Electronic – Lionel Model Train Transformers: Request for explanations of how things inside work

\$\operatorname{rect}(\cdot)\$ is a function that has value \$1\$ if whatever appears inside those parentheses (it's called the argument of the rect function and I used \$\cdot\$ instead of some algebraic variable as a place holder) has value between \$-\frac 12\$ and \$+\frac 12\$. Otherwise, when the argument is strictly smaller than \$-\frac 12\$ (or strictly larger than \$+\frac 12\$), \$\operatorname{rect}(\cdot)\$ has value \$0\$. In your instance, you need to ask

For what values of \$\omega\$ does \$\displaystyle \frac{\omega - 10}{2\pi}\$ equal a number between \$-\frac 12\$ and \$\frac 12\$?

and a little thought will show, I hope, that \$\omega\$ must be in the interval from \$10-\pi\$ to \$10 + \pi\$. If you have trouble deriving this, try and find the value of \$\omega\$ that makes \$\displaystyle \frac{\omega - 10}{2\pi}\$ equal exactly \$-\frac 12\$ and then, lather, rinse and repeat for exactly \$\frac 12\$.

A variac works just fine. I have a medium sized AC fan hooked directly up to a small, 1A variac which I made years ago as a quick and dirty solder fume extractor, but it worked so well that quick and dirty turned into 'what I've used for years'. The speed control works exactly like you'd expect.

What the amazon seller says is correct. You're also correct. The key difference is that virtually all AC fan motors are a special kind of induction motor called a shaded pole induction motor. These motors do not need any external capacitors to introduce phase lag, and instead rely on copper rings around each of the poles that act as single turn transformers, introducing a delay in the magnetic flux for that pole, and the net effect is a second phase.

This makes for a fairly inefficient, but extremely reliable motor. Due to their construction, they generate torque over their entire RPM range, including stopped, and their torque is mostly dependent on voltage, except for some nonlinearities at very slow speeds where reduced torque is generated.

You can see this in most fans - you turn them on but they take a while to get up to speed. The load from the fan blades and the air they move usually determines the final speed, which is whatever speed where the generated torque and the load torque match.

Since the load on a fan increases with rotation speed, this means that simply by varying the voltage, and thus the torque, this will also vary the speed the fan will spin, as less torque generated means the fan will reach that matching load torque at a lower speed.

Also, just to be clear, there is no magic going on with the TRIAC driver, it's nothing more than a dimmer circuit. It's chopping up the AC waveform to generate a lower Vrms. It's just a very noisy and harmonic filled way of reducing the average voltage, so using a TRIAC driver isn't any different from using a variac.

The variac would be much less noisy, and not add any harmonics, and ignoring cost, size, and weight, would be the preferred solution. The primary advantage of TRIAC drivers are cost. Either will work, but a small variac would be the more elegant and low-noise solution.