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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