Table of contents

0. Math Review31m
- Math Review
  31m
1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

31. Alternating Current

Series LRC Circuits

Physics

31. Alternating Current

Series LRC Circuits

Previous problemNext problem

Problem 58

Textbook Question

Show that the power factor of a series RLC circuit is cos ϕ=R/Z.

Verified step by step guidance

Start by understanding the power factor in an AC circuit. The power factor is defined as the cosine of the phase angle (ϕ) between the voltage and the current in the circuit. Mathematically, it is expressed as cos(ϕ).

In a series RLC circuit, the total impedance (Z) is given by the formula:

Z = √(R² + (X

_L - X_C)²), where R is the resistance, X_L is the inductive reactance, and X_C is the capacitive reactance.

The phase angle (ϕ) in a series RLC circuit is determined by the relationship between the resistive and reactive components of the impedance. Specifically, tan(ϕ) = (X_L - X_C) / R.

The power factor is the cosine of this phase angle, so cos(ϕ) = adjacent / hypotenuse in the impedance triangle. Here, the adjacent side corresponds to R (resistance), and the hypotenuse corresponds to Z (total impedance). Thus, cos(ϕ) = R / Z.

Conclude by noting that the power factor of a series RLC circuit is indeed cos(ϕ) = R / Z, as derived from the relationship between the resistive and total impedance components.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Power Factor

The power factor is a dimensionless number between -1 and 1 that represents the ratio of real power flowing to the load to the apparent power in the circuit. It indicates how effectively electrical power is being converted into useful work output. A power factor of 1 (or 100%) signifies that all the power is being effectively converted, while a lower power factor indicates inefficiencies.

Impedance (Z)

Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. It combines both resistance (R) and reactance (X), where reactance arises from inductors and capacitors in the circuit. The magnitude of impedance is crucial for calculating the power factor, as it determines how much current flows for a given voltage.

Phase Angle (ϕ)

The phase angle (ϕ) in an RLC circuit represents the phase difference between the voltage across the circuit and the current flowing through it. It is a critical factor in determining the power factor, as it affects how much of the total power is being used effectively. The cosine of the phase angle (cos ϕ) directly relates to the power factor, illustrating the relationship between resistance and impedance.

Watch next

Master LRC Circuits in Series with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Multiple Choice

An inductor, a capacitor, and a resistor are in series with an AC source. If the capacitor is 470 nF, the inductor is

24 mH

, and the resistor is 800 Ω

800 Ω

, what is the circuit's resonant (angular) frequency?

602

views

Multiple Choice

An inductor, a capacitor, and a resistor are in series with a

300 Hz

AC source. If the capacitor is

470 nF

, the inductor is

24 mH,

and the resistor is

800 Ω

. what is the impedance?

652

views

Multiple Choice

An inductor, a capacitor, and a resistor are in series with an AC source. If the capacitor is

470 nF

, the inductor is

24 mH

, and the resistor is

800 Ω

. what is the circuit's resonant frequency?

522

views

Textbook Question

A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the peak current I?

views

Textbook Question

The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. The receiver is tuned to a station at 104.3 MHz. What is the value of the capacitor in the tuning circuit?

views

Textbook Question

FIGURE CP32.68 shows voltage and current graphs for a series RLC circuit. What is the resistance R?

views

Textbook Question

Prove that the energy dissipation is a maximum at ω = ω₀.

views

Textbook Question

Show that the peak inductor voltage in a series RLC circuit is maximum at frequency $ω_{L} = {(\frac{1}{ω_{0}^{2}} - \frac{1}{2} R^{2} C^{2})}^{- 1 / 2}$ .

views

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap