Consider the network shown in the following figure. The state equation of the system will be

 

11) Consider the network shown in the following figure. The state equation of the system will be:

Answer: A

Solution (State Equations)

In state-space analysis of circuits, we typically choose the voltage across the capacitor (V_c) and the current through the inductor (i_L) as our state variables.

Step 1: Apply KCL at the top node

The input current I splits into the capacitor current (i_c) and the inductor branch current (i_L):

I = i_c + i_L
Since i_c = C(dV_c/dt):
I = C(dV_c/dt) + i_L

Rearranging for the derivative of the first state variable:

dV_c/dt = (0)V_c - (1/C)i_L + (1/C)I   --- (Eq. 1)

Step 2: Apply KVL to the right-hand branch

The voltage across the capacitor (V_c) is equal to the voltage across the series combination of L and R:

V_c = V_L + V_R
Since V_L = L(di_L/dt) and V_R = i_LR:
V_c = L(di_L/dt) + i_LR

Rearranging for the derivative of the second state variable:

di_L/dt = (1/L)V_c - (R/L)i_L + (0)I   --- (Eq. 2)

Step 3: Convert to Matrix Form (State-Space)

The standard form is ẋ = Ax + Bu. Combining Eq. 1 and Eq. 2:

d/dt

V_c

i_L

=

0

-1/C

1/L

-R/L

V_c

i_L

+

1/C

0

[I]

Conclusion

Comparing our derived matrices to the given choices, the values in the A matrix [0, -1/C; 1/L, -R/L] and the B matrix [1/C; 0] match perfectly with Option 1.

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