## 2 Materials and Methods
### Processing of Measurement Data  
Impedance spectra are usable for analysis if the measured system was linear and time-invariant (_LTI system_). The time invariance can be checked with a method named _zHit_ after completion of one spectrum. For this, the impedance amplitude \(Z(\omega_{0})\) at a defined frequency was calculated from the measured phase shift \(\phi(\omega_{0})\)using Equation (1), with \(Z(0)\) as arbitrary, but with a fixed impedance amplitude and with \(\gamma_{k}\) as a correction factor. The starting frequency for integration \(\omega_{s}\) is selected arbitrarily out of the measured frequencies. At least two impedance values are required to verify the validity of one impedance value because of the derivative of the phase shift in the equation. More details on the _zHit_ method are described in [20]. \[ln|Z(\omega_{0})|=ln|Z(0)|+\frac{2}{\pi}\int_{\omega_{s}}^{\omega_{0}}\phi( \omega)dln(\omega)+\sum_{k\geq 1}\gamma_{k}\frac{d^{k}\phi(\omega_{0})}{(dln( \omega))^{k}}. \tag{1}\]  
The calculated value was compared with the measured amplitude. If the amplitudes were different, the system was time-variant during the measurement, which can be induced by diffusion processes, for example, during the charging or discharging with a superimposed DC-current. The difference was never zero due to noise in the measurement, so a maximum difference of \(0.15\,\mathrm{m}\Omega\) as a validity limit was defined, and impedance values with greater differences were defined as invalid. This criterion is named as _TI criterion_ here. The linearity of the impedance spectra could not be verified after the measurement. To ensure linearity, the amplitude of the sinusoidal voltage amplitude has to be as small as possible with an adequate signal-to-noise ratio. Here, an amplitude of \(1.5\,\mathrm{m}\mathrm{V}\) was used, which ensured linearity [21]. To ascertain the usability of impedance spectroscopy for the _SoH_ detection, the impedance spectra were analyzed in two ways. First, the change of single impedance values at fixed frequencies during the aging test was analyzed to check if single impedance measurements were sufficient for diagnostic measurements. The impedance of the battery could be approximated with an equivalent electrical circuit (_EEC_) containing a resistance (\(R_{ohmic}\)) in series to an induction \(L\) and an RC-element (resistance \(R_{2}\) parallel to a capacitance \(C\)). Single frequency impedance measurements can determine the ohmic resistance \(R_{ohmic}\) of the battery [3]. For the determination of \(R_{ohmic}\), the correct measurement frequency \(\omega\) has to be as high as possible. Thereby, the second term in Equation (3) becomes infinitely small and the measured impedance should contain only the ohmic resistance and an inductive imaginary part (see Equation (2)). Either the imaginary part is determined and can be separated from the real part or the impedance value at lower frequencies, where the spectrum crosses the \(x\)-axis, has to be taken as \(R_{ohmic}\). Here, the impedance amplitude at the point where the spectrum crosses the \(x\)-axis was extracted together with the corresponding measurement frequency. It is verified if the frequency varies during the aging tests and if it affects the _SoH_ estimation using \(R_{ohmic}\). \[\underline{Z}=R_{ohmic}+j\omega L+\frac{R_{2}-j\omega R_{2}^{2}C}{1+(\omega R _{2}C)^{2}} \tag{2}\]  
\[Re\{\underline{Z}\}=R_{ohmic}+\frac{R_{2}}{1+(\omega R_{2}C)^{2}}. \tag{3}\]  
As a second method, the impedance spectra were fitted to an EEC with an inductance \(L\), an ohmic resistance \(R_{ohmic}\), and one RC-element connected in series. The impedance spectrum is illustrated as a Nyquist diagram, with the real part on the \(x\)-axis and the imaginary part on the \(y\)-axis. An example is shown in Figure 3. In this representation, the spectrum of lead-acid batteries appears as half-circles and tails at high and low frequencies, respectively.