Directional DC Charge-Transfer Resistance on an Electrode–Electrolyte Interface in an AC Nyquist Curve on Lead-Acid Battery

Battery management system design on lead-acid batteries with electrochemical impedance spectroscopy and ﬁrst principle linear circuit model. Abstract: Both the frequency domain Nyquist curve of electrochemical impedance spectroscopy (EIS) and time domain simulation of DC equivalent ﬁrst principle linear circuit ( FPLC DCequ ) are some of the fundamentals of lead-acid batteries management system design. The Nyquist curve is used to evaluate batteries’ state of health (SoH), but the curve does not distinguish charging / discharging impedances on electrode–electrolyte interfaces in the frequency domain. FPLC DCequ is used to simulate batteries’ terminal electrical variables, and the circuit distinguishes charging / discharging impedances on electrode–electrolyte interfaces in the time domain. Therefore, there is no direct physical relationship between Nyquist and FPLC DCequ This paper proposes an AC equivalent ﬁrst principle linear circuit ( FPLC ACequ ) by average switch modeling, and the novel circuit distinguishes charging / discharging impedances on electrode–electrolyte interfaces in Nyquist. The novel circuit establishes a physical bridge between Nyquist and FPLC DCequ for lead-acid batteries management system design.


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 DCequ ) [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 lumped Directional linear impedances on electrode-electrolyte interfaces of FPLC DCequ are solid in theory [2,3], and its transient-state and steady-state responses under DC-input have also been experimentally verified [2,3]. Furthermore, in this paper, average switch modeling is proposed to explore responses of FPLC DCequ under pure AC-input and to analyze AC equivalent linear impedances on electrode-electrolyte interfaces.

Vector Analysis of Responses of Electrode-Electrolyte Interfaces by Average Switch Modeling
Assuming that amplitudes of sinusoidal currents excited from the electrochemical workstation are small enough, voltage responses of batteries' terminals are also trigonometric waveforms as the same frequency as excitation currents [24]. At any certain frequency, internal impedances of batteries are equivalent to linear-constant-time-invariant impedances, and impedances on electrode-electrolyte interfaces of both positive and negative electrodes are also considered as linear-constant-time-invariant [1,3].

