Equivalent Circuit Model - Battery Design

An equivalent circuit model (ECM) is a phenomenological model widely used in industry to simulate the voltage response for subsequent Battery Management System control and state estimation.

By conducting experiments to measure the battery voltage at various SoCs and temperatures it is possible to develop an ECM that relates the applied current and the voltage.

first order equivalent circuit model — A first order Equivalent Circuit Model describing the terminal voltage (V_l) of a battery to a given applied current (I). U(z) is the OCV vs SoC look-up-table and the losses are modelled via resistor and capacitor elements that change with SoC (z) and temperature (T).

Battery Cell Modelling

The primary reason why one is interested in having a model of a battery is it allows us to make informed decisions, optimise or control a battery to deliver on its intended applications. For instance, we may be interested in knowing how different material choices, electrode compositions or cell packaging, will influence the electrochemical and thermal performance of the cell, a model can serve this purpose.

Two Related Families of Equivalent Circuit

The ECM above, and the Battery Cell Modelling post it links to, both describe one specific style of equivalent circuit: a low-order network of resistors and capacitors, fitted in the time domain from a pulse or multisine test, whose element values are tabulated against SoC and temperature so a BMS can run it in real time. That is the workhorse ECM for online state estimation, but it is not the only kind of equivalent circuit this site covers, and the two are easy to conflate because they share a name and a family resemblance.

The other branch comes out of electrochemical impedance spectroscopy (EIS). Instead of fitting a pulse response in the time domain, EIS measures the cell’s complex impedance across a frequency sweep, typically from single-digit mHz up to several kHz, and fits an equivalent circuit to the resulting Nyquist plot. The standard starting topology for that fit is the Randles circuit: a solution resistance in series with a charge-transfer resistance and double-layer capacitance in parallel, usually extended with a Warburg element to capture diffusion. Because each element in a Randles-type circuit maps onto a distinct physical process rather than a black-box RC pair, it is the model of choice when the goal is characterisation and diagnosis rather than real-time control.

Time-domain, Thevenin-style ECM (this page, Battery Cell Modelling): low element count, parameters fitted from pulse or multisine current profiles, tabulated against SoC and temperature, run online inside a BMS. See also DCIR of a Cell for the single-resistance, time-domain view of the same internal resistance that shows up as R0 here and as Rohmic on a Nyquist plot.
Frequency-domain, EIS-derived ECM: built around the Randles circuit and its extensions, parameters fitted by complex non-linear least squares against a measured impedance spectrum, run offline for characterisation, diagnosis, and research. Start with Electrochemical Impedance Spectroscopy and the Idealised Nyquist Plot of a Lithium-Ion Battery for how to read the spectrum, then Warburg Impedance for the diffusion tail specifically.

In practice the two do talk to each other. The RC time constant in a Thevenin ECM and the semicircle in a Randles-type fit are both trying to describe the same charge-transfer and double-layer behaviour, just sampled a different way, and a cell’s DCIR and its Rohmic intercept on a Nyquist plot are, to a first approximation, the same physical resistance measured in two domains. Where they diverge is fitting effort and intended use: a two- or three-parameter Thevenin ECM is deliberately cheap enough to run every control-loop tick on embedded BMS hardware, while a Randles-type fit trades that speed for a model whose parameters are worth interpreting individually, which is why it stays offline as a lab and diagnostic tool rather than replacing the BMS-resident ECM.

If you are trying to automate the EIS side of that fitting process, or want a primer on the extra distribution-of-relaxation-times (DRT) step that often follows it, see How well can an LLM Interpret Electrochemical Impedance Spectroscopy (EIS) Data?

Applications

There are many different uses for an ECM including: pack electrical design and optimisation, Battery Management System and also within the manufacturing process.

circuit schematic for HiPot test modelling

Using an ECM to Model HiPot Tests

With the RC model, we can predict the expected dynamic response (leakage current) through the system for a given capacitance. The nominal capacitance and insulation resistance of the cell can be estimated by fitting this model to empirical current response.

Parameter Estimation

Particle Swarm Optimization Based Battery Parameter Identification

Parameter identification of a battery equivalent circuit model (ECM) using Particle Swarm Optimization (PSO). The ECM includes a series resistance R0, an RC branch with R1 and C1, and a polynomial OCV-SOC relationship.

Beyond Equivalent Circuits

An ECM, of either family, is a lumped approximation: it fits the voltage or impedance response without directly representing the electrode particles, porous structure, or transport paths generating that response. When those physical details matter, for instance when you need to know whether a fast-charge protocol is pushing an electrode toward lithium plating, rather than just fitting the resistance that results, a physics-based model is the usual next step:

Doyle Fuller Newman (DFN) Battery Model, the standard porous-electrode model, is what most physics-based fast-charge and plating-risk work builds on. Our piece on Concentration Polarization is a worked example of the kind of transport-limited behaviour that an ECM’s Warburg or diffusion element only summarises, and that a DFN-type model resolves directly.
Single Particle Model (SPM) sits between the two: still physics-based, but simplified enough to run faster than a full DFN model.