Q:

Is the homebrew weapon attack cantrip 'Blizzard Blow' balanced?

I have created some new cantrips which include making a weapon attack as there are only two official ones to choose from. I've tried to follow these principles based on Booming Blade and Green-Flame Blade:

Melee weapon attack only, 5 foot range
No unavoidable extra damage to the primary target at level 1
Damage no more than 2 targets
Spellcasting ability modifier doesn't have to affect the spell at all
Only available on the Warlock, Sorcerer and Wizard spell lists

But with such a small amount of official content to compare them against, it's hard to say if they're balanced.
Is this spell balanced with Booming Blade and Green-Flame Blade?

Blizzard Blow
Evocation cantrip
Casting Time: 1 action
Range: 5 feet
Components: V, M (a weapon)
Duration: 1 round
As part of the action used to cast this spell, you must make a melee
attack with a weapon against one creature within the spell's range,
otherwise the spell fails.
On a hit, the target suffers the attack's normal effects, and an icy
blast from the impact fills the target’s space with frigid air until
the start of your next turn. A creature that ends its turn in the
frigid air takes 1d6 cold damage.
At 5th level, the melee attack deals an extra 1d6 cold damage to the
target, and the damage from the frigid air increases to 2d6. Both
damage rolls increase by 1d6 at 11th level and 17th level.

A:

Pretty Comparable to Booming Blade, But Probably a Little Better
Essentially this is Booming Blade except with d6's of cold damage instead d8's of thunder damage, and damage triggering on staying still instead of moving. So let's consider what that means.
In terms of the amount of damage it is obviously an average of one less damage per die and there are a lot more enemies with resistance or immunity to cold than to thunder. So this part clearly favors Booming Blade's d8's of thunder.
However it is more common to not be able to or not want to move than to be forced to move, especially since in both cases they are now in melee with the caster. When a low HP mage comes and stands toe to toe with them creatures are often content to stay right where they are and keep hitting them, and this cantrip punishes them for that. In a crowd or tunnel they may be unable to move or unable to move without triggering someone's opportunity attack, and there are plenty of ways to magically keep someone still. Booming Blade often requires some gimmicky way to force or incentivize them to move, often involving burning some sort of resource to make them chase you without triggering an opportunity attack.
There are also times when you could use it to incentivize an enemy to move to one or two specific available spots in a manner more predictable than Booming Blade, setting up all sorts of cool options for you or a teammate under various circumstances. Probably not as useful as using Booming Blade to incentivize them to even more predictably stay just where they are, but still lots of uses.
It will also matters how your DM plays enemy spell knowledge. Logically many enemies would not recognize that a Booming Blade attack would subsequently punish them for moving, but even a dumb beast would sense that "it's really cold here; I should move."
I would argue that your Blizzard Blow has the superior damage trigger to Booming Blade for the most part, but, obviously, it has objectively inferior damage. While I think the prior somewhat more than makes up for the latter, and would almost certainly take this over Booming Blade for most characters of mine, it is really hard to say without some serious playtesting.
On the whole it seems balanced. It also would lend itself to some interesting strategic decisions, which is one of the things that make for a really great cantrip.

A:

Balanced, probably underpowered.
This seems very similar to BB except that goal is have the target move away from you. The key difference here is that while a target may want to chase you (and suffer BB effects). This is situational and won't provoke an OA as long as they stay in range but change their space. Bu
Unlike BB where if you can move away, the target is likely to follow (and therefore trigger the boom), they really have no reason to not stay engaged. Simply moving out of the icy area can be done and they remain safe. The only time this works is if there are no other paths for them to escape but taking an OA.

