If R^2= L/C, show the total power is independent of frequency

[justtryingtopass]

Homework Statement

You have a circuit supplied with AC voltage, it has 2 parallel branches, 1 with R and L, the other with R and C.

a) If (frequency) w^2 = 1/LC, prove that the power in each branch is equal.
b) If R^2 = L/C, show that the power is the circuit is independent of frequency.

Homework Equations

The Attempt at a Solution

I am pretty sure I got a) wL=1/wC, so XL=XC, then I know currents are equal in each branch, and it's a parallel circuit so voltage is equal, therefore power is equal.

But, I don't even know where to begin for b). I can see that R^2 = w/w * L * 1/C (and that is independent of w) but I really think there is more to it than that, some type of proof that R^2 = XL/XC??

Please help!

 

[cnh1995]

type of proof that R^2 = XL/XC??

R² is not X_L/X_c. Find the correct relation among R, X_L and X_c from R²=L/C. Find the equivalent impedance of the circuit and use the obtained relation in it.

 

[justtryingtopass]

R² is not X_L/X_c. Find the correct relation among R, X_L and X_c from R²=L/C. Find the equivalent impedance of the circuit and use the obtained relation in it.

Ok, so I initially tried to work the result backwards. R^2 = L/C is the same as R^2 = L*1/C, this formula is independent of frequency, so I added frequency back in.
w/w*L*1/C. Which gives wL*1/wC. and then wL = XL and 1/wC = XC. Therefore R^2 = XL/XC. The only way I could get ohms = ohms. I'm not sure how Henry/Farad = ohms^2.

I understand that inductance of an inductor is determined by the details of its construction and is independent of the frequency of the circuit, same principle applies for the capacitor. But, finding the equivalent impedance of the circuit requires frequency...still lost...I don't know the correct relation among R, XL and Xc from R2=L/C.

 

[cnh1995]

R^2 = XL/XC.

No. Your RHS is unitless while the LHS has the unit ohm².

Which gives wL*1/wC. and then wL = XL and 1/wC = XC.

Correct. So R²=??
Also, are you familiar with the complex form of impedance?

 

[justtryingtopass]

No. Your RHS is unitless while the LHS has the unit ohm².

Correct. So R²=??
Also, are you familiar with the complex form of impedance?

Response part 2 - I'm still drawing a blank for R^2. I don't think I know the complex form of impedance. I know that in a parallel circuit Z^-1 = 1/Z+1/Z+1/Z+...but trying to do that in terms of R, C, and L got really messy and the result didn't give me any "ah-ha" moments.

 

[rude man]

So Z = jwL for the impedance of an inductor is not something you've seen before?
If not you will have to deal with the trigonometric relations between applied voltage v and current i for inductors and capacitors:

Let v = v₀ sin(wt) across an inductor L or a capacitor C, then
v = L di/dt so i_L = v₀/wL sin(wt - pi/2)
and i_C = C dv/dt = wC sin(wt + pi/2)

Then show that the sum of the squares of the magnitudes of the branch currents |i_L|² + |i_C|² add up to a constant independent of w.
Then wish you had covered complex electrical parameters!