AutoEZ:    Transmission Lines - Tuners - Stacks - Stubs

In the "Getting Started - What's New in v. 5.0" section of the EZNEC Help you'll find this paragraph:

EZNEC v. 5.0 marks the transition of EZNEC from an antenna analysis tool to an antenna system analysis tool. The new Transformer and L Network objects, in addition to the ability to include transmission line loss, extend the modeling capability for the first time from the antenna all the way back to the transmitter. Array phasing networks, tuners, impedance matching networks, along with realistic effects from transmission line loss, can now be part of the model. For the first time, the performance of the entire system can be seen.
Have you used the new EZNEC features to analyze your entire antenna system yet? If not, AutoEZ can help you get started.

Model a Complete Feed System, Step-by-Step

This example shows how to model a complete feed system for the BYDIPOLE.ez antenna. Starting at the transmitter, the model includes a Hi Pass T (C-L-C) configuration tuner, then a 1:4 step-up balun/transformer, then 100 feet of Wireman Type 553 "window" line between the balun and the antenna feedpoint, and finally the antenna wire itself. EZNEC "Virtual Wires" (aka "Virtual Segments") connect all the non-antenna portions of the feed system.

It's always a good idea to draw a sketch beforehand to avoid the inevitable "Now what gets connected to what?" questions. Here's what the feed system might look like.

Simulating the complete feed system must be done in two stages. First, model everything from the output port of the tuner (V2 in this example) all the way through the antenna. After that be sure to do a complete calculation, then add the tuner to the mix. The reason for this two-stage process will become clear shortly.

Open model BYDIPOLE.ez. Tab to the Calculate sheet and click Generate Test Cases. In the dialog window click Clear All if any variables are being set and then make appropriate entries for a frequency sweep over the 20M band.

Click OK and then Calculate All Rows. Then tab to the Custom chart sheet and plot "SWR at Src" vs "Frequency - Line". That gives you the SWR at the antenna feedpoint, just as a basis for reference. Take a snapshot and change the descriptive text to "At feedpoint". Note that when you take a snapshot the trace color will change from red to blue (or other color).

Now add the balun/transformer and the transmission line. On the Insr Objs sheet, in the Sources table, change the position of Source 1 (S1) from "Wire 1 / 50%" to "V2". You can clear the "% From E1" field, that's not used with virtual wires.

In the Transformers table define the balun/transformer like this:

Note that the relative impedance ratio numbers do not have to be "1" and "4". Entering "25" and "100" does the same thing. (For more information on this topic enter "Transformers" on the Index tab of the EZNEC Help.)

Finally, make appropriate entries in the Transmission Lines table. For the End 1 Connection enter "V3" which is the same as the port 2 connection of the transformer. For End 2, enter "1" and "50" for the "Wire #" and "% From E1" fields. That's the center of the antenna wire, originally where the source was placed. Then enter "100" in the "Len" field and click the Set Zo, VF, and Loss for Selected Row(s) button.

Did nothing happen when you clicked the button? That's because you probably forgot to complete the entry of 100 for "Len". In Excel, when you click on a cell and then type something (such as a number or a formula starting with an "=" sign), everything else is "frozen" until you complete the entry. You can do that by pressing Enter, by pressing one of the keyboard arrow keys, or by pressing the Tab key. But until you complete the entry, nothing else will happen. And that includes clicking on any buttons.

If ever AutoEZ seems to be unresponsive, look up just to the left of the Excel formula bar near the top of the window. If you see a red X and green checkmark there, or if you look down in the lower-left corner of the Excel window at the Excel status bar and see "Enter" there, it means that Excel is waiting for you to complete the entry. In easy-to-remember graphical terms:

So now you've completed the entry of 100 in the "Len" field, clicked the button, and selected Wireman 553 from the drop-down list of 100 available standard line types. AutoEZ will fill in the rest of the fields for this line. Here's what the results should look like:

Note that the characteristic impedance for Wireman 553 at 14.35 MHz is well below the "nominal" value of 450 ohms. That's not an error. AutoEZ makes detailed calculations to account for the fact that Zo and VF change as the frequency changes, as well as the expected change in loss (dB/100 ft). Also, concerning the line length, you may have noticed that when you entered "100" the value was displayed as "100.000". That's because the "Len" cells in the Transmission Lines table follow the same "number of decimals" formatting as the XYZ coordinate cells on the Wires sheet. (As of v. 2.0.16 that is no longer the case.)

Important: At this point tab to the Calculate sheet and do a calculation of the model as built so far.

This is the reason for building the complete feed system in two stages. Doing a calculation at this point will make available the impedance values at port 1 of the transformer. That's the same place where the output port of the tuner will be. When you create an impedance matching network, like a tuner, AutoEZ has to "know" what the impedance value is that you want to "match to" 50 ohms. That impedance (RjX) must be available on the Calculate sheet before the matching network is built. Otherwise AutoEZ has no way of calculating what component values for capacitors and inductors are needed in the matching network.

Now you are ready to create the Hi Pass T network. On the Insr Objs sheet, in the L Networks table, click the Create Impedance Matching Network button. Enter "V1" and "V2" for the network input and output terminals, and select Hi Pass T as the network type. In the lower section of the dialog choose the "Single R/X Pair" option. (The other option will be covered in a later example.) Assuming you want to match to the middle of the band use the spinner to select 14.175 MHz. Click OK.

AutoEZ can automatically create Hi Pass T, Lo Pass Pi, and Hi and Lo Pass L impedance matching networks, retrieving the most recently calculated RjX values from the Calculate sheet and then determining the appropriate values for the L and C components of the network.

In the case of Hi Pass T and Lo Pass Pi networks, the network actually consists of two back-to-back L networks. AutoEZ will automatically choose the correct "direction" for each L network (that is, place Port 1 on the input or output side depending on which side has the series component) and will automatically create an L net to L net connection via an unused Virtual Wire number.

Here's what you'll see in the L Networks table (or similar, depending on the current Options settings for capacitor and inductor Q, covered in the next section):

Note that comments may be entered in the various tables on the Insr Objs sheet, just like on the Wires sheet. In this case AutoEZ has created a two-line comment and two back-to-back L networks connected via Virtual Wire V900. The first comment line shows the component values for the Hi Pass T. Note that the specified inductance of 0.462981 H has been spread across the two inductors of the two separate L networks. The two inductors are in parallel so placing 0.925962 H on each yields a parallel combination of 0.462981 H. This is done to provide the greatest accuracy possible when using the EZNEC mixed-precision calculating engine. (For more details on this subject see the "L Network Limitations" section of the EZNEC Help.) The second comment line shows which cells are used for the input and output sides of the Hi Pass T. It's very easy to get these confused.

You'll also see a reminder asking if you want to change the source position, something that you may forget to do after the network is built.

Now tab to the Calculate sheet and do a full calculation again. Remember, you have added the matching network to the model and you have changed the source position, but you must now recalculate in order to show the new results.

After recalculating tab to the Custom sheet. Now you can compare the original "at feedpoint" SWR curve (was red, now blue) to the "with matching network tuned to 14.175 MHz" curve (red).

Take a second snapshot and then tab back to the Insr Objs sheet. This last step will demonstrate one additional AutoEZ feature.

Suppose, merely for illustration, that your interest lies in just the upper part of the 20M band. Click the Create Impedance Matching Network button again. You'll see this prompt.

The R/X values that are now on the Calculate sheet represent the input (matched) side of the current matching network, hence they are not suitable for use in creating a new network. However, AutoEZ has saved a copy of the R/X values that existed before a matching network was added.

You can "match to" a different frequency, or even pick a different type of network, as long as nothing has changed in the model which would affect the impedance as seen at the output port of the network. If that is not the case then it will be necessary to repeat the "stage 1" steps as described above so that a new set of R/X values will be available on the Calculate sheet.

In this example you have not changed the length of the antenna wire or its height above ground, you have not changed the length or type of transmission line, you have not changed the impedance ratio of the transformer, and you have not changed the range of frequencies. As long as all those conditions are met you can rebuild different matching networks just by clicking the Create Impedance Matching Network button again. This is a convenient way to experiment with different networks.

This time spin to 14.250 MHz. When you click OK notice that the new network has slightly different component values for the inductor and C2 capacitor.

As before, do a recalculation and then tab to the Custom sheet.

Now you have a lower SWR over the upper portion of the band, simulating what would happen if you "set the knobs" of your tuner for a perfect match at 14.250 MHz.

That's it for this example. You can erase any snapshots on the Custom chart sheet. If in the future you want to review the process of modeling a complete feed system you can open "Complete Feed System.weq". That model shows the end result of the above steps.

Simulate an Auto-Tuner

Auto-Tuners have become very popular in recent years, both built-in with commercial gear and as separate products offered by companies such as ICOM, Palstar, SGC, LDG, MFJ and others. Most of them are in the form of a Low Pass L network with the capacitor bank automatically switched from the input to output side depending on the load to be matched.

In the previous example you saw how to simulate a Hi Pass T (C-L-C) type tuner. Let's expand on that example to simulate an Auto-Tuner, and let's expand the frequency range to cover all bands from 20M to 10M (14 to 30 MHz).

If you worked through the previous example you can pick up where you left off. If not, open the "Complete Feed System.weq" sample model. In either case make the following changes.

  1. On the Wires sheet use the AutoSeg button to set the number of segments per wavelength to 20. That way as the frequency changes from 14 to 30 MHz the number of segments will change but the segmentation density will stay the same and you will avoid the EZNEC segmentation warning.

  2. On the Insr Objs sheet change the source position from "V1" back to "V2", the port 1 side of the balun/transformer. Clear the existing matching network.

  3. On the Calculate sheet use the Generate Test Cases button to create a frequency sweep from 14 to 30 MHz with 0.1 MHz steps.

Calculate and tab to the Custom sheet. Since the source has been placed where the output port of the "to be built" auto-tuner will be, what you see are the SWR values that the tuner will (try to) bring down to 1. (Band highlighting added for illustration.)

Another way of looking at things is on the Smith sheet. The job of the auto-tuner will be to move all the points to the center of the chart.

Back on the Insr Objs sheet click Create Impedance Matching Network. In the dialog window set the input to "V1", the output to "V2", and choose "Lo Pass L" as the Network Type. In the "R/X to be Matched" frame select the "All R/X pairs" option. Don't click OK just yet. Instead, click the Options... button.

First of all, notice that the AutoEZ Lo Pass L topology is the same as a typical auto-tuner. The inductor is in series between the input and output and the capacitor will be in parallel (shunt) either on the input or output side depending on the load.

Second, in the "Impedance Match Options" window change the "Round all Capacitor" and "Round all Inductor" options as shown below. The "No Rounding" choices allow you to simulate traditional variable capacitors and roller inductors, where the amount of capacitance or inductance is fully and continuously adjustable. On the other hand, the "To Nearest" choices allow you to simulate a typical auto-tuner, where the amount of capacitance or inductance varies in discrete steps.

Different companies use different capacitor and inductor step sizes in the capacitor bank and inductor bank. In some cases the step size is stated in the user manual, in other cases you'll have to study the schematic diagram for your particular model. The values shown are typical but the step size for capacitors in particular can vary from 10 pF to 50 pF.

The Qu(C) and Qu(L) settings determine the amount of series resistance that will be added to capacitors and inductors to simulate component loss, using the formula R = |X| / Q where X is the component reactance at any given calculation frequency. Q is typically (although not always) inversely proportional to frequency for capacitors and directly proportional to the square root of frequency for inductors.

If the Reference Freq is greater than zero, the frequency dependencies are:

Qu(C) at Calc Freq = Qu(C) at Ref Freq * Ref Freq / Calc Freq
Qu(L) at Calc Freq = Qu(L) at Ref Freq * SQRT(Calc Freq / Ref Freq)
Using the values shown in the illustration above, at 14 MHz the effective Q values would be:
Qu(C) at 14 MHz = 1000 * 1 / 14 = 71.4    (well below 1000)
Qu(L) at 14 MHz = 200 * SQRT(14 / 1) = 748    (well above 200)
On the other hand, if the Reference Freq is equal to zero the Q values will stay constant over all frequencies. That is:
Qu(C) at 14 MHz = 1000    (and 1000 at all other frequencies)
Qu(L) at 14 MHz = 200    (and 200 at all other frequencies)

After you click OK on both dialog windows AutoEZ will build the required entries in the L Networks table. This time the entries are a little more complicated. Because the component values will change for each frequency they are set via variables K and L rather than using fixed capacitor and inductor values. And because the load impedance (at the tuner output port) varies over such a wide range it will be necessary to use a "Series Input" network configuration for some test cases and a "Shunt Input" configuration for others. The choice of which topology to use for any given test case is controlled via a third variable M. This mimics the action of the auto-tuner logic circuits, where for some load conditions the capacitor bank is switched to the output side of the inductor bank and for other load conditions the capacitor bank is switched to the input side.

If you tab to the Variables sheet you can see that three variables K, L, and M have been automatically added. The initial values shown represent the conditions needed for a match at the first test case frequency, 14 MHz.

Variable M is used in the cell formulas in column A of the network table to "mark out" one network or the other.

For the first test case at 14 MHz a "Series Input" network is needed so the "Shunt Input" network defined in the second four rows of the table is "marked out" and will not be included in the temporary model file sent to EZNEC. For other test cases a "Shunt Input" network is needed so the "Series Input" network defined in the first four rows of the table will be "marked out".

For the "Shunt Input" (M=2) case note that the input is Port 2, not Port 1. When you define an L network for EZNEC the series branch (Port 1) is always the first row of the two-row definition and the shunt branch (Port 2) is always the second row. Hence for a shunt input type network the input is the second row of the two-row definition, the Port 2 branch. This can cause a lot of confusion so AutoEZ explicitly states in the comments which row in the table is the input and which row is the output.

On the Calculate sheet you can see that not only are the capacitor and inductor values set via variables K and L but also variable M is set for each test case row to control the network topology.

Moment of truth. Calculate all rows then tab to the Custom chart sheet.

Not perfect but not bad. Although the new SWR is not 1:1 across the entire frequency range it is perfectly acceptable in the ham bands.

On the Smith chart:

Again, although not all the points are in the center of the chart, almost all of them are well within the green 2:1 SWR circle.

Why isn't the match "perfect" at all frequencies?

It has to do with the "step sizes" of the capacitor pF values and the inductor H values. If the components were "infinitely adjustable" as would be the case with a variable capacitor and a roller inductor it would be possible to get a perfect match at every frequency. But in an auto-tuner the capacitor and inductor values can only be changed in discrete steps.

Recall from the previous example that you can experiment with different tuner configurations without repeating all the preliminary set-up steps. So return to the Insr Objs sheet, click Create Impedance Matching Network again, answer "Yes" to the "Use Saved R/X?" prompt, then click the Options... button. This time set the capacitor stepping level to 10 pF (was 25) and the inductor to 0.05 H (was 0.125).

Calculate and tab to the Smith sheet. Now all the points have an SWR of 1.5 or less. (Note the size of the green SWR circle has been changed.)

If you revert all the way back to "No Rounding" (that is, infinitely adjustable components) you would see a perfect match at every frequency but that would no longer be a realistic simulation of an auto-tuner.

Transmission Line Matching Networks

Suppose you intended to use the dipole only on the 20 meter band. You probably wouldn't feed it with ladder line in that case, and you would prefer to not use a tuner. What other alternatives do you have for reducing the SWR at the rig?

Open model BYDIPOLE.ez again to start with a fresh copy. Calculate the single test case row at the initial frequency of 14 MHz. The source impedance, at the center of Wire 1, is approximately 79-j45 ohms. Obviously the antenna is not resonant at 14 MHz so first let's find at what frequency it is resonant.

Make sure cell B11 is still selected and then click the Resonate on Selected Cell button. You'll see that at the current length of 33.43 ft and the current height of 30 ft the antenna is resonant at 14.464 MHz and also has a slightly higher radiation resistance.

Now let's make the antenna resonant at the center of the 20 meter band, 14.175 MHz. On the Wires sheet, enter the formula "=L" in cell F11. Since variable L has not been given a specific value yet, and since all variables have an initial value of 0, the wire now has an End 2 Y coordinate of 0 and hence a length of 0. We'll remedy that shortly.

On the Calculate sheet, change the frequency in cell B11 to 14.175. Enter L in cell C10 and give L an initial value of 33.43 in cell C11. Make sure cell C11 is still selected and then click the Resonate on Selected Cell button.

At the new length of about 34.1 ft the antenna is now resonant at 14.175 MHz. The value for variable L has been set on the Variables sheet and will retain its current value until it is manually reset or used again in a test case row.

Now let's run a calculation over the entire 20 meter band. Click the Generate Test Cases button. You'll see a warning message.