• Analysis of Steady-State DC Static Responses by Tafel law
Under stable AC current injection, steady-state DC static responses of impedances on electrode-electrolyte interfaces should be zero in EIS operation. If there are any steady-state DC static voltage responses of batteries' terminals, there should exist non-zero steady-state DC static current injections through batteries' terminals according to Tafel law [25]. This is contradictory to EIS practice.
• Vector Analysis of Steady-State AC Fundamental Voltage Response by average switch modeling Firstly, linear decomposition of equivalent steady-state AC fundamental response vector on electrode-electrolyte interfaces exists. According to the electrical vector analysis principle [26], space vector is regarded as a static vector in rotating the coordinate as the same rotating frequency as stable excitation, and each space vector (V O ) has a clear amplitude and phase in the rotating coordinate. In the rotating coordinate, the target vector can be equivalent to two vectors (V 1 and V 2 ) by linear superposition. As shown in Figure 1a, amplitudes and phases of the two vectors can be different, and there are infinite sets of two vectors (V 1i and V 2i ) equivalent to the target space vector (V O ). In the following sections, the target vector is set as steady-state AC fundamental voltage response of impedances on electrode-electrolyte interfaces of both positive and negative electrodes. Secondly, the linear decomposition of steady-state AC fundamental voltage response vector on electrode-electrolyte interfaces is unique. When stable AC current injections flow through charging impedances on electrode-electrolyte interfaces of both positive and negative electrodes, the charging vector of steady-state the AC voltage fundamental response (V Cha ) is obtained as Equation (1), so be discharging vector of steady-state AC voltage fundamental responses (V Dis ), as Equation (2). During one period of stable AC current injections, both charging and discharging vectors appear for exactly half a period, and their available time is mutually exclusive and collectively exhaustive. According to average switch modeling [23,26], a new charging vector (V Cha ' ), as Equation (3), is exactly half an amplitude and the same phase of charging vector in the coordinate plane. So be new discharging vector (V Dis ' ), as Equation (4) because, during stable AC excitation, both charging and discharging vectors are only available at a half-cycle of stable AC injection. The target vector (V eei ) can only be formed by new charging (V Cha ' ) and discharging (V Dis ' ) vectors by linear superposition as Equation (5) and Figure 1b. Figure 2 is an AC equivalent impedance (Z eei e jθ eei ) similar to Equation (6) by removing the current (I EIS e jθ EIS ) from Equation (5). The physical meanings of elements in the above equations are explained in Table S1 of the Appendix.
V Cha =(I EIS e jθ EIS )(Z Cha e jθ Cha ), V Dis =(I EIS e jθ EIS )(Z Dis e jθ Dis ), V Cha ' =0.5×(I EIS e jθ EIS )(Z Cha e jθ Cha ), V Dis ' =0.5×(I EIS e jθ EIS )(Z Dis e jθ Dis ), Z eei e jθ eei =0.5×(Z Cha e jθ Cha ) + 0.5×(Z Dis e jθ Dis ), During charging, the voltage response of impedances on electrode-electrolyte interfaces of both positive and negative electrodes is regarded as a linear transfer function [3,20], and positive halfwave sinusoidal current excites the transfer function in its charging half-cycle. So be during Secondly, the linear decomposition of steady-state AC fundamental voltage response vector on electrode-electrolyte interfaces is unique. When stable AC current injections flow through charging impedances on electrode-electrolyte interfaces of both positive and negative electrodes, the charging vector of steady-state the AC voltage fundamental response (V Cha ) is obtained as Equation (1), so be discharging vector of steady-state AC voltage fundamental responses (V Dis ), as Equation (2). During one period of stable AC current injections, both charging and discharging vectors appear for exactly half a period, and their available time is mutually exclusive and collectively exhaustive. According to average switch modeling [23,26], a new charging vector (V Cha ), as Equation (3), is exactly half an amplitude and the same phase of charging vector in the coordinate plane. So be new discharging vector (V Dis ), as Equation (4) because, during stable AC excitation, both charging and discharging vectors are only available at a half-cycle of stable AC injection. The target vector (V eei ) can only be formed by new charging (V Cha ) and discharging (V Dis ) vectors by linear superposition as Equation (5) and Figure 1b. Figure 2 is an AC equivalent impedance (Z eei e jθ eei ) similar to Equation (6) by removing the current (I EIS e jθ EIS ) from Equation (5). The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.
Z eei e jθ eei = 0.5× Z Cha e jθ Cha + 0.5× Z Dis e jθ Dis , Secondly, the linear decomposition of steady-state AC fundamental voltage response vector on electrode-electrolyte interfaces is unique. When stable AC current injections flow through charging impedances on electrode-electrolyte interfaces of both positive and negative electrodes, the charging vector of steady-state the AC voltage fundamental response (V Cha ) is obtained as Equation (1), so be discharging vector of steady-state AC voltage fundamental responses (V Dis ), as Equation (2). During one period of stable AC current injections, both charging and discharging vectors appear for exactly half a period, and their available time is mutually exclusive and collectively exhaustive. According to average switch modeling [23,26], a new charging vector (V Cha ' ), as Equation (3), is exactly half an amplitude and the same phase of charging vector in the coordinate plane. So be new discharging vector (V Dis ' ), as Equation (4) because, during stable AC excitation, both charging and discharging vectors are only available at a half-cycle of stable AC injection. The target vector (V eei ) can only be formed by new charging (V Cha ' ) and discharging (V Dis ' ) vectors by linear superposition as Equation (5) and Figure 1b. Figure 2 is an AC equivalent impedance (Z eei e jθ eei ) similar to Equation (6) by removing the current (I EIS e jθ EIS ) from Equation (5). The physical meanings of elements in the above equations are explained in Table S1 of the Appendix.
V Dis =(I EIS e jθ EIS )(Z Dis e jθ Dis ), Z eei e jθ eei =0.5×(Z Cha e jθ Cha ) + 0.5×(Z Dis e jθ Dis ), During charging, the voltage response of impedances on electrode-electrolyte interfaces of both positive and negative electrodes is regarded as a linear transfer function [3,20], and positive halfwave sinusoidal current excites the transfer function in its charging half-cycle. So be during • Time Domain Simulation of FPLC ACequ Impedances During charging, the voltage response of impedances on electrode-electrolyte interfaces of both positive and negative electrodes is regarded as a linear transfer function [3,20], and positive half-wave sinusoidal current excites the transfer function in its charging half-cycle. So be during discharging. Two linear transfer functions responses are linearly superposed to simulate a one-cycle voltage response of impedances on electrode-electrolyte interfaces.
Vector analysis has a strict time-sharing requirement; the sinusoidal current excitation is naturally divided into positive and negative half. Amplitude superposition is carried out by linearly adding up charging and discharging voltage responses. Figure 3 is an FPLC DCequ [3,20] for steady-state simulation with divided half-wave sinusoidal currents, and is intuitive compared with Figure 2.
Appl. Sci. 2019, 9, x FOR PEER REVIEW  5 of 22 discharging. Two linear transfer functions responses are linearly superposed to simulate a one-cycle voltage response of impedances on electrode-electrolyte interfaces. Vector analysis has a strict time-sharing requirement; the sinusoidal current excitation is naturally divided into positive and negative half. Amplitude superposition is carried out by linearly adding up charging and discharging voltage responses. Figure 3 is an FPLCDCequ [3,20] for steady-state simulation with divided half-wave sinusoidal currents, and is intuitive compared with Figure 2. However, in FPLCDCequ simulation, steady-state DC static voltage response of impedances on electrode-electrolyte interfaces appear in Figure 4. Under positive half-wave sinusoidal current injection, periodic voltage responses of charging impedances on electrode-electrolyte interfaces are excited; According to average switch modeling principles, an equivalent DC positive response 〈v Cha 〉 is on electrode-electrolyte interfaces, as Equation (7). Similarly, an equivalent DC negative response 〈v Dis 〉 is on electrode-electrolyte interfaces, as Equation (8). Then equivalent DC response V DC on electrode-electrolyte interfaces is the sum of 〈v Cha 〉 and 〈v Dis 〉, as Equation (9). Steady-state DC static voltage responses are shown in FPLCDCequ simulation under DC-input [2,3,20], but do not exist in experiments under AC-input. The physical meanings of elements in the above equations are explained in Table S1 of the Appendix.
〈v Dis 〉= Linear superposition of terminal transient voltage responses in FPLCDCequ, excited by timedivision input, has been verified on physical-electrochemical derivation, numerical calculations and experiments under DC-input [20]; linear sub-models are switched by a low-pass filter and hysteresis relay [20]. In Figure 4, the black solid line is the output of transient voltage responses in FPLCDCequ simulation under AC-input. The black dashed line cancels the steady-state DC voltage response from the black solid line, and the DC component is exactly equal to Equation (9). The blue dotted line filters the steady-state AC fundamental voltage response out of the black solid line by Equation (10), and the blue dotted line is very close to the black dashed line, which has a little distortion from higher harmonics. The pink dot-dash line is the steady-state AC voltage response in the FPLCACequ simulation. The physical meanings of elements in the above equations are explained in Table S1 of the Appendix.
In Figure 4, the steady-state AC fundamental voltage response of both FPLCACequ and FPLCDCequ in the simulation are equal, and electrode-electrolyte interface impedances in FPLCACequ and FPLCDCequ are one-to-one equivalent. The equivalent relationship is unique, and equivalent results in vector analysis are available for both steady and transient states in electrical circuits [27]. The possibility of distinguishing charging/discharging impedances is lost in BHLCACequ because BHLCACequ obtains a steady-state AC fundamental voltage response by filtering out DC responses in FPLCDCequ as the blue dotted line. Therefore, the composition of steady-state AC fundamental voltage is not studied in However, in FPLC DCequ simulation, steady-state DC static voltage response of impedances on electrode-electrolyte interfaces appear in Figure 4. Under positive half-wave sinusoidal current injection, periodic voltage responses of charging impedances on electrode-electrolyte interfaces are excited; According to average switch modeling principles, an equivalent DC positive response v Cha is on electrode-electrolyte interfaces, as Equation (7). Similarly, an equivalent DC negative response v Dis is on electrode-electrolyte interfaces, as Equation (8). Then equivalent DC response V DC on electrode-electrolyte interfaces is the sum of v Cha and v Dis , as Equation (9). Steady-state DC static voltage responses are shown in FPLC DCequ simulation under DC-input [2,3,20], but do not exist in experiments under AC-input. The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.
Linear superposition of terminal transient voltage responses in FPLC DCequ , excited by time-division input, has been verified on physical-electrochemical derivation, numerical calculations and experiments under DC-input [20]; linear sub-models are switched by a low-pass filter and hysteresis relay [20]. In Figure 4, the black solid line is the output of transient voltage responses in FPLC DCequ simulation under AC-input. The black dashed line cancels the steady-state DC voltage response from the black solid line, and the DC component is exactly equal to Equation (9). The blue dotted line filters the steady-state AC fundamental voltage response out of the black solid line by Equation (10), and the blue dotted line is very close to the black dashed line, which has a little distortion from higher harmonics. The pink dot-dash line is the steady-state AC voltage response in the FPLC ACequ simulation. The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.
Appl. Sci. 2020, 10,1907 6 of 23 In Figure 4, the steady-state AC fundamental voltage response of both FPLC ACequ and FPLC DCequ in the simulation are equal, and electrode-electrolyte interface impedances in FPLC ACequ and FPLC DCequ are one-to-one equivalent. The equivalent relationship is unique, and equivalent results in vector analysis are available for both steady and transient states in electrical circuits [27]. The possibility of distinguishing charging/discharging impedances is lost in BHLC ACequ because BHLC ACequ obtains a steady-state AC fundamental voltage response by filtering out DC responses in FPLC DCequ as the blue dotted line. Therefore, the composition of steady-state AC fundamental voltage is not studied in BHLC ACequ , but research on the composition of DC components is necessary to study FPLC DCequ and FPLC ACequ .

