Damping in RLC Circuits

quality factor Q

0.1

This Demonstration shows the variation with time of the current I in a series RLC circuit (resistor, inductor, capacitor) in which the capacitor

C

is initially charged to a voltage

V

. The resonant frequency of the circuit is

f=

8

10

Hz

and the plotted normalized current is

I

2πfVC

. There are three types of behavior depending on the value of the quality factor

Q

: overdamping when

Q<1/2

(no oscillation); critical damping when

Q=1/2

, (no oscillation and the most rapid damping); and underdamping when

Q>1/2

(damped oscillations).

Details

Snapshot 1: overdamping

Snapshot 2: critical damping

Snapshot 3: underdamping

The capacitor charge

q

satisfies the differential equation

..

q

+

ω

0

Q



q

+

2

ω

0

q=0

, where the resonance frequency

f

is given by

ω

0

=2πf=

1

LC

and the quality factor

Q=

ω

0

L

R

. It is convenient to work with a normalized current

I=

-



q

ω

0

VC

. Initially the capacitor is charged to voltage

V

and the current is 0.

For critical damping,

Q=1/2

and the current is

I=

ω

0

t

-

ω

0

t

e

.

For overdamping and underdamping,

Q≠1/2

and the current is

I=

ω

0

r

1

-

r

2

(

r

1

t-

r

2

t

e

)

, where

r

1

,

r

2

=-

ω

0

2Q

±

ω

0

-1+

1

4

2

Q

. So when

Q>1/2

, both

r

1

and

r

2

are complex, leading to a damped oscillating current. But when

Q<1/2

, both

r

1

and

r

2

are real and negative, so that the current is damped without any oscillations.

References

J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, New York: Addison–Wesley, 1993.

External Links

Series RLC Circuits

Unforced, Damped, Simple Harmonic Motion

Frequency Response of an LRC Circuit

Permanent Citation

James Burgess

"Damping in RLC Circuits"
http://demonstrations.wolfram.com/DampingInRLCCircuits/
Wolfram Demonstrations Project
Published: March 7, 2011