Antennas and Wireless Propagation

8/6/2019 Antennas and Wireless Propagation

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Tom Penick [email protected] www.teicontrols.com/notes AntennasAndWirelessPropagation.pdf 1/30/2003 Page 1 of 31

ANTENNAS AND WIRELESS PROPAGATION EE325K

(DIR) directivity.............181/4-wave monopole .........21

A vector magnetic potential....................................10

AF array factor ...............22AM .................................... 4amp....................................3Ampere's law.....................8anechoic...........................31angle of incidence............15angle of intromission.......16angle of reflection............15angle of transmission.......15angstrom............................3anisotropic.......................31antenna ............................19

1/4-wave monopole.....21circular aperture ..........22half-wave dipole..........20infinitesimal dipole .....19isotropic ......................18log-periodic...........20, 21

aperture theory.................21area

ellipse..........................28sphere..........................28

array factor ..........21, 22, 23array steering...................23arrays...............................22atmospheric attenuation.....5Avogadro's number............3axial ratio.........................11azimuth............................21

B magnetic flux dens.3, 8, 9beamwidth.......................26

circular aperture ant. ...22

Boltzmann's constant.........3boundary conditions..14, 15,16

Brewster angle.................16CDMA...............................4cellular...............................4characteristic impedance....5charge density....................3circular aperture...............22circular polarization.........11communications freqs........4complex conjugate...........26complex numbers ............26complex permittivity........12complex phase constant.....6

conductivity.....................12conservation of charge.......9conservative field law........9constants............................3constitutive relations .........9coordinate systems...........29cosmic rays........................4coulomb.............................3Coulomb's law...................9Courant stability condition24critical angle....................14cross product ...................28

curl .................................. 28current density................... 3

surface................... 14, 17current distribution.......... 20

D electric flux dens. . 3, 8, 9dBm.................................18del.................................... 27dielectric.......................... 31diff. eqn. 2nd order....... 7, 10dipole ..............................19dipole power.................... 19directivity .................. 18, 23

circular aperture ant. ... 22dispersion relationship 6, 13divergence .......................27dot product...................... 28E electric field...... 3, 5, 8, 9efficiency......................... 18EHF................................... 4electric

charge density ...............3current density .............. 3

field............................... 3flux density ................... 3

electromagnetic spectrum4, 5electromagnetic wave ........ 5electron mass..................... 3electron volt.......................3element pattern................ 30elevation.......................... 21ELF ................................... 4ellipse .............................. 28elliptical polarization....... 11empirical.......................... 31equation of continuity ....... 9evanescent wave.............. 31

far field...................... 18, 21farad .................................. 3Faraday's law..................... 8farfield............................. 30FDTD finite difference time

domain ........................25fields.................................. 3finite difference time domain

.................................... 25finite dipole .....................19FM.....................................5Fourier transform ...... 20, 30frequency domain.............. 8frequency spectrum ........... 4Friis formula.................... 24

gain.................................. 18gamma rays........................4Gauss' law......................8, 9geometrical optics

approximation ............. 25geometry.......................... 28glossary ........................... 31grad operator ...................27graphical method............. 23grating lobes.................... 23GSM.................................. 4

H magnetic field.... 3, 5, 8, 9

half-wave dipole.............. 20Helmholtz equation......... 10henry ................................. 3HF ................................. 4, 5horsepower........................ 3identities

vector .......................... 30image theory.................... 21impedance......................... 5incident plane.................. 31incident wave ............ 15, 16induced voltage

Faraday's law ................ 8infinitesimal dipole ......... 19instantaneous................... 10intrinsic wave impedance5, 6ionosphere................... 5, 12ISM ................................... 5isotropic .............. 18, 19, 31

J current density....... 3, 8, 9joule .................................. 3JS sfc. current dens. .. 14, 17

k phase constant ........... 6, 7k wave vector ............. 6, 13k0 phase constant in free

space........................... 13k'-jk'' complex ph. const... 6kxi, etc. ............................. 13kxi, etc. wave vector

components........... 15, 16Laplacian......................... 27LF...................................... 4light ................................... 4

speed of......................... 3line current................ 19, 30linear ............................... 31

linear polarization ........... 11lithotripsy........................ 16log-periodic antenna.. 20, 21loss tangent ..................... 12lossless materials............. 12lossy materials................. 12lossy medium............ 12, 16LPDA.............................. 20magnetic

field............................... 3flux density ................... 3

magnetic energy .............. 13magnetic potential........... 10magnitude........................ 26Maxwell's equations.......... 8

MF..................................... 4monopole ........................ 21nabla operator ................. 27near field ......................... 18newton............................... 3numerical methods.... 24, 25parallel polarization .. 15, 16PCS ................................... 4penetration depth............. 13perfect conductor ............ 17permeability ...................... 3permittivity........................ 3

complex .......................12permittivity of the ionosphere

.....................................12perpendicular polar. ...14, 15phase................................26phase constant................6, 7phase matching cond..15, 16phasor domain....................7phasor notation ..................7Planck's constant................3plane of incidence......15, 16plane wave .........................6plane wave solution ...........8plasma..............................12polarization ......................11

parallel.........................15perpendicular...............14

polarizing angle................16power ...............................18

dBm.............................18dipole...........................19time-average ..................7

power conservation..........10power transfer ratio..........24Poynting vector....10, 18, 19Poynting's theorem...........10propagating wave...............7radar.............................4, 24radial waves .....................11radiated power..................18radiation pattern.........17, 23reflected wave............15, 16reflection coefficient ..14, 17Rydberg constant ...............3

S Poynting vector ...........10sense.................................11

SHF....................................4sidelobe levelcircular aperture ant.....22

sinc function.....................26skin depth.........................13SLF ....................................4Snell's law........................15source-free .....................6, 8sources ...............................3space derivative................27space factor ................20, 30spectrum.........................4, 5sphere...............................28standing wave ratio ......8, 17Stokes' theorem................28

surface current dens. ..14, 17SWR standing wave ratio17tan (loss tangent)..........12tesla....................................3tilt angle...........................11time domain ...................7, 8time-average.................7, 10time-average power............7time-dependent...................8time-harmonic....................8TL transmission loss........24total field..........................21


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transmission coefficient...14transmission loss .............24transmitted wave........15, 16transverse.........................31traveling wave ...................7trig identities....................26TV ..................................... 4UHF...................................4ULF ................................... 4ultraviolet ..........................4units...................................3vector diff. equation ........27vector Helmholtz equation10vector identities ...............30vector magnetic potential 10velocity of propagation....10VHF...................................4visible region...................23

visualizing theelectromagnetic wave.... 5

VLF................................... 4volt .................................... 3voltage standing wave ratio

.................................... 17volume

sphere.......................... 28vp velocity of propagation10VSWR voltage standing

wave ratio....................17watt.................................... 3wave equation....................7wave impedance................ 5wave number .....................6wave refl./trans................ 15wave vector .......................6

in 2 dimensions...........13wavelength ....................3, 7

weber................................. 3X-ray ................................. 4Yee grid........................... 25

y param. of discontinuity14y|| param. of discontinuity14 reflection coefficient... 17 reflection coeff. ......... 14|| reflection coeff........... 14J() space factor............ 20

phase constant........... 6, 7 skin depth.................... 13new complex permit. ...... 16new complex permittivity120 array steering.............. 23 intrinsic wave impd... 5, 60 characteristic impedance

of free space.................. 5 wavelength................ 3, 7

loss tangent ..................12 tilt angle.......................11B Brewster angle............16c critical angle ...............14i angle of incidence .......15r angle of relection ........15t angle of transmission...15 charge density................3 volume charge dens......8 conductivity...........12, 16

transmission coeff.......14|| transmission coeff. ......14 array phase angle.........23 curl............................28 divergence..................272 Laplacian ...................27


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http://www.ece.utexas.edu/courses/fall_00/user: harrypotter, password: muggles

SOME BASICS

COULOMB

A unit of electrical charge equal to one amp second,the charge on 6.2110

18electrons, or one joule per

volt.

UNITS, electromagnetics

2

2

2

electric field [V/m]

magnetic field [A/m]fields

electric flux density [C/m ]

magnetic flux density [Wb/m ]

electric current density [A/m ]

electric

v

E

H

D

B

J

v

v

v

v

v

3sources

charge density [C/m ]

These vectors are a function of space and time ( ),r tv

.

UNITS, electrical

I(current in amps) =

q W J N m V C

s V V s V s s= = = =

q(charge in coulombs) = J N m W s

I s V CV V V

= = = =

C(capacitance in farads) =2 2

2

q q q J I s

V J N m V V = = = =

H(inductance in henrys) = V s

I (note that 2HF s= )

J(energy in joules) =2

2 q

N m V q W s I V s C V C

= = = = =

N(force in newtons) =2

J q V W s kg m

m m m s= = =

T(magnetic flux density in teslas) =2 2 2

Wb V s H I

m m m= =

V(electric potential in volts) =

W J J W s N m q

I q I s q q C = = = = =

W(power in watts) = 2 1

746 J N m q V C V

V I HPs s s s

= = = = =

Wb(magnetic flux in webers) = J

H I V sI

= =

where s is seconds

WAVELENGTH [m]The distance that a wave travels during one cycle.

2pv

f k

= =

For complex k,( jk k k = ):

2

k

=

vp = velocity of propagation (speed oflight 2.99810

8m/s in free space)

f= frequency [Hz]

k= 2 =

, the wave number or

propagation constant [m-1]

The relation at right can be used toquickly approximate radio frequencywavelengths. For example, at 300 MHz itis easily seen that the wavelength is 1m.

( )

300MHzf

=

CONSTANTS

Avogadros number

[molecules/mole]231002.6 =AN

Boltzmanns constant231038.1 =k J/K5

1062.8

= eV/KElementary charge

191060.1 =q C

Electron mass31

0 1011.9=m kg

Permittivity of free space12

0 1085.8= F/m

Permeability constant7

0 104= H/m

Plancks constant341063.6 =h J-s151014.4 = cV-s

Rydberg constant 678,109=R cm-1

kT @ room temperature 0259.0=kT eV

Speed of light810998.2 =c m/s

1 (angstrom) 10-8 cm = 10-10m

1 m (micron) 10-4cm1 nm = 10 = 10-7cm

1 eV = 1.6 10-19J


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ELECTROMAGNETIC SPECTRUM

FREQUENCY WAVELENGTH(free space)

DESIGNATION APPLICATIONS

< 3 Hz > 100 Mm Geophysical prospecting

3-30 Hz 10-100 Mm ELF Detection of buried metals

30-300 Hz 1-10 Mm SLF Power transmission, submarine communications0.3-3 kHz 0.1-1 Mm ULF Telephone, audio

3-30 kHz 10-100 km VLF Navigation, positioning, naval communications

30-300 kHz 1-10 km LF Navigation, radio beacons

0.3-3 MHz 0.1-1 km MF AM broadcasting

3-30 MHz 10-100 m HF Short wave, citizens' band

30-300 MHz54-7276-8888-108174-216

1-10 m VHF TV, FM, policeTV channels 2-4TV channels 5-6FM radioTV channels 7-13

0.3-3 GHz470-890 MHz915 MHz800-2500 MHz1-22.452-4

10-100 cm UHF

"money band"

Radar, TV, GPS, cellular phoneTV channels 14-83Microwave ovens (Europe)PCS cellular phones, analog at 900 MHz, GSM/CDMA at 1900L-band, GPS systemMicrowave ovens (U.S.)S-band

3-30 GHz4-88-1212-1818-27

1-10 cm SHF Radar, satellite communicationsC-bandX-band (Police radar at 11 GHz)Ku-band (dBS Primestar at 14 GHz)K-band (Police radar at 22 GHz)

30-300 GHz27-4040-6060-8080-100

0.1-1 cm EHF Radar, remote sensingKa-band (Police radar at 35 GHz)U-bandV-bandW-band

0.3-1 THz 0.3-1 mm Millimeter Astromony, meteorology

1012-1014 Hz 3-300 m Infrared Heating, night vision, optical communications

3.951014-7.71014 Hz

390-760 nm625-760600-625577-600

492-577455-492390-455

Visible light Vision, astronomy, optical communicationsRedOrangeYellow

GreenBlueViolet

1015-1018 Hz 0.3-300 nm Ultraviolet Sterilization

1016-1021 Hz X-rays Medical diagnosis

1018-1022 Hz -rays Cancer therapy, astrophysics

> 1022 Hz Cosmic rays Astrophysics


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ELECTROMAGNETIC WAVES

THE ELECTROMAGNETIC SPECTRUM

The HF band is useful for longrange communicationsbecause these frequencies tend to bounce off theionosphere (a atmospheric layer whose lowerboundary is about 30 miles up) and the earth's

surface. Only three transmitters would be required forglobal coverage. The effect varies with time of daydue to the effects of sunlight on the ionosphere. Theionosphere is transparent to the FM band.