AC Equivalent Frist Principle Impedances on Electrode-Electrolyte Interfaces
Equation (6) implies that equivalent AC impedances on electrode-electrolyte interfaces is equal to half of the sum of charging and discharging impedances on electrode-electrolyte interfaces of both positive and negative electrodes.
Each element in AC equivalent first principle impedances is non-directional and totally linear. This advantage simplifies the simulation, and it does not need to determine the direction of excitation in advance, nor does it need auxiliary circuits to switch sub-models in simulations [20]. Further, the charging impedances ( Z Cha e jθ Cha ) are strictly separated from discharging impedances ( Z Dis e jθ Dis ). Equations (11) to (13) expand the impedances ( Z Cha e jθ Cha , Z Dis e jθ Dis , Z eei e jθ eei ) to charge-transfer resistors (R ct,pc , R ct,nc , R ct,pd and R ct,nd ) and double-layer capacitors (C dl,pc , C dl,nc , C dl,pd and C dl,nd ); Equation (14) is an equivalent transformation for the Nyquist fitting. The physical meanings of elements in the above equations are explained in Table S1

AC Equivalent First Principle Linear Elements on Batteries
R C , L S and C bulk are typical equivalent first principle linear elements on batteries [3,20]. Under AC current injections, steady-state DC static and AC fundamental voltage responses of each element are formulated in Equations (15)- (20). These equivalent elements only have AC fundamental voltage responses, and do not have DC static voltage responses; therefore, it is not necessary to determine the direction of excitation in advance, nor does it need to switch sub-models in simulations. The physical meanings of elements in the above equations are explained in Table S1 of the Appendix.
Linear C bulk , replacing the nonlinear Warburg impedance (z Warburg ) in Equation (21), represents the diffusion process of electrolyte. z Warburg usually characterizes the diffusion process of electrolyte [19],

AC Equivalent Frist Principle Impedances on Electrode-Electrolyte Interfaces
Equation (6) implies that equivalent AC impedances on electrode-electrolyte interfaces is equal to half of the sum of charging and discharging impedances on electrode-electrolyte interfaces of both positive and negative electrodes.
Each element in AC equivalent first principle impedances is non-directional and totally linear. This advantage simplifies the simulation, and it does not need to determine the direction of excitation in advance, nor does it need auxiliary circuits to switch sub-models in simulations [20]. Further, the charging impedances (Z Cha e jθ Cha ) are strictly separated from discharging impedances (Z Dis e jθ Dis ). Equations (11) to (13) expand the impedances (Z Cha e jθ Cha , Z Dis e jθ Dis , Z eei e jθ eei ) to charge-transfer resistors (R ct,pc , R ct,nc , R ct,pd and R ct,nd ) and double-layer capacitors (C dl,pc , C dl,nc , C dl,pd and C dl,nd ); Equation (14) is an equivalent transformation for the Nyquist fitting. The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.

AC Equivalent First Principle Linear Elements on Batteries
R C , L S and C bulk are typical equivalent first principle linear elements on batteries [3,20]. Under AC current injections, steady-state DC static and AC fundamental voltage responses of each element are formulated in Equations (15)- (20). These equivalent elements only have AC fundamental voltage responses, and do not have DC static voltage responses; therefore, it is not necessary to determine the direction of excitation in advance, nor does it need to switch sub-models in simulations. The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.
Linear C bulk , replacing the nonlinear Warburg impedance (z Warburg ) in Equation (21), represents the diffusion process of electrolyte. z Warburg usually characterizes the diffusion process of electrolyte [19], so researchers [28] propose to connect z Warburg with impedances on electrode-electrolyte interfaces in series to simplify the equivalent circuit. Warburg impedances of both negative and positive electrodes are combined as one equivalent z Warburg . In this paper, this equivalent z Warburg is replaced by one C bulk . Firstly, linear C bulk has a clear electrochemical meaning under small-signal EIS operation [20]; secondly, linear C bulk is a special nonlinear Constant Phase Element (z CPE ) with P equal to one in Equation (22).

Experiment of Steady-state DC static Voltage Responses on Battery Terminals
The purpose of the experiment is to clarify that there are no steady-state DC static voltage responses on batteries' terminals under stable AC sinusoidal current excitation. Nowadays, board-level BMS is used to operate at frequencies as low as 0.5 Hz or 1.0 Hz. So, the electrochemical workstation (Solartron ModulabXM) sends high-precision pure trigonometric (sinusoidal) current-source excitations (5A rms) to the tested battery at two fixed frequencies, 0.5 Hz and 1.0 Hz. The tested battery, manufactured by Zhejiang Narada Power Source Co., Ltd., as shown in Figure 5, is a 500 Ah lead-acid cell with model name GFM-500R and number E-08#. The detection circuit, in Figure 6, is designed to sample battery terminal voltage and to amplify errors between terminal voltage and reference voltage (2.132 V) with DC gain (40.14). The overall configuration of the experimental platform is in Figure 7. AC current injections of the electrochemical workstation are intermittently closed and open, and the outputs of the detection circuit are measured by a digital multimeter (UNI-T UT33A+). When AC current injection is closed, the multimeter display is read at a fixed time delay; when AC current injection is open, the multimeter display is read at the same fixed time delay. After measuring two cycles by the same time delay, length of the next time delay is increased by 30 s. reference voltage (2.132 V) with DC gain (40.14). The overall configuration of the experimental platform is in Figure 7. AC current injections of the electrochemical workstation are intermittently closed and open, and the outputs of the detection circuit are measured by a digital multimeter (UNI-T UT33A+). When AC current injection is closed, the multimeter display is read at a fixed time delay; when AC current injection is open, the multimeter display is read at the same fixed time delay. After measuring two cycles by the same time delay, length of the next time delay is increased by 30 s.

Experiment of Steady-State AC fundamental Voltage Responses on Battery Terminals
Voltage response in the frequency domain reflects the steady-state amplitude and phase electrical characteristic of batteries' terminals. The experimental platform is set as Figure 7, but excludes the detection circuit.

Experiment of Extracting Directional Charge-Transfer Resistance in AC Nyquist Curve
Four batteries were paralleled in 70 °C accelerated aging tests for seven months, part of the twelve batteries stack multi-objective experiments in Figure 8. All batteries were designed and manufactured by Zhejiang Narada Power Source Co., Ltd. with model name GFM-500R; these four samples were numbered as E-52#, E-11#, E-02# and E-20#. After float charging with regulated terminal voltage (N*2.23 V, N is the number of testing batteries in the stack) at 70 °C for a month, the

Experiment of Steady-State AC fundamental Voltage Responses on Battery Terminals
Voltage response in the frequency domain reflects the steady-state amplitude and phase electrical characteristic of batteries' terminals. The experimental platform is set as Figure 7, but excludes the detection circuit.

Experiment of Extracting Directional Charge-Transfer Resistance in AC Nyquist Curve
Four batteries were paralleled in 70 °C accelerated aging tests for seven months, part of the twelve batteries stack multi-objective experiments in Figure 8. All batteries were designed and manufactured by Zhejiang Narada Power Source Co., Ltd. with model name GFM-500R; these four samples were numbered as E-52#, E-11#, E-02# and E-20#. After float charging with regulated terminal voltage (N*2.23 V, N is the number of testing batteries in the stack) at 70 °C for a month, the

Experiment of Steady-State AC fundamental Voltage Responses on Battery Terminals
Voltage response in the frequency domain reflects the steady-state amplitude and phase electrical characteristic of batteries' terminals. The experimental platform is set as Figure 7, but excludes the detection circuit.

