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. This study focuses on the nonlinear dynamics of a 2H-bridge cascaded three-level single phase inverter controlled by a quasi-Proportional-Resonant (quasi-PR) regulator. The single phase inverter is a key component in AC drive systems, and understanding its bifurcation phenomena is essential for ensuring reliable operation. We employ stroboscopic mapping to derive a discrete mathematical model and analyze bifurcation and folding diagrams to identify stable operating regions. This analysis provides insights into the design and tuning of single phase inverter systems for enhanced stability.
The single phase inverter topology examined here consists of two series-connected H-bridges, enabling multi-level output voltages. By comparing the load current \( i_L \) with a reference current \( i_{\text{ref}} \), a modulation signal \( u_{\text{con}} \) is generated through a proportional gain. This signal is then modulated using Carrier Phase Shift-Sinusoidal Pulse Width Modulation (CPS-SPWM) with bipolar triangular waves. The inverter can operate in four distinct states based on the AC-side voltage levels: State 1 (+0 and +E), State 2 (+E and +2E), State 3 (-0 and -E), and State 4 (-E and -2E). Each state involves nine switching modes per switching cycle, with detailed voltage levels and switch states summarized in Table 1.
| Mode | State 1 Voltage | State 1 Bridge 1 Switches | State 1 Bridge 2 Switches | State 2 Voltage | State 2 Bridge 1 Switches | State 2 Bridge 2 Switches | State 3 Voltage | State 3 Bridge 1 Switches | State 3 Bridge 2 Switches | State 4 Voltage | State 4 Bridge 1 Switches | State 4 Bridge 2 Switches |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | E | 1,4 | 1,3 | E | 1,4 | 1,3 | -E | 2,3 | 1,3 | -E | 2,3 | 1,3 |
| 2 | 0 | 2,4 | 1,3 | 2E | 1,4 | 1,4 | 0 | 2,4 | 1,3 | -2E | 2,3 | 2,3 |
| 3 | E | 2,4 | 1,4 | E | 2,4 | 2,4 | -E | 2,4 | 2,4 | -E | 2,4 | 2,4 |
| 4 | 0 | 2,4 | 2,4 | 2E | 1,4 | 1,4 | 0 | 2,4 | 2,4 | -2E | 2,3 | 2,3 |
| 5 | E | 1,4 | 2,4 | E | 1,4 | 1,4 | -E | 1,3 | 2,4 | -E | 1,3 | 2,3 |
| 6 | 0 | 1,3 | 2,4 | 2E | 1,3 | 1,4 | 0 | 1,3 | 2,4 | -2E | 1,3 | 2,3 |
| 7 | E | 1,3 | 1,4 | E | 1,3 | 1,4 | -E | 1,3 | 2,3 | -E | 1,3 | 2,3 |
| 8 | 0 | 1,3 | 1,3 | 2E | 1,4 | 1,4 | 0 | 1,3 | 1,3 | -2E | 2,3 | 2,3 |
| 9 | E | 1,4 | 1,3 | E | 1,4 | 1,3 | -E | 2,3 | 1,3 | -E | 2,3 | 1,3 |
The state equations for each operating state of the single phase inverter are derived based on the circuit dynamics. For State 1, the equations are:
$$ \frac{di_L}{dt} = \frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = -\frac{i_L R}{L} $$
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 mode \( n \). For State 2, the equations become:
$$ \frac{di_L}{dt} = \frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = \frac{2E}{L} – \frac{i_L R}{L} $$
with the discrete model:
$$ I_{n+1} = \left( I_n – a + \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} + \left( a – \frac{a d_n}{2} \right) $$
For State 3, the 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 model:
$$ I_{n+1} = \left( I_n + \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} – \frac{a d_n}{2} $$
Finally, for State 4:
$$ \frac{di_L}{dt} = -\frac{E}{L} – \frac{i_L R}{L} \quad \text{and} \quad \frac{di_L}{dt} = -\frac{2E}{L} – \frac{i_L R}{L} $$
with the discrete model:
$$ I_{n+1} = \left( I_n + a – \frac{a d_n}{2} \right) e^{-\frac{T_s}{\tau}} + \left( -a + \frac{a d_n}{2} \right) $$
These discrete models form the basis for analyzing the nonlinear behavior of the single phase inverter. The parameters used in this study are listed in Table 2, which are essential for simulations and bifurcation analysis.
