LCR Circuits
Akshaya Annavajhala
Department of Physics, Case Western Reserve University
Cleveland, OH 44016
Abstract:
Filler
Introduction and Theory:
The picture to the left shows a circuit with an inductor (L),
a resistor (R), and a capacitor (C), and a voltage source (V),
whose properties can be controlled by the two switches S1 and
S2.
When S1 is closed and S2 is left open, a voltage is placed
across the circuit (excluding L, as S2 is left open), charging
the capacitor. When S2 is closed and S1 is left open, A circuit
consisting only of C, R, and L is formed, thus creating an
LCR circuit. In this circuit, the capacitor discharges across the
resistor and inductor, oscillating back and forth as the charge is flipped until all of the
energy is dissipated across the resistor. This action is illustrated in the following figure:
As one can see, this oscillatory motion is very similar to that of a mass on a spring, or a
mechanical oscillator. In a mathematical
sense, this electronic oscillator and its
mechanical analogue are the same, as is
evident from the following formulas:
2
2
0
d Q
dQ
Q
L
R
dt
C
dt
+
+
=
2
2
0
d y
dy
m
b
ky
dt
dt
+
+
=
The analogy is thus made very clear:
L
is
the same as
m
, which is the inertial factor
in the equation, and
b
, which is the
damping force, translates into
R
, while the inverse capacitance is the analogue of the
spring constant
k
, or the restorative force. Using many different types of solution
methods, one can come up with a solution to the equation as:
2
0
1
Q=Q
sin(
)
2
t
R
e
t
LC
L
τ
ϕ


+
where Q
0
is the initial charge and phi is the phase shift of the sine wave. Because Q =
CV, the following equation, which is far more helpful to us, is also true: