How to Calculate Inverter Power Rating and Inverter Battery Backup Time

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Inverter systems are a common feature in our homes and workplace where they play a prominent role in the ensuring uninterruptible power to sensitive loads and devices. For home applications, there is the need to adequately size your inverter to be able to meet the expected load demand. 

Inverters convert DC voltage to AC voltage. They have a battery system which provide adequate backup time to provide continuous power in the home. The inverter system then converts the battery voltage to AC voltage through electronic circuitry. The inverter system also has some charging system that charges the battery during utility power. During utility power, the battery of the inverter is charged and at the same time power is supplied to the loads in the house. When utility power fails, the battery system begins to supply power via the inverter to the loads in the home as shown below:

How to Size and Calculate the Inverter Power Requirement

Inverter power is rated in VA or KVA.

Power in VA = AC Voltage x AC Current in Amps

Power in KVA = AC Voltage x AC Current in Amps/1000

Power in Watts = AC Voltage x AC Current in Amps x PF

Where PF = power factor 

Power in KW = AC Voltage x AC Current in Amps x PF/1000

Also  Power in W = Power in VA x PF

         Power in KW = Power in KVA x PF

Suppose we want to size an inverter to carry the following loads:

1. Lighting load, 300W

2. 3 Standing fans of 70W, each

3. 2 LCD TV, 100W

4. 1 Home Theatre Music System, 200W

5. 1 Juice extractor, 150W

Applying Power in KW = Power in KVA x PF

Power in KVA  = Power in KW/PF = Power in KW/0.8    (Nominal PF = 0.8, which is standard for homes)

Total load in Watts = 300 + (3 x 70) + 200 + 200 + 150 = 1060W = 1.06KW

Power in KVA = 1.06/0.8 = 1.325

An inverter of standard rating 1.5KVA is required to carry the loads above.

How to Calculate Inverter Battery Backup Time

The backup time for batteries in an inverter system depends on the number of batteries as well as their capacity in Amp-hours.

Inverter battery backup time is calculated as:

Back up time = Battery Power in Watt hour (Wh)/Connected Load in Watts (W)

Battery Power in Wh = Battery Capacity in AH x Battery Voltage (V) x Number of Batteries

Let us shorten the formula by using the following Symbols:

Let BUT = battery backup time in hours

            C = battery capacity in AH 

    V = battery voltage in volts

    N = Number of batteries in series or parallel as the case may be.

    $P_L$ = connected load in watts (W)

Now

$$BUT = {\frac{C*V*N}{ P_L}}$$

In our example above, suppose we have selected a 24V, 1.5KVA inverter system that is to use two 12V batteries in series connection and suppose further that the capacity of our batteries are 200AH each, then :

C = 200AH

V = 12V

N = 2

$P_L$   = 1,060W

$$BUT = {\frac{200 * 12 * 2}{1060}} = 4.53 hrs$$

Common Causes of Battery Failures

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All batteries have a limited life span. However the life span can be considerably shortened by certain factors which tend to cause premature battery failure. The factors discussed below are some of the most common causes of battery failure. Given the roles batteries play and will continue to play in our everyday life, a thorough understanding of these factors will enable engineers and technicians involved in the maintenance of batteries prevent the occurrence of some of these factors in order to prolong battery life.

The factors discussed below are mostly applicable to VRLA (Valve Regulated Lead Acid) batteries which have almost completely replaced conventional lead acid batteries with refillable liquid sulphuric acid electrolyte. VRLA batteries are designed to be maintenance free and the hydrogen that is emitted is recombined internally so that the electrolyte (essentially a paste) does not need replacing over the life of the battery, a valve is installed to release any excess pressure that may build up if the battery were failing. A key drawback of VRLA batteries is the short service life which was procured at the expense of low maintenance.

