Power Factor of an L-C-R curcuit

[whitejac]

Homework Statement

A series L-C-R circuit Comprises of a 250mH coil, 0.01 MFD capacitor and a 5W resistor. A dependent voltage defined by e = 2.5cos(80000t) is applied across it. Determine the power factor. [/B]

Homework Equations

Power Factor = cos(x) = R/Z
Z = sqrt^[R^2 + (XL - XC)^2]

The Attempt at a Solution

Z = sqrt^[R^2 + (wL - 1/(wC))^2] , where w = angular frequency
w = 1/sqrt^[LC] = 1/sqrt^[0.25H * 0.1MFD]

Power Factor = 5W/Z

This was where I came to. I'm having trouble most with the units. I do not know how to utilize a 5W resistor and I don't know what a MFD Capacitor is. At first I thought it could be a Mega-Ferrad, but I don't know how MFD converts to Capacitance. I believe my equation manipulation is correct if I can get those straightened out though.

 

[gneill]

Hi whitejac, Welcome to Physics Forums.

MFD means micro-Farad (or μF). It's a fairly old term from the early days of radio electronics. A related term is MMFD, or micro-micro-Farad, equivalent to the modern pico-Farad (pF).

In electronics W is usually reserved for units of Watts. Resistance is given in Ohms (Ω).

Does that help?

 

[whitejac]

Thank you! The MFD is certainly a bit of trivia I'm glad to know now. I don't understand then how to solve this problem... My book actually says 5W resistor. It might be a printing error.

I'll consider it a 5ohm resistor and see if I receive the answer in the back: cos(x) = 1/25 x 10^-5.

One thing I don't understand is why the voltage is given, however. If I consider it a misprint then it's a useless extra.

 

[gneill]

Thank you! The MFD is certainly a bit of trivia I'm glad to know now. I don't understand then how to solve this problem... My book actually says 5W resistor. It might be a printing error.

Yes, probably an issue with fonts. If you change a W in "standard" font to a math font, it converts to Ω.

I'll consider it a 5ohm resistor and see if I receive the answer in the back: cos(x) = 1/25 x 10^-5.

One thing I don't understand is why the voltage is given, however. If I consider it a misprint then it's a useless extra.

The voltage is given so that you will know what frequency (ω) the circuit is being driven at. Note that it is not the resonant frequency!

 

[whitejac]

My set up is wrong then.

If I assume what I said in the OP was correct, then:
Power Factor = R/Z
Z = sqrt^[R^2 + (ωL - 1/ωC)^2]

Using the given data and a conversion in my book: ω = 1/sqrt[LC] = 0.89.

Subbing into Z, we're given sqrt^[25 + ((0.89*0.25) - 1/(0.89 * 0.1x10^-7))^2]
This gives an answer close to 4.99 which would imply that the Power Factor is almost 1, and that does not come anywhere near my posted solution. Am I using a wrong formula for finding ω and/or the Power Factor?

 

[gneill]

My set up is wrong then.

If I assume what I said in the OP was correct, then:
Power Factor = R/Z
Z = sqrt^[R^2 + (ωL - 1/ωC)^2]

Using the given data and a conversion in my book: ω = 1/sqrt[LC] = 0.89.

That formula gives you the natural resonant frequency of the RLC circuit. It is NOT the frequency at which the circuit is being driven by the voltage source! Look in the given data to find the frequency at which it is being driven.

 

[whitejac]

I'm sorry. I can't see how you acquire the frequency from the voltage without knowing time prior. The problem question is all of the information I'm allowed to have. If I use v = e = 2.5cos(80000t) then I have 2 unknowns. If I use another formula, say...
v = Vcos(ωt +∅) then I have a whole bunch of unknowns. The only sure thing I know how to find is ω at resonance. I don't know how to find it when it's not.

 

[gneill]

I'm sorry. I can't see how you acquire the frequency from the voltage without knowing time prior. The problem question is all of the information I'm allowed to have. If I use v = e = 2.5cos(80000t) then I have 2 unknowns. If I use another formula, say...
v = Vcos(ωt +∅) then I have a whole bunch of unknowns. The only sure thing I know how to find is ω at resonance. I don't know how to find it when it's not.

In the expression v = Vcos(ωt +∅), what do the terms ω, t, and ∅ represent?

 

[whitejac]

ω = frequency, which is integral in finding Z.
t = time
∅ = the angle between the voltage phasor and the current phasor (phase angle)

 

[gneill]

ω = frequency, which is integral in finding Z.
t = time
∅ = the angle between the voltage phasor and the current phasor (phase angle)

Okay, so you were given the driving voltage as: e = 2.5cos(80000t).

Can you pick out the (angular) frequency?

 

[whitejac]

Oh. Wow. Yes I can.
ω = 80000 which will give me a what i need to find Z which will allow for me to find P.
I can't believe it was staring me in the face that blatantly.