I could see how a loose hot line could cause AC noise under load.
Not talking about AC noise. I'm talking about DC riding on the AC causing the transformer core to act like an electromagnet with a fixed north and south pole. The core's not supposed to do that. The primary side is supposed to create an alternating magnetic field that induces an alternating EMF into the secondary side. The magnetic pole flips at 50hz/60 hz to do that. A fixed DC voltage from the primary side wrapped around the core will turn the core into an electromagnet no different than wrapping a wire around a nail and connecting it to a battery. This causes heat which can damage the insulation of the transformer's windings.
This is why DC riding on AC is called a transformer killer...it causes unnecessary heating of the core and the insulated wires that make up the transformer winding's. Eventually, the insulation on the winding's will fail causing a transformer short circuit and death. The added stress can cause a transformer to hum.
The safety ground is not tied to the variac secondary.
Not necessarily true. It depends on the Variac.
This is bad.
The input and output ground sockets of the variac appear to be connected to something, possibly to the cabinet. Same for the RFI filter. The schematic says they're connected to the house ground.
At the service panel (fuse box) the Neutral buss is connected to the Ground buss.
Therefore every neutral wire of every wall outlet in the house is tied together as well as to ground at the service panel. The neutral on the variac's input and output are therefore directly connected via ground in the service panel. No isolation at all as DC can still travel from anywhere in the house along the neutral through the ground buss/Neutral Buss connection in the service panel. Every neutral in the house is connected regardless of whether or not the outlet or appliance has a ground plug.
In the diagram above the RFI filter only passes AC noise to ground via a capacitor. Since capacitors block DC, no DC on the primary side can pass to ground. Any DC on the primary side (can be neutral or hot wire) will therefore magnetize the core like a nail.
Grounding the transformer's core only means if a wire loses it's insulation and contacts the core and goes to ground, the fuse will blow. This is an example of transformer death. The insulation on the windings has failed.
This is still not good.
Neutral is still connected to ground at the service panel. If the AC output in the above diagram has a ground, it's still connected to neutral. No isolation.
This is ideal.
A two wire isolation transformer. No pathway from primary to secondary for DC to travel. The Secondary is pure AC. If any DC is riding on the primary side, only the isolation transformer will be at risk, and possibly hum. No ground for DC to travel via the Neutral to Ground buss connection at the service panel. No way for DC to pass even if a light dimmer or flat screen TV puts DC on the Neutral/Ground or hot wire.
If a variac has 3 wires in and out...I don't see how it can serve as an isolation transformer if the output ground is connected.