Problem 55-7    Source:   Problem 9 - List of RLC Problems - Discipline Electrical Circuits of the School of Engineering - UFRGS - 2011 - Prof. Dr. Valner Brusamarello.

In the circuit shown in the Figure 55-7.1, we have   I = 20∠0° mA,    R₁ = 1,2 kΩ,    - jX_C1 = - j1,8 kΩ,   jX_L1 = j1,2kΩ   and   jX_L2 = jX_L3 = j2,4 kΩ. Determine:

a) the values of   I₁  and   I₂:

b) the values of   V_aT  and   V_dT.

Solution of the Problem 55-7   

Item a   

To calculate the currents I₁ and I₂, we chose to use a current divider. In this way, for I₁:

Substituting for numeric values:

Performing the calculation:

And to I₂:

Substituting for numeric values:

Performing the calculation:

The negative sign of I₂, means that the direction of the current is contrary to that indicated in the figure of the circuit.

Item b   

To find the value of V_dT, we must calculate the parallel of X_L2 and X_L3.

Now applying the Ohm's law:

Note that j1.2 = 1.2∠90°. To calculate V_aT, we chose to find the equivalent impedance of the whole circuit and multiply by the value of I (Ohm's law). For this, it is necessary to calculate the value of the parallel between X_L1 and X_C1. Assim:

Then summing all the impedances between the points a-T, we obtain:

As we can see, the circuit has inductive predominance. We can write Z_eq = 4.95∠75.96° in polar format. Then the value of V_aT will be:

Finally, performing the calculation:

In the Figure 55-7.2 we see the graph of currents and voltages in the circuit. Notice that V_R is in phase with I, because it is the voltage on the resistor R₁.  V_dT, that is the tension over L₂ and L₃, is advanced by 90° in relation to the current I that flows through them. In its turn, V_bc is advanced by 90° in relation to the current I₁ that circulating by L₁ and, at the same time, is delayed by 90° in relation to I₂, current that is circulating by C₁.   And   V_aT   is the phasor sum between V_R and V_bc + V_dT. Notice, in the chart, how all the phasors are perfectly in agreement with the circuit.