Transcript Slide 1

Time - Varying
Fields
EEL 3472
Time-Varying Fields
Time-Varying Fields
Stationary charges
electrostatic fields
Steady currents
magnetostatic fields
Time-varying currents
electromagnetic fields
Only in a non-time-varying case can electric and magnetic fields be considered
as independent of each other. In a time-varying (dynamic) case the two fields
are interdependent. A changing magnetic field induces an electric field, and vice
versa.
2
EEL 3472
Time-Varying Fields
The Continuity Equation
Electric charges may not be created or destroyed (the principle of conservation
of charge).
Consider an arbitrary volume V bounded by surface S. A net charge Q exists
within this region. If a net current I flows across the surface out of this region,
the charge in the volume must decrease at a rate that equals the current:
I 
 J  dS  
S
Divergence theorem
dQ
d

dt
dt
   Jdv  
V
V
V
dv
t

V
V
dv
Partial derivative
because V may be
a function of both
time and space
This equation must hold regardless of the choice of V, therefore the integrands
must be equal:
the equation of
3
V
(
A
/
m
)
J  
continuity
t
For steady currents
J 0
Kirchhoff’s current
law
follows from this
that is, steady electric currents are divergences or solenoidal.
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EEL 3472
Time-Varying Fields
Displacement Current
For magnetostatic field, we recall that
 H  J
Taking the divergence of this equation we have


  H  0   J
However the continuity equation requires that
J  
V
0
t
Thus we must modify the magnetostatic curl equation to agree with the
continuity equation. Let us add a term to the former so that it becomes
 H  J  J d
where J is the conduction current density J   E E , and J d is to be determined
and defined.
4
EEL 3472
Time-Varying Fields
Displacement Current continued
Taking the divergence we have


  H  0   J   J d
  J d    J
In order for this equation to agree with the continuity equation,
Gauss’ law
  J d    J 


V

D

D  
t
t
t
Jd 
D
t
Ñ ÑH = J +
Ñ ¶D Ñ
ÑÑxH Ñds= ÑÑÑJ + ¶t ÑÑÑds
S
S
displacement current density
¶D
¶t
¶D
× ds
¶t
S
ò H × dl = I + ò
L
Stokes’ theorem
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EEL 3472
Time-Varying Fields
Displacement Current continued
A typical example of displacement current is the current through a capacitor
when an alternating voltage source is applied to its plates. The following
example illustrates the need for the displacement current.
Using an unmodified
form of Ampere’s law
dQ

H

dl

J

dS

I

I

enc


dt
L
S1

 H  dl   J  dS  I enc  0
 L
S2
Jd  0
(no conduction current
flows through S2
( J =0))
To resolve the conflict we need to include J d in Ampere’s law.
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EEL 3472
Time-Varying Fields
Displacement Current continued
Charge -Q
Charge +Q
Jd 
I
Surface S1
Charge -Q
Path L
Charge +Q
dD
dt
The total current density is J  J d .In
the first equation J d  0 so it
remains valid. In the second
equation J  0 so that
Q
 H  dl   Jd  d S 
L
S2

dQ
I
dt
d
DdS
dt S2
J 0
 DdS  Q
S1  S 2
 DdS  0
S1
I
Surface S2
So we obtain the same current for
either surface though it is
conduction current in S1 and
displacement current in S2 .
Path L
7
EEL 3472
Time-Varying Fields
Faraday’s Law
Faraday discovered experimentally that a current was induced in a conducting
loop when the magnetic flux linking the loop changed. In differential (or point)
form this experimental fact is described by the following equation
x E  
B
t
Taking the surface integral of both sides over an open surface and applying
Stokes’ theorem, we obtain
Integral form
 E  dl  
L


B

dS


t S
t
where  is the magnetic flux through the surface S.
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EEL 3472
Time-Varying Fields
Faraday’s Law continued
Time-varying electric field is not conservative.
Suppose that there is only one
unique voltage VAB  VA  VB .
Then
B
V AB   E  d l   E  d l
L1
L2
However,
Path L1
 E  dl 
L
Path L2
A
The effect of electromagnetic induction. When
time-varying magnetic fields are present, the
value of the line integral of E from A to B
may depend on the path one chooses.
9
 E  dl 
L1



