Transcript Fields and Waves I Lecture - Rensselaer Polytechnic Institute

Fields and Waves I
Lecture 19
Maxwell’s Equations & Displacement Current
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. Maréchal
Power Engineering Department
Institut National Polytechnique de Grenoble, France
These Slides Were Prepared by Prof. Kenneth A. Connor Using
Original Materials Written Mostly by the Following:
 Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic
Institute, Troy, NY
 J. Darryl Michael – GE Global Research Center, Niskayuna, NY
 Thomas P. Crowley – National Institute of Standards and
Technology, Boulder, CO
 Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic
Institute, Troy, NY
 Lale Ergene – ITU Informatics Institute, Istanbul, Turkey
 Jeffrey Braunstein – Chung-Ang University, Seoul, Korea
Materials from other sources are referenced where they are used.
Those listed as Ulaby are figures from Ulaby’s textbook.
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Overview
 Usual approximations of Maxwell’s equations
 Displacement Current
 Continuity Equation and boundary conditions
 Quasi-Statics approximation
 Conductors vs. Dielectrics
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Maxwell’s Equations & Displacement Current
Usual approximations
Usual models in physics
Maxwell’s equations Models all electromagnetism
• Models can vary according to
 Time
• Steady state
• Phasor
• Transient
 Frequency
• No
• Low
• High
 Material
• Linear / non linear
• Isotropic / anisotropic
• Hysteretic
 Scale
• Microscopic
• Usual
Maxwell’s equations
• Can be simplified for each model
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Maxwell’s Equations – static models
 
 D  dS   dv  Qencl
 
 E  dl  0

 D  

 E  0
 
 
 H  dl   J  dS  I encl
 
 H  J
 
 B  dS  0

 B  0
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Maxwell’s Equations – quasi static models
 
 
 H  dl   J  dS  I encl
 
 H  J
 
 B  dS  0

 B  0
 
d  
 E  dl   dt  B  dS


B
E  
t
Added term in curl E equation for time varying
current or moving path that gives an electric field
from a time-varying magnetic field.
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Full Maxwell’s Equations
Added term in curl H equation for time varying
electric field that gives a magnetic field.

 
 
D 
 H  dl   j  ds   t  ds
 
 B  dS  0
 
d  
 E  dl   dt  B  dS
 
 D  dS   dv  Qencl

  D
H  j 
t

 B  0


B
E  
t

 D  
First introduced by Maxwell in 1873
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Maxwell’s Equations & Displacement Current
Displacement current
Displacement Current
Ampere’s Law – Curl H Equation
(quasi) Static field
 
 H  j
Time varying field

  D
H  j 
t
Displacement
current density
Integral Form of Ampere’s Law for time varying fields



D 
C H  dl  I c  S t  ds
Displacement
current
IC – Conduction Current [A] linked to a conductivity property

D – Electric Flux Density (Electric Displacement) [in C/unit area]

jc
– Conduction Current Density (in A/unit area)
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Displacement Current


 H  dl  I c  I d  I
Total current
C
Conduction current density
 


I c   jc  ds   E  ds


( jc  E)



D 
I d   jd  ds  
 ds
t
S
S
Displacement current density
Connection between electric and magnetic fields under
time varying conditions
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Example: Parallel Plate Capacitor
What are the meanings of these currents ?
+
Vs (t )
-
Imaginary
surface S1
++++++++++++++++++++++++++++
Imaginary
surface S2
---------------------------
E-Field
Vs (t )  V0 cost
S1=cross section of the wire
S2=cross section of the capacitor
I1c, I1d : conduction and displacement currents in the wire
I2c, I2d : conduction and displacement currents through the capacitor
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Example: Parallel Plate Capacitor
The wire is considered as a perfect conductor
 
D E 0
I1d = 0
+
From circuit theory:
Vc (t )
Vs (t )
Vc  Vs (t )
-
dVC
d
I 1c  C
 C (V0 cost )  CV0 sin t
dt
dt
Total current in the wire:
I1  I1c  CV0 sin t
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Example: Parallel Plate Capacitor
The dielectric is considered as perfect (zero conductivity)
Electrical charges can’t move physically through a perfect
dielectric medium
I2c= 0
no conduction between the plates
The electric field between the capacitors
 VC
V
E
aˆ y  0 cos taˆ y
d
d
d :spacing between the plates
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Example : Parallel Plate Capacitor
The displacement current I2d
I 2d

D 

 ds
t
S
   V0

  
costaˆ y   aˆ y ds
t  d

A
A
  V0 sin t  CV0 sin t
d
I 2d  I1c
Displacement current doesn’t carry real charge, but behaves
like a real current
If wire has a finite conductivity σ then both wire and dielectric
have conduction AND displacement currents
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Order of magnitude
 Consider a conducting wire
• Conductivity = 2.107S/m
• Relative permittivity = 1
• Current = 2 . 10-3 sin(t) A
•  = 109 rad/s
 Find the value of the displacement current
jd  0.8851012 cos(t )
Phase quadrature
9 order of magnitude
Negligible in
conductors
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Maxwell’s Equations & Displacement Current
Maxwell’s equations, boundary conditions
Maxwell’s Equations
 
 H  dl 
  d  
 J  dS  dt  D  dS
 
 B  dS  0
 
d  
 E  dl   dt  B  dS
 
 D  dS   dv  Qencl

  D
 H  J 
t

 B  0


B
E  
t

 D  
Note that the time-varying terms couple electric and magnetic
fields in both directions. Thus, in general, we cannot have one
without the other.
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Fully connected fields

  D
H  j 
t

D

H

 D  


D  E

Material property

E
Sources

j


j  E


B
E  
t


B  H
Material property

B
Maxwell’s equations are fully coupled.
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Continuity Equation
Begin by taking the divergence of Ampere’s Law



D
  H  0   J  
t



 J  
D
t


where we have used the vector identity that the divergence of the curl
of any vector is always equal to zero.
Now from Gauss’ Law,




 J  
D  
t
t

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
or
Fields and Waves I


 J  
t
20
Continuity Equation : integral form
 
j  ds
Now, integrate this equation over a volume.


