Analysis of Lithium-Ion Battery Voltage

Voltage Parameters
In lithium-ion battery charge-discharge testing or practical use, key voltage parameters include **platform voltage**, **median voltage**, **average voltage**, and **cutoff voltage**. A typical discharge curve is shown in Figure 1.

### Platform Voltage
This is the voltage corresponding to a period where the voltage changes minimally while the capacity changes significantly. Batteries like lithium iron phosphate and lithium titanate exhibit distinct platform voltages, which can be clearly identified in charge-discharge curves. For most batteries with less obvious voltage platforms, data is collected at voltage intervals during testing, and the voltage curve is differentiated. The platform voltage is determined by the peak of the dQ/dV curve.

### Median Voltage
This is the voltage corresponding to half of the battery’s capacity. For materials with prominent platforms, such as lithium iron phosphate and lithium titanate, the median voltage typically equals the platform voltage.

### Average Voltage
This is calculated as the effective area under the voltage-capacity curve (i.e., the battery’s charge/discharge energy) divided by the capacity, with the formula:
**Ü = ∫U(t)*I(t)dt / ∫I(t)dt**.
In charge-discharge test data, the average voltage is obtained by dividing the charge or discharge energy by the capacity. Conversely, the battery’s energy density is estimated using the average voltage, where **energy = capacity * average voltage / battery mass (or volume)**.

### Cutoff Voltage
This refers to the minimum voltage allowed during battery discharge and the maximum voltage allowed during charging. If discharge continues below the discharge cutoff voltage, the positive electrode potential continues to decrease, while the negative electrode potential rises rapidly, leading to over-discharge. Over-discharge may damage the electrode’s active materials, rendering them inactive and shortening battery life. It can also cause decomposition of the copper foil in the negative electrode, which may precipitate on the positive electrode, posing a short-circuit risk. If the charging voltage exceeds the charge cutoff voltage, the positive electrode potential continues to rise, causing excessive lithium extraction, crystal structure failure, and electrolyte decomposition, which depletes lithium ions. Meanwhile, the negative electrode potential continues to drop, leading to excessive lithium intercalation, disintegration of the graphite layered structure, and lithium plating on the electrode surface.

## Battery Voltage Determination
In practice, the battery voltage **U_battery** is determined by the difference between the electrode potential of the positive electrode **E_cathode** and the negative electrode **E_anode**, as expressed by Equation (1):
**U_battery = E_cathode – E_anode (1)**.
In battery systems, the standard lithium electrode is commonly used as the reference electrode. The electrode potentials of the positive and negative electrode materials are generally the potentials generated by reactions between the reactants, products, and the reference lithium electrode. As shown in Figure 2, during the charge-discharge process, the positive and negative electrode materials undergo lithium extraction or intercalation, causing changes in their electrode potentials. The battery voltage is the difference between these two potentials.

To understand a battery’s voltage, it is essential to first comprehend the electrode potentials of various electrode materials. Understanding the equilibrium electrode potential curves of materials helps in better grasping the battery’s voltage characteristics.

## Open Circuit Voltage
The **open circuit voltage (OCV)** is the potential difference between the positive and negative electrodes of a battery in a non-working state, i.e., when no current flows through the circuit. When electrode materials are assembled with metallic lithium into a coin half-cell, the open circuit voltage corresponds to the equilibrium potential of the electrode material.

### Open Circuit Voltage Testing Methods
The equilibrium potential of electrode materials is tested as follows: The electrode material is prepared into an electrode sheet and assembled with metallic lithium into a coin half-cell. The open circuit voltage of the half-cell is measured at different states of charge (SOC), and a mathematical expression for the OCV curve is determined using polynomial or Gaussian fitting. The main OCV testing methods include:

1. **Galvanostatic Intermittent Titration Technique (GITT)**:
The basic principle is to apply a constant current to the system under a specific environment for a set duration, then interrupt the current and observe the potential change over time during the current application period, as well as the equilibrium voltage (i.e., OCV) after relaxation. An example of a GITT test procedure is as follows:
(i) Charge at C/50 until the voltage reaches the upper limit, e.g., 4.2 V;
(ii) Rest for 2 hours;
(iii) Discharge at 1C for 6 minutes, record the discharge capacity;
(iv) Rest for 15 minutes, record the voltage;
(v) Repeat steps (iii) and (iv) nine times;
(vi) Discharge at C/50 until the voltage reaches the lower limit, e.g., 3.0 V;
(vii) Normalize the capacity-voltage curve recorded in steps (iii) and (iv), create an SOC-voltage curve, and fit it to obtain the mathematical expression of the OCV curve.

2. **Low-Current Charge-Discharge Curve**:
Charge and discharge at an extremely low rate (e.g., 0.01C) with constant current, setting upper and lower voltage limits to obtain the low-current charge-discharge curve. Use points with consistent capacity as the curve’s starting point, average the voltages in the charge-discharge curves, normalize the x-axis based on theoretical capacity, and then use curve fitting to obtain the OCV curve.

## Battery Polarization
When current passes through an electrode, the phenomenon where the electrode deviates from its equilibrium potential is called **battery polarization**, which generates an overpotential. Polarization can be classified into **ohmic polarization**, **concentration polarization**, and **electrochemical polarization**, as shown in Figure 2.

