Question

Derive an expression for the output voltage of the following circuit:

Short answer

The expression for output voltage of the given circuit is,

$v_o=(-20v_1-5v_2)t$

Step-by-step solution

Step 1. Given Information

Given an integrator circuit with two input signals.

We are asked to find the expression for output voltage.

Step 2. Explanation

The circuit has two input signals $v_1$and $v_2$. So the corresponding currents are $i_1$and $i_2$.

We know that there is no input current in the circuit.

Let $i_f$be the feedback current.

$i_1+i_2=i_f$

But, $i_1=\frac{v_1}{R_1}=\frac{v_1}{5M\Omega }$

Also, $i_2=\frac{v_2}{R_2}=\frac{v_2}{20M\Omega }$

The feedback current is given by;

$i_f=-C\frac{d{v}_0}{dt}=-0.010\frac{d{v}_0}{dt}$

Step 3. Substitution

Substituting the values of $v_1,v_2,v_f$in the equation;

$v_1+v_2=v_f$

$\frac{v_1}{5}+\frac{v_2}{20}=-0.010\frac{d{v}_0}{dt}$

$\frac{d{v}_0}{dt}=-20v_1-5v_2$

$\Rightarrow v_o={\int }_0^t(-20v_1-5v_2)dt$

localid="1645771581722" $\Rightarrow v_o=(-20v_1-5v_2)t$