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I have a LED strip with a wonderful no touch dimmer on the bottom of the kitchen cabinets. What I would like to do is to "link" the inner lighting of the cabinets to the same switch, in a manner that when the dimmer is turned on, the cabinet lighting is also turned on, but without any dimming.

My first thought was to add a relay in series to the dimmer, which would turn on the strip directly from the power supply. But the dimmer adjusts the voltage between 7 and 12 V. Therefore the voltage would kill a 5 V relay, and the 12 V relay would only work on max dimmer brightness.

My question is, is there a good way of making this work (haven't found relays that work with multiple DC trigger voltages), without adding an additional switch?

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Why don't you use a transistor to control the relay instead of directly controlling the relay?

Transistors can handle enough volts if you chose the right one. The transistor output will give 12V (or 5V if you put a DC/DC converter in between the power and transistor), which controls the relay.

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Relays are not static things. They work based on current through a coil to actuate a contact. The coils are fairly forgiving. You can often turn on a relay coil at a percentage of its nominal voltage. Most relays will pick up at 70/75% of nominal Up to 125% for safe operation. For a 12V relay that's roughly 9V to 16V. For a 9V relay you are looking at 6ish to 12ish volts.

And a relay typically has a much lower release voltage. And it's fairly slow. So a fast PWM signal at 70% would be enough to keep the relay on.

Using a 9V relay, paired with a diode (1n4001) and capacitor (1000uF) to smooth out the pwm, you should be able to turn that dimming output into a usable on/off signal. But even a 12V relay will probably work with enough capacitance. See this answer https://electronics.stackexchange.com/a/386573/17178

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