Mathematical Modelling of Z-source Inverter Using State-Space Model

Abstract

In this lab report, we investigate the Z-source inverter under two states: shoot-through state and non-shoot-through state. The Z-source inverter is analyzed using state space equations and small signal mathematical models. We explore the average large signal model and derive transfer functions for the dynamic response of the system. The equations are presented and analyzed to understand the behavior of the Z-source inverter.

Introduction

The Z-source inverter is a crucial component in power electronics, used to convert direct current (DC) into alternating current (AC).

It is known for its ability to boost the voltage levels in the circuit, making it valuable in various applications. This lab report focuses on understanding the behavior of the Z-source inverter in both shoot-through and non-shoot-through states. State space equations, small signal models, and transfer functions will be derived and analyzed to gain insights into its dynamic response.

Methods

We begin by considering two states of the Z-source inverter: shoot-through state and non-shoot-through state. The shoot-through state is characterized by a duty cycle (D), while the non-shoot-through state is characterized by ((1-D)) duty cycle.

State Space Equations

The state space equations for the shoot-through state are represented as follows:

[ dot{x} = a_1 cdot x + b_1 cdot u ]

[ y = c_1 cdot x + d_1 cdot u ]

Similarly, for state 2 (non-shoot-through state):

[ dot{x} = a_2 cdot x + b_2 cdot u ]

[ y = c_2 cdot x + d_2 cdot u ]

Average Large Signal Model

We obtain the average large signal model of the Z-source system as follows:

[ dot{x} = A cdot x + B cdot u ]

[ y = C cdot x + D cdot u ]

Where

[ A = a_1 cdot D + a_2 cdot (1-D) ]

[ B = b_1 cdot D + b_2 cdot (1-D) ]

[ C = c_1 cdot D + c_2 cdot (1-D) ]

[ D = d_1 cdot D + d_2 cdot (1-D) ]

These equations involve weighted averages based on the duty cycle (D).

Small Signal Mathematical Model

To study the dynamic response of the Z-source inverter, we establish a small signal mathematical model through perturbation of variables. We use equation 3 to achieve this:

[ X + hat{x} = [a_1 cdot (D + hat{d}) + a_2 cdot (1 - (D + hat{d})) cdot (X + hat{x})] + [b_1 cdot (D + hat{d}) + b_2 cdot (1 - (D + hat{d})) cdot (U + hat{u})] ]

Here, (hat{d}) is a perturbation of variable (D).

Assumptions

Several assumptions are made in this analysis:

1. The product of small signal values is negligible compared to the steady-state values ((x̂/X⋅d̂/D ≪ 1))
2. dx/dt = 0

Using these assumptions, we simplify equation (dot{x} = A cdot x + B cdot u = 0) to:

[ dot{x̂} = bar{A} cdot x̂ + bar{B} cdot û + [(a_1 - a_2) cdot x + (b_1 - b_2) cdot u] cdot d̂ ]

Where (bar{A}) represents (a_1 cdot D + a_2 cdot (1-D)) and (bar{B}) represents (b_1 cdot D + b_2 cdot (1-D)).

The output equations for small signal become:

[ ŷ = bar{C} cdot x̂ + bar{D} cdot û + [(c_1 - c_2) cdot x + (d_1 - d_2) cdot u] cdot d̂ ]

Results

Let's analyze the circuit during the shoot-through state. In this state, switch S1 is in the ON state. Applying Kirchhoff's voltage law (KVL) to the circuit, we can derive the following equations:

[ L_1 cdot frac{dI_{L1}}{dt} - V_{c1} = 0 quad text{(9)}

[ L_2 cdot frac{dI_{L2}}{dt} - V_{c2} = 0 quad text{(10)}

[ C_1 cdot frac{dV_{c1}}{dt} - I_{L1} = 0 quad text{(11)}

[ C_2 cdot frac{dV_{c2}}{dt} - I_{L2} = 0 quad text{(12)}

Using these equations, we can form the state matrix for the shoot-through state.

[ begin{bmatrix} dot{I}_{L1} \ dot{I}_{L2} \ dot{V}_{c1} \ dot{V}_{c2} end{bmatrix} = begin{bmatrix} 0 & 0 & frac{1}{L_1} & 0 \ 0 & 0 & 0 & frac{1}{L_2} \ 0 & -1 & 0 & 0 \ 0 & 0 & 0 & -1 end{bmatrix} begin{bmatrix} I_{L1} \ I_{L2} \ V_{c1} \ V_{c2} end{bmatrix} quad text{(13)}

During the non-shoot-through state (when switch S1 is in the OFF state), the following KVL and KCL equations apply:

[ V_{dc} = L_1 cdot frac{dI_{L1}}{dt} + V_{c2} quad text{(14)}

[ V_{dc} = L_2 cdot frac{dI_{L2}}{dt} + V_{c1} quad text{(15)}

[ C_1 cdot frac{dV_{c1}}{dt} - I_{L2} + I_{dc} = 0 quad text{(16)}

[ -C_2 cdot frac{dV_{c2}}{dt} + I_{L1} - I_{dc} = 0 quad text{(17)}

Now, we can form the state matrix for the non-shoot-through state.

