RL Circuit
description
Transcript of RL Circuit
RL Circuit
0
0
iRdt
diL
iRVL
iR
didt
L
dt
diLiR
1
t=0, i=0 LRteR
i /1
R/L
Switch to position a
Switch to position bLRtLRt eie
Ri /
0/
Initially, i change is max, thus largest VL. After t>>all voltage is on R, di/dt=0, so VL =0
In a dc circuit, inductor behaves like a short circuit
Inductor & Capacitor in DC Circuit
If there is a sudden change in current or Voltage occurs in a circuit such as close or open a switch, then
Inductor Capacitor
Current (iL) must be continuous,
i.e. i+=i-
Voltage (Vc) must be continous,
i.e. V+=V-
At t>>
0dt
diLVL Short circuit 0
dt
dqiC Open circuit
Magnetic field energy stored in an inductor:
Ridt
diiLi
iRdt
diL
2
Power supplied by battery
Dissipated power
Work stored
2
2
2
1LiU
dt
dU
dt
diL
dt
diiLP
B
B
Concept Check
A battery is connected to a solenoid. When the switch is opened, the light bulb
1. Remain off2. Goes off3. Slowly dims out4. Keeps burning as brightly as it did before
the switch was opened.5. Flares up brightly, then dims and goes out
Answer 5
LC Circuita) Charged C connected L
Vmax=qmax/C, i = 0, di/dt: max
UE=qmax2/2C, max
UB=Li2/2=0b) U=UB+UE
c) imax, q=0, UB max
LC oscillationVmax=qmax/C, i=0
UE=qmax2/2C, max
UB=Li2/2=0
Speed of charging depends on L, C
UE=q2/2C, UB=Li2/2
q=0, imax
UE=q2/2C=0UB=Limax
2/2, max
Vmax=qmax/C, i=0
UE=qmax2/2C, max
UB=Li2/2=0
The charge starts to flow back the other way, resulting opposite current
LC oscillation
LC oscillation The oscillations continuous indefinitely in the absence of loss (R=0)
The Vc (or charges) is out of phase with i, i.e. Vc max. at i=0, vice versa.
LC circuit Oscillating block-spring systems
q Displacement: x
i=dq/dt v=dx/dt
L m
C 1/k
UB=Li2/2 Uk=mv2/2
UE=q2/2C U=kx2/2
mkLC
1
0d
m
energyconstant for 0/2
1
2
1
2
2
22
kxdt
x
dtdU
kxmvU
tXx cos
tQqC
q
dt
q
cos
0d
L2
2
LC oscillation Circuit
0/
22
1 22
dt
dq
C
q
dt
diLidtdU
C
qLiU
0 Q,q 0, tif
cos
0d
L2
2
tQq
C
q
dt
q
fTf
LC
C
tQtLQ
1,
2,
1
0cos
cos2
Concept CheckWhich Circuit takes the least time to fully discharge the capacitors during the oscillation
(a)
LCf
TfLC
2
1,
2,
1
Answer: (b) has smaller Ceq, thus smaller T, fast discharge
(b)
Example: RC circuit
33-19P, In an oscillating LC circuit, L=3.0 mH and C=2.60 F. At t=0 the charge on the capacitor is zero and the current is 2.00 A. (a) what is the maximum charge that will appear on the capacitor? (b) In terms of the period T of the oscillation, how much time will elapse after t=0 until the energy stored in the capacitor will be increasing at its greatest rate? c) What is this greatest rate at which energy is transferred to the capacitor?
sec41.14/
(sec)1065.5
LC2/2/1
1080.1
1070.21000.32
LC1 Q.i 0, t
cos
sin
4
4
63
Tt
fT
C
LCIQ
at
tQdt
dqi
tQq
tC
Q
C
ttQ
dt
dU
C
tQ
C
qU
E
E
2sin2
cossin
2
sin
2
2
2
222
Damped and Forced Oscillations
Damped Oscillation Forced Oscillation