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GATE 2026 EC – Question 23

Networks, Signals and Systems · Time and Frequency Domain Analysis of Linear Circuits · 1 mark · Multiple choice

The relation between the input current ($I$) and the output voltage ($V$) of a circuit is governed by the equation: $C\frac{dV}{dt}=I(t)-m(t)$. The circuit is excited by $I(t)=q\delta(t)$, where $q$ is a real valued constant. $V$ at $t=0^-$ is $V_0$. Which of the following is an equivalent representation of the above case?

  1. $C\frac{dV}{dt}=-m(t)$, with $V(t=0^-)=V_0+q/C$
  2. $C\frac{dV}{dt}=-m(t)$, with $V(t=0^-)=V_0+q/C+m(t=0^-)$
  3. $C\frac{dV}{dt}=-m(t)$, with $V(t=0^-)=V_0-q/C+m(t=0^-)$
  4. $C\frac{dV}{dt}=-m(t)$, with $V(t=0^-)=q/C$

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Correct answer: (A) $C\frac{dV}{dt}=-m(t)$, with $V(t=0^-)=V_0+q/C$

Explanation

The impulse $q\delta(t)$ integrates to a voltage jump of $q/C$ across the capacitor. It can therefore be replaced by an initial condition $V_0+q/C$ with only $-m(t)$ as the forcing term.