In this article, I present a comprehensive design and analysis of a single phase inverter for photovoltaic (PV) grid-connected systems. The single phase inverter serves as a critical interface between PV arrays and the AC grid, converting DC power generated by solar panels into AC power suitable for grid injection. I focus on a two-level bridge voltage source inverter utilizing IGBTs, which offers a balance between efficiency, cost, and controllability. The design process involves developing mathematical models, selecting key parameters, designing control strategies, and validating performance through simulation. Throughout this work, the term “single phase inverter” is emphasized to highlight its centrality in PV system integration.
The structure of a typical PV system includes PV panels, a DC-DC converter (e.g., Boost converter) for voltage regulation, and a DC-AC inverter for grid connection. The single phase inverter discussed here employs a full-bridge topology with IGBTs and anti-parallel diodes, as illustrated in the following figure, which shows the main circuit topology. This single phase inverter configuration enables bidirectional power flow and high-frequency switching, making it suitable for grid-tied applications.
The mathematical modeling of the single phase inverter is fundamental to understanding its behavior and designing effective controls. In the time domain, the single phase inverter circuit can be analyzed using Kirchhoff’s laws. Define the bridge arm states: for arm a, \( S_a = 1 \) when T1 is on and T3 is off, and \( S_a = 0 \) when T1 is off and T3 is on. Similarly for arm b. The voltage at point a is given by \( u_a = S_a U_{dc} \), where \( U_{dc} \) is the DC input voltage. The output voltage \( u_{ab} \) can be \( U_{dc} \), 0, or \( -U_{dc} \), depending on the switching states. The inductor voltage equation is:
$$ u_L = L \frac{di_L}{dt} = u_{ab} – u_o $$
where \( L \) is the filter inductance, \( i_L \) is the inductor current, and \( u_o \) is the output voltage across the capacitor \( C \). The capacitor current equation is:
$$ i_C = C \frac{du_o}{dt} = i_L – i_o $$
where \( i_o \) is the output current. Transforming these equations to the s-domain facilitates control system design. The s-domain models are:
$$ u_L(s) = L s i_L(s) = u_{ab}(s) – u_o(s) $$
$$ i_C(s) = C s u_o(s) = i_L(s) – i_o(s) $$
Solving for inductor current and output voltage:
$$ i_L(s) = \frac{u_{ab}(s) – u_o(s)}{L s} $$
$$ u_o(s) = \frac{i_L(s) – i_o(s)}{C s} $$
These expressions form the basis for designing the control loops in the single phase inverter. The dynamics of the single phase inverter are influenced by the LC filter parameters, and proper selection is crucial for performance.
Current paths in the single phase inverter depend on the inductor current direction and switching states. For instance, when \( S_a = 0 \) and \( S_b = 0 \), and inductor current is positive, the path is through T4 and D3. When \( S_a = 0 \) and \( S_b = 1 \), the path involves D2, the DC source, and D3. Understanding these paths ensures safe operation and informs the design of protection circuits in the single phase inverter.
Parameter selection for the single phase inverter is critical for reliability and efficiency. Consider a design with DC input voltage of 800 V, output AC voltage amplitude of 311 V (220 V RMS), and maximum load current of 30 A. The IGBTs and diodes must withstand the maximum reverse voltage. With a safety margin of 2, the rated voltage is 1600 V. The current rating is determined from the load current. The RMS current through each IGBT and diode is:
$$ I_{VD} = I_T = \frac{1}{\sqrt{2}} I_o = 15 \, \text{A} $$
The average current for thermal design is:
$$ I_{VD_{aver}} = I_{T_{aver}} = \frac{15}{1.57} \approx 9.6 \, \text{A} $$
With a 1.5 times safety margin, the selected current rating is 14.3 A. Thus, IGBTs and diodes with 1600 V and 15 A ratings are chosen. The LC filter parameters are selected to attenuate switching harmonics. For a switching frequency of 10 kHz, the filter cutoff frequency should be much lower, e.g., 1 kHz. The inductance and capacitance can be calculated based on the desired ripple current and voltage. For instance, with \( L = 5 \, \text{mH} \) and \( C = 470 \, \mu\text{F} \), the cutoff frequency is:
$$ f_c = \frac{1}{2\pi \sqrt{LC}} \approx 1.04 \, \text{kHz} $$
This ensures effective filtering for the single phase inverter. The table below summarizes key parameters for the single phase inverter design.
