What is the equation for the discharge of a lead acid battery?
2 Answers
The discharge of a lead-acid battery can be described by a number of equations depending on the specifics of the analysis you are looking for, such as voltage, capacity, or current. Here are some key equations and concepts involved:
### 1. **Battery Discharge Current (Ohm's Law)**
The current \( I \) drawn from the battery during discharge can be determined using Ohm’s Law:
\[ I = \frac{V_{battery} - V_{load}}{R_{load}} \]
Where:
- \( V_{battery} \) is the voltage of the battery.
- \( V_{load} \) is the voltage across the load.
- \( R_{load} \) is the resistance of the load.
### 2. **Battery Capacity (Ampere-Hour Rating)**
The capacity of a lead-acid battery, denoted as \( C \), is usually given in ampere-hours (Ah). During discharge, the battery's capacity decreases over time, which can be approximated by:
\[ C_{remaining} = C_{initial} - I \times t \]
Where:
- \( C_{remaining} \) is the remaining capacity.
- \( C_{initial} \) is the initial capacity of the battery.
- \( I \) is the discharge current.
- \( t \) is the time elapsed.
### 3. **Voltage Drop during Discharge**
The voltage \( V \) of a lead-acid battery decreases as it discharges. A simple model to estimate this voltage drop is:
\[ V_{discharge} = V_{nominal} - \frac{I \times R_{internal}}{N} \]
Where:
- \( V_{discharge} \) is the voltage during discharge.
- \( V_{nominal} \) is the nominal voltage of the battery (e.g., 2.0 V per cell for a fully charged lead-acid battery).
- \( R_{internal} \) is the internal resistance of the battery.
- \( N \) is the number of cells in the battery.
### 4. **Peukert’s Equation**
To account for the non-linear relationship between discharge rate and battery capacity, Peukert’s equation can be used:
\[ t = \frac{C_{nominal}}{I^k} \]
Where:
- \( t \) is the time to discharge.
- \( C_{nominal} \) is the nominal capacity of the battery.
- \( I \) is the discharge current.
- \( k \) is Peukert’s constant, which depends on the battery design and discharge conditions.
### 5. **State of Charge (SoC)**
The State of Charge of a lead-acid battery can be estimated based on the voltage and capacity:
\[ SoC = \frac{V_{current} - V_{min}}{V_{full} - V_{min}} \times 100\% \]
Where:
- \( V_{current} \) is the current voltage of the battery.
- \( V_{min} \) is the minimum voltage of the battery (fully discharged).
- \( V_{full} \) is the maximum voltage of the battery (fully charged).
These equations and models help in understanding and predicting the performance of lead-acid batteries during discharge.
### 1. **Battery Discharge Current (Ohm's Law)**
The current \( I \) drawn from the battery during discharge can be determined using Ohm’s Law:
\[ I = \frac{V_{battery} - V_{load}}{R_{load}} \]
Where:
- \( V_{battery} \) is the voltage of the battery.
- \( V_{load} \) is the voltage across the load.
- \( R_{load} \) is the resistance of the load.
### 2. **Battery Capacity (Ampere-Hour Rating)**
The capacity of a lead-acid battery, denoted as \( C \), is usually given in ampere-hours (Ah). During discharge, the battery's capacity decreases over time, which can be approximated by:
\[ C_{remaining} = C_{initial} - I \times t \]
Where:
- \( C_{remaining} \) is the remaining capacity.
- \( C_{initial} \) is the initial capacity of the battery.
- \( I \) is the discharge current.
- \( t \) is the time elapsed.
### 3. **Voltage Drop during Discharge**
The voltage \( V \) of a lead-acid battery decreases as it discharges. A simple model to estimate this voltage drop is:
\[ V_{discharge} = V_{nominal} - \frac{I \times R_{internal}}{N} \]
Where:
- \( V_{discharge} \) is the voltage during discharge.
- \( V_{nominal} \) is the nominal voltage of the battery (e.g., 2.0 V per cell for a fully charged lead-acid battery).
- \( R_{internal} \) is the internal resistance of the battery.
- \( N \) is the number of cells in the battery.
### 4. **Peukert’s Equation**
To account for the non-linear relationship between discharge rate and battery capacity, Peukert’s equation can be used:
\[ t = \frac{C_{nominal}}{I^k} \]
Where:
- \( t \) is the time to discharge.
- \( C_{nominal} \) is the nominal capacity of the battery.
- \( I \) is the discharge current.
- \( k \) is Peukert’s constant, which depends on the battery design and discharge conditions.
### 5. **State of Charge (SoC)**
The State of Charge of a lead-acid battery can be estimated based on the voltage and capacity:
\[ SoC = \frac{V_{current} - V_{min}}{V_{full} - V_{min}} \times 100\% \]
Where:
- \( V_{current} \) is the current voltage of the battery.
- \( V_{min} \) is the minimum voltage of the battery (fully discharged).
- \( V_{full} \) is the maximum voltage of the battery (fully charged).
These equations and models help in understanding and predicting the performance of lead-acid batteries during discharge.
