Calculating Impedance Using Capacitive Reactance

In summary: You could have also used the phasor diagram to find the magnitude of the impedance, since it is the hypotenuse of the triangle formed by R and XL.
  • #1
PeachBanana
191
0

Homework Statement



A 29kΩ resistor is in series with a 62mH inductor and an ac source. Calculate the impedance of the circuit if the source frequency is 80Hz .

Homework Equations



XL = ωL
Z = [(R^2 + (XL - XC)^2)] ^ 1/2


The Attempt at a Solution



R = 29,000 Ω

XL = 0.062 H (2∏ * 80 Hz)
XL = 31.16 Ω

I'm having difficulty figuring out how to find XC. XC = 1 / ωC but what is "C?"
 
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  • #2
PeachBanana said:

Homework Statement



A 29kΩ resistor is in series with a 62mH inductor and an ac source. Calculate the impedance of the circuit if the source frequency is 80Hz .

Homework Equations



XL = ωL
Z = [(R^2 + (XL - XC)^2)] ^ 1/2

The Attempt at a Solution



R = 29,000 Ω

XL = 0.062 H (2∏ * 80 Hz)
XL = 31.16 Ω

I'm having difficulty figuring out how to find XC. XC = 1 / ωC but what is "C?"
Your formula for Z applies to RLC circuits. You have an RL circuit. Do you know how that formula for Z was derived? It was probably using a phasor diagram. If you understand the derivation, it's pretty easy to see what you need to do.
 
  • #3
Was the formula for "Z" derived using the Pythagorean theorem? If so, I still do not see the connection to capacitance.
 
  • #4
Describe how the formula was derived.
 
  • #5
It looks as if it uses the following relations:

Vrms = Irms * Z
Vnaught = InaughtZ

The book also shows a phasor diagram where Vnaught is the sum vector and acts as the hypotenuse of the triangle.
 
  • #6
And where does the triangle come from?
 
  • #7
The triangle comes from a phasor diagram. I'm attaching an image made in paint of the one I see in the textbook. It's a tad messy, but it is readable.
 

Attachments

  • phasor.png
    phasor.png
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  • #8
In your picture, I should actually point in the same direction as VR=IR.

Do you understand why the one leg is VL-VC and why you add all the voltages?
 
  • #9
Are the voltages being added together to find the peak voltage source, Vnaught?
Is Vc subtracted from Vl because Vc lags the current by 90°?
 
  • #10
PeachBanana said:
Are the voltages being added together to find the peak voltage source, Vnaught?
Yes, it's Kirchoff's voltage law applied to the elements in series.

Is Vc subtracted from Vl because Vc lags the current by 90°?
And because VL leads the current by 90°. The two phasors point in opposite directions, so when you add them vector-wise, you subtract their magnitudes.

So back to your original problem, if you follow the same analysis, what do you get for the impedance?
 
  • #11
Since this is an RL circuit then

Z = [(R^2 + XL^2)]^1/2
 
  • #12
Yup, you have it.
 

1. What is impedance and why is it important?

Impedance is the measure of opposition to the flow of an alternating current in a circuit. It is important because it helps us understand how much resistance a circuit has to an AC current and how much energy is being consumed or lost in a circuit.

2. What is capacitive reactance and how is it related to impedance?

Capacitive reactance is the measure of opposition to the flow of an alternating current in a capacitor. It is related to impedance because it is a component of impedance in AC circuits, along with resistance and inductive reactance.

3. How do you calculate capacitive reactance?

Capacitive reactance can be calculated using the formula Xc = 1/(2πfC), where Xc is the capacitive reactance in ohms, π is the mathematical constant pi, f is the frequency of the AC current in hertz, and C is the capacitance of the capacitor in farads.

4. How do you calculate impedance using capacitive reactance?

Impedance can be calculated using the formula Z = √(R² + Xc²), where Z is the impedance in ohms, R is the resistance in ohms, and Xc is the capacitive reactance in ohms.

5. Can capacitive reactance and impedance be negative?

Yes, both capacitive reactance and impedance can be negative. This occurs when the circuit has an inductive component, which can create a phase shift in the current and result in a negative reactance and impedance value.

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