Lesson 15Wave Equations

楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan

Outline

Wave equations of fields (propagation)Wave equations of potentials (generation)Electromagnetic spectrum (applications)

Sec. 15-1Wave Equations of Fields

1. Homogeneous equations in time domain

2. Homogeneous equations in frequency domain

3. EM fields in lossy media4. Microwave ovens

Homogeneous wave equation in time domain-1

In a simple (linear, isotropic, homogeneous), charge free (ρ = 0), nonconducting ( , ) medium, Maxwell’s equations reduce to:

0=σ 0=Jv

tHE∂

∂−=×∇

vv

μ

0=⋅∇ Ev

tEH

∂∂

=×∇v

vε

0=⋅∇ Hv

tBE

∂∂

−=×∇v

v

ρ=⋅∇ Dv

tDJH

∂∂

+=×∇v

vv

0=⋅∇ Bv

(1)

(2)

(3)

(4)

Homogeneous wave equation in time domain-2

tHE∂

∂−=×∇

vv

μ

0=⋅∇ Ev

tEH

∂∂

=×∇v

vε

( )

2

2

tE

tE

t

Htt

HE

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−=

×∇∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−×∇=×∇×∇

vv

vv

v

μεεμ

μμ

( ) AAA vvv ×∇×∇−⋅∇∇≡∇2( ) EEE vvv 22 −∇=∇−⋅∇∇=

022

2 =∂∂

−∇tEEv

vμε⇒

Homogeneous 2nd-order PDE of vector field E

v

(1) (3)

(2)

Homogeneous wave equation in time domain-3

0=⋅∇ Hv

( )

2

2

tH

tH

t

Ett

EH

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

=

×∇∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

×∇=×∇×∇

vv

vv

v

μεμε

εε

( ) AAA vvv ×∇×∇−⋅∇∇≡∇2( ) HHH vvv 22 −∇=∇−⋅∇∇=

⇒ 022

2 =∂

∂−∇

tHHv

vμε

tEH

∂∂

=×∇v

vε t

HE∂

∂−=×∇

vv

μ

Homogeneous 2nd-order PDE of vector field H

v

(3) (1)

(4)

Homogeneous wave equation in time domain-4

Compare with the equations of lossless TX lines:

022

2 =∂

∂−∇

tHHv

vμε

022

2 =∂∂

−∇tEEv

vμε),(),( 2

2

2

2

tzvt

LCtzvz ∂

∂=

∂∂

),(),( 22

2

2

tzit

LCtziz ∂

∂=

∂∂

Scalar waves

Velocity:LC

vp1

=

Vector waves

Velocity:με1

=pu

Comments-1

We have assumed to derive:

022

2 =∂

∂−∇

tHHv

vμε

022

2 =∂∂

−∇tEEv

vμε

i.e., these equations deal with how EM waves propagate, instead of how they are generatedfrom time-varying sources .

0 ,0 == Jv

ρ

( )Jv,ρ

Comments-2

We have assumed simple media to derive:

022

2 =∂

∂−∇

tHHv

vμε

022

2 =∂∂

−∇tEEv

vμε

i.e., these equations (standard waves properties) have to be modified if the medium is nonlinear, anisotropic, nonhomogeneous.

Why are we interested in time-harmonic (sinusoidal) fields?

Any periodic (aperiodic) function → superposition of discrete (continuous) sinusoidal functions by Fourierseries (integral).

Maxwell’s equations are linear PDEs. ⇒ (1) Sinusoidal sources produce sinusoidal fields of the same frequency in steady state. (2) Total field can be derived by superposition of individual sinusoidal responses.

