Battery Capacity

Capacity

The capacity of Batteries is the total amount of energized charge available, $C_{\text{cell}}$.

• This is different from the capacitance of Capacitors!

Capacity is expressed as:

$$\text{Capacity}=\text{charge}\times \text{time}$$

For example, if we have a battery that has a capacity of 1000 $\text{mA}\cdot \text{h}$, it can do $100$ milliamps of current for 10 hours, or 200 milliamps of current for 5 hours.

Example

Energizer Max AA battery has $C_{\text{cell}}=2500\text{ mAh}$. How long can it power a 20 mA LED circuit? 2 A circuit?

$$\begin{array}{rl}\text{Capacity}& =\text{Current}\times \text{Time}\\ & =I\cdot t\\ t_1& =C/I_1=\frac{2500}{20}=125\text{ h}\\ t_2& =C/I_2=\frac{2500}{2}=1\text{h }15\text{min}\end{array}$$

Q (Total Charge)

Capacity is a proxy for $Q_{\text{cell}}$, the total charge that can be delivered at $V_{\text{cell}}$.

For example, we can have:

$$1\text{ mA}\times \text{h}=10^{-3}\times 60\frac{\text{s}}{\text{min}}\times 60\frac{\text{min}}{\text{h}}=3.6\text{ C}$$

Basically the charge we can get (number of coulombs), is the 3.6 times the milli-amp hours.

Example

What is the $Q_{\text{cell}}$ for the Energizer AA?

Since we have $C_{\text{cell}}=2500\text{ mAh}$, we have:

$$Q=3.6\times 2500=9\text{ kC}$$

E (Energy)

The total usable energy stored in a battery is $E_{\text{cell}}=Q_{\text{cell}}{V}_{\text{cell}}$. It quantifies how much “work” the battery can do. We can relate energy to power – energy is how much power you can deliver, such that:

$$P=IV,\int IVdt⟹E=QV$$

Example

How much How much energy is stored in the Energizer Max AA battery given $V_{cell}=1.5\text{ V}$?

We have:

$$E=QV=9000\text{ [C]}\times 1.5[\text{J}/\text{C}]=13.5\text{ kJ}$$