Experiment of Extracting Directional Charge-Transfer Resistance in AC Nyquist Curve
Four batteries were paralleled in 70 • C accelerated aging tests for seven months, part of the twelve batteries stack multi-objective experiments in Figure 8. All batteries were designed and manufactured by Zhejiang Narada Power Source Co., Ltd. with model name GFM-500R; these four samples were numbered as E-52#, E-11#, E-02# and E-20#. After float charging with regulated terminal voltage (N*2.23 V, N is the number of testing batteries in the stack) at 70 • C for a month, the batteries stack was discharged for checking capacities and recharged for capacities recovering at room temperature. Twenty-four hours after recharging, each battery was individually scanned by the electrochemical workstation (Solartron ModulabXM) with 5A rms pure sinusoidal currents excitation and frequencies bands were from 1 MHz to 1 kHz. After EIS scanning, batteries were put back into a high-temperature container for the next test round. Battery capacities checking are plotted in Figure 9a

Results of Steady-State DC Static Voltage Responses on Battery Terminals
All measurement points are shown in Table 1. Some points data, before closed AC injection, are off records because they are almost the same as the last check point. The time domain outputs of the detection circuit are shown in Figure 10. Procedures of pulse and step were used to reduce sampling points and to act as low-pass filtering. In Figure 10, outputs are mainly in the transient process at the beginning, and gradually approaching to 660 mV after one hour. Then, the outputs are nearly stable around 660-670 mV, and the maximum voltage fluctuation, 10 mV in the secondary side, will not exceed 0.25 mV in the primary side, which is significantly less than the magnitude (3.2 mV in the primary side) obtained by FPLCDCequ simulation in Figure 4. Therefore, the experiment can prove that steady-state DC static overpotential on electrode-electrolyte interfaces is indeed zero under fixed frequency AC current inputs.

Results of Steady-State DC Static Voltage Responses on Battery Terminals
All measurement points are shown in Table 1. Some points data, before closed AC injection, are off records because they are almost the same as the last check point. The time domain outputs of the detection circuit are shown in Figure 10. Procedures of pulse and step were used to reduce sampling points and to act as low-pass filtering. In Figure 10, outputs are mainly in the transient process at the beginning, and gradually approaching to 660 mV after one hour. Then, the outputs are nearly stable around 660-670 mV, and the maximum voltage fluctuation, 10 mV in the secondary side, will not exceed 0.25 mV in the primary side, which is significantly less than the magnitude (3.2 mV in the primary side) obtained by FPLC DCequ simulation in Figure 4. Therefore, the experiment can prove that steady-state DC static overpotential on electrode-electrolyte interfaces is indeed zero under fixed frequency AC current inputs.  Under sinusoidal AC current injections, steady-state DC static voltage responses on electrodeelectrolyte interfaces are nearly zero in this detection experiment. Results of the detection circuit are almost ideal; some small errors can be tolerated and may be caused by ambient temperature fluctuations and battery self-heating by injection currents. Maybe a little steady-state DC static voltage response does appear on battery terminals, although it does not affect FPLCACequ in engineering applications. Figure 11a is FPLCACequ on battery, and Figure 11b is BHLCACequ on battery. FPLCACequ is shown in Equation (23) and BHLCACequ is shown in Equation (24). The Nyquist curve from the EIS of the same 500 Ah battery (E-08#) is plotted with diagonal cross points in Figure 12. In Figure 12, the fitting curve of FPLCACequ is plotted with cross points, and the fitting curve of BHLCACequ is plotted with dot points. The physical meanings of elements in the above equations are explained in Table S1 of the Appendix. Under sinusoidal AC current injections, steady-state DC static voltage responses on electrode-electrolyte interfaces are nearly zero in this detection experiment. Results of the detection circuit are almost ideal; some small errors can be tolerated and may be caused by ambient temperature fluctuations and battery self-heating by injection currents. Maybe a little steady-state DC static voltage response does appear on battery terminals, although it does not affect FPLC ACequ in engineering applications. Figure 11a is FPLC ACequ on battery, and Figure 11b is BHLC ACequ on battery. FPLC ACequ is shown in Equation (23) and BHLC ACequ is shown in Equation (24). The Nyquist curve from the EIS of the same 500 Ah battery (E-08#) is plotted with diagonal cross points in Figure 12. In Figure 12, the fitting curve of FPLC ACequ is plotted with cross points, and the fitting curve of BHLC ACequ is plotted with dot points. The physical meanings of elements in the above equations are explained in Table A1

