CALC-098

The thermal limit said to have sized the 1978 transmit antenna, closed at its own design point

Feeds CLM-0891. Answers QST-004. Basis: kW/m2 of RF power density at the array centre (waste heat as the alternative reading).

Why it was needed. VC05's governing relationship is not the efficiency chain, and SRC-085 says so at PDF p.124 (printed A-25), quoted in CLM-0023: "Thermal dissipation limits set a heat density value on the transmit antenna of approximately 21 kW/m2. For the operating frequency of 2.45 GHz, studies have indicated that the power density of the SPS beam should not exceed 23 mW/cm2. ... With these two limits, and a desired value of power output at the rectenna, transmit antenna size and rectenna size can be traded." TWO CONSTRAINTS TRADED AGAINST EACH OTHER, COUPLED BY DIFFRACTION. The efficiency chain is an input to that trade, not the relationship itself.

Relationship.

transmit side   q_waste = (P_rf / A_t) x (peak/average) x (1 - eta) / eta   <=  21 kW/m2
ground side     beam power density at the rectenna                          <=  230 W/m2
coupling        diffraction fixes rectenna size from aperture and wavelength

Inputs.

Working.

aperture area          pi x 500^2                  = 785,398 m2
average RF density     6.72e+09 W / area          = 8,556 W/m2 = 8.56 kW/m2
average/peak           (1 - e^-ln10) / ln10        = 0.39087, so peak/average = 2.5584
peak RF density        8.56 x 2.5584            = 21.89 kW/m2
waste heat at centre   21.89 x (1 - 0.85) / 0.85   = 3.863 kW/m2

Result. THE PEAK RF DENSITY AT THE ARRAY CENTRE, 21.89 kW/m2, SITS AT THE STATED LIMIT. SRC-085 states the limit at the array centre as "22 kw/m2rf" (PDF p.281, via CLM-0204; also PDF p.40 Figure 15 and PDF p.222), that is, as RF power density, not waste heat. Read that way, the design point is at its limit, which fits the statement that the limit sized the antenna.

The alternative reading, kept for completeness. If the 21 kW/m2 were waste heat, the centre would carry 3.86 kW/m2 against it, about 5.4 times below the limit; the two limits would then be mutually consistent only at a DC-to-RF efficiency of 0.510, against the 0.85 of the same document's Figure 25. The "kw/m2rf" wording on PDF p.281 favours the RF reading.

Sensitivity. If 6.72 GW is measured after the 0.9653 pointing and tolerance factor, aperture RF is 3.6 per cent higher and peak becomes 22.68 kW/m2, just above the 22 kW/m2 centre limit. The peak/average factor is the assumption that matters: a uniform aperture would give 1.0 and a heavier taper more than 2.56, so the match at the limit holds only for the assumed 10 dB Gaussian.

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