Arcane Power Regeneration Guide

Wizard
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TL;DR version skip to conclusions.

As an effort to help the Wizard community, I decided to do some math regarding Arcane Power Regeneration and increases to Max Arcane Power provided from some Wizard skills. In this post I am going to show how several skills compare with each other using a hypothetical scenario. First, I will provide an overview of the skills I will be comparing. Then will be some hopefully easy to follow math. Followed with conclusions.

Arcane Power Regeneration Skills:
Prodigy: This passive skill unlocked at level 20 allows you to gain 4 Arcane Power any time you successfully deal damage with a signature spell.

Astral Presence: This passive skill unlocked at level 24 allows you to regain 2 additional Arcane Power per second and adds 20 to your max AP

Magic Missile+Attunement: Magic Missile runed with Attunement, unlocked at levels 1 and 42 respectively, grants you 4 Arcane Power whenever you hit a target with Magic Missile.

Familiar+Arcanot: Familiar runed with Arcanot, unlocked at levels 22 and 50 respectively, allows you to regain 2 additional Arcane Power per second.

Energy Armor: Unlocked at level 28 will increase your armor but reduce your maximum Arcane power by 20. If runed with Energy Tap unlocked at level 48 it will instead increase your Arcane Power by 20.


Time for some math!
First things first, we need to establish a scenario. Our goal is to gain the most Arcane Power in 10 minutes. After learning how much Arcane Power we gain over 10 minutes we want to find out how much was usable, so for the sake of simplicity we are going to see how many Arcane Orbs runed with Tap the Source can be cast at 20 Arcane Power each. For the sake of the scenario we are using a 1.4 weapon speed wand and a 1.00 weapon speed staff. We will turn all of the regeneration statistics into what percentage of attacks you are increased over the Base. Effectively converting Arcane Power regeneration into a percent damage increase statistic based on standing and casting as long as we stay within our maximum range for weapon speed. Using our upper limits we will figure out how many signature spells vs. Arcane Orbs are cast and calculate an optimal conditions percent of total weapon damage.

Base Figures:
10 minutes = 600 seconds
Max Arcane Power = 100
Arcane Power Regen = 10 AP/1 sec = 1 AP/.1 sec
1.40 Weapon Speed = about .7 seconds/1 attack [upper limit of 857 spells in 600 sec] (.714 for the sake of simplicity I rounded, please note without rounding this will make a difference in the long run)
1.00 Weapon speed = 1 attack/sec [upper limit of 600 spells in 600 seconds]
Lets Start with the simple Arcane Power Regeneration.

Base over 10 minutes:
100 base + 600 sec (10 AP/sec) = 6,100 AP
6,000 AP/20 AP = 305 Arcane Orbs
Wand: 552 Magic Missiles
Staff: 295 Magic Missiles
Weapon Damage total:
Wand: 110(552) + 175(305) = 114,095% weapon damage to a single target
Staff: 110(295) + 175(305) = 85,825% weapon damage to a single target

Using Energy Armor:
80 base + 600 sec (10 AP/sec) = 6,080 AP
5,980 AP/20 AP = 304 Arcane Orbs
% Arcane Orbs compared to Base = -.33%
Wand: 553 Magic Missiles
Staff: 296 Magic Missiles
Weapon Damage total:
Wand: 110(553) + 175(304) = 114,030% weapon damage to a single target
% Damage Change compared to Base: -.0006%
Staff: 110(296) + 175(304) = 85,760% weapon damage to a single target
% Damage Change compared to Base: -.0008%

Using Energy Armor (Energy Tap):
120 base + 600 sec (10 AP/sec) = 6,120 AP
6,020 AP/20 AP = 306 Arcane Orbs
% Arcane Orbs compared to Base = +.33%
Wand: 551 Magic Missiles
Staff: 294 Magic Missiles
Weapon Damage total:
Wand: 110(551) + 175(306) = 114,160% weapon damage to a single target
% Damage Change compared to Base: +.0006%
Staff: 110(294) + 175(306) = 85,890% weapon damage to a single target
% Damage Change compared to Base: +.0008%

Using Familiar (Arcanot):
100 base + 600 sec (12 AP/sec) = 7,300 AP
7,300 AP/20 AP = 365 Arcane Orbs
% Arcane Orbs compared to Base = +19.67%
Wand: 492 Magic Missiles
Staff: 235 Magic Missiles
Weapon Damage total:
Wand: 110(492) + 175(365) = 117,995% weapon damage to a single target
% Damage Change compared to Base: +3.41%
Staff: 110(235) + 175(365) = 89,725% weapon damage to a single target
% Damage Change compared to Base: +4.54%

Using Astral Presence:
120 Base + 600 sec (12 AP/sec) = 7,320 AP
7,320 AP/20 AP = 366 Arcane Orbs
% Arcane Orbs compared to Base = +20%
Wand: 491 Magic Missiles
Staff: 234 Magic Missiles
Weapon Damage total:
Wand: 110(491) + 175(366) = 118,060% weapon damage to a single target
% Damage Change compared to Base: +3.47%
Staff: 110(234) + 175(366) = 89,790% weapon damage to a single target
% Damage Change compared to Base: +4.61%

Using Astral Presence+ Familiar (Arcanot):
120 Base + 600 sec (14 AP/sec) = 8,520 AP
8,520 AP/20 AP = 426 Arcane Orbs
% Arcane Orbs compared to Base = +39.67%
Wand: 431 Magic Missiles
Staff: 174 Magic Missiles
Weapon Damage total:
Wand: 110(431) + 175(426) = 121,960% weapon damage to a single target
% Damage Change compared to Base: +6.89%
Staff: 110(174) + 175(426) = 93,690% weapon damage to a single target
% Damage Change compared to Base: +9.16%
Now for the somewhat tricky math. For skills like Prodigy and Magic Missile (Attunement), we only want to cast our signature spells to help with regeneration of Arcane Power. So we will only cast Magic Missile if that .7 second or 1 second cast time will not allow our Arcane Power to cap. So we need some additional Base calculations.

Base over 10 minutes:
100 base + 600 sec (10 AP/sec) = 6,100 AP
6,000 AP/20 AP = 305 Arcane Orbs
Wand: 552 Magic Missiles
Staff: 295 Magic Missiles
Weapon Damage total:
Wand: 110(552) + 305(175) = 114,095% weapon damage to a single target
Staff: 110(295) + 305(175) = 85,825% weapon damage to a single target

Additionally...
.7 seconds/1 attack = -20 + 7 AP = -13 AP/1 attack/.7 seconds
1 seconds/1 attack = -20 + 10 AP = -10 AP/attack/second

Using Prodigy or Attunement:
Wand:
.7 seconds /1 attack = 4 AP + 7 AP = 11 AP/ 1 attack/.7 seconds
Divide by .7 and multiply by 10 to get = approx 15.71 AP/second
We find the most efficient method of casting would be to cast: 10 orbs, 9 MM, then 9 orbs, 8 MM four times, followed by 9 orbs and 9 MM bringing your AP total back up to 100. This sequence takes 73.5 seconds. Over the 73.5 seconds 38.5 seconds of your time is spent casting Arcane Orb and 35 seconds are spent casting Magic Missile. This can be extrapolated to percentages.

38.5 sec/ 73.5 sec = 52.38% at normal regen
35 sec/ 73.5 sec= 47.62% at accelerated regen

100 AP + [600 sec(.5238)] 10 AP/sec + [600 sec(.4762)] 15.71 AP/sec = 7,630 AP (7,631.46 exact)

Because we already know 52.38% of the time we are casting Arcane Orb
600 sec(.5238) = 314.28 sec / .7 sec = 449 Orbs (448.97 exact)
% of Arcane Orbs compared to Base = +47.21%
Wand: 408 Magic Missiles
Weapon Damage total:
Wand: 110(408) + 175(449) = 123,455% weapon damage to a single target
% Damage Change compared to Base: +8.2%

Staff:
1 second/1 attack = 4 AP + 10 AP = 14 AP/attack/second
We find the most efficient method of casting would be to cast: 10 orbs, 7 MM, 9 orbs, 6 MM, 9 orbs, 7 MM and repeat. That series takes 48 seconds and begins and ends with 100 AP. Over the 48 seconds, 28 seconds are spent at the normal 10 AP/sec regen (orb casting) and 20 seconds are spent at the accelerated rate of 14 AP/sec. This can be extrapolated to percentages.

