In modern power electronics, the single phase inverter serves as a critical component in applications such as uninterruptible power supplies, renewable energy systems, and motor drives. Achieving high-quality output voltage with robust dynamic and static performance remains a significant challenge due to inherent issues like residual oscillations caused by LC filter resonances. Traditional control methods, including proportional-integral (PI) and proportional-resonant (PR) controllers, often fall short in fully suppressing these oscillations, leading to compromised system reliability. This article presents a comprehensive analysis and design of a dual-loop plus time-delay hybrid control strategy for single phase inverters, addressing these limitations through a novel integration of Posicast control. The focus is on enhancing the performance of single phase inverter systems by combining voltage and current loops with a time-delay compensator.
The mathematical model of a single phase inverter is foundational to understanding its behavior. A typical half-bridge topology includes an LC filter with inductance L, capacitance C, and an equivalent resistance r accounting for parasitic losses. The output voltage \(v_o\) and inductor current \(i_L\) are key variables. The state-space representation or transfer function model can be derived as follows. For a single phase inverter, the relationship between the output voltage and the input modulation signal can be expressed using a second-order system due to the LC filter. The open-loop transfer function from the modulation signal to the output voltage is given by:
$$ G_{ol}(s) = \frac{1}{LCs^2 + rCs + 1} $$
This model highlights the resonant peak at frequency \(f_r = \frac{1}{2\pi\sqrt{LC}}\), which often leads to instability or oscillations in the output. To address this, a dual-loop control structure is employed, consisting of an inner current loop and an outer voltage loop. The inner loop uses a proportional controller to improve stability, while the outer loop utilizes a proportional-integral-resonant (PIR) controller to achieve zero steady-state error at the fundamental frequency, typically 50 Hz or 60 Hz for single phase inverter applications.
The current inner loop controller is designed as a proportional controller with gain \(k_{cp}\). The closed-loop transfer function for the current loop, considering the PWM delay and system parameters, is derived to ensure adequate phase margin and bandwidth. For a single phase inverter with parameters such as input voltage \(V_{dc} = 750V\), \(L = 1mH\), \(C = 80\mu F\), and \(r = 0.1\Omega\), the current controller gain must satisfy constraints from PWM linearity. Specifically, the gain is bounded by:
$$ k_{cp} \leq \frac{V_{dc}}{2A_c f_s L} $$
where \(A_c\) is the carrier amplitude and \(f_s\) is the switching frequency. Setting \(k_{cp} = 0.5\) for a 10 kHz switching frequency, the Bode plot of the current loop shows reduced gain at the resonant frequency, but a significant peak remains, causing residual oscillations. This underscores the need for additional compensation in single phase inverter systems.
For the voltage outer loop, a PIR controller is implemented with the transfer function:
$$ G_{PIR}(s) = k_{vp} + \frac{k_{vi}}{s} + \frac{k_{vr} s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \(k_{vp}\), \(k_{vi}\), and \(k_{vr}\) are the proportional, integral, and resonant gains, respectively, \(\omega_0\) is the resonant frequency (e.g., 314 rad/s for 50 Hz), and \(\omega_c\) is the cutoff frequency. The parameters are designed to achieve a bandwidth of 800 Hz, which is one-tenth of the switching frequency, ensuring effective harmonic suppression. The proportional gain \(k_{vp} = 1.2\), integral gain \(k_{vi} = 267\), and resonant gain \(k_{vr} = 98\) are calculated based on steady-state error requirements, targeting less than 1% error at 50 Hz. The voltage open-loop transfer function becomes:
$$ G_{u}(s) = G_{i}(s) G_{PIR}(s) \frac{1}{LCs^2 + rCs + 1} $$
where \(G_{i}(s)\) is the closed-loop current transfer function. Analysis of the Bode plot for the voltage loop reveals a persistent resonant peak near the LC filter frequency, indicating that dual-loop control alone cannot eliminate oscillations in single phase inverter systems. This residual peak motivates the incorporation of a time-delay based compensator.
Posicast control, a time-delay method, is introduced to suppress the resonant peak. It works by splitting a step input into two sub-signals with specific time delays, effectively canceling oscillations in the response. For a second-order system with damping ratio \(\zeta\) and natural frequency \(\omega_n\), the Posicast controller transfer function is:
$$ G_{po}(s) = \frac{1}{1 + \sigma} + \frac{\sigma}{1 + \sigma} e^{-s t_d} $$
where \(\sigma\) is the overshoot of the original system, and \(t_d = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}\) is the delay time corresponding to the peak time of the step response. In single phase inverter applications, \(\omega_n = \frac{1}{\sqrt{LC}}\) and \(\zeta = \frac{r}{2} \sqrt{\frac{C}{L}}\). For the given parameters, \(t_d = 0.89ms\) and \(\sigma = 72.9\%\). The Posicast controller introduces zeros at odd multiples of the resonant frequency, canceling the poles of the LC filter and thus eliminating oscillations.
The hybrid control scheme integrates the dual-loop with Posicast control by placing the Posicast block in the current loop output. The modified control structure enhances the voltage loop performance. The closed-loop transfer function for the voltage with hybrid control is:
$$ G_{cu}(s) = \frac{G_{po}(s) G_{i}(s) G_{PIR}(s)}{1 + G_{po}(s) G_{i}(s) G_{PIR}(s) G_{plant}(s)} $$
where \(G_{plant}(s)\) represents the inverter plant model. Frequency domain analysis shows that the hybrid control reduces the gain at the resonant frequency from over 8 V to below 0.5 V, significantly improving tracking performance. Additionally, the phase margin remains above 40 degrees even when accounting for computational and PWM delays, ensuring stability in single phase inverter systems.
To quantify the performance, several key metrics are evaluated. The steady-state error, total harmonic distortion (THD), and dynamic response under load changes are critical for single phase inverter applications. The following table summarizes the controller parameters and their impact on system performance:
| Parameter | Symbol | Value | Effect on Single Phase Inverter |
|---|---|---|---|
| Current Controller Gain | \(k_{cp}\) | 0.5 | Reduces current loop oscillations |
| Voltage Proportional Gain | \(k_{vp}\) | 1.2 | Improves transient response |
| Voltage Integral Gain | \(k_{vi}\) | 267 | Eliminates DC offset |
| Voltage Resonant Gain | \(k_{vr}\) | 98 | Enforces zero error at 50 Hz |
| Posicast Delay Time | \(t_d\) | 0.89 ms | Cancels resonant peak |
| Posicast Overshoot Factor | \(\sigma\) | 72.9% | Determines step signal splitting |
System performance is further analyzed through error transfer functions and sensitivity analysis. The error transfer function for the voltage loop is:
$$ E(s) = \frac{1}{1 + G_{po}(s) G_{i}(s) G_{PIR}(s)} R(s) $$
where \(R(s)\) is the reference signal. The magnitude of \(E(s)\) at the resonant frequency is drastically reduced with hybrid control, confirming its efficacy. Moreover, robustness to parameter variations is assessed by evaluating the system under component tolerances. For instance, variations in L and C by ±10% show that the hybrid control maintains a THD below 2% and overshoot under 5% in single phase inverter outputs, whereas dual-loop control alone exceeds these limits.
Experimental validation on a 15 kW single phase inverter prototype demonstrates the practical benefits. The system parameters align with the theoretical design: \(V_{dc} = 750V\), \(L = 1mH\), \(C = 80\mu F\), \(r = 0.1\Omega\), and switching frequency \(f_s = 10 kHz\). The controllers are implemented on a TMS320F2812 DSP. Steady-state tests under no-load and full-load conditions reveal that the hybrid control achieves a THD of less than 1.2% and output voltage accuracy within 0.8%, outperforming traditional dual-loop control. Dynamic tests involve load switching between no-load and full-load. The hybrid control reduces voltage overshoot to below 2% with no oscillations, while dual-loop control exhibits over 5% overshoot and sustained ringing. The following table compares key performance indicators:
| Performance Metric | Dual-Loop Control | Dual-Loop + Posicast Control |
|---|---|---|
| Steady-State THD (%) | 1.2 | 1.1 |
| Output Voltage Error (%) | 0.8 | 0.7 |
| Overshoot in Load Transition (%) | >5 | <2 |
| Settling Time (ms) | 20 | 10 |
| Resonant Peak Magnitude (V) | 8 | 0.5 |
The experimental results validate the superiority of the hybrid approach in single phase inverter systems. The Posicast control effectively cancels the resonant peak without compromising stability, and the dual-loop structure ensures precise tracking and disturbance rejection. Further analysis includes the impact of time delays from digital implementation. The total delay \(T_d = 1.5 T_s\), where \(T_s\) is the switching period, is incorporated into the model as \(H_d(s) = e^{-s T_d}\). The open-loop transfer function with delays becomes:
$$ G_{ol,delay}(s) = G_{po}(s) G_{i}(s) G_{PIR}(s) H_d(s) $$
Bode plots indicate that the phase margin decreases from 55 degrees to 40 degrees but remains sufficient for stability. This highlights the robustness of the hybrid control in practical single phase inverter implementations where delays are inevitable.
In conclusion, the dual-loop plus time-delay hybrid control strategy offers a significant advancement for single phase inverter performance. By integrating a Posicast controller into the current loop, the method eliminates residual oscillations at the LC resonant frequency, enhancing both dynamic and static characteristics. Extensive analysis and experimental results on a 15 kW system confirm reduced THD, minimal overshoot, and improved robustness. This approach is particularly beneficial for high-power single phase inverter applications in renewable energy and UPS systems, where voltage quality and reliability are paramount. Future work could explore adaptive versions of the Posicast control to handle parameter variations in real-time for single phase inverter systems.
The mathematical formulations and performance data presented herein provide a comprehensive framework for designing and optimizing single phase inverter controls. The use of Laplace domain models, frequency response analysis, and time-domain validations ensures a holistic understanding. As power electronics evolve, such hybrid strategies will play a crucial role in advancing the efficiency and stability of single phase inverter based systems.