When should active Perception use a lower difficulty than passive?

Question

There are a few places throughout the officially published adventures where the Perception DC to spot something differs depending on whether it is an active or passive check. However, this doesn't seem to be consistent. For instance (spoilers for Lost Mine of Phandelver):

The two starting traps on the Goblin Trail have the same active and passive DC:

"If the characters are searching for traps, the character in the lead spots the trap automatically if his or her passive Wisdom (Perception) score is 12 or higher. Otherwise, the character must succeed on a DC 12 Wisdom (Perception) check to notice the trap." (Phandelver, pg.7, my emphasis)

"The character in the lead spots the hidden pit automatically if his or her passive Wisdom (Perception) score is 15 or higher. Otherwise the character must succeed on a DC 15 Wisdom (Perception) to spot the hidden pit." (Phandelver, pg.7, my emphasis)

Whereas in these cases, passive Perception is given a significantly higher DC:

Rebrand Hideout: "Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check." (Phandelver, pg.20, my emphasis)

Cragmaw Castle: "Spotting the tripwire requires a passive Wisdom (Perception) score of at least 20, or a successful DC 10 Wisdom (Perception) check if the characters are actively searching for traps in the area." (Phandelver, pg.36, my emphasis)

The latter makes a lot of sense to me - a character focusing their attention (active check) should gain some sort of benefit over a character who might simply happen to notice something (passive). I like this a lot, if anything because it seems to help assuage the desire to keep both active and passive Perception relevant.

1. Firstly, are there any rules (or just reasonable) reasons as to why this doesn't seem to remain consistent in the published adventures?
2. Is there a standard (official?) way to determine what the DC difference between the active and passive checks should be when designing my own Perception situations?

Answer 1

In the cases you cited, the change in difficulty appears to be a result of more focused investigations.

"If the characters are searching for traps, the character in the lead spots the trap automatically if his or her passive Wisdom (Perception) score is 12 or higher. Otherwise, the character must succeed on a DC 12 Wisdom (Perception) check to notice the trap." (Phandelver, pg.7, my emphasis)

"The character in the lead spots the hidden pit automatically if his or her passive Wisdom (Perception) score is 15 or higher. Otherwise the character must succeed on a DC 15 Wisdom (Perception) to spot the hidden pit." (Phandelver, pg.7, my emphasis)

What appears to occur is that the passive and active Perception DCs are based on similar circumstances. Either both are searching for traps, or neither is. In contrast:

Rebrand Hideout: "Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check." (Phandelver, pg.20, my emphasis)

Cragmaw Castle: "Spotting the tripwire requires a passive Wisdom (Perception) score of at least 20, or a successful DC 10 Wisdom (Perception) check if the characters are actively searching for traps in the area." (Phandelver, pg.36, my emphasis)

What we see here are different assumptions. In the case of the door, we have the difference being passive catching it without paying any particular attention to the wall versus actively scanning the wall. The passive Perception has a higher DC because the user has other distractions to focus on, other walls to check. The active roll is paying especial attention to this one wall, so the lack of distractions makes the check easier.

In the case of the tripwire, we again see a difference in expectations behind the check. In the case of the passive check, the characters are simply being alert and not paying particular attention to any traps. The active roller is instead forgoing looking for other stimuli to focus on potential traps. This focus makes his check easier.

To answer your final questions, there doesn't appear to be much documentation regarding how to decide these DCs. It becomes a great matter of guessing.

Answer 2

Never

Specifically for traps the DMG (p.120) is clear:

A trap's description specifies the checks and DCs needed to detect it, disable it, or both. A character actively looking for a trap can attempt a Wisdom (Perception) check against the trap's DC. You can also compare the DC to detect the trap with each character's passive Wisdom (Perception) score to determine whether anyone in the party notices the trap in passing.

Even though this refers to "DCs", the context is that there are two: one to detect and the other to disable.

Further:

You should allow a character to discover a trap without making an ability check if an action would clearly reveal the trap's presence.

Obviously, as DM you can build your traps however you like with 16 different checks if you like but the assumption is that the trap is this difficult to find and while things the characters do (like actively looking instead of glancing around, or upping the lighting to remove disadvantage - don't forget dim light causes disadvantage to perception) can improve (or reduce) their senses, they don't change the trap itself. A side note is that an active check can never be worse than a passive - if you roll 1-9 on your active check you can always fall back on your passive result.

Lost Mine of Phandelver (and the Tyranny of Dragons) was written before the DMG using rules that were still evolving - it contains a number of things that are not in the final rules.