Doorbell replacement

That's an ideal transformer, a simplified model of a transformer for classroom usage.

Real transformers are non-ideal devices. More elements in the transformer represent significant effects, and must be modeled.

Let's test.

I plug in the Brother LabelMaker AC transformer, into the Kill-A-Watt meter. This represents a transformer suspected to be "in good shape". I do have a transformer here, a small one, in failure state, but I can't find it now (if that remains plugged in for an hour, it becomes too hot to touch).

Unloaded transformer (primary connected, secondary disconnected)

The 2.5 watts will be delivered as heat, to the room. I would be billed for that vampire load.

I have a 200 watt transformer here, that runs ice cold, and it is less wasteful than the failed 20 watt transformer. There is some variation in "transformers in good shape", as to what level of power they waste when unloaded. Some are wasteful, even when brand new.

Look at a practical transformer design. There are laminations. Thin steel plates coated in varnish, then slapped together. The purpose of the varnish, is as an insulator, good to a certain temperature. The insulation in that case, prevents the flow of eddy currents. We want flux to flow in our transformer core, but without wasteful eddy currents at the same time. That's why we use laminations.

"Putting it all together gives us a model for a transformer shown in Figure 4a.

There is a resistance and inductance in series with the primary and secondary windings of the ideal transformer.

Rp and Rs are the winding resistances, Ls and Lp are the leakage inductances due to imperfect flux coupling.

There will also be a resistance and inductance across the primary;

Rc, which represents the core losses (hysteresis and eddy current) and Lm which represents the magnetising current. "

The winding short... remains unmodeled in diagrams like that one. A winding short needs yet another model.

Since I've owned a transformer that was unloaded, and too hot to touch, I would say that transformers do fail (via one or more failure mechanisms). And you can likely think of various mechanisms where a failure can occur. The visible evidence on the failed transformer, is surface rust on the outside. That makes you think about where else the rust could be hiding.

The bell ring transformer in question, is an energy limited transformer. Does such a designation make a difference to the analysis ? Yes, partially. You would have to carefully examine the magnetic circuit, how the laminations are arranged, to develop an opinion which failure modes are likely. Your bell push is energy-limited. Your shaving outlet in the bath, is energy limited. And the transformer in the furnace, in the thermostat circuit, is also energy limited (to cover cases where a home owner carelessly shorts something there). I could tell my spiffy new furnace had one, because of the hum that became evident a couple of months after taking delivery. Energy limited transformers, seem to hum a bit more. Because they don't know the words.

There is a kind of transformer, which does not use laminations. How lucky is that ? It is the toroidal transformer, which suggests all the flux travels in a useful direction. The core of those, is not necessarily made out of what you would think. And you have to be careful with toroidal transformers, as they have an "inrush problem". When you energize one, there is a fairly violent current flow at first. The line voltage in the house could drop, glitching some of your electronics. Hammond Manufacturing, on their toroid sales page, warns about the inrush problem, but does not provide a practical circuit diagram of how designers choose to fix that! Which I consider to be pretty careless. If you're going to sell an item with an obvious problem like that, a suggested application circuit would be nice. To design a counteracting circuit for that, we need to know the magnitude and duration of the excess currents involved (don't want to blow our NTC resistor to hell and back). The inrush current happens, even when the secondary of the toroidal transformer is unloaded.

Paul

The key is the magnetic leakage. And the primary resistance. Putting all those wattless VAs through resistance generates heat, as does core hysteresis,

BIG toroids are very very good.

Small 'E & I' laminated ones, not so much

In the distant past, I have used small 50Hz transformers, before switch-modes were a practical price. Just looking at specs, the smaller the transformer, the less efficient it was. Since the volume was also smaller, they tended to run hotter than the larger transformers, even when unloaded.

They use far fewer winding turns than conventional transformers, due to better control of the magnetic circuit, so the primary DC resistance of a 300VA transformer is a fraction of an Ohm. 240V AC mains hits about 340V at the peaks, so turning on the transformer at just the wrong time results in a stupidly high surge current, enough to weld switch contacts. The various fuses in circuit do limit the current a bit, but not predictably.

I needed to use one, of around 300VA, to drive a 3-channel power amplifier assembly. This was maybe thirty years ago, and proper mains semiconductor switching was still fairly expensive, and I didn't want thyristor noise on the power rails. I wanted the mains as clean as possible.

I used a relay and (big) series resistor to keep the surge down to about 10A, and a spreadsheet to analyse the startup and work out when to turn on the shorting relay so that the surge wasn't too high. With the output rectified and feeding big smoothing capacitors, calculation was pointless and a numerical method was called for. Given the power involved and the value of the load, I didn't try for zero-volt switching, just brute force and failsafe ignorance.

It's just another tool a professional engineer has in his toolbox :-)