Battery Runtime

Calculating the Battery Runtime

If the battery were a perfect power source and behaved linearly, the discharge time could be calculated according to the in-and-out flowing currents. “What is put in should be available as an output in the same amount” goes the argument, and “a one-hour charge at 5A should deliver a one-hour discharge at 5A, or a 5-hour discharge at 1A." This is not possible because of intrinsic losses. The output is always less than what has been put in, and the losses escalate with increasing load. High discharge currents make the battery less efficient. To learn about the coulomb counter, see Inner Workings of a Smart Battery.

The efficiency factor of a discharging battery is expressed in the Peukert Law. W. Peukert, a German scientist (1897), was aware of this loss and devised a formula that expresses the loss at a given discharge rate in numbers. Because of sluggish behavior of lead acid, the Peukert numbers apply mostly to this battery chemistry and help in calculating the capacity when loaded at various discharge rates.

The Peukert Law takes into account the internal resistance and recovery rate of a battery. A value close to one (1) indicates a well-performing battery with good efficiency and minimal loss; a higher number reflects a less efficient battery. The Peukert Law of a battery is exponentialand the readings for lead acid are between 1.3 and 1.4. Nickel-based batteries have low numbers and lithium-ion is even better. Figure 1 illustrates the available capacity as a function of ampere drawn with different Peukert ratings.

Figure 1: Available capacity of a lead acid battery at Peukert numbers of 1.08–1.50 A value close to 1 has the smallest losses; higher numbers deliver lower capacities. Source: von Wentzel (2008)

The lead acid battery prefers intermittent loads to a continuous heavy discharge. The rest periods allow the battery to recompose the chemical reaction and prevent exhaustion. This is why lead acid performs well in a starter application with brief 300A cranking loads and plenty of time to recharge in between. All batteries require recovery, and with nickel- and lithium-based system, the electrochemical reaction is much faster than with lead acid. Read more about the Basics About Charging.

The runtime of batteries in portable devices relates to the specific energy marked in Ah (mAh in personal devices). Ah as a performance indicator works best at low discharge currents. At higher loads, the internal resistance begins to play a larger role in the ability to deliver power. Resistance acts as the “gatekeeper.” Energy in Ah presents the available storage capacity of a battery and is responsible for the runtime; power governs the load current. These two attributes are critical in digital devices that require long runtimes and must deliver high-current pulses.

Ah alone is not a reliable runtime indicator and the relationship between capacity and the ability to deliver current can best be illustrated with the Ragone Chart. Named after David V. Ragone, the Ragone chart evaluates batteries not on energy alone but also represents power. 

Figure 2 illustrates the Ragone chart on a digital camera that is powered by an Alkaline, Lithium (Li-FeS2) or NiMH battery drawing 1.3W. (1.3W at 3V draws 433mA.) The horizontal axis displays energy in Watt/hours and the vertical axis displays power in Watts. The scale is logarithmic to allow a wide selection of battery sizes.

Figure 2: Ragone chart illustrates battery performance with various load conditions. Digital camera loads NiMH, Li-FeS2 and Alkaline with 1.3W pulses according to ANSI C18.1 (dotted line). The results are: - Li- FeS2 690 pluses - NiMH 520 pulses - Alkaline 85 pulses Energy = Capacity x V Power = Current x V Courtesy of Exponent

The dotted line represents the power demand of the digital camera. All three batteries have similar Ah rating: NiMH delivers the highest power but has the lowest specific energy. This battery works well at high loads such as power tools. The Lithium Li-FeS2 offers the highest specific energy but has moderate loading conditions. Digital cameras and personal medical instruments suit the system well. Alkaline offers an economic solution for lower current drains such as flashlights, remote controls and wall clocks, but a digital camera is stretching the capability of Alkaline. Read more about the Choices of Primary Batteries.