INSULATED ANTENNAS

R. P. Haviland, W4MB
1035 Green Acres Circle, North
Daytona Beach, Florida 32119

INTRODUCTION

By an insulated antenna, I mean one which has enough thickness of
insulation on its surface to affect its characteristics. The more
accurate term found in the technical literature is "dielectric
coated antenna".

There isn't much about insulated antennas in the Amateur
literature. I could find nothing in the bibliography in the 1988
edition of the ARRL Antenna Handbook (1). Rosen's
cross-referenced bibliography, From Beverage to Oscar (2),
produced no search matches. I do remember a short note in one of
Bill Orr's columns to the effect that use of insulated wire in
quads caused detuning, but I havn't been able to find the exact
column. There is a little more in the technical literature, but
we'll get to that later.

CONCEPTUAL APPROACH TO INSULATION EFFECTS

We can get a general idea of the effect of an insulating coating
on an antenna by taking a simple dipole and considering only
major (first-order) effects. As sketched in Fig. 1, let Rw be the
radius of the conductor of an antenna, and Rd be the radius of a
surrounding insulation, a dielectric, where Rd is always greater
than Rw. The dielectric constant of the insulation is K times
that of the surrounding air. For simplicity assume that the
magnetic permeability of the wire and insulation are the same as
for air, and that the losses in the insulation can be neglected.

A short length of such insulated conductor will have resistance,
inductance and capacitance. To the first order, we can write:

Z=XL-XC+R+A*f*f+()+()
(1)

Where Z is a per-unit impedance
f is the operating frequency
XL is the reactance due to inductance of the wire
XC is the reactance due to capacitance of the wire
R is the RF resistance of the wire
A is a resistance representing energy loss by radiation
from the wire,
and the () represents smaller terms which are neglected.

Since the simplifying assumptions are that there is no inductive
or resistive effects, the presence or absence of the insulation
can only affect the C term, and possibly the A term. Let us
concentrate on the first.

If the insulation were essentially absent, the C term must be the
same as for a bare wire. And if the the insulation extended to
infinity, the unit capacity would be increased by the amount of
the dielectic constant. With practical amounts of insulation, the
C term will be greater than for a bare wire, but by a small
amount.

The square root of the quantity L*C is the reciprocal of a
frequency, and taken over the entire length of the wire is the
resonant frequency of the antenna. Since the effect of the
insulation is to increase C, it is also changing the resonant
frequency, moving it lower. The amount of change is small for a
thin layer of insulation. The ratio of no-insulation frequency to
coated frquency is always less than the square root of the
dielectric constant, K.

We can get an idea of the way the resonant frequency varies by
further simplifying assumptions. Let the capacitance be composed
of two capacitors in series. C1 is that of a coaxial capacitor
formed of the wire and a conductive cylinder of radius Rd, with
the space between filled with the dielectric. C2 is the capacity
between the conductive cylinder and infinite space, with air as
the dielectric.

A look at the equations in Terman's Radio Engineer's Handbook (3)
quickly shows that a lot of "number shoving" can be avoided if it
is assumed that the antenna is spherical rather than cylindrical,
since the resulting symmetry simplifies the equations for C.
While this changes the details somewhat, the principle is exactly
the same.

The equations needed are:

C1=1.412*K*(Dw*Dd)/(Dw-Dd) (2)
C2=1.412*Dd (3)

Diameter being used rather than radius. Combining these by the
series capacitance formula and taking the square root of the
result gives the relative variation in frequency as the amount of
insulation (Dd/Dw or Rd/Rw ratio) is changed. This quantity is
plotted in Fig. 2 for a dielectric constant of 3.

With no insulation, the resonant frequency is that of an isolated
wire. As the insulation thickness increases, the frequency
decreases linearly with the thickness ration for a time. When the
thickness is equal to the wire size, the rate of change is about
half the initial. With insulation of ten times wire size, the
relative resonant frequency has reduced to nearly the limiting
value, the reciprocal of the square root of the dielectric
constant, 0.58 times the free space value in this case.

METHOD OF MOMENTS ANALYSIS

This is a very approximate analysis, but it shows the overall
trend to be expected, and some of the intermediate details. More
accurate results needs a much more complex analysis.

