Perfect Conductivity Lecture 2
Terry P. Orlando
Dept. of Electrical Engineering
MIT
September 13, 2005
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Outline
1. Persistent Currents
2. Parts of a Physical Theory
3. Circuits and Time Constants
4. Distributive Systems and Time
constants
A.Quasistatics
B.MagnetoQuasiStatics (MQS)
Massachusetts Institute of Technology
6.763 2005 Lecture 2
1
Persistent Currents
L
R
If the field is turned off, then
If the loop is made out of a superconductor,
Experimentally the dc resistivity of a superconductor is at least as small
. The superconducting state is “truly” zero dc resistance.
as
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Perfect Conductivity: t << τRL= L/R,
sytem looks like R is zero
Superconductivity: for all time,
R is zero
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6.763 2005 Lecture 2
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Charging up a superconducting loop
I
L
R
Superconducting material
This Persistent Mode is the basis of MRI magnets, SMES, flux memory….
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Parts of a Physical Theory
1. Governing Laws:
Maxwell’s Equations, Newton’s equations,
2. Constitutive Laws:
Models of the system
like ohm’s law,
3. Summary Relations:
Transfer functions, Dispersion relations
Massachusetts Institute of Technology
6.763 2005 Lecture 2
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LRC Circuit
+
v
iC
i
vC
C
iL.vL
L
R
1 jωL
jωC
R
iR.vR
1. Governing Equations
Current conservation: i=iC + iL
iL =iR
Energy Conservation v = vC = vR + vL
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6.763 2005 Lecture 2
2. Constitutive Relations
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6.763 2005 Lecture 2
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3. Summary Relation
1
jωL
jωC
R
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6.763 2005 Lecture 2
Simpler Circuits and Time Constants
L
C
L
C
R
R
Energy stored
in inductor
Resonant transfer of
energy between L and C
Energy stored in
capacitor
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6.763 2005 Lecture 2
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Reducing the Circuit to a simpler form ?
1
jωL
jωC
R
?
?
?
C R
L
C L
R
Massachusetts Institute of Technology 6.763 2005 Lecture 2
Order of time constants
a
b
a
ω
1/τLR
τLR > τRC
Low
frequency
circuit
1/τLC
1/τRC
1/τRC
τLR < τRC
Low R
a
L
R
b
ω
1/τLC
High R
a
R
b
L
Low
frequency
circuit
1/τLR
CR
b
R
C
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6.763 2005 Lecture 2
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Moral of time constants
If you know what frequency range you
want to study or what physics dominates
the problem,
then you can solve a simpler problem.
Useful, especially in more complex situations.
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Distributed Systems
1. Governing Equations: Maxwell’s Equations
Faraday’s Law
Ampere’s Law
Gauss Law
Gauss’ Magnetic Law
Conservations laws
Charge conservation
Also Poynting’s
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6.763 2005 Lecture 2
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Distributied systems con’t
2. Constitutive Relations
Local in space,
linear time invariant
Ohm’s Law
3. Summary relations
Complex: Search first for first order in time approximation
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6.763 2005 Lecture 2
Quasistatic Limit
Speed
of light
Length scale
of system
Frequency
(angular)
Wavelength of
E&M wave
If the dimensions of a structure are much less than the
wavelength of an electromagnetic field interacting with it, the
coupling between the associated electric and magnetic fields is
weak and a quasistatic approximation is appropriate.
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6.763 2005 Lecture 2
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Time Constants
Electromagnetic coupling time
Charge relaxation time
Magnetic diffusion time
Image removed for copyright reasons.
Please see: Figure 2.9, page 34, from Orlando, T., and
K. Delin. Foundations of Applied Superconductivity.
Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234.
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Order of time constants
a
b
a
ω
1/τm
1/τem
τm > τc
1/τc
ω
1/τc
τm < τe
High
conductivity
b
1/τem
Low
conductivity
EQS
MQS
Low
frequency
circuit
1/τm
a
L
R
a
R
b
L
Low
frequency
circuit
CR
b
R
C
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6.763 2005 Lecture 2
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MagnetoQuasiStatics
Solve first
Image removed for copyright reasons.
Please see: Figure 2.9, page 34, from Orlando, T., and
K. Delin. Foundations of Applied Superconductivity.
Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234.
Solve for E once B is found
Boundary conditions:
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6.763 2005 Lecture 2
MQS: Magnetic Diffusion Equation
For a metal B = µ0 H, D = ε0 E and J = σ0E, so that
Image removed for copyright reasons.
Please see: Figure 2.9, page 34, from Orlando, T., and
K. Delin. Foundations of Applied Superconductivity.
Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234.
Magnetic Diffusion Equation
Massachusetts Institute of Technology
6.763 2005 Lecture 2
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