Very few LEDs will dim completely to off. Part of the way LEDs are built prevents this. The internal power supply in an LED lamp is typically built to maintain a constant light output, and this design works against the function of a dimmer that throttles the power down that the lamp receives. There are a few fixtures that are built with this in mind, and they are nearly always labelled as 1% dimming, or .1% dimming, fixtures. These typically cost a lot more than a typical LED.
The way in which the LED power supply interacts with the electronics in a dimmer depends on the construction of both. There are no standards, so many manufacturers provide use case testing to show the results with different products. I have linked an excerpt from a test report for one manufacturer and a specific Philips lamp (Philips
409904). As you can see, the interaction is also not completely linear, and is totally different from the way incandescent lamps work.
As far as general types of dimming, there are at least 6.
The first, a simple Rheostat. Rheostatic dimmers divide the incoming voltage into two rails via a resistor, some of the voltage goes to the lamp, and some is directed away from the lamp. Most of this lost voltage is dissipated as heat. These are rarely used today, and not suitable for use with LED.
The second type is Triac. A triac is a type of electronic gate, that can latch open and closed really fast, and is able to let very tiny amounts of power through. Most old rotary and slide dimmers are of this type. These dimmers date back to the late 1950's. These dimmers work in an unpredictable fashion with LEDs.
The 3rd type is PWM. PWM works via a very fast switch, that can stay on for a specified amount of time, letting a specific amount of power through. PWM dimmers are available in two common types, called Leading Edge or Forward Phase, and Trailing Edge or Reverse Phase. These describe the portion of the power wave that they intend to let through. Different LED lamps behave differently when attached to these different types. Many LED manufacturers will specify which type works best with their product.
The other types are not commonly found in residential applications, and involve separate wired or wireless control signals in addition to power to control the lamp.
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