The RLC Circuit

AP Physics C

Montwood High School

R. Casao

• A more realistic circuit consists of an inductor, a capacitor, and a resistor connected in series.

• Assume that the capacitor has an initial charge Qm before the switch is closed.

• Once the switch is closed and a current is established, the total energy stored in the circuit at any time is given by:

220.5

2

C LU U U

QU L I

C

• The energy stored in the capacitor is and

the energy stored in the inductor is 0.5·L·I2.

• However, the total energy is no longer constant, as it was in the LC circuit, because of the presence of the resistor, which dissipates energy as heat.

• Since the rate of energy dissipation through the resistor is I2·R, we have:

– the negative sign signifies that U is decreasing in time.

2

2QC

2dUI R

dt

• Substituting this equation into the time derivative of the total energy stored in the LC circuit equation:

2 2

2

2 2

2

2 2

12 21

2 22 2

Q L Id d

CdU dUI R

dt dt dt dt

d Q d IdU Ldt C dt dtdU dQ L dI

Q Idt C dt dtdU Q dQ dI

L Idt C dt dt

Q dQ dII R L I

C dt dt

• Using the fact that

2

2

dQ dI d QI and

dt dt dt

2

22

2

Q dQ dII R L I

C dt dt

Q d QI R I L I

C dt

• Factor out an I and set up the resulting quadratic equation:2

22

2

2

2

2

2

( )

0

0

Q d QI R I L I

C dt

Q d QI I R I L

C dt

d Q Q dQL I R I

C dtdt

d Q dQ QL R

dt Cdt

• The RLC circuit is analogous to the damped harmonic oscillator.

• The equation of motion for the damped harmonic oscillator is:

• Comparing the two equations:

– Q corresponds to x; L corresponds to m; R corresponds to the damping constant b; and 1/C corresponds to 1/k, where k is the force constant of the spring.

• The quantitative solution for the quadratic equation involves more knowledge of differential equations than we possess, so we will stick with the qualitative description of the circuit behavior.

2

20

d x dxm b k x

dtdt

2

20

d Q dQ QL R

dt Cdt

• When R = 0 , reduces to a simple LC circuit and the charge and current oscillate sinusoidally in time.

• When R is small, the solution is:

• The charge will oscillate with damped harmonic motion in analogy with a mass-spring system moving in a viscous medium.

2

20

d Q dQ QL R

dt Cdt

2

12 2

cos

12

R tL

m d

d

Q Q e t

Rwhere w

L C L

• The graph of charge vs. time for a damped RLC circuit.

• For large values of R, the oscillations damp out more rapidly; in fact, there is a critical resistance value Rc above which no oscillations occur.– The critical value is given by 4

cL

RC

• A system with R = Rc is said to be critically damped.

• When R exceeds Rc, the system is said to be overdamped.

• The graph of Q vs. t for an overdamped RLC circuit, which occurs when the value of 4 L

RC