Transcript No Slide Title

Fundamentals of Electromagnetics:
A Two-Week, 8-Day, Intensive Course for
Training Faculty in Electrical-, Electronics-,
Communication-, and Computer- Related
Engineering Departments
by
Nannapaneni Narayana Rao
Edward C. Jordan Professor Emeritus
of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, India
Amrita Viswa Vidya Peetham, Coimbatore
August 11, 12, 13, 14, 18, 19, 20, and 21, 2008
6-1
Module 6
Statics, Quasistatics, and
Transmission Lines
Gradient and electric potential
Poisson’s and Laplace’s equations
Static fields and circuit elements
Low-frequency behavior via quasistatics
The distributed circuit concept and the transmission line
6-2
Instructional Objectives
25. Find the static electric potential due to a specified charge
distribution by applying superposition in conjunction
with the potential due to a point charge, and further find
the electric field from the potential
26. Obtain the solution for the potential between two
conductors held at specified potentials, for onedimensional cases in the Cartesian coordinate system
(and the region between which is filled with a dielectric
of uniform or nonuniform permittivity, or with multiple
dielectrics) by using the Laplace’s equation in one
dimension, and further find the capacitance per unit
length (or capacitance in the case of spherical
conductors) of the arrangement
6-3
Instructional Objectives (Continued)
27. Perform static field analysis of arrangements consisting of
two parallel plane conductors for electrostaic,
magnetostatic, and electromagnetostatic fields
28. Perform quasistatic static field analysis of arrangements
consisting of two parallel plane conductors for
electroquastatic and magnetoquasistatic fields
29. Understand the development of the transmission-line
(distributed equivalent circuit) from the field solutions
for a given physical structure
Gradient and Electric Potential
(FEME, Sec. 6.1; EEE6E, Secs. 5.1, 5.2)
6-5
Gradient and the Potential Functions
ax

×A 
x
Ax
ay

y
Ay
az

z
Az



  × A   × A x   × A y   × A z
x
y
z

x
 
x
Ax
0

y

y
Ay

z

z
Az
6-6
Since  B  0,
B can be expressed as the curl of a vector.
Thus
B= A
A is known as the magnetic vector potential.
Then

 × E =   × A 
t
A
  ×
t
6-7
A 

 ×E+
0
t 

A
E+
 
t
A
E =  
t
 is known as the electric scalar potential.
 

 
   ax
 ay
+ az  
y
z 
 x




ax 
ay 
az
x
y
z
is the gradient of .
6-8
ax

 ×  
x
   x
ax

 x

x
0
ay

y
   y
ay

y

y
az

z

z
az

z
   z
6-9
Basic definition of 
Q  x  dx, y  dy, z  dz 
  d
dl

d    d l
P  x, y, z 
From this, we get
d
 
an
dn
   Maximum rate of increase of 
an  direction of the maximum rate of increase, which
occurs normal to the constant  surface.
B
×E= 
t
6-10
D 
×H =J+

t 

 D=
 B=0
B=×A
A
E =  
t

and using  A =  
t
2



 2    2  
t
 Potential function
2

A
2
 A   2    J
t
equations
6-11
    
2
Laplacian of scalar
 A =   A   ×  ×A Laplacian of vector
2
In Cartesian coordinates,
  
  2  2  2
x
y
z
2
2
2
2
 2 A =   2 Ax  ax    2 Ay  a y    2 Az  az
6-12
   0   ×A

 

×E= 
  ×E +
0
t
t 


E+
 
t

E =  
t

For static fields,
0
t
6-13
E   

B
A
B
E d l     d l
A
B
  d 
A
   B
A
  A  B
But,

B
A
E d l  VA  VB
 voltage between A and B
also known as the potential difference between A and
B, for the static case.
6-14
  V
E  V
Given the charge distribution, find V using superposition.
Then find E using the above.
For a point charge at the origin,
V
Since
Q
4  r
V
E  V  
ar
r
  Q 
 
 ar
 r  4  r 

Q
ar
4  r
agrees with the previously known result.
2
6-15
Thus for a point charge at an arbitrary location P
V
Q
4  R
Q
P5.9
P
R
a
dz 
1

z
z
r
a
2
6-16
Considering the element of length dz at (0, 0, z), we have
dV 
L 0 dz 
4  r   z  z  
2
2
Using
z  z   r tan 
2

d z  r sec  d
 L 0 r sec  d
dV 
4  r sec 
 L 0

sec  d
4 
2
6-17
a
V  1
z  a
dV
L 0 

sec  d




4 

L 0

1n  sec   tan   
4 
L 0
sec 2  tan 2

1n
4 
sec 1  tan 1
1
2
1
2
r   z  a   z  a
L 0

1n
2
2
4 
r   z  a   z  a
2
2
for z  a
6-18
Magnetic vector potential due to a current element
 I dl
A
4 R
Analogous to
V
Q
4  R
P
R
I dl
Poisson’s and Laplace’s
Equations
(FEME, Sec. 6.2; EEE6E, Sec. 5.3)
6-20
Poisson’s Equation
For static electric field,
B
×E= 
0
t
E = V
Then from
 D= 
  V   
If  is uniform,   V   

 V  



V 

Poisson’s
equation
6-21
If  is nonuniform, then using
       ,
  V    V  V  
Thus
  V  V    
 V   V   
2V 2V 2V

Assuming uniform , we have
 2  2 
2
x
y
z

2

For the one-dimensional case of V(x), V   
x2

6-22
D5.7
43
x
V  V0  
d 
Anode, x = d
V = V0
Vacuum Diode
Cathode, x = 0
V=0
(a)
V  x d
8
43
43
 1 
 1 
 V0 
  V0  81 2 


 8
1 V0
 V0 2 3 
8
4
6-23
(b)
E x d 8   V  x d 8
 V 
 
ax 
 x  x  d 8
 4V0  x 1 3 
 
  ax 
 3d  d 
 x  d 8
13
4V0  1 

  ax
3d  8 
2V0

ax
3d
6-24
(c)
   x d 8
  0   V  x  d 8
 2V 
 0  2 
 x  x  d 8
 4V0  x  2 3 
 0  2   
 9d  d   x  d 8
2 3
4V0  1 
  0 2  
9d  8 
4V0
23
  0 2  8 
9d
16  0V0

9d 2
Laplace’s Equation
6-25
If   0, Poisson’s equation becomes

 V    0 for uniform 

 V   V    0 for nonuniform 

Let us consider uniform  first
Parallel-plate capacitor
x = d, V= V0
x = 0, V = 0
6-26
Neglecting fringing of field at edges,
V  V  x
2

V

 V  2  0
x
V
A
x
V  Ax  B General solution
Boundary conditions
6-27
V  0 at x  0
V  V0 at x  d
0  0 B  B  0
V0
V0  Ad  0  A 
d
V0
V  x Particular solution
d
V
E   V  
ax
x
V0
  ax
d
6-28
ax   Ex 0 for x  0
S  an D  
ax   Ex  d for x  d
 V0

ax  d ax for x  0

a  V0 a for x  d
 x d x


  V0
 d for x  0




  V0 for x  d
 d


ax
ax

x=d
x

x=0
6-29
Q  S A

For nonuniform ,
area of plates
 V0 A
d
Q A
C 
V0
d
 2V   V  0
For
V  V  x ,
2V    V
 2   
x  x  x

0

  V 

0
x  x 
6-30
Example
x = d, V = V0
x 

   0 1  
 2d 
x = 0, V = 0
  
x  V 
 0 1     0

x   2d  x 
 
x  V 
0
1 



x  2d  x 
x  V

A
1 

 2d  x
6-31
V
A

x 1  2xd
x 

V  2 Ad 1n 1 
 B
 2d 
V  0 for x  0
0  2 Ad 1n 1  B  B  0
V  V0 for x  d
V0
3
V0  2 Ad 1n  A 
2
2d 1n 32
V0
x 

