Download The RLC Circuit

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:
U  UC  U L
2
Q
2
U 
 0.5  L  I
2 C
2
Q
• The energy stored in the capacitor is
and
2 C
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: dU
2
dt
 I  R
– the negative sign signifies that U is decreasing in time.
• Substituting this equation into the time derivative of the
total energy stored in the LC circuit equation:
 Q2 
 L I2 
d 
 d 

2 C 
2 
dU
dU


2
 I  R


dt
dt
dt
dt
 
2
 
2
dU
1 d Q
L d I


 
dt
2 C
dt
2
dt
dU
1
dQ L
dI

 2 Q 
 2 I 
dt
2 C
dt
2
dt
dU Q dQ
dI
 
L I 
dt
C dt
dt
Q dQ
dI
2
I  R  
L I 
C dt
dt
• Using the fact that
dQ
I 
dt
and
2
dI d Q

2
dt
dt
Q dQ
dI
I  R  
L I 
C dt
dt
2
2
Q
d Q
I  R   I  L  I 
2
C
dt
2
• Factor out an I and set up the resulting quadratic equation:
2
Q
d Q
I  R   I  L  I 
2
C
dt
2 
Q
d Q
I ( I  R )  I    L 

C

dt


2
L
L
2
d Q
2
dt
2
d Q
dt
2
Q
  I R  0
C
dQ Q
R

0
dt C
dQ
I 
dt
• The RLC circuit is analogous to the damped harmonic
oscillator.
• The equation of motion for the damped harmonic
oscillator is:
2
m
d x
dt 2
dx
b 
k x  0
dt
• Comparing the two equations:
L
d 2Q
dt 2
dQ Q
R

0
dt C
– 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.
d 2Q
dQ Q
• When R = 0 , L 
reduces to a

R



0
dt C
dt 2
simple LC
circuit and
the charge and current oscillate sinusoidally in time.
• When R is small, the solution is:
Q  Qm
R t
 e 2L
 cos d  t 
1
2 2
 1
 R 
where wd  



 L  C  2  L  
• The charge will oscillate with damped harmonic motion
in analogy with a mass-spring system moving in a viscous
medium.
• 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L
Rc 
C
• 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
R
C