Know about hidden battery losses when estimating the energy reserve.
If the battery were a perfect power source and behaved linearly, charge and discharge times could be calculated according to in-and-out flowing currents. “What is put in should be available as 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.
The Peukert Law expresses the efficiency factor of a battery on discharge. W. Peukert, a German scientist (1897), was aware of this and devised a formula to calculate the losses in numbers. They apply mostly to lead acid and help estimate the runtime under different discharge loads.
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. Peukert’s law is exponential and the readings for lead acid are between 1.3 and 1.5. 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
A value close to
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 most other systems provide a faster electrochemical reaction than lead acid. (See BU-501: Basics About Discharging)
Lithium and nickel-based batteries are more commonly evaluated by the Ragone plot. Named after David V. Ragone, the Ragone plot looks at the battery’s capacity in Wh and the discharge power in W The big advantage of the Ragone plot over Peukert is the ability to read the runtime in minutes and hours presented on the diagonal lines.
Figure 2 illustrates the Ragone plot of four lithium-ion systems in 18650 cells. The horizontal axis displays energy in watt-hours (Wh) and the vertical axis is power in watts (W). The diagonal lines across the field reveal the length of time the battery cells can deliver energy at given loading conditions. The scale is logarithmic to allow a wide selection of battery sizes. The battery chemistries featured in the chart include lithium-iron phosphate (LFP), lithium-manganese oxide (LMO), and nickel manganese cobalt (NMC). (See BU-205: Types of Lithium-ion)
Figure 2: Ragone plot reflects Li-ion 18650 cells. Four Li-ion systems are compared for discharge power and energy as a function of time. Courtesy of Exponent
Legend: The A123 APR18650M1 is a lithium iron phosphate (LiFePO4) Power Cell rated at 1,100mAh, delivering a continuous discharge current of 30A. The Sony US18650VT and Sanyo UR18650W are manganese–based Li-ion Power Cells of 1500mAh each delivering a continuous discharge of 20A. The Sanyo UR18650F is a 2,600mAh Energy Cell for a moderate 5A.discharge. This cell provides the highest discharge energy but has the lowest discharge power.
The Sanyo UR18650F  in Figure 2 has the highest specific energy and can power a laptop or e-bike for many hours at a moderate load. The Sanyo UR18650W , in comparison, has a lower specific energy but can supply a current of 20A. The A123  has the lowest specific energy but offers the highest power capability by delivering 30A of continuous current. Specific energy defines the battery capacity in weight (Wh/kg); energy density is given in volume (Wh/l).
The Ragone plot helps choosing the best Li-ion system to satisfy maximum discharge power and optimal discharge energy as a function of discharge time. If an application calls for very high discharge current, the 3.3 minute diagonal line on the chart points to the A123 (Battery 1); it can deliver up to 40 Watts of power for 3.3 minutes. The Sanyo F (Battery 4) is slightly lower and delivers about 36 Watts. Focusing on discharge time and following the 33 minute discharge line further down, Battery 1 (A123) only delivers 5.8 Watts for 33 minutes before the energy is depleted whereas the higher capacity Battery 4 (Sanyo F) can provide roughly 17 Watts for the same time; its limitation is lower power.
Figure 3 illustrates the Ragone plot featuring a alkaline, lithium (Li-FeS2) and NiMH battery, each drawing 1.3W to power a digital camera (1.3W at 3V is 433mA). The dotted horizontal 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. The Lithium Li-FeS2 offers the highest specific energy but has moderate loading conditions. Alkaline offers an economic solution for lower current drains such as flashlights and remote controls, but a digital camera is stretching its capability. (See BU-106a: Choices of Primary Batteries)
Figure 3: 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
Energy = Capacity x V
Courtesy of Exponent
The design engineer should note that the Ragone snapshot taken by the battery manufacturers represents a new cell, a condition that is temporary. When calculating power and energy needs, engineers must take into account the battery fade caused by cycling and aging. Battery operated systems must still function with a battery that will eventually drop to 70 or 80 percent. A further consideration is low temperature as a battery momentary loses power when cold. The Ragone plot does not show these decreased performances.
The design engineer should also prevent loading a battery to its full power capability as this increases stress and shortens life. If high current is needed on a continued basis, the battery size should be increased. An analogy is a truck equipped with a large diesel engine that is durable as opposed to installing a small souped-up engine of a sports car with similar horsepower.
The Ragone plot can also calculate the power requirements of capacitors, flywheels, flow batteries and fuel cells. However, a conflict develops with the internal combustion engine and fuel cell that draw fuel from a tank. Re-fuelling cheats the system. Similar plots are used to find the optimal loading ratio in renewable power sources, such as solar cells and wind turbines.
Presentation by Quinn Horn, Ph.D., P.E. Exponent, Inc. Medical Device & Manufacturing (MD&M) West, Anaheim, CA, 15 February 2012
Last updated 1/12/2015