5 Wave Equation - PolyUem/em05pdf/5 Wave Equation.pdf · 2005. 3. 31. · Plane waves in lossless...
Transcript of 5 Wave Equation - PolyUem/em05pdf/5 Wave Equation.pdf · 2005. 3. 31. · Plane waves in lossless...
Intro.1
Reference
Wave Equation (2 Week)6.5 Time-Harmonic fields7.1 Overview7.2 Plane Waves in Lossless Media7.3 Plane Waves in Lossy Media7.5 Flow of Electromagnetic Power and the Poynting Vector7.6 Normal Incidence of Plane Waves at Plane Boundaries
Intro.2
PLANE ELECTROMAGNETIC WAVES
Wireless applications are possible because electromagnetic fields can propagate in free space without any guiding structures.
Plane waves are good approximations of electromagnetic waves in engineering problems after they propagate a short distance from the source.
Intro.3
Time-Harmonic Electromagnetics
In past chapter, field quantities are expressed as functions of time and position. In real applications, time signals can be expressed assuperimposition of sinusoidal waveforms. So it is convenient to use the phasor notation to express fields in the frequency domain, just as in a.c. circuits.
tjezyxtzyx ω),,(Re),,,( EE ≡
tj
tj
ezyxj
tezyx
ttzyx
ω
ω
ω ),,(Re
),,(Re),,,(
E
EE
=
∂∂
=∂
∂
Intro.4
Time-Harmonic Electromagnetics
Differential form of Maxwell’s equations
Time-harmonic Maxwell’s equations
0=⋅∇=⋅∇
∂∂
+=×∇
∂∂
−=×∇
BD
DJH
BE
v
t
t
ρ
0=⋅∇=⋅∇
+=×∇−=×∇
BD
DJHBE
v
jj
ρω
ω
Intro.5
Wave equations in Source-Free Media
In a source-free media (i.e.charge density ρv=0), Maxwell’s equation becomes:
00
)(
=⋅∇=⋅∇
+=×∇−=×∇
HE
EHHEωεσ
ωµj
j
We will derive a differential equation involving E or H alone. First take the curl of both sides of the 1st equation:
EEEHE
)(2 ωεσωµ
ωµ
jjj
+−=∇−⋅∇
×∇−=×∇×∇
Intro.6
Since in source-free media, we obtain
where
0=⋅∇ E
022 =−∇ EE γ
)(2 ωεσωµγ jj +=
The above equation is called Helmholtz equation, and γ is called the propagation constant. In rectangular coordinates, Helmholtz equation can be decomposed into three scalar equations:
022 =−∇ ii EE γ zyxi ,,=
Similarly for the magnetic field,
022 =−∇ HH γ
Intro.7
Helmoltz Equation and Transmission Line equations
Helmoltz equation Transmission Line equations
In rectangular co-ordinates
022 =−∇ EE γ
022 =−∇ HH γI
zI
Vz
V
22
2
22
2
γ
γ
=∂∂
=∂∂
022 =−∇ ii EE γ zyxi ,,=
Intro.8
Plane waves in lossless media
In a lossless medium, the conductive current is zero so σ is equal to zero; there is no energy loss and µ, ε are both real numbers. Therefore in lossless media
µεωγ j=
Plane wave is a particular solution of the Maxwell’s equation, where both the electric field and magnetic field are perpendicular to the propagation direction of the wave. Let the wave propagate along the z-direction, and the electric field is along the x-direction.
Intro.9
E- and H-field vectors for a plane wave propagating in z-direction
Intro.10
xaE )(zEx=
Substituting this into Holmholtz equation, and since
, we obtain
022
2
=− xx E
dzEd γ
This second order linear differential equation has two independent solutions, so the solution of Ex is
zxo
zxox eEeEzE γγ +−−+ +=)(
The 1st term is a wave travelling in the +z direction, and the 2nd term is a wave travelling in the -z direction.
