Electricity and Magnetism

41–268 Electricity and Magnetism

UNDERDAMPED IF the value of the discriminant is negative,

γ

2

− ω

2

0

< 0 ,

THEN the RCL circuit is underdamped. Therefore, the square ro ot term is imaginary-

valued, and the r

±

are complex conjugates. Thus, the general solution of the underdamped

equation of motion is

q(t) = A

+

exp



(−γ + i eω) t



+ A

−

exp



(−γ − i eω) t



= exp(−γ t)

h

A

+

e

+i eω t

+ A

−

e

−i eω t

i

,

for constant coefficients A

±

and eω =

p

|γ

2

− ω

2

0

| = +

p

ω

2

0

− γ

2

< ω

0

.

There are several aspects of the underdamped case deserving of mention.

U-D An overall exponential envelope, with decay constant γ, suppresses the two

terms in the bracket. The exponentials with imaginary argumen ...