CFA Calculating Dimensions
I have been following the various bulletin board entries and gathering bits and pieces of information about the workings of the CFA antenna for some time, before building one. I have found many gaps, particularly in the relationships between the size of the CFA, the proportions of its parts, and the frequency of operation. This written piece is an attempt to find some of the relationships and reduce our understanding of their operation to that of more familiar things in circuit theory.
In my speculation, the CFA antenna is essentially two coaxial capacitors occupying a relative large volume of space. The D plate capacitor has a low reactance, because the displacement current through it generates the H field of the antenna. It works in a series tuned circuit that at resonance is purely resistive at its terminals. The voltage across the capacitance of this tuned circuit is phase shifted from the current by 90 degrees. The E plate has a high reactance value, because the voltage across it generates the E component of the radiated field. It works in a parallel tuned circuit because at resonance it is purely resistive and the voltage across the tuned circuit is not phase shifted from the current. When both tuned circuits are resonant at the operating frequency and driven from the same source, the displacement current (H field) is in phase with the E field. When the ratio of E to H is 377 ohms, all the energy in these fields can combine to form a radiating field and no non-radiating inductive fields, according to the claims of the CFA’s inventors. The configuration of the capacitors is intended to force E x H, the radiation Poynting Vector in the ring-shaped region of space around the axis of the capacitors.
The requirement that the impedance, or ratio of E to H, be 377 ohms imposes a ratio restriction on the voltage on the E plate and the voltage (which causes the displacement current ) on the D plate. This requirement is in addition to the phasing requirement that is needed to force H to be in phase with E. The articles and notes I have read have not mentioned this amplitude restriction, but have emphasized the phase restriction.
It seems to me that the capacitance of the D plate is a good place to start for a CFA design. As capacitance is increased for a given frequency, or as frequency is increased for a fixed capacitance, the displacement current and therefore H field is increased, requiring a higher voltage on the E plate to keep up and preserve the 377 ohm requirement. Any radiated energy attributable to the D plate comes from the displacement current, so I would assign a (probably 377 ohm) loss resistance across the D plate. The L-C ratio of the E plate parallel tuned circuit must be adjusted for the proper voltage ratio.
Derivation of the Voltage Ratio Equation
The E field magnitude from a point on the cylindrical E plate to the ground plane is equal to the voltage on the E plate divided by the distance along the field line between the point and the ground plane. The field line will leave both end points in a direction 90 degrees from the each surface. Assume that at some reasonable radius r (it doesn’t matter much how big a radius because this distance will cancel out later in the E/H ratio) the field line is an arc of a circle centered on the ground plane at the axis of the cylinder and center of the D plate disc. (in a vertical plane). If this arc is a quarter of a circle, as it might be for many of the configurations mentioned in the various notes, then :
Where:
V1 = the E plate voltage
(2*pi*r)/4 = the circumference of the quarter circle
The D plate is a disc with an area A, which is the area of the disc less the area of a hole in the center. The area is placed a distance d1 above the ground plane. The displacement current at some point in A is the time derivative of the displacement field, D at that point. The displacement field is the E field (a voltage/a distance) but in a dielectric material, which in this case is air and has a dielectric constant of e0. If V2 is the voltage between the area and the ground plane, and d1is the separation of the disc and ground, then:
D = e0*V2/d1 at each point in A;
e0*A*V2/d1 is the total field contributing to the current.
The time derivative of a sine wave of V2 amplitude and frequency f is:
(note the 90 degrees phase shift from a sine to a cosine wave).
The H field is a closed line that surrounds a current, which crosses through the plane of the line. Maxwell’s equation says that integrating the value of H around its line of direction gives a value equal to the current enclosed by the line. At our distance r from the center of the disc, in all planes parallel to the ground plane that fall inside the D plate, the integration of H is (2*pi*r)*H. Putting it all together:
Note that 2*pi*f*C*V2 is the current through a capacitor C caused by a voltage V2 across C.
Finally, the desired ratio E/H:
The (2*pi*r) ‘s cancel, so:
But, E/H = 377 ohms, the impedance of free space, so:
And V1/V2 = 377*(2*pi*f)*C/4
Results
Table 1 gives the voltage ratio for a representative selection of ham bands and D plate capacitance. This table is intended to show how frequency, capacitance, and V1/V2 relate.