Results of Steady-State AC Fundamental Voltage Responses on Battery Terminals
z BHLC ACequ = R CBH + jωL SBH + 1 jωC bulkBH + R ct,p 1 + jR ct,p ωC dl,p + R ct,n 1 + jR ct,n ωC dl,n ,   The Nyquist curve is fitted by FPLCACequ and BHLCACequ through software Zview, and element values in circuits are compared in Table 2. According to test conditions, SoC on batteries are nearly 100%; charge-transfer resistances (R ct,pc R ct,nc ) for charging of FPLCACequ, are much higher than charge-transfer resistances ( R ct,pd R ct,nd ) for discharging [3,20]. Charge-transfer resistors and double-layer capacitors are directional different. In Figure 4, elements values for simulation are also from Table 2.    The Nyquist curve is fitted by FPLCACequ and BHLCACequ through software Zview, and element values in circuits are compared in Table 2. According to test conditions, SoC on batteries are nearly 100%; charge-transfer resistances (R ct,pc R ct,nc ) for charging of FPLCACequ, are much higher than charge-transfer resistances ( R ct,pd R ct,nd ) for discharging [3,20]. Charge-transfer resistors and double-layer capacitors are directional different. In Figure 4, elements values for simulation are also from Table 2.  The Nyquist curve is fitted by FPLC ACequ and BHLC ACequ through software Zview, and element values in circuits are compared in Table 2. According to test conditions, SoC on batteries are nearly 100%; charge-transfer resistances (R ct,pc and R ct,nc ) for charging of FPLC ACequ , are much higher than charge-transfer resistances (R ct,pd and R ct,nd ) for discharging [3,20]. Charge-transfer resistors and double-layer capacitors are directional different. In Figure 4, elements values for simulation are also from Table 2.

Results of Extraction Directional Charge-Transfer Resistance in AC Nyquist Curve
In Figure 13c,d, both magnitude and phase bode plots have obvious fitting errors at low frequencies. The errors are mainly due to C bulk . z Warburg is supposed to reduce these errors, but is not convenient for BMS design.

Results of Extraction Directional Charge-Transfer Resistance in AC Nyquist Curve
In Figure 13c,d, both magnitude and phase bode plots have obvious fitting errors at low frequencies. The errors are mainly due to C bulk . z Warburg is supposed to reduce these errors, but is not convenient for BMS design. The sums of resistances in FPLCACequ are shown in Equations (25)- (28), and their amplitudes in magnitude bode plots are marked in Figure 13a. Based on contacting resistance, each step is corresponding to one charge-transfer resistance, and 0.5R ct,nd , 0.5R ct,nc , 0.5R ct,pd and 0.5R ct,pc orderly add up. The sums of resistances in BHLCACequ are shown in Equations (29)-(30), and R ct,n and R ct,p orderly add up. They are also marked in Figure 13b. In magnitude bode plots, the asymptotes which roughly outline measured and fitting curves are drawn according to poles and zeros points.  The sums of resistances in FPLC ACequ are shown in Equations (25)- (28), and their amplitudes in magnitude bode plots are marked in Figure 13a. Based on contacting resistance, each step is corresponding to one charge-transfer resistance, and 0.5R ct,nd , 0.5R ct,nc , 0.5R ct,pd and 0.5R ct,pc orderly add up. The sums of resistances in BHLC ACequ are shown in Equations (29)-(30), and R ct,n and R ct,p orderly add up. They are also marked in Figure 13b. In magnitude bode plots, the asymptotes which roughly outline measured and fitting curves are drawn according to poles and zeros points.
In experiments, batteries' capacities are almost full-charging; Therefore, charge-transfer resistances for discharging (R ct,pd or R ct,nd ) are smaller than charge-transfer resistances for charging (R ct,pc or R ct,nc ). Furthermore, if batteries' SoH is good, R ct,pd or R ct,nd for discharging are less than R ct,pc or R ct,nc for charging across the broad SoC range; otherwise, when batteries begin to discharge, R ct,pd and R ct,nd are used to increase significantly, and the batteries' terminal voltages drop obviously. Extracting the DC directional charge-transfer resistance from the AC Nyquist curve has engineering significance. In Figure 14, charge-transfer resistances of both FPLC ACequ and BHLC ACequ fluctuate during accelerated aging tests.  Figure 14, charge-transfer resistances of both FPLCACequ and BHLCACequ fluctuate during accelerated aging tests. However, increasing the decibel value of the sum of internal resistances, including contacting resistance and charge-transfer resistances, is relatively stable. Each decibel value of each ascending step in the magnitude bode plot is related to one charge-transfer resistance and one asymptotic line configured by the pole-zero pair in Figure 13a,b. Bode plots change during accelerated aging tests, However, increasing the decibel value of the sum of internal resistances, including contacting resistance and charge-transfer resistances, is relatively stable. Each decibel value of each ascending step in the magnitude bode plot is related to one charge-transfer resistance and one asymptotic line configured by the pole-zero pair in Figure 13a,b. Bode plots change during accelerated aging tests, but orders of FPLC ACequ and BHLC ACequ are fixed. Therefore, the scale ratio of the decibel value of each ascending step should be stable. Three ratios of decibel value of ascending steps in FPLC ACequ , as Equations (31)-(33), are stable between 0.2 and 2.5 in Figure 15; these three steps are different and the ratios decrease from DelatdB FP1 to DelatdB FP3 . The ratio of decibel value of ascending steps in BHLC ACequ , as Equation (34), is stable between one and two in Figure 15.
DelatdB FP2 = 20log 10 R C + 0.5R ct,nd + 0.5R ct,nc + 0.5R ct,pd DelatdB FP3 = 20log 10 R C + 0.5R ct,nd + 0.5R ct,nc + 0.5R ct,pd + 0.5R ct,pc Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 22 but orders of FPLCACequ and BHLCACequ are fixed. Therefore, the scale ratio of the decibel value of each ascending step should be stable. Three ratios of decibel value of ascending steps in FPLCACequ, as Equations (31)-(33), are stable between 0.2 and 2.5 in Figure 15; these three steps are different and the ratios decrease from DelatdB FP1 to DelatdB FP3 . The ratio of decibel value of ascending steps in BHLCACequ, as Equation (34), is stable between one and two in Figure 15.
DelatdB FP1 =20 log 10 ( DelatdB BH =20 log 10 ( R C +R ct,n +R ct,p In magnitude bode plots of Figure 13, the curves drawing from northwest at low frequency to southeast at high frequency are roughly straight lines. Assuming there is a four-bit analog-to-digital converter for ideal straight-line input, its interpretation steps should be defined as 2 0 : 2 1 : 2 2 : 2 3 . So does a two-bit analog-to-digital converter, as 2 0 : 2 1 . The scale ratios of adjacent steps are two to maximize the measurement range. However, the curves in magnitude bode plots are not ideal straight lines, so the scale ratio of adjacent sums of resistances is not as ideal as two. Figure 15 shows a similar law, as the numbers of steps are strictly limited by FPLCACequ and BHLCACequ. From this perspective, expansions and contractions of magnitude bode plots are reflecting on the sum of internal resistances, and resistances obtained in Nyquist are viable to characterize batteries' SoH.  In magnitude bode plots of Figure 13, the curves drawing from northwest at low frequency to southeast at high frequency are roughly straight lines. Assuming there is a four-bit analog-to-digital converter for ideal straight-line input, its interpretation steps should be defined as 2 0 : 2 1 : 2 2 : 2 3 . So does a two-bit analog-to-digital converter, as 2 0 : 2 1 . The scale ratios of adjacent steps are two to maximize the measurement range. However, the curves in magnitude bode plots are not ideal straight lines, so the scale ratio of adjacent sums of resistances is not as ideal as two. Figure 15 shows a similar law, as the numbers of steps are strictly limited by FPLC ACequ and BHLC ACequ . From this perspective, expansions and contractions of magnitude bode plots are reflecting on the sum of internal resistances, and resistances obtained in Nyquist are viable to characterize batteries' SoH. In Figure 16, contacting resistances and stray inductances of both FPLC ACequ and BHLC ACequ are very close. High frequency zero points, determined by contacting resistances and stray inductances as Equation (35), are very close in both FPLC ACequ and BHLC ACequ .
Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 22 In Figure 16, contacting resistances and stray inductances of both FPLCACequ and BHLCACequ are very close. High frequency zero points, determined by contacting resistances and stray inductances as Equation (35), are very close in both FPLCACequ and BHLCACequ.
The FPLCACequ provides DC directional charge-transfer resistances for charging and discharging. Characteristic charge-transfer resistances on the electrode-electrolyte interface are calculated in [29] by FPLCACequ. During the first four accelerated aging tests rounds, positive characteristic chargetransfer resistances of E-02# are always high, and positive characteristic charge-transfer resistances of E-20# are climbing up in Figure 17a [29]. The positive characteristic charge-transfer resistances of E-52# reach a maximum at the fifth test round in Figure 17a [29]. The increasing of positive characteristic charge-transfer resistances represents non-cohesion of active mass of the positive electrode [4]; all these changes are early warnings of subsequent capacity degradation in Figure 9a. These early warnings are not be interpreted in BHLCACequ; the positive/negative behavioral chargetransfer resistances and contacting resistances of E-20# and E-11# fluctuate close to each other in Figures 14e,f and 16b. Charge-transfer resistances in FPLCACequ, compared in BHLCACequ, characterize inside physical and electrochemical changes.  The FPLC ACequ provides DC directional charge-transfer resistances for charging and discharging. Characteristic charge-transfer resistances on the electrode-electrolyte interface are calculated in [29] by FPLC ACequ . During the first four accelerated aging tests rounds, positive characteristic charge-transfer resistances of E-02# are always high, and positive characteristic charge-transfer resistances of E-20# are climbing up in Figure 17a [29]. The positive characteristic charge-transfer resistances of E-52# reach a maximum at the fifth test round in Figure 17a [29]. The increasing of positive characteristic charge-transfer resistances represents non-cohesion of active mass of the positive electrode [4]; all these changes are early warnings of subsequent capacity degradation in Figure 9a. These early warnings are not be interpreted in BHLC ACequ ; the positive/negative behavioral charge-transfer resistances and contacting resistances of E-20# and E-11# fluctuate close to each other in Figure  r ct,n of negative electrode. [29].

Boundary between First Principle Circuits on Batteries in Field Applications
The steady-state DC static voltage responses appear in experiments in the literature [2,3,20], and the steady-state DC static voltage response is nearly zero in experiments of this paper. In field applications, injections of battery terminals are generally constant DC or stable AC currents; engineering separation between FPLCACequ and FPLCDCequ can mainly be determined by time constants of injection currents. Time constants of currents are compared with time constants of impedances on electrode-electrolyte interfaces, especially of positive electrodes. If the time constants of impedances are much larger than the time constants of currents, FPLCACequ is suitable; if the time constants of impedances are much smaller than the time constants of currents [20], FPLCDCequ is more suitable.
The practical rule of the boundary is considered in perspective of limit, but not from the perspective of arbitrary injections. This practical rule of option between these two first principle circuits has not been mentioned before. Boundaries of separation between FPLCACequ and FPLCDCequ may be gradual or abrupt, and the boundaries patterns still need further research.
The purpose of this article is to establish a one-to-one relationship between the Nyquist curve and FPLCDCequ. The steady-state DC static voltage response in Nyquist is zero after software and hardware filtering. The FPLCACequ satisfies requirements in Section 2.1 and matches experimental results.

