electromagnetics and applications lecture 3 waves in conducting / lossy medium. electromagnetic...
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ELECTROMAGNETICS AND APPLICATIONS
Lecture 3Waves in Conducting / Lossy Medium.
Electromagnetic Power & Energy.
Luca Daniel
L3-2
• Review of Fundamental Electromagnetic Laws
• Electromagnetic Waves in Media and Interfaceso Waves in homogeneous lossless and lossy media
o Power flow and energy balance (Poynting Theorem)
o Waves at interfaces
• Digital & Analog Communications
• Microwave Communications
• Optical Communications
• Wireless Communications
• Acoustics
Course Outline
L3-3
• Course Overview and Motivations• Maxwell Equations (review from 8.02)• EM waves in homogenous media
– EM Wave Equation– Solution of the EM Wave equation
Uniform Plane Waves (UPW) Complex Notation (phasors)
– EM Waves in homogeneous conducting/lossy media
• Electromagnetic Power and Energy– The Poynting Theorem– Wave Intensity– Poynting Theorem in Complex Notation
• EM Fields at Interfaces between Different Media
Today’s Outline
TodayToday
L3-4
Waves in Conducting/Lossy Medium
E j H
H j E
2 2E Ek 0
effj E eff 1 jj
the imaginary part is the “lossy” party
2 2 2
effk 1 j
jkzE 0xE e
For example wave in good conductor
1 j
k j
1 jnote: j
2
2
where skin/penetration depth
z
0E
0E
e
x
E
2 m for copper at 1GHz
L3-5
• Course Overview and Motivations• Maxwell Equations (review from 8.02)• EM waves in homogenous media
– EM Wave Equation– Solution of the EM Wave equation
Uniform Plane Waves (UPW) Complex Notation (phasors)
– EM Waves in homogeneous conducting/lossy media
• Electromagnetic Power and Energy– The Poynting Theorem– Wave Intensity– Poynting Theorem in Complex Notation
• EM Fields at Interfaces between Different Media
Today’s Outline
TodayToday
L3-6
Power and Energy
d
p(t) wdt
Units of Power:[Joule]=[W s]=[V A s][Watts]=[V A]
Units of Energy:
at steady state: 0d
dt
0i dissp p dissp
1p
2p3p
non-steady state:
i diss storedi
dp p w
dt
dissp
1p
2p
storedw
3p
Net power flowINTO the surface
Power dissipatedinside volume
Non-zero power balance generatesan increase of stored energy
What is the relation between Power p(t) and Energy w(t)?
L3-7
Electromagnetic Power Flow
E H
E
H
dan
propagation direction: E H
E H nda
S
Net power flow INTO the surface:
ˆE H nda
2has the units of power: [V/m][A/m][m ]=[V A]
i diss storedi
dp p w
dtNon-zero INCOMING power balancegenerates an increase of stored energy
i diss storedi
dp p w
dt
dissp
1p
2p
storedw
3p
L3-8
Electromagnetic Power and Energy
Vector Identity
dissp
sˆIf we compute - E H nda we will have an expression
for the power dissipated and for the energy stored in the volume
s VˆE H nda E H dv
E H E H H E
using Faradayand Ampere’s Laws
using Gauss Divergence Theorem
E HE E H
t t
2 2 2d 1 d 1
E E H dt 2 dt 2
2 2 2
S V V V
d 1 1ˆE H n da E dv E dv H dv dt 2 2
d 1 dENote: E E E
dt 2 dt
storedd
wdt
ii
p
L3-9
The Poynting Theorem
2 2 2 3d 1 d 1E H E E H [W/m ]
dt 2 dt 2
Energy Stored in Magnetic Field
wm
Energy Storedin Electric Field we
Power dissipated
wd
Net power flow INTO the surface
E
HS S E HThe Poynting vector: gives both the magnitude
of the power density and the direction of its flow.
2 2 2
S V V V
d 1 d 1ˆE H n da E dv E dv H dv [W]dt 2 dt 2
i diss storedi
dp p w
dtdissp
1p
2p
storedW
3p
L3-10
Uniform Plane Wave: EM fields
EM Wave in z direction:
0E
ˆH z,t x cos t kz
Linearity implies superposition many wave solutions for different ,k,
Magnetic energy density
Electric energy density
y
z
z
x
E z,0
H z,0 2 c
k f
Wavelength
2
e1W E2
2
m1W H2
0ˆE z,t y E cos t kz ,
L3-11
Power Flow in Uniform Plane Waves
oˆE yE cos t kz
oEˆH x cos t kz
2 2e o
1W E cos t kz
2
2 2
m o2
1W E cos t kz
2
Note: is typically called “intensity” [W/m2] of the wave S
22 2oE
ˆS(t) E H z cos t kz (W/m )
2o
T
E1 1ˆ S S(t) dt zT 2
2
2
oEˆS(t) z cos t kz 2 [W/m ]
0S(z at t )
0
0E(z at t )
eW
z
L3-12
Poynting Vector in Complex Notation
Defining a meaningful and relating it to is not obvious.It is easier to relate it to the intensity (time average):
S S(t)
Thus, we can define and
S
S Re E H 12
S E H
j tr i r iE E jE H H jH e cos t jsin tNote:
j t j tS(t) E(t) H(t) Re E e Re H e
r i r i[E cos( t) E sin( t)] [H cos( t) H sin( t)]
1
2 r r i i S E H E H
S (by definition)
1
2 = Re E H
L3-13
• Course Overview and Motivations• Maxwell Equations (review from 8.02)• EM waves in homogenous media
– EM Wave Equation– Solution of the EM Wave equation
Uniform Plane Waves (UPW) Complex Notation (phasors)
– EM Waves in homogeneous conducting/lossy media
• Electromagnetic Power and Energy– The Poynting Theorem– Wave Intensity– Poynting Theorem in Complex Notation
• EM Fields at Interfaces between Different Media
Today’s Outline
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