0=∂∂
=∂∂
yE
xE xx
Intro.11
γ is called the propagation constant. In general, γ is complex
where α, β are the attenuation and phase constants. But in lossless media,
βαγ j+=
µεωβ
α
=
= 0
So the electric field in lossless media is given by:
( ) xaE zjxo
zjxo eEeE ββ +−−+ +=
(µ, ε are real)
Intro.12
Let up be the phase velocity with which either the forward or backward wave is travelling, and λ is the wavelength.
Since , therefore
βπλ
µεβω
2
1
=
==pu
In free space (vacuum), ε = εo, µ = µo and the phase velocity is the same as the velocity of light in vacuum (3×108 m/s)
.
µεωβ =
Intro.13
Similarly for the magnetic field, H can be found by solving
under the conditions
We obtain
ωµj−×∇
=EH
[ ] yaH zjxo
zjxo eEeE
zjββ
ωµ+−−+ +
∂∂−
=1
[ ] yazjxo
zjxo eEeE ββ
ωµβ +−−+ −=
0=∂∂
=∂∂
yE
xE xx
Intro.14
We notice that the magnetic field is in the y-direction, i.e. perpendicular to the electric field, and the ratio of magnitudesof electric and magnetic fields is:
εµ
βωµη
η
==
== −
−
+
+
xo
xo
xo
xo
HE
HE
η is called the intrinsic (or wave) impedance of the medium. In free space η has a value approximately equal to 377Ω.
Intro.15
The above plane wave is called TEM (transverse electromagnetic) wave, because both the electric and magnetic fields only exist in the transverse directions perpendicular to the propagation direction.
Other waves (e.g. in waveguides) can be also:
(a) TE (transverse electric) wave - electric field exists only in transverse direction, i.e. zero electric field in the propagation direction.
(b) TM (transverse magnetic) wave - magnetic field exists only in transverse direction, i.e. zero magnetic field in the propagation direction.
Intro.16
Example: A uniform plane wave with E=Exax propagates in a lossless medium (εr=4, µr=1, σ=0) in the z-direction. If Ex has a frequency of 100MHz and has a maximum value of 10-4
(V/m) at t=0 and z=1/8 (m),
a) write the instantaneous expression for E and H, b) determine the locations where Ex is a positive maximum when t=10-8s.
Solution:
jj3
44103102
8
8 ππγ =××
=
rrroro cjj εµωεεµµωγ ==
since velocity of light 81031×==
oo
cεµ
Intro.17
)3
4102cos(10 84 φππ +−= − ztxaE(a)
Setting Ex=10-4 at t=0 and z=1/8,
6πφ == kz
πεεµµη
η
60==
==
or
or
xy
EH yy aaH
(b) At t=10-8, for Ex to be maximum,
nz
nz
m
m
23
813
281
34)10(102 88
±=
±=
−−− πππ
,....2,1,0=n
The H-field:
Intro.18
7.4 Plane Waves in Lossy Media
The electric field in the x-direction is given by:
zxo
zxox eEeEzE γγ +−−+ +=)(
where γ is in general complex for lossy media
βαγωεσωµγ
jjj
+=
+= )(
So E-field can be written as:
xx aaE zjzxo
zjzxo eeEeeE βαβα ++−−−+ +=
Intro.19
Substitute into Maxwell’s equations, the magnetic field is:
[ ] yaH zxo
zxo eEeE γγ
η+−−+ −=
1
where the intrinsic impedance η is complex.
ωεσωµ
γωµη
jjj+
==
The loss can be due to (i) conduction loss where σ is non-zero, (ii) dielectric loss where ε is complex,
''' εεε j−=
Intro.20
Example: The E-field of a plane wave propagating in z-direction in sea water is E=ax100 cos(107πt) V/m at z=0. The parameters of sea water are εr=72, µr=1, σ=4 S/m. (a) Determine the attenuation constant, phase constant, intrinsic impedance,and phase velocity. (b) Find the distance where the amplitude of E is 1% of its value at z=0.
Solution: (a))( ωεσωµγ jj +=
In our case σ/ωε =200>>1 so that α, β are approximately
89.8=== µρπβα f
βα j+=
Intro.21
σωµ
ωεσωµη j
jj
≈+
=
4/76
4)104)(105()1()1( ππππ
σµπη jejfj =
××+=+=
−
Phase velocity: m/s67
1053.389.8
10×===
πβωu
(b) Distance z1 where wave amplitude decreases to 1%:
518.089.8
605.4100ln101.0
1
1
===
=−
α
α
z
e z
Intro.22
7.5 Polarization: Polarization describes how the E-field vector varies with time as the wave propagates. Let Ex, Ey be the components of E-field in the x and y directions.
• Linear polarization: The E-field vector is always in the same direction. Ex, Ey are either in phase or out of phase.
yx
yx
aaE
aaE
)cos()cos()(
)cos()cos()(
21
21
ztEztEt
ztEztEt
mm
mm
βωβω
βωβω
−−−=
−+−=
Intro.23
• Circular polarization: The E-field vector rotates around the axis of propagation and has constant amplitude. Ex, Ey have equal magnitudes and phase angle difference of π/2.
• Elliptic polarization: The E-field vector rotates around the axis of propagation but has time-varying amplitude. Ex, Ey can have any values of magnitude and phase angle difference.
yx aaE )2/cos()cos()( πβωβω ±−+−= ztEztEt mm
yx aaE )cos()cos()( 21 δβωβω +−+−= ztEztEt mm
Intro.24
7.6 Power transmission - Poynting’s Theorem
Power is transmitted by an electromagnetic wave in the direction of propagation. It can be derived from Maxwell’s equation in the time domain that:
dvHEt
dvdVVS ∫∫∫
+
∂∂
+⋅=⋅×−22
22 µεJESHE
On the right hand side, the 1st term represents the instantaneous ohmic power loss, the 2nd and 3rd term represent the rate of increase of energies stored in the electric and magnetic fields respectively. So the left hand side represents the net instantaneous power supplied to the enclosed surface, or
Intro.25
)()()( ttt
dWSout
HEP
SH.E
×=
×= ∫
The instantaneous Poynting vector P (in watt per sq.m) represents the direction and density of power flow at a point.
In time-varying fields, it is more important to find the average power. We define the average Poynting vector for periodic signals as Pavg
∫ ∫ ×==T T
dtttT
dtT 0 0
)()(11 HEPPavg
Using complex phasor notation of E, H,
( )*Re21 HEPavg ×=
Intro.26
For a uniform plane wave, E & H field vectors are given by:
xaE zjzxo eeE βα −−=
[ ] yaH zjzxo eeE βα
η−−=
1
ηθηη je=
Substituting in equ. of Pavg gives
zavg aP ηα θ
ηcos
22
2zxo eE −=
Intro.27
7.7 Reflection of Plane Waves at Normal Incidence
x
z
Medium 2 (σ2,ε2,µ2)Medium 1(σ1,ε1,µ1)
(incident wave)
(reflected wave)
(transmitted wave)
Ei
ak
ak
ak
Et
Er
Hi
Hr
Ht
ak – propagation direction
y direction
-y direction
Intro.28
Incident wave:
yzio
yz
ioi
xz
ioi
eEeH
eE
aaH
aE
11
1
1
γγ
γ
η−−
−
==
=
Reflected wave:
yzro
yz
ror
xz
ror
eEeH
eE
aaH
aE
11
1
1
)( γγ
γ
η−=−=
=
Transmitted wave:
yzto
yz
tot
xz
tot
eEeH
eE
aaH
aE
22
2
2
γγ
γ
η−−
−
==
=
Intro.29
Boundary conditions at the interface (z=0):
Tangential components of E and H are continuous in the absence of current sources at the interface (refer to tutorials), so that
( )21
1ηη
toroio
toroio
toroio
EEE
HHHEEE
=−
=+=+
Reflection coefficient12
12
ηηηη
+−
==Γio
ro
EE
Transmission coefficient21
22ηη
ητ+
==io
to
EE
Intro.30
Note that Γ and τ may be complex, and
101
≤Γ≤
=Γ+ τ
Similar expressions may be derived for the magnetic field.
In medium 1, a standing wave is formed due to the superimposition of the incident and reflected waves. Standing wave ratio can be defined as in transmission lines.
Intro.31
END