V 
1n 1 

1n 1.5  2d 
6-32
V
ax
x
V0
1

ax
x
2d 1n 1.5 1  2d
Ε  V  
 0 V0
D Ε= 
S
x 0
 S
xd
2d 1n 1.5
 0 V0

2d 1n 1.5
Q  S A 
C
ax
 0 V0 A
2d 1n 1.5
0 A
Q

V0 2d 1n 1.5
0
C

A 2d 1n 1.5
Static Fields and Circuit
Elements
(FEME, Sec. 6.3; EEE6E, Sec. 5.4)
6-34
Classification of Fields
Static Fields ( No time variation;    t = 0)
Static electric, or electrostatic fields
Static magnetic, or magnetostatic fields
Electromagnetostatic fields
Dynamic Fields (Time-varying)
Quasistatic Fields (Dynamic fields that can be
analyzed as though the fields are static)
Electroquasistatic fields
Magnetoquasistatic fields
6-35
Static Fields
For static fields,    t = 0, and the equations reduce to
E  dl  0
x E0
H  dl   J  dS
xH  J
C
C
S
D  dS   dv
S
v
B  dS  0
S
S
J  dS  0
D = 
 B 0
J  0
6-36
Solution for Potential and Field
1
(r) 
4

V
(r )
r  r
dv
Solution for
charge distribution
Q(r )
(r) 
4 r  r 
Solution for
point charge
Q(r )(r  r )
E(r) 
3
4 r  r 
Electric field due
to point charge
6-37
Laplace’s Equation and
One-Dimensional Solution
For   0, Poission’s equation reduces to
2
   0 Laplace’s equation
2
d 
2 0
dx
  Ax + B
6-38
Example of Parallel-Plate Arrangement;
Capacitance
l
w
y
d
V0
z
(a)
x= 0
, 
x
x= d
z
z – l
z=0
S
    
V0
 

x  0,  = V0
E, D
– – – –
z= –l
– x  d,  = 0
– – –
z
z= 0
S
V0
(x) 
(d  x)
d
y
z
(b)
x
6-39
Electrostatic Analysis of Parallel-Plate
Arrangement
V0
E   
ax
d
D
 V0 
wl V
Q 
(wl)

d 
d 0
Q wl
C

V0
d
 V0
d
ax
C
wl
d
Capacitance of the arrangement, F
1 2

We    Ex  (wld) 
2
1  wl  2 1
2
V0  CV0


2 d
2
6-40
Magnetostatic Fields
C
H  dl   J  dS
S B  dS  0
2
 A  – J
S
xH  J
 B 0
Poisson’s equation for
magnetic vector potential
6-41
Solution for Vector Potential
and Field
A(r) 
A(r) 

4

V
J(r )
dv
r  r
I dl( r )
4 r  r 
Solution for
current distribution
Solution for
current element
I dl( r ) x (r  r)
B(r) 
3
4 r  r 
Magnetic field due
to current element
2A = 0
For current-free region
6-42
Example of Parallel-Plate Arrangement;
Inductance
l
y
w
d
z
I0
x= 0
(a)
x
, 
x=d
z – l
z
z= 0
x=0
I0
y
z
H, B
x d
JS
z – l
z
z=0
x
(b)
6-43
Magnetostatic Analysis of
Parallel-Plate Arrangement
(I0 w)az

J S  (I0 w)a x
(I w)a
 0
z
ax

x
Hx
on t he plat ex  0
on t he plat ez  0
on t he plat ex  d
ay
az
0
0 0
Hy
Hz
Bx
0
x
I0
H  ay
w
6-44
Magnetostatic Analysis of
Parallel-Plate Arrangement
B
I0
w
ay
  I0 
 dl 
 
(dl)

