In recent years, the rapid development of distributed generation, energy storage, and islanded microgrid systems has positioned the three phase inverter as a critical component in renewable energy systems. As a key power conversion device, the three phase inverter must ensure high-quality output voltage with low total harmonic distortion (THD) under various load conditions. Traditional control strategies, such as proportional-integral (PI) control, are widely used in inverter systems due to their simplicity and effectiveness in regulating DC signals. However, PI controllers exhibit limitations in tracking sinusoidal references and suppressing high-frequency disturbances, leading to compromised power quality. To address these issues, proportional resonance (PR) control has emerged as a promising alternative, offering superior sinusoidal tracking capabilities and harmonic rejection. In this paper, we explore the application of quasi proportional resonance (QPR) control in three phase inverter systems, focusing on its digital implementation, phase compensation techniques, and performance validation through simulation and experimental results.
The three phase voltage source inverter topology, as illustrated in the context of LC filtering, is fundamental to achieving sinusoidal output waveforms. A typical three phase inverter consists of a bridge circuit driven by sinusoidal pulse width modulation (SPWM), where the modulation wave is generated by comparing a reference signal with a carrier wave. When the switching frequency (carrier frequency) is much higher than the fundamental frequency of the signal wave, the output of the inverter bridge contains harmonics that must be filtered to produce a clean sinusoidal voltage. The LC filter, with inductors in star configuration and capacitors in delta, is commonly employed for this purpose. The cutoff frequency of the filter is given by $$f_0 = \frac{1}{2\pi \sqrt{3LC}}$$, which must be carefully selected to balance harmonic attenuation and system stability. For instance, with a fundamental frequency of 50 Hz and a switching frequency of 3 kHz, the cutoff frequency should satisfy $$3f_r < f_0 < \frac{1}{5}f_c$$ to effectively suppress harmonics while maintaining performance. In our analysis, we consider filter parameters such as $L_f = 6.3\,\text{mH}$ and $C_f = 35\,\mu\text{F}$ for a three phase inverter system.
Control system design for a three phase inverter typically involves a dual-loop structure to enhance both steady-state and dynamic performance. The inner loop, often based on instantaneous value control, utilizes QPR control to achieve accurate tracking of sinusoidal references and rejection of disturbances. The outer loop, employing PI control, regulates the RMS value of the output voltage to maintain stability under load variations. This combination leverages the strengths of both controllers: QPR provides high gain at the fundamental frequency and its harmonics, while PI ensures zero steady-state error for DC components. The control system structure includes feedback coefficients $K_{f1}$ and $K_{f2}$ for signal normalization, a feedforward coefficient $K_{Pf}$ to compensate for disturbances, and a phase compensator to address delays introduced by sampling circuits. The phase compensator is crucial in practical implementations, as hardware-induced phase lags can degrade control performance. We propose a digital phase compensation algorithm derived from a passive lead network, whose transfer function is given by $$\frac{U_2(s)}{U_1(s)} = \frac{1}{a} \frac{1 + aTs}{1 + Ts}$$, where $a = \frac{R_1 + R_2}{R_2} > 1$ and $T = \frac{R_1 R_2}{R_1 + R_2} C$. The maximum lead angle $\phi_m$ is calculated as $$\phi_m = \arcsin \frac{a – 1}{a + 1}$$, and for a phase lag of approximately 18° (1 ms) in our system, we obtain $a \approx 1.8944$ and $T = 2.313 \times 10^{-3}$ s. Using bilinear transformation with a sampling period $T_s = \frac{1}{3000}$ s, the discrete-time transfer function becomes $$\frac{U_2(z)}{U_1(z)} = \frac{0.9683 – 1.6999z^{-1}}{1 – 0.8656z^{-1}}$$, leading to the microprocessor implementation: $y(k) = 0.9683u(k) – 1.6999u(k-1) + 0.8656y(k-1)$.
The core of our control strategy lies in the QPR controller, which improves upon the ideal PR controller by introducing a bandwidth factor to enhance robustness. The transfer function of the QPR controller is expressed as $$G_{QPR}(s) = K_p + \frac{K_r \omega_c s}{s^2 + 2\omega_c s + \omega_o^2} = \frac{K_p s^2 + 2(\omega_c K_p + K_r \omega_c) s + K_p \omega_o^2}{s^2 + 2\omega_c s + \omega_o^2}$$, where $K_p$ is the proportional coefficient, $K_r$ is the resonant coefficient, $\omega_o = 2\pi f_c$ is the resonant angular frequency, and $\omega_c$ is the bandwidth factor typically ranging from 5 to 15 rad/s. The parameters $K_r$ and $\omega_c$ influence the resonant gain and bandwidth, respectively, and their selection is critical for system performance. For digital implementation, we apply bilinear transformation with $s = \frac{2}{T_s} \frac{z-1}{z+1}$ to discretize the controller, resulting in $$G_{QPR}(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}$$, where the coefficients are derived as follows:
$$b_0 = K_p + \frac{4K_r \omega_c T_s}{\omega_o^2 T_s^2 + 4\omega_c T_s + 4}$$
$$b_1 = \frac{K_p (2\omega_o^2 T_s^2 – 8)}{\omega_o^2 T_s^2 + 4\omega_c T_s + 4}$$
$$b_2 = \frac{K_p (\omega_o^2 T_s^2 – 4\omega_c T_s + 4) – 4K_r \omega_c T_s}{\omega_o^2 T_s^2 + 4\omega_c T_s + 4}$$
$$a_1 = \frac{2\omega_o^2 T_s^2 – 8}{\omega_o^2 T_s^2 + 4\omega_c T_s + 4}$$
$$a_2 = \frac{\omega_o^2 T_s^2 – 4\omega_c T_s + 4}{\omega_o^2 T_s^2 + 4\omega_c T_s + 4}$$
For example, with control parameters $K_p = 0.2$, $K_r = 0.1$, $T_s = \frac{1}{3000}$ s, and $\omega_c = 1$ rad/s, the discrete-time QPR controller becomes $$G_{QPR}(z) = \frac{0.2003 – 0.3997z^{-1} + 0.1995z^{-2}}{1 – 1.9984z^{-1} + 0.993z^{-2}}$$, and the digital realization in a microprocessor is given by $y(k) = 0.2003u(k) – 0.3997u(k-1) + 0.1995u(k-2) + 1.9984y(k-1) – 0.993y(k-2)$. This approach ensures precise tracking of sinusoidal signals in the three phase inverter while mitigating the effects of non-ideal components and dead-time distortions.
To validate the proposed control strategy, we conducted simulations using MATLAB/Simulink, modeling the discrete topology of the three phase inverter with a sampling time of $T_s = 3.33 \times 10^{-4}$ s. The simulation parameters are summarized in Table 1, which includes key values such as output frequency, switching frequency, filter components, and control coefficients. The inner loop employs QPR control, while the outer loop uses PI control for RMS voltage regulation. The carrier-based SP modulation is implemented with a carrier amplitude range of 0 to 25,000, corresponding to the hardware system’s time-base frequency of 150 MHz, and the signal wave has an amplitude of 12,500 with an offset of 12,500 for normalization. Feedback signals are scaled using coefficients $K_{f1}$ and $K_{f2}$ to match the digital control domain.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Output Frequency (Hz) | 50 | Switching Frequency (Hz) | 3000 |
| Filter Inductance (mH) | 6.3 | Filter Capacitance (μF) | 35 |
| Load Resistance (Ω) | 75 | DC Bus Voltage (V) | 300 |
| Output Voltage RMS (V) | 110 | Rated Current (A) | 3 |
| $K_{f1}$ | 77.85 | $K_{f2}$ | 110.09 |
| PI Proportional Gain | 1.7 | PI Integral Gain | 100 |
| $\omega_c$ (rad/s) | 5.0 | QPR Proportional Gain | 8.0 |
| $K_r$ | 30 | Feedforward Coefficient $K_{Pf}$ | 1.0 |
Simulation results demonstrate the effectiveness of QPR control in the three phase inverter. Figure 1(a) and 1(b) compare the startup processes under proportional control and QPR control, respectively. With proportional control, the output waveform exhibits significant distortion until approximately $t = 0.06$ s, whereas QPR control achieves a smoother transition within $t = 0.04$ s. The steady-state error comparison in Figure 1(c) highlights the superior tracking performance of QPR, with minimal deviation from the reference. Additionally, the outer PI loop ensures robustness against disturbances; Figure 2(a) shows the dynamic response to a DC bus voltage step change, where the output remains stable, and Figure 2(b) illustrates the response to load resistance variations from 75 Ω to 37.5 Ω and back, confirming the system’s ability to maintain voltage regulation under dynamic conditions.
For experimental validation, we constructed a hardware platform centered on a DSP controller, incorporating a single-phase uncontrolled rectifier, an IPM intelligent power module, LC filters, sampling circuits, signal conditioning, and isolated drivers. The experimental parameters, listed in Table 2, align with the simulation setup, with additional details on transformer ratios and sampling frequencies. The feedback signals are attenuated and conditioned to 0–3 V sinusoids before ADC conversion, and digital filtering techniques, such as median-average filtering, are applied to reduce noise. For instance, 360 raw data points are processed to yield 60 effective samples per cycle, ensuring smooth feedback signals without phase shift, as shown in Figure 3(a) and 3(b). The phase compensation algorithm is implemented in the DSP to correct hardware-induced delays, resulting in aligned sampling and output signals for accurate control, as depicted in Figure 3(c).
