# 14.6 Rlc series circuits

 Page 1 / 4
By the end of this section, you will be able to:
• Determine the angular frequency of oscillation for a resistor, inductor, capacitor $\left(RLC\right)$ series circuit
• Relate the $RLC$ 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

$\frac{dU}{dt}=\frac{q}{C}\phantom{\rule{0.2em}{0ex}}\frac{dq}{dt}+Li\frac{di}{dt}=\text{−}{i}^{2}R$

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

$L\frac{{d}^{2}q}{d{t}^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=0.$

This equation is analogous to

$m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+kx=0,$

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 $\left(\sqrt{k\text{/}m}>b\text{/}2m\right)$ , critically damped $\left(\sqrt{k\text{/}m}=b\text{/}2m\right)$ , or overdamped $\left(\sqrt{k\text{/}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 $\sqrt{1\text{/}LC}>R\text{/}2L$ , we obtain

$q\left(t\right)={q}_{0}{e}^{\text{−}Rt\text{/}2L}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega \text{′}t+\varphi \right)$

where the angular frequency of the oscillations is given by

${\omega }^{\prime }=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^{2}}$

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, $L=5.0\phantom{\rule{0.2em}{0ex}}\text{mH},C=6.0\mu \text{F},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R=200\phantom{\rule{0.2em}{0ex}}\text{Ω}.$ (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of $3.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{C}$ on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?

a. overdamped; b. 0.75 J

## Summary

• The underdamped solution for the capacitor charge in an RLC circuit is
$q\left(t\right)={q}_{0}{e}^{\text{−}Rt\text{/}2L}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega \text{′}t+\varphi \right).$
• The angular frequency given in the underdamped solution for the RLC circuit is
${\omega }^{\prime }=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^{2}}.$

## Key equations

 Mutual inductance by flux $M=\frac{{N}_{2}{\text{Φ}}_{21}}{{I}_{1}}=\frac{{N}_{1}{\text{Φ}}_{12}}{{I}_{2}}$ Mutual inductance in circuits ${\epsilon }_{1}=\text{−}M\frac{d{I}_{2}}{dt}$ Self-inductance in terms of magnetic flux $N{\text{Φ}}_{\text{m}}=LI$ Self-inductance in terms of emf $\epsilon =\text{−}L\frac{dI}{dt}$ Self-inductance of a solenoid ${L}_{\text{solenoid}}=\frac{{\mu }_{0}{N}^{2}A}{l}$ Self-inductance of a toroid ${L}_{\text{toroid}}=\frac{{\mu }_{0}{N}^{2}h}{2\pi }\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{R}_{2}}{{R}_{1}}.$ Energy stored in an inductor $U=\frac{1}{2}L{I}^{2}$ Current as a function of time for a RL circuit $I\left(t\right)=\frac{\text{ε}}{R}\left(1-{e}^{\text{−}t\text{/}{\tau }_{L}}\right)$ Time constant for a RL circuit ${\tau }_{L}=L\text{/}R$ Charge oscillation in LC circuits $q\left(t\right)={q}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega t+\varphi \right)$ Angular frequency in LC circuits $\omega =\sqrt{\frac{1}{LC}}$ Current oscillations in LC circuits $i\left(t\right)=\text{−}\omega {q}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\omega t+\varphi \right)$ Charge as a function of time in RLC circuit $q\left(t\right)={q}_{0}{e}^{\text{−}Rt\text{/}2L}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega \text{′}t+\varphi \right)$ Angular frequency in RLC circuit $\omega \text{′}=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^{2}}$

A 40cm tall glass is filled with water to a depth of 30cm. A.what is the gauge pressure at the bottom of the glass? B.what is the absolute pressure at the bottom of the glass?
A glass bottle full of mercury has mass 50g when heated through 35degree, 2.43g of mercury was expelled. Calculate the mass of the mercury remaining in the bottle
Two electric point charges Q=2micro coulomb are fixed in space a distance 2.0cm apart. calculate the electric potential at the point p located a distance d/2 above the central point between two charges
what is wave
What is charge bodies
which have free elections
Usman
Show that if a vector is gradient of a scaler function then its line around a closed path is zero
Pak
Charge bodies are those which have free electons
Pak
the melting point of gold is 1064degree cencius and is boiling point is 2660 degree cenciu
is Thomas's young experiment interference experiment or diffraction experiment or both
An aqueous solution is prepared by diluting 3.30 mL acetone (d = 0.789 g/mL) with water to a final volume of 75.0 mL. The density of the solution is 0.993 g/mL. What is the molarity, molality and mole fraction of acetone in this solution?
eugene
A 4.0kg mess kit sliding on a fractionless surface explodes into two 2.0 kg parts.3.0 m/s due to north and 0.5 m/s 30 degree north of east. what is the speed of the mess kit
Shahid
it's a line used to represent a complex electrical quantity as a vector
what is the meaning of phasor?
The electric field inside a sphere of radius is given by the expression for some constants and. Find the charge density and the total charge contained in this sphere.
what is motion?
is the change of position of body
Kate
it's the process of moving something.
the state of a body in which it change its position with respect to sorrounding is known as motion for example player of football change its position with respect to spectator
Ilyas
it is a state of body in which it changes it,s position with respect to their time and immediate surrounding
Harish
a particle change with respect to time and position
Babak
it is a stat of body in which it changes its position with respect to mean and exstrem position
Manzoor
A 4.0 kg mass kit sliding on a fractionless surface explodes into two 2.0 kg parts,3.0 m/s due to north and 0.5 m/s 30 degree north of east. what the speed of mass kit
Shahid
momentum conservation
Mehmet
3.5 m/s north of east.
Mehmet
give me a compete solution
Shahid
where the solving of questions of this topic?
According to Nernst's distribution law there are about two solvents in which solutes undergo equilibria. But i don't understand how can you know which of two solvents goes bottom and one top? I real want to understand b'coz some books do say why they prefer one to top/bottom.
I need chapter 25 last topic
What is physics?
Abdulaziz
physics is the study of matter and energy in space and time and how they related to each other
Manzoor
interaction of matter and eneegy....
Abdullah
thanks for correcting me bro
Manzoor
What is electrostatics bassically?
study of charge at rest
wamis
A branch in physics that deals with statics electricity
Akona
what is PN junction?
Manzoor