One approach to this can be developed from the Method of Moments,
widely used in antenna analysis. You will recall that this
divides the antenna into segments, establishes an impedance
matrix Zmn of the self and mutual impedances of the segments, and
calculates the matrix values from the fields generated. The
matrix is then inverted to calculate the currents in the
segments, the pattern and the drive conditions.

This approach was followed by J. H. Richmond of Ohio State
University in a study done for NASA (4). He found that adequate
accuracy could be had if the impedance matrix was considered to
be composed of two parts,

Z=Zmn+Z'mn (4)

where Zmn is the matrix for for the antenna with no insulation
present. The additive matrix Z'mn contains all of the effects of
the insulation.

The elements of this add-on matrix are of the form:

Z'mn=P*F(l) (5)

where P is a function of the insulation dielectric constant and
dimension, and it's multiplier F(l) is solely a function of
segment length factors multiplied by constants.

The complete expression for P is:

P=(K-1)/K*LN(Rd/Rc) (6)

Where the LN term is the natural logarithm of the radii ratio.

The relations involved in obtaining the impedance matrices and
their solutions have been written as a Fortran computer program
by Richmond. This is available as a well documented paper copy in
a NASA Contractors report (5) or alternately from the NTIS (6).
(While well documented, it is not always easy to follow the
translation from the equations to the computer code).The program
is quite lengthy, about 1000 lines, and uses the complex
arithmetic common in Fortran. Input and output are by the punched
cards common in large computer installations of the 70's. I have
not checked, but the source and compiled code should also be
available on tape from NTIS.

The Fortran code is standard, and anyone with a small computer
Fortran compiler, 1977 version, could get the program running at
the cost of some typing effort. Alternatively, the nine line
Fortran subprogram which generates the matrix Z'mn could be
translated to BASIC, and be included in Mininec, adding to it's R
and X impedance matrices in the same fashion as is done for
lumped loads. However, for the purpose of this article, even this
seemed to be more work than was justified.

Looking at equation 5 again, it is clear that all solutions
involving insulation must include as one endpoint the
corresponding value with no insulation. Further, even though the
wire part of one antenna is different from that of another,
equation 6 shows that the effect of adding insulation to the
second will be proportional to that of the first. This means that
we can take a particular example solution, and develop from this
values for many design situations. Let us do this for resonant
frequency, confining our effort to wire antennas.

To do this simply, we need three curves. The first is a curve of
resonant length with no insulation. As has been shown many times,
this is a function of the length to diameter ratio of the antenna
conductor, or alternatively, the wavelength to diameter ratio.
Fig 3 shows this as a fraction of free-space wavelength for
monopoles and dipoles (7). Fig. 4 gives the resonant loop
circumference for equal sided quads and for equilateral delta
loops (8). The resonant length can also be determined by using
one of the available versions of Mininec.

The second curve needed is the P-factor, given by equation 6.
This is plottted in Fig. 5 as a function of the ratio of
insulation to wire diameters (or ratio of radii). Dielectric
constant is used as a parameter, and intermediate values should
be interpolated betweeen the curves. The P-factor can also be
determined with a calculator using Eq. 6.

Finally, the length factor with added insulation is secured from
Fig. 6 for dipoles/monopoles, or from Fig. 7 for quads and
deltas. The correct curve is selected using the "no-insulation"
value from Figs. 3 or 4, interpolating between curves as
necessary. Then the "with insulation" length factor is read off
at the intersection of the curve and the P-factor from Fig. 5.
These last curves are developed from a curve in Richmond and
Newman (9), and from the example data given.

The results of using the ensemble of curves will be reasonably
accurate for small amounts of low-K insulation, but of lesser
accuracy as the insulation size grows. For such situations, the
more exact Fortran program of Richmond could be used. However, he
warns that its calculation accuracy decreases as the insulation
becomes large compared to wire size. In such cases, full scale
measurements of a working prototype are indicated if an
application is critical with respect to element length, as in
parasitic arrays.

DRIVE RESISTANCE AND OTHER FACTORS

Returning to the discussion following equation (1), the square
root of the quantity L/C is a resistance, from which the drive
resistance is developed. Since The C term is increased by the
addition of insulation, the drive resistance decreases as the
thickness of insulation increases. The maximum change is of the
same order as for the resonant frequency, i.e., a reduction by
the square root of the dielectric constant K.

A three-graph proceedure could be also used to develop the drive
resistance of the element. This has not seemed worthwhile, since
it is so variable in arrays, and since any change is easily
matched out. The reduction is approximately by the same ratio as
for the resonant frequency.

Richmond and Newman (9) state that another factor will appear if
the more exact analysis of the computer program is used,
specifically that the bandwidth of the antenna decreases. The
statement is also made by Lammensdorf (10) in his report of
experimental measurements on insulated antennas . Both refer to
the fact that adding insulation narrows the antenna conductance
response curve. In Amateur terms, the SWR bandwidth will be less
for insulated than for bare antennas, even though they both are
matched at the center frequency. In most situations, the effect
can be neglected.

SOME EXAMPLE SITUATIONS

Before closing, let us look at a few typical situations, for some
using both the approximate graphical method described here and
the more exact and measured results of the references.

For the first, assume that a dipole is made of an 8" length of
RG-59/U coax with the outer cover and the braid outer conductor
stripped off. What is the resonant frequency?

For this, the length/diameter ratio is 8/.025 or 320. The ratio
of insulation to conductor diameter is .146/.025, or 5.8, and the
dielectric constant is 2.3. From Fig. 3 the bare wire length
factor is about .48. From Fig. 4, the P-factor is .9. The
resonant frequency with insulation is .425 wavelengths by
interpolation. Ref. 9 shows a value of .425 for both theoretical
and measured resonance.

Suppose the same wire were formed into a quad loop. The
perimeter/ radius ratio is about 600, and the free space length
factor is about 1.18 (Fig. 3). with insulation this reduces to
.93 (Fig 4). Ref (7) shows a theoretical value of .92, and a
measured value of .95 for such a loop.

Going outside of the range for which test data is available,
suppose that a 10 meter dipole is made from a piece of old RG8/U
with the jacket and braid stripped off. Does the antenna need
shortening from standard design length?

The insulation ratio for this wire is about .285/.082 or 3.5
average. Dielectric constant is 2.26. Normal length is about 16
feet, for a L/D ratio of 2350, giving a length factor of .455.
The P-factor is about .69. The resonant frequency occurs at a
length factor of .415. The dipole should be shortened to 15 feet
2 inches length if the length is critical, as for a Yagi
director. Otherwise, the change probably can be ignored, since it
is of the same order as changes due to height above ground, or to
nearby objects.

In contrast, what is the effect of Formvar insulation of a 40
meter dipole of #12 wire? Wire tables show this gives 12 turns
per inch, for an insulation diameter of 82.5 mils, compared to
80.8 for bare wire. The diameter ratio is 1.025, By calculation,
the P-factor is about 0.06. Fig.- shows that this amount of
insulation does move the resonant point slightly, but this small
chamge is negligible in practical situations.

CONCLUSIONS

The overall conclusions are:

-Adding insulation to the wire does change its performance,
especially if the insulation is thick or of high dielectric
constant.

-In most real-world situations involving use of thin layers of
insulation on wires, the effect is not likely to be seen, since
it is about the same as that due to other causes.

-In critical applications, as in Yagi parasitic elements, the
efects of insulation require compensation, by the approximatiate
method shown here, or preferably by more exact calculation.

-The state of theory is such that extreme conditions of thickness
or high dielectric constant will need full scale measurements for
accuracy.

Haviland/high gain yagi

Legend for Program

Computer program to generate dimensions for a optimum High Gain
Yagi for any frequency, and to give a shape plot. Tabulated
dimensions should be transfered to a full size pattern and used
to bend the elements to proper shape.

The program is in Commodore 64 Simons' Basic, but the numerical
part will run on most small computers without change. The plot
routine will require re-writing.

Legend for plot

A high resolution plot of the position of the element centers.
The director is the element with the deep V at the center. All
elements are 1.5 wavelengths long at the design frequency.

hHaviland,High Gain Yagi, Legends#
fR.P.Havland,1035 Green Acres Circle,N, Daytona Beach FL 32019

Fig. 1
General principle of a shaped element. A half wave dipole section
is formed into a V. The center of radiation is away from the apex
of the V. Radiation from this point arrives at the center of
radiation of the outer sections at least partly in phase with
their radation, despite the fact that the outer sections are
end-driven out of phase with the center. This produces gain.

Fig. 2
Computer program to give dimensions of a three element version of
the shaped element array. Written in Commodore 64 Simons' BASIC,
translation for other computers should not be difficult.

Fig. 3
Tabular output of the computer program for a freqency of 147 MHz.
X and Y are normal geometric coordinates. The 0's indicate that
the end of the element has been passed. These values may be
scaled for other frequencies.

Fig. 4
High resolution computer plot of the data of Fig. 3, showing the
center line of the elements. Although each element is 1.5
wavelength long, the shaping makes the reflector appear longer
and the director shorter than the radiator.hHaviland,High Gain
Yagi, Legends#
fR.P.Havland,1035 Green Acres Circle,N, Daytona Beach FL 32019

Fig. 2
Computer program to give dimensions of a three element version of
the shaped element array. Written in Commodore 64 Simons' BASIC,
translation for other computers should not be difficult.

fR.P.Haviland,1035 Green Acres Circle,N, Daytona Beach FL 32019

References

(1) F. M. Landstorfer, "A new type of directional antennas",
Antennas Propagat. Soc. Int. Symp. Dig.,pp. 169-172, 1976

(2) Chang-Hong Liang and David K. Cheng, IEEE Trans. Antennas and
Propagation, v 31 n 3, May 1983

Insulated Antennas
Captions

Fig. 1
Cross section of an insulated antenna. Rw- radius of wire, Rd-
radius of dielectric, K- dielectric constant of insulation. The
diameter ratio may be used.

Fig.2
Very approximate effect of insulation with dielectric constant of 3 on a spherical antenna, showing reduction of resonant frequency as insulation thickness increases. The limit of reduction is 1 over the square root of the dielectric constant.

Fig. 3
Resonance length factor of dipoles and monopoles as a function of its length to diameter ratio. For quarter-wave verticals over an infinite ground, use the dipole curve with twice the monopole length to diameter ratio. "Half-wave" dipoles are always shorter than the free-space one-half wavelength at resonance.

Fig. 4
Resonance length factor of a square quad loop as a function of its circumference to radius ratio. Use for equilateral delta loops with negligible error. "One-wavelength" bare conductor loop perimeters are always longer than the free-space wavelength at resonance.

Fig. 5
Insulated antenna "P-factor" as a function of the insulation to wire diameters (or radii). Interpolate between curves for the exact dielectric constant. Use equation 6 for values outside the curves.

Fig. 6
Approximate shortening effect of insulation on dipoles. Select the uninsulated length factor curve using data from Fig. 3, and for the P-factor from Fig. 5, interpolating as necessary. For monopoles use one half this indicated length. See text for values beyond the data shown.

Fig. 7
As Fig. 6, but for loops. Enter with data from Figs. 4 and 5. Insulated loops may have a perimeter less than one wavelength at resonance.

hR.P.Haviland,1035 Green Acres C N, Daytona Beach FL 321119

cInsulated Antennas
REFERENCES

(1) Topical Bibliography on Antennas, The ARRL Antenna Book, 15th ed., ARRL, Newington, CT. 1988

(2) From Beverages thru Oscar - A Bibliography 1908-1988, Didah Publishing, Nashua, NH. Available on disk or microfiche.

(3) F. E. Terman, Radio Engineer's Handbook, McGraw-Hill, NY, 1943

(4) J. H. Richmond, Radiation and Scattering by Thin-wire Structures in the Complex Frequency Domain, NASA Report CR-2396, May 1974. Available from NTSC and from depository libraries.

(5) J. H. Richmond, Computer Program for Thin-wire Structures in a Homogenous Conducting Medium, NASA Report CR-2399, June, 1974. Availability as (4)

(6) National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161

(7) The ARRL Antenna Book, Current ed., ARRL, Newington, CT.

(8) R.. P. Haviland, The Quad Antenna, CQ Books, Hicksville, NY, Summer, 1992

(9) J. H. Richmond and E. H. Newman, Dielectric Coated Antennas, Radio Science, v11, n1, pp. 13-20, National Bureau of Standards

(10) David Lammersdorf, An Experimental Investigation of Dielectric Coated Antennas, IEEE Trans. Ant. Prop., AP-15, N.6, Nov. 1967, pp 767-771