Illumination efficiency

Last updated October 06, 2023

Antenna [aperture] illumination efficiency is a measure of the extent to which an antenna or array is uniformly excited or illuminated. It is typical for an antenna [aperture] or array to be intentionally under-illuminated or under-excited in order to mitigate sidelobes and reduce antenna temperature. It is not to be confused with radiation efficiency or antenna efficiency.^[1]

Definition

Antenna [aperture] illumination efficiency is defined as "The ratio, usually expressed in percent, of the maximum directivity of an antenna [aperture] to its standard directivity." It is synonymous with normalized directivity. Standard [reference] directivity is defined as "The maximum directivity from a planar aperture of area A, or from a line source of length L, when excited with a uniform-amplitude, equiphase distribution."^[1] Key to understanding these definitions is that "maximum" directivity refers to the direction of maximum radiation intensity, i.e., the main lobe. Therefore, illumination efficiency is not a function of angle with respect to the antenna [aperture], but rather is a constant of the aperture for all aspect angles.

Standard directivity

The distinction between maximum directivity and standard directivity is subtle. However, one can infer that, if an antenna [aperture] were excited [illuminated] uniformly with no phase difference (equiphase) over the entire aperture, then the illumination efficiency would be equal to unity. It is very typical for an antenna [aperture] to be intentionally under-excited [illuminated] with a "taper" in order to reduce radiation pattern sidelobes and antenna temperature. In such a design, the maximum directivity is reduced because the full aperture is not being used to the full extent possible, and the illumination efficiency will be less than unity. IEEE's choice of words is somewhat confusing, because "maximum" directivity is always less than or equal to "standard" directivity. The word maximum, in this case, is used to mean the maximum radiation intensity of the overall directivity pattern, which is otherwise defined for all aspect angles.

Relationship to antenna efficiency

There are critical differences in how various authors and IEEE define antenna efficiency and effective area of an antenna. IEEE defines the antenna efficiency of an aperture-type antenna as, "For an antenna with a specified planar aperture, the ratio of the maximum effective area of the antenna to the aperture area."^[1]

\eta _{a}={\frac {A_{e,max}}{A}}

and under effective area of an antenna, IEEE states, "The effective area of an antenna in a given direction is equal to the square of the operating wavelength times its gain in that direction divided by 4π." Gain is also defined to be less than directivity by the radiation efficiency, $\eta$ ^[1]

A_{e}=G{\frac {\lambda ^{2}}{4\pi }}=\eta D{\frac {\lambda ^{2}}{4\pi }}

However, other reputable authors define the effective area in terms of the directivity:^[2]^[3]

A_{e}=D{\frac {\lambda ^{2}}{4\pi }}

Either way, the standard directivity cannot exceed:

D_{std}\leq A{\frac {4\pi }{\lambda ^{2}}}

since $\eta _{a}\leq 1$ .

Per the IEEE definitions:

D_{max}=\eta _{i}D_{std}\leq {\frac {\eta _{i}}{\eta _{a}}}A_{e,max}{\frac {4\pi }{\lambda ^{2}}}={\frac {\eta _{i}}{\eta _{a}}}\eta D_{max}

where $\eta _{i}$ is the illumination efficiency.

However, per the definition of other authors:^[2]

D_{max}=\eta _{i}D_{std}\leq {\frac {\eta _{i}}{\eta _{a}}}A_{e,max}{\frac {4\pi }{\lambda ^{2}}}={\frac {\eta _{i}}{\eta _{a}}}D_{max}

So clearly there is a problem. If the IEEE definitions are true, then ${\frac {\eta _{i}}{\eta _{a}}}\eta =1$ and therefore $\eta ={\frac {\eta _{a}}{\eta _{i}}}$ . Or, if the other authors are correct, then $\eta _{a}=\eta _{i}$ .

Related Research Articles

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In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

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References

1 2 3 4 IEEE Standard for Definitions of Terms for Antennas. IEEE. Std 145-2013.
1 2 Cheng, David K. (1992). Field and Wave Electromagnetics. Reading, MA: Addison-Wesley. pp. 634–637. ISBN 0-201-12819-5.
↑ Balanis, Constantine A. (2016). Antenna theory: analysis and design (4th ed.). Hoboken, New Jersey. p. 86. ISBN 978-1-119-17898-9. OCLC 933291646.{{cite book}}: CS1 maint: location missing publisher (link)

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[IEEE-1] 1 2 3 4 IEEE Standard for Definitions of Terms for Antennas. IEEE. Std 145-2013.

[Cheng-2] 1 2 Cheng, David K. (1992). Field and Wave Electromagnetics. Reading, MA: Addison-Wesley. pp. 634–637. ISBN 0-201-12819-5.

[Balanis-3] Balanis, Constantine A. (2016). Antenna theory: analysis and design (4th ed.). Hoboken, New Jersey. p. 86. ISBN 978-1-119-17898-9. OCLC 933291646.{{cite book}}: CS1 maint: location missing publisher (link)

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