Problem + Hard 47-2.    Source:   Exam Question for Electronic Engineer - Petrobras - 2012.

Resistive transducers are used in many industrial control systems. One way to measure the electrical resistance of a resistive transducer R_S is illustrated in the circuit shown in the Figure 47-02.1, where we consider the amplifiers ideal operations. Determine the value of electrical resistance R_S of the transducer through from a mathematical expression that is a function of the values ​​of the resistors that make up the circuit and the output voltage V_o.

Solution of the Problem 47-2

In the circuit shown in the Figure 47-02.1, names were given to the nodes and in the input circuit we calculated the Thevenin equivalent. Knowing that the input current of the ideal operational amplifier is zero and that the input voltages of OP 1 are virtually equal, we can find a relationship between V_b and V_S. Using the nodal voltage method we have:

And from there, we get the desired relation, or:

Let's save this relationship because we will need it later. On the other hand, we can find a relation between V_a and V_x making the input mesh of OP 1. Note that the current passing through the feedback resistor (2 R₁) is the same current flowing through R₁ and the input source. Then, we can write:

We can eliminate R₁ and after some algebraic manipulation, we find:

Note that we used the eq. 47-2a to arrive at eq. 47-2b. Since we have two variables, we need one more equation that relates V_x and V_S to solve the system. From the circuit we can easily see that i_x = i_b + i_S. Thus, We can relate the two variables of interest as follows:

Where i_b = V_S / 2 R₁ and i_S = V_S / R_S. And after some algebraic manipulation we arrive at the desired partial result, or:

Note that we now have the eq. 47-2b and the eq. 47-2c relating the two variables of interest. So, equating the two equations we arrive at:

And once again doing proper algebraic work we find the relation:

Note that so far we have only worked with OP 1. At this point, how do we need to relate R_S depending on V_o and other components, let's direct our attention to OP 2. Note that the same is in the non-inverting configuration. Therefore, the following relation between V_o and V_S holds:

And since we need the value of V_S we can transform the above equation into:

Substituting this value in eq. 47-2d we will obtain the relation requested by the problem statement.