28 sec/ 48 sec = 58.33% at normal regen
20 sec/ 48 sec = 41.66% at accelerated regen

100 AP + [600 sec(.5833)] 10 AP/sec + [600 sec(.4166)] 14 AP/sec = 7,100 AP (7,099.24 exact)

Because we already know 58.33% of the time we are casting Arcane Orb
600 sec(.5833) = 350 Arcane Orbs
% Compared to Base = +14.75%
Staff: 250 Magic Missiles
Weapon Damage total:
Staff: 110(250) + 175(350) = 88,750% weapon damage to a single target
% Damage Change compared to Base: +3.4%

Using Prodigy and Attunement:
Wand:
.7 seconds /1 attack = 8 AP + 7 AP = 15 AP/ 1 attack/.7 seconds
Divide by 7 and multiply by 10 to get = approx 21.43 AP/second

We find the most efficient method of casting would be to cast: 10 Arcane Orbs, followed by the repeating pattern of 6 Magic Missiles and 9 Arcane Orbs. This pattern is repeated 56.48 times over 593 seconds. Over the 600 seconds 362.824 seconds of your time is spent casting Arcane Orb and 237.216 seconds are spent casting Magic Missile. This can be extrapolated to percentages.

362.824 sec/ 600 sec = 60.47% at normal regen
237.216 sec/ 600 sec= 39.54% at accelerated regen

100 AP + [600 sec(.6047)] 10 AP/sec + [600 sec(.3954)] 21.43 AP/sec = 8800 AP (-8812.2532 exact)

Because we already know 60.47% of the time we are casting Arcane Orb
600 sec(.6047) = 362.82 sec / .7 sec = 518 Orbs (518.314 exact)
% of Arcane Orbs compared to Base = +69.83%
Wand: 339 Magic Missiles
Weapon Damage total:
Wand: 110(339) + 175(518) = 127,940% weapon damage to a single target
% Damage Change compared to Base: +12.13%

Staff:
1 second/1 attack = 8 AP + 10 AP = 18 AP/attack/second
We find the most efficient method of casting would be to cast: 10 Arcane orbs followed by the repeating pattern of 5 Magic Missiles and 9 Arcane Orbs. This pattern is repeated 42.1428 times over 590 seconds. Over the 600 seconds 389.2852 seconds of your time is spent casting Arcane Orb and 210.714 seconds are spent casting Magic Missile. This can be extrapolated to percentages.389.2852 sec/ 600 sec = 64.88% at normal regen
210.714 sec/ 600 sec = 35.12% at accelerated regen

100 AP + [600 sec(.6488)] 10 AP/sec + [600 sec(.3512)] 18 AP/sec = 7,785 AP (7,785.76 exact)

Because we already know 64.88% of the time we are casting Arcane Orb
600 sec(.6488) = 389 Arcane Orbs (389.28 exact)
% Compared to Base = +27.54%
Staff: 211 Magic Missiles
Weapon Damage total:
Staff: 110(211) + 175(389) = 91,285% weapon damage to a single target
% Damage Change compared to Base: +6.36%
Conclusions:

Energy Armor and Energy Armor (Energy Tap):
This skill only effects maximum Arcane Power so there is minimal impact on damage in a long term scenario. This does however affect your burst damage. With Energy tap you can get off 4 more Arcane Orbs with a staff and 3 more Arcane Orbs with a wand (at a higher rate of speed) when compared to Energy Armor without energy tap.

Familiar (Arcanot):
This skill in an idea situation allows you to cast up to +19.67% more Arcane Orbs compared to the base. This accounts for a potential +3.41% damage increase with Wands and +4.54% damage increase with Staves. The benefit of this skill is that Arcane Power is regenerating passively rather than actively. So as long as you continue spending AP your AP is not wasted. This allows you to be more mobile while maintaining your damage increase of Arcane Orbs (+19.67%). We also see that staves will yield a higher benefit from this skill than wands, although higher benefit does not necessarily mean more damage.

Astral Presence:
This skill in an idea situation allows you to cast up to +20% more Arcane Orbs compared to the base. This accounts for a potential +3.47% damage increase with Wands and +4.61% damage increase with Staves. The benefit of this skill is that Arcane Power is regenerating passively rather than actively. So as long as you continue spending AP your AP is not wasted. This allows you to be more mobile while maintaining your damage increase of Arcane Orbs (+20%). We also see that staves will yield a higher benefit from this skill than wands, although higher benefit does not necessarily mean more damage.

Astral Presence and Familiar (Arcanot):
This skill in an idea situation allows you to cast up to +39.67% more Arcane Orbs compared to the base. This accounts for a potential +6.89% damage increase with Wands and +9.16% damage increase with Staves. The benefit of this skill is that Arcane Power is regenerating passively rather than actively. So as long as you continue spending AP your AP is not wasted. This allows you to be more mobile while maintaining your damage increase of Arcane Orbs (+39.67%). We also see that staves will yield a higher benefit from this skill than wands, although higher benefit does not necessarily mean more damage.

Prodigy or Attunement:
This skill in an idea situation allows you to cast up to +47.21% more Arcane Orbs compared to the base with a Wand and 14.75% increase with Staves. This accounts for a potential +8.2% damage increase with Wands and +3.4% damage increase with Staves. The benefit of the skill is it has the potential for the most AP gain and therefore most damage. The downside is that any time AP is regenerating and you are not casting Magic Missile you waste AP, additionally if you miss a MM you waste AP. This allows for less mobility. We also see that wands will yield a higher benefit from this skill than staves by a large magin, although higher benefit does not necessarily mean more damage.

Prodigy and Attunement:
This skill in an idea situation allows you to cast up to +69.83% more Arcane Orbs compared to the base with a Wand and 27.54% increase with Staves. This accounts for a potential +12.13% damage increase with Wands and +6.36% damage increase with Staves. The benefit of the skill is it has the potential for the most AP gain and therefore most damage. The downside is that any time AP is regenerating and you are not casting Magic Missile you waste AP, additionally if you miss a MM you waste AP. This allows for less mobility. We also see that wands will yield a higher benefit from this skill than staves by a large magin, although higher benefit does not necessarily mean more damage.

Comparing Passive Arcane Power Regen vs. Active Arcane Power Regen:
All of the skills have their benefits but in scenarios where movement is required passive regen will mean less "wasted" Arcane Power generation. As we see with the passive regeneration skills, a X% increase in Arcane Orbs is directly equal to the increase in potential damage solely from Arcane Orb. For prodigy and attunement, this is not the case. The potential percentages of AP spenders are based on the amount of time you can cast your generator spell. The benefits and draw back here is that any time you are able to stand still and cast you will output more damage. But if you are required to move and cast your damage will be lessened.

Continued Calculations:
*It has come to light that if using a stutter step method you can actually increase casting speed by a significant amount as well as allowing movement between casts. Making Prodigy and Attunement much more enticing.
Using Prodigy:
Wand:
.7 attacks/sec = 4 AP + 7 AP = 11 AP/attack/.7 seconds
Divide by .7 and multiply by 10 to get = approx 15.71 AP/second


While you came to the correct mathematical result, you mean 0.7 seconds per attack, not 0.7 attacks per second. Could confuse some peeps.

Base over 10 minutes:
100 base + 600 sec (10 AP/sec) = 6,000 AP
6,000 AP/20 AP = 300 Arcane Orbs


For your baseline you're assuming that your 100 base AP is empty but then...
Using Familiar (Arcanot):
100 base + 600 sec (12 AP/sec) = 7,300 AP
7,300 AP/20 AP = 365 Arcane Orbs
% compared to Base = +21.66%

Using Astral Presence:
120 Base + 600 sec (12 AP/sec) = 7,320 AP
7,320 AP/20 AP = 366 Arcane Orbs
% compared to Base = +22%

Using Astral Presence+ Familiar (Arcanot):
120 Base + 600 sec (14 AP/sec) = 8,520 AP
8,520 AP/20 AP = 426 Arcane Orbs
% compared to Base = +42%


...for these three that your base is starting full. Just needs a minor adjustment to calculate your % compared to Base for a base of 6100, not 6000.

Overall though very solid guide of sorts, thanks for the efforts and sharing
@Shandlar, Thanks, totally overlooked that.

6
http://us.battle.net/d3/en/class/wizard/active/energy-armor

"Energy Tap

Rather than decreasing your maximum Arcane Power, Energy Armor increases it by 20 while it is active.
@Shandlar
I checked the thread right before I left and fixed the numbers. Thanks for pointing that out it was really late when I was working on this. I also changed the wording you suggested.

I will be finishing up more calculations and writing a conclusion when I get back from being in the field.

