Energy and power models

Battery capacity is measured in amp-hours. A simplified equation to estimate the life of a battery power source is given as:

In the equation, C_p is Peukert's capacity, I represents the discharge current, and n is Peukert's exponent. The Peukert effect, as it's known, helps predict the life of a battery where the capacity of a battery decreases at a different rate as discharge increases. The equation shows how discharging at higher rates removes more power from the battery. Alternatively, discharging at lower rates will increase the effective runtime of the battery. One way of thinking of this phenomenon is a battery manufacturer rated at 100 Ah, and discharging it fully in 20 hours (5A, for this example). If one were to discharge it faster (say, in 10 hours), the capacity would be lower. If one were to discharge it more slowly (say, over 40 hours), it would be greater. However, when represented on a graph, the relationship is nonlinear. Peukert's exponent usually lies between 1.1 and 1.3. As n increases, we move further from a perfect battery to one that discharges faster as the current increases. Peukert's curve applies to lead-acid battery performance, and an example is shown in this next graph:

Peukert curves from 1.1 to 1.5 for a battery rated at 100A over 20 hours. The curves show the declining capacity as the Peukert coefficient grows.  

One can see the difference in discharge rates for various types of batteries. One advantage of alkaline batteries is that the discharge rate is almost linear in a large portion of the graph. Lithium-ion has a stair-step function in performance, thus making battery charge predictions more difficult. That said, Li-ion provides a near steady and continuous voltage level over the life of the charge, and consistently powers electronics over the span of its charge:

Example of relative discharge rates for various batteries. Li-ion provides a nearly constant voltage over its life, but has a precipitous drop towards the end of its storage capacity. 

The graph also illustrates that lead acid and Ni-Cd have less voltage potential and a curvilinear degradation in power that can be computed more reliably. The trailing slopes are also indicative of Peukert's capacity.

Temperature greatly affects battery life, and specifically the electro-active carriers in a cell. As temperature increases, the internal resistance of a battery decreases when being discharged. Even when batteries are stored, they can self-discharge, which impacts the total lifetime of a battery. 

A Ragone plot is a useful way to show the relationship between energy storage systems when trading off energy capacity and power handling. It is a log-based scale where the energy density (Wh/kg) of a power source is plotted against the power density (W/kg). This relationship shows devices that tend to have a greater lifetime (batteries) versus devices that tend to store more energy (supercapacitors):  

Ragone plot depicting capacitor, supercapacitors, battery, and fuel cell differences in energy capacity versus lifetime supply of energy. 

Batteries like Li-ion have higher energy density and discharge rates than nickel-cadmium and nickel-hydride batteries. Capacitors produce very high-power densities, but are relatively weak energy density. Note the plot is log-based, and also shows the discharge time for various storage systems. Image courtesy of C Knight, J. Davidson, S. Behrens "Energy Options for Wireless Sensor Nodes", Sensors, 2008, 8(12), 8037-8066.