GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 43 of 65

Marks: +1, -0

For the given circuit, which one of the following is the correct state question?

$\;{ {d}}/{ {dt}}[\table {v} ; {i}]=[\table -4 , 4 ; -2 , -4][\table {v} ; {i}]+[\table 0 , 4 ; 4 , 0][\table {i}_{1} ; {i}_{2}]$
$\;{ {d}}/{ {dt}}[\table {v} ; {i}]=[\table -4 , -4 ; -2 , -4][\table {v} ; {i}]+[\table 4 , 0 ; 0 , 4][\table {i}_{1} ; {i}_{2}]$
$\;{ {d}}/{ {dt}}[\table {v} ; {i}]=[\table 4 , -4 ; -2 , -4][\table {v} ; {i}]+[\table 0 , 4 ; 4 , 4][\table {i}_{1} ; {i}_{2}]$
$\;{ {d}}/{ {dt}}[\table {v} ; {i}]=[\table -4 , -4 ; -2 , 4][\table {v} ; {i}]+[\table 4 , 4 ; 4 , 0][\table {i}_{1} ; {i}_{2}]$

Solution:

KVL in Loop

-0.5 \;{d i}/{d t}-V+(i_{1}-i) 2=0

\;{d i}/{d t}=-2 V+(-4 i)+4 i_{1} . . .(i)

{KCL}

at node

{V}

- {i}+ {C} \;{ {dV}}/{ {dt}}+\;{ {V}}/{1}- {i}_{2}=0

-i+0.25 \;{d V}/{d t}+V-i_{2}=0

0.25 \;{ {dV}}/{ {dt}}=- {V}+ {i}+ {i}_{2}

\;{d V}/{d t}=-4 V+4 i+4 i_{2} \;\; .

.......(ii)
write eqn. (i) and (ii) in matrix from

\;{ {d}}/{ {dt}}[\table {V} ; {i}]=[\table -4 , 4 ; -2 , -4][\table {V} ; {i}]+[\table 0 , 4 ; 4 , 0][\table {i}_{1} ; {i}_{2}]

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