Problem 53-6    Source:  Problem prepared by the site's author.

In Figure 53-06.1 we have V = 50∠0° and we know that I₁ = 10∠- 90° . Determine the value of I₃ and Z.

Solution of the Problem 53-6

As the value of I₁ has been given, then we can calculate the value of voltage between points a - d i.e. V_ad. Like this:

Where we use the fact that - j2 = 2∠-90°. As we know the values of V and V_ad, we can easily find the value of V_dc, or:

Now, knowing the value of V_dc we can calculate the value of I₃.

Note que I₃  =  I₁ + I₂. Logo, temos:

Carrying out the calculation, we find:

Knowing the value of I₂ we can calculate the value of V_bd, or:

Carrying out the calculation, we find:

With the knowledge of V_ad and V_bd we can calculate the value of V_ab , or:

Carrying out the calculation, we find:

Knowing the value of V_ab it is possible to calculate the value of I, or:

Using the circuit, we see that I = I₂ + I_Z. Then the value of I_Z will be:

Carrying out the calculation, we find:

And from the circuit, we see that - V + V_ab + V_bc = 0. Then the value of V_bc will be:

Carrying out the calculation, we find:

Therefore the value of Z will be:

Carrying out the calculation, we find:

Important Note

The impedance Z can be represented by a series circuit between a resistor and an inductor. However, NOTE that the resistor value is negative. So, for this circuit to be implemented in practice, it is necessary that the imedance Z contains active circuits, such as transistors, operational amplifiers, etc...