Elevated Temperatures

Anticipated battery life is specified by the manufacturer for batteries installed in an environment at or near the reference temperature of 25°C (77°F). Above this temperature, battery life is reduced. The chief aging mechanism is accelerated corrosion of the positive plates, grid structure, and strap, which increases exponentially as a function of temperature. Elevated temperatures reduce battery life. An increase of 8.3°C (15°F) can reduce lead-acid battery life by 50% or more. 

Repeated Cycling

Repeated cycling from fully charge to fully discharge and back may cause loss of active materials from the positive plates. This reduces battery capacity and its useful life.

Overcharging

Overcharging by the battery charging system causes excessive gassing and high internal heat. Too much gassing can lead to the removal of active material from the plates. Too much heat can also oxidize the positive plate material and warp the plates.

Undercharging

A faulty charging system will not maintain the battery at full charge. Severe undercharging allows sulfate on the plates to become hard and impossible to remove by normal charging. The undercharged battery may fail to deliver the required power needed for its application.

Over discharge

Over discharge leads to hydration. Hydration occurs in a lead-acid battery that is over discharged and not promptly recharged. Hydration results when the lead and lead compounds of the plates dissolve in the water of a discharged cell and form lead hydrate, which is deposited on the separators. When the cell is recharged, multiple internal short circuits occur between the positive and negative plates. Once hydration is evident, the cell is permanently damaged. Hydration is not visible in VRLA cells because the containers are opaque

Vibration

A battery must be mounted securely. Vibrations can loosen connection, crack the case and damage internal components.

DC Ripple Current

Excessive DC ripple current might contribute to battery aging. VRLA batteries are extremely susceptible to ripple current since it can lead to cell heating and will accelerate the degradation of cells which are at risk from thermal runaway.

Improper Storage

Storing wet cells beyond the manufacturer’s recommended duration promotes sulfation, and decreases cell capacity and life.

Misapplications

Batteries are commonly designed for a specific use. If the battery is not designed for a given application, it might not meet its life or performance expectations.

Understanding Battery Technical Specifications.

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Commonly in a specification sheet for a typical battery, you have all kinds of technical terms that need to be understood so as to be able to use the battery in the right way to get maximum benefit from  the battery in a particular application. Summarized below are some of the key technical terms used in battery specifications:

Nominal Voltage (V)

This is the reference voltage of the battery, also sometimes thought of as the “normal” voltage of the battery.

Cut-off Voltage (V)

This is the minimum allowable voltage of a battery. It is this voltage that generally defines the “empty” state of the battery.

Capacity or Nominal Capacity (AH for a specific C rate)

This is the total Amp-hours available when the battery is discharged at a certain discharge current (specified as a C-rate) from 100 percent state-of-charge to the cut-off voltage. Capacity is calculated by multiplying the discharge current (in Amps) by the discharge time (in hours) and decreases with increasing C-rate.

State of Charge (% SOC)
SOC is defined as the remaining capacity of a battery and it is affected by its operating conditions such as load current and temperature. It is calculated as:

$$  SOC  = {\frac {Remaining \ Capacity}{Rated \ Capacity}} $$

Depth of Discharge
DOD is used to indicate the percentage of the total battery capacity that has been discharged.

$$ DOD = 1 - SOC $$

Energy or Nominal Energy (Wh for a specific C rate)

This is the “energy capacity” of the battery, the total Watt-hours available when the battery is discharged at a certain discharge current (specified as a C-rate) from 100 percent state-of-charge to the cut-off voltage. Energy is calculated by multiplying the discharge power (in Watts) by the discharge time (in hours). Like capacity, energy decreases with increasing C-rate.
The rated Wh capacity of a battery can be calculated as:

$Rated \ Wh = Rated \ Ah \ Capacity \ * \ Rated \ Battery \ Voltage$

Cycle Life (Number for a specific DOD)

This is the number of discharge-charge cycles the battery can experience before it fails to meet specific performance criteria. Cycle life is estimated for specific charge and discharge conditions. The actual operating life of the battery is affected by the rate and depth of cycles and by other conditions such as temperature and humidity. The higher the DOD, the lower the cycle life.

Specific Energy (Wh/Kg)

This is the nominal battery energy per unit mass, sometimes referred to as the gravimetric energy density. Specific energy is a characteristic of the battery chemistry and packaging. It is expressed in Watt-hours per kilogram (Wh/kg) as:

$$Specific \ Energy = {\frac {Rated \ Wh \ Capacity}{Battery \ Mass \ in \ Kg}}$$

Specific Power (W/Kg)

This is the maximum available power per unit mass. Specific power is a characteristic of the battery chemistry and packaging. It determines the battery weight required to achieve a given performance target. It is expressed in W/kg as:

$$Specific \ Power = {\frac {Rated \ Peak \ Power}{Battery \ Mass \ in \ Kg}}$$

Peak Power
The peak power of a battery is defined as:
$$P = {\frac{2V_{oc}^2}{9r}}$$
Where:
$V_{oc}$ is the open circuit voltage of battery
$r$ is the internal resistance of the battery

Energy Density (Wh/L)

This is the nominal battery energy per unit volume, sometimes referred to as the volumetric energy density. Specific energy is a characteristic of the battery chemistry and packaging. Along with the energy consumption of the vehicle, it determines the battery size required to achieve a given electric range.

Power Density (W/L)

The maximum available power per unit volume. Specific power is a characteristic of the battery chemistry and packaging. It determines the battery size required to achieve a given performance target.

Maximum Continuous Discharge Current

This is the maximum current at which the battery can be discharged continuously. This limit is usually defined by the battery manufacturer in order to prevent excessive discharge rates that would damage the battery or reduce its capacity. 

Maximum 30-sec Discharge Pulse Current

This is the maximum current at which the battery can be discharged for pulses of up to 30 seconds. This limit is usually defined by the battery manufacturer in order to prevent excessive discharge rates that would damage the battery or reduce its capacity.

Charge Voltage (V)

This is the voltage that the battery is charged to when charged to full capacity. Charging schemes generally consist of a constant current charging until the battery voltage reaches the charge voltage, then constant voltage charging, allowing the charge current to taper until it is very small.

Float Voltage (V)

This is the voltage at which the battery is maintained after being charge to 100 percent SOC to maintain that capacity by compensating for self-discharge of the battery.

(Recommended) Charge Current

The ideal current at which the battery is initially charged (to roughly 70 percent SOC) under constant charging scheme before transitioning into constant voltage charging.

Internal Resistance (Maximum)

This is the resistance within the battery, generally different for charging and discharging.

Basic Battery Terminology

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There are certain  basic battery terminology that tends to be misunderstood in practice. These terms commonly refers to the condition of the battery as well as capacity of the battery:

State of Charge (SOC) %

SOC defines the present battery capacity as a percentage of maximum capacity. SOC is generally calculated using current integration to determine the change in battery capacity over time.

Depth of Discharge (DOD) %

This expresses the percentage of battery capacity that has been discharged expressed as a percentage of maximum capacity. A discharge to at least 80 % DOD is referred to as a deep discharge.

Terminal Voltage (V)

This is the voltage between the battery terminals with load applied. Terminal voltage varies with SOC and discharge/charge current.

Open Circuit Voltage (V)

This is the voltage between the battery terminals with no load applied. The open-circuit voltage depends on the battery state of charge (SOC), increasing with state of charge.

Internal Resistance

This is the resistance within the battery, generally different for charging and discharging, also dependent on the battery state of charge. As internal resistance increases, the battery efficiency decreases and thermal stability is reduced as more of the charging energy is converted into heat.

Consider the battery circuit above, where E is the open circuit voltage, I is the load current flowing in amps, V is the terminal voltage across a load resistance $R_L$, r is the battery internal resistance

Power  input to battery is given by: $$P_i = IE$$ $$E = I (R+r)$$ $$P_i = I *I(R+r) = I^2(R+r)$$ power output: $$P_o = IV = I*IR = I^2R$$ Battery Efficiency  $$Eff = {\frac{P_o}{P_i}} = {\frac{I^2R}{ I^2(R+r)}} = {\frac{R}{(R+r)}}$$ As can be seen, the more the internal resistance of the battery, the less efficient the battery becomes due to increasing conversion of useful battery energy to heat.

