In modern power electronics, the three phase inverter plays a critical role in converting direct current (DC) to alternating current (AC), particularly in applications such as electric vehicle traction systems and industrial drives. The stability and efficiency of the three phase inverter are paramount for ensuring reliable operation of AC motors and grid-connected systems. Among various modulation techniques, Sinusoidal Pulse Width Modulation (SPWM) has gained prominence due to its ability to generate high-quality output waveforms with reduced harmonic distortion. This paper presents a comprehensive simulation study of a three phase inverter based on SPWM control, utilizing MATLAB/Simulink to model and analyze its performance. We focus on evaluating output voltage characteristics, harmonic spectrum, and conversion efficiency under different conditions, with an emphasis on the impact of LC filtering. The insights derived from this analysis aim to support the design and optimization of three phase inverter systems for enhanced performance in real-world scenarios.
The foundation of a three phase inverter lies in its ability to synthesize three-phase AC voltages from a DC source through controlled switching of power devices. In a standard configuration, the three phase inverter consists of six switching devices, typically IGBTs, arranged in a bridge topology. Each phase leg comprises an upper and lower switch, which are operated in a complementary manner to prevent short-circuiting the DC supply. The output voltages are derived from the midpoint of each phase leg, and the switching patterns determine the quality of the AC waveform. The SPWM technique enhances this by comparing a high-frequency triangular carrier wave with three sinusoidal modulation waves that are 120 degrees apart in phase. This comparison generates pulse signals that control the IGBTs, resulting in output pulses whose widths vary sinusoidally. The mathematical representation of the SPWM process involves the modulation index, defined as the ratio of the amplitude of the sinusoidal modulation wave to that of the triangular carrier wave. For a three phase inverter, the modulation index m is given by:
$$ m = \frac{A_m}{A_c} $$
where \( A_m \) is the peak amplitude of the sinusoidal modulation wave and \( A_c \) is the peak amplitude of the triangular carrier wave. The output phase voltage in a three phase inverter under SPWM control can be expressed as a function of the DC input voltage \( U_{DC} \) and the modulation index. For ideal conditions, the fundamental component of the output voltage is approximated by:
$$ V_{\text{fund}} = \frac{m \cdot U_{DC}}{2} $$
However, in practical implementations, non-ideal factors such as switch dead times and device voltage drops introduce distortions. To mitigate these, the three phase inverter often incorporates filters, such as LC networks, to smooth the output waveform. The design of these filters is crucial for attenuating harmonics, particularly those arising from the switching frequency. The cutoff frequency of an LC filter is determined by:
$$ f_c = \frac{1}{2\pi\sqrt{LC}} $$
where L is the inductance and C is the capacitance. Proper selection of L and C values ensures that the filter attenuates high-frequency components while passing the fundamental frequency unchanged. In our simulation, we adhere to these principles to model a three phase inverter that achieves low total harmonic distortion (THD) and high efficiency.
To construct the three phase inverter model in Simulink, we utilized the Simscape Electrical library to assemble the main circuit components. The core of the model includes six IGBT/diode modules, a DC voltage source, a three-phase resistive load, and LC filter circuits for each phase. The IGBTs are configured in a standard bridge arrangement, with anti-parallel diodes to provide freewheeling paths during switching transitions. The DC source is set to 700 V, representing a typical input for traction applications, and the three-phase load consists of pure resistors with a value of 48.4 Ω per phase. This load value is chosen to achieve an output power of approximately 1,000 W per phase when the output voltage is 220 V RMS. The LC filter parameters are calculated based on the switching frequency of 15 kHz, with the inductor and capacitor values selected to position the cutoff frequency at one-tenth of the switching frequency to effectively suppress harmonics. The maximum inductor value is derived from the current ripple requirement, using the formula:
$$ L_{\text{max}} = \frac{U_{DC}}{8 f_s \Delta I_L} $$
where \( f_s \) is the switching frequency and \( \Delta I_L \) is the allowable current ripple, set to 15% of the rated current. Through iterative calculation, we determined L = 18 mH and C = 0.62 μF for each filter stage. These values ensure that the filter minimizes ripple without excessively increasing the system size or cost. The following table summarizes the key parameters used in the main circuit of the three phase inverter:
| Parameter | Value | Description |
|---|---|---|
| DC Input Voltage (\( U_{DC} \)) | 700 V | Supply voltage for the three phase inverter |
| Switching Frequency (\( f_s \)) | 15 kHz | Frequency of the triangular carrier wave |
| Load Resistance per Phase | 48.4 Ω | Pure resistive load for each phase |
| Filter Inductance (L) | 18 mH | Inductor value in LC filter |
| Filter Capacitance (C) | 0.62 μF | Capacitor value in LC filter |
| Modulation Index (m) | 0.9 | Ratio of modulation to carrier amplitude |
For the control circuit, we implemented the SPWM pulse generation using Simulink blocks, including sinusoidal wave generators, a triangular wave generator, relational operators, and logical gates. Three sinusoidal waves with amplitudes of 0.9 V and frequencies of 50 Hz are generated with phase shifts of 0°, 120°, and -120° to correspond to the U, V, and W phases, respectively. The triangular carrier wave has an amplitude of 1 V and a frequency of 10 kHz, though for detailed waveform analysis, we sometimes reduced this to 1 kHz to visualize pulse patterns clearly. The relational operators compare the sinusoidal and triangular waves, producing high-level outputs when the sinusoidal wave exceeds the triangular wave. These outputs are then processed through logical NOT gates to create complementary signals for the upper and lower IGBTs in each phase leg. This arrangement ensures that the three phase inverter operates in 180-degree conduction mode, with each IGBT conducting for 180 degrees per cycle and phase shifts of 120 degrees between phases. The following table outlines the parameters for the SPWM pulse generation module:
| Component | Parameter | Value |
|---|---|---|
| Sinusoidal Wave | Amplitude | 0.9 V |
| Sinusoidal Wave | Frequency | 50 Hz |
| Sinusoidal Wave | Phases | 0°, 120°, -120° |
| Triangular Wave | Amplitude | 1 V |
| Triangular Wave | Frequency | 10 kHz (1 kHz for detailed analysis) |
| Relational Operator | Function | Output high when sine > triangle |
Simulation results for the three phase inverter reveal significant improvements in output waveform quality with the incorporation of LC filters. Without the LC filter, the output phase voltages exhibit a stepped waveform characterized by high harmonic content, as shown in the simulated waveforms. The voltage peaks align with the DC input level, but the transitions between levels introduce substantial distortion. In contrast, with the LC filter applied, the output voltages become smooth sinusoidal waves with minimal ripple. Measurements from the Simulink scope indicate that the RMS value of each phase voltage is approximately 222.8 V, which closely matches the theoretical value of 220 V, demonstrating the efficacy of the filter in producing clean AC power. The phase difference between U, V, and W voltages is consistently 120 degrees, confirming proper operation of the three phase inverter. This visual assessment is supported by quantitative analysis of the harmonic spectrum.
Harmonic analysis using Fast Fourier Transform (FFT) highlights the impact of filtering on the three phase inverter output. Without the LC filter, the FFT plot shows a dominant fundamental component at 50 Hz, but significant harmonics at multiples of the switching frequency, leading to a THD of 79.47%. This high distortion level is unacceptable for most applications, as it can cause overheating in motors and interference in sensitive equipment. With the LC filter, the harmonic amplitudes are drastically reduced; the component at 10 kHz is attenuated to only 0.75% of the fundamental, and the overall THD drops to 1.16%. This reduction is achieved because the LC filter acts as a low-pass network, shunting high-frequency currents through the capacitor while the inductor limits the rate of current change. The THD calculation is based on the formula:
$$ \text{THD} = \frac{\sqrt{\sum_{n=2}^{\infty} V_n^2}}{V_1} \times 100\% $$
where \( V_1 \) is the RMS value of the fundamental frequency component and \( V_n \) represents the RMS values of the nth harmonic components. The following table compares the harmonic performance for the three phase inverter with and without the LC filter:
| Condition | Fundamental Amplitude (V) | THD (%) | Dominant Harmonic Frequency |
|---|---|---|---|
| Without LC Filter | 315.4 | 79.47 | 10 kHz |
| With LC Filter | 315.0 | 1.16 | 50 Hz (fundamental) |
Efficiency is another critical metric for the three phase inverter, influenced by losses in switching devices and passive components. In our simulation, we assumed ideal capacitors and inductors, so losses primarily originate from the IGBTs. The IGBT parameters include an internal resistance (Ron) of 1 mΩ, snubber resistance (Rs) of 100 kΩ, and snubber capacitance (Cs) of 0, minimizing additional losses. The output power is calculated from the load voltages and currents, while the input power is derived from the DC source voltage and current. Specifically, the average DC current measured is 4.408 A, yielding an input power of:
$$ P_{\text{in}} = U_{DC} \times I_{\text{DC}} = 700 \, \text{V} \times 4.408 \, \text{A} = 3,085.6 \, \text{W} $$
The output power for the three-phase system is computed as the sum of powers in each phase. For a balanced resistive load, the phase voltage \( V_{\text{phase}} = 222.8 \, \text{V} \) and phase resistance \( R = 48.4 \, \Omega \), so the power per phase is:
$$ P_{\text{phase}} = \frac{V_{\text{phase}}^2}{R} = \frac{(222.8)^2}{48.4} \approx 1025.5 \, \text{W} $$
Total output power for three phases is:
$$ P_{\text{out}} = 3 \times P_{\text{phase}} = 3 \times 1025.5 \, \text{W} = 3,076.5 \, \text{W} $$
Thus, the conversion efficiency is:
$$ \eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% = \frac{3076.5}{3085.6} \times 100\% \approx 99.7\% $$
This high efficiency underscores the effectiveness of the SPWM control and filter design in the three phase inverter. However, in practical scenarios, factors such as non-ideal components, switching losses, and thermal effects would reduce this value. For instance, switching losses in IGBTs can be estimated using:
$$ P_{\text{sw}} = f_s \cdot (E_{\text{on}} + E_{\text{off}}) $$
where \( E_{\text{on}} \) and \( E_{\text{off}} \) are the energy losses during turn-on and turn-off, respectively. Future work could involve incorporating these losses to refine the efficiency model for the three phase inverter.
In conclusion, our simulation demonstrates that the SPWM-controlled three phase inverter, when paired with an appropriately designed LC filter, achieves excellent performance in terms of waveform quality, harmonic distortion, and efficiency. The THD reduction from 79.47% to 1.16% highlights the filter’s crucial role in producing sinusoidal outputs suitable for sensitive loads like traction motors. The three phase inverter model developed in Simulink provides a robust platform for further optimization, such as adapting to variable load conditions or integrating advanced control strategies like space vector modulation. While the current study assumes ideal conditions, real-world implementations of the three phase inverter must account for component tolerances, environmental factors, and operational dynamics. Continued research will focus on enhancing the three phase inverter’s robustness and adaptability, ensuring its reliability in diverse applications ranging from electric vehicles to renewable energy systems.