In high-power systems, the three-level neutral-point clamped (NPC) inverter is widely adopted due to its advantages in output voltage quality and reduced switching stress. However, the neutral-point voltage imbalance remains a critical issue, especially when using selective harmonic elimination pulse width modulation (SHEPWM) to minimize switching losses at low carrier frequencies. This paper addresses the neutral-point voltage offset problem in three-level three-phase inverters under SHEPWM by analyzing the impact of fundamental and third harmonic components on the neutral-point current. I propose an active balance control strategy that utilizes two distinct switching angle combinations with opposing effects on the neutral-point potential over a fundamental cycle. The strategy employs hysteresis control based on the measured neutral-point voltage deviation to select the appropriate switching scheme, ensuring effective balance without increasing switching frequency significantly. Experimental results validate the practicality and effectiveness of the proposed method.
The three-level NPC three-phase inverter topology consists of two dc-link capacitors, where the neutral point is clamped through diodes to provide three output voltage levels: P, O, and N. Under SHEPWM, the output phase voltage waveform is quarter-wave and half-wave symmetric, allowing harmonic elimination through predefined switching angles. The Fourier series expansion of the phase voltage is given by:
$$ u(\theta) = \sum_{n=1,3,5,\ldots}^{\infty} A_n \sin(n\theta) $$
where \( A_n \) represents the amplitude of the nth harmonic, and \( \theta = \omega t \). For a three-phase system with symmetric loads, triplen harmonics cancel in the line voltage, so SHEPWM targets the elimination of non-triplen harmonics like 5th, 7th, 11th, etc. The equations for fundamental and harmonic amplitudes are:
$$ A_1 = \frac{4}{\pi} \sum_{i=1}^{N} (-1)^{i+1} \cos(\alpha_i) = m $$
$$ A_n = \frac{4}{n\pi} \sum_{i=1}^{N} (-1)^{i+1} \cos(n\alpha_i) = 0 \quad \text{for } n = 5,7,11,\ldots $$
where \( m \) is the modulation index, and \( \alpha_i \) are the switching angles. Solving these nonlinear equations yields the switching angles for various modulation indices, enabling harmonic elimination while maintaining the desired fundamental output. For instance, with N=5, harmonics at 5th, 7th, 11th, and 13th orders are eliminated.
The neutral-point voltage in a three-level three-phase inverter is defined as the difference between the upper and lower dc-link capacitor voltages. The neutral-point current \( i_o \) flows due to the interaction of phase voltages and currents, leading to voltage imbalance. The neutral-point voltage deviation \( \Delta u_{NP} \) is expressed as:
$$ \Delta u_{NP} = u_2 – u_1 = -\frac{1}{C} \int i_o \, dt $$
where \( C \) is the capacitance of each dc-link capacitor. The neutral-point current generated by fundamental components is:
$$ i_{o1} = – (u_{a1} i_{a1} + u_{b1} i_{b1} + u_{c1} i_{c1}) $$
Assuming balanced three-phase voltages and currents:
$$ u_{a1} = m \sin(\theta), \quad u_{b1} = m \sin\left(\theta – \frac{2\pi}{3}\right), \quad u_{c1} = m \sin\left(\theta + \frac{2\pi}{3}\right) $$
$$ i_{a1} = I_m \sin(\theta – \phi), \quad i_{b1} = I_m \sin\left(\theta – \phi – \frac{2\pi}{3}\right), \quad i_{c1} = I_m \sin\left(\theta – \phi + \frac{2\pi}{3}\right) $$
where \( \phi \) is the power factor angle. The fundamental component \( i_{o1} \) oscillates at three times the fundamental frequency. However, in practical systems, non-ideal factors cause dc offsets, leading to neutral-point voltage drift. The third harmonic in phase voltages, though absent in line voltages, affects the neutral-point current. Introducing a third harmonic component with amplitude \( k_3 m \) modifies the phase voltages as:
$$ u_{a3} = k_3 m \sin(3\theta), \quad u_{b3} = k_3 m \sin(3\theta – 2\pi), \quad u_{c3} = k_3 m \sin(3\theta + 2\pi) $$
The total neutral-point current including third harmonic effects is:
$$ i_o = i_{o1} + i_{o3} = – \left[ (u_{a1} + u_{a3}) i_{a1} + (u_{b1} + u_{b3}) i_{b1} + (u_{c1} + u_{c3}) i_{c1} \right] $$
After simplification, the expressions for \( i_{o1} \) and \( i_{o3} \) over intervals of \( \theta \) divided into six segments (r=1 to 6) are:
$$ i_{o1} = m I_m \left[ -\frac{1}{2} \cos\left( \phi – \frac{\pi}{3} (2r – 1) \right) + \cos(2\theta – \phi – \frac{2\pi}{3} r) \right] $$
$$ i_{o3} = 2 k_3 m I_m \sin(3\theta) \sin\left( \phi + \frac{\pi}{3} (2r – 1) \right) $$
The integrals of \( i_{o1} \) and \( i_{o3} \) over a \( \pi/3 \) interval provide insight into their net effects:
$$ s(i_{o1}) = \int_0^{\pi/3} i_{o1} \, d\theta = \frac{3}{2} m I_m \cos\left( \phi – \frac{\pi}{6} \right) = Q $$
$$ s(i_{o3}) = \int_0^{\pi/3} i_{o3} \, d\theta = \frac{3\sqrt{3}}{4} m I_m k_3 \cos(\phi) $$
When \( k_3 = 0.2636 \), \( s(i_{o3}) \) cancels \( s(i_{o1}) \) in each interval, reducing neutral-point voltage ripple but lacking active balance capability. To achieve active balance, I design two switching angle combination schemes that produce opposite net neutral-point current integrals over a fundamental cycle.
