 # Energy Density or Power Density for your Application

Different battery applications have different requirements in terms of rechargeability, safety and biocompatibility. Aside from these factors, there is one fundamental question for any application: is the application better suited for high energy or high power batteries?  Answering this question will help determine the cell chemistry and design the application needs. When you look at commercial batteries, they are normally described as primary or rechargeable.  They are sometimes referred to as being a nominal trade size (AA, AAA, AAAA, C, D, CR2016, 18650, etc.).  They are also normally described as having a given battery chemistry (NiMH, Ni-Zn, alkaline, Zn-MnO2, carbon-zinc, lithium, lithium-ion, and so on), and manufacturers normally provide nominal values for voltage and capacity.  It’s relatively easy to say how much energy the battery provides, but different chemistries can have different form factors (e.g. you don’t see AA lithium-ion cells very often), complicating apples-to-apples comparisons between chemistries. All else equal, a larger battery will generally have more energy. However, battery manufacturers are often vague on power. Sometimes batteries are referred to as “high power” without really disclosing exactly what this means.

In this article, we will discuss what power and energy are. What is the difference between them? What do energy density, power density, specific energy, and specific power mean, exactly?  This article will help you understand these concepts and tell you how to calculate the energy density, power density, specific energy, and specific power your application needs. It will provide some educated guess on how difficult your requirements are to be met.

#### 1. What Are Energy and Power?

The amount of charge a battery contains is its energy, generally expressed in terms of Watt-hours or Ampere-hours. This tells us how long a battery can last. The relationship between Power (normally measured in Watts) and Energy can be expressed according to the  equation (1a), while the relationship between capacity and energy is given in equation (1b):

Power (W) × Time (h) = Energy (W-h) —— (1a)

Average voltage (V) x Capacity (A-h) = Energy (W-h) —— (1b)

The Power can be expressed as in equation (2):

Power (W) = U (V) × I (A) —— (2)

Where U is the battery’s discharge voltage, and I is the associated discharge current. Power is an instantaneous measure of how much energy flows through an electrical circuit at a given time; it does not, however, tell us how long the battery can be used at this power level. In general, a battery can supply less energy for a higher power level.

#### 2. Energy Density (W-h/L) and Specific Energy (W-h/kg)

Energy density is the maximum available energy (W-h) per unit of volume (L), sometimes referred to as the volumetric energy density, a characteristic of the battery chemistry and its packaging. It tells us how much energy the battery can store in a given size. Energy density is commonly used to compare different batteries. Normally, this value is taken from a discharge at low current rate, and it is determined by the quantity of electrode materials within the cell. A battery with a higher energy density will be smaller than a battery with the same energy and a lower energy density. In other words, a battery with a higher energy density can power a given application for a longer time than one with a lower energy density and the same physical size.

To determine the practical energy density of a battery under a given application load (to a specific test cut-off voltage, and at a given temperature), one needs to multiply the capacity (in Amp-hours) that the battery delivers under those conditions by the average discharge voltage, and divide it by the cell volume, as shown in equation (3). ——(3)

Note that a battery’s nominal voltage is always different from its average discharge voltage. The nominal voltage refers to the open-circuit voltage of a fresh or fully-charged battery.  The battery’s discharge voltage will decrease under load, and in many batteries the voltage will also decrease over the course of the discharge due to the chemical nature of the reactive materials.

The energy density can be estimated using the manufacturer’s published product data sheet for a given battery. Table 1 shows the data sheet of a typical Alkaline-Manganese Dioxide AA battery, which we will use as an example throughout this article.

Table 1: Specific data sheet of a typical alkaline-manganese dioxide AA battery (Cut-off voltage: 0.8 V at 21 ⁰C; Based on 1.2 V average operating voltage at 21 ⁰C).

 Product Number Size Nominal Voltage Rated Capacity Load Weight Volume volts Ampere-hours Ohms Kilograms Liters MN 1500 AA 1.5 2.85 43 0.024 0.008

—–(4)

According to equation (3), one can calculate the energy density of this AA size alkaline MnO2 battery, like the example in equation (4) below.

Energy density is an important factor when battery size or weight are the primary design considerations and when anticipated discharge currents are relatively low. In portable devices, energy density is a critical parameter, but in stationary applications such as photovoltaic storage, energy density may be less important.

It has to be stressed that the calculated energy density is related to a given discharge rate, temperature, battery size, average discharge voltage, and cut-off voltage.  A battery can, for instance, have a higher energy density when discharged at a lower discharge rate or to a final lower cutoff voltage.

