We present a detailed spectroscopic analysis of the luminous blue variable AG Carinae during the last two visual minimum phases of its S-Dor cycle (1985-1990 and 2000-2003). The analysis reveals an overabundance of He, N, and Na, and a depletion of H, C, and O, on the surface of AG Car, indicating the presence of CNO-processed material. Furthermore, the ratio N/O is higher on the stellar surface than in the nebula. We found that the minimum phases of AG Car are not equal to each other, since we derived a noticeable difference between the maximum effective temperature achieved during 1985-1990 (22,800 K) and 2000-2001 (17,000 K). While the wind terminal velocity was 300 km/s in 1985-1990, it was as low as 105 km/s in 2001. The mass-loss rate, however, was lower from 1985-1990 (1.5 x 10^(-5) Msun/yr) than from 2000-2001 (3.7 x 10^(-5) Msun/yr). We found that the wind of AG Car is significantly clumped (f=0.10 - 0.25) and that clumps must be formed deep in the wind. We derived a bolometric luminosity of 1.5 x 10^6 Lsun during both minimum phases which, contrary to the common assumption, decreases to 1.0 x 10^6 Lsun as the star moves towards maximum flux in the V band. Assuming that the decrease in the bolometric luminosity of AG Car is due to the energy used to expand the outer layers of the star (Lamers 1995), we found that the expanding layers contain roughly 0.6 - 2 Msun. Such an amount of mass is an order of magnitude lower than the nebular mass around AG Car, but is comparable to the nebular mass found around lower-luminosity LBVs and to that of the Little Homunculus of Eta Car. If such a large amount of mass is indeed involved in the S Dor-type variability, we speculate that such instability could be a failed Giant Eruption, with several solar masses never becoming unbound from the star.(abridged)
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