will be far less than had we designed an inductor at the feedpoint to compensate for the capacitive reactance, followed by a 2:1 transformer to bring the resistive component of the source impedance closer to 50 ohms.</p>
<p>This example, I hope, illustrates the utility of judiciously employing stubs in antenna models beyond their function to load antenna wires. I recommend that you rebuild the model using each of the other three combinations of match line and stub just for the practice in using TL entries in models (if not out of curiosity over whether those values "work").</p>
<h3>E. Physically Modeling Stubs</h3>Like regular transmission lines, we can also model stubs as physical elements within the antenna design. The advantage is that any interactions of the stubs with other parts of the antenna structure will show up in the model's output data. The disadvantage is that modeling the stubs will occupy wires and segments, thus slowing the core calculations. For complex designs, modeling all stubs can drive the model size to the program limits.
<p>In addition, we must always be aware of basic NEC limitations. Primary among these limits is the possible unreliability of results where wires of different radii (diameters) meet, especially at angular junctions. If possible, one should model such geometries using the same wire size throughout, and this may require the development of several different models to obtain all of the desired output data, such as element lengths, stub lengths, gain, and source impedance. In addition, segments at and near angular junctions should be consistent in length and short.</p>
<p>Let's sample the process by replacing the loads and transmission lines in the half-length dipole with physically modeled transmission line stubs. To do this, we must restructure the entire model. Instead of one wire, we shall need 9, in accord with the upper diagram of the following figure.</p>
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<img src="ltl4-5.gif" width="438" height="479" border="2" vspace="5" hspace="9">
</center>
<p>The wires follow the following table:</p>
<pre>Wire Conn. End 1 (x,y,z : in) Conn. End 2 (x,y,z : in)
1 -101.21, 0.000, 0.000 W2E1 -54.000, 0.000, 0.000
2 W1E2 -54.000, 0.000, 0.000 W3E1 -54.000, 0.000, 76.300
3 W2E2 -54.000, 0.000, 76.300 W4E1 -50.000, 0.000, 76.300
4 W3E2 -50.000, 0.000, 76.300 W5E1 -50.000, 0.000, 0.000
5 W4E2 -50.000, 0.000, 0.000 W6E1 50.000, 0.000, 0.000
6 W5E2 50.000, 0.000, 0.000 W7E1 50.000, 0.000, 76.300
7 W6E2 50.000, 0.000, 76.300 W8E1 54.000, 0.000, 76.300
8 W7E2 54.000, 0.000, 76.300 W9E1 54.000, 0.000, 0.000
9 W8E2 54.000, 0.000, 0.000 101.210, 0.000, 0.000</pre>
<p>The source is at the center of wire 5. Dimensions have been translated into inches for ease of seeing the modeled structure. The stubs are composed of the same #14 copper wire as the antenna element and are 4" wide by 76.3" long. The stubs are centered at the approximate points where the loads and transmissions lines had been assigned in previous models.</p>
<p>The model returns a gain of 1.47 dBi and a source impedance of 18.02 - j0.55 ohms. The gain figure is less than the 1.76 dBi returned by the earlier models. However, remember that the stub now accounts for wire resistance losses. In fact, if we add a resistance of 0.93 ohms to the reactive load of 870.4 ohms in previous models, we find that the R-X model returns the same gain figure of 1.47 dBi. 0.93 ohms resistance in series with a reactance of 870.4 ohms represents a Q of about 936, well higher than what we can achieve with real inductors.</p>
<p>The model is also instructive with respect to the length of the stub, which is considerably shorter than what a standard calculation of a shorted stub would yield. #14 wire with 4 inch spacing has an approximate Zo of 580 ohms, which would call for a length of about 130" for the stub. The actual stub is 76.3" long. However, the theoretical stub presumes equal but opposite currents in each leg. Examining the current table returned for this model will show the currents for any given position across the stub lines to be significantly different.</p>
<p>(<b>Note:</b> Modelers should be very cautious at this point with respect to the accuracy of the stub length as modeled in NEC-2 or NEC-4. Even 4" apart at 14.15 MHz, the stub wires are very close. NEC may become inaccurate with very closely spaced wires, even when the modeler has observed all cautions regarding using the same wire diameter at angular junctions and regarding the alignment of segments in the parallel wires. This same antenna was modeled in MININEC 3.13 (which has limitations of its own, but not in this area) via ELNEC, and the requisite stub lengths were 1.3 to 1.6 inches longer, depending upon the precise technique used to construct the model. Since the stub length differential approaches 2%, the variance must be considered significant.)</p>
<p>Stubs close to the feed point, where the current changes slowly as one moves along the antenna wire, will model physically in close accord with standard parallel transmission line calculations. However, for closest equality of current magnitude on the wires, they must be positioned symmetrically relative to the antenna wire, as shown in the lower left of the figure. If they are positioned linearly, the currents will be unequal due to the unequal distances from the main antenna element. Likewise, stubs positioned further out along the antenna wire, even if positioned symmetrically relative to the antenna wire, will show considerable differences of current on the two parallel wires. At a certain indefinite point along the wire, stubs act less like transmission line sections and more like simple wire extensions necessary to lengthen an antenna element to resonance.</p>
<p>The differences in currents along the two stub wires tend to be somewhat less if the stub wires are folded back or folded outward, parallel to the antenna wire and equally spaced from it. When positioned straight down, the unequal currents in the stubs yield a small vertically polarized field that in one plane is down from the main lobe by only -18 dB. It is largely the interaction of this field with the main element field that reduces the source impedance to 18 ohms from the 26 ohms found in the other models.</p>
<p>A physical model of the stub-loaded half-length dipole also provides an interesting lesson in operating band width. Relative to the resonant source impedance of 18 ohms, the SWR at 14.0 MHz is 2.50 and at 14.3 MHz is 2.30, well below the modeled figures for lossless transmission lines and lossless inductors. However, the physical model includes its losses, which increase the operating bandwidth.</p>
<p>The point of generating this sample physical model of transmission line stubs--usually called "linear loads" among antenna designers--is to illustrate some, but by no means all, of the differences between using lossless and non-interacting mathematical antenna parts and physical lossy and fully interactive antenna parts. Each provide valuable information about anticipated antenna performance and each withholds valuable information. Physical modeling is limited by certain program limitations that force the modeler to depend in part on the use of mathematical substitutes. However, only by thorough modeling in both modes can one develop a good sense for what to expect by way of field adjustment in real antennas.<br>
<br></p>
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<br>
This last point is as good as any for bringing to a close this inexhaustive but hopefully useful set of exercises in modeling with loads and transmission lines. One could multiply the examples almost endlessly to draw out innumerable subtleties regarding both antennas and models of them. However, I hope that the examples and notes in this series allow you to carry on the process on your own. There is much we can learn about antennas, and antenna modeling can go a little way toward instructing us-- even about its own limitations.<br>
<br>
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<p><i>Updated 1-3-98. &COPY; 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.</i><br>
<br></p>
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