Power Factor Correction: Calculating True and Apparent Power in a Circuit

[billmccai]

Homework Statement

This is the circuit:

http://img129.imageshack.us/img129/3566/pfcog8.jpg

Assume voltage is in rms and f is 50Hz.

I have to do three things.

First, calculate and compare true power vs apparent power in the load.

Second, find the value of a capacitor placed in parallel that would bring the LOAD to unity power factor.

Finally, calculate the real power in the circuit (both transmission and load impedances) with and without this capacitor, and compare them.

Homework Equations

The Attempt at a Solution

I didn't have a problem with the first part.

The load impedance is (8 + j3) ohms.

Power worked out to be 6212.33 + j2367.12. So true power was 6312.33 W and apparent was 6741.57I'm having trouble with the second part. I've been told that I need to use:

Z_{corrected load} = Z_c || (Z_r + Z_L)

which becomes Z_{corrected load} = [Z_c * (Z_r + Z_L)] / [Z_c + (Z_r + Z_L)]

But I'm not sure how to manipulate this to give me the value of Z_c.And I can't even start the third part until I've worked this out.

Please help.

 

[jhicks]

find Zr + Zl first. A corrected load is one such that (Zr + Zl) || Zc has no imaginary component. Stick a capacitor on to cancel out the imaginary part added by Zr and Zl and you're good to go. Since it's two inductive loads you need to correct, you can assume it's a capacitor with impedance 1/(j*2*pi*f*C).

 

[billmccai]

I'm meant to bring the pf to unity at the load only, so I'm assuming i just forget the line impedance for now. In this case, i thought Zr + Zl just referred to (8 + j3), which is my load impedance. I don't know how to work out what value for Zc will cancel out the imaginary part in the formula Zcorrected load = [Zc * (Zr + ZL)] / [Zc + (Zr + ZL)].

 

[jhicks]

I'm meant to bring the pf to unity at the load only, so I'm assuming i just forget the line impedance for now.

You have to have the impedance have 0 phase (i.e. the voltage and current through the load have the same phase). Find the load impedance as it is now. What is the imaginary part? Set this equal to some impedance created by a capacitor with capacitance C and solve for it.

 

[billmccai]

Still not following.

My load impedance is 8 + 3j, right? I understand that I need to get rid of the imaginary part.. i just don't get how I mathematically determine the correct C value. I know it's going to be -(something)j ohms. But how to i manipulate Zcorrected load = [Zc * (Zr + ZL)] / [Zc + (Zr + ZL)] to spit out this value.

 

[jhicks]

Oh I finally see what you meant by "Zr" and "Zl". There's no such capacitor that I can tell that will satisfy that equation which implies placing it in parallel with the entire load. You have to place it either in series with the load or in parallel to the inductor. Sorry for the misunderstanding.

 

[jhmz]

this might help:
http://img516.imageshack.us/my.php?image=pfsd0.jpg
im doing the same question and am in same situation as op

My load impedence is 3 + j3
i started out using:
$$\frac{1}{Z_{L}}=\frac{1}{Z_{C}}+\frac{1}{Z_{PL}}$$
where $$Z_{PL}$$= Impedance of previous RL load which was 3+j3
and $$Z_{L}$$ is the impedance of the new load with capacitor
the frequency is 50Hz

i have got it down to:
$$Z_{L}=\frac{3+j3}{(j100\pi)(3) + (j100\pi)(j3)}$$
im not sure if this is right at all tho and where i go from here if it is correct, i can't find any similar examples in the textbook or online as other power correction questions use the apparent,reactive and power factor formulas to work out the power factor correction but the pf never has to equal 0 in these examples, which is what unity power factor is? The reactive power equation cannot be used for calculating the unity power factor as sin(0) = 0

so this equivalent impedance way seems to be the only method of calculating the size of the capacitor for power factor correction