In recent years, the adoption of renewable energy sources has gained significant attention due to their sustainability and environmental benefits. The proportion of renewable energy generation in the overall power system has been increasing annually. However, the rapid development of renewable energy generation faces several challenges. The electrical energy output from renewable sources cannot be directly integrated into the grid using a three phase inverter without proper filtering, as harmonic distortions can cause severe damage to the grid. The filter plays a critical role in attenuating high-frequency harmonics and is an essential component of the system model. Different filters require specific grid-connected control algorithms. Moreover, an oversized filter can reduce the power density of the system. The LCL filter has been widely adopted for its superior attenuation of high-frequency harmonics. This article investigates the parameters of the LCL filter for a three phase inverter and provides a design basis for these parameters.
The topology of a three phase inverter is fundamental to understanding its operation. A three phase inverter converts DC voltage into AC power, necessitating a DC-link capacitor on the input side. The main circuit consists of six power switches that facilitate power transmission and voltage conversion. The output voltages of the three phase inverter contain substantial high-frequency harmonics, making direct grid connection infeasible. An LCL filter is inserted between the inverter and the grid to mitigate these harmonics. The inductor closer to the inverter is termed the inverter-side inductor, while the one near the grid is the grid-side inductor. A filter capacitor connects between them. Parasitic resistances from components like IGBTs and inductors are often simplified in analysis to focus on core performance. The inverter output currents, capacitor currents, and grid currents are critical for evaluating filter effectiveness. The grid voltages represent the utility supply that the three phase inverter synchronizes with.
The high-frequency filtering performance of the LCL filter is a key advantage over traditional L filters. By neglecting small parasitic resistances, the transfer functions for both filters can be derived. For the LCL filter, the transfer function relating the grid current to the inverter output voltage is given by:
$$ G_1(s) = \frac{i_{2k}(s)}{u_k(s)} = \frac{1}{s^3 L_1 L_2 C_2 + s(L_1 + L_2)} $$
For the L filter, the transfer function is:
$$ G_2(s) = \frac{i_{2k}(s)}{u_k(s)} = \frac{1}{sL} $$
Under equivalent inductance conditions, where the sum of the LCL filter inductors equals the L filter inductance, the Bode plots demonstrate that the LCL filter provides 3 times greater attenuation for harmonics above the resonant frequency compared to the L filter. Below the resonant frequency, both filters exhibit similar performance, allowing the LCL filter to be approximated as an L filter. This enables the use of smaller total inductance in the LCL filter, reducing losses, improving efficiency, and enhancing power density. These benefits make the LCL filter highly suitable for high-frequency PWM applications in three phase inverter systems.
To analyze the LCL filter in the frequency domain, substitute \( s = j h \omega \) into the transfer function, where \( h \) is the harmonic order and \( \omega \) is the angular frequency. The frequency-domain transfer function becomes:
$$ \frac{i_{2k}(j h \omega)}{u_k(j h \omega)} = \frac{-j}{h \omega (-h^2 \omega^2 L_1 L_2 C_2 + L_1 + L_2)} $$
The magnitude of the LCL filter response is:
$$ |H_{\text{LCL}}(j h \omega)| = \frac{1}{h \omega |-h^2 \omega^2 L_1 L_2 C_2 + L_1 + L_2|} $$
The resonant frequency of the LCL filter is a critical parameter and is defined as:
$$ \omega_{\text{res}} = \sqrt{\frac{L_1 + L_2}{C_2 L_1 L_2}} $$
By drawing an analogy to parallel circuit branch currents, a simplified method for determining the inverter-side inductor harmonic current magnitude avoids complex transfer function solving. The relationship between the grid-side current and the inverter-side current is derived as:
$$ i_{2k}(j h \omega) = \frac{1}{1 – h^2 \omega^2 C_2 L_2} i_{1k}(j h \omega) $$
Thus, the inverter-side current magnitude is:
$$ i_{1k}(j h \omega) = \frac{1 – h^2 \omega^2 C_2 L_2}{h \omega (-h^2 \omega^2 L_1 L_2 C_2 + L_1 + L_2)} $$
At harmonic frequencies much higher than the resonant frequency, the capacitive reactance is much smaller than the grid-side inductive reactance, making the inverter-side inductor the primary determinant of high-frequency harmonic currents in the three phase inverter.
