BU501a: Calculating the Battery Runtime
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 inandout flowing currents. “What is put in should be available as output in the same amount” goes the argument, and “a onehour charge at 5A should deliver a onehour discharge at 5A, or a 5hour 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.
Peukert Law
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 wellperforming 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 BU501: Basics About Discharging)
Ragone Plot
Lithium and nickelbased 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 lithiumion systems in 18650 cells. The horizontal axis displays energy in watthours (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 lithiumiron phosphate (LFP), lithiummanganese oxide (LMO), and nickel manganese cobalt (NMC). (See BU205: Types of Lithiumion)
Figure 2: Ragone plot reflects Liion 18650 cells. Four Liion 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 Liion 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 [4] in Figure 2 has the highest specific energy and can power a laptop or ebike for many hours at a moderate load. The Sanyo UR18650W [3], in comparison, has a lower specific energy but can supply a current of 20A. The A123 [1] 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 Liion 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 (LiFeS2) 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 LiFeS2 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 BU106a: Choices of Primary Batteries)

Figure 3: Ragone chart illustrates battery performance with various load conditions. Digital camera loads NiMH, LiFeS2 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 soupedup 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. Refuelling cheats the system. Similar plots are used to find the optimal loading ratio in renewable power sources, such as solar cells and wind turbines.
Reference
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
Comments
please i want to install solar system,and i will use flood deep cycle battery,
i will use four battery 12 volt DC, can i use two battery each other 24 volt and what advantage to use 4 battery.
thanks a lot.
Dear Sir/Madam,
I am Susanta Kumar Sahu to inform you that i have no knowledge about battery discharge (EX: 180AH battery, connecting with load that depends on customer how much it give back up like this ).Can you tell BACK UP concept of all bettery?
Thank You.
Susanta Kumar Sahu
Is there any empirical formula to calculate battery DEPTH of Discharge for a given lifetime?
Hi,
Can somebody tell me how to model peukart law im modelling please?
Thanks alot
Peukart (title and sidebar menu) or Peukert (article)?
1) Can Peukert’s constant be used across all battery chemistry’s ?
2) Is there any equation where we can determine the remaining time in a battery upon
knowing the discharge voltage during operation?
3) What exactly do you mean by efficiency of the battery ?
4) What is the effect of soldering on a Liion cell for paralleling by our self ?
can anyone answer question number one from pier ? can peukerts equation be used for lithium type batteries?
You may have missed this, gary: “The Peukert Law of a battery is exponentialand the readings for lead acid are between 1.3 and 1.4. Nickelbased batteries have low numbers and lithiumion is even better.” Missing space in there. Different chemistries have different values.
Soldering directly onto a Liion cell is not recommended. They can explode.
thanks ian. i saw that on the second read thru. i have never been a good student
Dear Sir,
I live in Pakistan and i am working in lead acid battery company.
My question is, How can i calculate Peukert’s number by theoretical or practical and each battery (size wise) has a different peukert number.
Regards,
TAHIR
Please infrom me how to calculate the time for discharging a traction battery with 560Ah & 420
how to calculate the charging and discharging time of Liion battery with specification of 100 Ah, 12 V.and applying 2 A constant current.?????
Yes Peukert’s Equation can be used for Lithium batteries. Their exponents are closer to 1.0 than lead acid Batteries.
How exactly do you define one cycle in a LiIon cell ? I find cycle count in Lenovo thinkpads but not sure as to how they compute it !!!
Hello,
How to calculate the charging and dischargin time of a Li ion polymer battery of capacity 1230mAh, 3.7V applying 250mA constant current?
Thank You
Hello,
How do we calculate the no. of hours a battery can provide on continuous discharge of 250mA of capacity 1320mAh?
Also providing the voltage the battery can provide after every hour of discharge of 250mA would be good.
Note:
Nominal voltage of the battery is 3.7V.
max operating range is 2.75V to 4.2V.
max continuous discharge current is 1000mA
internal resistance is 150milli ohm
Thanks
My Dear friends;
I designed a solar system in which appliance load is 500 Watt and appliance watt per day is 1500.
So I used 150watt 2 solar panel, 24volt and 12 A Solar charge controller, 2 batteries of 125Ah each.
Normally 7 hours sunlight.
My problem is that I want to calculate in how much time period batteries will again recharge.
Please help me in this problem
Thanks
my cell battry is bulging , it’s a liion, and i was wondering how i make it safe(er) to dispose of, can you help with that? the phone says “use authentic battery” so i’m afraid it might explode,
im using 12v 7.4Ah SLA battery x 3 pcs.. the load 19w 12v dc.. By calculation :
(12V x 7.2Ah )/ 19W = 4.5 hour x 3 pcs battery in parallel connection can i get 13 hour ??
but by actual run i got 57 hour only. i’m i using wrong battery or my calculation wrong..
7Ah is the capacity at a rate that flattens the battery in 20 hours. Your load is much higher than that so you need to take the Peukert effect into consideration. SLA batteries have quite high Peukert effect. The Peukert calculation is much more complicated than what you have done, but you could use the top graph instead to make it easier.
If inverter efficiency is around 80%, current from 12 volts for 19W load is approx. 2 amps.
When 3 batteries are in parallel, You have a 7.2 x 3 i.e.21.6AH (@ 20 Hr. rate.)
Using Peukert equation calculation, you should get 8Hr. 30Min. approx. This makes the battery fully discharged. For better life, batteries are not discharged beyond 80%. So inverters’ cut off volts are adjusted to higher value. Probably, the inverter efficiency may be even less than 80%. If you find exact current from the batteries, Peukert formula will give you quite accurate duration. There is no much complication.
If possible, kindly share the info on making an
SMPS charger of 4000 Watts and above
Zahid wrote:
im using 12v 7.4Ah SLA battery x 3 pcs.. the load 19w 12v dc.. By calculation :
(12V x 7.2Ah )/ 19W = 4.5 hour x 3 pcs battery in parallel connection can i get 13 hour ??
but by actual run i got 57 hour only. iām i using wrong battery or my calculation wrong..
———————————————————————————
The Peukert effect. Your 7ah battery is too small. The rate in the above chart for 120Ah compared to you 7Ah*3=21Ah battery. Lead acid is 1.3 to 1.4. Your is 19w /12v = 1.58 Amp draw. 120/21 for battery in above chart *1.58 = proportional equivalent~ 9.0 amp so look the 9 amp and it gives capacity reduction of about 50% in the Peukert chart so you will get 20 hours run time down to 10 hours just from that, and even worst as your batteries age. SLA rating is only good for long slow discharge 20AH is 1 Amp for 20 Hours. Take 10x times (10Amps) that out in 2 Hours you get only roughly half to 1/3 that amount if your lucky. That is why my project ebike will have NiMH battery instead of SLA. The higher cost 5x is justified in lifetime use and performance. LiPo4 is still too much expense and funny picky charging/discharge/ quality.
Sorry correction, your example only 13 hours use * 50% = 6.5 hours as you discovered, I was using 20 hour run for 10 hours use.
SLA rating is only for low current draw. Peukert charts are very much relevant. NiMH is twice that of SLA in rating for real world application using any kind of Medium to Heavy use draw. Medium use is >10% current rating of battery capacity. Low is less than 1<%.
Your example of three 7aH batery for 21 Ah array pulling anything more than 23 Amps is going to cut your real amp hour capacity to 50% for SLA. You really need to buy 10x more battery or get NiMH—though NiMH self discharge over time quickly over several days in a solar array.
SIR PLEASE TELL ME HOW THE HRD END CRURRENT CALCULATE
Dear sir
I have One 3.7V,600mAh Liion battery that battery have DoD is 27% but i want to DoD 70% above that. So what is Procedure for this any circuit have built or other..Please Suggest me.
Give me process for improve battery DoD.
i want to know about the normal c20 and c10 rated battery and how to calculate, how much kWh(unit) can be obtained form particular battery like C10 26Ah dc12V battery how much kWh can be obtained at 12V and for C20 Rating Battery.
how can i calculate 50%volt shutdown on a 24vdc/1400amps inverter batterry.
NB: Inverter capacity is 30kva
i Have 24 numbers of battery, each having 2V,600Ah connected in series with only 400 Watt load that means 48V of my system So can you help me how to calculate the back up time of this system.
What is the correct formula to determine how many hours I can draw .028 amps from a 12 volt 700 amp hour lead acid battery before the battery voltage drops to 11 volts?
A lead acid battery can be transformed to a non acid one. Remove the acid, clean several times with distiled hater and then fill with a solution of KAl(SO4)2 Potassium Alum. This kind of conversion will result in a new kind of energy source: 1) Less voltage; 2) Allows 100% discharge (even shot circuit) with no damage; 3) Fast recharge, but with pulsed DC; 4)Non toxic.
A battery of 12v 40ah and 12v 80 ah in parallel , which battery would discharge first?
hello sir, i have 12v 150ah lead acid tubular battery, solar panel giving 4.5A/ hr and avg 6hrs is sunshine available. Load drawing around 20A/ day. battery charging through mppt controller of 30A rated. then, how much time required to fully charge my battery.
what is 400 cca to amp / hr how to conbert as i have 4 x 12 volt lithium polymer batteries here in my home for a project but need to know the amp/hr on my batteries can you help?
hi, I need some clarification of UPS battery run time calculation. The detail as follow:
UPS Size : 10kVA
Battery Voltage : 12V
Battery Ah Rating: 100
Battery Qty: 32
Inverter Eff.: 93.5%
I need to know what is the actual battery run time during the failure of main source? does the calculation load shall take on the total load of equipment or the ups rating 10kVA?
Kindly please advise how to calculate and what is the best formula to used.
battery full lod discharging chart