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Totally newbie in eletronics here, but could not find the answer for this one.

When a battery is nearly-exhausted, it's voltage and or amperage dims, right?

My question is if this kind of scenario (running low battery) would affect the arduino internal clock using only it.

I'm using arduino to a device that will operate remotely (with a 9V battery), and also logs intervals using the time function.

If battery is near-exhausted, then will this skew the numbers I get?

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Microprocessor clock crystal frequencies can vary a few parts per million when microprocessor supply voltages vary significantly. For example, a Czechoslovakian blog article (which was mentioned in avrfreaks.net and is shown in automatic-translated form at translate.google.com) shows a nearly-negligible drop of about 14 PPM as supply voltage for an Atmel ATtiny2313 rises from 2.4 V to 3.5 V or drops from 5 V to 3.5 V. In the results, crystal frequency was about 4000000 Hz near 2.4 V and 5 V, and drops to 3999945 Hz near 3.5 V.

I refer to that 14 PPM frequency change as nearly negligible because typical inexpensive crystals have ±50 PPM or worse accuracy and stability specifications, and ±50 PPM or worse allowance over temperature changes of a few degrees, and somewhat larger allowances as the crystal's load capacitance varies by a few pF. Thus, crystal frequency changes due to supply voltage changes are negligible compared to changes due to other causes.

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Let's assume by "internal clock" you mean the internal oscillator, as the arduino's processor does not have a built-in real-time clock (something that is meant to keep track of real time like seconds/minutes/hours.)

The arduino is not like a motor where it runs slower if a lower voltage is applied. If the voltage goes too low, the arduino will just reset. Once it resets, the program running in it just starts over again, unless the battery voltage is so low it can't even start up.

If you're feeding it with a 9V battery, you're going through the 5V linear regulator on-board, which is not very energy efficient. Keep that in mind if your goal is to create a project that is meant to run a long time on the 9V battery. See the notes at http://www.gammon.com.au/power for tips on reducing the power used by an arduino and/or ATMega-based project.

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