Unique Passive – Insight: Gain ability power equal to 3% of your maximum mana.
Unique Passive – Mana Charge: Each time you cast a spell or spend mana, you gain 6 maximum mana (3 second cooldown). Bonus caps at +750 bonus mana.
Assuming a charge is gained every 3 seconds, the 750 mana Unique Passive takes 7 minutes and 30 seconds to max. (The Tears of the Goddess takes 9 minutes and 20 seconds).
There is a 3 charge banking system to help burst casters.
Archangel's Staff has great synergy with a Rabadon's Deathcap, as it increases ability power by 30%. This means that whenever your mana is increased, the Archangel's Staff will increase your ability power, and then the Rabadon's Deathcap will increase it again due to the 30% ability power passive.
By building a full set of items giving mana, (fully stacked Archangel's Staff, maxed Rod of Ages,Banshee's Veil,Frozen Heart and Lich Bane ) you will acquire total bonus 2950 mana, which will grant total 88.5 bonus ability power, not counting for champion's base mana rating and other stats given by those items indefinitely.
To gain more AP from the Archangel's Staff than a single Rabadon's Deathcap (which gives 150 ability power if alone) a champion would need more than 3334 mana. However, a champion would only need 2334 mana for the Archangel's Staff to provide more ability power than a second Rabadon's Deathcap.
Rammus'sPowerball adds one stack upon cast and another stack if ended early without a collision. Similarly, Janna'sHowling Gale adds one stack upon activation, and a second if activated early.
Singed'sPoison Trail adds one stack whenever the ability's toggle is active and the Tear/Archangel's Staff stack is not cooling down (ie, continuously).