Easy to operate if phasors are used:

,ωjt

→∂∂

ωjdt 1→∫

From scalar phasors to vector phasors

Scalar phasors of voltages & currents are sufficient to describe steady-state response of TX lines:

{ },)(Re),( tjezVtzv ω⋅= { }tjezItzi ω⋅= )(Re),(

Vector phasors of E-field and M-field are required to describe time-harmonic EM fields:

{ }tjezyxEtzyxE ω),,(Re),,,( vv ={ }tjezyxHtzyxH ω),,(Re),,,( vv =

Homogeneous wave equation in frequency domain-1

In a simple, source-free (ρ = 0, ) medium, phasors of EM fields satisfy with the equations

0=Jv

tHE∂

∂−=×∇

vv

μ

0=⋅∇ Ev

tEH

∂∂

=×∇v

vε

0=⋅∇ Hv

HjEvv

ωμ−=×∇

0=⋅∇ Ev

EjHvv

ωε=×∇

0=⋅∇ Hv

)(),( rEtrE vvvv →

)(),( rHtrH vvvv →

ωjt

→∂∂

(1)

(2)

(3)

(4)

Homogeneous wave equation in frequency domain-2

HjEvv

ωμ−=×∇ EjHvv

ωε=×∇

( ) ( )

( ) ( ) EEEEEjj

HjHjE

vvvvv

vvv

222 −∇=∇−⋅∇∇==−=

×∇−=−×∇=×∇×∇

μεωωεωμ

ωμωμ

( ) AAA vvv ×∇×∇−⋅∇∇≡∇2

022 =+∇ EkEvv

⇒

λπωμεω 2===

puk Wavenumber (spatial

angular frequency)

022 =+∇ HkHvv

(1) (3)

0=⋅∇ Ev (2)

με1=pu

Homogeneous wave equation in frequency domain-3

Compare with the equations of lossless TX lines:

Scalar waves, ODEs Vector waves, PDEs

)()( 222

zVzVdzd β−=

)()( 222

zIzIdzd β−=

LCωβ =

022 =+∇ EkEvv

022 =+∇ HkHvv

μεω=k

EM fields in lossy media-complex permittivity

If the medium is conducting (σ ≠ 0), the presence of results in conduction currents

if time-harmonic, simple medium

EJvv

σ=Ev

,EjEj

jEjEH cvvvvv

ωεεω

σωωεσ =⎟⎟⎠

⎞⎜⎜⎝

⎛+=+=×∇⇒

,tDJH

∂∂

+=×∇v

vv

ωσεε jc −= is the complex permittivitywhere

EM fields in lossy media-complex wavenumber

⎟⎠⎞

⎜⎝⎛ −=−=

ωεσε

ωσεε jjc 1

Loss tangent

( )ccc jk δμεωμεω tan1−==

ωεσδ =ctan

If , ⇒ , the medium behaves like a dielectric

1tan cδ { }ckIm

μεω=k

EM fields in lossy media-propagation loss

are associated with the power loss when the wave propagates through the medium.

1. , ⇒ describes is not in phase with the driving due to the inertia of the bound charges, ⇒ frictional damping.

EP cvv

ε=

{ } { }cc kIm ,Im ε

{ }cεIm Pv

Ev

2. Ohmic power loss due to collision among free charges and atoms.

( )∫ ⋅= V dvJEP vv

Example 15-1: Property of a moist ground

At f =1 kHz, , like a conductor

,10 0εε = ),mS( 10 2−=σ

110tan 4 >>≈cδωεσδ =ctan

At f =10 GHz, , like a dielectric110tan 3

Microwave oven-history

Invented by Percy Spencer (1945), chocolate bar melt when building magnetrons(磁控管) for radar sets with Raytheon, tested popcorn & egg.

First commercial model Radarange (1947).