Parameter selection for the LCL filter in a three phase inverter is crucial for optimal performance. Assuming a switching frequency of 10 kHz, a harmonic order of 200, and a target attenuation of 0.1, the relationships between key parameters can be explored. The following table summarizes the interactions between the inductance ratio \( L_1/L_2 \), total inductance \( L_1 + L_2 \), and filter capacitance \( C_2 \):
| Parameter | Effect on Filter Performance |
|---|---|
| \( L_1/L_2 \) | Influences resonant frequency and harmonic attenuation; values between 3 and 7 are optimal. |
| \( L_1 + L_2 \) | Larger total inductance reduces harmonic currents but increases size and losses. |
| \( C_2 \) | Larger capacitance reduces total inductance requirement but affects sensitivity to inductance ratio. |
The three-dimensional relationship between \( L_1/L_2 \), \( \omega_{\text{res}} \), and \( C_2 \) shows that for a fixed \( L_1/L_2 \), the resonant frequency decreases as \( C_2 \) increases. If \( C_2 \) is small, the resonant frequency increases with \( L_1/L_2 \). Controlling \( L_1/L_2 \) is essential to prevent an excessively low resonant frequency, which could amplify mid-low frequency harmonics and complicate controller design in the three phase inverter.
Assuming a constant total inductance \( L_1 + L_2 \) and a harmonic voltage of 1V at the 200th harmonic, the inverter-side harmonic current magnitude \( |i_1(j h \omega)| \) varies with \( L_1/L_2 \) and \( C_2 \). The following table illustrates this relationship:
| \( L_1/L_2 \) | \( C_2 \) Trend | \( |i_1(j h \omega)| \) Behavior |
|---|---|---|
| Constant | Increasing | Increases to a maximum then decreases; larger \( C_2 \) shifts resonance away from switching frequency. |
| 0 to 1 | Constant | Rapid decrease in harmonic current. |
| 3 to 7 | Constant | Stable low harmonic current; optimal range. |
| >7 | Constant | Slight increase in harmonic current. |
Similarly, the grid-side harmonic current magnitude \( |i_2(j h \omega)| \) exhibits comparable trends. For a fixed \( L_1/L_2 \), \( |i_2(j h \omega)| \) peaks when the resonant frequency aligns with the switching frequency and decreases with further increases in \( C_2 \). When \( C_2 \) is constant, \( |i_2(j h \omega)| \) is minimized at \( L_1/L_2 = 1 \), indicating the best harmonic suppression for grid current. However, very low \( L_1/L_2 \) can lead to a low resonant frequency, increasing mid-low frequency harmonics and challenging controller design. Therefore, a balanced approach is necessary for the three phase inverter system.
The analysis indicates that for effective LCL filter design in a three phase inverter, the inductance ratio \( L_1/L_2 \) should be maintained between 3 and 7, with a moderately large \( C_2 \) to enhance filtering. The resonant frequency must be sufficiently lower than the switching frequency to avoid instability but not too low to prevent amplification of lower-order harmonics. Practical considerations such as current ripple requirements, system capacity, and switching frequency must also be integrated into the design process for the three phase inverter.
In conclusion, this research provides a simplified method for analyzing LCL filter parameters in a three phase inverter by leveraging analogies to parallel circuits. Through computational simulations and parameter relationships, design guidelines are established. The optimal inductance ratio and capacitance selection ensure effective harmonic attenuation while maintaining system stability and performance for the three phase inverter. Future work could explore adaptive control strategies to further optimize LCL filter behavior under varying grid conditions.