Problem 45-2    Source:  Problem 4-34 - page 213 -    THOMAS, Roland E. , ROSA, Albert J. , THOUSSAINT, Gregory J. -   Book: The Analysis & Design of Linear Circuits - 6ª Edição - Ed. John Willey & Sons, Inc. - 2009.

Use the nodal analysis method to show that in the circuit of the Figure 45-02.1 i_o = - V_s / 2 R , independent of the load R_L. In other words, show that the circuit of the figure below is a current source voltage controlled (V_s).

Solution of the Problem 45-2

Using the ideal model of an operational amplifier, it is known that no electric current circulates in the inputs of the amplifier. Then, based on the figure above, we can write to the left side of the circuit the equation:

Working algebraically this equation and rewriting it:

On the other hand, on the right side of the circuit R_L is in parallel with 2 R. The result of this parallel is shown in the equation below:

Using this result, the equation for the right side of the circuit results.

Not forgetting that for an ideal operational amplifier, we have V_a = V_b.Thus, taking this into consideration and replacing in the previous equation, R_eq by the value of the parallel and by simplifying similar terms, we obtain:

Note that, the equations eq. 45-02.1 and eq. 45-02.2 above, express V_o in function of V_a e V_s. Then, equating these equations:

We can cancel the terms (3/2) V_a. So expressing V_a in function of V_s:

In possession of the value of V_a, and remembering again that V_a = V_b, we can write i_L as:

Replacing the value of V_a nthis equation and carrying out the appropriate simplifications, the desired result is achieved, or:

Therefore, it is clear that the value of i_L is totally independent of the value of R_L. In this way, i_L depends only on the input voltage V_s and the value of resistor R. Thus, with the choice of these values there is how to design a source of constant current controlled by voltage with the desired value, provided that the limitations of the operational amplifier used in the design are obeyed.