Problem 45-1    Source:  Problem 4-29 - page 212 -   THOMAS, Roland E. , ROSA, Albert J. , THOUSSAINT, Gregory J. -    Book: The Analysis & Design of Linear Circuits - 6ª Edição - Ed. John Willey & Sons, Inc. - 2009.

Suppose that in circuit show in Figure 45-01.1, S key is initially open.

a) Find V_o in terms of V₁ and V₂.

b) Repeat the calculation for the closed S key.

Solution of the Problem 45-1

Item a

In this circuit there is how to do a "source explosion" in the input circuit, because the two voltage sources are in series and can be separated, as can be seen in the Figure 45-01.2. Of course, the new value of the voltage source at the negative input is V'₁ = V₁ + V₂ and in the positive entry only V₂.

Notice that the circuit has been transformed and remained exactly like the circuit we have in the theoretical part of this chapter. Therefore, it is already known which equation determines the output voltage of the circuit shown above. See below the equation:

Comparing with the circuit, the values of the resistances and the voltages of the sources are:

R₁ = 30 kΩ  -  R₂ = 60 kΩ   -   R₃ = 10 kΩ   -  R₄ = 20 kΩ  -   V'₁ = V₁ + V₂   and   V₂ = V₂

Making the substitution by the respective numerical values, we will find the equation that determines the output voltage V_o in function of V₁ and V₂, when the key S it's open.

The meaning of this equation relates to the V₂, which is present at both the inverter and the non-inverting inputs, and results in its overriding, because this is the function of a differential amplifier: reduce zero all the voltages that are in phase when they appear simultaneously in the two inputs (inverter and non-inverter). Therefore, this circuit acts as a common inverter circuit, as if it had only V₁ feeding its input.

Practical Addendum

This differential configuration is widely used in professional sound desks, especially at the input to microphone. This entry is known as balanced input. Microphones have long cables which connect the microphone to the sound table. Along the way there is noise induction, even if they use shielded cables in order to reduce them. However, as noise induction occurs at both inputs, theoretically with the same intensity and same phase, noise upon reaching the input of the differential amplifier is neutralized, ensuring that the sound table can amplify the original noise-free signal.

Item b

In this item, the relationship between the output voltage and the input voltages in the case of the closed S will be determined. Keep in mind that when the key S is closed, the positive input of the operational amplifier is grounded. Thus, the circuit becomes an inverter circuit. The resistance of 10 kΩ is in parallel with the voltage source V₁. However, this in no way affects the operation of the circuit and can be discarded. Therefore, the input voltage is V₁ + V₂. And for an inverter circuit, the output voltage is given by:

In this case:

R_i = 30 kΩ   -   R_f = 60 kΩ. Then, making the substitution by the respective numerical values and remembering that V_i = V₁ + V₂:

In other words, the signal that appears on the output is the input signal, (V₁ + V₂) amplified by a factor 2 and inverted.