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. 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. 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