The discharge of a lead-acid battery can be described through several key equations that involve different aspects of the battery's performance. Here’s a detailed explanation:
### 1. **Peukert's Law**
One common way to describe the discharge behavior of lead-acid batteries is Peukert's Law. This law provides a way to estimate the battery’s capacity based on the discharge rate.
**Peukert's Law Equation:**
\[ C_{d} = C_{n} \times \left(\frac{I_{n}}{I_{d}}\right)^{k} \]
Where:
- \( C_{d} \) is the capacity of the battery at the discharge current \( I_{d} \).
- \( C_{n} \) is the nominal capacity of the battery at a standard discharge current \( I_{n} \) (typically the 20-hour rate).
- \( I_{d} \) is the discharge current.
- \( k \) is Peukert's constant, which is specific to the battery and reflects its discharge characteristics.
### 2. **Capacity Equation**
The capacity of a lead-acid battery can also be expressed as:
\[ C = I \times t \]
Where:
- \( C \) is the capacity of the battery (in ampere-hours, Ah).
- \( I \) is the discharge current (in amperes, A).
- \( t \) is the discharge time (in hours, h).
This equation is more straightforward but doesn’t take into account variations in discharge rate, which is where Peukert’s Law becomes useful.
### 3. **Battery Voltage During Discharge**
The voltage of a lead-acid battery during discharge can be approximated with the following formula, which considers the internal resistance and the rate of discharge:
\[ V_{d} = V_{oc} - I \times R_{int} \]
Where:
- \( V_{d} \) is the voltage of the battery during discharge.
- \( V_{oc} \) is the open-circuit voltage of the battery (voltage when not under load).
- \( I \) is the discharge current.
- \( R_{int} \) is the internal resistance of the battery.
### 4. **State of Charge (SOC) and Depth of Discharge (DOD)**
The state of charge and depth of discharge are related to the battery’s capacity. For lead-acid batteries:
- **State of Charge (SOC)** can be estimated from the voltage, but a more accurate method is to measure the specific gravity of the electrolyte or use a battery monitor.
- **Depth of Discharge (DOD)** is given by:
\[ DOD = \frac{C_{n} - C_{d}}{C_{n}} \times 100\% \]
Where:
- \( DOD \) is the percentage of capacity used.
- \( C_{n} \) is the nominal capacity of the battery.
- \( C_{d} \) is the remaining capacity.
### Summary
To fully understand and predict the discharge behavior of a lead-acid battery, you need to consider several factors, including the Peukert’s Law for capacity estimation, internal resistance for voltage drop, and SOC/DOD for state monitoring. These equations together help in modeling the battery's performance under different conditions.
### 1. **Peukert's Law**
One common way to describe the discharge behavior of lead-acid batteries is Peukert's Law. This law provides a way to estimate the battery’s capacity based on the discharge rate.
**Peukert's Law Equation:**
\[ C_{d} = C_{n} \times \left(\frac{I_{n}}{I_{d}}\right)^{k} \]
Where:
- \( C_{d} \) is the capacity of the battery at the discharge current \( I_{d} \).
- \( C_{n} \) is the nominal capacity of the battery at a standard discharge current \( I_{n} \) (typically the 20-hour rate).
- \( I_{d} \) is the discharge current.
- \( k \) is Peukert's constant, which is specific to the battery and reflects its discharge characteristics.
### 2. **Capacity Equation**
The capacity of a lead-acid battery can also be expressed as:
\[ C = I \times t \]
Where:
- \( C \) is the capacity of the battery (in ampere-hours, Ah).
- \( I \) is the discharge current (in amperes, A).
- \( t \) is the discharge time (in hours, h).
This equation is more straightforward but doesn’t take into account variations in discharge rate, which is where Peukert’s Law becomes useful.
### 3. **Battery Voltage During Discharge**
The voltage of a lead-acid battery during discharge can be approximated with the following formula, which considers the internal resistance and the rate of discharge:
\[ V_{d} = V_{oc} - I \times R_{int} \]
Where:
- \( V_{d} \) is the voltage of the battery during discharge.
- \( V_{oc} \) is the open-circuit voltage of the battery (voltage when not under load).
- \( I \) is the discharge current.
- \( R_{int} \) is the internal resistance of the battery.
### 4. **State of Charge (SOC) and Depth of Discharge (DOD)**
The state of charge and depth of discharge are related to the battery’s capacity. For lead-acid batteries:
- **State of Charge (SOC)** can be estimated from the voltage, but a more accurate method is to measure the specific gravity of the electrolyte or use a battery monitor.
- **Depth of Discharge (DOD)** is given by:
\[ DOD = \frac{C_{n} - C_{d}}{C_{n}} \times 100\% \]
Where:
- \( DOD \) is the percentage of capacity used.
- \( C_{n} \) is the nominal capacity of the battery.
- \( C_{d} \) is the remaining capacity.
### Summary
To fully understand and predict the discharge behavior of a lead-acid battery, you need to consider several factors, including the Peukert’s Law for capacity estimation, internal resistance for voltage drop, and SOC/DOD for state monitoring. These equations together help in modeling the battery's performance under different conditions.