In the course of modeling, analyzing models, looking at modeling software, and sundry activities, I run across bits of information that might be useful to other modelers. Normally, I wait until I have a sufficient collection of information of a particular theme and then incorporate my scraps of information into the broader column. However, some useful items seem never to have a thematic home. So they sit forlornly on note-pad sheets, pieced by the old-fashioned spindle file that I use. Some scraps of paper have multiple holes, because I use the notes myself. Eventually, the paper begins to yellow and grow crisp around the edges. That is my warning to either use the data or lose it in a fit of housekeeping.

The notes in this column are my attempt to pass along some items that I have found useful. Perhaps the one common thread is that all of the small entries share is an involvement with numbers. None of these matters is particularly new. Instead, each matter represents something useful that we do not happen to find in common handbooks, especially those directed toward radio amateurs. A few are matters that we forget with age and distance from school. And some are items that we all too easily overlook on the first go-around and never return for a second look.

**1. Pseudo-Brewster Angles**

Chapter 3 of *The ARRL Antenna Book*
contains an excellent introduction to the effects of ground upon
antennas of various sorts. An interesting section of the treatment
concerns the pseudo-Brewster angle (PBA) as applied to the radiation
patterns of vertically polarized antennas (pp. 3-15 through 3-15 of the
20th edition). The PBA rests upon adapting optical concepts to the
reflection of radio waves by the ground as a lossy medium. Let's assume
that the radio source is a point that shines equally in all directions.
It shines both downward and upward. Therefore, radio waves (or rays) at
various angles above ground will sometimes add and sometimes subtract
from each other, creating the lobes and nulls that become familiar in
elevation patterns we draw from antenna modeling software.
Unfortunately, the current chapter does not illustrate the PBA with any
such patterns, so many folks just pass over the material without
realizing its implications. Therefore, let's approach the PBA from the
other direction.

**Fig. 1** shows the elevation patterns of a
single vertical antenna (a J-pole), but over 3 different levels of
ground quality. Very good ground has a standard definition of a
conductivity of 0.0303 S/m ad a relative permittivity (dielectric
constant) of 20. The values for average ground are 0.005 S/m and 13;
while for very poor ground, the values are 0.001 S/m and 5. The red
lines in the figure indicate an elevation region in which the lobes
diminish in strength and the nulls diminish in depth. The result is a
gain level that is close to the free-space gain of the antenna. This is
the PBA, the angle at which the incident and reflected rays neither add
nor subtract from each other.

The next notable feature of **Fig. 1**
is that the angle changes as we change the ground quality. The better
the ground quality, the lower the elevation of the PBA. The effect
occurs at all frequencies, although it has a major impact on lower HF
antenna installations. As well, the effect results largely from waves
already in the far field, so adding radials to a monopole to improve
ground quality has minimal consequence for the PBA. The figure uses
antennas a many wavelengths above ground so that the effect shows
clearly in the elevation pattern that consists of many lobes and nulls.

The *Antenna Book* account
provides a very precise and equally forbidding set of equations for
calculating the PBA. They involve calculating the complex relative
permittivity of the soil with the required frequency adjustments and
then using this calculation in a long equation. In many instances, we
only need to know the general vicinity of the PBA for the soil quality
that we estimate beneath our feet. Over dry land, the soil conductivity
plays only a small role in determining the PBA. Therefore, we may use
the relative permittivity alone to calculate an estimated PBA that is
good within perhaps +/-2 degrees of elevation over poorer soils and
within perhaps +/-1 degree for better soils, all from the mid-HF region
upward. One shortcut equation follows:

Since we are only estimating the PBA, you might
use either the arcsin or the arctan of the expression in parentheses,
where epsilon is the symbol for relative permittivity. For very good
soil, we obtain an angle of about 12.5-13 degrees. Average soil gives a
value between 15.5 and 16 degrees. Very poor soil yields 24 to 26
degrees. You may compare these values to the angles of the lines in **Fig. 1**.

The precise value of the PBA rests on
several factors, such as the source being a true point, knowing with
quite good accuracy the soil conductivity and permittivity at depths to
which radio waves penetrate, and the frequency of operation. The crude
estimate, which does not depart too far from the calculated values in
the *Antenna Book* (Chapter 3, Table 3), bypasses most of these
factors. Hence, its use is limited, mostly to partially explaining the
depression region we find in elevation angles of all vertically
polarized antennas and to setting an expectation that finds the
depression region in these radiation patterns normal rather than odd.
If your operating frequency is quite low or you are over salt water,
use the entire equation in the *Antenna Book*.

One final note: the PBA is not dependent upon antenna height. As shown in **Fig. 2**,
antennas a different heights show just about the same PBA or depression
region of the elevation pattern. However, the patterns that we
encounter for lower vertical antenna heights become ambiguous. Very
often, we cannot tell if the depression is a PBA effect or a simple
shallow null between lower-angle lobes. As we raise the antenna--in
this case to a height of 5 wavelengths--the picture grows clearer.
Without PBA, the null between the lower two lobes in the upper portion
of the figure would be deeper. Still, since we are never without PBA
when using a vertical antenna, we cannot achieve the deeper null
between lobes, at least not over dry land.

For modelers, the introduction of the crude PBA calculation should take the mystery out of elevation patterns for vertical antennas, especially those well above ground. Those who have only modeled a few such antennas may wonder whether the depression region is a software problem or a problem with their particular model. Actually, it is neither. Rather it is part of the normal propagation of far-field radiation. (Horizontal antennas do not share this effect, at least not in the same way or to the degree shown by vertical antennas.)

**2. The Velocity Factor of Coaxial Cables**

Most introductory texts for amateur
radio operators present the concept of a coaxial transmission line or
coaxial cable in an abbreviated and incomplete manner. **Fig. 3**
outlines the essential dimensions of the cable necessary for
calculating the characteristic impedance (Zo) of the line. The
dimension D is the diameter of the inner surface of the shell or braid,
also generally called the outer conductor. Dimension d is the outer
diameter of the center or inner conductor. Due to skin effect at HF and
above, current do not penetrate deeply into either conductor.
Therefore, the outer surface of the braid or shell does not plat a role
in the transmission-line function of the cable. However, it may play a
role in radiation or common-mode currents.

Between the inner and outer conductors of a coaxial transmission line, there is a space that cable makers fill with an insulating or dielectric material. The material may range from dry air to an inert gas to a foam material to a solid plastic material. The dielectric plays a role in the determination of the cable's Zo according to the relative dielectric constant or permittivity (epsilon) of the material. Therefore, the complete equation for calculating the Zo of a coaxial cable requires that we include the factor:

In most lower-level treatments, we do not find epsilon. Rather, the treatments assume the improbable case of using dry air as the dielectric. Dry air has a relative permittivity of about 1.0, and so the term drops out. In fact, most of the cables used by radio amateurs (and professionals) use a dielectric other than air.

The missing term then creates a mystery. Independently, we are told that all transmission lines have a velocity factor (VF) such that the cables electrical length is always longer than its physical length. The ratio of physical length to electrical length defines the VF value. Hence, a cable with a VF of 0.67 is electrically 1 wavelength long when the cable is physically about 2/3-wavelength long. We may acquire a sense of the source of the velocity factor's source by noting differences in the dielectric material used on cables with the same designation but using different dielectric materials.

The unknown might go away if we began with the more complete equation for calculation a cables Zo. Then we could set up the relationship between epsilon and VF:

This version of the relationship requires that we know the dielectric constant for the material used between cable conductors. In fact, cable specification tables are more likely simply to name the dielectric material and to list a velocity factor value. Therefore, we may turn the simple equation around.

Using this equation, we may easily derive the dielectric constant for the cable material. For example, one common dielectric material used in cables is called FPE or foamed polyethylene. A typical cable VF using this dielectric is 0.78. Our handy equation yields a permittivity of about 1.6. Foamed polyethylene is a mixture of solid polyethylene and trapped dry air. Therefore, our exploration of the dielectric material must end here.

Cables with solid dielectrics are
historically older and use a solid polyethylene material. A typical VF
value is 0.66. By our equation, the resulting permittivity value is
about 2.3. A table in Kraus' *Antennas* shows the value as 2.2,
very close to our calculation. (In fact, a value of 2.2 yields a VF of
0.67, another value found in some tables for older coaxial cables.)