Table 1: Ratio of voltage on the E plate (cylinder) to the voltage on the D plate (disc) for various D plate capacitance
A/dl (m) | C pF | 2 MHz | 3.5 MHz | 4 MHz | 7 MHz | 14 MHz | 21 MHz | 28 MHz |
---|---|---|---|---|---|---|---|---|
225.9887 | 2000 | 2.36756 | 4.14323 | 4.73512 | 8.28646 | 16.57292 | 24.85938 | 33.14584 |
112.9944 | 1000 | 1.18378 | 2.071615 | 2.36756 | 4.14323 | 8.28646 | 12.42969 | 16.57292 |
56.49718 | 500 | 0.59189 | 1.035808 | 1.18378 | 2.071615 | 4.14323 | 6.214845 | 8.28646 |
28.24859 | 250 | 0.295945 | 0.517904 | 0.59189 | 1.035808 | 2.071615 | 3.107423 | 4.14323 |
11.29944 | 100 | 0.118378 | 0.207162 | 0.236756 | 0.414323 | 0.828646 | 1.242969 | 1.657292 |
75 | 0.088784 | 0.155371 | 0.177567 | 0.310742 | 0.621485 | 0.932227 | 1.242969 | |
50 | 0.059189 | 0.103581 | 0.118378 | 0.207162 | 0.414323 | 0.621485 | 0.828646 | |
25 | 0.029595 | 0.05179 | 0.059189 | 0.103581 | 0.207162 | 0.310742 | 0.414323 |
Table 2 gives the reactance of the D plates in Table 1, and is intended to suggest the magnitudes of the impedance levels involved relative to the impedance of free space, 377 ohms, and typical cable impedance such as 50 ohms.
Table 2: The capacitive reactance for each value and frequency in table 1(plus 10 pF)
C pF | 2 MHz | 3.5 MHz | 4 MHz | 7 MHz | 14 MHz | 21 MHz | 28 MHz |
---|---|---|---|---|---|---|---|
2000 | 39.78877 | 22.73644 | 19.89438 | 11.36822 | 5.68411 | 3.789407 | 2.842055 |
1000 | 79.57754 | 45.47288 | 39.78877 | 22.73644 | 11.36822 | 7.578813 | 5.68411 |
500 | 159.1551 | 90.94576 | 79.57754 | 45.47288 | 22.73644 | 15.15763 | 11.36822 |
250 | 318.3102 | 181.8915 | 159.1551 | 90.94576 | 45.47288 | 30.31525 | 22.73644 |
100 | 795.7754 | 454.7288 | 397.8877 | 227.3644 | 113.6822 | 75.78813 | 56.8411 |
75 | 1061.034 | 606.3051 | 530.5169 | 303.1525 | 151.5763 | 101.0508 | 75.78813 |
50 | 1591.551 | 909.4576 | 795.7754 | 454.7288 | 227.3644 | 151.5763 | 113.6822 |
25 | 3183.102 | 1818.915 | 1591.551 | 909.4576 | 454.7288 | 303.1525 | 227.3644 |
10 | 7957.754 | 4547.288 | 3978.877 | 2273.644 | 1136.822 | 757.8813 | 568.411 |
Table 3 gives the required series inductance to resonate the D plate capacitance at the operating frequency. In my experiments, this series tuned circuit has a high Q which drops rapidly when a resistor is placed across the D plate to simulate radiation. My scope shows the 90 degrees phase shift between D plate voltage and the series tuned circuit driving voltage.
Table 3: Inductance to resonate each capacitance in Table 2 at each indicated frequency
C pF | 2 Mhz | 3.5 Mhz | 4 Mhz | 7 Mhz | 14 Mhz | 21 MHz | 28 MHz |
---|---|---|---|---|---|---|---|
2000 | 3.166292 | 1.033891 | 0.791573 | 0.258473 | 0.064618 | 0.028719 | 0.016155 |
1000 | 6.332585 | 2.067783 | 1.583146 | 0.516946 | 0.129236 | 0.057438 | 0.032309 |
500 | 12.66517 | 4.135566 | 3.166292 | 1.033891 | 0.258473 | 0.114877 | 0.064618 |
250 | 25.33034 | 8.271131 | 6.332585 | 2.067783 | 0.516946 | 0.229754 | 0.129236 |
100 | 63.32585 | 20.67783 | 15.83146 | 5.169457 | 1.292364 | 0.574384 | 0.323091 |
75 | 84.43446 | 27.57044 | 21.10862 | 6.892609 | 1.723152 | 0.765845 | 0.430788 |
50 | 126.6517 | 41.35566 | 31.66292 | 10.33891 | 2.584728 | 1.148768 | 0.646182 |
25 | 253.3034 | 82.71131 | 63.32585 | 20.67783 | 5.169457 | 2.297536 | 1.292364 |
10 | 633.2585 | 206.7783 | 158.3146 | 51.69457 | 12.92364 | 5.743841 | 3.230911 |
Table 4 gives the capacitance of various diameter D plates at a selection of spacing. I have cut out a 12 inch diameter D plate to go with a 20-inch ground plane (size set by what sheet metal I have in the junk box).
