my two main relay identification are marked in blue and purple:
What does the blue part means? that it should be powered from 5V dc?
what does the purple marked means?
generaly asking - if I need to use relay with a load that working with +12V I should have a relay marked at 12V and if I need to use a relay with a load that working with +5V I should have a relay marked 5V ?
Yes, although often a jumper needs to be added/removed/moved on the 12V relay board.
Yes, otherwise the signal won't be received by the relay board.
Another solution is to power your relay board and the arduino with a DC buck converter like this DC DC converter.
It have enough output power (3A) to supply the arduino and the 4 relays at 5v. You can power it with your 12volts supply already available for your solenoid.
Since you already have the relay board, you would only need to find a converter. Not sure what is cheaper, new converter or new 4 relay 12v board?
No. The cyan box you put in your image shows what voltage is needed to run the relay coil, 5V in this case, which is handy because that's what your Arduino can provide.
The magenta box specifies what the load side is capable of switching, which basically amounts to 10A whether its AC or DC and irrespective of the voltage. I assume most of that is there because lawyers.
Note though that on the side powering the relay coils (the Arduino), you may be surprised by how much current those coils need. That is why your relay module allows you to power them separately.
I see. How can I know how much current the relay spring is needed? No I start to understand why in some cases is better to power the relay itself from a external power supply and not from the arduino itself.
The data sheet for the relay should tell you. Sadly, those numbers may be optimistic. They will likely claim ~30mA or less but may actually be as much as three times that. Use your DMM to be sure.
So if that's true in your case, the coils may pull as much as 360mA when all are activated. If you're powering the Arduino through the barrel jack, that's too much to be sustained and you'll get thermal shutdown.
OK, now I have taken a closer look at that relay board and noted the "JD-VCC" jumper.
Here is the standard recommendation for wiring it:
Given that you have the 5 V version, you need a 5 V supply capable of providing four times 70 mA (given the "SLC" version of the relay) or say 300 mA. The one cdb101 cites would suit. There is I believe a 12 V version of the board so the "separate supply" for that version could then be your same 12 V supply.
Perhaps also note that the two wires from the 5 V "separate supply" to the relay board should run together as a pair from one device to the other, and similarly the connections from relay board to Arduino should travel as a group and the wires to the power supply, relay board and solenoid likewise. Do not leave any open loops of wiring.
I that wiring diagram the relay coil gets is power from the external power supply. no GND is connected between arduino and the relay and the expternal psu therefore now path for current between relay and arduino - therefore more 'secure' connection?
The main concern is that impulses from the solenoid switching - or the relay switching - will be coupled to the Arduino inputs or at least, some part which will interfere with its operation and potentially crash it. If you have no direct connection between the Arduino and the relay, that can make things a trifle more "secure" in that regard.
Note that you can still use the same power supply to power the three different parts - the solenoid, the relays and the Arduino. What is required is that the supply and ground for each part are run as I described, as a pair, from that part directly to the power supply (or buck regulator) and only connect together at the output terminals of that power supply, where the main decoupling capacitor is.
Proper routing of all wiring in this manner, avoiding open loops, is generally much more important than using completely separate supplies.
Put another way, the relay module is designed to be used in two ways - first is easy mode where the Arduino powers the relay coils and you accept the risk of interference crashing it. The other takes full advantage of the opto-isolators to ensure that there is no electrical connection between the Arduino and the coils. This second method is safer, but marginally less convenient.
What does the term 'open loop' mean?
Where the two "legs" of a circuit - power and ground, or signal and ground - take different paths.
So to say avoiding open loops is same as connect all grounds together?
No, completely unrelated. 
Hello together , hello hk_jk
First some general things of elektronics.
If you want to control whatever you want to control by a signal you always need a reference point to find out which level the signal has.
This means for this Board that from the arduino you need GND and the signal output. this has to be conneted to the GND of the Board and the input connetor.
The Bouard itself is powered by a DC Voltage to control and activate the relays. This power is taken out of a seperate DC power supply. the ground of this power supply is connected to the ground Power connector of the relay board and can be f.e. in your case 12V DC or maybe another higher DC votage if the relay board requires this.
For the Arduino you use a seperate Power supply to put power to the Arduino. This is again connected to GND and the +5V DC Input or the +Vcc input at the Arduino board.
This is in general the way this two board are working together.
In your case with that 12V DC you have aspecial situation and you can use the 12V DC directly to power up the Arduino board if you use the +Vcc input bevause the Arduin board has an own Power unit on Board do generate 5V DC out of higher DC voltages but due to the components un the Arduino board the maximum possible DC input is 12V otherwise you will overheat the power converter.
I have not been talking abolut the relay contacts for the output till now.
The used relay have no electrical connetion to the power of the board. there is only a isolated mechanical connection that is fixed to the magnet in the coil that that moves the contacts here yopu have a relay that has a general contact and two toggle contacts. the funktion is as follows:
You put pover to the general contatct and if the relay is not activated this contact is connected to the one of the toggle contacts that is normally marked with "NC". as soon as the relay is activated the "NC" output is switched off and the other output, normally marked with "NO" is witched on.
Restrictions of the Voltage and the amount of current, this relay is relating to the constructinon of the contacts istself. thers is a difference if you switch AC or DC Voltage. That is why there are some diffent values for the Voltage.
Your relay says (in that purple field that it is sutable to switch 10A at 250V AC and also 10A 125V AC. For DC the values are much smaller there it is agan 10 A but only 30V DC maximum. The same restriction of 10A is at 28V DC.
In the blue field you have the "name" (type) of the rela. If you use this at a list of the manifactoring company you wii find a data sheet of that relays with all information about the relay. Also in the internet you will find this data sheet if you just enter: "SRD-5VDC-SL-C"
Sorry for some typing mistakes
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