You can click Yes. The value for variable L has already been transferred to the Variables sheet and we don't care about the other results. In the "Generate Test Cases" dialog click Clear All if any variables are being set and then make appropriate entries for the desired frequency range.

Click OK and then Calculate All Rows. Then tab to the Custom chart sheet and perform these steps:

  1. Plot "SWR at Src" (Y axis) vs "Frequency - Line" (X axis).
  2. Click the Snapshots button. In the dialog click the first Take Snapshot button and then the Hide box. Close the dialog.

  1. Change the Y axis selection to plot "Slice Max Gain".
  2. Again click the Snapshots button. This time click the second Take Snapshot button and the Hide box. Close the dialog.
  3. Change the Y axis selection to re-plot "SWR".

Why you did all those steps will become clear in a moment. For now, let's see how to create a different kind of matching network. Tab to the Insr Objs sheet. In the Transmission Lines table click the Create T Line Matching Network button.

A busy-looking dialog window will appear but most of the fields will be pre-set to the previously-used choices. In this example we're going to create a Series Section matching network using an algorithm developed by Frank Regier, OD5CG. The main feedline from the station is RG-8X and the matching section is RG-59/U. After you've chosen which R/X pair is to be matched you can click the Calculate Lengths button. AutoEZ will determine the precise lengths for the final line segment and the series section segment, including line loss in the calculations.

There will always be two solutions possible, denoted by Option A and Option B. Use whichever best meets your needs. In this example it is assumed that the total line length needed to reach from the antenna feedpoint back to the station is 100 ft. The Option A solution gives line lengths (~5.8 ft and ~4 ft) which total ~9.8 ft so 90.2 ft is the remaining length needed for the run back to the station.

If a series section solution is not feasible you can also experiment with solutions using a shunt stub.

For now, stay with the series section and click OK. AutoEZ will make all the necessary entries in the Transmission Lines table and will also offer to change the source from the existing "Wire 1 50%" position to the network input of V1 which is what you entered in the dialog.

Note: You may notice that the line lengths are displayed with a different number of decimals than are shown in the illustration above. That's because the "Len" cells in the Transmission Lines table follow the same "number of decimals" formatting as the XYZ coordinate cells on the Wires sheet. (As of v. 2.0.16 that is no longer the case.)

In a previous example, when AutoEZ computed the component values for a Hi Pass T impedance matching network, the goal was to be very close to 50+j0 ohms on the input side of the network. That is not the case with transmission line matching networks like a Regier series section. Here the goal is to have a perfect match to the main transmission line. Only if the main line has a characteristic impedance (Zo) of exactly 50+j0 ohms, which never happens for real lines, would the SWR(50) be 1:1. In this case the SWR(50) is 1.016 but the main line is perfectly matched, hence will have minimum loss.

Now let's see what we've got. On the Calculate sheet click Calculate All Rows and then tab to the Custom chart sheet. Plot "SWR" vs "Frequency" (probably the way you left the plot choices), click the Snapshots button, and "Unhide" the Snapshot #1 trace.

Click the Snapshots button and Erase Snapshot for the Snapshot #1 trace. Close the dialog. Change the Y axis selection to plot "Slice Max Gain". Click the Snapshots button and "Unhide" the Snapshot #2 trace. The difference between the two traces is the total loss of the complete transmission line, including the ~5.8 ft and ~4 ft sections at the antenna end which do the actual impedance matching.

Erase the Snapshot #2 trace.

Phasing Lines for a 4 Square Array

Enough with impedance matching. Another way you can make use of the new EZNEC features is to model array phasing networks, either with transmission lines only or with a combination of transmission lines and phase shift L networks.

Open model "4Square TL Separate Sources.weq". This model has four vertical elements placed in a square with sides of length λ/4 at 7.150 MHz. All elements are fed with equal amplitude currents. Element 4 is 180 out of phase with element 1. Elements 2 and 3 are in phase with each other and lag element 1 by 90.

This model and its feed system (in a slightly modified format) are described in detail in the "Phased Array Design Examples" section of Chapter 8 of the 21st and later editions of The ARRL Antenna Book. (The AutoEZ version has been modified to use virtual segments and with the transmission lines reordered for clarity. Otherwise it is exactly like the Antenna Book version, as created by W7EL.) The diagram below shows the first step in the design process, placing a source at each element feedpoint in order to determine the feedpoint impedance when all elements are being fed at the proper amplitude and phase.

Adapted from The ARRL Antenna Book.

Do a calculation and then record the feedpoint impedance for Src1, Src2, and Src4. (You can skip Src3 since it is identical to Src2.) Note that when a model contains multiple sources AutoEZ will show a dropdown which can be used to select the desired source. Also, tab to the Patterns sheet and take a snapshot of the radiation pattern.

In this example we'll perform the steps needed to design an "all transmission line" feed system, also called a "simplest" feed system. In addition we'll see how transmission line loss affects the results. The feed system will be designed using the Arrayfeed1 tool developed by W7EL. Arrayfeed1 is included on the Antenna Book CD and is also available for download from the EZNEC web site.

The four separate sources of the initial model will be replaced with a single source, S1, at the main feedpoint. Two phasing lines (T1 and T2 in the diagram below) will be used to provide the 90 lag to junction V3 in reference to V2. From the V2 junction, element 1 will be fed via a λ/4 "current forcing" line (T3). Element 4 will be fed with a similar λ/4 "current forcing" line plus an additional λ/2 section to provide the required 180 phase shift (T4). Since elements 2 and 3 are in phase with each other they are both fed with identical λ/4 "current forcing" lines (T5 and T6). (The concept of "current forcing" is explained in detail in Chapter 8 of the Antenna Book.)

