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By the end of this section, you will be able to:
  • Determine the angular frequency of oscillation for a resistor, inductor, capacitor ( R L C ) series circuit
  • Relate the R L C circuit to a damped spring oscillation

When the switch is closed in the RLC circuit    of [link] (a), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate i 2 R . With U given by [link] , we have

d U d t = q C d q d t + L i d i d t = i 2 R

where i and q are time-dependent functions. This reduces to

L d 2 q d t 2 + R d q d t + 1 C q = 0 .
Figure a is a circuit with a capacitor, an inductor and a resistor in series with each other. They are also in series with a switch, which is open. Figure b shows the graph of charge versus time. The charge is at maximum value, q0, at t=0. The curve is similar to a sine wave that reduces in amplitude till it becomes zero.
(a) An RLC circuit. Electromagnetic oscillations begin when the switch is closed. The capacitor is fully charged initially. (b) Damped oscillations of the capacitor charge are shown in this curve of charge versus time, or q versus t . The capacitor contains a charge q 0 before the switch is closed.

This equation is analogous to

m d 2 x d t 2 + b d x d t + k x = 0 ,

which is the equation of motion for a damped mass-spring system (you first encountered this equation in Oscillations ). As we saw in that chapter, it can be shown that the solution to this differential equation takes three forms, depending on whether the angular frequency of the undamped spring is greater than, equal to, or less than b /2 m . Therefore, the result can be underdamped ( k / m > b / 2 m ) , critically damped ( k / m = b / 2 m ) , or overdamped ( k / m < b / 2 m ) . By analogy, the solution q ( t ) to the RLC differential equation has the same feature. Here we look only at the case of under-damping. By replacing m by L , b by R , k by 1/ C , and x by q in [link] , and assuming 1 / L C > R / 2 L , we obtain

q ( t ) = q 0 e R t / 2 L cos ( ω t + ϕ )

where the angular frequency of the oscillations is given by

ω = 1 L C ( R 2 L ) 2

This underdamped solution is shown in [link] (b). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. [link] can be confirmed experimentally by measuring the voltage across the capacitor as a function of time. This voltage, multiplied by the capacitance of the capacitor, then gives q ( t ).

Try an interactive circuit construction kit that allows you to graph current and voltage as a function of time. You can add inductors and capacitors to work with any combination of R , L , and C circuits with both dc and ac sources.

Try out a circuit-based java applet website that has many problems with both dc and ac sources that will help you practice circuit problems.

Check Your Understanding In an RLC circuit, L = 5.0 mH , C = 6.0 μ F , and R = 200 Ω . (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of 3.0 × 10 −3 C on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?

a. overdamped; b. 0.75 J

Got questions? Get instant answers now!

Summary

  • The underdamped solution for the capacitor charge in an RLC circuit is
    q ( t ) = q 0 e R t / 2 L cos ( ω t + ϕ ) .
  • The angular frequency given in the underdamped solution for the RLC circuit is
    ω = 1 L C ( R 2 L ) 2 .

Key equations

Mutual inductance by flux M = N 2 Φ 21 I 1 = N 1 Φ 12 I 2
Mutual inductance in circuits ε 1 = M d I 2 d t
Self-inductance in terms of magnetic flux N Φ m = L I
Self-inductance in terms of emf ε = L d I d t
Self-inductance of a solenoid L solenoid = μ 0 N 2 A l
Self-inductance of a toroid L toroid = μ 0 N 2 h 2 π ln R 2 R 1 .
Energy stored in an inductor U = 1 2 L I 2
Current as a function of time for a RL circuit I ( t ) = ε R ( 1 e t / τ L )
Time constant for a RL circuit τ L = L / R
Charge oscillation in LC circuits q ( t ) = q 0 cos ( ω t + ϕ )
Angular frequency in LC circuits ω = 1 L C
Current oscillations in LC circuits i ( t ) = ω q 0 sin ( ω t + ϕ )
Charge as a function of time in RLC circuit q ( t ) = q 0 e R t / 2 L cos ( ω t + ϕ )
Angular frequency in RLC circuit ω = 1 L C ( R 2 L ) 2

Questions & Answers

What is differential form of Gauss's law?
Rohit Reply
help me out on this question the permittivity of diamond is 1.46*10^-10.( a)what is the dielectric of diamond (b) what its susceptibility
OLUWA Reply
a body is projected vertically upward of 30kmp/h how long will it take to reach a point 0.5km bellow e point of projection
Abu Reply
i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
Lord Reply
kya ye b.sc ka hai agar haa to konsa part
MPL Reply
what is charge quantization
Mayowa Reply
it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
it is just like bohr's theory. Which was angular momentum of electron is intral multiple of h/2π
Aditya
determine absolute zero
OFERE Reply
The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
Opeyemi Reply
U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
Gwam Reply
what is copper loss
timileyin Reply
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
Ifeoma Reply
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe
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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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