# What is the Phase Angle of an LRC Circuit with Given Components and Frequency?

• ooohffff
In summary, a 35 mH inductor with 1.0 ohm resistance is connected in series to a 20 µF capacitor and a 60 Hz, 40-V (rms) source. The phase angle for the impedance is -89.5°, while the phase angle for the current with respect to the voltage is +89.5°. The exact wording of the question is not specified, but it can be assumed that the phase angle being referred to is the phase angle of the current with respect to the voltage.
ooohffff

## Homework Statement

A 35 mH inductor with 1.0
resistance is connected in series to a 20 µF capacitor and a 60 Hz, 40-V (rms) source. Calculate the phase angle.

## Homework Equations

tan φ = (XL - XC) / R

## The Attempt at a Solution

Solving for φ:

φ = tan -1 [(XL - XC) / R]

XL = 2πfL = 13.194 Ω
XC = 1/(2πfC) = 132.63 Ω
R=1Ω

Plugging those values in I get
Φ = -89.5° = 270.48°

I submitted the 270.48 but it's wrong?

What you've found is the phase angle of the impedance. You want the phase angle of the current. What law relates the current to the voltage and impedance?

gneill said:
What you've found is the phase angle of the impedance. You want the phase angle of the current. What law relates the current to the voltage and impedance?

Ohm's law I=V/Z ?

ooohffff said:
Ohm's law I=V/Z ?
Yup. What will be the current's angle if your Z's angle is -89.5° ?

gneill said:
Yup. What will be the current's angle if your Z's angle is -89.5° ?
Would it be tan φ = (VL - VC) / Z ? If not then I'm thoroughly confused.

No. The only voltage of consequence here is the supply voltage. If this was a DC circuit with resistors you'd write I = V/R. In this AC circuit you write I = V/Z.

V is the source voltage that also serves as the reference for the phase angle. As such its phase angle is 0°. Your Z has a phase angle of -89.5°. So what's the phase angle of V/Z? (How do you handle angles when you do a division)?

gneill said:
No. The only voltage of consequence here is the supply voltage. If this was a DC circuit with resistors you'd write I = V/R. In this AC circuit you write I = V/Z.

V is the source voltage that also serves as the reference for the phase angle. As such its phase angle is 0°. Your Z has a phase angle of -89.5°. So what's the phase angle of V/Z? (How do you handle angles when you do a division)?

Would you break it into components like:
I = (40/(Zcosφ)) i + (40/(Zsinφ)) j

and then find the angle of I?

No need. You have the angles already. There's a simple rule for dividing two complex numbers when you know the angles. What's the rule?

gneill said:
No need. You have the angles already. There's a simple rule for dividing two complex numbers when you know the angles. What's the rule?

I just derived this because I don't think I know what you're talking about or I'm misguided, but I got tan-1 (cotφ)

No, you're getting way too complicated. When you divide two numbers in complex polar form you simply subtract the angle of the denominator from the angle of the numerator:

##\frac{a ∠ θ}{b ∠ φ} = \left(\frac{a}{b}\right) ∠ (θ - φ)##

gneill said:
No, you're getting way too complicated. When you divide two numbers in complex polar form you simply subtract the angle of the denominator from the angle of the numerator:

##\frac{a ∠ θ}{b ∠ φ} = \left(\frac{a}{b}\right) ∠ (θ - φ)##
But then the resulting angle would be at 89.5°?

ooohffff said:
But then the resulting angle would be at 89.5°?
I've tried that before initially but that angle was also incorrect.

ooohffff said:
But then the resulting angle would be at 89.5°?
Yes. That should be the phase angle of the current with respect to the voltage.

ooohffff said:
I've tried that before initially but that angle was also incorrect.
You tried +89.5° and it was considered incorrect? Perhaps they're being picky about significant figures?

gneill said:
Yes. That should be the phase angle of the current with respect to the voltage.You tried +89.5° and it was considered incorrect? Perhaps they're being picky about significant figures?

Yup. That was the first one I tried with the equation: φ = cos -1 (R/Z) = 89.52028539°

Yes, maybe I should try with more sig figs.

How many significant figures does the given data suggest?

gneill said:
How many significant figures does the given data suggest?

Ah I figured out the problem! My original answer in this post -89.5° was correct. The mistake I made was that I should not have converted it to 270.48°, since technically φ should be negative since XC > XL, and you should generally take the smaller angle of the angles between two vectors.

What was the exact phrasing of the question as you received it? Generally "phase angle", unless qualified, refers to the phase angle of the current with respect to the voltage. Were they only looking for the phase angle of the impedance?

gneill said:
What was the exact phrasing of the question as you received it? Generally "phase angle", unless qualified, refers to the phase angle of the current with respect to the voltage. Were they only looking for the phase angle of the impedance?

That is the exact phrasing of the question. That phase angle, according to the formula that I used, should be the angle between voltage and current, not impedance and current.

tanΦ = (VL-VC )/ VR = (XL - XC)/ R

Yes, it yields the angle associate with the impedance which is also the phase angle of the current with respect to the voltage.

But as I mentioned previously, unless otherwise specified, generally when one talks about phase angle one is referring to the phase angle of the current with respect to that of the voltage, not the voltage with respect to the current. Basically, I'm not very happy with the problem as it is presented. But if you reached the answer that they're looking for, not much more can be said

## 1. What is an LRC circuit?

An LRC circuit is a type of electrical circuit that consists of inductors (L), resistors (R), and capacitors (C). The name LRC comes from the symbols used to represent these components in circuit diagrams.

## 2. How does an LRC circuit work?

An LRC circuit works by using the properties of inductors, resistors, and capacitors to create an oscillating current. The inductor stores energy in the form of a magnetic field, the capacitor stores energy in the form of an electric field, and the resistor dissipates energy in the form of heat. Together, these components create a cycle of energy storage and release that results in an alternating current.

## 3. What is the phase angle in an LRC circuit?

The phase angle in an LRC circuit is the measure of the difference in timing between the voltage and current in the circuit. It is represented by the Greek letter phi (φ) and is measured in degrees. The phase angle can be used to determine the behavior of the circuit, such as whether it is inductive or capacitive.

## 4. How do you calculate the phase angle in an LRC circuit?

The phase angle in an LRC circuit can be calculated using the formula φ = arctan(XL - XC / R) where XL is the inductive reactance, XC is the capacitive reactance, and R is the resistance. Alternatively, it can also be calculated using the power triangle, where the phase angle is equal to the inverse tangent of the ratio of the reactive power to the apparent power.

## 5. What is the significance of the phase angle in an LRC circuit?

The phase angle in an LRC circuit is significant because it can be used to analyze the behavior of the circuit. A phase angle of 0 degrees indicates a purely resistive circuit, a positive phase angle indicates an inductive circuit, and a negative phase angle indicates a capacitive circuit. It can also be used to calculate the impedance of the circuit and determine the resonance frequency.

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