| Parameter | Value |
|---|---|
| DC Input Voltage (\( U_{dc} \)) | 800 V |
| Output Voltage Amplitude | 311 V |
| Output Voltage RMS | 220 V |
| Maximum Load Current (\( I_o \)) | 30 A |
| IGBT Voltage Rating | 1600 V |
| IGBT Current Rating | 15 A |
| Filter Inductance (\( L \)) | 5 mH |
| Filter Capacitance (\( C \)) | 470 μF |
| Switching Frequency | 10 kHz |
Control strategy for the single phase inverter employs a dual-loop approach with an outer voltage loop and an inner current loop, both using PI regulators. The current loop ensures fast tracking of the reference current, while the voltage loop maintains the output voltage stability. The s-domain model for the current loop is shown in the block diagram, where \( u_{ab}^* \) is the reference inverter output voltage, and the PI controller is \( k_{ip} + \frac{k_i}{s} \). The inductor current is:
$$ i_L(s) = \frac{u_{ab}^*(s) – u_o(s)}{L s} $$
and the reference voltage is:
$$ u_{ab}^*(s) = \left( k_{ip} + \frac{k_i}{s} \right) (i_L^*(s) – i_L(s)) $$
Combining these, the closed-loop transfer function for the current loop is:
$$ i_L(s) = \frac{ k_{ip} s + \frac{k_i}{L} }{ s^2 + \frac{k_{ip}}{L} s + \frac{k_i}{L} } i_L^*(s) – \frac{ s/L }{ s^2 + \frac{k_{ip}}{L} s + \frac{k_i}{L} } u_o(s) $$
This represents a second-order system. Comparing with the standard form \( s^2 + 2\xi \omega_n s + \omega_n^2 \), we have:
$$ \frac{k_i}{L} = \omega_n^2 $$
$$ \frac{k_{ip}}{L} = 2\xi \omega_n $$
Choosing \( \xi = 0.707 \) and \( \omega_n = 2\pi \times 1000 \, \text{rad/s} \), the parameters are:
$$ k_i = \omega_n^2 L = (2000\pi)^2 \times 0.005 \approx 19700 $$
$$ k_{ip} = 2\xi \sqrt{k_i L} = 2 \times 0.707 \times \sqrt{19700 \times 0.005} \approx 44.4 $$
For the voltage loop, the PI controller parameters are selected as \( k_{up} = 5 \) and \( k_{ui} = 200 \) based on simulation tuning. The output voltage in terms of reference voltage and disturbance is complex; a simplified expression is:
$$ u_o(s) = \frac{ k_{up} s^2 + (k_{up} k_i + k_{ui} k_{ip}) s + k_{ui} k_i }{ L C s^4 + k_{ip} C s^3 + (1 + k_{up} + k_i C) s^2 + (k_{up} k_i + k_{ui} k_{ip}) s + k_{ui} k_i } u_o^*(s) – \frac{ L s^3 + k_{ip} s^2 + k_i s }{ L C s^4 + k_{ip} C s^3 + (1 + k_{up} + k_i C) s^2 + (k_{up} k_i + k_{ui} k_{ip}) s + k_{ui} k_i } i_o(s) $$
This ensures that at steady state, the output impedance is low, and voltage regulation is achieved. The control of the single phase inverter is vital for grid synchronization and power quality.
Implementation of the PI controllers using analog circuits is practical for hardware realization. The current loop PI controller circuit consists of an operational amplifier with resistors and capacitors. Using virtual short and open principles, the transfer function is:
$$ \frac{u_{out}(s)}{u_{in}(s)} = -\left( \frac{R_2}{R_1} + \frac{1}{R_1 C_1 s} \right) = -\left( k_{ip} + \frac{k_i}{s} \right) $$
Thus,
$$ k_{ip} = \frac{R_2}{R_1} $$
$$ k_i = \frac{1}{R_1 C_1} $$
Setting \( R_1 = 1 \, \text{k}\Omega \), we get \( R_2 = k_{ip} R_1 = 44.4 \, \text{k}\Omega \) and \( C_1 = \frac{1}{R_1 k_i} \approx 0.51 \, \mu\text{F} \). Similarly, for the voltage loop PI controller, with \( R_3 = 1 \, \text{k}\Omega \), \( R_4 = k_{up} R_3 = 5 \, \text{k}\Omega \) and \( C_2 = \frac{1}{R_3 k_{ui}} = 5 \, \mu\text{F} \). An inverting stage may be added to correct the sign. This analog implementation ensures robust control for the single phase inverter.
Simulation analysis validates the single phase inverter design. Using MATLAB/Simulink, a model is built with a Boost converter input (120 V to 800 V) and the single phase inverter with LC filter (\( L = 5 \, \text{mH} \), \( C = 470 \, \mu\text{F} \)). Switching frequency is 10 kHz for the inverter and 20 kHz for the Boost. The control system uses voltage and current loops with PI regulators, and a carrier-based PWM scheme. The simulation time is 0.6 s. Key waveforms include output current, PCC voltage, and grid voltage.
For static performance, with reference current amplitude set to -20 A (inverting mode, unit power factor), the output current is sinusoidal and 180 degrees out of phase with the grid voltage, confirming proper operation of the single phase inverter. The current THD is low, indicating good power quality. When the reference current is positive (rectifying mode, not desired), the current is in phase with voltage, demonstrating controller flexibility but highlighting the need for correct mode setting in the single phase inverter.
Dynamic performance is tested with a step change in reference current from 20 A to 30 A at 0.2 s. The output current tracks the reference quickly with minimal overshoot, showing excellent dynamic response of the single phase inverter control system. The table below summarizes simulation parameters and results.
| Simulation Parameter | Value |
|---|---|
| Boost Input Voltage | 120 V |
| Boost Output Voltage | 800 V |
| Inverter Switching Frequency | 10 kHz |
| Boost Switching Frequency | 20 kHz |
| Reference Current Step | 20 A to 30 A at 0.2 s |
| Output Current Response Time | Less than 0.1 s |
| Steady-State Error | Negligible |
In conclusion, the design of a single phase photovoltaic grid-connected inverter involves detailed modeling, careful parameter selection, and robust control design. The single phase inverter based on IGBT bridge topology with LC filtering and PI control demonstrates stable performance under static and dynamic conditions. The mathematical models in time and s-domains provide a foundation for analysis, and the analog PI implementation offers a practical solution. Simulation results confirm the effectiveness of the single phase inverter in grid-tied applications, with fast response and low distortion. Future work could explore advanced control techniques and higher switching frequencies for improved efficiency in single phase inverter systems.
The single phase inverter is a key component in renewable energy systems, and its design optimization contributes to grid stability and power quality. By repeatedly focusing on the single phase inverter, this article underscores its importance in modern power electronics. The use of formulas and tables aids in clarifying design choices and performance metrics for the single phase inverter.