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Transformer Ratio | 2:3 | Sampling Frequency (Hz) | 18,000 |
| Algorithm Execution Frequency (Hz) | 3,000 | QPR Proportional Gain | 0.05 |
| Resonant Coefficient | 1.0 | Feedforward Coefficient | 1.0 |
| PI Proportional Gain | 0.9 | PI Integral Gain | 0.01 |
Steady-state performance under different control laws is evaluated experimentally. Figure 4(a) shows the output voltage waveform with proportional control, where the V-phase exhibits noticeable distortion with a THD of 2.08%. In contrast, Figure 4(b) demonstrates the output with QPR control, reducing THD to 1.40% and improving sinusoidal tracking. The dynamic response to load disturbances, as shown in Figure 5(a) for loading and Figure 5(b) for unloading, corroborates the simulation findings, indicating strong anti-disturbance capability and stable output in the three phase inverter system.
In conclusion, the application of quasi proportional resonance control in three phase inverter systems offers significant advantages in terms of sinusoidal tracking, harmonic suppression, and dynamic performance. The digital implementation of QPR control, coupled with phase compensation algorithms, addresses practical challenges such as hardware delays and discrete processing. Simulation and experimental results confirm that the proposed strategy achieves low THD (below 1.5%) and robust operation under varying loads, making it suitable for modern renewable energy applications. Future work could explore adaptive tuning of QPR parameters and integration with advanced modulation techniques to further enhance the performance of three phase inverters.
The three phase inverter is a versatile power electronic device that plays a pivotal role in converting DC power from sources like solar panels or batteries into AC power for grid connection or standalone use. Its performance is critical in ensuring energy efficiency and power quality in systems such as microgrids and electric vehicles. The control of a three phase inverter involves managing the switching of power devices to generate three-phase voltages with desired magnitude, frequency, and phase. Traditional methods like space vector modulation (SVM) and SPWM are commonly employed, but the control algorithm determines the overall efficiency and reliability. In this context, QPR control provides a robust solution by offering high gain at specific frequencies, which is essential for rejecting harmonics and compensating for nonlinearities in the three phase inverter.
Moreover, the design of the LC filter for a three phase inverter must consider factors such as resonance damping and component sizing to prevent instability. The transfer function of the output filter can be modeled as $$H(s) = \frac{1}{s^2 L C + 1}$$ for a single-phase equivalent, but for a three-phase system with delta-connected capacitors, the dynamics are more complex. The impedance of the filter affects the control loop gain, and thus, the QPR controller parameters must be tuned to ensure stability. Using root locus or Bode plot analysis, we can determine the appropriate $K_p$ and $K_r$ values. For instance, increasing $K_r$ enhances the resonance peak, improving harmonic rejection but potentially leading to overshoot. Similarly, $\omega_c$ widens the bandwidth, allowing the controller to handle frequency variations in the three phase inverter output.
In practical implementations, digital signal processors (DSPs) are preferred for controlling three phase inverters due to their high-speed computation capabilities. The discrete-time control laws derived via bilinear transformation ensure that the algorithms can be executed in real-time. However, quantization errors and limited resolution of ADCs may introduce additional distortions. To mitigate this, oversampling techniques and higher-bit ADCs can be employed. The three phase inverter system also requires protection circuits against overcurrent and overvoltage conditions, which can be integrated with the control software. For example, the DSP can monitor current sensors and trigger shutdown mechanisms if thresholds are exceeded, ensuring the safety and longevity of the three phase inverter.
Another aspect to consider is the impact of dead time in the switching devices of the three phase inverter. Dead time is introduced to prevent shoot-through currents in the bridge legs, but it causes voltage distortions and increases THD. The QPR controller compensates for these effects by providing high gain at the fundamental frequency and its harmonics. The transfer function of the dead-time effect can be approximated as a phase lag, which aligns with the need for phase compensation in our control design. By combining QPR control with dead-time compensation algorithms, the performance of the three phase inverter can be further improved.
In summary, the integration of QPR control into three phase inverter systems represents a significant advancement in power electronics. Its ability to handle sinusoidal references and disturbances makes it ideal for applications requiring high power quality. Through detailed modeling, simulation, and experimental validation, we have demonstrated that QPR control, when properly implemented, enhances the stability and efficiency of three phase inverters. As renewable energy systems continue to evolve, the role of advanced control strategies like QPR will become increasingly important in achieving reliable and clean power conversion.