Problem 43-2.    Source:   Exercicie 4.19 - page 181 -   CARLSON, A. Bruce -    Book: Engineering Concepts and Analysis of Linear Eletric Circuits - Ed. Preliminary - Ed. John Wiley & Sons, Inc. - 1996.

In the circuit of the Figura 43-02.1, make    R₁ + R₂ = R₄,   R₁ = R₄ / K₁    and    R₃ = R₄ / K₂. Assume an ideal operational amplifier to show that:    V_o / V_i   =  K₁ (K₂ + 2)

Solution of the Problem 43-2

Assuming that the operational amplifier is ideal, then the input current of it is null. In addition, the input voltage V_i = V_a. By applying a resistive voltage divider we can calculate V_a in function of V₁, that is:

However, we need a relationship between V₁ and V_i (or V_a). Making an algebraic rearrangement:

On the other hand, it is also easy to find V₁ in function of V_o applying a resistive voltage divider. For this, we must take into account the series circuit between R₁ and R₂, which in turn is in parallel with R₃. All this circuit is under the action of V₁. So we can write that:

At the moment there are two equations that are related to each other. Thus, by equating these two equations and working algebraically, we obtain:

From the problem statement there are the following relations:    R₁ + R₂ = R₄,   R₁ = R₄ / K₁    and    R₃ = R₄ / K₂.

Making the necessary substitutions, we find the following equation.

After an algebraic work, it is possible to eliminate the term R₄ and find a relationship with K₁ and K₂, or:

A last algebraic arrangement and we get the result intended by the problem, that is: