White Paper - Short Discharge Time

A tool to compare cells and batteries for power applications

Introduction

When selecting cells for a power battery, it is useful to have the ability to quickly compare various cell technologies, and calculate the resulting pack resistance and efficiency, independently of capacity and voltage. This article proposes a way of doing so, using the "Short Discharge Time", the theoretical time required to discharge a full cell (or battery) through a short circuit. This constant is a characteristic of each battery cell technology, regardless of capacity or voltage.

Characterization

The short discharge time of a battery technology can be derived from specification sheets or empirically.

From specifications

Given a cell's or battery's DC resistance , capacity and voltage, the short discharge time is:

short_discharge_time [h] = capacity [Ah] * resistance [Ω] / voltage [V]

In practice, the short discharge time or actual cells ranges from 0.004 to 0.06 hours (15 to 220 s). Therefore, seconds is a more practical measure of short discharge time than hours:

short_discharge_time [s] = 3600 * capacity [Ah] * resistance [Ω] / voltage [V]

For example, the specifications of a 26650 size, LiFePO4 cell from A123 are: 3.3 V, 2.3 Ah, 10 mΩ. Therefore, the short discharge time of those cells (and of batteries built from those cells, regardless of the arrangement) is:

short_discharge_time [s] = 3600 * 2.3 [Ah] * 10 m [Ω] / 3.3 [V] = 25 s

Graphically

Unfortunately, very few manufacturers specify true DC resistance. Many specify impedance at 1 kHz instead, which easy to measure, but is useless to the user, and is quite unrelated to DC resistance. In that case, if discharge curves are available, they may be used to derive the short discharge time. These graphs plot the cell voltage versus SOC at various specific currents (such as 0.5 C, 1C, 2C, 5 C…). From such a set of curves, pick two points at 50 % SOC (for example, at 0.5 C and 2 C). Note the difference in the cell voltage at those two points; that's the delta-voltage, in Volts. Take the difference of the two specific currents; that's the delta-specific-current, in 1/ hours. Then, use those values to calculate the short discharge time:

short_discharge_time [s] = 3600 * delta_voltage [V] / delta_specific_current [1/h] / cell_voltage [V]

or

short_discharge_time [s] = 3600 * delta_voltage [V] * capacity [Ah] / delta_current [A] / cell_voltage [V]

For example, we can pick two points in the following discharge curve for a LiFePO4 cell.

The delta voltage is 3.28 - 2.98 = 0.3 [V]; the delta specific current is 5 - 1 = 4 [1/h]; the cell voltage is 3.3 V. Then, the short discharge time of this cell (and of cells of any size using the same technology, and of batteries using these cells) is:

short_discharge_time [s] = 3600 * 0.3 [V] / 4 [1/h] / 3.3 [V] = 82 s

Empirically

Having access to an actual cell, one can derive the short discharge time empirically;

1. Fully charge the cell
   
2. Discharge the cell to empty, in the span of 1 hour, while integrating the current
   
3. The cell capacity is the final value of the integral
   
4. Charge the cell to 50 %
   
5. Measure the cell's open circuit voltage
   
6. Apply a load to the cell to draw approximately 1 C of current
   
7. Measure the load current
   
8. Wait for the cell voltage to settle and measure the loaded cell voltage
   
9. Calculate the cell resistance = (Open circuit voltage - loaded cell voltage) / load current

At this point, the cell voltage, cell capacity and cell resistance are known, and therefore the cell's short discharge time can be calculated:

short_discharge_time [s] = 3600 * capacity [Ah] * resistance [Ω] / voltage [V]

Sample short discharge times

Using the methods above, the short discharge time of various cell technologies was calculated, and is listed below.

This table compares the short discharge time of various cell and battery chemistries.

Technology	Model	Short discharge time [s]
Lead acid	Panasonic VRLA	65~86
	Xtreme Power	18
	EnerSys Cyclon	22~54
Li-ion	See table below	14~220
NiMH	Energizer	111~155
	Panasonic HHR	40
NiCd	Sanyo Cadnica	25

You will note that all chemistries offer some cells that have a low resistance; and that Li-ion covers the entire range, from the best to the worst.

More specifically, this graph compares the short discharge time of a select few cell and battery technologies.

Applications

The short discharge time can be used to rapidly select cell technology for power applications, to rapidly calculate battery resistance, and round trip efficiency.

Cell selection

Having the short discharge time of various cell technologies, one may immediately select the one that will result in the most efficient battery, by selecting the one with the lowest short discharge time. Of the cells analyzed in this study, the Kokam SLPB-H5 series LiPo cells have the best short discharge time, and should be selected to manufacture batteries (of a given capacity and voltage) with the lowest resistance.

Obviously, battery resistance is not the only criterion used in cell selection; cost, energy, weight and volume are also important. Specifically, if considering cost, one may select Enerdel cells over Kokam cells: Enerdel 16 Ah cells are not as expensive as Kokam SLPB-H5 cells, and provide a better value, in the sense that a battery using $ 1000 worth of Enerdel cells,will have a lower resistance that a battery of the same voltage using $ 1000 worth of Kokam cells (it will also have a higher capacity).

Battery resistance calculation

Or, having the short discharge time of the cell technology used in a battery, one may rapidly calculate the nominal internal resistance of that battery.

Given the battery voltage and capacity:

resistance [Ω] = short_discharge_time [s] * voltage [V] / capacity [Ah] / 3600

Or, given the voltage and energy:

resistance [Ω] = short_discharge_time [s] * (voltage [V])^2 / energy [Wh] / 3600

Or, given the energy and capacity:

resistance [Ω] = short_discharge_time [s] * energy [Wh] / (capacity [Ah])^2 / 3600

For example, using a cell technology that has a short discharge time of 36 seconds, given the battery voltage and capacity:

• 10 V, 100 Ah -> resistance = 36 * 10 / 100 / 3600 = 1 mΩ
  
• 100 V, 100 Ah -> resistance = 36 * 100 / 100 / 3600 = 10 mΩ
  
• 10 V, 10 Ah -> resistance = 36 * 10 / 10 / 3600 = 10 mΩ
  
• 100 V, 10 Ah -> resistance = 36 * 100 / 10 / 3600 = 100 mΩ

Which makes sense: as the voltage increases, having more cells in series results in a higher resistance; conversely, as the capacity increases, having more cells in parallel results a lower resistance.

Efficiency calculation

Given the short discharge time of a battery, the efficiency is easily derived.

efficiency [%] = 100 * (1 - (short_discharge_time [s] / actual_discharge_time [s]) )

This table lists the efficiency for various full discharge (100 % to 0 %) times. That same data are shown in the graph.
Here we see that, if a battery is fully discharged 10 times slower than its short discharge time, the efficiency is 91 %; if 100 times, 99 %.
The Peak Power Point occurs when the battery is fully discharged for a time equal to its SDT: the efficiency is 50 % (half the power goes to the load, half is wasted in heat within the cells).

Relative discharge time Efficiency 1 (disch. time = SDT) 0% 1.5 33.3% 2 (twice as long) 50% 3 66.7% 5 80% 7.5 86.6% 10 90% 15 93.3% 20 95% 30 96.7% 50 98.0% 75 98.7% 100 (very slow disch.) 99.0%

Standardizing power density

Today, power density is the preferred measure of a cell's or battery's technology ability to provide power for a given volume or mass. Volumetric power density is measured in Wh / liter, and gravimetric power density is measured in Wh / kg.

Lack of a standard

Energy density and short discharge time are well defined physical characteristics. On the contrary, power density is a nebulous measure: who is to say how much power a cell can deliver? Is it the continuous power that will not result in a damaging temperature rise? Is it the peak power that the conductors can handle? Is it the maximum power delivered when the load has the same resistance as the battery (at 50 % efficiency)? It is up to the manufacturer to define what the power that a cell technology can deliver: on one side, a conservative manufacturer may define a low value, for the sake of improving the cycle life of the cell; on the other side, and aggressive manufacturer may define a high value, for the sake of impressing the market. That may be the reason why so few manufacturers specify power density.

Proposed standard

The industry would be well served if power density were specified at a standard point. Just as the industry picked C/20 or C/1 as the current used to measure capacity, the industry may pick a point to measure power density. Using a point that is a particular multiple of the short discharge time would be convenient, as it would make that point independent of voltage and capacity. Possible candidates for such factors include:

• e (~2.718), where the efficiency is 1-e^-1 (~63.21%)
  
• 10, where the efficiency is 90 %
  
• 100, where the efficiency is 99 %.

The factor of e is mathematically elegant, while a factor of 10 is more easily described. Ultimately, we chose a factor of e because it results in values that are more in line with the practice of the few cell manufacturers who do specify specific power (A123 among them).

Sample power densities

An analysis of power density of the same 20 Li-ion cell technologies was performed, at a current that will discharge the cell in e times the short discharge time. This analysis reveals that the short discharge time is a close indicator of power density.

This graph lists the cell technologies in order of short discharge time, the same order as the previous graph. You will note that the decrease of power density is nearly monotonic (with the notable exception of Boston Power), indicating a close relationship between short discharge time and power density.

A scatter plot of power density versus inverse short discharge time makes that point even clearer.

Current limits

Short discharge time is not a replacement for the current limits specified by the manufacturer. For one thing, short discharge time is a theoretical time: you certainly to not want to discharge a cell into a short circuit! For another thing, short discharge time does not consider any limitations on the cell current that are imposed by the chemistry and by the interconnections.

Therefore, when selecting a cell for a given application, short discharge time must be used in combination with the manufacturer's specified current limits.

Energy density vs short discharge time

Classically, we use Ragone charts to correlate the energy density and the power density of a cell or battery technology. Given the impreciseness of power density, let us create a new type of chart, replacing power density with inverse short circuit time.

Series and parallel

The SDT of a battery that uses elements (cells or batteries) that all have the same SDT, is equal to that same SDT.
e.g.: if two cells (even of different capacity) with SDT of 100 seconds are placed in parallel, the resulting battery also has an SDT of 100 s.

If the elements have unequal SDT (which may not be advisable from a technical standpoint), then calculating the total SDT of the battery is not straightforward.
To calculate the total SDT one must convert the SDT of each element to resistance (using the capacity and voltage of the element) then use standard methods to calculate the total resistance of the battery and finally convert that back to SDT, using the total capacity and voltage of the battery.

Davide Andrea, Elithion, 10/21/12; Updated 12/22/16