Problem 78-12    Source:    Problem 49 on the list of Electrical Circuits II - UFRGS School of Engineering - 2017 - Prof. Dr. Valner Brusamarello.

Knowing that K = 0,5, determine V_ab on the circuit shown in Figure 78-12.1, when Z assumes the following values:

a) Z = - j15  Ω

b) Z = j25  Ω

c) Z = 60 - j75  Ω

Solution of the Problem 78-12   

First, let's make some simplifications in the circuit shown in Figure 78-12.1 . Note that the 10 ohm resistor is in parallel with the voltage source. Soon, we can take it out the circuit. We also have a capacitor in series with an inductor with the same values, in module, impedance. Therefore, they can be replaced by a short circuit. In this way, we reduced the circuit, after a modification in the topology, to that shown in Figure 78-12.2.

The value of K has been provided, so we can calculate the value of jωM, remembering the eq. 78-09, or:

Substituting for the numerical values, we find:

Now we are going to redesign the circuit taking into account the influences of the magnetic coupling. Figure 78-12.3 shows the circuit that we are going to use to write the equations that solve the problem. Note that we assume the currents entering the side of the coils with a point. Thus, the positive pole of the dependent sources faces the side of the coil without a point.

So, the equations for the mesh I₁ and I₂ are:

a) When Z = - j15   

In this case we have the equations:

Using Octave, we find:

And from the circuit we have to:

b) When Z = j25   

In this case we have the equations:

Using Octave, we find:

And from the circuit we have to:

c) When Z = 60 - j75   

In this case we have the equations:

Using Octave, we find:

And from the circuit we have to:

Performing the calculation, we find