(May 2 2006)
Resonance frequency for an element inside the cavity without any boom tube The first experiment is to measure the resonance frequency of several director type metal rods inside the cavity. ELEMENT.DAT contains the dimensions for the elements used in the measurements and the measured resonance frequencies.
The experimental data fit well to this expression: F = 0.5*C/(0.9453*L + 0.733*sqrt(D) + 0.13405) F = resonance frequency inside cage C = speed of light=299.729458 L = element length in meters D = element diameter in meters The formula is purely empirical. The fit to the experimental data is shown below in table 1. The RMS error is 12kHz while the shift range is 8.3MHz. If you want to do something similar, here is the computer program to fit resonance in cage
Table 1 Fit of experimental resonance frequencies to three parameter formula for element inside cage. No boom tube present. Effect of using short pieces of boom tubes rather than a long tube It is obvious that a boom tube section that is very short will not screen the magnetic field caused by the element part inside the tube much. (Imagine a boom tube of length similar to the element diameter) With increasing length the effect of the boom tube will gradually increase until it reaches it's asymptotic value for a very long tube. NEC simulations on element plus boom tube in free space show an oscillatory behaviour that we do not see experimentally inside the cavity. To get an estimate how limited lengths of boom tubes affect our boom correction measurements we made a series of measurements as in fig 1. Fig 1 Two sections of 50 mm boom tubes placed symmetrically around an antenna element. The gap between the two (200 mm here) and their length are entries in FARSHIFT.DAT. Our experimental data for the frequency shift caused by two equal pieces of boom tube placed symmetrically around an element as shown in fig. 4 can be found in FARSHIFT.DAT The entries in FARSHIFT.DAT are the following: 1. Element number. (Dimensions in ELEMENT.DAT) 2. Round or quadratic cross section (0=round, 1=quadratic) 3. Boom tube outer diameter in mm. (Side length if quadratic) 4. Boom tube length in mm. (Both tubes have the same length) 5. Gap length in mm. The element is at the center of this gap. 6. Frequency shift in MHz. We have adopted a simple formula for the influence of the two equal boom tube sections in this experiment. We can regard the two tubes as a short boom tube with a gap at the center. F=2*G + 2*E F=frequency shift in MHz 2*G=frequency shift caused by a very long boom tube with a gap surrounding the element. 2*E=difference in frequency shift between a very long boom tube and a short boom tube through which the elemented is mounted at the midpoint. E=( 23 * D^{2} ) / ( dist^{0.887} ) D=Boom tube diameter in wavelengths for round boom tube. D=1.18*Boom tube side length for quadratic boom cross section. dist=DE=Distance from element to end of boom tube in wavelengths. For G we use the same formula as for E but then dist has a slightly different meaning: dist=DG+0.592*D0.698*a=effective distance to end of gap in the boom tube. It is obvious that the frequency shift must have a finite value for DG=0 (no gap) so we must divide by DG plus something. dist is some kind of average distance between points on the element surface and points on the boom tube end surface. a=Element diameter in wavelengths. The wavelength is the wavelength in free space at the frequency of resonance for the element inside the cage. The crude approximation gives a good fit to our measured data. The frequency shifts range from 15 to 760kHz and the RMS error is 10kHz. Table 2 below gives a few typical examples of the data. FARFIT.DAT contains all 73 measured frequency shifts. The FORTRAN program FAREXP.FOR was used for the least squares fit.  
N Shape D Gap Length Fshift Error 1 0 0.02800 0.5000 0.3180 0.020 0.006 1 0 0.02800 0.2000 0.3180 0.070 0.006 1 0 0.02800 0.0700 0.3180 0.185 0.014 1 0 0.02800 0.0400 0.3180 0.285 0.008 1 0 0.01500 0.1000 0.2970 0.045 0.004 1 0 0.07625 0.5000 0.2160 0.150 0.029 1 0 0.07625 0.1000 0.2160 0.610 0.000 1 0 0.07625 0.7000 0.1575 0.055 0.001 1 0 0.07625 0.2000 0.1575 0.265 0.014 1 0 0.04980 0.5000 0.2910 0.070 0.001 1 0 0.04980 0.3000 0.2910 0.150 0.018 1 0 0.04980 0.2000 0.2910 0.205 0.002 1 1 0.02000 0.3000 0.4250 0.035 0.004 1 1 0.02000 0.0400 0.4250 0.240 0.007 1 1 0.03000 0.3000 0.2780 0.065 0.006Table 2 Typical fit for frequency shifts in cavity caused by two equal boom tube sections "length" long separated by "gap". N is element number and Shape 0 or 1 for round or quadratic cross section. Example from FARFIT.DAT  
Through hole mounted elements inside cage For these measurements we have used 14 different metal boom pieces. All of them are sufficiently long for the formula for end corrections from the previous experiment to be accurate enough. The boom pieces have different wall thicknesses, from 1mm to 35mm solid rod and the cross section is round or quadratic. BOOMS.DAT describes all the short boom tube sections we used to measure the frequency shift for through hole mounting of yagi antenna elements inside the cavity. The entries in BOOMS.DAT are the following: 1. Boom tube number. 2. Round or quadratic cross section (0=round, 1=quadratic) 3. Boom tube outer diameter in mm. (Side length if quadratic) 4. Wall thickness in mm. (no 8 and 12 are solid rods.) 5. Distance from hole to one tube end. 6. Distance from hole to the other tube end. The different boom sections were measured several times for the different elements. Each time with a larger hole for the element. FREQ.DAT contains the resonance frequencies for various combinations of boom tubes, elements and hole diameters. The entries in FREQ.DAT are the following: 1. Element number. (defined in ELEMENT.DAT) 2. Boom tube number. (defined in BOOMS.DAT. 0 means no boom tube) 3. Hole diameter in mm. 4. Frequency in MHz. 

Shape D ELD H XL TH Fshift Error 0 40.1 6.1 8.0 898.3 3.200 0.895 0.012 1 30.0 6.1 8.0 898.3 1.900 0.695 0.002 0 25.0 6.1 8.0 898.3 12.499 0.955 0.049 0 25.0 6.1 8.0 898.3 1.300 0.520 0.001 1 25.2 6.1 8.0 898.3 1.600 0.605 0.019 0 35.0 6.1 8.0 898.3 7.500 1.225 0.009 0 35.0 6.1 8.0 898.3 17.499 1.645 0.012 0 50.0 6.1 8.0 898.3 1.500 0.795 0.020 0 76.3 6.1 8.0 898.3 2.500 1.310 0.010Table 3 A few typical lines from the boom correction least squares fit from HOLEFIT.DAT  
The BOM subprogram gives a length correction in meters. The three parameter formula of the first experiment is used to convert the length change to a frequency shift. The parameters of the subprogram BOM are adjusted for this frequency shift to agree with Fshift, the corrected experimental value that would apply for a very long boom tube inside the cage. The FORTRAN program HOLE.ZIP was used for the least squares fit. The frequency shifts range from 350 to 1700 kHz and the RMS error of the fit is 16 kHz. Computer program for boom corrections The subprogram BOM gives the boom correction for an element in our cage experiment at 144MHz. We can not say with what accuracy these corrections will be correct for a 144MHz yagi antenna. In case there are any discrepances they will be independent of the boom tube interior (wall thickness and hole diameter). In case there is any difference at all between measurements in a cage and in a yagi structure, the difference should be a function of the boom tube outer diameter and the element length allone. The computer program BC can be used to calculate boom corrections with the parameters we have deduced from the cage experiments. 