Problem 55-4    Source:   Problem 7 - page 511 -    BOYLESTAD, Robert - Book: Introdução à Análise de Circuitos - Ed. Pearson Education do Brasil - 10ª edição - 2004.

For the circuit shown in the Figure 55-4.1, calculate:

a) the current I.

b) the voltage V_L.

c) the average power supplied to the circuit.

Solution of the Problem 55-4   

Item a

In order to calculate the current I, we must calculate the impedance that the circuit offers to the voltage source. To do this, based on the circuit shown in the Figure 55-4.2, note that the capacitor is in parallel with the voltage source. It is therefore very easy to calculate the current flowing through it. Thus, we try to calculate the impedance that the two resistors and the inductor offer to the voltage source.

In the point a, we have the parallel of the inductor with the 20 ohms resistor. Then the impedance Z_a will be:

Performing the calculation:

Adding the 10 ohms resistor (which is in series with Z_a) with Z_a, we have the impedance that the circuit highlighted in blue offers to the voltage source, impedance is called Z_fon. Then:

We can calculate the current I₁, that is:

Doing the calculation we found to I₁ the value of:

To calculate the current flowing through the capacitor, applying the law of Ohm:

we knowing that I = I₁ + I_C, then:

Note that the angle is positive, meaning that the current is advanced in relation to the voltage. So the circuit has capacitive predominance.

Item b

To calculate V_L, we must calculate the voltage at the point a. As we know the value of Z_a and the value of I₁, applying the Ohm's law:

Performing the calculation:

Item c

To calculate the average power, we can find the apparent power.

We must find the angle of lag between the voltage V and the current I. Since the voltage angle is zero and the current angle is 18,43°, then the value of φ is equal to φ = 18.43°. Therefore, using the equation of average power and performing the calculation: