As someone trying to learn electronics, I found the following diagram where there are these two electrolytic capacitors (470uF).
Could someone explain the use of these components? How does the number/size 470 get chosen? Is their inclusion meeting a particular requirement?
In any electronic circuit, especially digital ones like this, current drawn from the power supply will vary, as things like LEDs and logic gates switch on and off. Sudden changes like these can cause momentary spikes or dips in the voltage provided by the supply, which propagate around the circuit.
The long wires usually used to distribute voltage (and carry current) to the various modules of the circuit, have inductance. The effect of inductance is to make it hard for electric current to change quickly. Think of it as like momentum/inertia of a moving object, which prevents the object's velocity from changing quickly. In a circuit which has things switching on and off quickly and frequently, causing sudden increases or decreases in current demand, that long wiring makes it difficult for the power supply and its distribution system to "keep up". If the power supply is suddenly asked to provide an extra 10mA for an LED for example, inductance makes it impossible for the supply to instantly comply. The consequence is a momentary drop in supply voltage, while the power supply and its wiring "catches up".
There are usually several elements connected to that same supply, and they will all "see" these fluctuations, and can suffer to some degree. For example, such a supply voltage disturbance might cause an audible "pop" in an audio system. In your circuit here, you have a very complex microcontroller and display system, in which a moderate supply disturbance could flip a memory bit or even cause a reset, or any number of glitches.
The presence of capacitance between power supply rails acts as a short-term reservoir of energy, able to supply extra current when demand spikes suddenly (and temporarily), or absorb excess energy when demand drops suddenly. This has the effect of "smoothing out" the potential difference between the two supply rails, keeping that voltage much steadier than it would be without the capacitors.
Capacitor C10, to the left of the 7805 regulator in your circuit, performs this function. That regulator is very good at preventing voltage disturbances at its input from reaching its output, but not that good. C10 lends a helping hand to the regulator, so that severe fluctuations in voltage (from an essentially untrusted 12V source) are quenched to some degree prior to regulation by the 7805.
On the output side of the regulator, you also have capacitor C9, which performs a similar function, on behalf of the the circuitry following the regulator. But C9 also has a second purpose. The 7805 (especially older models) actually requires some capacitance across its output in order to be stable. Without at least a few microfarads there, it's possible for the regulator output to oscillate, which can be devastating to any sensitive circuitry connected to it. All regulator datasheets tell you the minimum (and sometimes maximum) capacitance that can/should be present across its output.
Fat electrolytic capacitors, like those 470μF devices C9 and C10, are good at accommodating moderately fast changes in current demand, but often you will also find much smaller (1nF to 100nF) capacitors across the supply too. These would be present to mitigate really fast spikes and dips.
The actual values and construction of these capacitors would be chosen according to the conditions they are expected to operate under. 470μF seems excessive to me, for such a small system, but it could be that the designers are anticipating that you will be adding additional loads, perhaps badly behaved ones like relays or motors, later on. Typical values to find immediately either side of a regulator such as the 7805 would range from 10μF to 220μF, depending on the expected current through the regulator, and the nature of the loads it is supplying. Again, the datasheet usually provides recommendations for choosing capacitor values.
If we don't know what the source of 12V is (at the regulator's input), a large capacitance there (well above 220μ) is not uncommon, since we have no idea of how clean that supply is. However, if you knew that the 12V supply was very clean, and very stable, you could probably reduce C10 to 10μF. Lower value capacitors are physically smaller and less expensive, which might be important to you.
It's difficult to say how big these capacitors should be, because every application and regulator has its quirks and idiosyncrasies, but values you choose for C9 and C10 here would range from tens to hundreds of microfarads, for current demands ranging from tens to hundreds of milliamps. That's a really "ballpark" guideline though.
Nothing wrong with using those values. Maybe they had them around . Much larger and the capacitors rated for adequate voltage will get physically relatively large, but for 100-220uF or less they're relatively small.
Larger capacitors provide a bit more reservoir in case of a momentary surge of current or blip in the input (especially when isolated with a diode in regard to the latter consideration) but that does not appear to be much of a factor here given the components shown.
Electrolytic caps as shown have relatively high ESR, a few ohms maybe for ordinary parts, but the LM7805 does not require very low ESR and there are low-ESR bypass caps on the Arduino Nano module.
The reason why there are capacitors in the first place is that the regulator just needs them to work properly.
Without the capacitors, the 7805 might not be a regulator any more, but it might ne unstable and start oscillating at some frequency.
Usually there are two small caps to handle high frequency ripples and two larger caps to handle lower frequency ripples.
You would generally use simple maths to figure out lower or upper limit for your capacitor and then take some value that is above the minimum you need or less than the maximum you can have.
For the value of 470 uF as the only capacitor on input and output, the value is likely chosen at random or not changed from some default value. 470uF is likely a poor, maybe even unsuitable or just wrong. Or it might be the value you happened to have and it worked and solved some problem. At least it is likely better to have 470uF than no capacitor, but it again depends how long input or output wires there are to regulator and if there are capacitors at the end of the wires already in the power supply or load.
Those capacitors are bypass capacitors, and are normally chosen pretty much according to the datasheet of the regulator; these are much larger than recommended. Under some conditions they can be omitted but you can get instability in the voltage. Consider them as just part of "how you use a 7805 regulator".
Capacitors at these places are required by the 7805 specs and the other answers explain why.
In regard to their value: The datasheet requires much less of a values at both places, say 1uF.
On the other hand, when one creates hobby kits, some important factors get into play when selecting elements:
Ease of handling. 1uF capacitor is too small and has tiny leads. 470uF is much more finger-friendly and also has readable markings on the package. Small elements tend to have cryptic abbreviations or simply too small markings.
The circuit has to be simple, but not necessarily efficient. If a single big capacitor can save few other elements, let be it.
The kit has to be cheap, but not necessarily reliable. If the "proper" part is expensive and the off-spec, but working part is available at lower price, the cheaper one is used.