What we learn from this primitive little exercise is that the Zo and the VF of a cable are intimately related and not simply independent facts about cables. Of course, we also need to learn that cable specification tables list nominal or typical values for a line's VF. If the length of a line is critical to a given application, such as the construction of a matching section, then the builder must determine the VF of the line to be used. The precise VF of a line varies from one manufactured batch to the next. For example, some foam lines listed as having a VF of 0.78 actually turned to have a VF in the 0.70 to 0.72 range under tests in the high HF range.

**3. Modeling a Transformer Using NT**

In episodes 95 and 96 of this series, I examined the NEC NT or network command and introduced some rudimentary applications. Since writing those notes, I have acquired another application, but not a sufficient number of others to form an entire episode. So I shall note it in this collection.

The NT or network command is less
familiar to entry-level modelers, although its special application, the
TL or transmission-line facility is very familiar. In NEC, a network is
a 2-port y-parameter admittance network, as suggested by **Fig. 4**.
The network has two ports, designated externally as end 1 and end 2.
These ends simply attach to two different segments within the same
model. For most applications, these segments require different wires,
although the command allows the ends to be different segments on the
same wire.

Internally, we must enter values for three shunt or parallel admittance values: y11, y12, and y22. See the earlier episodes for more details on the meaning of these entries. Here we wish to focus on a special application: the creation of a rudimentary impedance transformer. To create the transformer, we must know the end-1 impedance (Z1) and the end-2 impedance (Z2). As well, we must decide upon a value for the Q of the end-1 side of the transformer. In our simplification, we may ignore the imaginary components (susceptances) of each entry and simply calculate values for the real components or conductances. We must also take two simple calculation steps before creating input values for the NT command. First, we calculate the ratio of the end-1 impedance to the end-2 impedance (Zr), an easy task. Second, we calculate the transformer turns ratio (Tr), which is simply the square root of the impedance ratio. Under these conditions, we can easily calculate the required real values for the NT command.

To see how this works, let's examine a real model
before and after installation of the transformer. Suppose that we have
a resonant folded dipole, such as shown in the upper part of **Fig. 5.**. It .NEC-format model might look like the following lines.

CM fd #18 cu 1" 28.5

CE

GW 1,199,-2.513,0.,0.,2.513,0.,0.,5.119E-4

GW 2,1,2.513,0.,0.,2.513,0.,.0254,5.119E-4

GW 3,199,-2.513,0.,.0254,2.513,0.,.0254,5.119E-4

GW 4,1,-2.513,0.,0.,-2.513,0.,.0254,5.119E-4

GE 0

LD 5,1,0,0,5.7471E+7,1.

LD 5,2,0,0,5.7471E+7,1.

LD 5,3,0,0,5.7471E+7,1.

LD 5,4,0,0,5.7471E+7,1.

FR 0,1,0,0,28.5

GN -1

EX 0,1,100,0,1,0.

RP 0,1,361,1000,90.,0.,0.,1.,0.

EN

If we run this model, we obtain a source impedance of 288.7 + j0.7 Ohms. Now let's suppose that we wish to install a transformer that yields a perfect match to 50 Ohms. The impedance ratio (Zr) is 5.774, while the turns ratio (Tr) is 2.403. Selection of Q is arbitrary with the user, but should reflect a reasonable component value. Let's use 500 as the value. Under these conditions, the value of Y11-real is 1.7319; the value of Y12-real is -4.1616; and the value of Y22-real is 10.0.

Now we may rebuild the model to incorporate the NT-transformer. We begin by adding a new wire at some arbitrary distance from the antenna wires. The new wire is short and thin and serves as the segment for end 2 of the NT. As well, we move the antenna source (EX) to this wire to show the transformed source impedance. The NT command places end-1 on the former source segment at the center of wire (or tag) 1 in the model at segment 100. The revised model has the following appearance.

CM fd #18 cu 1" 28.5

CM NT transformer

CE

GW 1,199,-2.513,0.,0.,2.513,0.,0.,5.119E-4

GW 2,1,2.513,0.,0.,2.513,0.,.0254,5.119E-4

GW 3,199,-2.513,0.,.0254,2.513,0.,.0254,5.119E-4

GW 4,1,-2.513,0.,0.,-2.513,0.,.0254,5.119E-4

GW 5,1,-.03048,0.,.9144,.03048,0.,.9144,5.119E-4

GE 0

LD 5,1,0,0,5.7471E+7,1.

