## 1 Introduction  
Lead-acid batteries are widely used as a backup in power systems, e.g., in telecommunication stations and power plants. On lead-acid batteries, both the frequency domain Nyquist curve of EIS [1] and time domain simulation of DC equivalent first principle linear circuit (\(FPLC_{DCqu}\)) [2] are widely used in battery management system (BMS) design. In BMS design, linear electrical circuits, including AC and DC equivalent circuits, are more competitive than nonlinear electrochemical circuits. Firstly, linear electrical circuits, with electrical equivalent lumped elements, are intuitive for electrical engineers to understand the physical and electrochemical characteristics of batteries [3]. Batteries' physical and electrochemical parameters are mapped to electrical lumped elements, which are used to estimate batteries' state of health (SoH) or state of charge (SoC) [4]. Secondly, linear impedances are more convenient for simulating batteries' terminal electrical variables in software. Advanced digital signal processing requires that elements should be linear functions of frequency [5], and electrical lumpedimpedances are nature linear functions of frequency. However, lumped electrochemical impedances are nonlinear functions of frequency [6], and are not convenient for programming. Unless special mention is given, the following circuits in this paper only refer to linear electrical circuits. Elements in a circuit, fitting for EIS, are simplified from governing equations [7,8] in lead-acid battery and have clear physical/electrochemical meanings; so, they are the first principle and basis of classic threshold-based BMS design. Charge-transfer resistance and double-layer capacitance are related to non-cohesion of active mass of electrodes; contacting resistances represent the existence of stratification and corrosion of electrodes; nonlinear diffusion impedances characterize sulfating of electrodes [4,9]. Multi-scale SoC/SoH judgment based on EIS is an extension of threshold-based management [10,11]; circuits elements are nonlinearly influenced by various factors, e.g., SoC, charging or discharging, superimposed DC current [12]; other form signals, such as short step response, can take place of sinusoidal excitation to demonstrate dominant aging mechanisms on battery capacity [8]. The leading edge of BMS to forecast and optimize batteries' SoC/SoH is usually based on time domain simulation [13,14]; therefore, extracting directional DC charge-transfer resistances from an AC Nyquist curve is significant in time domain research on BMS. The Nyquist curve, describing AC steady-state volt-ampere characteristics of batteries, is used to evaluate SoH [1], and AC equivalent behavioral linear circuit (\(BHLC_{ACqu}\)) is used for the curve fitting [15]. In certain frequency bands of Nyquist, impedances on electrode-electrolyte interfaces dominate curve shape [1,3]; at a certain frequency point of Nyquist, \(BHLC_{ACqu}\) has a corresponding time domain waveform to approach the measurement point. Then, the behavioral relationship between \(BHLC_{ACqu}\) and Nyquist is established. \(BHLC_{ACqu}\) is an interpretation basis of the Nyquist curve [16]. The physical and electrochemical meanings of contacting resistor (\(R_{C}\)), stray inductor (\(L_{S}\)) and electrolyte bulk capacitor (\(C_{bulk}\)) in \(BHLC_{ACqu}\) are relatively clear [3]. Individual elements in \(BHLC_{ACqu}\) rely on the first principle [17], simplified from governing equations [7,8], but the elements' combination does not consider the first principle. Linear impedance on positive (or negative) electrode-electrolyte interfaces is generally modeled by a charge-transfer resistor (\(R_{ct}\)) and a double-layer capacitor (\(C_{dl}\)) [3,18]. The \(R_{ct}\) describes the DC steady-state volt-ampere characteristics of an electrode-electrolyte interface according to Tafel law [19]; the \(C_{dl}\) describes the DC transient-state volt-ampere characteristics of an electrode-electrolyte interface according to spatial anisotropy [19].