Adapted from The ARRL Antenna Book.

Since V3 lags V2 by 90 you might think you can just use a length for T2 that is 90 longer than T1 but that is not the case. If you did that the pattern would look like this.

Instead, you need to make T1 and T2 special lengths and that's where Arrayfeed1 comes into play.

In this example all transmission lines will be Belden 9258 RG-8X. At 7.15 MHz the Zo for this type of line is 50.8 ohms and the VF is 0.807. That information and the feedpoint impedances recorded above are used as input to Arrayfeed1.

Now it's time to replace the four separate sources with one common source and add the transmission lines. Initially the lines will have zero loss. To save you the trouble of transcribing all the information you can just open model "4Square TL With Feed System.weq".

Calculate and tab to the Patterns sheet. You'll see the blue snapshot trace that you captured earlier but what happened to the red trace for the calculation you just did?

It's "underneath" the snapshot trace. You can see it by clicking the Snapshots button and "Hiding" the snapshot trace. "Unhide" the snapshot to confirm that the radiation pattern produced by the model with transmission lines is identical to the model where each element was fed directly with the required current magnitude and phase.

You can further verify proper operation via the Currents chart sheet. Click the small All button to show all four wires of the model, then click in turn on segment 1 for each wire. (Segment 1 is where the transmission lines connect. It is also the segment on each wire with the highest current magnitude.) As you click on each segment dot a small "Info Box" will appear showing the Wire (W) number and Segment (S) number along with the magnitude and phase of the current on that segment.

As expected, the current magnitudes are all the same. Note that the absolute value doesn't matter as long as the currents are identical. If you changed the amplitude of the single source from 1 amp to an arbitrary 10 amps and recalculated, the currents at the base of each element would increase by a factor of 10 but would still be identical.

The phases also have the expected differences. At the base of wire 1 the phase is -140.7. Wires 2 and 3 should lag that by 90 and they do for all practical purposes. (-140.7 - 90 = -230.7, plus 360 to "wrap around" = 129.3) Just like with the magnitudes, if you changed the phase of the single source from 0 to an arbitrary 140.72 and recalculated, the phases at the base of each element would change but the phase differences would remain the same.

Now let's see how things change when loss is added to the transmission lines. In the Transmission Lines table select cells in any column for all rows except the comment row. Then click the Set Zo, VF, and Loss for Selected Row(s) button and choose Belden 9258 RG-8X from the list of standard lines.

Recalculate and tab to the Patterns sheet. You'll see that the max gain is down by 0.77 dB due to the loss in the transmission lines. That's to be expected.

But look at the rear lobe. If you move the green marker to 225 and then switch it back and forth between the red trace (-27.26 dBi, with loss) and blue trace (-22.17 dBi, without loss) you'll see that not only has the shape changed but in this case the Front/Back ratio is actually better by 4.32 dB (-30.02 dB vs -25.70 dB). Shouldn't the use of transmission lines with loss just "shrink" the previous pattern without changing the shape?

No. The voltage/current as well as impedance transformations along a line with loss are slightly different compared to a line without loss. And that leads to differences in the current amplitude and phase ratios at the base of each element as shown below. And because the currents in the four elements no longer have the same relationships the pattern shape changes.

So let's try adjusting the lengths for the two phasing lines T1 and T2. In the Transmission Lines table replace the T1 length 7.491 with the formula "=A". Replace the T2 length 50.797 with the formula "=B". Then set up a series of test cases like the example shown below.

Calculate. On the Patterns sheet set the "Outer Ring is Frozen at" box to -10 dBi to magnify the shape of the rear lobe. (In the animation below, look in the lower right corner to see the A and B values for each frame.)

(Press Esc to stop the animation, F5 to restart.)

Notice that the Max Gain varies only slightly, from 2.77 dBi to 2.69 dBi. As W7EL states in the "Practical Aspects of Phased Array Design" section of Chapter 8 of the Antenna Book:

"If a phased array is constructed only to achieve forward gain, adjusting it is seldom worthwhile. This is because the forward gain of most arrays is quite insensitive to either the magnitude or phase of the relative currents flowing in the elements."
However, the Fr/Back ratio varies from over 40 dB to under 20 dB for just a 3 ft change in the length of T2.

Of course you can also experiment with changing variable A for the length of the T1 line. But don't let yourself get too carried away if you intend to use the array across the entire 40 meter band. Resetting the T1 and T2 lengths to the original values of 7.491 and 50.797 and then running a frequency sweep from 7.0 to 7.3 MHz shows that the pattern and Fr/Back ratio vary considerably across the band. (In the animation below you can read the frequency in the lower right corner. The outer ring has been frozen at 2.88 dBi. The blue snapshot trace has been deleted since it applied only to 7.150 MHz.)

(Press Esc to stop the animation, F5 to restart.)

Since you just did a frequency sweep we might as well take a look at the SWR across the band. Tab to the Custom chart sheet, plot "SWR at Src" (which is V1, the main feed point for the array), and take a snapshot. Then back in the Transmission Lines table click the Create T Line Matching Network button.

You'll find that a Series Section (Regier) match is not practical since it requires a non-standard line for the middle segment. But a stub match using RG-8X will work.

Note that the input is set to V10. There's no need to have virtual wires be in any particular numerical order. Any unused V number between V1 and V999 will do.

A bit of experimenting with the Calculate Lengths button shows that the least amount of line is used with "Option B" for the Match Line Segment along with an "Open" stub. Since the match line is ~8.5 ft long that leaves 41.5 ft for the main feedline if we assume a total run of 50 ft is needed to reach back to the station.

Recalculate and return to the Custom chart sheet.

