I got a homework exercise about the calculation of the power factor in a RL-circuit and in a RLC-circuit.

The question consists of 3 sub questions, I have managed to calculate the first 2 however I am struggling with the third one.

For the calculation of the third question, the other 2 questions are also needed.

**Question 1**

In the circuit below the given values are:

V_dc = 24V, R = 6.37 ohm, L = 115,76 mH, R_L = 30 ohm

Assume that all circuit elements are ideal components.

The question is to calculate the impedanze Z_eq, as seen by the source, and the current I.

__Solution:__

Since there is a DC voltage, the inductor L behaves as a wire. The impedance of L is X_L = 2*pi*0*115,76*10^-3 = 0, because of the DC voltage.

Thus Z_eq = 6.37 + 30 = 36.37 ohm

The current I is then I = 24/36.37 A = 0.65988 A

**Question 2**

The DC voltage source is replaced by an AC voltage source, that generates a sinusoidal vol andtage has a frequency of 50 Hz and a peak voltage of 325V.

The question is to calculate the impedance Z_eq, the current i and the power factor.

__Solution__

First of all, calculating X_L. X_L = 2*pi*50*115,76*10^-3 ohm

Z_eq = sqrt((X_L)^2+(R+R_L)^2) giving Z_eq = 51.43288 ohm

Then calculating the V_rms:

V_rms = V_peak / sqrt(2) = 325/sqrt(2)

i = 325/sqrt(2) / Z_eq = 4.46815 A

Eventually the power factor:

P = I^2*R = (4.46815)^2*36.37 = 726.10393 W

S = V^2/Z = 1026.82370 W

Power factor: P_f = P/S = 726.10393/1026.82370 = 0.70714

**Question 3**

Now to improve the power factor, a capacitor is added to the circuit.

C = 52 µF

The question is to calculate the improves power factor.

__Solution__

I think the steps to take are:

1) Calculate impedance of C

2) Calculate the total impedance of C and R_L (parallel)

3) Calculate the total impedance of the circuit by summing the answer of step (2) and the answer of X_L + R

4) Since the AC voltage did not change, the V_rms can be reused from question 2 and the current can be calculated with the answer of step (3).

i = V_rms/(answer step 3)

5) With all the calculated information the equation P_f = P/S can be calculated.

I don't know if this is the right way to calculate it, because I think at step 3 a difficult answer will be derived with fractions. I know that the power factor has improved by adding the capacitor C.

Please help me with the calculation. I don't want fully written out answers, but rather something that pushes me in the right way.

Thanks for the effort and time in advance!