 #### Passive damping technology of LCL filter grid-connected inverter

The series or parallel approach of passive resistors increases system damping to address the resonance problems inherent in LCL filters. As shown in Figure 1, the passive damping method of series resistance on the capacitor branch can attenuate the LCL resonant peak to a large extent, so that the amplitude-frequency curve is below the 0dB line near the resonant frequency, so that the open-loop phase crosses -180° The amplitude at the frequency is no longer greater than 0dB, thus satisfying the stability criterion. Note that the resistance value of the capacitor branch in series is Rd, taking the inverter side current feedback as an example, the transfer function from uinv to iL1 becomes

Figure 2 shows the open-loop amplitude-frequency and phase-frequency characteristic curves of inverter-side current feedback control using passive damping (parameters are the same as in Section 5.3). When the amplitude gain is greater than 0dB, there is no intersection between the phase-frequency curve and the -180° curve, and the closed-loop system is stable. In addition, the phase margin is maintained at 45° and the amplitude margin is 5~9dB, and the dynamic response of the system is good. The main cost of this passive damping method is additional power loss, and the following will introduce several methods to reduce losses and their principles.

For the passive damping scheme in Figure 1, the losses on Rd are mainly caused by the current flowing through the resistor. The main components of the current flowing through Rd are the fundamental current and the switching frequency sub-harmonic current. Therefore, how to reduce the magnitude of these two components of the current flowing through the resistor is an effective way to reduce the additional loss caused by passive resistance.

First, Figure 3 shows a passive damping method for reducing losses by paralleling inductors. Its essence is to shunt the fundamental wave current through the inductance, that is, by selecting an appropriate inductance value to ensure that the impedance value of the inductance Ld at the fundamental frequency is much smaller than the impedance value of Rd. In addition to the above conditions, the parallel inductance should not be too small to affect the suppression of the resonance peak, because if the inductance is too small, the impedance of the inductance Ld at the LCL filter resonant frequency will be close to or even smaller than the impedance of Rd value, this passive damping method will not be able to use the resistor Rd to dampen the resonant current at the resonant frequency. To sum up, the size of the parallel inductance Ld should satisfy:

Figure 4 shows a passive damping method for reducing losses by paralleling inductors and capacitors, which is an improvement on the method in Figure 3. Its essence is to shunt the switching frequency sub-harmonic current through a capacitor, that is, by selecting an appropriate capacitor value to ensure that the impedance value of the capacitor Cd at the switching frequency and above is much smaller than the impedance value of Rd. In addition, similar to the method in Figure 3, the parallel inductance and capacitance should not be too large to affect the damping characteristics near the resonant frequency, that is, the size of the parallel inductance Ld should satisfy the above formula, and the size of the parallel capacitor Cd should satisfy:

Two relatively simple methods for further reducing the loss of passive resistors are given above, and other circuit modification methods will not be repeated. It should be noted that the essence of this type of method is to construct a shunt branch to reduce the current components flowing through Rd except for components near the resonance frequency. Although the loss is reduced, the circuit structure becomes complicated.