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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

what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
Electric density formula
Biren Reply
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Shikhar Reply
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
Antares Reply
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
which topic now u studying?
jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
Why?
Arzoodan
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
sahoo u don't seem to understand relative physics, plz study that first.
Antares
exatly...
Arzoodan
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
Ali brother ur exactly spelling is wrong,,,,,
jyotirmayee
why the colour of tube light white?
jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
all formula for calculate specific latent heat of any substance
Idowu Reply
hello
Nigar
Hi, your real Nigga...? Or this jock?
Arzoodan
real
Nigar
You from Pakistan or India?
Arzoodan
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
What you doing about Physics
Arzoodan
you both?
jyotirmayee
in which class?
jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
Bilal Reply
it is the product of electric charges and distance between the two charges
Shikhar
yeap. this correct answer...
Arzoodan
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
Brad
brad is separation and distance ,are they different?
Ssempala
What is dielectric
Ashis Reply
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
Brad
What is the difference between specific heat capacity and heat capacity? Give the equations
elly Reply
presentation on power
Dyutee Reply
relation between Celsius and Kelvin
Anish Reply
0" degree Celsius=273kelvin
jyotirmayee
Newton's second laws is call with
Dyutee Reply
what is mean by thermodynamics
Prasad Reply
it is study about temperature and it's equilibrium
thiru
Its the study of heat and its relation with others kind of energy
Antonio
state caulombs law clearly
constand Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
ezra Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
srikanta
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
Sikandar
Practice Key Terms 3

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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