## 2 AC Equivalent First Principle Linear Circuit and Experiments
### Analysis of AC Equivalent Impedances on Electrode-Electrolyte Interfaces by Average Switch Modeling
#### 2.2.1 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 [PERSON] 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 (\(V_{Chu}\)) 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 (\(V^{\prime}_{Chu}\)), as Equation (3), is exactly half an amplitude and the same phase of charging vector in the coordinate plane. So be new discharging vector (\(V^{\prime}_{Dis}\)), 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 (\(V_{exi}\)) can only be formed by new charging (\(V^{\prime}_{Chu}\)) and discharging (\(V^{\prime}_{Dis}\)) vectors by linear superposition as Equation (5) and Figure 1b. Figure 2 is an AC equivalent impedance (\(Z_{exi}e^{i\theta_{cn}}\)) similar to Equation (6) by removing the current (\(I_{EIS}e^{i\theta_{ES}}\)) from Equation (5).