Microwave oven-working principle (1)

Pass EM wave at 2.45 GHz (λ=12.2 cm) through food. Polarized molecules (water, fat, ..) will rotate with the AC field, motion(~heat) dispersed to other non-polarized molecules by collision. (Not due to resonance, for water vapor fres=20 GHz)

_

+Ev

E-torque Epvv //

Microwave oven-working principle (2)

Microwave oven-working principle (3)

Microwave oven-working principle (4)

Microwaves can penetrate dry non-conducting substances, inducing initial heat more deeply(~cm) than other methods.(Not cook from the inside out)

The oven door usually has a window with a layer of conductive mesh for shielding. The grid size is

Example 15-2: Microwave oven

to cook a beef steak of , and

A microwave oven generates an AC E-field of:

0.35tan c =δ

)m(V )10 2.45 (2cos250)( 9 ttE ⋅×⋅⋅= π

040εε =

⇒= ,tanωεσδ c == cδωεσ tan

( ) ( )mS 9.135.0361040 )10 2.45 (2

99 =⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅×⋅

−

ππ

( ) ⇒=⋅= ,)()()( 2tEtJtEtpvvv

σ ( )320 cmmW 6021

≈= EPavg σ

Sec. 15-2Wave Equations of Potentials

1. Non-homogeneous equations in time domain

2. Solutions to homogeneous equations3. Non-homogeneous equations in

frequency domain

Potentials in time-varying cases (1)

In the presence of time-varying fields:

0=⋅∇ Bv

ABvv

×∇=remains, ⇒ remains valid.M-field vector potential

, ⇒0≠∂∂

−=×∇tBEv

vVE −∇≠

v

Instead, ( ),At

Evv

×∇∂∂

−=×∇ ,0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+×∇tAEv

v V∇−

tAVE

∂∂

−−∇=v

v⇒

Potentials in time-varying cases (2)

tAVE

∂∂

−−∇=v

v

scalar potential, conservative component, ~charge distribution

vector potential, nonconservative component, ~time-varying current

Nonhomogeneous wave equations of vector potential in time domain-1

In simple media:

tDJH

∂∂

+=×∇v

vv

tEJB

∂∂

+=×∇v

vvμεμ

ABvv

×∇=tAVE

∂∂

−−∇=v

v

( ) ( ) AAAB vvvv 2∇−⋅∇∇=×∇×∇=×∇⇒

⇒

2

2

tA

tV

tE

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

−∇=∂∂

vv

( ) 22

2

tA

tVJAA

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∇−=∇−⋅∇∇v

vvvμεμεμ

Nonhomogeneous wave equations of vector potential in time domain-2

⇒ ⎟⎠⎞

⎜⎝⎛

∂∂

+⋅∇∇+−=∂∂

−∇tVAJ

tAA μεμμε

vvv

v2

22

0=∂∂

+⋅∇tVA με

vBy Lorentz gauge:

JtAA

vv

vμμε −=

∂∂

−∇ 22

2

is decoupled with V,Av

⇒Nonhomogeneouswave equation

Nonhomogeneous wave equations of scalar potential in time domain

In simple media:tAVE

∂∂

−−∇=v

v

⇒

ρ=⋅∇ Dv

ερ

=⋅∇ Ev

( ) ( ) ( )At

VAt

VEvvv

⋅∇∂∂

−−∇=⋅∇∂∂

−∇⋅−∇=⋅∇ 2⇒

⇒

0=∂∂

+⋅∇tVA με

v

ερμε =

∂∂

+−∇=⋅∇ 22

2

tVVE

v

ερμε −=

∂∂

−∇ 22

2

tVV

Nonhomogeneouswave equation

Lorentz gauge

Comments-1

Given charge and current distributions , ),( trJ vv

Solve

),( trvρ

),( trA vv

),( trV vερμε −=

∂∂

−∇ 22

2

tVV

JtAA

vv

vμμε −=

∂∂

−∇ 22

2

Derive fields by:tAVE

∂∂

−−∇=v

v

ABvv

×∇=

TBD…

Comments-2

In static cases:

Lorentz gauge

0=∂∂

+⋅∇tVA με

v

Coulomb’s gauge

0=⋅∇ Av

Nonhomogeneouswave equations Poisson’s equations

ερμε −=

∂∂

−∇ 22

2

tVV

JtAA

vv

vμμε −=

∂∂

−∇ 22

2

ερ

−=∇ V2

JAvv

μ−=∇2

Solutions to nonhomogeneous wave equations in time domain (1)

