The rapid growth in global photovoltaic (PV) capacity underscores the pivotal role of solar power in the future energy mix. Central to harnessing this energy is the power conversion unit—the solar inverter. Among various topologies, the three-level T-type non-isolated inverter has gained significant traction in PV systems due to its advantages of lower switching losses, reduced harmonic distortion, and higher efficiency. A critical challenge in non-isolated solar inverters is the generation of leakage currents caused by the parasitic capacitance between the PV panels and the ground. To mitigate this, a common and cost-effective approach is to connect the common point of the LCL filter’s capacitors to the DC-link midpoint, forming what is known as a mid-point back-connected LCL filter.
While this connection effectively suppresses high-frequency leakage current by providing a low-impedance path, it introduces a significant drawback: it can easily excite common-mode (CM) resonance within the inverter’s CM loop. This resonant current manifests as severe oscillations in the inverter-side bridge currents, degrades grid current quality, and can even lead to system instability. Traditional damping methods, such as adding passive resistors in series with the filter capacitor, increase losses and reduce filtering performance. This paper delves into the mechanism behind this CM resonant current and investigates the influence of different pulse-width modulation (PWM) strategies on it. Subsequently, we propose a novel adaptive third harmonic injection pulse width modulation (THIPWM) algorithm designed to suppress the CM resonant current effectively.
The topology of a T-type three-level non-isolated PV grid-connected inverter with a back-connected LCL filter is shown in the figure. Each bridge leg can output three voltage levels (P, O, N) relative to the DC-link midpoint O. Applying Kirchhoff’s voltage law, the per-phase mathematical model of the system can be established as:
$$
\begin{cases}
u_{xo} = L_1 \frac{di_{x1}}{dt} + R_1 i_{x1} + \frac{1}{C_f}\int i_{fx} dt + u_{n1o} \\
u_{xo} = L_1 \frac{di_{x1}}{dt} + R_1 i_{x1} + L_2 \frac{di_{x2}}{dt} + R_2 i_{x2} + e_x + u_{no}
\end{cases}
$$
where \(x = a, b, c\); \(u_{xo}\) is the inverter output voltage; \(i_{x1}\) and \(i_{x2}\) are the inverter-side and grid-side currents, respectively; \(i_{fx}\) is the filter capacitor current; \(e_x\) is the grid voltage; \(u_{n1o}\) and \(u_{no}\) are the voltages of points n1 and n relative to O; \(L_1, L_2, R_1, R_2\) are filter inductances and their parasitic resistances; \(C_f\) is the filter capacitance.
Summing the three-phase equations yields the CM model:
$$
\begin{cases}
3u_{zo} = 3L_1 \frac{di_{z1}}{dt} + 3R_1 i_{z1} + \frac{1}{C_f}\int 3i_o dt + 3u_{n1o} \\
3u_{zo} = 3L_1 \frac{di_{z1}}{dt} + 3R_1 i_{z1} + 3L_2 \frac{di_{z2}}{dt} + 3R_2 i_{z2} + 3u_{no}
\end{cases}
$$
where \(u_{zo} = (u_{ao}+u_{bo}+u_{co})/3\) is the CM voltage, \(i_{z1} = i_{a1}+i_{b1}+i_{c1}\) is the inverter-side CM current, \(i_{z2}=i_{a2}+i_{b2}+i_{c2}\) is the system leakage current, and \(i_o = i_{fa}+i_{fb}+i_{fc}\) is the back-connection line current.
For a back-connected LCL filter, \(u_{n1o} = 0\). The parasitic capacitances \(C_{pv}\) between the PV array and ground introduce the leakage current paths. The final simplified CM equivalent circuit reveals a parallel RLC branch formed by \(L_1\), \(R_1\), and \(C_f\). The resonant frequency of this CM loop is approximately:
$$
f_r \approx \frac{1}{2\pi \sqrt{L_1 C_f}}
$$
For typical solar inverter parameters (e.g., \(L_1 = 500 \mu H\), \(C_f = 4.7 \mu F\)), \(f_r\) is around 3.2 kHz. The primary excitation source for the CM resonant current is the CM voltage \(u_{zo}\), whose harmonic spectrum is directly determined by the PWM strategy. Therefore, selecting a modulation strategy that minimizes \(u_{zo}\) harmonics around \(f_r\) is crucial for resonance suppression.
Analysis of Modulation Strategies and Proposed Adaptive THIPWM
Common modulation strategies for three-level solar inverters include Space Vector PWM (SVPWM), Saddle PWM (SAPWM), and Third Harmonic Injection PWM (THIPWM). All can be implemented by injecting a specific zero-sequence voltage \(V_{zs}\) into the three-phase sinusoidal reference waves \(V_a, V_b, V_c\):
$$
\begin{aligned}
V_a &= V_m \cos(\omega t + \theta_0) \\
V_b &= V_m \cos(\omega t + \theta_0 – 2\pi/3) \\
V_c &= V_m \cos(\omega t + \theta_0 + 2\pi/3)
\end{aligned}
$$
where \(V_m = m U_{dc}/2\), \(m\) is the modulation index, and \(U_{dc}\) is the DC-link voltage.
The injected zero-sequence voltages for the three strategies are:
1. SVPWM: $$V_{zs}^{sv} = -\frac{1}{2}[\max(V_x + k_x) + \min(V_x + k_x)] + \frac{U_{dc}}{4}$$ where \(k_x=0\) if \(V_x \ge 0\), else \(k_x=U_{dc}/2\).
2. SAPWM: $$V_{zs}^{sa} = -\frac{1}{2}[\max(V_a, V_b, V_c) + \min(V_a, V_b, V_c)]$$
3. THIPWM: $$V_{zs}^{thi} = -\lambda V_m \cos(3\omega t + 3\theta_0)$$ where \(\lambda\) is the third harmonic injection coefficient.
Simulation analysis of the CM voltage \(u_{zo}\) spectrum under these strategies reveals significant differences. The results, summarized in the table below, show that THIPWM generates the lowest CM voltage magnitude both at low frequencies (e.g., 150 Hz) and, most importantly, around the critical resonant frequency \(f_r\). This characteristic makes THIPWM the most suitable candidate for mitigating CM resonance in back-connected LCL solar inverters.
| DC Voltage (V) | Modulation Strategy | Low-f (≤2kHz) (Vrms) | Near fr (3.2-3.8kHz) (Vrms) |
|---|---|---|---|
| 600 | SVPWM | 5.00 | 0.426 |
| SAPWM | 5.00 | 0.391 | |
| THIPWM | 0.46 | 0.276 | |
| 760 | SVPWM | 11.00 | 2.106 |
| SAPWM | 5.00 | 0.729 | |
| THIPWM | 0.48 | 0.439 |
While THIPWM is effective, its performance hinges on the proper selection of the injection coefficient \(\lambda\). An incorrect \(\lambda\) can lead to over-modulation or increased low-frequency CM current. We first derive the valid range for \(\lambda\) to ensure correct modulation. The modulation waves after injection are:
$$
V_{x\_thi} = \frac{mU_{dc}}{2} \left[ \cos(\omega t + \theta_x) – \lambda \cos(3\omega t + 3\theta_0) \right]
$$
To avoid over-modulation, the condition \(\max(|V_{x\_thi}|) \le U_{dc}/2\) must hold. Through analysis of the extremum points of \(V_{a\_thi}\), we derive the constraint relationship between \(m\) and \(\lambda\):
$$
\begin{cases}
m \le \frac{1}{1-\lambda}, & \text{for } 0 \le \lambda < \frac{1}{9} \\
m \le f(\lambda) = \frac{1}{\sqrt{\lambda(1+3\lambda)/3} (2\sqrt{\lambda} + \sqrt{(1+3\lambda)/3})}, & \text{for } \frac{1}{9} \le \lambda < \frac{1}{3}
\end{cases}
$$
The function \(f(\lambda)\) reaches its maximum \(2/\sqrt{3}\) at \(\lambda = 1/6\), which is the maximum achievable modulation index for linear modulation, identical to SVPWM and SAPWM. Based on this, we propose an adaptive law for \(\lambda\) that balances DC-link voltage utilization and low-frequency CM current reduction:
$$
\lambda = \frac{\sqrt{3}}{12} m
$$
This law ensures \(\lambda\) smoothly increases with \(m\) (i.e., when \(U_{dc}\) is low, to maintain high voltage utilization) up to \(1/6\), and decreases when \(m\) is lower (i.e., when \(U_{dc}\) is high), thereby minimizing low-frequency CM components. The adaptive third harmonic injection voltage is thus:
$$
V_{zs}^{adap} = -\frac{\sqrt{3}}{12} m V_m \cos(3\omega t + 3\theta_0)
$$
Closed-Loop Control Implementation for Practical Solar Inverters
Direct implementation of the adaptive THIPWM in a closed-loop controlled solar inverter faces practical challenges. Calculating \(V_{zs}^{adap}\) precisely requires knowledge of \(V_m\) and \(\theta_0\), which are derived from the dq-axis controller outputs \(V_d\) and \(V_q\):
$$
V_m = \sqrt{V_d^2 + V_q^2}, \quad \theta_0 = \arctan\left(\frac{V_q}{V_d}\right)
$$
Substituting into the exact formula leads to a computationally intensive expression involving square roots, divisions, and arctangent functions, which burdens the digital controller and introduces noise sensitivity.
To facilitate engineering application, we derive a simplified, robust implementation suitable for current closed-loop control. Noting that for grid-connected operation, \(V_q^2 << V_m^2\) and thus \(\sin^2\theta_0 \approx 0\), we can use the approximations \(\cos 3\theta_0 \approx 1\) and \(\sin 3\theta_0 \approx 3\sin\theta_0\). Leveraging the phase-locked loop (PLL) angle \(\theta\), the simplified adaptive zero-sequence voltage for modulation is:
$$
V_{zs}^{simp} = -\lambda \left[ V_d \cos(3\theta) – 3 V_q \sin(3\theta) \right]
$$
where \(\lambda = \frac{\sqrt{3}}{12} m\), and \(m\) is known from the system operating point. The terms \(\cos(3\theta)\) and \(\sin(3\theta)\) can be efficiently computed from the PLL outputs \(\cos\theta\) and \(\sin\theta\) using trigonometric identities. This method avoids complex function calls, significantly reduces computational load, enhances real-time performance, and improves system stability, making it ideal for practical solar inverter products.
Simulation and Experimental Verification
The proposed adaptive THIPWM method was validated through simulations and experiments on a 20kW T-type three-level solar inverter prototype. Key parameters are listed below.
| Parameter | Value |
|---|---|
| Rated Power | 20 kW |
| Grid Voltage (L-N RMS) | 230 V |
| DC-link Voltage \(U_{dc}\) | 600 / 680 / 760 V |
| Switching Frequency \(f_s\) | 16 kHz |
| Inverter-side Inductor \(L_1\) | 500 μH |
| Grid-side Inductor \(L_2\) | 50 μH |
| Filter Capacitor \(C_f\) | 4.7 μF |
Simulation results demonstrated the adaptive behavior of \(\lambda\) with changing \(U_{dc}\). More importantly, the spectral analysis of the CM current \(i_{z1}\) and the inverter-side phase currents under different modulations confirmed the superiority of the proposed method. The adaptive THIPWM consistently yielded the lowest current magnitude around the resonant frequency \(f_r\) and at 150 Hz.
| \(U_{dc}\) (V) | Modulation | Low-f (≤2kHz) mArms | Near \(f_r\) (3.2-3.8kHz) mArms |
|---|---|---|---|
| 600 | SVPWM | 638 | 410 |
| SAPWM | 638 | 368 | |
| Adaptive THIPWM | 500 | 274 | |
| 760 | SVPWM | 1352 | 2443 |
| SAPWM | 639 | 633 | |
| Adaptive THIPWM | 393 | 365 |
Experimental results on the 20kW prototype corroborated the simulations. Under various DC voltages and power factors, the adaptive THIPWM strategy effectively suppressed CM resonant current oscillations, resulting in clean inverter-side and grid-side current waveforms. In contrast, SVPWM showed severe current distortion at high \(U_{dc}\), and SAPWM still exhibited noticeable resonance harmonics.
The improvement in grid current quality is quantifiable. The Total Harmonic Distortion (THD) of the grid current was measured under different loads and DC voltages.
| \(U_{dc}\) (V) | Modulation | THD at Output Power Percentage | |||
|---|---|---|---|---|---|
| 30% | 50% | 70% | 100% | ||
| 600 | SVPWM | 6.10 | 4.24 | 3.76 | 2.79 |
| SAPWM | 5.95 | 4.22 | 3.58 | 2.68 | |
| Adaptive THIPWM | 5.60 | 3.91 | 3.37 | 2.50 | |
| 760 | SVPWM | 10.52 | 7.73 | 6.37 | 4.91 |
| SAPWM | 5.70 | 3.83 | 3.35 | 2.43 | |
| Adaptive THIPWM | 5.30 | 3.52 | 3.12 | 2.34 | |
The proposed adaptive THIPWM strategy consistently achieved the lowest grid current THD across all tested conditions, confirming its benefit for enhancing the power quality of solar inverters.
Conclusion
This paper addresses the critical issue of common-mode resonant current in non-isolated T-type three-level solar inverters employing midpoint back-connected LCL filters. Analysis reveals that the harmonic spectrum of the common-mode voltage, dictated by the PWM strategy, is the key excitation source for this resonance. Among standard modulation techniques, Third Harmonic Injection PWM (THIPWM) inherently generates a CM voltage with the lowest spectral energy at both low frequencies and the critical resonant frequency.
We proposed an adaptive THIPWM algorithm where the third-harmonic injection coefficient \(\lambda\) is dynamically adjusted based on the modulation index \(m\) according to the law \(\lambda = \frac{\sqrt{3}}{12} m\). This adaptive law ensures correct modulation across the operating range while optimizing the trade-off between DC-link voltage utilization and low-frequency CM current minimization. Furthermore, a simplified, computation-efficient implementation formula suitable for direct current closed-loop control in practical solar inverters was derived, enhancing real-time performance and robustness.
Comprehensive simulation and experimental results on a 20kW platform validate the method’s effectiveness. The proposed adaptive THIPWM successfully suppresses CM resonant current, reduces low-frequency CM components, and yields superior grid current quality with lower THD compared to conventional SVPWM and SAPWM strategies under various operating conditions, including different DC-link voltages and power factors. This work provides a theoretical foundation and a practical implementation pathway for applying advanced modulation techniques to enhance the stability and performance of modern solar inverter systems.