Method & interpretation
A small experiment with a visible chain of reasoning
This study asks how a single cell responds to a sustained discharge current under a stated thermal boundary. It connects three effects: charge leaves the cell, terminal voltage falls below its open-circuit value, and electrical resistance generates heat. The result is an estimate of the charge and energy available before a voltage or charge boundary, or the chosen time limit, ends the run.
Use it to understand a tradeoff, screen candidate cells or examine a cooling assumption. The useful output is both the trace and its explanation: which parameter changed, which equation carried that change, and why the run ended. A result without the input assumptions is incomplete.
Layers that open into real models
The canvas begins with a compact overview. Opening a block enters its internal model; it does not simply highlight a group in one flat diagram. Each scope has an interface, internal connections and a place in the hierarchy. Use the tree to jump between scopes, or the breadcrumb to return to a parent.
- Study overview. The load protocol, environment, cell and measurements are the top-level models. Each opens directly into its own implementation.
- Cell. Electrical response and thermal response receive the same cell-current input.
- Electrical / thermal. Electrical opens into SOC, OCV and terminal response. Thermal opens into heat generation, cooling and temperature evolution; its heat-generation model computes resistive dissipation from current.
- Primitive equations. Inspect the sources, gains, products, sums, limits and state integrators that implement those relationships.
The measurements model reads the solved states; it does not manufacture a second trajectory for the display. Likewise, expanding a model changes the view, not the physical assumptions. Changing a study input rebuilds the model and makes the previous run stale until it is run again.
| Model | What it owns | Primitive relationship |
|---|---|---|
| SOC | Charge state | Current scaling and integration |
| OCV | Voltage at rest | Piecewise-linear chemistry curve |
| Terminal response | Loaded voltage | Resistance gain and voltage sum |
| Heat generation | Resistive dissipation inside Thermal | Current product scaled by resistance |
| Cooling | Heat transfer to ambient | Temperature difference and conductance |
| Temperature state | Average cell temperature | Net heat divided by heat capacity, then integrated |
Electrical response: charge, voltage and energy
Discharge current is positive. With constant capacity and resistance, the model uses Coulomb counting for SOC, a representative open-circuit-voltage curve, and an ohmic voltage drop:
SOC rateds/dt = −I / (3600 Q)
Terminal voltageV = OCV(s) − I R
Resistive heatingPheat = I² R
Delivered energyEWh = ∫ V I dt / 3600
Here, s is the SOC fraction, Q is capacity in Ah, current is in A, resistance is in Ω and time is in seconds. The form accepts resistance in mΩ and converts it before calculation. Delivered energy integrates the loaded voltage, rather than multiplying capacity by one nominal voltage.
The OCV subsystem interpolates between declared SOC–voltage points. A chemistry label provides a representative shape, not a measured characterization of every cell with that chemistry. Its role is especially visible near a flat plateau or the lower-voltage knee. Open OCV to see its SOC segments, then open a segment to inspect its offset, interpolation, bounds and voltage contribution. These are nested subsystems built from the existing primitives. The electrical definitions are documented in Bessler (2024), equations 6–7.
Thermal response: one temperature, two heat paths
The thermal model treats the cell as one body with a single average temperature. Its heat-generation subsystem receives cell current and computes I²R using the declared DC resistance. The cooling subsystem exchanges heat with a fixed ambient:
Thermal capacityCth = m cp
Cooling conductanceUA = h A
Temperature ratedT/dt = [I²R − UA(T − Tamb)] / Cth
Mass is converted from grams to kilograms, and area from cm² to m². Specific heat is entered in J/(kg·K). A temperature difference has the same numerical value in kelvins and degrees Celsius. When the cell is cooler than ambient, the ambient term warms it; it is not clipped to “cooling only”.
Area means the area that participates in the chosen boundary. A cylinder’s geometric surface area is a starting estimate, but installed contacts, sleeves, neighboring cells and blocked surfaces change effective heat transfer. The coefficient h must match that interpretation. Setting h to zero creates an adiabatic comparison; it does not describe every insulated installation. See Bernardi, Pawlikowski and Newman for the general heat balance and the mechanisms omitted by this reduction.
