# AC impedance analysis

1 Principle

AC impedance is an electrochemical method to obtain the resistance, capacitance and inductance by detecting the current response under AC voltage. As shown in formula (1.1), the AC voltage changes periodically with time. From figure 1.11, we can see that there is a phase difference between voltage and current.

Where VM is the maximum voltage value; ω Is the angular frequency.

The phase difference between current response (I) and AC voltage is θ， It can be obtained from formula (1.3).

The amplitude of AC voltage and current can be expressed by complex exponential function, such as formula (1.4) and formula (1.5)

As shown in.

Impedance (z) is defined by equation (1.6), and its amplitude can be expressed by equation (1.7).

Based on the definition in formula (1.6), impedance can be divided into real part (z’) and imaginary part (Z “). The real part is resistance, and the imaginary part is reactance, including capacitance and inductance.

Using the phase difference, the real and imaginary parts of the above impedance can be expressed by equations (1.9) and (1.10). Phase difference（ θ） It is expressed by formula (1.11). The impedance amplitude is expressed by formula (1.12).

Based on Euler formula

, the above rectangular coordinates can be transformed into polar coordinates, and the transformation relationship is shown in Figure 1.12.

Figure 1.12 is the phasor diagram in the complex plane, that is, Nyquist diagram or Cole Cole diagram. Figure 1.13 shows the variation of z’and Z “with frequency.

2 equivalent circuit model

The flow of current in matter is related to resistance and capacitance. When only resistance is present, θ= 0, real part for impedance (z（ ω）= Z’（ ω）） express. As shown in formula (1.14), it is independent of frequency (see Figure 1.14 and figure 1.15).

When there is only capacitance, the electrostatic capacity (q) is expressed by formula (1.15). The substitutional formula (1.1) can be expressed as formula (1.16).

Since current represents the change of charge with time, it can be expressed by formula (1.17), where im is the maximum current.

From the above equation, we can see that voltage and current obey sine and cosine functions respectively. This means that the phase difference between them is π /2.

The capacitive reactance of capacitance (see Figure 1.16 and figure 1.17) is represented by X and defined by formula (1.18).

The inductance is generated by the magnetic field generated by the current passing through the coil (see Figure 1.18 and figure 1.19). It is expressed in L and the unit is Henry. The relationship between voltage, current and inductance can be expressed by equations (1.19) and (1.20). Inductive reactance XL, expressed by formula (1.21).

As shown in formula (1.20), the current lags behind the voltage π /2.

If the above capacitance (XC) and inductance (XL) are connected in series, the equivalent circuit can be shown in Figure 1.20, and the corresponding Nyquist diagram is shown in Figure 1.21.

In a system composed of resistance, capacitance and inductance, the total impedance x can be expressed by formula (1.22), where XC and XL belong to imaginary parts (see Figure 1.22 and figure 1.23). That is, the imaginary part Z “corresponds to capacitive reactance and inductive reactance.

If the inductance can be ignored, equation (1.22) can be simplified to equation (1.23).

The impedance of the resistance and capacitance connected in parallel (see Figure 1.24) can be expressed by equations (1.24) and (1.25)

express.

Multiply equation (1.25) by conjugate complex function (1-J ω RC), the formula (1.26) can be obtained. The real and imaginary parts of impedance are expressed by equations (1.27) and (1.28).

In formula (1.28), when ω When maxrc=1, Z “Max is the maximum value r/2, and the corresponding Nyquist diagram is shown in Figure 1.25.

Figure 1.26 shows the equivalent circuit of a resistor (RB) and a capacitor (CB) connected in parallel and then connected in series with another resistor (RS). From the Nyquist diagram in Figure 1.27, we can see that the capacitance semicircle moves rs. Figure 1.28 is the Nyquist diagram of a li/pan-spe/li battery, corresponding to an equivalent circuit (r1-c) in series and parallel with R.

The equivalent circuit diagram shown in Figure 1.29 is resistance Rb and capacitance CB. A circuit connected in parallel and then connected in series with capacitor CE. The Nyquist diagram in Figure 1.30 shows a semicircle corresponding to the capacitance in the high frequency region and a pure imaginary line corresponding to the series capacitance in the low frequency region.

When CB in the equivalent circuit shown in Figure 1.29 decreases, the semicircle in Figure 1.30 begins to twist. For negligible CB values, the circuit characteristics are similar to RB CE equivalent circuits.