A common misconception
The unit milli ampere-hour (mAh) is generally used to describe the capacity of a battery. One common misconception is that we can simply divide the capacity of a battery by the capacity of a smartphone/tablet to figure out how many cycles we can use this power bank to charge the smartphone/tablet. However, that is not the correct algorithm.
Capacity and energy are different concepts
Simply put, Ampere-Hour (mAh) is a measure of electric charge, which represents the capacity of a battery, while Watt Hour (Wh) is a measure of electric energy.
Watt-Hour = Ampere-Hour x Voltage
A battery of 10400mAh means this battery can provide a total charge of 10400mAh at a specific voltage. As to a Lithium-ion battery most of its charge is exchanged at around 3.7V. Thus the total energy of a 10400mAh battery is theoretically 10400mAh x 3.7V = 38480mWh, which is approximately 38Wh.
Estimate the charge cycles of a power bank
Take power bank TL-PB10400_V1 as an example.
TL-PB10400_V1 consists of a 10400mAh Lithium-ion battery. When we use a TL-PB10400_V1 to charge other devices, like using many other power bank products, the output charge voltage is converted to 5V.
Thus the total available output electric charge is 38480mWh / 5V = 7696mAh in theory. The internal circuit must consume some power so the efficiency cannot be 100%. Given that the actual discharge efficiency of TL-PB10400 is about 90% under 1A current, TL-PB10400 can actually "push out" a total electric charge of 7696mAh *0.9 = 6926mAh.
Note: The discharge efficiency is less than 90% under 2A current.
Now you might divide 6926mAh by your smartphone's capacity to figure out the possible charge cycles. For example, 6926mAh should be able to fully charge a 2600mAh device for 6926mAh / 2600mAh = 2.66 times. But that's still under the ideal conditions.
In fact, the internal conversion circuits inside the smartphone/tablet must consume some energy too. As a result only a part of the electric charge pushed from the power bank will eventually enter into the battery of the smartphone/tablet. So you may only get less than 2.4 cycles in the above example. Another thing is that different devices may have different charge efficiency depending on their different internal design, so the charge cycle may be different even two devices have the same battery capacity.
In addition, it consumes more energy by LED screen, Wi-Fi transceiver, CPU and other components if the smartphone is operating or running some background programs during the charge, which makes the charge efficiency seem even lower.
The final charge efficiency of other rechargeable devices (smartphones/tablets) are also subjected to their own design for the same principles as described above.
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