Find Frequency to Minimize Impedance of RLC Circuit

[jacksonwiley]

Homework Statement

For an RLC circuit with a resistance of 13.0 kΩ a capacitance of 7.0 µF, and an inductance of 35.0 H. What frequency is needed to minimize the impedance?
A) 0.064 kHz
B) 0.010 kHz
C) 12 kHz
D) 2.1 kHz

Homework Equations

Xc = 2∏ƒL
XL = 1/ (2∏ƒC)

The Attempt at a Solution

i'm really unsure if i need to use the XL or the Xc equation?
2(3.14)*frequency*L
or 1/(2(3.14)*frequency*capacitance
i've been stuck on this one forever.. any guidance is much appreciated!

 

[tms]

Since the circuit has both inductive and capacitive reactance, you need to use both in your calculations.

 

[jacksonwiley]

Since the circuit has both inductive and capacitive reactance, you need to use both in your calculations.

Would i just find the difference of the two?

 

[tms]

You would use the total reactance, which is a function of ##X_C## and ##X_L##. Then you have to do something with that equation to find the minimum.

 

[BiGyElLoWhAt]

series circuit?

 

[jacksonwiley]

series circuit?

Yeah I think it's assumed that this is a series RLC circuit

 

[jacksonwiley]

You would use the total reactance, which is a function of ##X_C## and ##X_L##. Then you have to do something with that equation to find the minimum.

if i set them equal to each other and then solved for frequency that would lead to the right answer, right?

 

[tms]

if i set them equal to each other and then solved for frequency that would lead to the right answer, right?

Set what equal to what? There is an expression for the total reactance, given ##X_C## amd ##X_L##; you need to use that. Once you get that, how do you find a minumum?

 

[NascentOxygen]

if i set them equal to each other and then solved for frequency that would lead to the right answer, right?

magnitudes of the reactances, yes.

 

[BiGyElLoWhAt]

Are you measuring over the resistor?

 

[BiGyElLoWhAt]

Also, I'm not sure if you can do this without considering the imaginary parts of the ractances. Maybe I'm wrong.

 

[BiGyElLoWhAt]

if i set them equal to each other and then solved for frequency that would lead to the right answer, right?

Spoiler

Basically, yes. This comes from mapping out your transfer function and solving for the minimum reactance.

Try writing your total reactance as a function of ##X_{c}## & ## X_{L}##.

But instead of using the equations you have, use ##X_{c} = \frac{1}{i\omega c}## & ## X_{L} = IL\omega i##

with I being current, omega angular velocity, i the imaginary number, and L and c inductance and capacitance.

Or alternatively if you want to solve for frequency and not angular velocity you can later substitute ##\omega = 2\pi f##

 

[CWatters]

Perhaps try plotting or sketching the impedance of the L and C on a graph. Add a line for the sum. Find where it's a minimum.

or write an equation for the curve of the sum and then find it's minimum (eg where the slope is zero). Example