Problem 53-4    Source:  Problem developed by the site author.

In the Figure 53-4.1 we have V = 2 sin (500 t - 30°). Determine the value of V_o.

Solution of the Problem 53-4

Initially we must calculate the equivalent circuit Thévenin for the point V_x, as show in the Figure 53-4.1. As we know, the inverter input of OP 1 does not circulate current. Therefore, we can calculate the voltage at the input of the circuit using a voltage divider. Thus, we have:

Notice that in the transformation was used - j5 = 5 ∠-90° in the above equation. Replacing V by its numerical value (in the phasor form) and performing the calculation, we obtain:

To determine the impedance of Thévenin, we shorted the voltage source V. Therefore, it turns out that the resistor of 5 ohms is in parallel with the capacitor - j5. Doing calculating this parallel we find:

So, we have the circuit shown in the Figure 53-4.2.

We clearly have an operational amplifier in the inverter configuration, and we know that the gain of this circuit, adapted to AC, is given by:

Where   Z_i = 2.5 - j2.5  and  Z_f = 5 - j5.   Therefore, replacing these values ​​in the above equation, we find:

Notice in the equation above that to eliminate the negative sign, we add the angle of 180° the result. Then, as we know the value of V_th = √2 ∠ -75°, we can calculate the value of V_o, or:

Substituting for the values ​​already found, we obtain:

It is also possible to write the final result in trigonometric form. See below.