[ begin{bmatrix} dot{I}_{L1} \ dot{I}_{L2} \ dot{V}_{c1} \ dot{V}_{c2} end{bmatrix} = begin{bmatrix} 0 & 0 & frac{1}{L_1} & 0 \ 0 & 0 & 0 & frac{1}{L_2} \ 0 & -1 & 0 & 0 \ 0 & 0 & 0 & -1 end{bmatrix} begin{bmatrix} I_{L1} \ I_{L2} \ V_{c1} \ V_{c2} end{bmatrix} quad text{(18)}

The state equations presented above are averaged while considering the shoot-through duty ratio (D), which represents the ON state of switch S1.

Averaging the two matrices, we obtain:

[ x(t) = D(a_1x + b_1u) + (1-D)(a_2x + b_2u) ]

The state space model for this equation is:

[ dot{x} = A cdot x + B cdot u ]

Since this equation contains non-linear terms, we perform a small signal analysis by introducing perturbations in the first equation for variables:

[ Lleft(frac{d(I_{L1} + î_{L2})}{dt}right) = (D + d̂)(V_{C1} + v̂_{C1}) - (D' - d̂)(V_{C2} + v̂_{C2}) + (D' - d̂)(V_{dc} + v̂_{dc}) quad text{(20)}

Where (D' = (1-D))

By neglecting higher-order terms and considering AC terms, we get:

[ Lleft(frac{dî_{L}}{dt}right) = 2d̂V_C + (2D-1)V̂_C + (D'V̂_{dc} - d̂v̂_{dc}) quad text{(22)}

Similarly for voltage (V_C):

[ Cleft(frac{dV̂_C}{dt}right) = -2d̂V_C + (2D-1)V̂_C + (D'V̂_{dc} - d̂V̂_{dc}) quad text{(23)}

Now, we take the Laplace transform of equation (22) to obtain:

[ SLî_L(s) = 2d̂V̂_C(s) + (2D-1)V̂_C(s) + D'V̂_{dc}(s) + V_{dc}d̂(s) quad text{(24)}

Similarly, for equation (23):

[ SCV̂_C(s) = -2I_Ld̂(s) + (1-2D)î_L(s) - D'Î_dc(s) + I_{dc}d̂(s) quad text{(25)}

Repositioning these equations into matrices, we get:

[ begin{bmatrix} î_L(s) \ V̂_C(s) end{bmatrix} = frac{1}{K} begin{bmatrix} SC & (2D-1) \ -(2D-1) & SL end{bmatrix} begin{bmatrix} 2V_C + V_{dc} \ I_{dc} - 2I_L end{bmatrix}d̂(s) + frac{D'}{K} begin{bmatrix} SC & (2D-1) \ -(2D-1) & SL end{bmatrix} begin{bmatrix} V̂_{dc}(s) \ -î_{dc}(s) end{bmatrix} quad text{(28)}

Where (K = S^2LC + (2D-1)^2)

Finally, the transfer functions for (V_C(s)) and (î_L(s)) can be derived as:

[ frac{V_C(s)}{d̂(s)} = frac{(2V_C + V_{dc})(1-2D) - SL(2I_L - I_{dc})}{S^2LC + (2D-1)^2} quad text{(30)}

[ frac{î_L(s)}{d̂(s)} = frac{(2V_C + V_{dc}) - (2D-1)(2I_L - I_{dc})}{S^2LC + (2D-1)^2} quad text{(31)}

Discussion

The analysis of the Z-source inverter in both shoot-through and non-shoot-through states provides valuable insights into its behavior. The derived state space equations, small signal models, and transfer functions allow us to understand how the system responds to small disturbances and variations.

One important aspect to note is the influence of the duty cycle (D) on the system's behavior. In the shoot-through state, when (D) is high, the equations reflect the behavior of the inverter during that state. Conversely, in the non-shoot-through state, when (D) is low, the equations represent the inverter's response under those conditions.

The transfer functions ((V_C(s)/d̂(s)) and (î_L(s)/d̂(s))) are essential for analyzing the dynamic response of the system to perturbations. These transfer functions allow us to understand how changes in input variables, such as (d̂(s)) (perturbations), affect the output variables, (V_C(s)) and (î_L(s)), in the frequency domain.

In equation (30), the transfer function (V_C(s)/d̂(s)) represents the relationship between the perturbation in the DC link voltage ((d̂(s))) and the output voltage (V_C(s)) across the capacitor. This transfer function depends on various parameters, including the duty cycle (D), inductance ((L)), and capacitance ((C)) values. It describes how changes in the input perturbation propagate to the output voltage and how the system's behavior is influenced by these parameters.

Similarly, in equation (31), the transfer function (î_L(s)/d̂(s)) shows the relationship between the perturbation in the DC link current ((d̂(s))) and the inductor current (î_L(s)). This transfer function also depends on the duty cycle (D) and the inductance ((L)) values. It provides insights into how variations in the input perturbation affect the inductor current and, consequently, the system's performance.

Understanding these transfer functions is crucial for designing and controlling the Z-source inverter. Engineers and researchers can use this information to optimize the inverter's performance, ensure stability, and meet specific application requirements. By analyzing the small signal behavior of the system, it becomes possible to fine-tune the control strategies and enhance the overall efficiency and reliability of the Z-source inverter.

Conclusion

This lab report has explored the behavior of the Z-source inverter under two important states: shoot-through and non-shoot-through. By deriving state space equations, small signal models, and transfer functions, we have gained valuable insights into how this power electronics component responds to perturbations and variations in its input variables. The duty cycle (D) plays a significant role in determining the system's behavior, and understanding these dynamics is essential for designing and optimizing Z-source inverters for various applications.