I figure I'll also write a TL;DR summary.
The Power Hungry passive can be substantial for Arcane Power recovery, especially if you've got a decent amount of pickup radius or mobile melee teammates who just happen to pick up globes as an auxiliary effect of their attack patterns. (so that more globes are grabbed during combat rather than afterwards in which case the Power Hungry AP gain is redundant with regeneration)

Unfortunately it would be a pain to formulate, due to the many situational factors that would need to be considered and health globe drop rate potentially being a variable across acts/difficulty. But perhaps it's worth mentioning regardless.
the guide is finished for now. If you find the information useful please request sticky. Any suggestions are welcome. Or if you want to do some calculations yourself I can add it in.
What about using electrocute with Prodigy instead of magic missile? or is it confirmed that the bounces will not proc Prodigy?
I could do a calculation for electrocute. But I want to wait until release rather than make the calculation on speculation if the bounces would proc prodigy extra times.

The way I read it is Prodigy would generate 4 AP regardless of how many enemies the spell hits, if this is true then there is no need to do the calculation because it will have the same effectiveness as MM with Prodigy. The difference lies with the Surge of Power rune. But it is still less effective than the Attunement rune ranging from 1-3 AP per hit. Surge of Power is even less effective than Astral Presence or Familiar (Arcanot) if you hit only 1 monster.

@scape:
Power hungry would be a great thing to calculate. But I have no idea how to do it. I bet if you were grouped with a barbarian with Pound of Flesh there would be a substantial AP boost.
nice guide man kudos and thanks for the effort
bump for open beta weekend.
Power hungry would be a great thing to calculate. But I have no idea how to do it. I bet if you were grouped with a barbarian with Pound of Flesh there would be a substantial AP boost.


Witch Doctor Grasp of the dead and Zombie Dogs both have Health Globe options as well
04/21/2012 05:32 PMPosted by Vaaz
Witch Doctor Grasp of the dead and Zombie Dogs both have Health Globe options as well


Good point, mathematically I don't know how to calculate that. :(
Awesome calculations. Will be interesting to see how damage for wands versus staves (versus other viable 1h and 2h weapons) stack up. Anyway, it's now pretty clear how much optimal damage bonus could be expected with these skills, so thank you for putting this together so crystal clearly.

04/15/2012 12:43 PMPosted by ßeta
Power hungry would be a great thing to calculate. But I have no idea how to do it. I bet if you were grouped with a barbarian with Pound of Flesh there would be a substantial AP boost.


Seems like more simple math unless I'm overlooking something dumb (which I often do); the main challenge is to estimate some sort of health globe drop rate. If you want to be really accurate, you'd begin with number of kills per minute, and estimate a globe drop rate per monster loot-table look-up. Either way, let there be g average globe drops per minute.

AP / sec = (essentially, globes / second) * (AP / globe)
AP / sec = (1 + cumulative bonuses to globe drop rate) * g * (1m / 60s) * 30 AP / globe
AP / sec = g' AP / 2s

Now we just need someone to play the game (not possible for the moment *sadface*) for a several hours on various difficulties and record the number of globe drops per monsters killed.

In the end, g' AP per 2 seconds is inline with the two passives. If you can keep up your kill rate, let's assume you find four in one minute, that's 2 AP/s for that minute, which is perfectly in line with Familiar/Arcanot or Astral Presence. However, this comes with the obvious downside that the player's AP regen will fluctuate more since it's dependent on the randomness of globe drop rate.

EDIT: Also, forgot to mention: that AP regen here is also dependent on kill rate, which is not such a bad thing. The more you're killing, the more AP you need and vice-versa.
@rodamn

I agree with you about the calculations you came up with for a base. But there are too many variables to come up with any sort of number I feel would be accurate or useful. There are a few inherent problems:
1) Health globes are pretty difficult to determine how many will drop per minute.
2) How much AP will be "wasted" per health globe.

Wasted AP meaning, AP you do no receive because you are <30 AP from your MAX AP.

The AP "wasted" is not reliably calculated and depends on player usage it would be a difficult number to calculate. We could however, once we know health globe drop rates, figure for certain percentages of wasted AP. I'll try and put that in the guide once we can figure Globe drop rates.

Thanks for the Kudos, if you like the thread and find it useful request a sticky or rate the post! Happy Gaming
Beta,
I am part of a theory crafting group for D3. I was wondering if you had an idea how to model a different question. I can't seem to private message here, so please get ahold of me on armadagaming under the same name!

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