## 2 Experimental Study
### Passivity Impedance-Based Stability Criterion  
Similar phenomena were also described in Reference [22], and called \"resonance instability\" or \"harmonic instability\". This can be explained by the impedance-based stability criterion [13] and passivity theory [14]. Considering the nonlinearity of an inverter-grid system, the linearization of the equivalent circuit is based on the small signal modeling, as illustrated in Figure 4.  
Most grid-connected inverters use the current tracking mode, so the inverter can be thought of as an ideal current source (H\({}_{\mathrm{c}}\)\({}_{\mathrm{ref}}\)) and an inverter output impedance (Z\({}_{\mathrm{inv}}\)) connected in parallel, as shown in Figure 4, in the grey box. The expression of the inverter output current is expressed in Equation (1). \[\mathrm{i}=\mathrm{H_{C}}(\mathrm{s})\mathrm{i_{ref}}+\mathrm{Y_{inv}}(\mathrm{ s})\mathrm{v} \tag{1}\]  
where i is the inverter output current; H\({}_{\mathrm{c}}\)(s) is the transfer function of the current control loop of the inverter; i\({}_{\mathrm{ref}}\) is the reference current; Y\({}_{\mathrm{inv}}\)(s) is the inverter output admittance and v is the inverter output voltage. The complete inverter output current expression of the inverter-grid system is obtained in Equation (2), by Equation (1) and Figure 4,  
\[\mathrm{i}=\underbrace{\frac{\mathrm{H_{C}}(\mathrm{s})}{1+\mathrm{Y_{inv}}( \mathrm{s})Z_{g}(\mathrm{s})}}_{\mathrm{H(s)}}\mathrm{i_{ref}}+\underbrace{ \frac{\mathrm{Y_{inv}}(\mathrm{s})}{1+\mathrm{Y_{inv}}(\mathrm{s})Z_{g}( \mathrm{s})}}_{\mathrm{Y(s)}}\mathrm{v} \tag{2}\]  
Z\({}_{\mathrm{g}}\)(s) represents the grid impedance; vg is the grid voltage, H(s) and Y(s) are the closed-loop transfer function and input admittance of the inverter-grid system, separately. Figure 4: The equivalent circuit diagram of the inverter-grid system. The stability of Y(s) may be analyzed by the Nyquist criterion with the open-loop transfer function \(\mathrm{Y_{OL}(s)}=\mathrm{Y_{inv}(s)Z_{g}(s)}\). When there is a resonance point in the grid impedance with a small damping coefficient, the grid impedance will intersect the output impedance of the inverter near the resonance frequency \(\omega_{r}\) with a large peak. Additionally, the phase angle of grid impedance will abruptly change from \(\pi\) to \(-\pi\) at \(\omega_{r}\). When the phase angle of the inverter output admittance meets \(-\pi\geq\arg[\mathrm{Yinv}(\mathrm{j}\omega)]\geq\pi\), then the phase angle of the open-loop transfer function \(\arg[\mathrm{Y_{OL}(\mathrm{j}\omega)}]\) will certainly cross over \(\pm 2\pi\) near the resonance frequency \(\omega_{r}\). Thus, \(\mathrm{Y(s)}\) will be unstable in this situation. Conversely, if the phase angle of the inverter output admittance meets \(-\pi\leq\arg[\mathrm{Y_{inv}(\mathrm{j}\omega)}]\leq\pi\), which can also be called passivity, \(\mathrm{Y(s)}\) can be stable. The passivity of a transfer function Y(s) is defined as [14]  
1. Y(s) is stable and;
2. \(-\pi\leq\arg[\mathrm{Y(\mathrm{j}\omega)}]\leq\pi\) and can also be expressed equivalently as;
3. Re(\(\mathrm{Y(\mathrm{j}\omega)}\)) \(\geq 0\), \(\forall\omega\in(0,\mathrm{fL}]\). which means that the transfer function is passive in the interval (0,fL). Y(s) is stable since the Nyquist curve of Y(s) cannot encircle \((-1,\mathrm{j}0)\) if it is located at the right half s plane, of which the real part is \(\geq\)0 in the interval (0,fL). To ensure that Y(s) is passive, \(\mathrm{Y_{inv}(s)}\) and \(Z_{g}(s)\) need to be proven to be passive. We assume that both \(\mathrm{Y_{inv}(s)}\) and \(Z_{g}(s)\) are passive, as shown in Equation (3). \[\left\{ \begin{array}{c}\Re\{\mathrm{Y_{inv}(j\omega)}\}\geq 0\\ \Re\{\mathrm{Z_{g}(j\omega)}\}\geq 0\end{array} ,\forall\omega\in(0,\mathrm{f}]\right. \tag{3}\]  
By bringing Equation (3) into the Y(s) of Equation (2) to get the expression of the real part of Y(s), it is shown that Y(s) is also passive (Equation (4)). \[