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I have a 5V 1A portable battery with USB port (used for charging smartphones). Is the current okay for arduino? Can I use it without worries it'll fry?
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the arduino, and devices hooked to it will draw what current it needs regardless of capabilities of the supply

drawing 10ma with a 1 amp supply, no problem, drawing 10 amps from a 1 amp supply, the supply will fry
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The battery also has 2.1A port... But I guess it's useless as max current through arduino is 200mA. Right?
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Arduino USB port also has 500mA series fuse, will be hard to draw more than that.

"drawing 10 amps from a 1 amp supply, the supply will fry"
Not necessarily - many (most?) switching power supplies are short circuit protected and will simply shut down until the short is removed.
Linear regulated supplies, agree, damage may occur.
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Ok so it means I can use it and be aware nit to draw more than 200mA total (maximum for Mega 2560 R3). Right?
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No, 500mA. Same as a PC supplies thru the USB port.
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What about the limit? If powering, for example, leds through digital pins, someone mentioned in another threads that he max total current through digital pins is 200ma, also 40ma each?
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1. Total current coming in thru the USB port is 500mA.
2. Digital pins have an absolute max of 40mA each, after which damage may occur.
2a. LEDs typically have max continuous current of 20mA. Some may be pulsed for short durations with more. Read the datasheet of the part you want to use.
2b. Further, the Ports of the Mega chip have max limits for sinking/sourcing current:

Although each I/O port can sink more than the test conditions (20mA at VCC = 5V, 10mA at VCC = 3V) under steady state
conditions (non-transient), the following must be observed:
ATmega640/1280/2560:
1.)The sum of all IOL, for ports J0-J7, A0-A7, G2 should not exceed 200 mA.
2.)The sum of all IOL, for ports C0-C7, G0-G1, D0-D7, L0-L7 should not exceed 200 mA.
3.)The sum of all IOL, for ports G3-G4, B0-B7, H0-B7 should not exceed 200 mA.
4.)The sum of all IOL, for ports E0-E7, G5 should not exceed 100 mA.
5.)The sum of all IOL, for ports F0-F7, K0-K7 should not exceed 100 mA.
If IOL exceeds the test condition, VOL may exceed the related specification. Pins are not guaranteed to sink current greater
than the listed test condition.

Although each I/O port can source more than the test conditions (20mA at VCC = 5V, 10mA at VCC = 3V) under steady
state conditions (non-transient), the following must be observed:
ATmega640/1280/2560:
1)The sum of all IOH, for ports J0-J7, G2, A0-A7 should not exceed 200 mA.
2)The sum of all IOH, for ports C0-C7, G0-G1, D0-D7, L0-L7 should not exceed 200 mA.
3)The sum of all IOH, for ports G3-G4, B0-B7, H0-H7 should not exceed 200 mA.
4)The sum of all IOH, for ports E0-E7, G5 should not exceed 100 mA.
5)The sum of all IOH, for ports F0-F7, K0-K7 should not exceed 100 mA.
If IOH exceeds the test condition, VOH may exceed the related specification. Pins are not guaranteed to source current
greater than the listed test condition.

2c. Further, the Mega has 4 Vcc pins, each pin can support 200mA. Thus the part can support 800mA of current, more than the USB port can supply.  I posted a definitive answer from Atmel about this sometime in January.
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Designing & building electrical circuits for over 25 years. Check out the ATMega1284P based Bobuino and other '328P & '1284P creations & offerings at  www.crossroadsfencing.com/BobuinoRev17.
Arduino for Teens available at Amazon.com.

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