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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

determining dimensional correctness
PATRICK Reply
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
akuribire Reply
what are the elementary compositions of a cell?
jackson Reply
when a current pass through a material does the velocity varies
lovet Reply
what is spin entropy ?and disorder in ferromagnetic material
Nepal Reply
diagram of an hall effect sensor
Aweda Reply
if a magnetised wire having dipole moment M is bent in the form of arc subtending angle of 45°at centre,new magnetic moment is
Pravin Reply
is this book for preparing IIT or neet?
jyotirmayee Reply
is it possible to increase the temperature of a gas without adding heat to it?
gurpreet Reply
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
Brad
when a current pass through a material does the velocity varies
lovet
yes at adiabatic compression temperature increase
Nepal
how to draw a diagram of a triode
Ssempala Reply
whate is fckg diagrame?
Arzoodan
why do we use integration?
Monalisa Reply
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
Heeran Reply
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
Eric Reply
what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
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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