That should keep your transmitter happy. Of course the complete Transmission Lines table now looks like this.

That's almost 300 ft of RG-8X. Hope you got a good price from your supplier.

2 Meter Yagi, Stack and Feed

Special lengths for transmission lines aren't restricted to just vertical arrays. Another example is feeding a stack of Yagis. Let's see what it takes to first build and then feed a 2x2 Yagi stack for 2 meters.

Open sample model file "W5TX 2M5WB375.ez". This is a 2 meter wide band design by W5TX. All elements are 0.375" diameter. The initial model is positioned 10 ft (120", about 1.5 λ at 146 MHz) above Real/Average ground.

Do an initial calculation then tab to the Patterns sheet and take a snapshot to use as a baseline reference. Note that the Max Gain is 15.29 dBi at 9 elevation.

We're going to build a 2x2 rectangular stack with variable spacing in both the vertical and horizontal directions so let's use variable names V for vertical and H for horizontal. Tab to the Variables sheet. Note that λ is about 20". Let's start with a very small λ/8 as the initial vertical separation so enter 10 as the starting value for V. We can't make the initial horizontal separation that small or the elements would overlap, so enter 50 as the starting value for H.

Of course a vertical separation of λ/8 and a horizontal separation of 5λ/8 are much too small but we'll expand from there. What makes these initial values too small? It has to do with the "capture area" of the antenna. Ian White, GM3SEK, well-known author of the "In Practice" series in the RadCom magazine, has written a very nice overview explanation of capture area.

On the Wires sheet click the Move/Copy button. Create a copy of all wires using "=V" as the Z axis delta. Don't yet create a second set of two antennas with a horizontal separation, that will come later. On the Insr Objs sheet add a source for new wire 7 using the same amplitude and phase as the original source. (As with almost all Yagi stacks, all antennas will be fed in phase.) Then on the Calculate sheet set up a series of test cases as shown below.

Calculate and tab to the Patterns sheet. In the animation below the outer ring has been frozen at 18.02 dBi to allow easy comparison of the patterns. Look in the lower right corner to see the V value for each frame.

(Press Esc to stop the animation, F5 to restart.)

The vertical separation which yields the largest gain increase is 70" (~7λ/8) with a Max Gain of 18.02 dBi, 2.73 dB greater than a single antenna and at 2 lower elevation. But there's a price to be paid for that increased gain, larger back lobes. (Below, the outer ring is set at 0 dBi to magnify the back lobes.)

Let's assume you're willing to make that trade-off. On the Variables sheet set the value for V to 70. Back on the Calculate sheet click the Clear All button, change the Plot Type to Azimuth at 7 elevation, and calculate the single row. On the Patterns sheet click the Snapshots button, erase the elevation slice snapshot, and take a new snapshot of the dual antenna vertical stack azimuth trace.

Now it's time to add two more antennas with a horizontal offset. On the Wires sheet click Move/Copy again but this time create a copy of all wires using "=H" as the Y (not Z) axis delta. On the Insr Objs sheet add two more sources for new wires 12 and 17. On the Calculate sheet set up a new series of test cases.

Calculate and tab to the Patterns sheet. In the animation below the outer ring has been frozen at 21.11 dBi. Look in the lower right corner to see the H value for each frame.

(Press Esc to stop the animation, F5 to restart.)

This time it's harder to justify the last few tenths of a dB in gain given the large sidelobes that develop. Let's compromise at 70" spacing (~7λ/8) which gives a Max Gain of 20.76 dBi, down 0.35 dB from the max at 100" but with a much cleaner pattern. Click Clear All on the Calculate sheet. On the Variables sheet set the value for H to 70, same as V.

Now let's build the transmission line system which will feed this array in place of the four separate sources. There are several different ways to feed Yagi stacks. Martin Steyer, DK7ZB, has created two very complete explanatory pages, Stacking with Coax and Stacking with Power Splitters. In this example we'll use 50 ohm transmission lines for the entire system including two back-to-back λ/4 sections serving as Quarter-Wave Transformers, also sometimes called Q-sections. All lines will be Times Microwave LMR-400.

How long should the lines be? The two Quarter-Wave Transformer sections are obviously λ/4 for LMR-400 at 146 MHz, about 17". More on that in a moment. For T3-T6 let's assume 24" coiled to make a choke balun at the antenna feedpoint, then 13" to reach the center of the boom, then 35" down (or up) the vertical support member, then 18" (35 minus 17) along the horizontal support to reach the outer end of the Q-section. That's 90".

Open model "W5TX 2M5WB375 Stack With Feed.weq".

To get the exact length for the Quarter-Wave Transformers you can use the Conversions button located below the Transmission Lines table. If you select a row in the table that already has entries for Frequency and Velocity Factor before clicking the button those two fields will be set in the dialog, but you are free to change any of the entry fields as desired.

Calculate and note the new Max Gain of 20.61 dBi (compared to 20.76 without transmission lines) and the SWR(50) at V1 of 1.102. Looks like we're good to go.

Or are we? To insure that all four antennas are fed in phase you must make lines T3-T6 identical. That's easy to do. And it's easy to make both Q-sections identical as well. But it would be tough to make each Q-section precisely 17.16 inches long. And what if the VF is not really 0.849? A bit of experimenting with the Conversions dialog shows that if the VF were 0.82 then λ/4 would be about 16.6". If the VF were 0.89 then λ/4 would be 18.0". Would it make much difference? Let's find out.

In the Transmission Lines table replace the T1 and T2 lengths of 17.16 with the formula "=A". Then set up a series of test cases on the Calculate sheet.

