It's not easy to find a voltage regulator that can handle 48V (and I assume that's why you tried to use a voltage divider).
To do this, I am taking the 36v input power, then use a voltage divider circuit, which utilizes 2 resistors (R1 = 1Kohm and R2=2.15Kohm). This circuit drops my input voltage to ~32.8V.
...Until you connect the load in parallel with the 2.15K resistor which lowers the effective resistance and the voltage drops to almost nothing. The effective resistance of active circuits is not constant... For example if you turn-on an LED that's more current (less resistance). So a regular voltage divider won't give you a constant-reliable voltage.
I then take this voltage an pass it through (1) 7805 Voltage Reducer
The proper description is voltage regulator, but you're right that a linear voltage regulator can only reduce the voltage. There are some switching regulators (or "DC-DC converters") that can boost the voltage.
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The "problem" with linear voltage regulators (78xx, etc.) is this:
Power (watts, related to heat) is calculated as Voltage x Current. The current that flows through the load (the Arduino) is the same current that flows through the regulator but the voltage is divided. So with 49V applied you have 5V across the load and 43V across the regulator. Let's say you have 1A (which is more than the Arduino should be using but it makes the example calculation easy) the Arduino is dissipating 5W and the voltage regulator is dissipating 43W. Even the regulator might be rated for up to 1A, it will overheat and shut-down. (The 7805 has a built-in temperature sensor and some "smarts" so it's not supposed to permanently burn-up.)
If you are regulating-down from 9V to 5V and/or if the current is very low, a linear regulator is not so bad, and they are very cheap & easy to use!
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Switching (AKA "switchmode") regulators or power supplies work differently:
The main transistor of MOSFET is switched on & off rapidly like a switch. (It's never partially-on.) When it's on, current flows through it but there is (almost) no voltage across it, and when it's off there voltage across it but no current through it. So it dissipates almost no power. The energy is stored in an inductor and capacitor and the final output voltage is filtered for constant DC (not switched on & off even though the regulator is switching internally).
The near 100% efficiency (almost no wasted power/heat) means you get more current out (at lower voltage) than goes-in! Back to the 5V x 1A = 5W example - 5W at 48V is slightly more than 0.1A so with all of the energy is going into the load you're only drawing about 0.1A from the 48V power supply! (Of course you never get 100% efficiency but more than 90% is not unusual.)
I am not using any capacitors within the 7805 or 7812 circuit.
That's not your problem this time but the manufacturer's data sheet tells you to use capacitors so if you don't use them and it doesn't work, you can't complain.