#### The structure and design basis of LLCL filter

**The structure**** ****of ****the ****LLCL filter**

The LLCL filter is a higher order filter. Its principle is to replace the capacitor C1 in the LCL filter with an LC resonant branch to achieve a better switching frequency harmonic filtering effect and reduce the volume. The inverter structure that adopts LLCL filter is shown as in Figure 1.

The transfer function of the output voltage u_{inv} to i_{g} between the arms of the LLCL filter:

From the point of view of the transfer function, the denominator s3 term and the numerator s2 term in the low frequency range can be ignored. The LLCL filter is equivalent to a single inductance filter with L1 in series with L2, which is an attenuation slope of -20dB/decade; The s2 term of the molecule in the middle and high frequency region can be ignored, and the LLCL filter is equivalent to an attenuation slash of -80dB/decade; in the high frequency region, due to the cancellation of the numerator and denominator s2 term, it is equivalent to a first-order filter, and its high-frequency attenuation characteristic is worse than that of the LCL filter, and its filtering characteristic is shown in Figure 2.

**Design Basis**** ****of ****the ****LLCL filter**

Compared with the LCL filter, the LLCL filter only inserts a small inductance in its capacitor branch. The LfCf resonant branch replaces the capacitor branch in the LCL to form a harmonic trap for the switching frequency harmonics. Therefore, its basic parameter design restriction conditions are similar to those of LCL filters.

1. Current ripple limit on the inverter side

Generally, the inverter side current ripple △I_{L1}_{_}_{max} is required to be 10%~30% of the rated current l_{rated}:

2. Network current ripple limit

The current ripple into the grid must meet the relevant restrictions in Table 1.1, and the ripple content of 33 times and above should be less than 0.3% of the rated value. Due to the introduction of the harmonic trap, the filter has a greater attenuation of the sideband harmonics of the switching frequency, the 33 times or more ripples of the grid current are mainly located near 2 times the switching frequency, that is, the ripple near 2 times the switching frequency is required to be less than 0.3% of the current rating.

3. The total inductance voltage drop limit of the filter

The design of the LCL filter is the same. In order to ensure the normal operation of the inverter, the impedance voltage drop generated by the LCL filter inductance under rated working conditions is less than 10% of the grid voltage, namely

U_{L}=ω_{0}(L_{1}+L_{2})·I_{rated}＜10%U_{g} (1.3)

4. Reactive power limit

Like the design of the LCL filter, the reactive power generated by the capacitor is generally restricted to not exceed 5% of the rated power of the inverter, namely

5. Frequency limit of harmonic trap

The voltage harmonics generated by PWM modulation are mainly the side frequency sub-harmonics of the switching frequency, so the harmonic trap must be able to achieve a better switching frequency sub-sideband harmonic filtering effect, namely

Among them, ω_{s} is the equivalent switching frequency of the inverter.

6. Resonance frequency limit

Similarly, the resonant frequency is designed to be greater than 10 times the fundamental frequency of the power grid and less than half of the switching frequency, namely

7. The quality factor limit of LC series resonant circuit

The quality factor of the LC series resonant circuit is:

Among them, Rf is the equivalent resistance of the resonant inductor Lf. The quality factor is expressed as the maximum voltage across the inductor and capacitor on the resonant circuit is twice the applied voltage. At the same time, the width of the passband of the resonant circuit is inversely proportional to the quality factor of the loop. The greater the quality factor of the loop, the narrower the passband, and the better the selectivity of the loop, which means that the resonant loop has less influence on harmonics other than the resonance frequency, and is closer to the ideal circuit. The ideal situation is Rf=0 and Q is infinite, but due to the existence of parasitic resistance on the resonance road, Q cannot be infinite, and a larger Q value will increase the peak voltage across the capacitor and the inductor and the resonance current. The voltage and current stress of the inductor and capacitor will also increase accordingly, so the Q value must be controlled within a reasonable range.