Problem 52-12    Source:  Problem prepared by the author of the site.

In the circuit of Figure 52-12.1 we know that:

1 - The A ammeter measures a current of 8 A.

2 - The V voltmeter measures a voltage of 160 V.

3 - The W wattmeter measures a real power of 916.5 W.

4 - The power factor of the circuit is 0.783.

Based on the information above, determine:

a)   the values ​​of X₂ and the power in the resistor R₂.

b)   the values ​​of R₁ and X₁

Solution of the Problem 52-12   

Item a   

Let's take the current measured by the ammeter as a reference. Then we can write:

As the measurement of the wattmeter and the power factor were given, then it is possible to calculate the apparent power and the reactive power that the circuit consumes. Soon:

From the PF it is possible to find the phase shift angle between the voltage applied to the circuit and the electric current that was measured by the ammeter A. Logo φ = cos^-1 0.783 = 38.46°. Hence, we have sin φ = 0.622. Then:

Writing the apparent power in its complex form, we have:

As we know the value of the current I flowing through the circuit, we can calculate the voltage V_cb supplying the circuit. So:

And in polar form, we have:

Note that we can easily calculate the power dissipated in the R₂ resistor, as:

On the other hand, we can calculate the apparent power consumed by the circuit formed by R₂ and X₂, which we will name it S₂. Soon:

Note that in the circuit formed by R₂ and X₂, the only element that has reactive power is the inductor represented by X₂. As we know the apparent power and the power dissipated in R₂, then it is possible to calculate this reactive power, because:

Doing the math, we find:

And with this value we can calculate the value of X₂, because:

Now we can write the circuit formed by R₂ and X₂, which we will call Z₂, in its polar and rectangular form, that is:

Knowing the angle of the impedance Z₂, we can determine the angle of the value measured by the voltmeter V, which let's call it V_1b, in its polar and rectangular form, that is:

Item b   

To find the values ​​of R₁ and X₁ let's calculate the voltage between the points c and 1, that is, calculate the voltage V_c1. To do this, we will use what we learned in the Chapter 51 in item 3.2 (the reader can review Here!). see what it is possible to calculate the value of V_c1, since we know two sides and the angle between them. Let's use the eq. 51-01 repeated below for clarity.

Note that x is the value of V_c1, a represents the voltage V_cb and b a voltage V_1b. So, substituting for numerical values ​​we have:

Doing the calculate, we find:

The angle of V_c1 was found using eq. 51-03. We leave this calculation to the reader. Note that the calculation performed was V_c1 = V_cb - V_1b.

Knowing the value of V_c1 it is possible to calculate the value of I₂, or:

Doing the calculate, we find:

Using Kirchhoff's law   for currents in the circuit of Figure 52-11.1, we have I = I₁ + I₂. So, I₁ = I - I₂. So:

Earlier, we calculated the power dissipated by R₂. So the power that the resistor R₁ must dissipate is equal to:

As we know the value of P_R1 and I₁ we can calculate the value of R ₁, or:

To find the value of X₁, let's calculate the reactive power of this element. First, we must calculate the capacitor reactive power - j20, given by:

Note the negative value in reactive power as it is a capacitor. Then Q_X1 is given by:

Carrying out the calculation, we get:

Now we can calculate the value of X₁, or:

Therefore, the impedance formed by R₁ and X₁, which we will call Z₁, can be written as: