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I have 10 bulbs that are like these: https://www.1000bulbs.com/product/63652/IND-0053-10PK.html (14v incandescent). They are in a hardware display that was created for other purposes. They light just fine if I connect a 9v battery, so perhaps they don't actually need the 14v.

I need to control them from the RPi GPIO.

Only 1 bulb ever needs to be on at any given time.

I would just use 10 GPIO pins, but the Pi only puts out the 3.3v, and so the bulbs just don't light.

Replacing these with a bright-enough LED might work, but the bases are already installed, and the hardware is nice, so I'd rather just use the existing wires and put something together.

I'm relatively new to electronics, so I'm not sure what I would use (some kind of external power supply?) or how to control the lights independently.

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What you need is something similar in function to a "voltage divider". And many ways of implementing it, eg using "Opamp".

A good intro to Opamp is here:

https://www.youtube.com/watch?v=kqCV-HGJc6A

And here is a setup using Opamp (Vin is your GPIO, and Vout is 14V)

enter image description here

More references:

http://henrysbench.capnfatz.com/henrys-bench/arduino-projects-tips-and-more/op-amp-basics-for-arduino-hobbyists/

https://electronics.stackexchange.com/questions/170764/the-purpose-of-voltage-dividers-in-op-amp-feedback

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You need to use an appropriate transistor and an external power supply. The transistor is a three-pin device that will (if you choose the right one) let you use a 3.3 V GPIO pin to switch current on and off on a higher voltage circuit.

Do not try to connect you GPIO directly in the 14 V circuitry. In the best case, that will result in nothing happening. In the worst case you'll burn out your Pi.

To get the right transistor, you'll need to shop around and check specifications. You probably want a MOSFET device for this application. If you search for those terms, you should find a lot of information from many sources giving details of how to connect this. When you're shopping, make sure that the device you choose is rated for the 14 V load at the current level that you expect and that it can be "saturated" (turned completely "on") with 3.3 V or less.

Here is a related but different question from EE.SE that includes a circuit diagram. In that case they are talking about 24 V, but the same basic principles apply.

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If you are not very familiar with Electronics, the easier way is to use one of those Relay Shields (just google for "relay shields raspberry pi") so that you don't need to figure out the circuitry and transistor yourself. A relay shield generally consists of a number of channels of relay circuitry, each with a control transistor, a relay plus a few components. Power the relay shield with 5v and common ground, and connect your GPIO to the relay shield control pin of one of the channel. You can connect your 14v to the NO (stands for Normal Open) of the relay, and light bulb to the Common of the relay. This basically adding a switch in serial to your light bulb power source.

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