LD 5,2,0,0,5.7471E+7,1.

LD 5,3,0,0,5.7471E+7,1.

LD 5,4,0,0,5.7471E+7,1.

LD 5,5,0,0,5.7471E+7,1.

FR 0,1,0,0,28.5

GN -1

EX 0,5,1,0,1,0.

NT 1,100,5,1,1.731902,0.,-4.161612,0.,10.,0.

RP 0,1,361,1000,90.,0.,0.,1.,0.

EN

In the model, note GW5, the new wire and the placement for the EX command. The NT command shows its connections and the parallel admittance values in order. The zeroes following each numerical value entry represent the imaginary values that we skipped for this simplified transformer. If we run this model, we obtain a source impedance of 50.1 + j0.1 Ohms, just what we wanted.

In using the NT command as a transformer, newer modelers should be cautious. First, be certain that you do not invert either the value of Zr or of Tr. Suppose that We had a case of up-transformation, say, from 50 to 200 Ohms. Zr is 0.25 and Tr is 0.5, working from end-1 to end-1 of the network. The temptation will be to think of the transformer as having a 4:1 Zr and a 2:1 Tr, when the situation requires just the obverse. Second, the use only of conductance components in the network provides an illusion of frequency-independence for the transformer. Hence, we might press it into service as a modeled substitute for a 4:1 balun, many of which are rated for 3.5-30-MHz use. However, over a wide frequency range, we may encounter at least two major variations from our simplified model. The first variation involves the Q of the device components, which may change over a wide frequency range. Remember that we only selected a plausible value, not an actual value. The second variation lies in the device we are modeling. Not all real transmission-line transformers handle highly reactive loads (or antenna element impedances) in the same way as our basic model of a transformer. I have directed these cautions to the newer modeler on the assumption that experienced modelers will automatically be duly cautious.

**4. The Center of a Triangle**

Our next bit of information arises out of numerous models that I have examined and e-mail questions. Suppose that we wish to model a geometric shape with the coordinate origin (X=Y=Z=0) at the center. The process is simple for squares and larger polygons. For example, if we have a horizontal square loop, we may take 1/2 the length of a side and use the number as the X and Y values (with suitable + and - designations) as the four corners of the loop. It is now centered on the coordinate system origin.

However, if we wish to set up an equilateral triangle with the same result, troubles begin. Most of the difficulties stem either from sleeping in trig class or from too many passing years since that lesson. Most of the models that come my way use eyeball measures for the corner positions. If it sort of looks equilateral, then it is close enough for the model.

The equilateral triangle is actually
a model of simplicity if we remember that the sine of 30 degrees is 0.5
and the cosine of 30 degrees is 0.866. Each corner angle of the
triangle will be 60 degrees, and we shall be interested in bisecting
each angle into 30-degree angles to obtain a center triangle. **Fig. 6** shows the general layout and the essential relationships among the parts.

Bisecting each corner angle produces three lines. Where they cross is the center of the triangle. The process creates 6 right triangles, with each composed of an a-, a b-, and a c-line. Because any long line (a + b) is the side adjacent to the hypotenuse (S) of the figure, its length of 0.866 times S. Every b-line is 1/2 a, or every a-line is twice b. Since the junction of every a-line with the extending b-line is the center of the triangle, the distance from the center to a corner (that is, a) must be 2/3 of a + b. 2/3 of 0.866 S is .577 S. If we set up the triangle so that one corner is parallel to a coordinate axis (arbitrarily X), and if the center is at the system origin, then one corner of the triangle will be at 0.577 S in that direction. The remaining corners will be at -0.289 S along the X-axis. The Y coordinates will be + and - half the value of S. You may check your work by using Pythagoras' theorem in which the square of the hypotenuse (c) of a right triangle is the sum of the squares of the other two sides (a and b).

Suppose that I needed an equilateral triangle that is 45' per side. The point along the +X-axis will be +25.97 (with a Y coordinate of zero, of course). The remaining corners will have coordinates of -13.01, -22.50 and -13.01, +22.50. If I apply the theorem to these "back" corner coordinates, I should come out with the value of the X-axis coordinate. Allowing for rounding, a result of 25.98 is close enough to ensure excellent symmetry and a circumference that is on target with the project. (Your hand calculator may give you further precision by adding decimal places to the process, but you will have to decide how many places you enter into your model.)

There is, of course, nothing startling or new in all of this. The entire function of this note is to remind the forgetful (like me) that we can set an equilateral triangle with good precision. We can then use program rotation facilities to align the triangle however we need it. If the triangle is vertical, we can construct it in free space around the origin and raise it to height before adding a ground command.

**5. Rho's Phase Angle**

Thirty years ago, radio amateurs knew only one member of a famous trio: SWR (aka VSWR). Like the Ames Brothers other than Ed, the remaining members languished in obscurity. Today, we know all of the names: SWR, reflection coefficient, and return loss. We also know that the three facets of transmission-line behavior are intimately interrelated. That interrelationship has come at a cost.

We ordinarily (in handbooks at least) begin with the reflection coefficient. The standard, slightly simplified equation has the following appearance, where Ro is the characteristic impedance of the feedline, and Rin and Xin are the resistive and reactive components of the antenna at the transmission line connecting point. |Rho| is the magnitude of the reflection coefficient.

The simplification in the equation consists of omitting the reactive component of the transmission-line Zo. The reactance for virtually all transmission lines is very small, and users often do not have access to the value without using manufacturer data sheets. Since the reactance is so small, we may omit it without incurring any significant error in the results.

Next, we often define VSWR based on the value of |rho|.

Of course, we may calculate the VSWR directly from the Zo of the transmission line and the resistive and reactive components of the antenna feedpoint impedance. We simply substitute the |rho| equation right side for each occurrence of |rho| in the VSWR equation. Because the reflection coefficient fits more readily into other post-measurement calculations a bit more readily that the VSWR, more and more laboratory instruments using software-based calculations provide their results in terms of |rho| rather than SWR. However, if we use the correct Y-axis scale for each, the curves will be identical.

The final member of the trio is return loss, RL.

We normally measure RL in dB. Since the reflection coefficient will be less than 1, its log will return a negative number. Older texts handle RL as a negative number, but modern instrumentation has multiplied the result by -1 (hence the - sign in the equation) to yield p[ositive results. We multiply by 20 because the terms are for voltage and we wish a result in dB.

Since we can derive any member of the trio from the basic terms (Zo, Rin, Xin), the three members of the group are mathematical manipulations, each developed to serve a useful purpose, but nevertheless indicating a set of relationships among the basic terms. If we go only this far, we lose sight of why many texts consider reflection coefficient to be the most fundamental of the three.

Reflection coefficient |rho| not only has a magnitude, but a phase angle as well. Many texts aimed at radio amateurs omit the derivation of the reflection coefficient phase angle. (If the coefficient did not have a phase angle, we would not need to designate clearly that the calculated value is a magnitude by the use of |rho|.) To fill the blank, you may calculate the phase angle of |rho| by the following equation:

From the magnitude and phase angle of |rho| you may derive the real and imaginary components just as you would for any other phasor quantity.

I note this equation to complete the picture, not
because you will have any major use for it. However, you may see the
reflection coefficient used instead or or in addition to a value for
VSWR, and the value of rho may include both a magnitude and a phase
angle. For example, the latest implementation of EZNEC has a frequency
sweep facility that lists all three members of the trio. **Fig. 7**
samples the sweep graph for a 5-element Yagi design in all three modes.
Note that the graph does not change, but the scale of the Y-axis does.
(Some instruments may reverse the vertical order of return-loss scale
markings.) The graphing line will only change with changes in the value
of Rin, Xin, or Zo.

**Conclusion**

Perhaps the only unifying theme within these notes is that all of them have generated questions from folks at various levels along the road to mastering the art and craft of modeling. Almost all of the answers that I have tried to give have also generated a similar response: "Of course! I should have thought of that!" In addition, I can now remove the papers from my spindle file and clear it for another batch of miscellaneous items that may be useful to one or another modeler.