I
 w  0
w 
L

I0

dl
w
L
 dl
w
Inductance of the arrangement, H
1
2
1   dl  2 1 2

Wm   H  (wld)  
I

LI
0
2
2 w 
2 0
6-45
Electromagnetostatic Fields
(J  J c   E)
C
E  dl  0
x E0
C
H  dl   J c  dS    E  dS
 x H  Jc   E
S
S D  dS  0
B  dS  0
S
S
 D  0
 B 0
6-46
Example of Parallel-Plate Arrangement
l
y
w
d
z
V0
x=0
, , 
x
(a)
x= d
z
z – l
z= 0
S
JS
Ic
      
H, B
V0
–
z= –l
– –
JS
E,
–
 x  0,   V y
0
z
Jc , D
– –
–
z
S
– x  d,   0
z= 0
x
(b)
6-47
Electromagnetostatic Analysis of
Parallel-Plate Arrangement
V0
E  ax
d
V0
Jc 
d
ax
 V0 

wl
Ic  
(wl)

V
d 
d 0
6-48
Electromagnetostatic Analysis of
Parallel-Plate Arrangement
Ic wl
G

V0
d
Conductance, S
V0
d
R

Ic  wl
Resistance, ohms
Pd  (E 2 )(wld)   wl  V02
d
2
V
2
 GV0  0
R
6-49
Electromagnetostatic Analysis of
Parallel-Plate Arrangement
H  H y (z)a y
H 
 V0
1
Li 
Ic
d
H y
V0

z
d
z ay
0
1
–
z




z l N d  Ic z  –l  l   Hy d(dz
0
1  dl

3 w
Internal Inductance
6-50
Electromagnetostatic Analysis of
Parallel-Plate Arrangement
Alternatively,
0
2
1
1 dl
Li  2 (dw)
H y dz 

z l
3 w
Ic
Equivalent Circuit
V0
C
 wl
d
R d
 wl
1  dl
Li 
3 w
Low Frequency Behavior via
Quasistatics
(FEME, Sec. 6.4; EEE6E, Sec. 5.5)
6-52
Quasistatic Fields
For quasistatic fields, certain features can be analyzed as though
the fields were static. In terms of behavior in the frequency domain,
they are low-frequency extensions of static fields present in a
physical structure, when the frequency of the source driving the
structure is zero, or low-frequency approximations of time-varying
fields in the structure that are complete solutions to Maxwell’s
equations. Here, we use the approach of low-frequency extensions
of static fields. Thus, for a given structure, we begin with a timevarying field having the same spatial characteristics as that of the
static field solution for the structure and obtain field solutions
containing terms up to and including the first power (which is the
lowest power) in w for their amplitudes.
63
6-53
Electroquasistatic Fields
Ig (t)
Vg t   V0 cos w t
S
JS

 


–
z = –l
–
–
–
 
 x 0
y
z
E0
, 
+
–
H1
–
z
– –
–
x d
x
z= 0
64
6-54
Electroquasistatic Analysis of
Parallel-Plate Arrangement
V0
E0 
cos w t a x
d
Hy1
Dx 0 wV0


sin w t
z
t
d
H1 
w V0 z
d
sin w t a y
6-55
Electroquasistatic Analysis of
Parallel-Plate Arrangement
 z l
Ig (t)  w Hy1
 wl 

V
sin
t
 w 
w
0
d 
C
dVg (t)
C
dt
Ig  jw CVg
where
C
 wl
d
wl
d
6-56
Electroquasistatic Analysis of
Parallel-Plate Arrangement

