Rint is the simplest model, Thevenin adds transient polarization, and PNGV adds an accumulator for long-term voltage behavior. The three models differ primarily in the number and purpose of their resistive and capacitive elements. Battery research equipment evaluates their parameters by applying controlled current profiles, recording voltage and current responses, and fitting the measured data to each model at defined SOC and temperature conditions.
The Rint model is suitable for static or slow-changing analysis, while the Thevenin model is generally the best baseline for dynamic lithium-ion battery testing because it captures transient polarization without the cumulative voltage drift associated with the PNGV model.
How the Three Equivalent Circuit Models Differ
Rint: An Ohmic Approximation
The Rint model consists of an open-circuit voltage source, (U_{OCV}), in series with an internal resistance, (R_0).
A simplified terminal-voltage relationship is:
[ V_t = U_{OCV} - I R_0 ]
The model represents the immediate voltage drop caused by current flowing through the cell's ohmic resistance. Its parameters are typically functions of SOC, temperature, and sometimes current direction.
Rint contains no capacitor and therefore has no internal state that can reproduce voltage relaxation after a current pulse. It cannot accurately represent charge-transfer polarization, double-layer effects, or diffusion-related transients.
Thevenin: A Dynamic Polarization Model
The Thevenin model extends Rint by adding a parallel resistor-capacitor branch in series with the voltage source and ohmic resistance.
Its main elements are:
- (U_{OCV}): open-circuit voltage source
- (R_0): instantaneous ohmic resistance
- (R_p): polarization resistance
- (C_p): polarization capacitance
The RC branch produces a time-dependent polarization voltage. A common representation is:
[ V_t = U_{OCV} - I R_0 - V_p ]
where (V_p) evolves according to the current and the RC time constant:
[ \tau = R_p C_p ]
This structure captures the voltage drop immediately after a load is applied and the subsequent voltage recovery when the load is removed. The model therefore tracks dynamic voltage behavior substantially better than Rint.
PNGV: A Thevenin Model with an Accumulating Voltage State
The Partnership for a New Generation of Vehicles (PNGV) model adds a bulk or series capacitance, commonly denoted (C_{pb}) or (C_b), to the Thevenin-type structure.
The model includes:
- (U_{OCV}): open-circuit voltage source
- (R_0): ohmic resistance
- (R_p) and (C_p): transient polarization branch
- (C_{pb}) or (C_b): cumulative charge-related voltage element
The polarization RC pair describes short-term dynamic behavior. The additional capacitance represents the cumulative change in cell voltage associated with integrating current over time.
Conceptually, the PNGV model separates two effects:
- Fast-to-moderate transients, represented by (R_p) and (C_p).
- Accumulated voltage change, represented by the bulk capacitance state.
This can improve short-pulse modeling, but the cumulative state can also introduce long-term voltage drift when small parameter errors or measurement offsets are integrated over extended testing.
What Each Component Represents Physically
Open-Circuit Voltage
The open-circuit voltage source represents the cell's equilibrium voltage at a specified SOC and temperature.
It is normally obtained from an OCV-SOC characterization test. Because equilibrium may require long rest periods, the measured value can depend on the selected rest duration and the cell's relaxation behavior.
Ohmic Resistance
(R_0) represents the immediate voltage response associated with electronic resistance, ionic resistance, current collectors, contacts, and other fast processes.
During a current step, it is commonly estimated from the instantaneous voltage change:
[ R_0 \approx \frac{\Delta V_{\text{instantaneous}}}{\Delta I} ]
This estimate must account for measurement bandwidth, switching delay, wiring resistance, and the distinction between charge and discharge behavior.
Polarization Resistance and Capacitance
(R_p) and (C_p) describe the slower voltage response that follows the initial ohmic drop.
The resistance controls the magnitude of the polarization voltage. The capacitance controls how quickly that voltage develops and relaxes. Their product defines the time constant:
[ \tau = R_p C_p ]
A small time constant produces a rapid transient, while a large time constant produces slower voltage relaxation.
Bulk or Accumulating Capacitance
The PNGV bulk capacitance represents a voltage state associated with accumulated current.
Because this state depends on current integration, it can reproduce longer-term voltage movement that a single Thevenin RC branch may not capture. However, it is also sensitive to current-sensor bias, inaccurate initial conditions, and changes in battery behavior that are not represented by a fixed capacitance.
How Battery Testing Equipment Generates the Required Data
Controlled Current Profiles
A battery cycler or precision battery test system applies a prescribed current profile while measuring terminal voltage, current, and often temperature.
Common profiles include:
- HPPC tests
- Constant-current pulses
- Charge and discharge pulse sequences
- Rate-capability tests
- Dynamic Stress Tests
- Drive-cycle or application-specific current profiles
The equipment must provide accurate current control, synchronized voltage and current sampling, programmable rest periods, and sufficient bandwidth to resolve the immediate ohmic response.
SOC and Temperature Control
Tests are usually repeated at multiple SOC values because (U_{OCV}), (R_0), (R_p), and (C_p) vary across the operating range.
A typical characterization sequence controls:
- Initial SOC.
- Cell temperature.
- Rest duration before each pulse.
- Pulse amplitude and duration.
- Recovery period after the pulse.
A thermal chamber or environmental test system is used when temperature dependence is being characterized. Consistent cell assembly and contact conditions are also important because fixture resistance and cell-to-cell variation can distort the extracted parameters.
HPPC Pulse Interpretation
An HPPC profile is particularly useful because it produces distinct voltage features for parameter extraction.
After a current step:
- The immediate voltage change is associated mainly with (R_0).
- The slower voltage change is associated with the polarization branch.
- The recovery response after the pulse provides information about (R_p), (C_p), and the relevant time constant.
- In PNGV identification, the longer-term voltage trend is also used to estimate the accumulating capacitance state.
The same approach can be applied to charge and discharge pulses, although the resulting parameters may not be identical.
How the Parameters Are Evaluated
Step 1: Establish the OCV-SOC Relationship
The cell is charged or discharged to a target SOC and allowed to rest until its terminal voltage approaches an equilibrium condition.
Repeating this process across the SOC range produces an OCV-SOC lookup table or fitted function. This table supplies (U_{OCV}) during subsequent model simulations.
Step 2: Estimate the Ohmic Resistance
The voltage immediately before and after a current transition is compared with the current change.
For a current step from (I_1) to (I_2):
[ R_0 \approx \frac{V(t_1^+) - V(t_1^-)}{I_2-I_1} ]
The exact sign convention depends on whether discharge current is defined as positive or negative. In practice, researchers also correct for delays and exclude samples affected by switching artifacts.
Step 3: Fit the Polarization Response
For a Thevenin or PNGV model, the transient portion of the pulse is fitted to an exponential response.
A single-RC branch has the general form:
[ V_p(t) = V_{p,0}e^{-t/\tau}
- R_p I\left(1-e^{-t/\tau}\right) ]
The measured voltage is compared with the model prediction while estimating (R_p) and (\tau). The capacitance is then calculated as:
[ C_p = \frac{\tau}{R_p} ]
This procedure can be performed independently at each SOC and temperature point.
Step 4: Estimate the PNGV Accumulating Capacitance
For PNGV identification, the measured voltage trend over longer pulse sequences is used to estimate the additional capacitance parameter.
The current is integrated over time, and the resulting accumulated charge is related to the voltage change attributed to (C_{pb}) or (C_b). Initial voltage conditions must be defined carefully because an incorrect initial capacitor voltage can appear as a parameter error.
The estimate is particularly sensitive to long-duration drift, current measurement bias, and the cell's changing OCV-SOC relationship.
Step 5: Use Least-Squares Parameter Estimation
Researchers commonly minimize the difference between measured and simulated terminal voltage:
[ J(\theta)=\sum_{k=1}^{N} \left[V_{\text{measured}}(k)-V_{\text{model}}(k,\theta)\right]^2 ]
where (\theta) contains the model parameters, such as:
[ \theta = {R_0, R_p, C_p, C_{pb}, U_{OCV}} ]
The time constant (\tau) can be scanned or optimized, and the value producing the best fit is selected. The coefficient of determination, (r^2), is one metric used to compare how well the fitted model explains the measured voltage response.