If there is no data sheet available, the information needed to calculate the energy density can also be obtained through direct battery testing. You can discharge a battery with a constant-resistance or constant-current discharge. By recording the load, the capacity (Ah), the average discharge voltage, the cut-off voltage, the temperature, and the volume of the battery, you can estimate the energy density through equation (3). Specific energy is the nominal battery energy (W-h) per unit of mass (kg), sometimes referred to as the gravimetric energy density as shown in equation (5). —–(5)

Specific energy becomes more important when the weight of the battery is critical, such as in portable electronics. Specific energy of a typical alkaline-manganese dioxide AA battery can be estimated as shown in equation (6): —–(6)

Energy density is a simple, straightforward way to compare batteries with different sizes.  It does not, however, tell the entire story, particularly at high current rates, as we discuss below.

Power Density (W/L) and Specific Power (W/kg)

Power density refers to the maximum available power (W) per unit of volume (L), and it measures how quickly the battery can deliver energy for a given size. Power density is sometimes an important consideration when space is constrained, as it determines the battery size required to achieve a given performance target, such as in an electrical vehicle.

Generally, power density is expressed as peak power density (maximum power per unit of volume) and specific power (maximum power per unit of weight). Note that a battery can only supply the maximum power for a very brief timeframe (seconds or less). Very often, batteries are discharged below their maximum power, thus the battery C-rate (discharge current divided by capacity) is often used to describe a battery’s power capability.

Peak Power Density is defined by the equation (7): According to the U.S. Advanced Battery Consortium (USABC)’s definition [1, 2], the peak power (P) is defined as equation (8):

Peak Power = 2(Voc)2/9R —— (8)

Where Voc is the battery open-circuit voltage and R is its internal resistance. For example, the typical effective internal resistance of a fresh alkaline AA cylindrical battery is approximately 0.1 Ω when tested with a 1000 Hz impedance test.

Therefore, the calculated peak power density for a typical fresh AA alkaline battery is:

Peak Power Density = [2 × (1.5 V)2/(9 × 0.1 W)]/0.008 L = 625 W/L   ——- (9)

The battery internal resistance can also be estimated from direct battery testing using a pulse method if there is no data sheet available from manufacturers. For example, in order to determine the internal resistance of a typical alkaline-manganese dioxide AA battery, one can apply a 5 mA stabilization drain for 2 minutes, followed by a 505 mA 100 millisecond pulse. The internal resistance can be calculated through equation (10):

R = DV/I          ——- (10)

Where DV is the voltage difference between start and end of the 100 millisecond pulse, I is the pulse discharge current of 505 mA.

Specific Power is the maximum available power (W) per unit of mass (kg), shown by equation (11). Similar to the power density, the specific power can be also divided into two different categories, specific peak power and specific power. Therefore, the specific peak power of a typical alkaline-manganese dioxide AA battery can be calculated as following in equation (12):

Specific Peak Power = [2 × (1.5 V)2/(9 × 0.1 W)]/0.024 kg = 208 W/kg   —— (12)

C-Rate

Peak power, as described above, is not the only value of interest when comparing batteries. In many applications, the battery is discharged at a power below the peak power and the discharge rate is often given for a battery discharged at a certain “C-rate.” The C-rate is the discharge rate divided by the battery nominal capacity. For instance, a 100 mA-h battery would have a discharge rate of 100 mA for a 1C discharge, and a 200 mA discharge rate for a 2C discharge. Generally, the battery capacity decreases with higher C-rate. The power of a battery is relative to the “C-rate” it is discharged at. For example, a typical 50 Ah – 3.2V LiFePO4 cell might have a continuous discharge rate of 2C and short duration bursts of 6 C. So its continuous current rating would be 2 × 50 A = 100 A. The continuous power is 100 A × 3.2 V = 320 W. The burst rate current is 6 × 50 A = 300 A, and the actual burst rated power = 300 A × 3.2 V = 960 W. If you know the battery volume in liters, then you can calculate the power density through equation (13) ——-(13)

Conclusion

Device designers can determine early in the product development the energy and power needed to operate the device. With a few calculations, one can determine quickly if the device is at all feasible as specified for weight and volume, based on its estimated operation mode.

 USABC, Electric Vehicle Battery Test Procedures Manual, revision 2, 1996.

 The 17th National Conference on Vehicle Engineering, Nov. 9, 2012, Nan Kai U. of Tech., Nantou, Taiwan, R.O.C., C -026. 