Increasing Power factor via series capicitance

[hsainz]

Homework Statement

problem: You're the operator of a 1.60×104 V rms, 60 Hz electrical substation. When you get to work one day, you see that the station is delivering 7.00 MW of power with a power factor of 0.920.
a) what is the rms current leaving the station?
b)how much series capacitance should you add to increase the power factor to 1 (give answer in microfarads)
c)how much power will the station then be delivering.

Homework Equations

P=IVcos(θ)(both rms)
tan(θ)=(Xl-Xc)/R
resonance frequency=1/[(LC)^.5]

The Attempt at a Solution

I got the first part of the question. But I am having trouble trying to solve part b. I believe when the power factor is equal to 1, the reactance of the inductor should be equal to that of the capacitor because they will be at the resonance frequency and the phase between the current and the voltage in he circuit will be 0 thus giving a power factor of one. However, i don't know how to find the capacitance required to reach the resonance frequency because i don't have the value of the inductance (L). All i know is that all the ciruit elements share the same current which is I=Vsource/(R^2+(Xl-Xc)^2)^1/2 but I am not sure how to go about solving for the capictance. I've been on theis problem for about 2 hours...can someone please help lol -___-

 

[MATLABdude]

You're overcomplicating things. What is the impedance (real and imaginary) which the substation sees?

 

[nlightner]

I would approach this problem by first calculating the current leaving the station using the information given. This would give me an idea of the current flow in the system and allow me to calculate the total impedance of the circuit.

Next, I would use the given equation for the power factor (tan(θ)=(Xl-Xc)/R) to determine the value of the reactance of the inductor and the capacitor at the current power factor of 0.920. This would give me an idea of how much the reactance of the capacitor needs to change in order to reach a power factor of 1.

To calculate the required capacitance, I would use the resonance frequency equation (1/[(LC)^.5]) and solve for C, using the known value of the inductance (which can be calculated using the impedance equation).

Once I have the required capacitance, I would add it in series with the circuit and recalculate the power factor to ensure it has reached 1. I would also recalculate the power being delivered by the station using the new power factor and current values.

In summary, increasing the power factor can be achieved by adding a series capacitor to the circuit, which will bring the reactance of the capacitor in line with the reactance of the inductor and improve the overall power factor. It is important to calculate the required capacitance using the resonance frequency equation and to verify the new power factor and power being delivered after the addition of the capacitor.