where
Pd = radiated power density at distance r from the antenna, W/m2
Pt = power input to the antenna, W
Gt = numerical gain of the antenna
r = distance from the antenna where the power density is evaluated, m
The second is Ohm’s Law5
where
P = power dissipated in a load, w
voltage across the dissipating element, V
R = resistance (impedance) of dissipating element or load, Ω
The third relationship is Ohm’s Law for Free Space6
where
Pd = power density of the incident wave, W/m2
E = electric field strength at that point in space, V/m
n = impedance of free space,= 377πΩ WΩ
Combining Equations (14) and (16) leads to a familiar expression
which relates the electric field strength at a point r away from the transmitting antenna having input power Pt and gain Gt.
By rearranging Equation (15) we have:
Recalling the definition of the transmit antenna factor, the ratio of the E-field developed to the input voltage to the antenna, we can find the TAF by taking the ration of the E-field produced from Equation (17) to the power dissipated in the antenna given in Equation (18)
Remembering that the transmitted power Pt is identical to the power dissipated in the load, Pin, and R is 50 W, Equation (19) simplifies to
This result is reasonable, as the TAF should be an inverse function of distance from the source, and a direct function of the gain of the transmitting antenna, and should be independent of power input. It should be noted that the gain value in Equation (17) is the effective gain of the antenna, calculated from the measured values of the AF. The TAF as used above incorporates antenna efficiency, the effect of antenna mismatch and other losses.
The TAF expression can be converted to dB form by taking 20 x log10 of both sides of Equation (20). This gives
Note that the TAF is proportional to the gain of the antenna and inversely proportional to the distance from the antenna. This is rational and suggests that the derivation is correct.
As can be seen from the derivations, although the AF and TAF have the same units, m-1, they are neither identical nor reciprocal. They are connected for both expressions. This fact allows the TAF to be computed from the AF by recalling that fM = c, and rewriting Equation (11) as
Substituting Equation (22) for Equation (23) gives
This conversion is valid for the conditions from which either the AF or TAF is measured. If the AF is measured over a ground plane (typical condition), then the TAF computed from the AF is valid for a similar condition.
Remember that the concept of reciprocity, as it applies to antennas, relates to the transmit and receive pattern. As such, the reciprocity does not include the effects of impedance mismatch, efficiency or other factors. These factors are included in the measured AF. Thus, if measured antenna factors are used, the TAF computed from these values will be accurate when the antenna is used under the same conditions, over a ground plane. A semi-anechoic chamber also fulfills the same conditions, subject to the constraint that, over the frequency range of the application of this concept, the RF absorber must be effective.
The above discussions have provided simple derivations of two parameters of an EMC antenna the AF and the TAF. These parameters are in daily use by many, but the source of the values is not well known. It is the purpose of this paper to provide the derivations of these parameters to illustrate the use of antennas and why they work as they do.
The author wishes to express appreciation to Edwin L. Bronaugh of EdB Consultants for technical input and reviews of portions of this paper, and to Dr. Thomas Chesworth of Electromagnetic News Report for permission to incorporate portions of a previously published discussion of TAF in this article.
1. Edwin L. Bronaugh and William S. Lambdin, Electromagnetic Interference Test Methodology and Procedures, Volume 6, A Handbook Series on Electromagnetic Interference and Compatibility (Gainesville, Virginia: Interference Control Technologies, 1988), p. 2.50.
2. John D. Kraus, Antennas (New York: McGraw-Hill Book Company, 1950), p. 43.
3. Reference Data for Radio Engineers, 6th Edition (Indianapolis: Howard W. Sams & Co., Inc, 1975), p. 27-8.
4. Kraus, p. 54.
5. N. Blabanian, Fundamentals of Circuit Theory (Boston: Allyn and Bacon, Inc., 1961), p. 13.
6. Kraus, p. 136.
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