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8/2/2019 Lecture 20 Plane Waves
1/23
EECS 117
Lecture 20: Plane Waves
Prof. Niknejad
University of California, Berkeley
Universit of California Berkele EECS 117 Lecture 20 . 1/
8/2/2019 Lecture 20 Plane Waves
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Maxwells Eq. in Source Free Regions
In a source free region = 0 and J = 0
D = 0
B = 0
E =
B
t
=
H
t
H = Dt
= E
t
Assume that E and H are uniform in the x-y plane sox = 0 and
y = 0
For this case the
E simplifies
Universit of California Berkele EECS 117 Lecture 20 . 2/
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Curl E for Plane Uniform Fields
Writing out the curl of E in rectangular coordinates
E = x y z
0 0 zEx Ey Ez
(E)x = Eyz = Hx
dt
(
E)y
=
Ex
z=
Hy
dt
(E)z = 0 = Hzdt
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Curl of H for Plane Uniform Fields
Similarly, writing out the curl H in rectangularcoordinates
H =
E
t
Hyz
= Ex
t
Hxz
= Ey
t
0 =
Ezt
Time variation in the z direction is zero. Thus the fieldsare entirely transverse to the direction of propagation.
We call such fields TEM waves
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Polarized TEM Fields
For simplicity assume Ey = 0. We say the field polarizedin the x-direction. This implies that Hx = 0 and Hy = 0
Exz
= Hyt
Hy
z =
Ex
t
2Exz2
= 2Hy
zt
2Hyzt
= 2Ext2
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One Dimensional Wave Eq.
We finally have it, a one-dimensional wave equation
2Ex
z2=
2Ex
t2
Notice similarity between this equation and the waveequation we derived for voltages and currents along a
transmission lineAs before, the wave velocity is v = 1
The general solution to this equation is
Ex(z, t) = f1(t zv
) + f2(t +z
v)
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Wave Solution
Lets review why this is the general solution
Ex
t= f
1+ f
2
2Ext2
= f1 + f2
Ex
z=
1
vf1
+1
vf2
2Exz2
=1
v2f1 +
1
v2f2
A point on the wavefront is defined by (t z/v) = cwhere c is a constant. The velocity of this point istherefore v
1 1v
zt
= 0
z
t= v
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Wave Velocity
We have thus shown that the velocity of this wavemoves is
v = c =1
In free-space, c 3 108m/s, the measured speed oflight
In a medium with relative permittivity r and relativepermeability r, the speed moves with effective velocity
v =c
rrThis fact alone convinced Maxwell that light is an EM
wave
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Sinusoidal Plane Waves
For time-harmonic fields, the equations simplify
dEx
dz
=
jHy
dHydz
= jEx
This gives a one-dimensional Helmholtz equation
d2Ex
d2z
=
2Ex
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Solution of Helmholtz Eq.
The solution is now a simple exponential
Ex = C1ejkz + C2ejkz
The wave number is given by k = = vWe can recover a traveling wave solution
Ex(z, t) = ExejtEx(z, t) =
C1ej(tkz) + C2ej(t+kz)
Ex(z, t) = C1 cos(t kz) + C2 cos(t + kz)The wave has spatial variation = 2k =
2v =
vf
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Magnetic Field of Plane Wave
We have the following relation
Hy =
1
j
dEx
dz
=
1
j jkC1ejkz + C2jkejkz
Hy =k
C1e
jkz C2jkejkz
By definition, k =
Hy =
C1ejkz C2ejkzThe ratio E+x and H
+y has units of impedance and is
given by the constant = /. is known as theimpedance of free space
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Plane Waves
Plane waves are the simplest wave solution ofMaxwells Eq. They seem to be a grossoversimplification but they nicely approximate real
waves that are distant from their source
source of radiation
wavefront at distant points
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Wave Equation in 3D
We can derive the wave equation directly in acoordinate free manner using vector analysis
E = Ht
= (H)t
Substitution from Maxwells eq.
H = Dt
= E
t
E = 2Et2
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Wave Eq. in 3D (cont)
Using the identity E = 2E + ( E)Since E = 0 in charge free regions
2E = 2Et2
In Phasor form we have k
2
=
2
2E = k2E
Now its trivial to get a 1-D version of this equation
2Ex =
2Ex
t2
2Ex
x2
= 2Ex
t2
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Penetration of Waves into Conductors
Inside a good conductor J = E
In the time-harmonic case, this implies the lack of freecharges
H = J + Et
= ( + j) E
Since H 0, we have( + j) E 0
Which in turn implies that = 0
For a good conductor the conductive currentscompletely outweighs the displacement current, e.g.
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C d i Di l C
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Conductive vs. Displacement Current
To see this, consider a good conductor with 107S/mup to very high mm-wave frequencies f 100GHzThe displacement current is still only
10111011 1
This is seven orders of magnitude smaller than theconductive current
For all practical purposes, therefore, we drop thedisplacement current in the volume of good conductors
Universit of California Berkele EECS 117 Lecture 20 . 16/
W E ti i id C d t
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Wave Equation inside Conductors
Inside of good conductors, therefore, we have
H = E
E = ( E)2E = 2E = jH2E = jE
One can immediately conclude that J satisfies the sameequation
2J = jJApplying the same logic to H, we have
H = ( H)2H = 2H = (j+) E
2H = jHUniversit of California Berkele EECS 117 Lecture 20 . 17/
Pl W i C d
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Plane Waves in Conductors
Lets solve the 1D Helmholtz equation once again forthe conductor
d
2
Ezd2x = jEz = 2Ez
We define 2 = j so that
=1 + j
2
Or more simply, = (1 + j)f = 1+jThe quantity = 1
fhas units of meters and is an
important number
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S l ti f Fi ld
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Solution for Fields
The general solution for the plane wave is given by
Ez = C1ex + C2ex
Since Ez must remain bounded, C2 0
Ez = E0ex = Exex/ mag
ejx/ phaseSimilarly the solution for the magnetic field and currentfollow the same form
Hy = H0ex/ejx/
Jz = J0ex/
ejx/
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T t l C t i C d t
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Total Current in Conductor
Why do fields decay in the volume of conductors?
The induced fields cancel the incoming fields. As
, the fields decay to zero inside the conductor.
The total surface current flowing in the conductorvolume is given by
Jsz =0
Jzdx =0
J0e(1+j)x/dx
Jsz =J0
1 + j
At the surface of the conductor, Ez0 =J0
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Internal Impedance of Conductors
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Internal Impedance of Conductors
Thus we can define a surface impedance
Zs =Ez0
Jsz=
1 + j
Zs = Rs + jLi
The real part of the impedance is a resistance
Rs =1
=
f
The imaginary part is inductive
Li = Rs
So the phase of this impedance is always /4
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Interpretation of Surface Impedance
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Interpretation of Surface Impedance
The resistance term is equivalent to the resistance of aconductor of thickness
The inductance of the surface impedance represents
the internal inductance for a large plane conductor
Note that as , Li 0. The fields disappear fromthe volume of the conductor and the internal impedance
is zero
We commonly apply this surface impedance toconductors of finite width or even coaxial lines. Its
usually a pretty good approximation to make as long asthe conductor width and thickness is much larger than
Universit of California Berkele EECS 117 Lecture 20 . 23/
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