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Is there a standard way of monitoring the current draw of a device like an Arduino using its own analogue inputs?
I was thinking of putting a resistor in the positive wire before the it connects to the Arduino then using two resistive dividers, one on each side connected to 2 analogue inputs and then measuring the voltage difference between the two. Is this the best approach? Can it be done using a single analogue input?


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Depends on power supply, if it provides regulated voltage or not. In first case you would not need one of the voltage dividers, as Vin = constant. If it's not regulated, you have to use differential amplifier ( OPA ) or more sophisticated current sensor (hall effect) instead of  a resistor .
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Using a current sense resistor and current sense amplifier, you could do what you want:

http://www.maxim-ic.com/products/amp_comp/current_sense/
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Or better still use a hall sensor powered by the system voltage which gives an isolated 0-5 volts output
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Using a current sense resistor and current sense amplifier, you could do what you want:

http://www.maxim-ic.com/products/amp_comp/current_sense/

Current sense amplifiers seem expensive, at least in small quantities.

I think I'll have a go at making a differential amp
http://www.allaboutcircuits.com/vol_3/chpt_8/9.html
as I already have some LM324 chips which have 4 op amps in the one package.

Resolution wise I only need to know the current to within about 10% so I can manage battery life. Hopefully with a 100mv drop across the sense resistor I will be able to ignore offsets and temperature effects.
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Hall effect sensors are not sensitive enough for small amounts of current that the arduino draws.
The problem is that the analogue input pins can't take a voltage higher than the supply rail which is what you have by definition from a sense resistor. Look at the data sheet but I don't think the LM324 can work at close to it's rails which is why i think you will struggle to get a reading across the sense resistor.
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I'll use resistive dividers to divide down the two voltages before feeding them to the op amps.
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See attached for a hall sensor with 250mA fullscale range
http://www.gmw.com/magnetic_sensors/ametes/documents/Ametes%20CS_Spec_020707.pdf
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http://focus.ti.com/lit/ds/symlink/ina219.pdf
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Quote
See attached for a hall sensor with 250mA fullscale range
The problem is that this is now a discontinued product. The TI one however might work for this application.
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