Consider a point charge at origin (spherical symmetry)

ερμε −=

∂∂

−∇ 22

2

tVV 01 2

22

2 =∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

tV

RVR

RRμε

(except for the origin)

( ) ( )tRVRtRU ,, ⋅=Define

022

2

2

=∂∂

−∂∂

tU

RU με⇒

⇒

…standard wave equation

( ) ),(, τftRU = ,puRt −=τμε1=pu

[ ] φθRRV )(2∇

( ) ,)(,R

ftRV τ=

Solutions to nonhomogeneous wave equations in time domain (2)

In static cases:R

qRV⋅

=πε4

)(

In time-varying ( at origin) cases:

( )R

vduRttRV p

⋅

′−=Δ

περ

4)(

,

vdt ′)(ρ)(τf

Solutions to nonhomogeneous wave equations in time domain (3)

Since is linear,ερμε −=

∂∂

−∇ 22

2

tVV

the potential due to over a volume V' is:),( tr ′vρ

pd

Vd

uRt

vdR

ttrtrV

=

′−′

=

∫ ′ ,),(

41

),(

v

v

ρπε

Solutions to nonhomogeneous wave equations in time domain (4)

Similarly, the vector potential due to over volume V' is:

),( trJ ′vv

∫ ′ ′−′

=V

d vdR

ttrJtrA

),(4

),(vv

vv

πμ

The potentials , are determined by the source at at an earlier time

),( trV v ),( trA vv

r ′v puRt −

⇒ Potential (& field) propagates with finite speed pu

pd uRt =

Nonhomogeneous wave equations of potentials in frequency domain

For time-harmonic waves:

ερμε −=

∂∂

−∇ 22

2

tVV

JtAA

vv

vμμε −=

∂∂

−∇ 22

2

ερ

−=+∇ VkV 22

JAkAvvv

μ−=+∇ 22

phasors,

λπωμεω 2===

puk

0=∂∂

+⋅∇tVA με

v0=+⋅∇ VjA ωμε

v

Solutions to nonhomogeneous wave equations of potentials in frequency domain

kRR

fR

uRt

pd

==

⋅==⋅

λπ

λωωω

2

For time-harmonic waves:

∫ ′ ′−′

=V

p vdR

uRtrtrV

),(4

1),(v

v ρπε

∫ ′ ′−′

=V

p vdR

uRtrJtrA

),(4

),(vv

vv

πμ

∫ ′−

′′

=V

jkR

vdRerrV

)(4

1)(v

v ρπε

∫ ′−

′′

=V

jkR

vdRerJrA

)(4

)(vv

vv

πμ

time retardation phase shift

( ) ,cos 00 φφω jeVtV →+

( )[ ][ ] ( )dtjd

d

eVttV

ttV⋅−→⋅−+

=+−ωφωφω

φω

00

0

cos

cosphasor

Justification

Sec. 15-3Electromagnetic Spectrum

Extremely low frequency (ELF): 0-300 Hz (1)

Global communications with deeply submergedsubmarines.

Why using ELF: attenuation of EM waves in sea water is lower for lower frequencies (TBD…).

Difficulties: (1) low data rate, (2) huge antenna size (in principle λ/2), only support unidirectional communications.

What’s the problem?

Crimson Tide(赤色風暴, 1995)

Not confirmed yet…

Launch nuke missiles now!

Extremely low frequency: 0-300 Hz (2)

US Navy Seafarer system: 76 Hz (λ≈3,900 km). Low-conductivity ground, ⇒ deeper penetration of current (use a part of the globe as antenna).

Medium frequency (MF): 0.3-3 MHz

Amplitude modulation (AM) broadcast: 0.53-1.61 MHz, antenna length λ/2≈150 m.

Max audio BW: 10.2 kHz (channel spacing: 20.4 kHz).

1906: 1st experiment (Canada).

(commons.wikimedia.org)

Signal range of AM broadcast

Day time: groundwave, diffracting around the curve of the earth, ~100 km.

Night: skywave(ionspherereflection), much longer.

(wikipedia.org)

Very high frequency (VHF): 30-300 MHz

In US, 101 channels, from 87.9 MHz to 107.9 MHz (spacing 200 kHz).

No ionsphere reflection, limited to line-of-sight range (~100 km).

1933: invented.

Frequency modulation (FM) broadcast: 88-108 MHz, antenna length λ/2≈1.5 m.

(commons.wikimedia.org)

Ultra high frequency (UHF): 300 MHz-3 GHz

TV broadcast: 530-596 MHz. (Audio:174-216 MHz)

Cell phone: Global System for Mobile communications (GSM), 2G: (900 MHz, 1.8 GHz); 3G: (850/880 MHz, 1.9, 2.0, 2.1 GHz).

Global positioning system (GPS): all satellites broadcast 1.58 GHz (L1), 1.23 GHz (L2).

2.45 GHz: Wi-Fi, Bluetooth, microwave oven.

Super high frequency (SHF): 3-30 GHz

Radar: L(1-2G), S(2-4G), C(4-8G, airborne weather), X(8-12G, missile guidance), Ku(12-18G).

Wireless Local Area Network (WLAN): provide access point to the internet.

λ ≈ centimeters.

Terahertz: 1012 Hz

Aircraft-to-satellite communications (low water vapor environment).

Security

Difficult to be generated/detected by conventional electronic/optical means.

Infrared (IR): 1013 -1014 Hz, λ=0.7-100 μm (1)

Fiber communications: λ≈1.55 μm (Near IR).

Corning SMF-28, 0.2 dB/km, ⇒50% power after 15 km propagation.

Infrared (IR): 1013 -1014 Hz, λ=0.7-100 μm (2)

Night vision devices (image intensifier): Photocathode converts weak visible & near IR photons into electrons, amplified by MCP, converted to visible photons by phosphor.

Infrared (IR): 1013 -1014 Hz, λ=0.7-100 μm (3)

Thermal imaging: universal black body radiation (live human λ=9.5 μm; missile λ=3−5 μm; mid-IR), sense temperature variation.

Passive house

Detectors: InSb(銻化銦, III-V semiconductor, 0.17 eV, sensitive to 1-5 μm), bolometer (測輻射熱計, sensitive to all λ’s)

(wikipedia.org)

Visible: (4-7)×1014 Hz, λ=0.4-0.7 μm

(nm)

Ultraviolet (UV): λ=10-100 nm

Light sources:

Mercury vapor lamp + filter: λ=365 nm.

Excimer lasers(准分子雷射): KrF λ=248 nm, ArF λ=193 nm(enable 30 nm feature size).

Photolithography

(cnx.org)

X-rays: λ=0.01-10 nm, E~keV (1)

X-ray tube:

Limit the max. photon energy (20-150 keV)

1. X-ray fluorescence: knock out inner shell e-, ⇒ e- at higher energy levels fall (discrete lines).

2. Bremsstrahlung: e- deflected by nucleus (cont. lines).

(www.clin-eng.net)

X-rays: λ=0.01-10 nm, E~keV (2)

Wilhelm C. Röntgen (1895)

Medical diagnostics:

Soft X-rays (λ>0.1 nm, E

γ-rays: λ

Appendix: Frequency chart

Human voice/cello: 60 Hz-1.5 kHz

Harp: 30 Hz-3 kHz

(audiokarma.org)

Middle C: 440 Hz

Human hearing: 20 Hz-20 kHz (2-4 kHz most sensitive)

Appendix: Microchannel plate (MCP, 微通道面板)

For detecting weak signals of ions or photons

(hea-www.harvard.edu)

Appendix: Bolometer(測輻射熱計)

Best choice for λ=200 μm - 1 mm (far-IR).