physics//electromagnetism//electric circuit//equivalent circuit model

An equivalent circuit model is a small network of ideal sources, resistors and capacitors chosen so that its voltage and current at the terminals behave like those of a real device, and its main use today is to represent a battery cell inside a battery management system. Nobody claims the cell contains those components. The chemistry inside is diffusion of ions and reactions at electrodes; the circuit only reproduces what the terminals show, which is what an estimator needs and what it can afford to compute a thousand times a second.

An equivalent circuit model is a small network of ideal sources, resistors and capacitors chosen so that its voltage and current at the terminals behave like those of a real device, and its main use today is to represent a battery cell inside a battery management system. Nobody claims the cell contains those components. The chemistry inside is diffusion of ions and reactions at electrodes; the circuit only reproduces what the terminals show, which is what an estimator needs and what it can afford to compute a thousand times a second.

The simplest version is a voltage source, the open-circuit voltage VocV_{oc}Voc​, in series with an internal resistance R0R_0R0​:

V=Voc−R0 I.V = V_{oc} - R_0\,I .V=Voc​−R0​I.

Under a discharge current III the terminal voltage sags by R0IR_0 IR0​I at once, and VocV_{oc}Voc​ itself depends on the state of charge through a curve measured for each chemistry. Adding one or two resistor-capacitor pairs in series reproduces the slower part of the response: after a load step the voltage keeps relaxing for seconds to minutes, and each pair's time constant RCRCRC stands for one of those slow processes.

Its value is that the parameters are few and trackable. With φ=[1, −I]⊤\varphi=[1,,-I]^{\top}φ=[1,−I]⊤ and θ=[Voc, R0]⊤\theta=[V_{oc},,R_0]^{\top}θ=[Voc​,R0​]⊤ the simplest model is linear in its parameters, so recursive least squares can follow both online as the cell ages and warms, and a rising R0R_0R0​ is itself a health indicator.

It is the model inside the filter. An extended Kalman filter that predicts the terminal voltage with this circuit and compares it with the measured one is how battery management systems estimate the charge left (state of charge estimation); the same circuit, kept calibrated to one particular pack, is the model of its digital twin.

Its limits are those of any grey box. It fits the conditions it was identified in; at extreme temperature, very high current or late in life the parameters move faster than the estimator follows, and an electrochemical model, far heavier to run, is what describes the physics there.