<< Chapter < Page Chapter >> Page >
By the end of the section, you will be able to:
  • Determine the peak ac resonant angular frequency for a RLC circuit
  • Explain the width of the average power versus angular frequency curve and its significance using terms like bandwidth and quality factor

In the RLC series circuit of [link] , the current amplitude is, from [link] ,

I 0 = V 0 R 2 + ( ω L 1 / ω C ) 2 .

If we can vary the frequency of the ac generator while keeping the amplitude of its output voltage constant, then the current changes accordingly. A plot of I 0 versus ω is shown in [link] .

Figure shows a graph of I0 versus omega. The curve ascends gradually, has one blunt peak at the centre and then gradually descends to its original value. The y-value at the peak is V0 by R and the x-value is omega 0.
At an RLC circuit’s resonant frequency, ω 0 = 1 / L C , the current amplitude is at its maximum value.

In Oscillations , we encountered a similar graph where the amplitude of a damped harmonic oscillator was plotted against the angular frequency of a sinusoidal driving force (see Forced Oscillations ). This similarity is more than just a coincidence, as shown earlier by the application of Kirchhoff’s loop rule to the circuit of [link] . This yields

L d i d t + i R + q C = V 0 sin ω t ,

or

L d 2 q d t 2 + R d q d t + 1 C q = V 0 sin ω t ,

where we substituted dq (t)/ dt for i (t). A comparison of [link] and, from Oscillations , Damped Oscillations for damped harmonic motion clearly demonstrates that the driven RLC series circuit is the electrical analog of the driven damped harmonic oscillator.

The resonant frequency     f 0 of the RLC circuit is the frequency at which the amplitude of the current is a maximum and the circuit would oscillate if not driven by a voltage source. By inspection, this corresponds to the angular frequency ω 0 = 2 π f 0 at which the impedance Z in [link] is a minimum, or when

ω 0 L = 1 ω 0 C

and

ω 0 = 1 L C .

This is the resonant angular frequency of the circuit. Substituting ω 0 into [link] , [link] , and [link] , we find that at resonance,

ϕ = tan −1 ( 0 ) = 0 , I 0 = V 0 / R , and Z = R .

Therefore, at resonance, an RLC circuit is purely resistive, with the applied emf and current in phase.

What happens to the power at resonance? [link] tells us how the average power transferred from an ac generator to the RLC combination varies with frequency. In addition, P ave reaches a maximum when Z , which depends on the frequency, is a minimum, that is, when X L = X C and Z = R . Thus, at resonance, the average power output of the source in an RLC series circuit is a maximum. From [link] , this maximum is V rms 2 / R .

[link] is a typical plot of P ave versus ω in the region of maximum power output. The bandwidth     Δ ω of the resonance peak is defined as the range of angular frequencies ω over which the average power P ave is greater than one-half the maximum value of P ave . The sharpness of the peak is described by a dimensionless quantity known as the quality factor     Q of the circuit. By definition,

Q = ω 0 Δ ω ,

where ω 0 is the resonant angular frequency. A high Q indicates a sharp resonance peak. We can give Q in terms of the circuit parameters as

Q = ω 0 L R .
Figure shows a graph of P bar versus omega. The curve ascends gradually, has one blunt peak at the centre and then gradually descends to its original value. The y-value at the peak is V squared subscript rms by R and the x-value is omega 0. The y value near the middle of the curve is V squared subscript rms by 2R. The width of the curve near the middle is labeled delta omega.
Like the current, the average power transferred from an ac generator to an RLC circuit peaks at the resonant frequency.

Resonant circuits are commonly used to pass or reject selected frequency ranges. This is done by adjusting the value of one of the elements and hence “tuning” the circuit to a particular resonant frequency. For example, in radios, the receiver is tuned to the desired station by adjusting the resonant frequency of its circuitry to match the frequency of the station. If the tuning circuit has a high Q , it will have a small bandwidth, so signals from other stations at frequencies even slightly different from the resonant frequency encounter a high impedance and are not passed by the circuit. Cell phones work in a similar fashion, communicating with signals of around 1 GHz that are tuned by an inductor-capacitor circuit. One of the most common applications of capacitors is their use in ac-timing circuits, based on attaining a resonant frequency. A metal detector also uses a shift in resonance frequency in detecting metals ( [link] ).

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 2' conversation and receive update notifications?

Ask