State Space Representation of the Circuit
Step 1: Choose State Variables
We select the following state variables:
- i(t): Current through the inductor (3 H).
- v(t): Voltage across the capacitor (3/2 F).
Step 2: Apply Kirchhoff’s Voltage Law (KVL)
Apply KVL to the left loop, which includes the voltage source (36 V), the resistor (12 Ω), the inductor (3 H), and the voltage across the capacitor (v(t)). The equation is:
36 - 12i - 3(di/dt) - v = 0
Rearranging for di/dt:
di/dt = (36 - 12i - v) / 3
Step 3: Apply Kirchhoff’s Current Law (KCL)
At the node where the capacitor (3/2 F) and the 6 Ω resistor are connected, the total current entering the node is:
i = C * (dv/dt) + (v / 6)
Rearranging for dv/dt:
dv/dt = (1 / C) * (i - v / 6)
Since C = 3/2, substitute the value:
dv/dt = (2 / 3) * (i - v / 6)
Step 4: Write the State Equations
The two state equations derived are:
-
di/dt = (36 - 12i - v) / 3 -
dv/dt = (2 / 3) * (i - v / 6)
Step 5: Matrix Representation
Write the equations in matrix form:
[di/dt] = [-12/3 -1/3] * [i] + [36/3]
[dv/dt] [ 2/3 -1/9] [v] [ 0 ]
Simplify the coefficients:
[di/dt] = [-4 -1/3] * [i] + [12]
[dv/dt] [2/3 -1/9] [v] [ 0 ]
Final Matrix Form
The state-space representation is:
d/dt [i] = [-4 -1/3] * [i] + [12]
[v] [2/3 -1/9] [v] [ 0 ]
If you need further clarification, feel free to ask!
Best regards,
Md. Anisur Rahman