The Tesla cannon is a pre-War designed shoulder-mounted heavy directed energy weapon, designed to replace conventional anti-tank weapons. Based on Nikola Tesla's designs, the weapon entered limited production before the war and remains a rare sight in the wasteland. However, it remains a powerful tool in the hands of a skilled user, as the focused energy blast will not only deliver massive damage to the target, but the nature of the blast will also deliver a potent electric shock.
The modified Tesla canon employed by Lyons' Brotherhood of Steel is far more powerful than the pre-war Tesla canon. Discharging a large electrical impulse upon impact with a much larger area of effect. This explosive damage makes the Lyon's model far more suitable for anti-Vertibird use.
As already mentioned the weapon is both identical to and uses the same ammunition as ordinary missile launchers, but still fires the same blast of electricity as the normal Tesla cannon. This Tesla cannon (it is just called "Tesla cannon" in the game) delivers less damage than the actual cannon, and uses up more A.P. per shot, but it can actually be repaired by ordinary missile launchers (but not vice versa). This means that unlike the regular Tesla cannon, the testPhil version can be repaired to 100% indefinitely (assuming the player's repair skill is high enough).
One copy of this weapon can be found outside the map boundaries at Satellite Relay Station. It is obtainable via some rather tricky jumping, building an ammunition staircase, using Dogmeat, or console commands.
Getting it Yourself
This glitch can be utilized during, or after the Liberty Prime destruction sequence. After Liberty Prime blows up and his head is propelled southwards use his head as reference and head southwest towards an outcropping of rocks. You should soon hit an invisible wall, denoting the map boundary, after trying to jump up onto a ledge. After you are in a position where you can see the Wasteland but you are blocked by the invisible wall, SAVE the game here.
Reload that same save, and while the game is loading, start moving forward until the game loads and your character has moved past the invisible wall (you may need to reload and try a number of times).
Now that you have made it past the invisible wall, head slightly south as you are going westwards. You will eventually come across a table with what appears to be a missile launcher levitating above it and 3 ammunition boxes underneath. This weapon is the Beta Tesla cannon, and the ammunition boxes contain some missiles.
Exit the cliffs from the way you got up. If you can't however, don't worry as there are a number of ways to get back inside the map boundary:
Skirt round to the north side of the map and jump down from on top of the building that got partially blown up during the Liberty Prime set-piece.
Find your way to the concrete support (with a communication transmitter on it) that is on the opposite side of the concrete support next to Liberty Prime's head. Jump onto it and again to ground level.
Jump at/onto ground-level trees from the rock ledge above (you should hear a wooden plank noise when you land), then jump again from the tree back onto the ground.
Using 5mm ammunition boxes as leverage in order to scale the rocks or the fence at the beginning of the area is also a possible method.
Using Dogmeat's weapon finding command, then waiting a few game hours, should glitch him to where the weapon is outside of the map, allowing it to be retrieved from him. This process can be sped up by already standing close to where the weapon is outside the map.
The Beta Tesla cannon is actually found levitating above a table with 3 ammunition boxes of missile ammunition underneath.
The Beta Telsa cannon can kill non-player character without Karma loss just fire near them (not too close and not too far) and the splash damage should trail them and drain most of their health.
Behind the scenes
A Tesla cannon was first planned for inclusion in Fallout: Brotherhood of Steel 2, as a "high-powered mass-cannon throwing tessellated plasma forward in an orgy of death and destruction", assembled from parts found during the game (including a Tesla coil).