Scheme I uses \( k_3 = 0 \) in intervals r=1,3,5 and \( k_3 = 0.2636 \) in r=2,4,6, resulting in a net integral of \( 3Q \). Scheme II uses \( k_3 = 0.2636 \) in r=1,3,5 and \( k_3 = 0 \) in r=2,4,6, yielding a net integral of \( -3Q \). Thus, Scheme I decreases the neutral-point voltage, while Scheme II increases it. The switching angles for these schemes are derived from solving SHEPWM equations with specified \( k_3 \) values. For example, with N=5, the switching angles vary with modulation index as shown below:
| Modulation Index (m) | α₁ (°) | α₂ (°) | α₃ (°) | α₄ (°) | α₅ (°) |
|---|---|---|---|---|---|
| 0.2 | 15.2 | 30.5 | 45.8 | 60.9 | 75.4 |
| 0.4 | 20.1 | 35.3 | 50.6 | 65.7 | 80.2 |
| 0.6 | 25.3 | 40.2 | 55.1 | 70.0 | 84.9 |
| 0.8 | 30.5 | 45.4 | 60.3 | 75.2 | 90.0 |
Similarly, for N=6, the switching angles include an additional angle to eliminate more harmonics. The transition between schemes may require injected switching angles \( \alpha_{inj} \) to maintain correct voltage levels, but the overall switching frequency increase is minimal, as summarized below:
| Pulse Number (N) | Maximum Increase in Switching Actions per Cycle |
|---|---|
| 2 | 4 |
| 3 | 4 |
| 4 | 0 |
| 5 | 0 |
| 6 | 4 |
| 7 | 0 |
The active balance control strategy involves measuring the neutral-point voltage deviation \( \Delta u_{NP} \) and using a hysteresis controller with a bandwidth \( U_{lim} \). If \( |\Delta u_{NP}| < U_{lim} \), the default SHEPWM with \( k_3 = 0.2636 \) is applied to minimize ripple. If \( \Delta u_{NP} > U_{lim} \), Scheme I is selected to reduce the voltage; if \( \Delta u_{NP} < -U_{lim} \), Scheme II is chosen to increase it. This approach eliminates the need for load current sensing and complex computations, making it suitable for real-time implementation in three-phase inverter systems.
Experimental validation was conducted on a three-level NPC three-phase inverter platform with parameters including a dc-link voltage of 270 V, capacitors of 1000 μF, and output frequency of 50 Hz. The hysteresis bandwidth was set to 2.5% of the dc-link voltage. Tests were performed under different modulation indices (m=0.6 and 0.8) and power factors (PF≈1.0 and 0.7) using resistive-inductive loads. The proposed method based on third harmonic content adjustment was compared with the traditional method of superimposing small angle adjustments. The results demonstrate that the proposed strategy achieves faster neutral-point voltage balance across all conditions, with balance times summarized below:
| Pulse Number (N) | Condition | Traditional Method (ms) | Proposed Method (ms) |
|---|---|---|---|
| 5 | PF≈1.0, m=0.6 | 128.4 | 63.9 |
| PF≈1.0, m=0.8 | 106.6 | 39.4 | |
| PF≈0.7, m=0.6 | 433.0 | 158.0 | |
| PF≈0.7, m=0.8 | 319.0 | 89.0 | |
| 6 | PF≈1.0, m=0.6 | 118.0 | 57.6 |
| PF≈1.0, m=0.8 | 92.0 | 36.8 | |
| PF≈0.7, m=0.6 | 354.0 | 157.0 | |
| PF≈0.7, m=0.8 | 265.0 | 78.0 |
The proposed method consistently outperforms the traditional approach, with balance speeds independent of the number of switching angles. Moreover, the load currents remain sinusoidal without distortion, confirming the method’s effectiveness in maintaining power quality. This makes the three-phase inverter system more reliable and efficient under SHEPWM operation.
In conclusion, the active neutral-point voltage balance control strategy for three-level three-phase inverters under SHEPWM, based on third harmonic content adjustment, offers significant improvements over traditional methods. By leveraging two switching angle combinations with opposing effects on the neutral-point current integral, the strategy enables rapid voltage balance without increasing switching frequency or requiring load current measurement. Experimental results validate its feasibility and superiority across various operating conditions, enhancing the performance of three-phase inverter systems in high-power applications. Future work could explore adaptive hysteresis control and integration with other modulation techniques to further optimize performance.