Post on 02-Apr-2015
transcript
1
Chapter 30
2
Mutual Inductance
Consider a changing current in coil 1We know that
B1=0i1N1
And if i1 is changing with time, dB1/dt=0N1 d(i1)/dt
But a changing B-field across coil 2 will initiate an EMF2 such that
EMF2=-N2A2 dB1/dtSince dB1/dt is proportional to di1/dt then the
dt
diMEMF
dt
diEMF
12
12
Where M is the mutual inductance which is based on the sizes of the coils, and the number of turns
3
Mutual-mutual Inductance
dt
diMEMF
dt
diEMF
21
21
But it could be that the changes are happening in coil 2. Then
It turns out that this value of M is identical to the previously discussed M so
2
11
1
2212
21 i
N
i
NMwhere
dt
diMEMFand
dt
diMEMF BB
4
My favorite unit—the henry
The Henry (H) is the unit of inductanceEquivalent to:
1H=1 Wb/A = 1 V*s/A = 1 *s = 1 J/A2
H is large unit; typically we use small units such as mH and H.
5
Self Inductance
But a coil of wire with a changing current can produce an EMF within itself.
This EMF will oppose whatever is causing the changing current
So a coil of wire takes on a special name called the inductor
6
Inductor
Definition of inductance is the magnetic flux per current (L)
For an N-turn solenoid, L is L= N/I N turns= (n turns/length)*(l length)
The near center solution of inductance depends only on geometry
Electrical symbol
Anl
LcenterNear
lAnL
i
NLIf
liAnN
inBwherenlBAN
20
20
20
0
7
Inductor
Electrical symbol
dt
diL
dt
dNEMF
ifand
NLii
NLIf
Again, the EMF acts to oppose the change in current
i (increasing)
VL
High potential
Low potential
acts like a
i (decreasing)
VL
Low potential
High potential
acts like a
8
RL Circuits
Initially, S is open so at t=0, i=0 in the resistor, and the current through the inductor is 0.
Recall that i=dq/dt
B
A
V
S
R
L
9
Switch to A
L
Rt
L
Rt
eL
V
dt
diande
R
Vi
Ansatzdt
diLiRV
Vdt
diLiR
1
0B
A
V
S
R
L
Initially, the inductor acts against the changing current but after a long time, it behaves like a wire
i
H
L
10
Voltage across the resistor and inductor
L
Rt
L
Rt
R eVReR
ViRV 11
Potential across resistor, VR
L
Rt
L
L
Rt
L
VeV
eL
VL
dt
diLV
Potential across capacitor, VC
At t=0, VL=V and VR=0
At t=∞, VL=0 and VR=V
B
A
V
S
R
L
11
L/R—Another time constant
L/R is called the “time constant” of the circuit
L/R has units of time (seconds) and represents the time it takes for the current in the circuit to reach 63% of its maximum value
When L/R=t, then the exponent is -1 or e-1
L=L/R
12
Switch to B
The current is at a steady-state value of i0 at t=0
L
Rt
eiti
dt
diL
dt
dqR
dt
diLiR
0)(
0B
A
V
S
R
L
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Energy Considerations
)2
1(
2
1 22 mvKErecallLiU
dt
diLi
dt
dUP
dt
diLiVi
ViPanddt
diLV
B
B
Rate at which energy is supplied from battery
Rate at which energy is stored in the magnetic field of the inductor
Energy of the magnetic field, UB
14
Energy Density, u
Consider a solenoid of area A and length, l
22
2
1
21
20
0
2
022
0
20
2
Euand
Bu
inBbutinu
Anl
Lbut
lA
Liu
lA
U
volume
Uu
EB
BB
Energy stored at any point in a magnetic field
Energy stored at any point in a magnetic field
15
L-C Oscillator – The Heart of Everything
CL
LCLC
tQLC
tQ
so
tQdt
qd
itQdt
dq
tQqAnsatz
qLCdt
qd
C
q
dt
qdL
dt
qd
dt
dq
dt
d
dt
di
dt
dqi
C
q
dt
diL
10
1
0cos1
cos
cos
sin
cos:
01
0
0
22
2
22
2
2
2
2
2
2
2
If the capacitor has a total charge, Q
16
Perpetual Motion?
17
Starting Points
Charge qCurrent i
t
The phase angle, , will determine when the maximum occurs w.r.t t=0The curves above show what happens if the current is 0 at t=0
18
Energy considerations
A quick and dirty way to solve for i at any time t in terms of Q & q
At t=0, the total energy in the circuit is the energy stored in the capacitor, Q2/2C
At time t, the energy is shared between the capacitor and inductor (q2/2C)+(1/2 Li2)
Q2/2C= (q2/2C)+(1/2 Li2)221
qQLC
i
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Oscillators is oscillators is oscillators
20
Give me an “R”!
Consider adding a resistor, R to the circuit
The resistor dissipates the energy. For example, consider a child on a swing. His/her father pushes the child and gets the child swinging. In a perfect system, the child will continue swinging forever.
The resistor provides the same action as if the child let their feet drag on the ground. The amplitude of the child’s swing becomes smaller and smaller until the child stops.
The current in the LRC circuit oscillates with smaller and smaller amplitudes until there is no more current
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Mathematically
2
22
2
2
2
2
2
4
cos)(
0
0
L
R
where
tQetq
AnsatzC
q
dt
dqR
dt
qdL
dt
qd
dt
di
dt
dqi
iRC
q
dt
diL
L
Rt
If R is small,underdamped
When oscillation stops due to R, critically damped
Very large values of R, overdamped
22
Why didn’t I use a voltage source?
The practical applications of the LC, LR, and LRC circuits depend on using a sinusoidally varying voltage source:An AC voltage source