# Balanis: *Antenna Theory — Analysis and Design* **Citation:** Balanis, Constantine A. *Antenna Theory: Analysis and Design*, 4th edition. John Wiley & Sons, 2016. **Author:** Constantine A. Balanis, Regents' Professor of Electrical Engineering at Arizona State University. Long-standing standard reference textbook on antenna theory used in graduate and upper-undergraduate electrical engineering programmes worldwide. Earlier editions (1st 1982, 2nd 1997, 3rd 2005) are also cited in the literature, with formulas and chapter numbering largely consistent across editions. ## Overview The standard modern textbook on antenna theory. Covers fundamental concepts (radiation integrals, directivity, gain, aperture efficiency), wire antennas (dipoles, loops, arrays), aperture antennas (horns, reflectors, apertures), broadband and frequency-independent antennas, and numerical methods. The BotB analysis uses Balanis as the canonical reference for the aperture-antenna directivity formula: $G = \frac{4\pi A_e}{\lambda^2}$ where $A_e$ is the effective aperture area and $\lambda$ is the wavelength. For a uniformly illuminated rectangular aperture the effective area equals the physical area, and the peak directivity is $G = 4\pi A/\lambda^2$. ## Used in the BotB null hypothesis Cited in the System specifications table of [[Knickebein_Propagation_Null]] for the directivity calculation of the Large Knickebein antenna: - Physical aperture: 99 m (width) × 29 m (height) = 2,871 m² - Wavelength at 31.5 MHz: λ = 9.517 m - Directivity: $G = 4\pi \cdot 2871 / 9.517^2 = 398.4$ (linear) = **26.0 dBi** The formula is in Balanis Chapter 12 (Aperture Antennas). The 26.0 dBi figure is the peak directivity of a uniformly illuminated rectangular aperture of the given dimensions at 31.5 MHz and is used as the TX gain in every flat-model and globe-model link budget calculation throughout the BotB analysis. ## The formula From Balanis, *Antenna Theory*, 4th ed., Chapter 12, the peak directivity of a uniformly illuminated aperture of physical area $A_p$ is: $D_0 = \frac{4\pi}{\lambda^2} A_{em}$ where $A_{em}$ is the maximum effective aperture area. For a uniformly illuminated aperture (no tapering of the illumination, which maximises gain at the expense of sidelobe suppression), $A_{em} = A_p$ and the formula reduces to: $G_{max} = \frac{4\pi A_p}{\lambda^2}$ In the BotB calculation: - $A_p = 99 \times 29 = 2{,}871$ m² (physical aperture of the Large Knickebein array) - $\lambda^2 = 9.517^2 = 90.57$ m² - $G_{max} = 4\pi \cdot 2871 / 90.57 = 398.4$ - In decibels: $10 \log_{10}(398.4) = 26.00$ dBi ## Assumptions and limitations The BotB analysis uses the **uniformly illuminated** directivity value, which is the maximum achievable directivity for a given aperture size. The actual Knickebein antenna used two rows of dipoles with outriggers to shape the equisignal corridor, which would reduce the effective aperture slightly and introduce tapering. The real directivity is therefore marginally below 26.0 dBi. The BotB analysis uses the uniform-illumination value as an **upper bound on the TX gain**, which is generous to the globe model (the globe model needs as much signal as possible to stand a chance of reaching the Midlands on a sphere). Using a lower effective directivity would make the flat model equally usable and the globe model worse, so the assumption is conservative relative to the null hypothesis conclusion. ## Related formulas from Balanis Other Balanis formulas relevant to the BotB analysis but not directly cited in the null doc: - Half-power beamwidth of a uniformly illuminated rectangular aperture: $\theta_{3dB} \approx 50.6°/L_{\lambda}$ where $L_{\lambda}$ is the aperture dimension in wavelengths. For the 99 m wide Knickebein aperture at $\lambda = 9.517$ m, this gives $\theta_{3dB} \approx 4.86°$ horizontal beamwidth, consistent with the measured Knickebein beam pattern. - Aperture efficiency: ratio of effective to physical area, typically 0.6-0.8 for practical antennas. The BotB analysis uses 1.0 (uniform illumination) as the conservative upper bound. ## Access notes Balanis is a widely-held university textbook available through major engineering libraries. The 4th edition (2016) is the current reference. Earlier editions contain the same aperture-antenna formulas in Chapter 12 with minor pagination differences. --- ## See Also - [[1946_Friis_Simple_Transmission_Formula]] — The link budget equation that uses the Balanis directivity as the TX gain input - [[Knickebein_Propagation_Null#The system]] — Where the 26.0 dBi directivity figure is cited