C and E – Rates

In describing batteries, discharge current is often expressed as a C-rate in order to normalize against battery capacity, which is often very different between batteries. A C-rate is a measure of the rate at which a battery is discharged relative to its maximum capacity.

C – Rate can be expressed as:

$$I = M * C_n $$     

Where:

I  = Discharge current in Amps

C = Numerical value of rated capacity of the battery in ampere – hours (AH)

n = Time in hours for which rated capacity of battery is declared

M = Multiple or fraction of C

Consider a battery rated at 200AH with a discharge current of 10Amps, the C rate is calculated as: 

M = I/Cn  = 10/200 = 1/20 = 0.05C or C/20 rate

E’’ Rate

An E-rate describes the discharge power. A 1E rate is the discharge power to discharge the entire battery in 1 hour. Just like the C rate, the E rate can be expressed as:

$$P = M * E_n $$    

Where:

I  = Discharge current in Amps

E = Numerical value of rated power of the battery in watt – hours (Wh)

n = Time in hours at which the battery was rated

M = Multiple or fraction of C

Consider a battery with rated at 1200mWh with rated power of 600mW, the E rate is calculated as :

M = P/En  = 600mW/1200mW  = 0.5E or E/2 rate

Basics of Battery Capacity Ratings

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With the rise in global warming due to excessive use of fossil fuels on the plant, calls are becoming increasingly louder for renewable energy sources that are less damaging to the environment and sustainable in the long run. Batteries are a common feature of renewable energy sources such as solar systems as well as a common feature in both cars using fossil fuel derivatives and electric cars which are being touted to replace fossil fuel cars in the feature.

Given the role batteries play in our everyday life, there is the need to understand battery capacity ratings which are commonly used.

What is the Capacity of a Battery?

Battery capacity is the amount of electrical energy a battery can deliver when fully charged. The capacity of a battery is determined by factors such as size, number of plates, the number of cells and the strength and volume of electrolyte.

Common battery capacity ratings in use are:

1. Cold Cranking Amperes (CCA)

2. Reserve Capacity (RC)

3. Amp-Hours (AH)

4. Power (Watts)

Cold Cranking Amperes (CCA)

This capacity rating applies to the ability of the battery to provide the required energy to drive a prime mover e.g. car engine. In this case, it refers to the ability of the battery to provide energy to crank an engine during starting. This will entail a large discharge in a short time. The CCA rating of a battery specifies in amperes the discharge load a fully charged battery at 0°F can deliver for 30 seconds while maintaining a voltage of at least 1.2 volts per cell 

Reserve Capacity (RC)

This describes the ability of a battery to provide emergency energy for a given time to meet certain load demands should the battery charging system fails. This will require adequate battery capacity at normal temperatures for certain period of time.  The RC rating of a battery specifies in minutes, the length of time a fully charged battery at 80°F (26.7°C) can be discharged at 25 Amps while maintaining a voltage of at least 1.75 volts per cell

Amp-Hours (AH)

The Amp-Hour (AH) rating of a battery is the most popular and commonly used rating of a battery. It is often called the 20-hour discharge rating. The Amp-Hour rating of a battery specifies in amp-hours, the current the battery can provide in 20 hours at 80°F (26.7°C) while maintaining a voltage of at least 1.75 volts per cell.

Amp-Hour (AH) = Current x Time (Hours)

A battery that delivers 10Amps for 20hrs has a capacity of = 10 x 20 = 200AH.

Power (Watts)

For prime mover applications where the power is required to provide cranking power, its capacity can also be rated in watts. The power rating of a battery in watt is determined by multiplying the current available by the battery voltage at 0°F (-17.8°C)