| Parameter | Value |
|---|---|
| DC Supply Voltage \( E \) (V) | 200 |
| Load Resistance \( R \) (Ω) | 30 |
| Load Inductance \( L \) (mH) | 20 |
| Reference Current \( i_{\text{ref}} \) (A) | 10 sin(100πt) |
| Switching Period \( T_s \) (μs) | 100 |
| Switching Frequency \( f_s \) (kHz) | 10 |
| Output Frequency \( f \) (Hz) | 50 |
The quasi-PR regulator plays a crucial role in controlling the single phase inverter. Its transfer function is given by:
$$ G(s) = K_p + \frac{2K_i \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
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 modulation signal \( u_{\text{con}}(s) = G(s) i_e(s) \), with \( i_e(s) \) being the error signal. Introducing state variables \( x_1 \) and \( x_2 \), and defining the state vector \( \mathbf{X} = [x_1, x_2]^T \), the state-space representation is:
$$ \dot{\mathbf{X}} = \mathbf{A} \mathbf{X} + \mathbf{B} i_e $$
$$ u_{\text{con}} = \mathbf{C} \mathbf{X} + K_p i_e $$
where
$$ \mathbf{A} = \begin{bmatrix} 0 & 1 \\ -\omega_0^2 & -2\omega_c \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad \mathbf{C} = \begin{bmatrix} 0 & 2K_i \omega_c \end{bmatrix} $$
The discrete-time model for the quasi-PR regulator, assuming constant error signal over a switching period, is:
$$ \mathbf{X}_{n+1} = e^{\mathbf{A} T_s} \mathbf{X}_n + (e^{\mathbf{A} T_s} – \mathbf{I}) \mathbf{A}^{-1} \mathbf{B} i_e(n) $$
$$ u_{\text{con}}(n) = \mathbf{C} \mathbf{X}_n + K_p i_e(n) $$
where \( \mathbf{I} \) is the identity matrix, and \( i_e(n) = i_{\text{ref}}(n) – i_L(n) \) with \( i_{\text{ref}}(n) = I_m \sin(\omega n T_s) \). This model integrates with the inverter dynamics to analyze system stability.
To investigate the bifurcation phenomena in the single phase inverter, we utilize bifurcation and folding diagrams. The bifurcation diagram is generated by varying the proportional gain \( K_p \) and plotting the peak values of the AC-side inductor current after the system reaches steady state. For each \( K_p \), we sample 50 consecutive peaks of the output current waveform. The resulting bifurcation diagram, shown conceptually, reveals the stability boundaries of the single phase inverter system. When \( K_p \) ranges from 0 to 0.03, the system exhibits multiple distinct current values, indicating instability. For \( 0.03 \leq K_p \leq 1.05 \), the current peaks converge to a single value, signifying stable single-period operation. At \( K_p = 1.06 \), the system undergoes period-doubling bifurcation, where the peaks split into two distinct values. Beyond \( K_p = 1.05 \), up to \( K_p = 2.00 \), the current peaks disperse over a range, indicating chaotic or multi-periodic behavior.
Folding diagrams provide further validation of the nonlinear dynamics. By iterating the discrete model and folding all sampling points over multiple stable cycles, we observe the system’s behavior at specific \( K_p \) values. For instance, at \( K_p = 0.02 \), the folding diagram shows a single cluster of points, confirming single-period stability. At \( K_p = 0.50 \), the points remain clustered, indicating sustained stability. However, at \( K_p = 1.06 \), the points split into two clusters, verifying the period-doubling bifurcation. These results align with the bifurcation analysis and demonstrate the accuracy of the discrete model for the single phase inverter.
The analysis highlights the sensitivity of the single phase inverter to control parameters. The quasi-PR regulator, while effective for tracking AC references, can introduce nonlinearities that lead to bifurcations. By understanding these phenomena, designers can optimize \( K_p \) and other parameters to avoid unstable regions. For example, maintaining \( K_p \) between 0.03 and 1.05 ensures stable operation of the single phase inverter, which is crucial for applications like electric trains where reliability is paramount.
In conclusion, this study comprehensively analyzes the bifurcation behavior of a three-level single phase inverter with a quasi-PR controller. Through stroboscopic mapping, we derived discrete models for various operating states and integrated them with the regulator dynamics. Bifurcation and folding diagrams revealed that the single phase inverter’s stability is highly dependent on the proportional gain \( K_p \), with stable operation occurring in a specific range. These findings provide a foundation for designing robust single phase inverter systems, ensuring stable performance in practical applications. Future work could explore the effects of other parameters, such as the integral gain \( K_i \) or load variations, on the nonlinear dynamics of single phase inverters.