In modern power electronics, the stability of inverter systems is critical for applications such as high-speed trains, where three-level inverters are widely used in AC drive systems. This study focuses on a 2H-bridge cascaded three-level single phase inverter, employing a quasi-Proportional-Resonant (PR) controller to regulate output current. The inherent nonlinearity due to switching actions necessitates a deep investigation into bifurcation and chaos phenomena. We establish a discrete mathematical model using stroboscopic mapping and analyze system stability through bifurcation diagrams and fold maps. Our findings delineate stable operating regions, providing insights for designing robust single phase inverter systems.
The proliferation of power electronic systems has heightened the need for reliable inverter topologies. Single phase inverters, particularly multi-level configurations, offer advantages like reduced harmonic distortion and improved efficiency. However, their nonlinear dynamics can lead to instabilities, manifesting as bifurcations or chaotic behavior. This research bridges theoretical modeling and practical implementation by examining how control parameters, especially the proportional gain of a quasi-PR regulator, influence the stability of a three-level single phase inverter. We employ numerical simulations and nonlinear analysis tools to predict and mitigate undesirable operational states.
The 2H-bridge cascaded three-level single phase inverter comprises two H-bridges connected in series, enabling multiple voltage levels at the output. Each H-bridge operates with bipolar Carrier-Based Phase-Shifted Pulse Width Modulation (CPS-PWM), where the modulation signal derives from the error between load current and reference current. The output voltage states are categorized into four modes: +0/+E, +E/+2E, -0/-E, and -E/-2E. Each switching cycle encompasses nine distinct modes, with specific switch states and voltage levels as summarized in Table 1.
| Mode | State 1 Voltage | Bridge 1 Switches | Bridge 2 Switches | State 2 Voltage | Bridge 1 Switches | Bridge 2 Switches |
|---|---|---|---|---|---|---|
| 1 | E | 1,4 | 1,3 | E | 1,4 | 1,3 |
| 2 | 0 | 2,4 | 1,3 | 2E | 1,4 | 1,4 |
| 3 | E | 2,4 | 1,4 | E | 2,4 | 1,4 |
| 4 | 0 | 2,4 | 2,4 | 2E | 1,4 | 1,4 |
| 5 | E | 1,4 | 2,4 | E | 1,3 | 1,4 |
| 6 | 0 | 1,3 | 2,4 | 2E | 1,4 | 1,4 |
| 7 | E | 1,3 | 1,4 | E | 1,3 | 1,4 |
| 8 | 0 | 1,3 | 1,3 | 2E | 1,4 | 1,3 |
| 9 | E | 1,4 | 1,3 | E | 1,4 | 1,3 |
The state equations for the single phase inverter are derived based on the operational modes. For State 1 (+0/+E), the dynamics are described by:
$$ \frac{di_L}{dt} = \frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = -\frac{i_L R}{L} $$
Applying stroboscopic mapping, the discrete model for State 1 is:
$$ I_{n+1} = \left( I_n – \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} + \frac{a d_n}{2} $$
where \( a = \frac{E}{L} \), \( \tau = \frac{L}{R} \), \( T_s \) is the switching period, and \( d_n \) is the duty cycle. For State 2 (+E/+2E), the equations become:
$$ \frac{di_L}{dt} = \frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = \frac{2E}{L} – \frac{i_L R}{L} $$
The discrete model for State 2 is:
$$ I_{n+1} = \left( I_n – a + \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} + \left( a – \frac{a d_n}{2} \right) $$
For State 3 (-0/-E), the state equations are:
$$ \frac{di_L}{dt} = -\frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = -\frac{i_L R}{L} $$
leading to the discrete form:
$$ I_{n+1} = \left( I_n + \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} – \frac{a d_n}{2} $$
Finally, for State 4 (-E/-2E):
$$ \frac{di_L}{dt} = -\frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = -\frac{2E}{L} – \frac{i_L R}{L} $$
with the discrete model:
$$ I_{n+1} = \left( I_n + a – \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} + \left( -a + \frac{a d_n}{2} \right) $$
The quasi-PR controller enhances tracking of sinusoidal references by providing high gain at the fundamental frequency. Its transfer function is:
$$ G(s) = K_p + \frac{2K_i \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \( K_p \) is the proportional gain, \( K_i \) is the integral gain, \( \omega_c \) is the cutoff frequency, and \( \omega_0 \) is the resonant frequency. The control signal \( u_{con}(s) = G(s) i_e(s) \), with \( i_e = i_{ref} – i_L \). Defining state variables \( x_1 \) and \( x_2 \), the state-space representation is:
$$ \dot{\mathbf{x}} = A \mathbf{x} + B i_e, \quad u_{con} = C \mathbf{x} + K_p i_e $$
where
$$ A = \begin{bmatrix} 0 & 1 \\ -\omega_0^2 & -2\omega_c \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 2K_i \omega_c \end{bmatrix} $$
Discretizing using sample-hold equivalence yields:
$$ \mathbf{X}_{n+1} = e^{A T_s} \mathbf{X}_n + (e^{A T_s} – I) A^{-1} B i_e(n) $$
$$ u_{con}(n) = C \mathbf{X}_n + K_p i_e(n) $$
where \( I \) is the identity matrix. The reference current is \( i_{ref}(n) = I_m \sin(\omega n T_s) \), and the error is assumed constant over each switching period.
To analyze bifurcation phenomena, we vary the proportional gain \( K_p \) while keeping other parameters fixed, as listed in Table 2. The single phase inverter’s stability is assessed through bifurcation diagrams and fold maps, which reveal transitions from periodic to chaotic behavior.
| Parameter | Symbol | Value |
|---|---|---|
| DC Supply Voltage | E | 200 V |
| Load Resistance | R | 30 Ω |
| Load Inductance | L | 20 mH |
| Reference Current | i_ref | 10 sin(100πt) A |
| Switching Period | T_s | 100 μs |
| Switching Frequency | f_s | 10 kHz |
| Output Frequency | f | 50 Hz |
The bifurcation diagram in Figure 2 plots the peak inductor current against \( K_p \). For \( 0 \leq K_p \leq 0.03 \), the system exhibits multiple distinct values, indicating instability. In the range \( 0.03 \leq K_p \leq 1.05 \), the points converge to a single line, signifying stable single-period operation. At \( K_p = 1.06 \), a period-doubling bifurcation occurs, and for \( 1.05 \leq K_p \leq 2.00 \), scattered points reflect chaotic dynamics. This analysis underscores the critical role of \( K_p \) in maintaining stability for the single phase inverter.
Fold maps further validate these findings. Figure 3(a) for \( K_p = 0.02 \) shows overlapping trajectories, confirming single-period stability. At \( K_p = 0.50 \) (Figure 3(b)), the fold map remains consistent with stable operation. However, at \( K_p = 1.06 \) (Figure 3(c)), the trajectories bifurcate, illustrating period-doubling. These visualizations align with the bifurcation diagram, affirming the accuracy of the discrete model and stability analysis.
The nonlinear dynamics of the single phase inverter are influenced by the interaction between the quasi-PR controller and the switching actions. The Jacobian matrix of the discrete system, derived from the state equations, helps quantify stability boundaries. For instance, the eigenvalues of the Jacobian must lie within the unit circle for asymptotic stability. Numerical computations reveal that as \( K_p \) increases, the dominant eigenvalue approaches unity, triggering bifurcations. This behavior is characteristic of nonlinear systems where parameter variations induce topological changes in phase space.
In practical terms, the single phase inverter’s design must avoid parameter regions prone to chaos. Our study identifies \( K_p \approx 1.05 \) as a critical threshold. Beyond this, the system exhibits subharmonics and increased total harmonic distortion (THD), compromising performance. Simulations in MATLAB/Simulink corroborate the theoretical predictions, showing distorted current waveforms for \( K_p > 1.05 \). Engineers can use these insights to tune quasi-PR controllers effectively, ensuring reliable operation of single phase inverters in applications like traction drives.
Further research could explore the impact of other parameters, such as the integral gain \( K_i \) or load variations, on bifurcation phenomena. Additionally, implementing advanced control strategies like sliding mode or adaptive control might suppress instabilities in wider operating ranges. The methodology presented here serves as a framework for analyzing nonlinearities in various power electronic converters, emphasizing the importance of dynamical systems theory in power electronics.
In conclusion, this work demonstrates the efficacy of stroboscopic mapping and nonlinear analysis tools in studying the stability of a three-level single phase inverter with a quasi-PR controller. By delineating stable operating regions through bifurcation and fold maps, we provide a foundation for robust inverter design. The single phase inverter’s performance is highly sensitive to control parameters, and our findings offer practical guidelines for avoiding chaotic regimes, thereby enhancing system reliability in critical applications.