Answer :
The state space averaged model is a mathematical representation used to analyze the behavior and control of power electronic converters such as the Buck converter and Boost converter.
1. Buck Converter:
The Buck converter is a step-down DC-DC converter that efficiently converts a higher input voltage to a lower output voltage. It consists of a power switch (MOSFET or BJT), an inductor (L), a diode (D), a capacitor (C), and a load resistor (RL). Let's derive the state space averaged model for the Buck converter.
a. Derivation:
The state space averaged model is derived by considering the average behavior of the circuit over a switching period. The key steps involved in the derivation are as follows:
Step 1: Define State Variables:
We define two state variables for the Buck converter:
- Inductor current: iL
- Capacitor voltage: vC
Step 2: State Equations:
The state equations describe the dynamics of the system in terms of the state variables and their derivatives.
- Inductor current equation:
Using Kirchhoff's voltage law, the equation for the inductor current can be derived as:
diL/dt = (v_in - vC)/L - (1 - D) x v_out / L
- Capacitor voltage equation:
Using the capacitor equation, the equation for the capacitor voltage can be derived as:
C x dvC/dt = (1 - D) x v_out - iL x RL
Step 3: Output Equation:
The output equation relates the output voltage to the state variables.
v_out = vC
Step 4: State Space Formulation:
By expressing the state equations in matrix form, we can obtain the state space averaged model for the Buck converter:
dx/dt = Ax + Bu
y = Cx + Du
Where:
- x = [iL, vC]T is the state vector.
- u = D is the duty cycle.
- y = v_out is the output voltage.
The matrices A, B, C, and D can be obtained by rearranging the state equations and output equation.
2. Boost Converter:
The Boost converter is a step-up DC-DC converter that converts a lower input voltage to a higher output voltage. It consists of a power switch (MOSFET or BJT), an inductor (L), a diode (D), a capacitor (C), and a load resistor (RL). Let's derive the state space averaged model for the Boost converter.
a. Derivation:
Similar to the Buck converter, the state space averaged model for the Boost converter can be derived using the same steps.
Step 1: Define State Variables:
- Inductor current: iL
- Capacitor voltage: vC
Step 2: State Equations:
- Inductor current equation:
diL/dt = (v_in - vC)/L - D x v_out / L
- Capacitor voltage equation:
C x dvC/dt = D x v_out - (1 - D) x iL x RL
Step 3: Output Equation:
v_out = D x vC
Step 4: State Space Formulation:
The state space averaged model for the Boost converter can be obtained in the same matrix form as the Buck converter:
dx/dt = Ax + Bu
y = Cx + Du
Where:
- x = [iL, vC]T is the state vector.
- u = D is the duty cycle.
- y = v_out is the output voltage.
To know more about boost converter, refer:
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