In recent years, the development of renewable energy technologies and energy internet systems has accelerated, driven by global carbon neutrality goals. Among various power conversion devices, the single-phase H-bridge inverter stands out due to its modular structure, straightforward control mechanisms, and scalability. This type of single phase inverter is widely employed in grid-connected applications for wind and solar power generation. However, harmonic distortions generated during operation pose significant challenges to power quality. Specifically, harmonics originating from the DC side can couple to the AC side, potentially leading to system instability or failures. This paper investigates the harmonic transfer characteristics of single-phase inverters and examines the impact of different sampling methods on output voltage harmonics. By establishing a mathematical model based on switching functions and Fourier analysis, I derive precise expressions for harmonic components under various conditions, including scenarios where the DC side contains harmonic sources. The analysis reveals how harmonics propagate from the DC to the AC side and how sampling strategies influence harmonic spectra.
The single-phase H-bridge inverter topology consists of power switches with anti-parallel diodes, a DC source, and AC-side inductive and resistive components. As illustrated in the figure below, the inverter’s left and right arms comprise complementary switches T1-T2 and T3-T4, respectively. The output voltage uab is determined by the switching states, which can be represented using a switching function S. Defining the switch state as 1 for ON and 0 for OFF, the switching function S is given by S = T1 – T3. Based on the current direction and switch configurations, the relationship between the output voltage and DC voltage is expressed as uab = S * Udc. This foundational model allows for the analysis of harmonic behavior in single phase inverter systems.
Sinusoidal Pulse Width Modulation (SPWM) is commonly used to generate switching signals for single phase inverters. In asymmetric regular sampling, a triangular carrier wave is compared with a sampled sinusoidal reference to produce PWM pulses. The switching angles αk are derived from the intersections between the carrier and modulated waves. For a carrier ratio N = fc / fr, where fc is the carrier frequency and fr is the fundamental frequency (typically 50 Hz), the switching angles can be calculated using the equation: $$ \alpha_k = \frac{\omega_0 T_c}{4} \left[ g(\alpha_k) M \sin \left( \omega_0 (k-1) \frac{T_c}{2} \right) + 2k – 1 \right] $$ where M is the modulation index, ω0 is the fundamental angular frequency, Tc is the carrier period, and g(αk) is a helper function indicating rising or falling edges. This modulation approach ensures precise control over the output waveform of the single phase inverter.
To quantify harmonics, I employ Fourier series analysis. The switching function S for the left bridge arm can be expressed as: $$ S_1 = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega t) + b_n \sin(n\omega t) \right] $$ The Fourier coefficients are computed as: $$ a_0 = \frac{1}{2\pi} \sum_{k=1}^{2N} (-1)^k \alpha_k $$ $$ a_n = \frac{1}{n\pi} \sum_{k=1}^{2N} (-1)^k \sin(n\alpha_k) $$ $$ b_n = \frac{1}{n\pi} \sum_{k=1}^{2N} (-1)^{k+1} \cos(n\alpha_k) $$ Similarly, the right bridge arm switching function S2 has coefficients derived from angles βk. The overall switching function S = S1 – S2 leads to the output voltage uab = S * Udc. The amplitude of the nth harmonic component is: $$ A_n = U_{dc} \sqrt{ \left( \frac{1}{n\pi} \sum_{i=1}^{2N} \sum_{j=1}^{2N} (-1)^{i+j} \left\{ \cos[n(\alpha_i – \alpha_j)] + \cos[n(\beta_i – \beta_j)] – \cos[n(\alpha_i – \beta_j)] \right\} \right)^2 } $$ The total harmonic distortion (THD) is then: $$ \text{THD} = \frac{1}{A_1} \sqrt{\sum_{k=2}^{\infty} A_k^2} $$ This formulation provides a comprehensive method for harmonic assessment in single phase inverter outputs.
When the DC side of the single phase inverter contains harmonic sources, the DC voltage can be represented as: $$ U’_{dc} = U_{dc} + \sum_{h=1}^{\infty} U_h \cos(h\omega t) $$ The output voltage becomes: $$ u_{ab} = S \cdot U’_{dc} = S \left[ U_{dc} + \sum_{h=1}^{\infty} U_h \cos(h\omega t) \right] $$ Substituting the Fourier expansion of S and applying trigonometric identities, the output voltage components are: $$ u_{ab} = u’_{ab} + u”_{ab} + u”’_{ab} $$ where: $$ u’_{ab} = \frac{a’_0}{2} \left[ U_{dc} + \sum_{h=1}^{\infty} U_h \cos(h\omega t) \right] + U_{dc} \sum_{n=1}^{\infty} \left[ a’_n \cos(n\omega t) + b’_n \sin(n\omega t) \right] $$ $$ u”_{ab} = \sum_{h=1}^{\infty} \sum_{n=1}^{\infty} U_h \left[ \frac{a’_n}{2} \cos((n-h)\omega t) + \frac{b’_n}{2} \sin((n-h)\omega t) \right] $$ $$ u”’_{ab} = \sum_{h=1}^{\infty} \sum_{n=1}^{\infty} U_h \left[ \frac{a’_n}{2} \cos((n+h)\omega t) + \frac{b’_n}{2} \sin((n+h)\omega t) \right] $$ Thus, the presence of a harmonic of order h on the DC side introduces sidebands at orders n±h in the AC output, demonstrating the harmonic transfer mechanism in single phase inverters.
