BU-501a: 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.

Available capacity of a lead acid battery at Peukert numbers of 1.08–1.50

 

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.

Ragone chart illustrates battery performance with various load conditions

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.

 

Reference

Presentation by Quinn Horn, Ph.D., P.E. Exponent, Inc. Medical Device & Manufacturing (MD&M) West, Anaheim, CA, 15 February 2012

Comments

On May 7, 2011 at 9:03pm
HITESH PATEL wrote:

battery full lod discharging chart

On May 25, 2011 at 12:17pm
Samuel Maher Youssef wrote:

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.

On June 6, 2011 at 10:06pm
Susanta Kumar Sahu wrote:

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

On June 9, 2011 at 11:42pm
Gaurav wrote:

Is there any empirical formula to calculate battery DEPTH of Discharge for a given lifetime?

On June 20, 2011 at 4:39am
CBT wrote:

Hi,
Can somebody tell me how to model peukart law im modelling please?
Thanks alot

On August 27, 2011 at 4:47am
Ian wrote:

Peukart (title and sidebar menu) or Peukert (article)?

On November 4, 2011 at 3:27am
Pier wrote:

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 Li-ion cell for paralleling by our self ?

On November 17, 2011 at 9:26pm
gary pione wrote:

can anyone answer question number one from pier ? can peukerts equation be used for lithium type batteries?

On November 17, 2011 at 9:33pm
Ian wrote:

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. Nickel-based batteries have low numbers and lithium-ion is even better.” Missing space in there. Different chemistries have different values.

Soldering directly onto a Li-ion cell is not recommended. They can explode.

On November 17, 2011 at 9:55pm
gary pione wrote:

thanks ian. i saw that on the second read thru. i have never been a good student

On November 25, 2011 at 9:53am
TAHIR HUSSAIN wrote:

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

On January 8, 2012 at 9:17pm
Terence Cheong wrote:

Please infrom me how to calculate the time for discharging a traction battery with 560Ah & 420

On February 8, 2012 at 4:30am
Komal wrote:

how to calculate the charging and discharging time of Li-ion battery with specification of 100 Ah, 12 V.and applying 2 A constant current.?????

On February 23, 2012 at 4:41am
Sheldon Patnett wrote:

Yes Peukert’s Equation can be used for Lithium batteries. Their exponents are closer to 1.0 than lead acid Batteries.

On February 23, 2012 at 9:42pm
Pier wrote:

How exactly do you define one cycle in a Li-Ion cell ? I find cycle count in Lenovo thinkpads but not sure as to how they compute it !!!

On May 8, 2012 at 6:13am
Syed wrote:

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

On May 8, 2012 at 6:19am
Syed wrote:

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

On July 24, 2012 at 1:20am
Muhammad Ahmed wrote:

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

On August 30, 2012 at 10:28am
Johnny wrote:

my cell battry is bulging , it’s a li-ion, 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,

On October 18, 2012 at 9:36pm
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 5-7 hour only. i’m i using wrong battery or my calculation wrong..

On October 18, 2012 at 9:45pm
Ian wrote:

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.

On February 11, 2013 at 3:48am
Suresh Patil wrote:

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.

On February 23, 2013 at 10:25pm
ashish wrote:

If possible, kindly share the info on making an
SMPS charger of 4000 Watts and above

On March 6, 2013 at 12:41pm
Slocket wrote:

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 5-7 hour only. i’m 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.

On March 6, 2013 at 12:50pm
Slocket wrote:

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 2-3 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.

On October 31, 2013 at 12:33pm
LALIT wrote:

SIR PLEASE TELL ME HOW THE HRD END CRURRENT CALCULATE

On November 15, 2013 at 12:38am
vishal wrote:

Dear sir
I have One 3.7V,600mAh Li-ion 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.

On January 12, 2014 at 11:44am
Anand patel wrote:

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.

On March 26, 2014 at 3:01am
okugbesan ibrahim wrote:

how can i calculate 50%volt shutdown on a 24vdc/1400amps inverter batterry.
NB: Inverter capacity is 30kva

On April 2, 2014 at 3:55am
Dj wrote:

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.

On May 31, 2014 at 1:46pm
Bob Sandor wrote:

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?

On June 26, 2014 at 10:13am
Thomas Soares wrote:

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.

On August 17, 2014 at 5:33am
Solid gold wrote:

A battery of 12v 40ah and 12v 80 ah in parallel , which battery would discharge first?

On September 20, 2014 at 5:55am
santosh wrote:

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.

On October 11, 2014 at 4:40pm
klaus wrote:

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?