Problema 42-3    Source:   Problem 4-33 - page 212 - THOMAS, Roland E. , ROSA, Albert J. , THOUSSAINT, Gregory J. -   Book: The Analysis & Design of Linear Circuits - 6ª ed. - Ed. John Willey & Sons, Inc. - 2009.

In the circuit shown in the Figura 42-03.1, determine:

a) Using nodal analysis, compute K or the relation V_o / V_s

b) Select values for the resistors in such a way that K= - 3.

Solution of the Problem 42-3

Item a

We identify the circuit nodes as shown in the figure above. In an ideal operational amplifier, the potential difference between the inverting and non-inverting inputs of the operational is equal to zero. In this way, we conclude that V₁ = 0. Doing the nodal analysis for the node 1:

As V₁ = 0 and rearranging the equation above:

For the node a, we get:

Substituting the penultimate equation in the above equation, after some algebraic work, we obtain the relation between V_o and V_s, which is the gain of the circuit to the closed loop, represented by the letter K.

Item b

In the equation of the gain to the closed mesh, obtained in the previous item, doing the following equality:

If we substitute these values in the gain equation to the closed loop, we obtain the desired gain of K = - 3. As an example, we can do:

This is just a possible example. There are endless solutions.