Table 4: Capacitance of D plate disc with various diameters and spacing from the ground plane
d1 | 4 | 6 | 10 | 12 | 15 | 20 | 36 |
---|---|---|---|---|---|---|---|
0.125 | 22.59834 | 50.84626 | 141.2396 | 203.385 | 317.7891 | 564.9584 | 1830.465 |
0.187 | 15.10584 | 33.98814 | 94.4115 | 135.9526 | 212.4259 | 377.646 | 1223.573 |
0.250 | 11.29917 | 25.42313 | 70.6198 | 101.6925 | 158.8946 | 282.4792 | 915.2326 |
0.375 | 7.532779 | 16.94875 | 47.07987 | 67.79501 | 105.9297 | 188.3195 | 610.1551 |
0.500 | 5.649584 | 12.71156 | 35.3099 | 50.84626 | 79.44728 | 141.2396 | 457.6163 |
1.000 | 2.824792 | 6.355782 | 17.65495 | 25.42313 | 39.72364 | 70.6198 | 228.8082 |
4.000 | 0.706198 | 1.588946 | 4.413738 | 6.355782 | 9.93091 | 17.65495 | 57.20204 |
Note: For 1″ Hole Reduced by 0.000507
Table 5 gives the voltage ratios I might need for various spacings of my 12 inch D plate at a selection of ham bands. (the capacitances shown are those in the 12 inch column of table 4. I hope to find a spacing where V1/V2 just equals the voltage ratio between the driving voltage on the 50 ohm cable and that which appears across the D plate (when loaded with the radiation losses.). That would eliminate the E plate tuned circuit!!
Table 5: Ratio of E plate (cylinder) voltage to D plate (disc) voltage for a 12-inch disc (4th column in table 4)
A/d1 (m) | C pF | 2 MHz | 3.5 MHz | 4 MHz | 7 MHz | 14 MHz | 21 MHz | 28 MHz |
---|---|---|---|---|---|---|---|---|
22.98136 | 203.385 | 0.240763 | 0.421335 | 0.481526 | 0.842671 | 1.685342 | 2.528013 | 3.370684 |
15.36187 | 135.9526 | 0.160938 | 0.281641 | 0.321876 | 0.563283 | 1.126565 | 1.689848 | 2.253131 |
11.49068 | 101.6925 | 0.120382 | 0.210668 | 0.240763 | 0.421335 | 0.842671 | 1.264006 | 1.685342 |
7.660453 | 67.79501 | 0.080254 | 0.140445 | 0.160509 | 0.28089 | 0.561781 | 0.842671 | 1.123561 |
5.74534 | 50.84626 | 0.060191 | 0.105334 | 0.120382 | 0.210668 | 0.421335 | 0.632003 | 0.842671 |
2.87267 | 25.42313 | 0.030095 | 0.052667 | 0.060191 | 0.105334 | 0.210668 | 0.316002 | 0.421335 |
0.718167 | 6.355782 | 0.007524 | 0.013167 | 0.015048 | 0.026333 | 0.052667 | 0.079 | 0.105334 |
It is hoped the above calculations will be of help to others who wonder about a more specific approach to calculating the various dimensions.
Formal Education:
- 1958, Bachelor of Science Degree at Massachusetts Institute of Technology
- 1962 Masters in Science at Purdue University
Worked for 16 years designing electronic countermeasures against monopulse and pulse compression radars, including remote means of bending the radar’s antenna pattern, and have designed and built an instrumentation radar.
Worked for about 15 years as an electronic consultant designing and troubleshooting microwave anechoic chambers using chamber simulation computer programs I wrote for the purpose. Designed and built (I have a machine shop) test fixtures to measure the electronic properties of materials and absorbers. Have written computer programs to do stock market analysis and to encrypt documents using simulated rotor machine coders.
Was first licensed as W9TPH in 1952, then K1ZLJ but let it lapse in the 1960’s. Retested and passed General class this summer, and Extra class this fall. After relocating from the East Coast to the West Coast in Oregon, USA, now, having put the ocean on the left instead of the right. Am interested in compact antennas to fit in my postage stamp of land looking out over the pacific ocean.
This was great work from Joel, Maxwell’s theory is a complete minefield and the deeper you go into it the deeper you go down the tunnel, and it is enough to stand on any braincell’s that you might have left, for anyone to attempt this it is and stay saine afterwards is a blummin challenge, A lot of people brush this off, because it is not an easy challenge to get it right, the problems are that this does work, I have seen the data, and on a few occausionswhen it was allowed , the antenna does perform very well, the problem is that this is very difficult to make it work, there is a so called enginner that has managed to repeat this on. afew occasions, but sadley the person is untrustworthy and will not allow the data to be let out of his hands, and especially when it counts, hopefully one day someone will work this out, but until then enjoy the ride.
Originally posted on the AntennaX Online Magazine by Joel C. Hungerford, KB1EGI
Last Updated : 3rd May 2024