– **Ohmic Polarization**: Caused by the resistance of various battery components, its voltage drop follows Ohm’s law. When the current decreases, polarization immediately reduces, and it disappears instantly when the current stops.
– **Electrochemical Polarization**: Caused by the sluggishness of electrochemical reactions on the electrode surface. It significantly decreases within microseconds as the current decreases.
– **Concentration Polarization**: Caused by the slow diffusion of ions in the solution, leading to a concentration difference between the electrode surface and the bulk solution under a given current. This polarization decreases or disappears on a macroscopic time scale (seconds to tens of seconds) as the current decreases.

The internal resistance of a battery increases with increasing discharge current, primarily due to the enhanced polarization trend at higher currents. The larger the discharge current, the more pronounced the polarization, as shown in Figure 2. According to Ohm’s law: **V = E0 – I × RT**, an increase in the total internal resistance **RT** reduces the time required for the battery voltage to reach the discharge cutoff voltage, thereby reducing the discharged capacity.

## Factors Affecting Polarization in Lithium-Ion Batteries
Lithium-ion batteries are essentially lithium-ion concentration cells, where the charge-discharge process involves the insertion and extraction of lithium ions at the positive and negative electrodes. Factors affecting polarization include:

### 2.1 Electrolyte Influence
Low electrolyte conductivity is a primary cause of polarization in lithium-ion batteries. Within typical temperature ranges, the conductivity of electrolytes used in lithium-ion batteries is generally 0.01–0.1 S/cm, only one-hundredth of aqueous solutions. Therefore, during high-current discharge, lithium ions cannot be replenished from the electrolyte quickly enough, leading to polarization. Improving electrolyte conductivity is key to enhancing the high-current discharge capability of lithium-ion batteries.

### 2.2 Influence of Positive and Negative Electrode Materials
Larger particle sizes in positive and negative electrode materials lengthen the diffusion path for lithium ions to reach the surface, which is unfavorable for high-rate discharge.

### 2.3 Conductive Agent
The content of the conductive agent is a critical factor affecting high-rate discharge performance. Insufficient conductive agent in the positive electrode formulation prevents timely electron transfer during high-current discharge, rapidly increasing polarization resistance and causing the battery voltage to drop quickly to the discharge cutoff voltage.

### 2.4 Electrode Design Influence
– **Electrode Thickness**: During high-current discharge, the reaction rate of active materials is fast, requiring rapid lithium-ion insertion and extraction. Thicker electrodes increase the diffusion path for lithium ions, creating a significant lithium-ion concentration gradient along the electrode thickness.
– **Compaction Density**: Higher compaction density reduces electrode porosity, lengthening the lithium-ion transport path. Additionally, excessive compaction density reduces the contact area between the material and electrolyte, decreasing the sites for electrode reactions and increasing internal resistance.

### 2.5 SEI Film Influence
The formation of the solid electrolyte interphase (SEI) film increases the resistance at the electrode/electrolyte interface, causing voltage lag, i.e., polarization.

## Operating Voltage
The **operating voltage**, also known as terminal voltage, is the potential difference between the positive and negative electrodes when the battery is in a working state, i.e., when current flows through the circuit. During discharge, the current overcomes the internal resistance of the battery, causing an ohmic voltage drop and electrode polarization, so the operating voltage is always lower than the OCV. During charging, the opposite occurs, with the terminal voltage being higher than the OCV. Polarization results in the terminal voltage being lower than the electromotive force during discharge and higher during charging.

Due to polarization, there is a slight deviation between the instantaneous voltage and the actual voltage during charging and discharging. During charging, the instantaneous voltage is slightly higher than the actual voltage, and the voltage drops back after polarization disappears at the end of charging. During discharge, the instantaneous voltage is slightly lower than the actual voltage, and the voltage rises back after polarization disappears at the end of discharge.

The composition of the battery’s terminal voltage is shown in Figure 3, with the expressions:
**Charging**: V_CH = (E_+ – E_-) + V_R = (E_+0 + η_+) – (E_-0 – η_-) + V_R
**Discharging**: V_D = (E_+ – E_-) – V_R = (E_+0 – η_+) – (E_-0 + η_-) – V_R

## Why Do Some Materials Have Distinct Voltage Platforms While Others Do Not?
In thermodynamics, the **degree of freedom (F)** is the number of independent variables (e.g., temperature and pressure) that can be changed without altering the number of phases in an equilibrium system. The relationship between the system’s degrees of freedom and other variables is:
**F = C – P + n**
Where:
– **F**: System’s degrees of freedom
– **C**: Number of independent components
– **P**: Number of phases
– **n**: External factors, typically n=2 (representing pressure and temperature)

For lithium-ion electrochemical systems, external factors **n=2** (voltage and temperature). Assuming constant temperature and pressure during the charge-discharge process, we discuss a binary system (**C=2**). If a particle contains one phase (**P=1**), then **F=1**, and the chemical potential varies with lithium concentration (e.g., lithium cobalt oxide, a solid solution with one phase and continuously varying lithium concentration).

If a particle contains two phases (**P=2**), then **F=0**. When two phases coexist in a binary electrode material, a flat voltage platform is observed (e.g., lithium iron phosphate, where two phases coexist, and the lithium concentration in each phase remains constant).