Self Inductance
To Calculate:
L = NΦ I
1.
2.
3.
4.
Assume a current I is flowing in your device
Calculate the B field due to that I
Calculate the flux due to that B field
Calculate the self inductance (divide out I)
The Effect: Back EMF:
ε
dI
≡ −L
dt
Inductors hate change, like steady state
They are the opposite of capacitors
P25- 3
LR Circuit
I (t ) =
ε
R
− t /τ
1
−
e
(
)
t=0+: Current is trying to change. Inductor works as
hard as it needs to to stop it
t=∞: Current is steady. Inductor does nothing.
P25- 4
LR Circuit: AC Output Voltage
Output Voltage
Output (V)
3
2
1
Inductor (V)
0
3
Voltmeter across L
2
1
0
-1
-2
-3
Current
I (A)
0.15
0.10
0.05
0.00
0.00
0.01
0.02
Time (s)
0.03
P25-
5
Non-Ideal Inductors
Non-Ideal (Real) Inductor: Not only L but also some R
=
In direction of current:
ε
dI
= − L − IR
dt
P25- 6
LR Circuit w/ Real Inductor
Due to Resistance
1. Time constant from I or V
2. Check inductor resistance from V just before switch
P25-
7
Mass on a Spring
(1)
(3)
(2)
What is Motion?
d 2x
F = − kx = ma = m 2
dt
2
d x
m 2 + kx = 0
dt
(4)
Simple Harmonic Motion
x(t ) = x0 cos(ω 0t + φ )
x0: Amplitude of Motion
ω0 =
φ: Phase (time offset)
k
= Angular frequency
m
P25- 10
Mass on a Spring: Energy
(1) Spring
(2) Mass
x(t ) = x0 cos(ω 0t + φ )
(3) Spring
(4) Mass
x '(t ) = −ω 0 x0 sin(ω 0t + φ )
Energy has 2 parts: (Mass) Kinetic and (Spring) Potential
2
1 ⎛ dx ⎞ 1 2 2
K = m ⎜ ⎟ = kx0 sin (ω 0t + φ )
Energy
2 ⎝ dt ⎠
2
sloshes back
1 2 1 2 2
and forth
U s = kx = kx0 cos (ω 0t + φ )
2
2
P25- 11
Simple Harmonic Motion
Amplitude (x0)
1
Period (T ) =
frequency (f )
2π
=
angular frequency (ω )
x(t ) = x0 cos(ω0t − φ )
Phase Shift (ϕ ) =
π
2
P25- 12
Analog: LC Circuit
Mass doesn’t like to accelerate
Kinetic energy associated with motion
2
dv
d x
1 2
F = ma = m = m 2 ; E = mv
2
dt
dt
Inductor doesn’t like to have current change
Energy associated with current
2
dI
d q
1 2
ε = − L = − L 2 ; E = LI
dt
dt
2
P25- 14
Analog: LC Circuit
Spring doesn’t like to be compressed/extended
Potential energy associated with compression
1 2
F = − kx; E = kx
2
Capacitor doesn’t like to be charged (+ or -)
Energy associated with stored charge
1
11 2
ε = q; E =
q
C
2C
F → ε ; x → q; v → I ; m → L; k → C −1
P25- 15
LC Circuit
1. Set up the circuit above with capacitor, inductor,
resistor, and battery.
2. Let the capacitor become fully charged.
3. Throw the switch from a to b
4. What happens?
P25- 16
LC Circuit
It undergoes simple harmonic motion, just like a
mass on a spring, with trade-off between charge on
capacitor (Spring) and current in inductor (Mass)
P25- 17
LC Circuit
Q
dI
dQ
−L
=0 ; I =−
C
dt
dt
2
d Q 1
Q=0
+
2
dt
LC
Simple Harmonic Motion
Q(t ) = Q0 cos(ω 0t + φ )
ω0 =
Q0: Amplitude of Charge Oscillation
φ: Phase (time offset)
1
LC
P25- 19
LC Oscillations: Energy
Notice relative phases
⎛ Q0 2 ⎞ 2
Q
=⎜
UE =
⎟ cos ω0t
2C ⎝ 2C ⎠
2
2
⎛
⎞
Q
1 2 1
U B = LI = LI 0 2 sin 2 ω0t = ⎜ 0 ⎟ sin 2 ω0t
2
2
⎝ 2C ⎠
Q 2 1 2 Q0 2
U = UE +UB =
+ LI =
2C 2
2C
Total energy is conserved !!
P25- 20
Damped LC Oscillations
Resistor dissipates
energy and system
rings down over time
⎛ R ⎞
Also, frequency decreases: ω ' = ω0 − ⎜ ⎟
⎝ 2L ⎠
2
2
P25- 22