the blue wire is usually AC for the feature lamp bridge and looks like it is going to the filter cap negative lead which is usually a black wire. I think someone has screwed around with that wiring so I would be double checking it.

Figure out the wiring first. Test the bridges. Add the safety fuses, see pinwiki or other guides. If the bridge is bad AC side fuse will sacrifice itself instead of burning the transformer.

Untitled (resized).png

Yes! I agree with what BarakandI said - the wiring looks to have been fanagled with so compare it to the drawing he included. And extremely important -- add the fuses!
https://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7#Additional_Fuses

The ground wires are normally black. Seeing a color that usually caries AC in the same area going the filter cap is concerning. It doesn't really matter what wire colors as long as the circuit wired up right. Hard to tell from your picture what exactly is going on but it looks like one of the wire insulation has melted off so that is a pretty good sign something is wired up very wrong or a bridge is shorted.

Hopefully no permanent damage has been done but don't power it up until you are sure it is wired up right, the bridges are not shorted, and the safety fuse added.

The blue wire going to BR1 is burnt. That is one AC leg from transformer. Second blue wire to BR1 is MIA, was apparently replaced by somebody with a yellow wire. Yellow may also be burnt since it looks a bit deformed at the top loop.
Take note of where the wires are, pull the quick connects from BR1 and test BR1 using a meter's diode test function. I'm leaning towards a shorted bridge rectifier. If so, replace with a GBPC3504 or GBPC3504L.

Before you power it up again - get yourself a couple inline fuse holders (e.g. Littelfuse 01550100Z) and 8A fuses and install them at the red "X" locations:

Fuse_Me (resized).jpg

With the mess in the wiring - do yourself a favor. Remove the mess that somebody put in there and rewire it properly.
Two new blue wires from transformer to bridge rectifier.
Two new red wires from transformer to bridge rectifier.
Add two new fuse holders and 8A fuses.
New violet wires from bridge rectifier to cap and connector 3P1.
New black wires from bridge rectifier to cap and connector 3P2.
New faston connectors (or solder direct to bridge).
New contacts for connections at 3P1 and 3P2.
Maybe new bridge(s) for BR1 and maybe BR2.
Maybe secondary side of bridge rectifier BR2 as well.

Wire size - depending on overall length of wire:

Fuse_Me_Wire_Size (resized).jpg

If you determine that one or both of the bridge rectifiers might be defective, you can get some here:
https://www.pinballlife.com/mm5/merchant.mvc?Screen=PROD&Product_Code=KBPC3504&Category_Code=
They are not very expensive.
Might as well replace them both.
You can keep one for a spare if they are both not bad.

Or better ones here:
https://www.greatplainselectronics.com/proddetail.asp?prod=GBPC3504L

If you are so inclined, you could post pictures of some of that special wiring that you encountered in this area:
https://pinside.com/pinball/forum/topic/worst-hackrepair-you-ever-saw
It will not be a really bad one, but it could qualify...

Leave the transformer in place (unless you can't reach the backside of transformer), you are only replacing the red and blue wires that go to lugs (solder points) on secondary side of the transformer. It takes a bit of heat to get the solder flowing on these so take your time.
The photo within post #1 shows the primary side of the transformer, opposite side not shown in photo is the secondary side which is where you would be replacing the wires.

Lugs #11 and 12 for lamps (blue wires), 13 and 14 for solenoids (red wires).

The transformer in photo of post number #1 actually looks pretty good. Only thing I would do is pull that brown primary wire away from lug #7 so that the lug doesn't chafe it's way through the insulation. And - in same photo, there appears to be a loose bit of wire under transformer that should be pulled out.

An option to inline fuse holders is to add a couple fuse blocks to the right of the two bridge rectifiers.
Here is an example for the fuse block:
https://www.greatplainselectronics.com/proddetail.asp?prod=354001GY

Oh, shit...

Yeah, that looks pretty nasty.
Perhaps those connections can be repaired.

Find a technician who will know how to repair that safely for you. It will be far easier and a lot cheaper than looking for a replacement - which may not be any better.

Like others have said - the very first job (forget the transformer for a moment) is to get some fuses installed before the bridges so things don't catch fire and burn your house down.

That transformer is

pasted_image (resized).png

Yes, perfect.

The transformer doesn't look cooked so I would try and find a technician to help connect new wires to it for you. $50 spent doing this should give piece of mind. A technician can also test the transformer on the bench to confirm it is OK.

To be blunt, if you have to ask this question you probably shouldn't be playing around with the mains. It can and does KILL people every day, even those who know what they are doing!

A friend of mine bought an early Stern with a suspect transformer and it came with a spare transformer.
The suspect transformer was faulty. We hooked up power to the spare transformer and it did exactly the same as yours. Got hot, made crackling noises and a smell after about a minute of power. Luckily he had his own spare working transformer which he ended up using.
The measurable DC resistances of the transformer windings including the input winding were lower than his good transformer.

If I were you, I would disconnect all transformer output connectors from the power/sound boards including the two bridge rectifiers and three fuses next to the bridges so the transformer outputs are totally isolated. Make sure the hacked bare wires are not touching anything they shouldn't.
If it still exhibits the same hot/smell/crackling symptom, it's time to look for a replacement used transformer. There are no reproductions yet.

Its rated 400V, 35A, so it is the same device.
They are pretty common.
I would not expect the photo to exactly resemble the device, but it is similar.

Where is the OP located? I would be willing to check this transformer on my impedance bridge and repair the wires. But would prefer he be local.

With the damage that close to the transformer itself, I'd be concerned that the windings may have also been cooked and damaged. I don't think I would really trust that transformer.

Personally, I would start putting feelers out for a replacement transformer.

There is this one on ebay, but shipping across the ocean might be a tad expensive: ebay.com link: Williams 15B6797 Transformer

An untested one, but it looks like it was in a flooded game, so it may be questionable: ebay.com link: Williams Pinball System 3 7 Transformer Assembly AS IS untested

This one may be promising: ebay.com link: Williams FLASH PINBALL Power Block Transformer 15B B797 Mepco 30000 uf Capacitor

You gotta snap up those hard-to-find items quickly before someone else nabs them.

I would guess that if there was an actual short in the transformer, there would have been a rather violent reaction, like an explosion or fire, or at least a blown 120V fuse on the input to the transformer. A problem with a bridge would create high current that would cause some heating in the transformer and possibly a buzzing sound. I think that is what is happening here, given the information that we can see at the moment. The same affect could be possible with a bad/damaged wire that has created a high resistance path to ground. My guess is that if the lead connections can be repaired, that the transformer is still ok. Some problem (like a bad bridge rectifier) on the low side of the transformer has caused it to get hot or buzz.

Yes, that would work.
But heat slug bridge rectifiers (phenolic body, metal top) have lower thermal resistance (lower is better) than full metal body bridge rectifiers.

If it’s shorted….you’ll know….