   Jdv    t dv
From the divergence theorem, the left hand side is
Ulaby

 
   Jdv   J  dS
For a fixed volume, we can move the derivative outside the integral on
the right to obtain the final form of this equation.
 
dQencl
d
 J  dS   dt  dv   dt
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Continuity Equation
Differential and integral forms of the Continuity Equation (Equation for
Charge and Current Conservation)
 
dQencl
d
 J  dS   dt  dv   dt


 J  
t
I3
I2
For statics, the current leaving some
volume must sum to zero
I1
I4
I5
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If the charge is time varying, sum of
currents is equal to this variation.
A general form of the Kirchoff Current Law.
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Summary

  D
H  j 
t

D

 D  


D  E


E


 J  
t

H

j


j  E


B
E  
t


B  H

B
Maxwell’s equations are fully coupled.
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Boundary conditions
Boundary conditions derived for electrostatics and magnetostatics
remain valid for time-varying fields:
- For instance, tangential Components of E
w
Material 1
h
h << w
Material 2
 
 E  dl  0
E2t  E1t  w  0
 E1t  E2t
Note: If region 2 is a conductor E1t = 0
Outside conductor E and D are normal to the surface
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Boundary Conditions
Case 1: REGIONS 1 & 2 are DIELECTRICS (Js = 0)
D1n  D2n   s
E1t  E2t
Bn1  Bn 2
Material 1
dielectric
s
js  0
Material 2
dielectric
H t1  H t 2
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Boundary Conditions
REGIONS 1 is a DIELECTRIC
REGION 2 is a CONDUCTOR, D2 = E2 =0
Case 2:
D2n  0
D1n   s
E1t  E2t  0
H t1  J s
Material 1
js
s
Material 2
conductor
Ht 2  0
Bn1  Bn 2  0
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Maxwell’s Equations & Displacement Current
Quasi static
A quasi-static approach
Because all four equations are coupled, in general, we must solve
them simultaneously.
We will see a general way to do this in the next lecture, which will
lead us to electromagnetic waves.
However, we will first look at the coupled equations as a perturbation
of what we have done so far in electrostatics and magnetostatics.
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Example
A parallel plate capacitor with circular plates and an air dielectric
has a plate radius of 5 mm and a plate separation of d=10 m.
The voltage across the plates is V  5cos t where  2 100kHz
a.
Find D between the plates.
b.
Determine the displacement current
density, D/t.
c.
Compute the total displacement
current,  D/t  ds , and compare it
with the capacitor current, I = C dV/dt.
d.
What is H between the plates?
e. What is the induced emf ?
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A quasi-static approach
The electric field for a parallel plate capacitor driven by a time-varying
source is

s ( t )
V (t )
E (t )  
z  
z
d

The time-varying electric field now produces a source for a magnetic
field through the displacement current . We can solve for the magnetic
field in the usual manner.
 
 H  dl 
  d  
 J  dS  dt  D  dS
0
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A quasi-static approach
The total displacement current between the capacitor plates
 a 2 V (t )
  V (t ) 
I D     
 z  dS 
t  d 
d
t
Using phasor notation for the voltage and current

V (t )  Re Vo e
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j t

I D  j
Fields and Waves I
a 2
d
Vo
31
A quasi-static approach
Applying Ampere’s Law to a circular contour with radius r < a, the
fraction of the displacement current enclosed is
r2
ID
a2
Ampere’s Law then gives us
 
r2
 H  dl  H 2r  I D a 2
r
a 2
r
r
H  I D
 j
Vo
 j Vo
2
2
d
2d
2a
2a
Thus, we now have both electric and magnetic fields between the plates.
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Example – Displacement Current
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Example – Displacement Current
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A quasi-static approach

 D
 H 
t

D

H
2


D  E
3
1

E
?


B
E  
t


B  H

B
In general, we should now use this magnetic field to find a correction to
the electric field by plugging it into Faraday’s Law. However, under what
we call quasi-static conditions, we only need to find this first term.
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Validity domain of quasi-static approach
Maxwell’s Equations.
Need a simultaneous solution for the electric and magnetic fields
Lead to a wave equation identical in form to the wave
equation found for transmission lines
Quasi static approach
Valid if the system dimensions are small compared to a
wavelength.
2 2
c 3 108

k



f

5
10
 300m
real meaning of low frequencies.
There is a reasonably complete derivation of this condition in Unit
9 of the class notes.
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Conductors vs. Dielectrics
The analysis of the capacitor under time-varying conditions assumed
that the insulator had no conductivity. If we generalize our results to
include both  and  we will have both a conduction and a
displacement current.
2
V

a
2 o
2 Vo
I  I C  I D  a
 j
Vo    j a
d
d
d
The material will behave mostly like a dielectric when
IC
ID
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

 1

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Conductors vs. Dielectrics
The material will behave mostly like a conductor when
IC
ID


 1

Loss tangent of the material.

tan  

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