Transcript Source-Free RLC Circuit
Parallel RLC Network
Objective of Lecture
Derive the equations that relate the voltages across a resistor, an inductor, and a capacitor in parallel as: the unit step function associated with voltage or current source changes from 1 to 0 or a switch disconnects a voltage or current source into the circuit.
Describe the solution to the 2 nd the condition is: order equations when Overdamped Critically Damped Underdamped
RLC Network
A parallel RLC network where the current source is switched out of the circuit at t = t o .
Boundary Conditions
You must determine the initial condition of the inductor and capacitor at t < t o and then find the final conditions at t = ∞s.
Since the voltage source has a magnitude of 0V at t < t o i L (t o ) = Is and v(t o ) = v C (t o ) = 0 V
v L (t o ) = 0 V and i C (t o ) = 0 A
Once the steady state is reached after the voltage source has a magnitude of Vs at t > t o , replace the capacitor with an open circuit and the inductor with a short circuit.
i L (∞s) = 0 A and v(∞s) = v C (∞s) = 0 V
v L (∞s) = 0 V and i C (∞s) = 0 A
Selection of Parameter
Initial Conditions
i L (t o ) = Is
and v(t o ) = v C (t o ) = 0V v L (t o ) = 0V and i C (t o ) = 0A Final Conditions
i L (∞s) = 0A
and v(∞s) = v C (∞s) = oV v L (∞s) = 0V and i C (∞s) = 0A Since the current through the inductor is the only parameter that has a non-zero boundary condition, the first set of solutions will be for i L (t).
Kirchoff’s Current Law
i R
(
t
)
i L
(
t
)
i C
(
t
) 0
v
(
t
)
v R
(
t
)
v L
(
t
)
v C
(
t
)
v R
(
t
)
i L
(
t
)
C R dv C
(
t
)
dt
0
v L
(
t
)
v
(
t
)
L di L
(
t
)
dt LC d
2
i L
(
t
)
dt
2
L R di L
(
t
)
i L
(
t
) 0
dt d
2
i L
(
t
)
dt
2 1
RC di L
(
t
)
dt i L
(
t
)
LC
0
General Solution
s
2 1
RC s
1
LC
0
s
1 1 2
RC
s
2 1 2
RC
1 2
RC
2 1
LC
1 2
RC
2 1
LC
s
1
s
2 2
o
2 2
o
2 1 2
RC
o
1
LC s
2 2
s
o
2 0 Note that the equation for the natural frequency of the RLC circuit is the same whether the components are in series or in parallel.
Overdamped Case
> o implies that L > 4R 2 C
i
s 1 and s 2
L
1 (
t
) are negative and real numbers
A
1
e s
1
t i L
2 (
t
)
t
t
A t o
2
e s
2
t i L
(
t
)
i L
1 (
t
)
i L
2 (
t
)
A
1
e s
1
t
A
2
e s
2
t
Critically Damped Case
o implies that L = 4R 2 C s 1 = s 2 = = -1/2RC
i L
(
t
)
A
1
e
t
A
2
te
t
Underdamped Case
< o implies that L < 4R 2 C
s
1
s
2 2 2
o
2
o
2
j
j
d d
d
o
2 2
i L
(
t
)
e
t
[
A
1 cos
d
t
A
2 sin
d
t
]
Other Voltages and Currents
Once current through the inductor is known:
v L
(
t
)
v L
(
t
)
di L
(
t
)
L v C
(
t dt
)
v R
(
t
)
i i C
(
t
)
R
(
t
)
C dv C
(
t dt
v R
(
t
) /
R
)
Summary
The set of solutions when t > t o for the current through the inductor in a RLC network in parallel was obtained.
Selection of equations is determine by comparing the natural frequency o to .
Coefficients are found by evaluating the equation and its first derivation at t = t o and t = ∞s.
The current through the inductor is equal to the initial condition when t < t o Using the relationships between current and voltage, the voltage across the inductor and the voltages and currents for the capacitor and resistor can be calculated.