A high (r^2) is useful, but it should not be the only criterion. Researchers should also inspect maximum voltage error, root-mean-square error, pulse recovery behavior, and performance under test profiles different from the identification data.
Step 6: Build Lookup Tables
Because battery parameters vary with operating conditions, the extracted values are commonly stored as lookup tables indexed by:
- SOC
- Temperature
- Charge or discharge direction
- Occasionally current level or aging state
The model can then interpolate parameters during simulation instead of using one fixed set of values for the entire operating range.
Understanding the Trade-offs
Rint Is Simple but Not Dynamic
Rint requires little computation and is easy to identify.
Its simplicity makes it useful for rough voltage estimation, steady-state analysis, and applications where transient accuracy is not important. It is unsuitable when the objective includes pulse-power prediction, transient voltage limits, or accurate SOC estimation during rapidly changing loads.
Thevenin Balances Accuracy and Stability
The Thevenin model captures the dominant transient polarization response with relatively few parameters.
It generally avoids the cumulative integration error associated with the PNGV bulk capacitance. This makes it a strong baseline for battery testing, dynamic simulation, and validation of fabricated lithium-ion cells.
Its limitation is that one RC branch cannot represent every electrochemical time scale. Cells with pronounced diffusion or multi-stage relaxation may require two or more RC branches.
PNGV Can Drift During Long Simulations
PNGV can represent both transient polarization and accumulated voltage change.
However, its accumulating capacitance can produce substantial long-term error if the parameter, initial condition, OCV map, or current measurement is inaccurate. Reports of large errors over repeated dynamic cycles are test-specific rather than universal, but they illustrate the model's sensitivity to cumulative effects.
More Parameters Increase Identification Risk
Adding resistors and capacitors does not automatically produce a better model.
Extra parameters can become correlated, meaning different parameter combinations produce similar voltage curves. Without sufficiently informative pulses, independent validation, and good measurement quality, a more complex model may fit the identification data while generalizing poorly.
Making the Right Choice for Your Goal
The appropriate model depends on the required time scale, accuracy, computational budget, and test duration.
- If your primary focus is simple steady-state voltage estimation: Use the Rint model and identify (U_{OCV}) and (R_0) across SOC and temperature.
- If your primary focus is dynamic pulse accuracy: Use the Thevenin model and identify (R_0), (R_p), and (C_p) from synchronized pulse and relaxation data.
- If your primary focus is short-duration behavior with cumulative voltage effects: Evaluate the PNGV model, but validate its bulk-capacitance state over the complete intended operating period.
- If your primary focus is real-time battery management: Use a Thevenin model or enhanced multi-RC derivative with lookup tables and recursive online parameter updates when operating conditions change.
- If your primary focus is research-grade parameter identification: Combine OCV-SOC tests, HPPC pulses, controlled temperature conditions, least-squares fitting, and independent dynamic validation.
The key is to select the simplest equivalent circuit that reproduces the voltage behavior required by the application and then verify its parameters under realistic operating conditions.
Summary Table:
| Model | Components | Dynamic Behavior | Parameter Evaluation |
|---|---|---|---|
| Rint | OCV + R0 | Static, no transient | R0 from instantaneous voltage change |
| Thevenin | OCV + R0 + RC branch | Captures transient polarization (RC time constant) | Fit transient response to obtain Rp, Cp; R0 from instantaneous drop |
| PNGV | OCV + R0 + RC + bulk capacitance | Captures transients and accumulated voltage change | Extend Thevenin fit; estimate bulk capacitance from long-term voltage trend |
Need reliable battery testing equipment to model your cells accurately?
At KINTEK, we provide advanced cyclers, HPPC-capable test systems, and environmental chambers designed to characterize Rint, Thevenin, and PNGV parameters efficiently. Our solutions support SOC/temperature mapping, pulse testing, and parameter fitting for lithium-ion and next-gen batteries. Contact our experts today to optimize your research workflow and get the precise data you need.