Problem 47-7    Source:  Adapted from the 2nd question on the list of Elementary Electronics I - Ufrgs - 2019.

For the circuit shown in the Figura 47-07.1, determine the value of N so that the circuit has a predominance of negative feedback. Afterwards, consider the value of N so that the circuit presents a predominance of positive feedback.

Solution of the Problem 47-7   

To solve this problem we will use the nodal voltage technique. Remembering that for ideal operational amplifiers, the following list applies:

Now using the nodal method for the point a, we get:

After some algebraic manipulation in the equation above, the following relation is obtained:

For the point x, we can write:

Replacing eq. 47-07.2 in the equation above and, after some manipulation algebraic, we get:

Now, for the point b, we can write:

Remembering the equalities in eq. 47-07.1 and substituting it in the equation above, we get:

And finally for point 2, we can write:

Again, remembering the equalities in eq. 47-07.1 and substituting it in the equation above, we get:

To obtain the circuit gain, just divide eq. 47-07.5 by eq. 47-07.4, that is:

Analyzing the above equation, it is easily concluded that to obtain a negative predominance, we must satisfy the condition below:

And to obtain a positive predominance, it is enough to satisfy the condition below:

Note that, N cannot take on a value less than zero, as this would imply a passive resistor with a value negative, which is physically impossible.