Read the provenance before the precision
Selecting a cell fills a starting parameter set and exposes its source notes. Manufacturer ratings, estimated fields and user overrides have different meanings. A catalog value can be useful without being measured under your operating conditions; a long decimal result does not improve the evidence behind it.
- Capacity and current rating: retain the source’s test temperature, cutoff and discharge conditions. A current command is a study input, not a new continuous-current rating.
- DC resistance: use a measurement with a stated SOC, temperature and pulse duration when available. AC impedance cannot be silently substituted for discharge DC resistance.
- Thermal inputs: specific heat, exposed area and heat transfer often require an explicit assumption. Record how each was obtained.
- Initial conditions: initial SOC and cell temperature need not represent a fully charged cell at ambient. They belong to the experiment, not solely to the cell record.
- Voltage bounds: distinguish the curve’s declared OCV range from the loaded-voltage cutoff used to end the discharge.
An edited field defines this study’s override; it does not revise the original cell record. Keep the exported model with the run so another engineer can reconstruct exactly which values were used.
Usable capacity is conditional on the load
At a given current, the cell reaches the terminal-voltage cutoff when OCV(s) = Vcut + I R. This study determines that crossing from the declared piecewise-linear curve and constant-current assumptions, then ends the native graph run at the earliest applicable boundary. It does not introduce a hidden BMS controller or claim a sampled protection-latch simulation.
The run can instead reach zero SOC or its time limit. If the initial loaded voltage already violates cutoff, there is no valid positive-duration discharge at that operating point. Always read the stated end condition: a time-limited run does not establish full available capacity.
To edit the exported hierarchy, choose Model JSON, open Lemonergy Studio, and use Import to load the file. The export’s cutoff horizon and end time were computed from this study’s constant-current inputs. Arbitrary internal parameter edits in Studio require recalculating that horizon and end time; editing this wiki’s input form regenerates them automatically.
Charge left behind at an early voltage cutoff is unextracted charge under this load. It is not permanent aging. A lower current or a rest period can make some charge accessible again. That distinction matters when comparing a high-resistance cell with a low-resistance cell of the same nominal capacity.
Change one question at a time
Equal current compares cells against the same load; equal C-rate compares current relative to capacity. These answer different questions. Equal heat transfer coefficient also does not imply equal cooling, because area differs. An imposed ambient temperature is not the same as a complete shared cooling system.
Save a baseline, change one parameter and save the next completed run. Compare delivered Wh alongside Ah, runtime and temperature. A larger cell may deliver more energy simply because it contains more active material. For a weight-constrained application, review mass as well as the discharge result. Preserve the initial SOC, cutoff convention and duration limit when those are intended to be common.
Check the model with simple limits
A useful verification starts with cases that can be calculated independently. The instantaneous voltage drop should equal I R, while resistive heating should equal I²R. Over a known uninterrupted interval, charge removal should equal I t / 3600. These checks test units and sign conventions before interpreting a more complicated curve.
For constant heating, the temperature has an analytical reference. With no ambient transfer, the rise is I²R t / Cth. With nonzero conductance, it approaches Tamb + I²R / UA exponentially with time constant Cth / UA. A cell initially at another temperature should approach the same equilibrium according to that initial condition.
Check cutoff against its curve segment, compare integrated energy with an independent integral, and repeat a run with tighter numerical settings when evaluating convergence. Export/import should preserve the model identity and result. The retained run evidence records prepared runs and their numerical identities. Focused analytical tests are separate from those run records; mathematical agreement is also distinct from comparison with measured cell data.
What this model leaves unresolved
Resistance and capacity stay fixed during a run. The electrical response does not identify diffusion dynamics, hysteresis, temperature-dependent resistance or cycle aging. The thermal response resolves neither a core-to-surface gradient nor internal hot spots. Its heat source omits reversible entropic heat and other mechanisms absent from the declared equations.
The calculation describes one cell and its imposed environment. It does not model pack interconnect resistance, uneven current sharing, coolant temperature rise, neighboring-cell heat transfer or a real controller’s response. A temperature output is an average under those assumptions, not a prediction of every location inside the cell. These boundaries define when a more detailed model or a measured discharge is needed.
Extend the subsystems in deliberate steps
The strategy is to keep a useful screening study and a transparent block model together. Both views use the same inputs and native calculation. Changes should preserve named interfaces so a more detailed electrical or thermal subsystem can replace the baseline without reorganizing the entire experiment.
| Stage | Change inside the hierarchy | Evidence before promotion |
|---|---|---|
| Current baseline | Existing sources, gains, products, sums, limits and integrators form the nested cell model shown above. | Analytical checks, native replay, charge and energy checks, export/import and declared data assumptions. |
| Characterized cell data | Replace representative OCV segments with measured SOC–voltage data and documented resistance measurements. Keep the cell interface. | Record test temperature, current, SOC, cutoff and measurement method; compare predicted voltage and delivered energy with held-out discharges. |
| Electrical dynamics | Add polarization states inside Electrical and support resistance varying with SOC and temperature. Introduce a validated stop/event mechanism before adding variable-current protocols. | Pulse and rest-response agreement, stable state initialization, and correct cutoff under changing load. The current precomputed horizon is insufficient for this stage. |
| Thermal detail | Replace the uniform temperature state with core and surface states, then an explicit cooling boundary. Couple electrical properties to the appropriate measured temperature. | Thermal energy balance and transient agreement with measurements across heating and cooling conditions. |
| Module or pack reuse | Compose separately parameterized cell instances inside module and pack subsystems, with interconnects and thermal paths made explicit. | Independent state per cell, unequal current sharing, electrical and thermal conservation, and controlled scaling of runtime. |
These later stages are a development plan, not capabilities claimed by this page. Each extension should retain the baseline for comparison and carry its own validation evidence. A new primitive is justified only when the existing blocks cannot express the required relationship or solve it reliably.
Research references and reproducibility
The references below support the equations, measurement definitions and limits of this study. The interactive hierarchy, model composition and interface are implemented with Lemonergy's own blocks. Citing a method does not establish agreement with measured behavior for every selectable cell.
- Electrical model and capacity definitions. Wolfgang G. Bessler (2024), Capacity and Resistance Diagnosis of Batteries with Voltage-Controlled Models, Journal of The Electrochemical Society 171, 080510. Equations 6–7 define current-based SOC evolution and the OCV-minus-ohmic-drop model with discharge-positive current. The paper also distinguishes capacity measured at a stated load and different resistance measurement methods. We cite those definitions; this study does not implement the paper's diagnosis algorithm.
- Battery heat balance and omitted mechanisms. D. Bernardi, E. Pawlikowski and J. Newman (1984), A General Energy Balance for Battery Systems, Lawrence Berkeley Laboratory report LBL-17776, April 1984. This original report develops a uniform-temperature energy balance and heat exchange with the surroundings. Its broader formulation includes reversible effects, mixing and other contributions beyond the resistive heating retained here. Equation 18 gives the ambient heat-transfer form used in our reduced model.
- Manufacturer data for the default cell. E-One Moli Energy, INR-21700-P42A product data sheet, version 1.7. It reports capacity, voltage limits, mass and discharge conditions, and distinguishes typical DC impedance of 16 mΩ at 10 A / 1 s from AC impedance of 10 mΩ at 1 kHz. The study's chemistry curve, exposed-area estimate and cooling coefficient remain declared assumptions. Other selectable cells retain their own record-level provenance.
Our implementation and verification. The source download contains the actual nested subsystem definitions, their interfaces, native runtime, cell-data adapter and focused tests. The retained evidence identifies prepared runs and their inputs. Reproducing a software result is separate from validating the model against a measured discharge.