VAB
 E  dl  
L2




t
VAB
VAB  VAB  0
Thus VAB can be unambiguously
defined only if  / t  0 .(in
practice, if   than the
dimensions of system in question)
EEL 3472
Time-Varying Fields
Faraday’s Law continued
According to Faraday’s law, a time-varying magnetic flux through a loop of wire
results in a voltage across the loop terminals:

V 
t
The negative sign shows that the induced voltage acts in such a way as to
oppose the change of flux producing it (Lenz’s law).
2
1
 E  dl   E  dl   E  dl  
Induced magnetic field Bind (t) (when
circuit is closed)
L
10

t

V12   N
t
2
 E  dl  V12  
1
The terminals are far away from
the time-varying magnetic field
No contribution from 2-3 and 4-1
because the wire is a perfect
conductor
2
Increasing time-varying
magnetic field B(t )
Direction of the
integration path
I
3
- 2
R
+ 1
4
A time-varying magnetic flux through a
loop wire results in the appearance of a
voltage across its terminals.
1

t
If N-turn coil instead
of single loop
Consider now terminals
3 and 4, and 1 and 2
2
3
1
4
 E  dl   E  dl  0
V12  V43
2
3
1
4
Integration path
EEL 3472
Time-Varying Fields
Faraday’s Law continued
Example: An N-turn coil having an area A rotates in a uniform magnetic field B
M  2n The speed of rotation is n revolutions per second. Find the voltage at
the coil terminals.
   B  dS
B
S
 BA sin
  2nt

v(t )  N
Rotation
Shaft
dS  dAen
90
o

cos(90   )  sin 
11

rad
 N

t

BA sin2nt 
t
 2

nNBA
nt )



 cos(2
amplitude

EEL 3472
Time-Varying Fields
Faraday’s Law continued
Boundary Condition on Tangential Electric Field
Using Faraday’s law,  E  dl   / t , we can obtain boundary condition on the
L
tangential component of E at a dielectric boundary.
 E  dl  
L
Medium 1
1
E 1 E 2
Medium 2
L2  0
2
S 0
L3
L1

t
L2  0 L4  0   0
(unless B   )
 E  dl   E  dl  0
L1
L3
E 1l  E 2l  0
E 1  E 2 - continuous at the boundary
L4  0
(For the normal component: DN1  DN2 )
at a dielectric boundary
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EEL 3472
Time-Varying Fields
Inductance
A circuit carrying current I produces a magnetic field B which causes a flux
  B  d S to pass through each turn of the circuit. If the medium

surrounding the circuit is linear, the flux
producing it   kI  .
+
I
-
I

is proportional to the current I
For a time-varying current,
according to Faraday’s law, we
have
 (voltage induced
V N
t across coil)
 Nk
I
I
L
t
t
L  Nk 
Magnetic field B produced by a circuit.
13
N


I
I
, H (henry)
Self-inductance L is defined as the
ratio of the magnetic flux linkage 
to the current I.
EEL 3472
Time-Varying Fields
Inductance continued
In an inductor such as a coaxial or a parallel-wire transmission line, the
inductance produced by the flux internal to the conductor is called the internal
inductance Lin while that produced by the flux external to it is called external
inductance Lext ( L  Lin  Lext )
Let us find Lext of a very long rectangular loop of wire for which R  0 ,a  w and
w  h . This geometry represents a parallel-conductor transmission line.
Transmission lines are usually characterized by per unit length parameters.
Wire radius = a
1
I
H
Lext 

I
w
S
h
Finding the inductance per unit length of a parallel-conductor transmission line.
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EEL 3472
Time-Varying Fields
Inductance continued
The magnetic field produced by each of the long sides of the loop is given
I
approximately by
H
2R
  w a
w a
 I  w
I
  2 B  d S  2   o Heˆn   1deˆn   2 o 
d  o ln  
2
 a
S
 a
a
Parallel – conductor
transmission line
Lext 

I

o  w 
ln 
 a
, H/m (per unit length)
The internal inductance Lin for nonferromagnetic materials is usually negligible
compared to Lext .
Let us find the inductance per unit length of a coaxial transmission line. We
assume that the currents flow on the surface of the conductors, which have no
resistance.
h
b
a
I
z
Finding the inductance per unit length of a coaxial transmission line.
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EEL 3472
Time-Varying Fields
Inductance continued
Resistance
 E  dl  
B
x E  
t
L

B  dS

t S
Since R=0, E has only radial component and therefore the segments 3 and 4
I
contribute nothing to the line integral.
2




E

dl

E

dl



H

d
S



H  e  hd e
1
2


t S
t 


v ( z ) v ( z  h )
h I

2 t
dI(z)
v(z  h)  v(z)
L

dt
h
b
1

a
d 
h  b  I
ln 
2  a  t
L
 b
ln 
2  a 
H/m
(from previous work)
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EEL 3472
Time-Varying Fields
General Forms of Maxwell’s Equations
Differential
1
Integral
ò D × ds= ò r
Ñ ÑD = rV
V
S
2
3
4
¶B
¶t
Ñ xH = J +
¶D
¶t
dv
S
¶
ò E × dl = - ¶t ò B× ds
L
Gauss’ Law
V
ò B × ds= 0
Ñ ÑB = 0
Ñ xE = -
Remarks
Nonexistence of
isolated magnetic
charge
Faraday’s Law
s
æ ¶D ö
H
×
dl
=
ò
ò çè J + ¶t ÷ø × ds
L
s
Ampere’s circuital law
In 1 and 2, S is a closed surface enclosing the volume V
In 2 and 3, L is a closed path that bounds the surface S
17
EEL 3472
Time-Varying Fields
Electric fields can originate on positive charges and can end on negative
charges. But since nature has neglected to supply us with magnetic charges,
magnetic fields cannot begin or end; they can only form closed loops.
18
EEL 3472
Time-Varying Fields
Sinusoidal Fields
In electromagnetics, information is usually transmitted by imposing amplitude,
frequency, or phase modulation on a sinusoidal carrier. Sinusoidal (or timeharmonic) analysis can be extended to most waveforms by Fourier and Laplace
transform techniques.
Sinusoids are easily expressed in phasors, which are more convenient to work
with. Let us consider the “curl H” equation.
 H  J 
D
t
H  f x, y, z, t 
Its phasor representation is
  H  J  j D  J  j E
H is a vector function of position, but it is independent of time. The three scalar
components of H are complex numbers; that is, if
H x, y, z, t   f1 x, y, z  cost  1 e x  f 2 x, y, z  cost  2 e y
then
H x, y, z   f1 x, y, z e j1 e x  f 2 x, y, z e j2 e y
19
EEL 3472
Time-Varying Fields
Sinusoidal Fields continued
j t
Time-Harmonic Maxwell’s Equations Assuming Time Factor e
.
Point Form
Integral Form
Ñ ÑD = rV
ò D × ds= ò r
Ñ ÑB = 0
Ñ ÑE = - jw B
Ñ ÑH = J + jw D
V
dv
ò B× ds= 0
ò E× dl = - jw ò B× ds
ò H × dl = ò ( J + jw D) × ds
H ( x, y, z, t ) = ReéëH ( x, y, z) ejwt ùû
20
EEL 3472
Time-Varying Fields
Maxwell’s Equations continued
Electrostatics
Electrodynamics
(a)
Magnetostatics
Free magnetic
charge density
m
(   0)
(c)
(b)
Electromagnetic flow diagram showing the relationship between the potentials and vector
fields: (a) electrostatic system, (b) magnetostatic system, (c) electromagnetic system.
[Adapted with permission from IEE Publishing Department]
21
EEL 3472
Time-Varying Fields
The Skin Effect
When time-varying fields are present in a material that has high conductivity, the
fields and currents tend to be confined to a region near the surface of the
material. This is known as the “skin effect”. Skin effect increases the effective
resistance of conductors at high frequencies.
Let us consider the vector wave equation (Helmholtz’s equation) for the electric
field:
 2 E  j  E  j E
 2 E   2 E x ex   2 E y e y   2 E z ez


If the material in question is a very good conductor, so that  E   , we can
write
 2 E  j E E
Since E  J /  E we also have
 2 J  j E J
Consider current flowing in the +x direction through a conductive material filling
the half-space z<0. The current density is independent of y and x, so that we
have
2






 Jx
 j E J x
z 2
2
J x  Ae

(1 j ) z

Current density at surface J0
(Air) Z>0
(Metal) Z<0
22
x
z
(1 j ) z
 Be

 Ae

(1 j ) z 0

 Be
(1 j ) z
  
Skin depth
y
Magnitude of current density decreases
exponentially with depth
    j


2
 E

1

1
 E f
EEL 3472
Time-Varying Fields
The Skin Effect continued
J x  Ae
 (1 j ) z / 
 Be
(1 j ) z / 

z
 Ae e

j
z

z
 Be e

j
z

As z   , the first term increases and would give rise to infinitely large
currents. Since this is physically unreasonable, the constant A must vanish. From
the condition J x  J o when z=0 we have
Current density magnitude
(0 ) (0 )
decreases exponentially with
Jo  Be e  B
depth. Its phase changes as
z
z
j
and
well.
J ( z)  J e  e 
x
o
At
z0
Jx
10mm
1mm
0.1mm
z 
1
 J o  0.37 J o
e
At z  
J x  0.043J o
10kHz    0.6mm

60 Hz    8.6mm
Skin depth  versus frequency for copper
23
EEL 3472
Time-Varying Fields
The Skin Effect continued
Skin Depth and Surface Resistance for Metals
 
Metal
24
1
E f

 E m
1
A
RS 
f
A
f
B f
E
B
Silver
6.81  10 7
6.10  10 2
2.41  10 7
Copper
5.91  10 7
6.55  10 2
2.59  10 7
Gold
4.10  10 7
7.86  10 2
3.10  10 7
Aluminum
3.54  10 7
8.46  10 2
3.34  10 7
Iron
1.02  10 7
1.58  10 1
6.22  10 7
EEL 3472
Time-Varying Fields
Surface Impedance
z
If J x ( z )  J o e e

j
, the total current per unit width is
0
Iw 
 J x dz 

Since Jo   E Eo we can write
Iw 
ZS 
1
1  j 4 5

e
1 j
2
z

0
 J oe
(1 j ) z

dz 

 E
1 j
Eo 
J o
1 j
1
Eo
ZS
Eo 1 j


 (1  j )
 RS  jX S
I w  E
2 E
Direction of current flow
45° out of phase with
Jo
2
 
 E
Voltage per unit length
Surface impedance
( RS  X S )
ZS   
or
“ohms per square”
I  I ww
V  E ol
w
l
Illustrating the concept of “ohms per square”
25
V E ol
l

 ZS  
I I ww
 w
If l=w (a square surface)
V
 ZS
I
EEL 3472
Time-Varying Fields
Surface Impedance continued

l 
 Rdc 


A
E


Copper
RS 
1
 E

f
E
RS is equivalent to the dc
resistance per unit length of
the conductor having crosssectional area 1 
Rac 
l

w

Rsl
w
A
For a wire of radius a,
w  2a
26
EEL 3472
Time-Varying Fields
Surface Impedance continued
  1cm  10 4 m
Wire (copper) bond
2r  d  0.1mm  100 m
f  10GHz  1010 Hz
 E  5.91  10 7 (  m) 1
z=?
Plated metal connections
   o  4  10 7 H / m
Substrate
1
d

 0.6m   50m
2
 E f
ZS 
(1  j)
 E
(a)
Thickness of current layer = 0.6 μm = δ
 (1  j)x0.026
w
 x 100 μm
 10 m 
l

Z  Z S    Z S 


100

m
 w


4
2 r
 0.86  j 0.86Ω
X  Lint
δ = 0.6 μm
Z  R  jX
100 μm
Lint  X / 
(Internal inductance)
d  2r
(b)
(c)
Finding the resistance and internal inductance of a wire bond. The bond, a length of wire connecting two
pads on an IC, is shown in (a). (b) is a cross-sectional view showing skin depth. In (c) we imagine the
conducting layer unfolded into a plane.
27
EEL 3472