Problem 44-1    Source:  Problem 4-26 - page 211 -   THOMAS, Roland E. , ROSA, Albert J. , THOUSSAINT, Gregory J. -   Book: The Analysis & Design of Linear Circuits - 6ª Edição - Ed. John Willey & Sons, Inc. - 2009.

a) Find V_o in terms of the inputs V₁ and V₂ of the circuit in the Figure 44-01.1.

b) If V₁ = 1 V what is the range of values V₂ can have without saturing the Opamp?

Solution of the Problem 44-1   

Item a   

The output voltage of an addition amplifier is given by the eq. 44-01 shown in below.

Comparing the equation with the circuit we easily conclude that:

Also, taking into account that by the equation, V_a = V₁ and V_b = V₂, we can write the equation that determines the voltage of circuit output, that is:

Item b   

It is understood by saturation of the output of an operational amplifier the value of the voltage of limit input where the output assumes the maximum value the value of the supply voltage of the circuit. From this value, increasing even more the input voltage, there will be no proportional increase in the output voltage. Therefore, if Vcc = ± 15 V the maximum output voltage will be ± 15 V.

This item of the problem admits two solutions: one is when the output reaches the limit voltage of - 15 volts; other, it is when the output voltage reaches the limit value of + 15 volts.

1ª Solution   

In this way, if V₁ = 1 V, and supposing V₂ = 0, we get in the output a voltage of:

So to reach the voltage of - 15 volts, which is the maximum voltage that the output can take, a contribution must be made by V₂ of:

Therefore, making V₂ ≤ 6 volts the objective of the problem is reached.

2ª Solution   

Note that in the previous solution a positive voltage was used to V₂. However, it is possible to place a negative voltage on V₂. Therefore, its value will be:

Therefore, the other value that V₂ can assume is V₂ ≥ - 9 volts.

Attention   

In order to obtain the complete solution of the problem, one must join the two solutions found. So, the correct answer is: