*Proposed Cell State Estimation Method*

Battery cell SoC can be estimated by continuously measured impedance during discharge. To estimate the cell state, a multi-sine signal with a small amplitude is applied to the cell operating current, and the cell impedance is measured through the amplitude of the voltage response. The multi-sine signal is the sum of two different frequencies, and each frequency is used to estimate the cell SoH and SoC.

If the current *i* expressed by Equation (1) flows through the battery cell, the cell voltage *e* can be expressed by Equation (2).

$$i = \mathbf{I}\_{\rm dc} + \Delta \mathbf{I}\_{\rm f} \cdot \sin(2\pi \mathbf{f} \mathbf{t}) \tag{1}$$

$$x = \mathbf{E}\_{\rm dc} + \Delta \mathbf{E}\_{\rm f} \cdot \sin(2\pi \mathbf{f}t + \Phi\_{\rm f}) \tag{2}$$

where Idc is direct current (DC) bias, ΔIf is the amplitude of the excited test frequency f, Edc is the offset voltage, ΔEf is the amplitude of the output voltage, and φ<sup>f</sup> is the phase difference.

Dividing the voltage by the current as Equation (3) produces a complex impedance Zf.

$$\mathbf{Z}\_{\mathbf{f}} = \frac{\Delta \mathbf{E}\_{\mathbf{f}}}{\Delta \mathbf{I}\_{\mathbf{f}}} \cdot \mathbf{e}^{\mathbf{j}\phi\_{\mathbf{f}}} = |\mathbf{Z}\_{\mathbf{f}}| \cdot \mathbf{e}^{\mathbf{j}\phi\_{\mathbf{f}}} = \mathbf{Z}\_{\mathbf{f}}^{\prime} + \mathbf{j} \cdot \mathbf{Z}\_{\mathbf{f}}^{\prime} \tag{3}$$

The electrochemical impedance of batteries depends on frequency and characterized by its modulus <sup>|</sup>Zf<sup>|</sup> and phase angle ej<sup>φ</sup>. Another expression is given as the real and imaginary parts of the complex impedance.

EIS measurements usually use a single-sine signal in which individual frequencies are measured sequentially, which is also known as stepped sine or frequency sweep. Therefore, single-sine EIS has the disadvantage that it takes a long time to acquire impedance in a wide frequency range. This disadvantage can be overcome by measuring several frequencies simultaneously. The method of measuring multiple frequencies at the same time is called multi-sine EIS. Multi-sine signals have already been used for impedance spectroscopy and transfer function measurements in biomedical applications [42,43], material characterization [44], and other fields such as battery measurements [26,28]. Nonetheless, multi-sine EIS is not a common method for estimating the in situ state of a battery cell. In general, the multi-sine EIS, like the single-sine EIS, requires the cell to be separated from the application circuit for impedance measurements. In the proposed battery SoC monitoring method, the sum signal of two test frequencies is excited to the cell operating current and its response voltage is measured. A Fourier transform is used on the sampled cell voltage to obtain the amplitude at each test frequency. The impedance at each test frequency is obtained by substituting the amplitude of each response voltage into Equation (3).

#### **2. Experiment**

#### *2.1. Measurement System*

The measurement system is configured to measure cell impedance by applying a multisine signal to the operating current. Table 1 shows the specifications of the Li-ion battery cell used and Figure 1 shows a block diagram and a picture of the measurement system.



<sup>1</sup> Consists of cobalt, nickel, and manganese.

**Figure 1.** (**a**) A block diagram of the measurement system; (**b**) a photo of the measurement system.

The test battery cell is placed in the temperature chamber of the Binder GmbH set to 25 ◦C. NXP's silicon temperature sensor KTY 81-110 is attached to the cell to measure the actual cell surface temperature. The voltage signal output port of USB-6212, which is a data acquisition (DAQ) module of the Natural Instruments, is connected to an electronic load to apply test frequencies to the cell operating current. This DAQ module also acquires cell voltage, current, and temperature data. Experiments are controlled by adjusting measurement parameters through a graphical user interface (GUI) created using LabVIEW from National Instruments. Measured and calculated data is displayed on the monitor and can also be written to the hard disk. The analog inputs of the DAQ module have 16-bit analogue to digital converter (ADC) resolution, a maximum sample rate of 400 kS/s, and an input range of ±10 V. The analog output has 16-bit digital to analog converter (DAC) resolution, an output range of ±10 V, and a maximum update rate of 250 kS/s. The operation of the electronic load is described in the author's previous paper [45].
