White Paper - Maximum Power Time

A tool to compare cells and batteries for power applications

Introduction

When selecting cells for a power application, 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 "Maximum Power Time", the time required to discharge a full cell (or battery) as it delivers the maximum power it can deliver. This constant is a characteristic of each cell technology, and of a battery composed of such cells, regardless of capacity or voltage.

Characterization

The Maximum Power 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 Maximum Power Time is:

maximum_power_time [h] = 2 * capacity [Ah] * resistance [Ω] / voltage [V]

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

maximum_power_time [s] = 7200 * capacity [Ah] * resistance [Ω] / voltage [V]

Calculator - Capacity: [Ah] Resistance: [mΩ]. Voltage: [V]. Max power time: [s]

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

maximum_power_time [s] = 7200 * 2.3 [Ah] * 10 m [Ω] / 3.3 [V] = 50 s (Beware: many specify impedance at 1 kHz instead, which is quite unrelated to DC resistance.)

Graphically

Unfortunately, very few manufacturers specify true DC resistance. If discharge curves are available, they may be used to derive the Maximum Power Time. Discharge curves 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. Take the difference of the two specific currents; that's the delta-specific-current [1/h]. Note the difference in the cell voltage at those two points; that's the delta-voltage [V]. Then, use those values to calculate the Maximum Power Time [s]:

maximum_power_time [s] = 7200 * delta_voltage [V] / delta_specific_current [1/h] / nominal_voltage [V]

Calculator:

Nom. voltage:

[V]:

Low specific current: High specific current:

[1/h] [1/h]

Voltage at low current: Voltage at high current:

[V] [V]

MPT: [s]

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 nominal voltage is 3.3 V. Then, the Maximum Power Time of this cell (and of cells of any size using the same technology, and of batteries using these cells) is:

maximum_power_time [s] = 7200 * 0.3 [V] / 4 [1/h] / 3.3 [V] = 164 s

Rarely, the discharge curves show actual current [A] (rather than specific current [1/h]). If so, use this equation instead:

maximum_power_time [s] = 7200 * delta_voltage [V] * capacity [Ah] / delta_current [A] / nominal_voltage [V]

Calculator:

Capacity: Nom. voltage:

[Ah] [V]

Low current: High current:

[A] [A]

Voltage at low current: Voltage at high current:

[V] [V]

MPT: [s]

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

The capacity is 10 Ah, the delta voltage is 3.28 - 2.98 = 0.30 [V]; the delta current is 50 - 10 = 40 [A]; the nominal cell voltage is 3.3 V. Then, the Maximum Power Time of this cell (and of cells of any size using the same technology, and of batteries using these cells) is:

maximum_power_time [s] = 7200 * 0.3 [V] * 10 [Ah] / 40 [A] / 3.3 [V] = 164 s

Empirically

Having access to an actual cell, one can derive the Maximum Power 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 Maximum Power Time can be calculated:

maximum_power_time [s] = 7200 * capacity [Ah] * resistance [Ω] / voltage [V]

Sample Maximum Power Times

Using the methods above, the Maximum Power Time of various cell technologies was calculated, and is listed below.

This table compares the Maximum Power Time of various cell and battery chemistries.

Technology	Model	Maximum Power Time [s]
Lead acid	Panasonic VRLA	170~172
	Xtreme Power	36
	EnerSys Cyclon	44~108
Li-ion	See table below	28~440
NiMH	Energizer	222~310
	Panasonic HHR	80
NiCd	Sanyo Cadnica	50

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

More specifically, this graph compares the Maximum Power Time of a select few cell and battery technologies.

Applications

The Maximum Power 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 Maximum Power 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 Maximum Power Time. Of the cells analyzed in this study, the Kokam SLPB-H5 series LiPo cells have the best Maximum Power 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 Maximum Power 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] = maximum_power_time [s] * voltage [V] / capacity [Ah] / 7200

Or, given the voltage and energy:

resistance [Ω] = maximum_power_time [s] * (voltage [V])^2 / energy [Wh] / 7200

Or, given the energy and capacity:

resistance [Ω] = maximum_power_time [s] * energy [Wh] / (capacity [Ah])^2 / 7200

For example, using a cell technology that has a Maximum Power Time of 72 seconds, given the battery voltage and capacity:

• 10 V, 100 Ah -> resistance = 72 * 10 / 100 / 7200 = 1 mΩ
  
• 100 V, 100 Ah -> resistance = 72 * 100 / 100 / 7200 = 10 mΩ
  
• 10 V, 10 Ah -> resistance = 72 * 10 / 10 / 7200 = 10 mΩ
  
• 100 V, 10 Ah -> resistance = 72 * 100 / 10 / 7200 = 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 Maximum Power Time of a battery, the efficiency is easily derived.

efficiency [%] = 100 * (1 - (maximum_power_time [s] / actual_discharge_time [s] / 2) )

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 Maximum Power 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 Maximum Power Time: 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 = MPT) 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%

Relationship to power density and specific power

Power density and specific power indicate the ability of a cell or battery technology ability to provide power for a given volume or mass. Volumetric specific power is measured in Wh / liter, and gravimetric power density is measured in Wh / kg.

These measures are not clearly defined because there is no standard operating point at which they are measured. Conversely, the operating point for the Maximum Power Point is well defined: operation at the maximum power the cell or battery can deliver at any given moment.

I did an analysis of power density of the same 20 Li-ion cell technologies, at a current that will discharge the cell at the Maximum Power Time. This analysis reveals that the Maximum Power Time is closely related to the cell's stated power density and specific power.

This graph lists the cell technologies in order of Maximum Power 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 Maximum Power Time and power density.

A scatter plot of power density versus inverse Maximum Power Time makes that point even clearer.

Current limits

Maximum Power Time is not a replacement for the current limits specified by the manufacturer because it 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, Maximum Power Time must be used in combination with the manufacturer's specified current limits.

Energy density vs Maximum Power 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 Maximum Power Time.

Batteries with cells of unequal Maximum Power Time

The Maximum Power Time of a battery composed of cells that all have the same Maximum Power Time, is equal to that same Maximum Power Time.
E.g., if two cells (even of different capacity) with Maximum Power Time of 100 seconds are placed in parallel, the resulting battery also has an Maximum Power Time of 100 s.

If the elements have unequal Maximum Power Time (which may not be advisable from a technical standpoint), then calculating the total Maximum Power Time of the battery is not straightforward.
To calculate the total Maximum Power Time one must convert the Maximum Power Time 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 Maximum Power Time, using the total capacity and voltage of the battery.