Problem + Hard 53-2    Source:   exercise 8-21 - page 407 - THOMAS, Roland E., ROSA, Albert J. , THOUSSAINT, Gregory J. -   Book: The Analysis & Design of Linear Circuits - 6th Edition - Ed. John Willey & Sons, Inc. - 2009.

Using superposition find the voltage V_o(t) in the circuit shown in the figure below, knowing that:      i₁(t) = 100 cos(10,000t) mA      and      v₁(t) = 20 cos(20,000t - 45°) V.

Solution of the Problem + Hard 53-2

This problem has the particularity of having two sources of energy that work on different frequencies. In this way, to solve the problem we will use superposition, analyzing the circuit independently for each source and, at the end, we add up the answers.

Due to this particularity, the capacitor will have two different reactances, one for each font. Let's start with the current source you have ω = 10,000 rad/s.

Considering only the current source, we must eliminate the voltage source. we do this short-circuiting this source. So we are left with a circuit where the capacitor and the resistor are in parallel. Calculating the equivalent impedance of this parallel, we have:

We can easily calculate the voltage v_o(t) because we just need to multiply the impedance electric current flowing through it. Like this:

Note that the value of the current source was in mA and we changed it to ampere. Carrying out the calculation, we get:

Now we must calculate v_o(t) devclass the voltage source. Therefore, we must eliminate the current source. This is achieved by eliminating it from the circuit because, in this case, current source is an open circuit. So our circuit boils down to to a capacitor in series with the resistor.

Note that the voltage source has a frequency of    ω = 20,000 rad/s. With this new frequency, let's calculate the new reactance of the capacitor.

So the impedance of the circuit is  Z_eq   =   100 - j 50   =   111.8 ∠-26.56°   Ω.   To calculate v_o(t), just apply a voltage divisor, that is:

Substituting for numerical values, we have:

In trigonometric form we have:

To find the final solution we must add these results. Then:

Veja na figura abaixo, a forma de onda que se desenvolve sobre o capacitor quando temos duas fontes de energia atuando sobre o circuito com frequências diferentes.