Calculate. Very little change in R/X/SWR at V1 and no perceptible change in the pattern from one end of the test case range to the other. If you tab to the Currents sheet, select All wires, and spin through the test cases you'll see that both the magnitudes and phases at the antenna feedpoints (Segment 6 on Wires 2/7/12/17, circled below) are all almost identical and change very little across the test case range.

So there's no need to lose any sleep worrying that the Q-sections aren't precise to a half-millimeter.

Stubs for Harmonic Suppression

So far the discussion has been about limiting transmission line losses and/or getting the best impedance match at the rig. But what if you wanted to do just the opposite? What if you wanted to get as much loss as possible by using a stub to attenuate the harmonics of your rig?

Open our old friend BYDIPOLE.ez again. For this example let's assume your transmission line is 50 ft of Belden 8267 RG-213. You want to use a stub to suppress the second harmonic at 28 MHz. How long should the stub be? Shorted or open? Should it be placed right at the rig output, at the antenna feedpoint, or somewhere in between? Let's find out.

On the Wires sheet change the number of segments from 11 to 21. On the Insr Objs sheet change the source to V1 and create a transmission line running from V1 to Wire 1/50% with a length of 50 ft. Use the Set Zo... button and pick Belden 8267 RG-213.

On the Calculate sheet create a second test case at 28 MHz and then calculate both rows. Note that the Max Gain at 28 MHz (1.78 dBi) is down only 4.62 dB from the Max Gain at 14 MHz (6.40 dBi). Tab to the Patterns sheet. Take a snapshot of the 14 MHz trace then spin to the second test case at 28 MHz.

Before moving on take a snapshot of the second trace (red, 28 MHz) as well.

Now let's experiment with a stub. What you want is a stub that will "short out" the transmission line at 28 MHz but at the same time have little effect at 14 MHz. A stub of length λ/2 (half-wave) at 28 MHz, shorted at the far end, would create (almost) a dead short at the near end. But at 14 MHz the same stub would be λ/4 (quarter-wave) and hence would be (almost) an open circuit at the near end. When placed across (in parallel with) the main feedline it would have no effect. So that answers the length and termination questions.

What about the placement? Rather than rely on something you might have read in a book or magazine let's do a test to find out if the placement matters. In the Transmission Lines table create a second and third transmission line, both as the same Belden 8267 RG-213. The first two lines (T1 and T2) will be combined such that we can "slide" the stub position along the total length of 50 ft. The third line T3 will be the shorted stub.

Note that the length for T1 is set via the formula "=A". The length for T2 is "=50-A". Hence if variable A has a value of 10, for example, T1 will be 10 ft long, the junction V2 where the stub is positioned will be at 10 ft, and T2 will be 40 ft to make the remainder of the run to the antenna. Enter a descriptive comment on the Variables sheet for A.

To calculate the length of the stub use the Conversions button. Change the electrical length to 0.5 wavelengths. At 28 MHz with a VF of 0.658, λ/2 is 11.557 ft.

(Aside: If you're sharp-eyed you'll see that this will not be a precise λ/4 at 14 MHz. At 14 MHz the VF for Belden 8267 is 0.657, not 0.658. So at 14 MHz a λ/4 line would be 11.539 ft long. You could split the difference but you'd be wasting your time.)

Now set up a series of test cases on the Calculate sheet with A ranging from 0.5 to 49.5, all at 28 MHz. You can't go from 0 to 50. That would result in either T1 or T2 being 0 ft long and EZNEC will not accept a zero length transmission line.

Calculate and then tab to the Patterns sheet. As you spin through the test cases this is what you'll see. (The value for A is shown in the lower right corner.)


(Press Esc to stop the animation, F5 to restart.)

Yes, it really does matter where you place the stub. For example, if you select test case 19 with A at 9.5 ft (left side below) and then switch the marker back and forth between the red trace (with stub) and green trace (no stub), you'll see that the stub has reduced the gain by 9.69 dB. That's not bad, but if you move on to test case 31 with A at 15.5 ft (right side below) the gain has been reduced by 34.90 dB. A difference in attenuation of 25.21 dB just by moving the stub 6 feet.

For a different way of looking at things tab to the Custom chart sheet and plot Slice Max Gain on the Y axis vs Variable 1 (A) on the X axis.

This makes it clear that a stub positioned at 15.5 ft would be much more effective than one at 9.5 ft. Another good choice would be 4 ft although you'd be giving up a few dB of attenuation compared to 15.5. Placement at 27, 38.5, or at the antenna feedpoint (50 ft) would be even better but may not be practical.

What's going on? Why such a big difference for a relatively small change in the stub placement? It has to do with the way the impedance changes along the line. Recall that the stub will be inserted in parallel with the main transmission line and that the goal is to have as little of the signal go into the main line (to the antenna) as possible at the harmonic frequency.

You can think of the junction where the stub and main line meet as being a power divider. Since the two branches are in parallel the voltage will be the same for both, hence the division of the power will be the same as the division of the current. If we assume for simplicity that E = 1 volt then:

Note that the total current is 1 over the total impedance of the stub and line combination, which is the parallel combination of the two branches. The math gets a little tricky because in general all the impedances are complex having both resistance and reactance. However, the absolute numbers aren't really important. What matters is the percentage of the total current (and hence power) that goes into the main line:
A few simple examples using resistance-only impedances will make it clear why the power into the main line changes. Assume the impedance at the input end of the stub is 2+j0 ohms and the impedance on the main line where the stub is placed is also 2+j0 ohms. Then:
That makes sense, half the power goes into each branch. Now suppose the impedance on the main line where the stub is placed is 200+j0 ohms. Then:
That also makes sense. The impedance looking into the main line is 100 times larger than the impedance looking into the stub, so (about) 1/100 of the power goes into the main line.

Now in reality the stub does not have an impedance of 2+j0 ohms. At 28 MHz, a VF 0.658 stub that is 11.557 ft long and shorted at the load end will have an impedance of 0.614-j0.001 ohms at the input end. As for the main line, right at the antenna feedpoint (Wire 1/50%) the impedance at 28 MHz is 3503+j423.7 ohms, a very high value. Moving back toward the rig by λ/4 the impedance has changed to 1.015-j0.091 ohms, a very low value. And moving back an additional λ/4 the impedance is back up to 1897+j116.9 ohms, a high value again.

Regardless of how overwhelming the math might be, from the above simple examples it seems that the connection points which permit as little power as possible to go to the antenna must be the points on the main line where the impedance is the highest. Let's verify that. To do so we'll need two pieces of information: 1) the RjX values at points along the main line, and 2) the equivalent Z magnitude for those values, which is a scalar number than can serve as a stand-in for "very high" or "very low" impedance. Z magnitude is defined as:

Z mag = SQRT(R2 + X2)

Note that for any given line at any given frequency, the RjX values along the line depend only on a) the impedance at the far (load) end of the line and b) the distance from the load back toward the source end of the line. Anything that might be connected to the line before a given point will have no effect on the RjX values at the given point.

To determine the RjX values along the line we need to create a "for testing" scenario. In the Transmission Lines table, temporarily "mark out" all three lines. (Recall that when you "mark out" an item, be it a wire or an insertion object, the item is not included in the model file that gets sent to EZNEC.) Then add a "for testing" line.

This temporary line will have the source (V1) temporarily positioned at the input end and the antenna feedpoint (Wire 1/50%) at the load end. The length is "=50-A" so if A is 10, meaning 10 ft from the rig, the line will be 40 ft long running from End 1 (the junction where the stub would be placed) to End 2 (the antenna). The reason the line has to be defined like this is because the input end has to have a source so we can get the RjX values at that point.

Perhaps a (moving) picture would be worth a thousand words. Here's what we are doing with the "for testing" scenario. Transmission lines shown with dashes are the ones that have been "marked out" temporarily.

  As A is made larger the length of the temporary line will become shorter.  

(Press Esc to stop the animation, F5 to restart.)

As A is changed from 0.5 to 49.5 under the "for testing" scenario the RjX values at the point where the stub would be placed will be shown on the results side of the Calculate sheet in columns G and H. However, there is no results column for Z magnitude, the second item of information needed.

You can define your own. Here's how.

On the Variables sheet enter a descriptive comment for an unused variable. Variable name Z is an obvious choice here. On the Calculate sheet in an unused "Variable Names and Values" column enter the formula for your variable then use the Excel "fill handle" to fill that formula down the page. Any cell references in your formula will be automatically adjusted for each test case row.

Now when you Calculate All Rows AutoEZ will, for each row, build a model with a (temporary) transmission line running from the point where the stub would be placed to the antenna feedpoint and send that model to EZNEC. EZNEC will calculate the impedance at the input end of the line because that's where the (temporary) source is. AutoEZ will read and parse the output file from EZNEC and put the R/X values in columns G/H. And Excel will use your formula to calculate the Z magnitude in column E.

Calculate and then tab to the Custom chart sheet. Change the Y axis selection to Variable 3 (your Z variable).

Recall that before we got side-tracked with all the "for testing" business, the previous Custom chart view (repeated below from earlier in this example) was:

Well what do you know. The points where the stub provides the best attenuation (4, 15.5, 27, and 38.5 ft from the rig) are exactly the points along the main line where the impedance magnitude is greatest.

You can also tab to the Smith chart sheet. The best choices for the stub position are on the right side of the circle, near 3:00 o'clock, the high impedance points. The worst choices for the stub position are on the left side of the circle, near 9:00 o'clock, the low impedance points.

Finally, now that we've determined the best place to put the stub and examined the reason behind why one place is better than another, let's take a look at the frequency response.

  1. On the Variables sheet, manually set the value for A to be 15.5. You can clear the Z variable or just ignore it.

  2. On the Insr Objs sheet, "un-mark" the original three lines and clear (delete) the "for testing" line. Remember, to clear a cell or group of cells use the Delete key. Don't use the Space bar to "blank out" the cell.

    You can experiment with different lengths for the stub by using the Conversions button again. In the example below the stub has been cut to be λ/2 at 28.35 MHz (11.414 ft) instead of the original 28 MHz (11.557 ft).

  3. On the Calculate sheet, click the Clear All button and then set up a new series of test cases with frequencies from 28 to 28.7 MHz. That is the range of second harmonic frequencies in the 10M band for primary frequencies over the entire 20M band. There is no need to set any variable values since A, the position of the stub, has been manually set to 15.5 ft already.

Calculate. On the Custom sheet change the Y axis selection back to Slice Max Gain and change the X axis selection to Frequency. Below are the results after doing a calculation with the stub in place and taking a snapshot, then "marking out" the stub and calculating again. The second calculation gives a base line to allow comparison of the Max Gain "with and without" a harmonic suppression stub.

And here are the results for the same steps but with the frequency range set as 14 to 14.350 MHz. This shows that the presence of the stub has only a very minor effect on the 20M band. (Note that the Y axis scale range is much smaller on this chart.)

Important: Don't forget to "Erase Snapshot" for the 10M frequencies or else the chart X axis will span from 14 to 28.7 MHz with a gap in the middle, probably not what you wanted. By design, snapshots persist until you manually erase them.

If your head is spinning after reading all this you might want to take a look at an explanation of stubs from a slightly different perspective. Jim Brown, K9YC, has created a very complete Some Q&A About Coax and Stubs for Your HF Station page with lots of interesting facts about coax in general and stubs in particular.