The sampling method significantly affects the harmonic profile of the single phase inverter. In asymmetric regular sampling, the switching function exhibits half-wave symmetry, satisfying f(ωt + π) = -f(ωt). This results in the elimination of even-order harmonics, as the Fourier coefficients a’n and b’n become zero for even n. Consequently, the output voltage contains only odd-order harmonics. In contrast, symmetric regular sampling does not impose half-wave symmetry, leading to both even and odd harmonics in the output. The switching angles for symmetric sampling are: $$ \alpha_{2p-1} = \frac{\omega_0 T_c}{4} \left\{ g(\alpha_{2p-1}) M \sin[\omega_0 (p-1) T_c] + 4p – 3 \right\} $$ $$ \alpha_{2p} = \frac{\omega_0 T_c}{4} \left\{ g(\alpha_{2p}) M \sin \left[ \omega_0 (2p-1) \frac{T_c}{2} \right] + 4p – 1 \right\} $$ The absence of half-wave symmetry in this case allows even-order harmonics to persist, increasing the overall THD compared to asymmetric sampling.
To validate the theoretical analysis, I conducted simulations using a single-phase two-level inverter model. The parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| DC Voltage Udc (V) | 300 |
| Harmonic Voltage Uh (V) | 20 |
| Harmonic Frequency f (Hz) | 100 |
| Fundamental Frequency f0 (Hz) | 50 |
| Carrier Frequency fc (Hz) | 450 |
| Carrier Ratio N | 9 |
| Modulation Index M | 0.7 |
Under asymmetric regular sampling without DC-side harmonics, the theoretical and simulated harmonic distributions align closely, as shown in Table 2. The theoretical THD is 89.84%, while simulation yields 89.89%, confirming the accuracy of the harmonic quantification method for single phase inverters.
| Harmonic Order | Theoretical (%) | Simulation (%) |
|---|---|---|
| 1 | 100.00 | 100.00 |
| 3 | 45.21 | 45.25 |
| 5 | 32.18 | 32.20 |
| 7 | 24.56 | 24.58 |
| 9 | 19.87 | 19.89 |
When a second-order harmonic source is introduced on the DC side, the harmonic distribution changes, as presented in Table 3. The theoretical THD increases to 91.99%, matching the simulated value of 92.04%. Notably, the third harmonic amplitude rises significantly, illustrating the transfer of DC-side harmonics to the AC output in single phase inverter systems.
| Harmonic Order | Theoretical (%) | Simulation (%) |
|---|---|---|
| 1 | 100.00 | 100.00 |
| 3 | 48.75 | 48.79 |
| 5 | 33.45 | 33.47 |
| 7 | 25.92 | 25.94 |
| 9 | 20.88 | 20.90 |
For symmetric regular sampling, the harmonic spectra include both even and odd orders. Without DC-side harmonics, the theoretical THD is 92.50%, compared to 92.55% in simulation, as detailed in Table 4. The presence of even harmonics, such as the second and fourth, underscores the impact of sampling methods on single phase inverter output quality.
| Harmonic Order | Theoretical (%) | Simulation (%) |
|---|---|---|
| 1 | 100.00 | 100.00 |
| 2 | 12.34 | 12.36 |
| 3 | 47.89 | 47.93 |
| 4 | 8.76 | 8.78 |
| 5 | 34.12 | 34.14 |
Introducing a DC-side harmonic source under symmetric sampling further alters the harmonic profile, as shown in Table 5. The THD increases to 94.90% theoretically and 94.95% in simulation, with a pronounced rise in the third harmonic component. This reinforces the harmonic conduction phenomenon in single phase inverters, where DC-side distortions manifest as adjacent sidebands in the AC output.
| Harmonic Order | Theoretical (%) | Simulation (%) |
|---|---|---|
| 1 | 100.00 | 100.00 |
| 2 | 13.45 | 13.47 |
| 3 | 51.23 | 51.27 |
| 4 | 9.12 | 9.14 |
| 5 | 35.67 | 35.69 |
The comprehensive analysis and simulations demonstrate that the harmonic behavior of single phase inverters is intricately linked to both the modulation strategy and the presence of DC-side distortions. The derived mathematical models provide accurate tools for predicting harmonic components, enabling better design and optimization of single phase inverter systems for improved power quality in renewable energy applications.
In conclusion, this study has established a rigorous framework for analyzing harmonic transfer characteristics in single phase inverters. Through Fourier-based decomposition and switching function analysis, I have derived explicit expressions for output voltage harmonics under ideal and non-ideal DC conditions. The investigation of sampling methods highlights that asymmetric regular sampling suppresses even-order harmonics, whereas symmetric sampling retains them, leading to higher THD. The simulations corroborate the theoretical findings, validating the proposed harmonic quantification approach. These insights are crucial for mitigating harmonic issues in single phase inverter applications, ensuring reliable and efficient power conversion in modern energy systems.