The Tesla cannon is a pre-War, portable, directable energy weapon. It was designed to be used as a lightweight, compact anti-tank weapon. The design was named after Nikola Tesla, whose designs for the technology were seized by the U.S. government after his death. When used, the cannon projects a blast that will damage the enemy on impact. It will continue to gradually reduce the enemy's health due to its ability to electrify the target and its surroundings.
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 test 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.
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 Tesla cannon can kill NPC's without Karma loss using the splash damage. Just fire near them (not too close and not too far) and the splash damage should trail them and drain most of their health.
The Tesla cannon was originally to be featured in the canceled Fallout: Brotherhood of Steel 2, bearing no resemblance to the weapon in Broken Steel. Described as a "high-powered mass-cannon throwing tessellated plasma forward in an orgy of death and destruction" in the Fallout: Brotherhood of Steel 2 design document, it was to be the so-called "uber-weapon" of player character Maxus. "Uber-weapons" were to be unique weapons for each player character which had to be assembled from 5 parts over the course of the game, or the player could have a NPC assemble one for them. One of the five parts for this weapon was a Tesla coil.
Nikola Tesla really did try to create a directed energy weapon but it would have been a particle beam weapon called a "Teleforce", something that would be immensely complicated to create - laser weapons would be simple by comparison.