How can I power a cellular phone for a week with a car battery or other type of battery?
Can I just connect a cigarette lighter USB plug to a car battery directly and plug the phone into the USB?


This is exactly what I thought of when I was reading your... text wall. Using an USB plug would be the cheapest way to power your phone, concerning both, costs and time.

The plug reduces the 12V from the battery to 5V for USB. There are still old plugs with a linear regulator out there, which waste more energy than the phone consumes. As they have to dissipate this as heat, they usually can not provide much power. So, pick one which can deliver 2-3 or more Amperes, as those plugs  contain a more efficient switching regulator.

About calculation: My phone has a 3100mAh battery (3.7V) and lasts 5 days when idle. So, the battery contains an energy of 3.1Ah * 3.7V = 11.5Wh

A small 12V 4Ah scooter battery contains 48Wh , which is four times the energy of my phone. So, it in principle it will last 20 days.

Note that this is a rough estimate. Switching power supplies are also not 100% efficient, so may be it's only 70-80%.

In your case, check out how long the battery lasts when doing the desired job. Keep in mind that signal strength also plays a role.

Finally, it may not be legal to keep your workers under surveillance. 



From the comments:


  Anyone know if it is more efficient to connect an external battery to the phone via USB and charge the internal battery or connect the external battery directly to the battery connector inside the phone? Lastly there are 4V 20AH lead acid batt. Can I connect direct to phone? 


It would be less efficient to charge your phone battery from the external battery, as charging always goes with some losses. However, if the phone battery is full at the beginning, you have the charge of the external battery plus the charge of the phone battery, and in total, the phone will last longer.

About the 4V: I doubt the phone will take the 4V from the external battery supplied via the USB port, as this is clearly below the USB standard. On the other side, a fully charged 3.7V LiIon battery has about 4.2V

This means it should be OK to remove the internal battery and connect to an external voltage of 4V. (It is not recommended to switch batteries, especially different types in parallel.)

On the other side, a lead-acid battery behaves similar: When it's full, it has up to 2.4V per cell, so 4.8V for your battery. This is only 0.6V above the maximum of the LiIon battery. I would guess, that it is OK, but I can't guarantee it. But if you measure the voltage of your battery, the actual voltage may not be that high, and then it's OK.