Pin  wd Ex 0 Hy1
z 0
 wl  2

V
sin
t
cos
t
 
w
w
w
0
d 
2
d
1

  CVg 
dt 2
6-57
Magnetoquasistatic Fields

Igt   I0 cos w t

Vg (t)
–
JS –
z– l


S
E1


, 
–
–
x=0
y
z
H0
–
x d
–
z
z= 0
x
6-58
Magnetoquasistatic Analysis of
Parallel-Plate Arrangement
I0
H0  coswt a y
w
By0 wI0
Ex1


sin w t
z
t
w
E1 
wI0z
w
sin w t a x
6-59
Magnetoquasistatic Analysis of
Parallel-Plate Arrangement
Vg (t)  dE x1 
z  l
 dl  I sin t
 w
w
0
 w 
L
dI g (t)
L
dt
Vg  jw L Ig where L 
 dl
w
 dl
w
6-60
Magnetoquasistatic Analysis of
Parallel-Plate Arrangement

Pin  wd Ex1 Hy 0
z l
 dl  2

I0 sin w t cos w t
 
w
w 
d  1 LI 2 
 
dt 2 g 
6-61
Quasistatic Fields in a Conductor
E 0, J c 0 H 0
Ig
y
x 0
Vg t   V0 cos wt
z
, , 
+–
x d
z
z = –l
z= 0
x
(a)
Ex 0
Hy 0
z
z = –l
Hyd1
z = –l
Hyc1
z= 0
(b)
E x1
z
z= 0
(c)
6-62
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
V0
E0 
cos w t a x
d
 V0
J c0   E 0 
cosw t a x
d
 V0z
H0  
cos w t a y
d
By0
Ex1
w V0 z


sin w t
z
t
d
Ex1  
w V0
2d

2
z l
2
sin w t
6-63
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
 H y1
 Ex 0
  Ex1  
z
t

Hy1 
w 2V0
2d
z

2
l
2
w 2V0 z3  3zl2
6d
sin w t 
wV0
d
sin w t
sin wt  wV0z sin wt
d
6-64
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
V0
wV0 2 2
Ex 
cos w t 
z  l sin w t
d
2d

Hy  
 V0 z
d
cos w t 
wV0 z
d

sin w t 

w 2V0 z3  3zl 2
6d
Vg
 2 2
Ex 
 jw
z  l Vg
d
2d

Hy  
z
d
Vg  jw
z
d

Vg  j w

 2 z3  3zl 2
6d
V
g
 sin wt
6-65
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
 z  l
Ig  w H y
2 3
 wl
 wl 
wl

Vg

 jw
 jw

d
3d 
 d
2

 wl
 l 
wl

Yin 
1 jw
 jw



d
d
3


Vg
Ig
wl

 jw

d
1
d
wl
1  jw 
 l 2
3
6-66
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
Yin  jw
 wl
d

1
1

j
w
C

d  jw  dl
R  jwLi
 wl
3w
Equivalent Circuit
V0
C
 wl
d
R d
 wl
1  dl
Li 
3 w
The Distributed Circuit Concept
and the Transmission Line
(FEME, Secs. 6.5, 6.6; EEE6E, Secs. 6.1)
6-68
Waves and the Distributed Circuit Concept
We have seen that quasistatic field analysis of a physical structure
Provides information concerning the low-frequency input behavior
of the structure. As the frequency is increased beyond that for
which the quasistatic approximation is valid, terms in the infinite
series solutions for the fields beyond the first-order terms need to
be included. While one can obtain equivalent circuits for
frequencies beyond the range of validity of the quasistatic
approximation by evaluating the higher order terms, no further
insight is gained through that process, and it is more straightforward to obtain the exact solution by resorting to simultaneous
solution of Maxwell’s equations when a closed form solution is
possible.
78
6-69
Wave Equation
D
E
xH

t
t
B
H
xE 
 
t
t
For the one-dimensional case of
E  Ex z,t  a x and H = Hy z,t a y ,
H y
Ex
 
z
t
2
 Hy
Ex
 
z
t
2
 Ex
 Ex
2
2  
z
t
One-dimensional wave equation
79
6-70
6-71
6-72
6-73
6-74
6-75
6-76
6-77
6-78
6-79
6-80
6-81
6-82
6-83
6-84
6-85
6-86
6-87
6-88
6-89
The End