Comparing with Nonlinear Models
Average switch modeling is proposed to obtain FPLCACequ, and circuit orders are increased to include directional charge-transfer resistances. Average switch modeling can be extended to nonlinear electrochemical circuits. Nonlinear electrochemical elements, such as the constant phase element and Warburg element, are totally nondirectional and frequency-dependent [1], and they mainly represent the electrolyte diffusion process [24]. So, linear charge-transfer resistances for charging/discharging in nonlinear electrochemical circuits are the same as in FPLCACequ. Figure 18a is an AC equivalent first principle nonlinear circuit (FPNLCACequ in Equation (36)), and Figure 18b is an AC equivalent behavioral nonlinear circuit (BHNLCACequ in Equation (37)) on batteries. The Nyquist of the EIS from the same 500 Ah battery (model GFM-500R and number E-8#), is plotted with diagonal cross points in Figure 19; in Figure 19, the fitting curve of FPNLCACequ is plotted with cross points, and the fitting curve of BHNLCACequ is plotted with dot points.
The Nyquist is fitted by FPNLCACequ and BHNLCACequ through software Zview, and element values in circuits are compared in Table 3. In Figure 19, both nonlinear circuits have obvious fitting errors at low frequencies. The errors are mainly due to z , and z Warburg is supposed to reduce these errors,

Boundary between First Principle Circuits on Batteries in Field Applications
The steady-state DC static voltage responses appear in experiments in the literature [2,3,20], and the steady-state DC static voltage response is nearly zero in experiments of this paper. In field applications, injections of battery terminals are generally constant DC or stable AC currents; engineering separation between FPLC ACequ and FPLC DCequ can mainly be determined by time constants of injection currents. Time constants of currents are compared with time constants of impedances on electrode-electrolyte interfaces, especially of positive electrodes. If the time constants of impedances are much larger than the time constants of currents, FPLC ACequ is suitable; if the time constants of impedances are much smaller than the time constants of currents [20], FPLC DCequ is more suitable.
The practical rule of the boundary is considered in perspective of limit, but not from the perspective of arbitrary injections. This practical rule of option between these two first principle circuits has not been mentioned before. Boundaries of separation between FPLC ACequ and FPLC DCequ may be gradual or abrupt, and the boundaries patterns still need further research.
The purpose of this article is to establish a one-to-one relationship between the Nyquist curve and FPLC DCequ . The steady-state DC static voltage response in Nyquist is zero after software and hardware filtering. The FPLC ACequ satisfies requirements in Section 2.1 and matches experimental results.

Comparing with Nonlinear Models
Average switch modeling is proposed to obtain FPLC ACequ , and circuit orders are increased to include directional charge-transfer resistances. Average switch modeling can be extended to nonlinear electrochemical circuits. Nonlinear electrochemical elements, such as the constant phase element and Warburg element, are totally nondirectional and frequency-dependent [1], and they mainly represent the electrolyte diffusion process [24]. So, linear charge-transfer resistances for charging/discharging in nonlinear electrochemical circuits are the same as in FPLC ACequ . Figure 18a is an AC equivalent first principle nonlinear circuit (FPNLC ACequ in Equation (36)), and Figure 18b is an AC equivalent behavioral nonlinear circuit (BHNLC ACequ in Equation (37)) on batteries. The Nyquist of the EIS from the same 500 Ah battery (model GFM-500R and number E-8#), is plotted with diagonal cross points in Figure 19; in Figure 19, the fitting curve of FPNLC ACequ is plotted with cross points, and the fitting curve of BHNLC ACequ is plotted with dot points.   Figure 19. Experimental Nyquist curve is fitted by FPNLCACequ and BHNLCACequ. Table 3. Values of elements in Figure 18.    Figure 19. Experimental Nyquist curve is fitted by FPNLCACequ and BHNLCACequ. Table 3. Values of elements in Figure 18.  The Nyquist is fitted by FPNLC ACequ and BHNLC ACequ through software Zview, and element values in circuits are compared in Table 3. In Figure 19, both nonlinear circuits have obvious fitting errors at low frequencies. The errors are mainly due to z CPE , and z Warburg is supposed to reduce these errors, but this is not mainly a concern in this paper.

Conclusions
Impedances on electrode-electrolyte interface of FPLC ACequ , established by average switch modeling, have a one-to-one relationship with impedances on the electrode-electrolyte interface of FPLC DCequ . Therefore, a unique physical bridge from AC Nyquist, fitting by FPLC ACequ , to the directional DC charge-transfer resistance of FPLC DCequ is proposed. Steady-state AC fundamental responses of FPLC ACequ are verified by theoretically derived time domain simulation and experiments in this paper.
This novel method provides an opportunity to review previous conclusions between BHLC ACequ and SoC/SoH with directional DC charge-transfer resistances, and this is going to be researched further by authors. linear impedances for discharging on electrode-electrolyte interfaces of both positive and negative electrodes, Z Dis is the amplitude and θ Dis is the static angle (rad) in rotating space vector plane  linear electrolyte bulk capacitance, reflecting capacity characteristics of bulk electrolyte; C bulk is the amplitude and − π 2 is the static angle (rad) in rotating space vector plane