Imagine I have a 10V battery with a capacity of 1Ahrs connected to a 100 Ohm resistor.

I'll get 0.1A (V/R) at a power of 1W (current x voltage) for a total of 10hrs. As a watt is defined as J/s my battery produces 3,600J (1x60x60) in total.

Now, imagine I get a second, identical battery and connect it in series to the first one. I will effectively have a battery of 20V with a capacity of 2Ahrs, all connected to the same 100 Ohm resistor.

I'll get 0.2A at a power of 4W (current x voltage) for a total of 10hrs. This will generate 14,400J (4x60x60).

So, from doubling my energy source I have quadrupled the energy output!

I've obviously made an error somewhere either in my calculations or conceptual understanding, and I would be most grateful to anyone who could set me right.

Thanks!

## Best Answer

Congratulations for having the wit to know something was wrong!

^{simulate this circuit – Schematic created using CircuitLab}Figure 1. Parallel and series arrangements of batteries will have the same VAh rating.That's the error.

This is one good reason to use Wh (watt-hours) for battery comparison. It makes it much easier to compare battery energy storage when the batteries have different voltages.

In your example you have \$ Capacity = V \cdot Ah = 10 \cdot 1 = 10~Wh \$. With two batteries we have \$ 20~Wh \$ capacity whether in series or parallel.

Our load power is given by \$ P = \frac {V^2}{R} \$.

^{simulate this circuit}Figure 2. An alternative view may help. (a) Is your original series circuit. (b) is a direct equivalent. Because the mid-point of the 50 Ω resistors is at the same potential as the mid-point of the battery stack it makes no difference if we connect them together as shown in (c). Now we have two 1 Ah circuits each feeding into a 50 Ω load.