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If a device says it needs a particular voltage, then you have to assume it needs that voltage. Both lower and higher could be bad.
At best, with lower voltage the device will not operate correctly in a obvious way. However, some devices might appear to operate correctly, then fail in unexpected ways under just the right circumstances. When you violate required specs, you don't know what might happen. Some devices can even be damaged by too low a voltage for extended periods of time. If the device has a motor, for example, then the motor might not be able to develop enough torque to turn, so it just sits there getting hot. Some devices might draw more current to compensate for the lower voltage, but the higher than intended current can damage something. Most of the time, lower voltage will just make a device not work, but damage can't be ruled out unless you know something about the device.
Higher than specified voltage is definitely bad. Electrical components all have voltages above which they fail. Components rated for higher voltage generally cost more or have less desirable characteristics, so picking the right voltage tolerance for the components in the device probably got significant design attention. Applying too much voltage violates the design assumptions. Some level of too much voltage will damage something, but you don't know where that level is. Take what a device says on its nameplate seriously and don't give it more voltage than that.
Current is a bit different. A constant-voltage supply doesn't determine the current: the load, which in this case is the device, does. If Johnny wants to eat two apples, he's only going to eat two whether you put 2, 3, 5, or 20 apples on the table. A device that wants 2 A of current works the same way. It will draw 2 A whether the power supply can only provide the 2 A, or whether it could have supplied 3, 5, or 20 A. The current rating of a supply is what it can deliver, not what it will always force thru the load somehow. In that sense, unlike with voltage, the current rating of a power supply must be at least what the device wants but there is no harm in it being higher. A 9 volt 5 amp supply is a superset of a 9 volt 2 amp supply, for example.
If you are replacing a previous power supply and don't know the device's requirements, then consider that power supply's rating to be the device's requirements. For example, if a unlabeled device was powered from a 9 V and 1 A supply, you can replace it with a 9 V and 1 or more amp supply.
The above gives the basics of how to pick a power supply for some device. In most cases that is all you need to know to go to a store or on line and buy a power supply. If you're still a bit hazy on what exactly voltage and current are, it's probably better to quit now. This section goes into more power supply details that generally don't matter at the consumer level, and it assumes some basic understanding of electronics.
Very basic DC power supplies, called unregulated, just step down the input AC (generally the DC you want is at a much lower voltage than the wall power you plug the supply into), rectify it to produce DC, add a output cap to reduce ripple, and call it a day. Years ago, many power supplies were like that. They were little more than a transformer, four diodes making a full wave bridge (takes the absolute value of voltage electronically), and the filter cap. In these kinds of supplies, the output voltage is dictated by the turns ratio of the transformer. This is fixed, so instead of making a fixed output voltage their output is mostly proportional to the input AC voltage. For example, such a "12 V" DC supply might make 12 V at 110 VAC in, but then would make over 13 V at 120 VAC in.
Another issue with unregulated supplies is that the output voltage not only is a function of the input voltage, but will also fluctuate with how much current is being drawn from the supply. A unregulated "12 volt 1 amp" supply is probably designed to provide the rated 12 V at full output current and the lowest valid AC input voltage, like 110 V. It could be over 13 V at 110 V in at no load (0 amps out) alone, and then higher yet at higher input voltage. Such a supply could easily put out 15 V, for example, under some conditions. Devices that needed the "12 V" were designed to handle that, so that was fine.
Modern power supplies don't work that way anymore. Pretty much anything you can buy as consumer electronics will be a regulated power supply. You can still get unregulated supplies from more specialized electronics suppliers aimed at manufacturers, professionals, or at least hobbyists that should know the difference. For example, Jameco has wide selection of power supplies. Their wall warts are specifically divided into regulated and unregulated types. However, unless you go poking around where the average consumer shouldn't be, you won't likely run into unregulated supplies. Try asking for a unregulated wall wart at a consumer store that sells other stuff too, and they probably won't even know what you're talking about.
A regulated supply actively controls its output voltage. These contain additional circuitry that can tweak the output voltage up and down. This is done continuously to compensate for input voltage variations and variations in the current the load is drawing. A regulated 1 amp 12 volt power supply, for example, is going to put out pretty close to 12 V over its full AC input voltage range and as long as you don't draw more than 1 A from it.
Since there is circuitry in the supply to tolerate some input voltage fluctuations, it's not much harder to make the valid input voltage range wider and cover any valid wall power found anywhere in the world. More and more supplies are being made like that, and are called universal input. This generally means they can run from 90-240 V AC, and that can be 50 or 60 Hz.
Some power supplies, generally older switchers, have a minimum load requirement. This is usually 10% of full rated output current. For example, a 12 volt 2 amp supply with a minimum load requirement of 10% isn't guaranteed to work right unless you load it with at least 200 mA. This restriction is something you're only going to find in OEM models, meaning the supply is designed and sold to be embedded into someone else's equipment where the right kind of engineer will consider this issue carefully. I won't go into this more since this isn't going to come up on a consumer power supply.
All supplies have some maximum current they can provide and still stick to the remaining specs. For a "12 volt 1 amp" supply, that means all is fine as long as you don't try to draw more than the rated 1 A.
There are various things a supply can do if you try to exceed the 1 A rating. It could simply blow a fuse. Specialty OEM supplies that are stripped down for cost could catch fire or vanish into a greasy cloud of black smoke. However, nowadays, the most likely response is that the supply will drop its output voltage to whatever is necessary to not exceed the output current. This is called current limiting. Often the current limit is set a little higher than the rating to provide some margin. The "12 V 1 A" supply might limit the current to 1.1 A, for example.
A device that is trying to draw the excessive current probably won't function correctly, but everything should stay safe, not catch fire, and recover nicely once the excessive load is removed.
Now sometimes a Voltage is required from our electrical setup that does not fit a Standard Power Supply. To deal with this we have two great options. Use a Variable Power Supply or Utilise Buck/Boost Converters. Variable Power Supplies have dials that we can rotate to provide the exact voltage we need. Buck/Boost Converters have Potentiometers on them that will allow us to alter their output voltage to exactly what is required. We will still be feeding the electricity from a standard Powersupply into these converters. Buck Converters decrease voltage (at the cost of inefficiencies (heat) but do increase the total MAX current coming out of them). Boost Converters increase the voltage (at the cost inefficiencies (heat) and a lower MAX Current coming out of them). Buck/Boost Converters are also referred to as Step-Down/Step-Up Converters. Linked Here is a great guide to diving into Buck Converters to drive High Power LEDs. See this linked Buck Converter in the image below.
Now DC Power Supplies are not all made equal. Some Power Supplies are made big, and some are made to fit a very small footprint. These have trade-offs. Often you will hear about Switching Power Supplies and Linear Power Supplies. The big difference between the two (without getting into the weeds) is the Switching Power Supply can be made significantly smaller (tinier footprint) however the electricity coming out of it will be noisier. Now to the human ear Maker level Power Supplies are silent, so what do I mean by Noise? In electronics, noise is an unwanted high-frequency disturbance or interference with the electrical signal. For advanced electronics, like a Raspberry Pi Single Board Computer, a Noisy Power Supply can cause problems. It can cause issues for analog or measurement systems as well. For simple electronics, like DC Motors or Lights, Noise at this level is not a problem. Learn more on this here.
Naturally making sure you have the right connector to easily incorporate the power supply into your system is crucial too. Be it to the wall, which changes depending on your country, or the other end. Make sure the other end will plug into your system (whether it terminates with a Micro-USB, USB-C, or DC Power Line). I often use DC Barrel Jack Adapter (Female) in my projects.
To be ultra clear with this guide's terminology (however full understanding of the below is not crucial or needed when it comes to picking a power supply, feel free to gloss over):
- Voltage (also known as electric pressure, electric tension, or (electric) potential difference) is the difference in electric potential between two points. One volt is defined as the electric potential between two points of a conducting wire when an electric current of one ampere dissipates one watt of power between those points.
- An electric Current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. One ampere of current represents one coulomb of electrical charge (6.24 x 10^18 charge carriers) moving past a specific point in one second.
- Electricity is measured in units of power called Watts named to honour the legendary James Watt (the inventor of the steam engine way back in 1764). A Watt is a unit of electrical power equal to one ampere under the pressure of one volt.
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