In modern power electronic systems, the three-phase inverter plays a critical role in various applications such as industrial drives, renewable energy integration, and aerospace systems. The widespread adoption of pulse width modulation (PWM) techniques has enhanced the performance of three-phase inverters, but challenges persist regarding harmonic distortions, particularly low-frequency harmonics on the DC side. This article investigates the influence of the output transformer, specifically a three-limb core-type transformer, on these harmonics. We analyze how asymmetric excitation currents and DC bias in the transformer contribute to low-frequency harmonic generation in the DC link of a three-phase inverter. By deriving mathematical models and conducting experimental validation, we provide insights into harmonic behavior under varying load conditions, offering guidance for mitigation strategies.
The output transformer in a three-phase inverter system is typically employed for electrical isolation and voltage level adjustment. However, the structural asymmetry of three-limb transformers leads to unbalanced magnetic paths, resulting in asymmetric excitation currents. This asymmetry, combined with DC bias effects, induces low-frequency harmonics—specifically even-order harmonics from unbalanced excitation and odd-order harmonics from DC bias. These harmonics can degrade system stability, especially in DC networks with limited capacitance. In this work, we first quantify the asymmetric excitation currents using magnetic circuit analysis. Then, we explore the mechanism through which these currents generate harmonics on the DC side, incorporating Fourier analysis of PWM switching signals. Finally, we validate our findings through experiments on a 5 kVA three-phase inverter platform, demonstrating the impact of load variations on harmonic amplitudes.
Analysis of Asymmetric Excitation Currents in Core-Type Transformer
The three-phase inverter with a Δ-Y connected output transformer is a common configuration, as shown in the equivalent circuit. The structural asymmetry of the transformer core—where three limbs are arranged in a plane—causes unequal magnetic reluctances, leading to unbalanced excitation currents. Let \( \phi_A \), \( \phi_B \), and \( \phi_C \) represent the main fluxes in phases A, B, and C, respectively. Under symmetric supply voltages, the fluxes are symmetric, satisfying:
$$ \phi_A + \phi_B + \phi_C = 0 $$
The magnetomotive forces (MMFs) \( F_A \), \( F_B \), and \( F_C \) are produced by the no-load currents \( i_{ao} \), \( i_{bo} \), and \( i_{co} \). Assuming negligible harmonics, the sum of these currents is zero:
$$ i_{ao} + i_{bo} + i_{co} = 0 $$
Thus, the MMFs also sum to zero:
$$ F_A + F_B + F_C = 0 $$
By applying magnetic circuit equations to the three limbs, with \( R_m \) as the reluctance of a single limb, we derive:
$$ F_A – F_B = 2R_m \phi_A – R_m \phi_B $$
$$ F_C – F_B = 2R_m \phi_C – R_m \phi_B $$
Solving these equations along with the flux symmetry condition yields:
$$ F_A = \left( -\frac{2}{3} + j\sqrt{3} \right) R_m \phi_B $$
$$ F_B = \frac{4}{3} R_m \phi_B $$
$$ F_C = \left( -\frac{2}{3} – j\sqrt{3} \right) R_m \phi_B $$
The ratio of no-load currents is then calculated as:
$$ K = \frac{i_{ao}}{i_{bo}} = \frac{F_A}{F_B} \approx 1.4 $$
This indicates that the excitation current in phase A is approximately 1.4 times that in phase B. The phase angles are also unbalanced, with \( i_{ao} \) leading \( i_{bo} \) by about 111°. The excitation currents in the Δ-connected primary can be expressed as:
$$ \begin{bmatrix} i_{ma} \\ i_{mb} \\ i_{mc} \end{bmatrix} = \begin{bmatrix} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} i_{ao} \\ i_{bo} \\ i_{co} \end{bmatrix} $$
Substituting the asymmetric current values, we obtain:
$$ i_{ma} = 1.3 i_o $$
$$ i_{mb} = i_o e^{-j131^\circ} $$
$$ i_{mc} = i_o e^{j131^\circ} $$
where \( i_o \) is the peak current in phases B and C. This confirms the magnitude and phase imbalance in excitation currents.
To assess the impact of load, consider the single-phase equivalent circuit including leakage inductance \( X_{sc} \) and magnetizing inductance \( X_m \). The phase current \( i_A \) is given by:
$$ i_A = \frac{U_{AN} – U_{A’N}}{X_{sc}} $$
where \( U_{A’N} \) is the voltage across the load referred to the primary. Using transformer equations, we derive:
$$ i_A = \left( \frac{\frac{N^2}{z}}{1 + \frac{N^2}{z} X_{sc} + \frac{X_{sc}}{X_m}} + \frac{\frac{1}{X_m}}{1 + \frac{N^2}{z} X_{sc} + \frac{X_{sc}}{X_m}} \right) U_{AN} $$
Since \( X_{sc} \ll X_m \), the term \( \frac{X_{sc}}{X_m} \) is negligible, simplifying to:
$$ i_A = \left( \frac{\frac{N^2}{z}}{1 + \frac{N^2}{z} X_{sc}} + \frac{\frac{1}{X_m}}{1 + \frac{N^2}{z} X_{sc}} \right) U_{AN} $$
This shows that \( i_A \) comprises load current and magnetizing current. The leakage inductance voltage drop increases with load, but its effect on the magnetizing current is minimal due to the small value of \( X_{sc} \). Thus, load variations do not significantly alter the unbalanced excitation currents.
| Parameter | Expression | Value/Description |
|---|---|---|
| Flux Symmetry | \( \phi_A + \phi_B + \phi_C = 0 \) | Fundamental condition |
| Current Ratio | \( K = i_{ao} / i_{bo} \) | ≈ 1.4 |
| Phase Angle | \( i_{ao} \) vs \( i_{bo} \) | 111° lead |
| Load Effect | \( i_A \) composition | Negligible on magnetizing current |
Low-Frequency Harmonic Analysis in DC Side
The DC input current of a three-phase inverter is influenced by the PWM switching patterns and the transformer’s behavior. To analyze harmonics, we use Fourier series expansion for the switching functions \( S_a \), \( S_b \), and \( S_c \):
$$ \begin{bmatrix} S_a \\ S_b \\ S_c \end{bmatrix} = \begin{bmatrix} 0.5 + \sum_{n=1}^{\infty} A_n \sin(n\omega t) \\ 0.5 + \sum_{n=1}^{\infty} A_n \sin(n(\omega t – 2\pi/3)) \\ 0.5 + \sum_{n=1}^{\infty} A_n \sin(n(\omega t + 2\pi/3)) \end{bmatrix} $$
For \( n = 0 \), \( A_n = 0.5 \); for \( n > 3 \), \( A_n \) is negligible. The DC input current \( i_{in} \) is given by:
$$ i_{in} = \sum_{x=a,b,c} S_x i_x $$
The phase current \( i_x \) includes components from the capacitor \( i_{cx} \), magnetizing current \( i_{mx} \), DC bias current \( i_{dx} \), and load current \( i_{load_x} \):
$$ i_x = i_{cx} + i_{mx} + i_{dx} + i_{load_x} $$
Symmetric currents \( i_{cx} \) and \( i_{load_x} \) contribute primarily to the DC component after transformation. However, the asymmetric magnetizing currents generate low-frequency harmonics. For the fundamental component (\( n = 1 \)), substituting the asymmetric magnetizing currents:
$$ i_{in1} = A_1 i_o \left[ \sin(\omega t) \cdot 1.3 \sin(\omega t + \theta) + \sin(\omega t – 120^\circ) \cdot \sin(\omega t – 131^\circ + \theta) + \sin(\omega t + 120^\circ) \cdot \sin(\omega t + 131^\circ + \theta) \right] $$
Assuming a purely inductive magnetizing branch, the power factor angle \( \theta = -90^\circ \). Solving this yields:
$$ i_{in1} = -0.33 A_1 i_o \sin(2\omega t) $$
This indicates a second-order harmonic in the DC side. Similarly, for even values of \( n \) (e.g., 2, 4, …), even-order harmonics are produced. Since the magnetizing current is independent of load, these even-order harmonics remain constant under load variations.
For DC bias currents, which sum to zero:
$$ i_{da} + i_{db} + i_{dc} = 0 $$
The contribution to \( i_{in} \) for \( n = 1 \) is:
$$ i_{in1} = i_{da} \sin(\omega t) + i_{db} \sin(\omega t – 120^\circ) + i_{dc} \sin(\omega t + 120^\circ) $$
Expressing in terms of \( i_{db} \) and \( i_{dc} \):
$$ i_{in1} = i_{db} \left( -\frac{\sqrt{3}}{2} \sin(\omega t) – \frac{3}{2} \cos(\omega t) \right) + i_{dc} \left( -\frac{\sqrt{3}}{2} \sin(\omega t) + \frac{3}{2} \cos(\omega t) \right) $$
As \( i_{db} \) and \( i_{dc} \) are DC quantities, this results in odd-order harmonics in the DC input current. For higher even \( n \), corresponding harmonic orders appear. The DC bias current increases with load due to enhanced magnetic saturation, leading to a rise in odd-order harmonics with load power.
| Source | Harmonic Order | Load Dependency |
|---|---|---|
| Asymmetric Excitation | Even-order (2nd, 4th, …) | Independent |
| DC Bias | Odd-order (1st, 3rd, …) | Increases with load |
Experimental Validation
To verify the theoretical analysis, we conducted experiments on a 5 kVA three-phase inverter system. The inverter parameters are listed in the table below:
| Parameter | Value |
|---|---|
| Rated Output Line Voltage | 220 V |
| Rated Output Power | 5 kVA |
| Output Frequency | 50 Hz |
| Filter Inductance (L) | 200 μH |
| Filter Capacitance (C) | 180 μF |
| DC Input Voltage | 400 V |
| Switching Frequency | 8 kHz |
| DC Link Capacitance (C_dc) | 2350 μF |
| DC Link Inductance (L_dc) | 2.2 mH |
First, we measured the no-load currents of the output transformer. The RMS values were approximately 0.860 A for phase A, 0.684 A for phase B, and 0.651 A for phase C, with phase angles showing a 131.6° lag between \( i_a \) and \( i_c \), and a 129.4° lead between \( i_a \) and \( i_b \). This aligns with the theoretical asymmetry.
Next, we analyzed the DC side current spectrum under no-load and 4 kVA load conditions. The results confirmed the presence of even-order harmonics (e.g., 2nd order) and odd-order harmonics (e.g., 1st, 3rd). The even-order harmonics remained relatively constant, while the odd-order harmonics increased by approximately 8 dB under load, consistent with DC bias effects. The experimental setup included a three-phase uncontrolled rectifier for DC input, which introduced additional 6th order harmonics, but the focus remained on low-frequency components.
The three-phase inverter system demonstrated that output transformer asymmetry directly influences DC side harmonics. This insight is crucial for designing filters and control strategies to suppress low-frequency harmonics in three-phase inverter applications.
Conclusion
In this study, we have analyzed the impact of a three-limb core-type output transformer on low-frequency harmonics in the DC side of a three-phase inverter. The asymmetric excitation currents due to magnetic path imbalances generate even-order harmonics, while DC bias currents produce odd-order harmonics. Load variations affect odd-order harmonics, increasing them with higher loads, but even-order harmonics remain stable. Experimental results on a 5 kVA three-phase inverter validate these findings, highlighting the importance of transformer design and bias mitigation in reducing harmonic distortions. This work provides a foundation for developing effective harmonic suppression techniques in three-phase inverter systems, enhancing stability and performance in DC networks.