Perfect Conductivity Lecture 2
Terry P. Orlando
Dept. of Electrical Engineering
MIT
September 9, 2003
Massachusetts Institute of Technology
6.763 2003 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 2003 Lecture 2
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
as
. The superconducting state is “truly” zero dc resistance.
Massachusetts Institute of Technology
6.763 2003 Lecture 2
Perfect Conductivity: t << τRL= L/R,
sytem looks like R is zero
Superconductivity:
R is zero
for all time,
Massachusetts Institute of Technology
6.763 2003 Lecture 2
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 2003 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 2003 Lecture 2
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
Massachusetts Institute of Technology
6.763 2003 Lecture 2
2. Constitutive Relations
Massachusetts Institute of Technology
6.763 2003 Lecture 2
3. Summary Relation
1
jωL
jωC
R
Massachusetts Institute of Technology
6.763 2003 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
Massachusetts Institute of Technology
6.763 2003 Lecture 2
Reducing the Circuit to a simpler form ?
1
jωL
jωC
R
?
?
L
?
C L
R
Massachusetts Institute of Technology
6.763 2003 Lecture 2
C R
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
b
R
L
Low
frequency
circuit
ω
1/τLC
1/τLR
High R
a
CR
Massachusetts Institute of Technology
6.763 2003 Lecture 2
b
R
C
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 2003 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
Massachusetts Institute of Technology
6.763 2003 Lecture 2
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
Massachusetts Institute of Technology
6.763 2003 Lecture 2
Quasistatic Limit
Speed
of light
Length scale
of system
Wavelength of
E&M wave
Frequency
(angular)
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.
Massachusetts Institute of Technology
6.763 2003 Lecture 2
Time Constants
Electromagnetic coupling time
Charge relaxation time
Magnetic diffusion time
i
v
h
σ0,µ0,ε
w
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6.763 2003 Lecture 2
d
Order of time constants
a
b
a
ω
1/τm
1/τem
τm > τc
1/τc
b
1/τc
τm < τe
High
conductivity
1/τem
1/τm
Low
conductivity
EQS
MQS
Low
frequency
circuit
ω
a
L
R
b
R
L
Low
frequency
circuit
a
CR
Massachusetts Institute of Technology
6.763 2003 Lecture 2
b
R
C
MagnetoQuasiStatics
Solve first
i
v
h
σ0,µ0,ε
w
Solve for E once B is found
Boundary conditions:
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6.763 2003 Lecture 2
d
MQS: Magnetic Diffusion Equation
For a metal B = µ0 H, D = ε0 E and J = σ0E, so that
i
v
h
σ0,µ0,ε
w
d
Magnetic Diffusion Equation
Massachusetts Institute of Technology
6.763 2003 Lecture 2