The attenuation effect of the atmosphere peaks atvarious frequencies, notably at 60 GHz due to oxygen.This frequency is used for intersatellitecommunications when it is desired that the signal notreach earth.

The attenuation effect of the atmosphere is at a lowpoint at 94 GHz, which is in the W-band, used forradar.

The frequency of U.S. microwave ovens (2.45 GHz) isan ISM frequency (Industrial, Scientific, Medical).

At the high frequencies, the electromagnetic spectrumbecomes more particle-like and less wave-like.

VISUALIZING THE ELECTROMAGNETICWAVE

The electric field Eand the magnetic fieldHare at rightangles to each other. With the electric field alignedwith thex-axis and the magnetic field on they-axis,

propagation is in thez-direction. Propagation is themovement of the effectof the electromagneticdisturbance. This is analogous to dropping a pebble ina pond. The ripples propagate outward from thesource of the disturbance but the water only moves invertical oscillation.

y

xzE

H

CHARACTERISTIC IMPEDANCE []The characteristic or intrinsic wave impedance is theratio of electric to magnetic field components, a

characteristic of the medium. 0 is the characteristic

impedance of free space with a value of 377.

=

= permeability [H/m]

=permittivity

[F/m]

The characteristic impedance can be used to relate theelectric and magnetic fields.

( ) E k H = v v

( )1 H k E =

v v

Other relations:

k = k =

k = a unit vector in the direction of propagation


6/31


k PHASE CONSTANT [m-1]The wave number or propagation constant or

phase constantk(sometimes called ) depends onthe source frequency and characteristics of the

medium. k= is called the dispersion

relationship.

2kc

= = = =

To obtain the complex phase constant, use the complex

permittivity new, see page 12. new jk k k = = Thereal part k (always positive) governs the propagation andthe imaginary part k (always subtracted) governs thedamping. The exponential term of the wave equation lookslike this:

+z propagating:( )j jj jk k zkz k z k ze e e e

= = -z propagating:

( )j jj jk k zkz k z k ze e e e + + + += =

Note that the exponent containing k will always have thesame sign as the exponent containing k , since wavedecay must occur in the same direction as wavepropagation. It is important to remember that when readingthe complex phase constant from a wave expression that

the k and k terms are themselves always positive andthe + or signs in the exponents are associated with z,determining the direction of propagation.

Complex phase constant in the time domain:

{ } ( ) {j j

dampspropagation in

direction

cosk z k z t k z

z z

e e e t k z e

+ +

= Re1442443

= angular frequency of the source [radians/s]0 = permittivity of free space 8.85 10-12[F/m]0= permeability of free space 410-7[H/m]= 2 c

k f

= , the wavelength [m]

kv

WAVE VECTOR [m-1]The phase constant kis converted to a vector. The

vector kv

is in the direction of propagation.

x y zk kk k x k y k z= = + +

v

k= 2 =

, the wave number or propagation constant

[rad./m]

PLANE WAVE

A pebble dropped in a pond produces a circular wave.A plane wave presents a planar wavefront. Functionvariables within a plane have uniform amplitude andphase values. Plane waves do not exist in nature butthe idea is useful as an approximation in somecircumstances. A radio wave at great distance from

the transmitting antenna could be considered a planewave. So treating a wave as a plane wave is to ignorethe source, hence they are also called "source-free"waves.

wavefront

z

Linearly polarized electromagnetic wave equations:

Electric field: ( )0 cosx E xE t kz= v

0kz

x E xE e=

v(phasor form)

Magnetic field: ( )0 cosyE

H y t kz=

v

j0 kzy

E H y e=

v(phasor form)

= angular frequency of the source [radians/s]E0 = peak amplitude[V/m]t= time[s]

k= 2 =


[rad./m]z = distance along the axis of propagation[m] = / intrinsic wave impedance, the ratio of electric to

magnetic field components, a characteristic of the

medium []


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TRAVELING/PROPAGATING WAVE

A wave traveling in the +z direction can be expressedjkze and a wave in the opposite direction would be

expressed+jkze .

z1

t1

z

to

z

k= 2 =


[m-1]

=2 ck f

= , the wavelength [m]

THE COMPLEX WAVE EQUATION

The complex wave equation is applicable when theexcitation is sinusoidal and under steady stateconditions.

22

2

( )( ) 0

d V zk V zdz + =

where0 0

2k

= =

is the phase constant and

is often represented by the letter .

The complex wave equationabove is a second-orderordinary differential equation commonly found in theanalysis of physical systems. The general solution is:

j j( ) kz kzV z V e V e+ += +

wherejkze and

jkze+ represent wave propagationin the +z and z directions respectively.

PHASOR NOTATION

When the excitation is sinusoidal and under steady-state conditions, we can convert between the timedomain and the phasor domain.

Wherez(t) is a function in the time domain and Zis itsequivalent in the phasor domain, we have

z(t) =Re{Zej

t

}.Time Domain Phasor Domain

Z(t) Z

cosA t A

( )0cosA t + 0j

Ae

( )0cosx

Ae t + 0jx Ae e

sinA t A

( )0sinA t + 0jjAe

( )d z tdt

Z

( ) z t dt 1

Z

Example, time domain to phasor domain:

( ) ( ) ( )

( ){ }j j j j , 2cos 4sin

2 j 4kz t kz t r t t kz x t kz y

e e x e e y

= + + +

= +

E

Re

v v

( ) j j 2 j4kz kzE r e x e y= v v

TIME-AVERAGE

When two functions are multiplied, they cannot beconverted to the phasor domain and multiplied.Instead, we convert each function to the phasordomain and multiply one by the complex conjugate ofthe other and divide the result by two.

For example, the function for power is:

( ) ( ) ( )P t v t i t = watts

Time-averaged power is:

( ) ( ) ( ) { }*0

1 12

T

P t v t i t dt V I T

= = Re wattsT= period [s]V= voltage in the phasor domain [V]I* = complex conjugate of the phasor domain current [A]


8/31


STANDING WAVE RATIO

+

===11

SWRmin

max

min

max

I

I

V

V

MAXWELL'S EQUATIONSMaxwell's equations govern the principles of guidingand propagation of electromagnetic energy andprovide the foundations of all electromagneticphenomena and their applications. The time-harmonicexpressions can be used only when the wave issinusoidal.

STANDARD FORM(Time Domain)

TIME-HARMONIC(Frequency Domain)

Faraday'sLaw t

B

E = -

vv

E B = -v v

Ampere'sLaw* t +

D H = J

v

v v j H D J +=v v v

Gauss'Law v D =

v vD =

v

no namelaw 0 B =

v 0B =

v

E = electric field [V/m]B = magnetic flux density [Wb/m2or T] B = 0Ht= time[s]

D = electric flux density [C/m2] D = 0E = volume charge density[C/m3]

H= magnetic field intensity [A/m]J= current density [

A/m

2]

*Maxwell added thet

D term to Ampere's Law.

MAXWELL'S EQUATIONSsource-free or plane wave solution

If we consider an electromagnetic wave at somedistance from the source, it can be approximated as aplane wave, that is, having a planar wavefront ratherthan spherical shape. In this approximation, thesource components of Maxwell's equations can be

ignored and the equations become:

SOURCE-FREE(Time-dependent)

t

B

E = -

vv

SOURCE-FREE(Time-harmonic)

jE B = -v v

t

D

H =

vv

j j H D E = =v v v

0 D =v

0D =v

0 B =

v

0B =

v

FARADAY'S LAW

When the magnetic flux enclosed by a loop of wirechanges with time, a current is produced in the loop.The variation of the magnetic flux can result from atime-varying magnetic field, a coil in motion, or both..

t

B

E = -

vv

Ev

= the curl of the electric field

B =0H magnetic flux density[Wb/m2or T]

Another way of expressing Faraday's law is that a

changing magnetic field induces an electric field.

indC S

dV dl ds

dt= = E B

vv v v

where S is thesurface enclosedby contour C.


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GAUSS'S LAWThe net flux passing through a surface enclosing a chargeis equal to the charge. Careful, what this first integral reallymeans is the surface area multiplied by the perpendicularelectric field. There may not be any integration involved.

0 encS

d Q = sEv

encS Vd dv Q= = sEv

0 = permittivity of free space 8.85 10-12F/mE = electric field [V/m]

D = electric flux density [C/m2]ds = a small increment of surface S

= volume charge density [C/m3]dv = a small increment of volume VQenc = total electric charge enclosed by the Gaussian

surface [S]

The differential version of Gauss's law is:

= Dv

or ( )div = Ev

GAUSS'S LAW an example problem

Find the intensity of the electric field at distance rfrom a

straight conductor having a voltage V.

Consider a cylindrical surface of length l and radius renclosing a portion of the conductor. The electric fieldpasses through the curved surface of the cylinder but notthe ends. Gauss's law says that the electric flux passingthrough this curved surface is equal to the charge enclosed.

2

0 0 0

r enc l lS

d E lr d Q l CVl

= = = = sE

so VCdrE lr = 2

00and

r

VCE lr

02

=

Er= electric field at distance rfrom the conductor [V/m]l = length [m]r d = a small increment of the cylindrical surface S[m2]l = charge density per unit length [C/m]Cl = capacitance per unit length [F/m]V= voltage on the line [V]

CONSTITUTIVE RELATIONS

( ) ( ), , B E H vv vv

Themagnetic field intensity vectorHv

is directly

analogous to the electric flux density vectorDv

in

electrostatics in that both Dv

and Hv

are medium-independent and are directly related to their sources.

In free space: ... 0= Evv

0= B H vv

D = electric flux density [C/m2]E= electric field [V/m]

B = magnetic flux density [Wb/m2or T]H= magnetic field intensity [A/m]0 = permittivity of free space 8.85 10-12[F/m]0= permeability of free space 410

-7[H/m]

Free space looks like a transmission line:

00

1 m

0

0 0 0

CONSERVATIVE FIELD LAW

0 =Ev

0S

d = lE E= vector electric field [V/m]dl = a small increment of length

EQUATION OF CONTINUITY

The differential form of the law of conservation of

charge.

t

=

J

v

In expanded phaser form: jyx zJJ J

x y z

+ + =

J= current density [A/m2] = volume charge density[C/m3]t= time[s]

COULOMB'S LAW

= Dv

S V

d dv=

sD

v

D = electric flux density [C/m2] = volume charge density [C/m3]ds = a small increment of surface S


10/31


POYNTING'S THEOREM

Poynting's theorem is about power conservation and isderived from Maxwell's equations.

( ) ( ) ( )Total power flowing Total power dissipatedRate at which the storedinto the surface . in volume .energy in volume is increasing.

m e eS V V

S VV

ds W W dv dvt

= + +

E H E E v v vv v

144424443 1442443144424443

Units are in watts.

Sv

POYNTING VECTOR [W/m2]The Poynting vector is the power density at a point inspace, i.e. the power flowing out of a tiny area ds.Units are in watts per meter squared. I haven't founda good font to do cursives with yet; the Poynting vectoris supposed to be a cursive capital S.

Instantaneous Poynting vector:

( ){ } ( ){ }

j21 1,2 2

t z t E H E H e = = + S E H Re Re

vv v v v vv

Hint: Convert to the time domain (sine and cosine), thenperform E H.

Time-averaged Poynting vector:

( ) ( ) { }0

1 1,

2

T

z t dt E H T

= S = E H Revv v vv

Hint: Either integrate the instantaneous Poynting vectoror use the simpler method involving the cross productE H*. Note thatH* is justHwith the signs reversed on

all thejs.

T= 2 1

f

=

the period [s]

Ev

= the electric field vector in phasor notation [V/m]

Hv

= the complex conjugate of the magnetic field intensity

in phasor notation [A/m]

Av

VECTOR MAGNETIC POTENTIAL[Wb/m]

The vector magnetic potential points in the direction ofcurrent.

( )0 0

1j

j=

E A + A

v v v = A B

v v

( )0

1=

H A v v

20

2 20 0 0

k

= A + A J v v v

123

In Cartesian coordinates:

2 20 0

2 20 0

2 20 0

x x x

y y y

z z z

A k A J

A k A J

A k A J

+ =

+ =

+ =

This statement says that currentJx produces only fluxAx,JyproducesAy, etc.

VECTOR HELMHOLTZ EQUATION2 2 0E k E + =

v v

In the case of a uniform plane wave whereEhas onlyanx component and is only a function ofz, theequation reduces to

2

22 0x xd E k Edz + =

Note that this is a second-order ordinary differentialequation (the same form as the wave equation) andhas the general solution

( ) j j1 2kz kz

xE z C e C e += +

C1 and C1 can be determined from boundaryconditions. The real or instantaneous electric field canbe found as

( ) ( ){ }

( ) ( )

j j j1 2

1 2

,

cos cos

kz kz t

xz t C e C e e

C t kz C t kz

+ = +

= + +

E Re

Ev

= the electric field vector in phasor notation [V/m]

k=2

=


[m-1] = angular frequency of the source [radians/s]0 = permittivity of free space 8.85 10 -12[F/m]0 = permeability of free space 410

-7[H/m]

vp VELOCITY OF PROPAGATION [m/s]The velocity of propagation is the speed at which awave moves through the medium. The velocityapproaches the speed of light but may not exceed thespeed of light since this is the maximum speed atwhich information can be transmitted.

1p

vk

= =

If the phase constant kis complex ( jk k k = ), thenthe relation holds using the real part of the phaseconstant k : /pv k= .

= permittivity of the material[F/cm]

= permeability of the material[H/cm] = frequency[radians/second]

k= 2 =

phase constant [m-1]


11/31


RADIAL WAVES

SENSE, AXIAL RATIO, AND TILT ANGLE

These three parameters determine whether anelectromagnetic wave has linear, circular, or ellipticalpolarization, define which direction the wave rotates intime, and describe the elongation and angular

orientation to the x-axis.Note that the polarization also depends on the angle ofobservation. For example, moving off axis from a circularlypolarized wave causes the polarization to become eliptical,and at 90off axis, it is linearly polarized.

sense: The rotation of the wave in time as viewed from thereceiving antenna. RH or LH. For linearly polarized waves,sense is irrelevant.

axial ratio: The length of the major axis divided by thelength of the minor axis. The value can range from 1(circularly polarized) to infinity (linearly polarized).

tilt angle: The tilting of the major axis with respect to the x-axis. For circular polarization the value is irrelevant.

To determine the three parameters, first determine thedirection of propagation, e.g. an e-jkz term indicatespropagation in the +z direction due to the negative sign inthe exponent. Next, determine the values forEL andER (twonew terms defined for this purpose) and convert thesecomplex values to polar notation.

jj

2L

x y

L L

E E E E e

+ = = jj

2R

x y

R R

E E E E e

+ += =

The sense is LH if |EL| > |ER| and RH if |ER| > |EL|. If|EL| = |ER|, then the polarization is linear. NOTE: Thisassumes a wave propagating in the +z direction. For a

wave traveling in thez direction, reverse the sensefound by this method.

The axial ratio is axial ratio R L

R L

E E

E E

+=

The tilt angle is ( )12tilt angle R L= Be careful to preserve the signs of the angles whenfinding the tilt.

LINEAR POLARIZATION

( ) kzx y E xE yE e= +

v

An linearly polarized wave is characterized byE

xandE

yin phase, a finite tilt angle, and an axial ratio of infinity.

major axis

y

tilt angle

x

CIRCULAR POLARIZATION

A wave is circularly polarized when 1)Ex andEy areequal in magnitude and 2) they are 90 out of phaseas in the example above. In the case of the equationbelow, propagation is in the +z direction (out of thepage) due to e being raised to a negative power.

Example: ( ) j 2 j 2 kz E x y e= v

propagationout of page

y

rotation

x

( ) 0 cos sinx y E z xE t yE t = = v

For the case wherez=0, at t=0 the electric field vectorpoints along the +x axis. At t= /2, the vector points in the-y direction. To determine sense, point the thumb in thedirection of propagation (out of the page in this case) andverify that the fingers curl in the direction of vector rotation.Whichever hand this works with determines the senseright-hand or left-hand. In this case it is the left hand. Thismethod of determining sense is an alternative to thepreviously mentioned method using theER andEL values.

ELLIPTICAL POLARIZATION

( ) j kzx y E xE yE e= +

v

An elliptically polarized wave is characterized by a

finite axial ratio greater than one.

major axis

minor axis

y

tilt angle

x


12/31


LOSSLESS MATERIALS

Characteristics of plane waves in common losslessmaterials:

Because ( ) ( )0 0, , > ,

k= gets larger

1pv

k

= =

gets smaller

2k

= gets smaller

so the dimensions of an antenna would be smaller

LOSSY/DISSIPATIVE MATERIALS

Lossy materials can carry some current so this

introduces a new term, conductivity (sigma) in units

of moes per meter [ J/m]. This new term results incomplex permittivity. To calculate for lossy materials,simply substitute the new value for complexpermittivity into the old equations. For example:

newj j H E H E = =v v v v

() PERMITTIVITY

OF THE IONOSPHERE [F/m]The permittivity of the earths ionosphere can bedescribed by an unmagnetized cold plasma and isdependent on the frequency of the incident radio

wave. For the FM band, the permittivity of theionosphere is about the same as that of free space sothe signal passes through it. For the AM band, thepermittivity of the ionosphere is much different fromfree space (its actually a negative value) so the waveis reflected.

2

0 2( ) 1 p

=

, where2

0p

Nq

m =

0 = permittivity of free space 8.85 10-12[F/m]p= plasma frequency? [radians/second]= radio frequency [radians/second]N= electron density, e.g. 1012[m-3]q = the electron charge, 1.602210

-19[C]

m = electron mass 9.109410-31

[kg]

new COMPLEX PERMITTIVITY [F/m]Complex permittivity is a characteristic of lossy

materials. The imaginary part of new accounts forheat loss in the medium due to damping of thevibrating dipole moments.

new 1 j j

= = ,

= permittivity [F/m]= (sigma) conductivity [ J/m or Siemens/meter]

= the real part of complex permittivity [F/m] = the imaginary part of complex permittivity [F/m]

tan LOSS TANGENTThe loss tangent, a value between 0 and 1, is the ratioof conduction current to the displacement current in alossy medium, or the loss coefficient of a wave after it

has traveled one wavelength. This is the way data isusually presented in texts.

tan

=

Graphical representation ofloss tangent:

For a dielectric, tan 1 = .

( )1

tan tan2

=

( )IImag.

( )IRe

is proportional to the amount of current going through

the capacitance C.

is proportional to the amount current going through theconductance G.


13/31


SKIN DEPTH [m]The skin depth or penetration depth is the distanceinto a material at which a wave is attenuated by 1/e(about 36.8%) of its original intensity.

1k

=

where 0 newjk k k = =

and new 1 j =

k= 2 =

phase constant [m-1]

= frequency[radians/second]0 = permeability of free space 410

-7[H/m]

= permittivity [F/m]= (sigma) conductivity [ J/m or Siemens/meter]

Skin Depths

60 Hz 1 MHz 1 GHz

silver

coppergoldaluminumiron

8.27 mm

8.53 mm10.14 mm10.92 mm

0.65 mm

0.064 mm

0.066 mm0.079 mm0.084 mm0.005 mm

0.0020 mm

0.0021 mm0.0025 mm0.0027 mm0.00016 mm

Wm MAGNETIC ENERGY

Energy stored in a magnetic field [Joules].

2

0

1'

2m VW dv=

B

Wm = energy stored in a magnetic

field [J]0= permeability constant

410-7 [H/m]B = magnetic flux density

[Wb/m2or T]

REFLECTION AND TRANSMISSION

kxi, etc. WAVE VECTOR COMPONENTS

kxi

kxr

zt

rk y

ki

kzi

i

kzrx

k

t

z

kxtk

1 1ik = 1 1 sin xi ik =

2 2tk = 1 1 cos zi ik =

1 = permeability of the medium of the incident wave [H/m]2 = permeability of the medium of the transmitted wave

[H/m]1 = permittivity of the medium of the incident wave [F/m]2 = permittivity of the medium of the transmitted wave [F/m]

kv

WAVE VECTOR IN 2 DIMENSIONS[m-1]

The vector kv

is in the direction of propagation.

sin cos x zk k x k z k x k z= + = +v

Dispersion Relationship:2 2 2

x zk k+ =

k0 PHASE CONSTANT IN FREE SPACE[m-1]

0 0 0k =


14/31


y,y|| PARAMETER OF DISCONTINUITY

The parameter is used in finding the reflection andtransmission coefficients.

1

2

zt

zi

ky

k

=

1

2

zt

zi

ky

k

=

P, where

2 2 2 2 2 xt zt k k+ = and xt xik k= 1 = permeability of the medium of the incident wave [H/m]2 = permeability of the medium of the transmitted wave

[H/m]1 = permittivity of the medium of the incident wave [F/m]2 = permittivity of the medium of the transmitted wave [F/m]kzt=z-component of the phase constant of the transmitted

wave [m-1]kzi=z-component of the phase constant of the incident

wave [m-1]

, || TRANSMISSION COEFFICIENT

The coefficient used in determining the amplitude ofthe transmitted wave.

21 y

=+

2

1 y =

+P P

y= parameter of discontinuity for perpendicular polarizedwaves

y||= parameter of discontinuity for parallel polarized waves

The reflection and transmission coefficients are related.

1 + =

, || REFLECTION COEFFICIENT

This value, when multiplied by the incident wave Eyand reversing the sign of the wave componentperpendicular to the plane of discontinuity, yields thereflected wave. For the reflection coefficient of atransmission line, see p17.

11

y

y

=

+

1

1

y

y

=

+P

P

P

y= parameter of discontinuity for perpendicular polarized

wavesy||= parameter of discontinuity for parallel polarized waves

c CRITICAL ANGLE

The minimum angle of incidence for which there istotal reflection. A critical angle exists for waves in adense material encountering a less dense material,

i.e. 2 < 1. Applies to both perpendicular and parallelpolarization.

1 2 2

1 1

sinc =

y

Incidentwaves

approachingfrom this region

are totally reflected.

i

c

k

z

xmedium 1

11 2 2medium 2

BOUNDARY CONDITIONS

The tangential components of the electromagneticwave are equal across the boundary (discontinuity) in

materials with finite (most materialsperfectconductors are the exception). Remember that theterm boundarymeans that we're talking about thecase wherez = 0. The tangential components are thex andy components, so the boundary conditions are:

Polarization:i r t

y y y E E E + = andi r t

x x x H H H + =

|| Polarization:i r t

x x x E E E + = and i r t y y y H H H + =

The only difference for a perfect conductor is that the sumof the incident and reflected magnetic components is notequal to the transmitted component (zero) but to the surfacecurrent densityJS in units of A/m.

Pol.:i r

x x S H H J + = and || Pol.: i r y y S H H J + =

The general expressions for boundary conditions are

( )1 2 0n E E =v v

and ( )1 2 Sn H H J =v v v

where n is a unit vector normal to the plane ofdiscontinuity ( n z= in the examples here). This alsogives us the direction of currentJS. These two equations

apply in all situations with the understanding that JS = 0 forall materials except perfect conductors and thatE2 =H2 = 0for perfect conductors (due to total reflection).


15/31


SNELL'S LAW

Relates the angles of incidence and transmission tothe properties of the materials. Don't use Snell's lawwith lossy materials.

sin sini i t t k k = or sin sini i i t t t =

ki= phase constant of the incident wave [m

-1

]kt= phase constant of the transmitted wave [m-1]i = angle of incidence []t= angle of reflection []

PERPENDICULAR POLARIZATION

PERPENDICULAR POLARIZATIONWave Reflection/Transmission

iEv

is perpendicular to the plane of incidence (thexzplane) with componentsEy andHx, propagating in the

ik

v

direction. The wave encounters a discontinuity atthexy plane with a reflection in the

rkv

direction and a

transmitted wave in thet

kv

direction. If the wave is

entering a denser medium (r2 > r1) then thetransmitted wave will bend toward the z-axis.

E

H

y

i

ki

ii

i

medium 1

kr

11x

z

t

k

medium 2

2 2

t

i = permeability of the medium of the incident wave [H/m]t= permeability of the medium of the transmitted wave

[H/m]i = permittivity of the medium of the incident wave [F/m]t= permittivity of the medium of the transmitted wave [F/m]i = angle of incidence []t= angle of reflection []

, ,i r tk k kv v v

= wave vectors for the incident, reflected, and

transmitted plane waves [rad./m]

PERPENDICULAR POLARIZATIONEQUATIONS

for incident, reflected, and transmitted waves

The form of the expressions for the electrical andmagnetic components of the perpendicularly polarizedwave encountering a discontinuity. Note theplacement of the k

ziand k

xiterms in the expression for

the magnetic field due to it's perpendicular orientationto the electric field.

Incident:j j

0xi zik x k zi E yE e e

=v

( ) j j0 xi zik x k zi zi xii

E H x k z k e e

= +

v

Reflected:j j

0xr zrk x k zr E yE e e

+=

v

( ) j j0 xr zrk x k zr zr xr i

E H x k z k e e

+= + +

v

Transmitted:j j

0xt zt k x k zt E yE e e

=

v

( ) j j0 xt zt k x k zt zt xt t

E H x k z k e e

= +

v

where:

Thex-components of the phase constant are equal for theincident, reflected, and transmitted waves. This is calledthe phase matching conditionand is determined by theboundary conditions.

xi xr xt k k k= = phase matching condition

Other relations: zi zr k k= and2 2 2

xt zt t t k k+ =

Also see Characteristic Impedance (p5) and Wave VectorComponents (p13) for help with these problems.

|| PARALLEL POLARIZATION

PARALLEL POLARIZATION

iEv

is parallel to the plane of incidence (the xz plane).

= 0

y

k

iH

Ei

i

z

i

i

medium 1

rk

1 1 x

tz

tk

medium 2

2 2


16/31


PARALLEL POLARIZATION EQUATIONSfor incident, reflected, and transmitted waves

The form of the expressions for the electrical andmagnetic components of the parallel polarized waveencountering a discontinuity.

Incident: ( )j j

01

1

xi zik x k zi

zi xiE k x k z H e e

=

v

j j0

xi zik x k zi H yH e e =

v

Reflected: ( )j j

01

1 xr zr k x k zr

zr xr E k x k z H e e +=

Pv

j j0

xr zr k x k zr H yH e e += P

v

Transmitted: ( ) j j02

1 xt zt k x k zt

zt xt E k x k z H e e =

P

v

j j0

xt zt k x k zt H yH e e = P

v

where:

Thex-components of the phase constant are equal for theincident, reflected, and transmitted waves. This is calledthe phase matching conditionand is determined by theboundary conditions.

xi xr xt k k k= = phase matching condition

Other relations: zi zr k k= and2 2 2

xt zt t t k k+ =

Also see Characteristic Impedance (p5) and Wave VectorComponents (p13) for help with these problems.

B BREWSTER ANGLENamed for Scottish physicist, Sir David Brewster, whofirst proposed it in 1811. For electromagnetic waves,the Brewster angle applies only to parallelpolarization. It is the angle of incidence at which there

is total transmission of the incident wave, i.e. || = 0.

For light waves it is the angle of incidence that results in anangle of 90between the transmitted and reflected waves.Also called the polarizing angle, this results in the reflectedwave being polarized, with vibrations perpendicular to theplane of incidence(in other words, perpendicular to thepage).

In acoustic applications such as lithotripsy, B is called theangle of intromission, used for blasting kidney stones.

2 2

1 1

tan tB

i

k

k

= =

y

90

k

B

i

medium 1

B

x

z

k

t

t

medium 2

LOSSY MEDIUMWhen a lossy medium is involved, use the same

equations but replace with new. The imaginary part ofnew accounts for heat loss in the medium due to damping ofthe vibrating dipole moments. Don't use Snell's law withlossy mediums.

kztmust have the form kzt'-jkzt'', positive real andnegative imaginary, i.e. change it if you have to. Thesolution will contain the term

j j xi zt zt k x k z k ze e e

Medium 1

1 1 Medium 2

{

new

2 2 2

1 j

=


17/31


PERFECT CONDUCTOR

Everything gets reflected. Since we don't know thesurface current densityJs, only one boundary conditionis useful. This is sufficient since we know thetransmitted waves are zero. Remember that the termboundarymeansz = 0.The tangential components of the incident and reflectedelectric fields are out of phase at the boundary so that theycancel.

The tangential components of the incident and reflectedmagnetic fields are in phase at the boundary creating a

strong magnetic field that produces surface currentJs [A/m].

Boundary Conditions:1

1

0S

n H J

n E

=

=

v v

v

where n z=

Polarized: 0i ry yE E+ = 1 = j j

0 0 0 xi xik x k x

E e E e

+ =

6 4zik kzi

E2i

02 kzi kzi

z

z

E0

The z component of the incident and reflected electric fieldsof the polarized wave produce a standing wave with

constructive peaks spaced at 2/kzi apart, beginning /kzifrom the conductor surface.

||Polarized: 0i rx xE E+ = 1 = +P

j j0 0

1 1

1 10 xi xik x k x zi zik H e k H e

= P

x

yz

ki

rk

0t tE H= =v v

j 0S SJ + =v

( )0

S z J n H ==

v v

where n z= in this example,andJS is the surface current[A/m]. Surface current ispresent only in a perfectconductor (see BoundaryConditions p14). Surface

meansz = 0.

ANTENNAS

REFLECTION COEFFICIENT [unitless]

This is the reflection coefficient for the transmissionline. For reflected waves in space, see p14.

j 0

0

L

L

Z Ze

Z Z

= =+

and 01

1L

Z Z+

=

= magnitude of the reflection coefficient [no units] = phase angle of the reflection coefficient [no units]ZL = load (antenna) impedance []Z0 = transmission line (characteristic) impedance []

SWR STANDING WAVE RATIO [V/V]

Also called the voltage standing wave ratioor VSWR.

1SWR

1+

=

= magnitude of the reflection coefficient [no units]

RADIATION PATTERN

The radiation pattern of an antenna is the relativestrength of the absolute value of the electric field as a

function of and . The radiation pattern is the samefor receiving antennas as for transmitting antennas.

Radiation Pattern: ( ), Er

The example on the left below is the radiation pattern for asingle dipole oriented along the verticle axis. The patternon the right is for a 2-dipole array with the elements lying in

the horizontal plane, one to the left and one to the right ofcenter, with their lengths extending into the page.

Radiation pattern |E()| for asingle dipole

Radiation pattern |E(90,)| for

2-element dipole with d=/2


18/31


POWER

Time-Averaged Poynting Vector [W/m2]

{ }*ff ff 1

2= S Re E H vv v

Total Time-Averaged Radiated Power [W]

( )2rad sinS SP ds r r d d = = S S rv v

dBm DECIBELS RELATIVE TO 1 mW

The decibel expression for power. The logarithmicnature of decibel units translates the multiplication anddivision associated with gains and losses into additionand subtraction.

0 dBm = 1 mW20 dBm = 100 mW

-20 dBm = 0.01 mW

( ) ( )dBm 10log mWP P=

( ) ( )dBm /10mW 10PP =

GAIN

Gain Directivity Efficiency=

EFFICIENCY

Power radiatedEfficiency

Power Power dissipatedradiated in conductor

=

+

(DIR) DIRECTIVITY [no units/dB]

The directivity is the gain in the direction of maximumradiation compared to an omnidirectional sphericalwave.

( )

max2

rad

Directivity

/ 4P r

=

Sv

[no units]

where { }2ff

*ff ff 12

02= =

E S Re E H

vvv v

and ( )2rad sinS

P r r d d = Sr

Directivity is often expressed in dB:

( ) ( )dB

DIR 10log DIR=

Directivity of Antenna Types

Type Directivity dB

IsotropicInfinitesimal dipoleHalf-wave dipoleWire-typeHornReflector

1.01.51.64

~10~100

103-109

01.762.15

~10~20

30-90

FAR FIELD APPROXIMATION

In general, we are only interested in the electric andmagnetic fields distant from the antenna. This allowsus to simplify the calculations by dropping the nearfieldcomponents. As a rule of thumb, the far fieldregion is defined as:

22Dr>

whereD is the diameter or size of the antenna

ISOTROPIC ANTENNA

An isotropic antenna is a theoretical antenna thatradiates equally in all directions. On any given"spherical shell" in the far field, EffandHff are in phaseand are equal in magnitude.

rad

24

P

r= S

( )DIR 1=


19/31


INFINITESIMAL DIPOLE AT ORIGIN

A theoretical model of a very short dipole antenna, themost basic of antennas. A points in the direction ofcurrent, in this casez. Oscillation occurs along thez-axis.

Vector magnetic potential

and magnetic field at a pointin space due to aninfinitesimal dipole antenna atthe origin.

( )0j

0 4

k re

z I zr

=

Av

x

z

zDipole

y

r

Point

( ) ( )0j

0

1

4

k re I z z

r

= =

H A v v

In the far field, only the slowest-decaying components are

significant.

( )

( )

0

0

jff

0

jff

0 0

j sin4

sin4

k r

k r

ek I z

r

ek I z

r

=

=

H

E

v

v

The radiation pattern of the infinitesimal dipole is non-isotropic. The directivity is 1.5, or 1.76 dBi.

z y

x x

Time-Averaged Poynting Vector [W/m2]

{ } ( )( )

2 2* 2ff ff 0 01

2 2

sin

2 4

kr I z

r

= =

S Re E H

vv v

Total Time-Averaged Radiated Power [W]

( )22

0 0rad 12S

k I zP ds

= =

Sv v

Directivity: ( )DIR 1.5= A= vector magnetic potential [Wb/m]H= magnetic field intensity [A/m]r= radial distance from the origin [m]

INFINITESIMAL DIPOLE ON Z-AXIS

The infinitesimal dipole at the origin is shifted toanother point on thez-axis.

r

r'

Point

z

y

1

x

z'

Dipole

z

r'

rz'

z'cos

In the far field, we can make these approximations.

1 1 1 cosr r z

From the expression for electric field of the dipole at theorigin, we get.

( )

( )0j cosff

0 0

This term hasless effect herethan when itappears in theexponent above.

sin4 ( cos )

k r ze

k I zr z

= E

v

14243

FINITE DIPOLE ON Z-AXIS

The finite dipole on thez-axis is calculated bysumming the infinitesimaldipoles over a finite length.

I(z') is a function describingthe current along the dipole.For a -wave dipolecentered at the origin, thiswould beI(z') =I0 cos(z'k0).

x

y

r

z Point

-z

+h

-h

FiniteDipole

From superposition

( ) 0 0

ff ff ff

0

j +j cos

0 0

lim

j sin4

h

hz

k r k zh

h

d

I z dz e ek

r

+

+

= =

=

E E E v v v

And finally the radiation due to an arbitrary line current I(z').We must understand this.

( )0

0

j+j cosff

0 0j sin

4

k rh

k z

h

ek I z e dz

r

+

=

Ev

( )ff ff 0

1r=

H E

vv

See the next section also.


20/31


J() SPACE FACTOR

From the previous section, here is the far-field radiation dueto an arbitrary line current:

( )

( )

0

0

j+j cosff

0 0

SPACE FACTORELEMENT PATTERN

the electric field due toan infinitesimal dipole

j sin4

k rh

k z

h

ek I z e dz

r

+

=

Ev

144424443144424443

J

The Space Factor depends on the length of the elementand the current distribution within the element. Note thatthe space factor resembles a Fourier transform of thecurrent distribution, the only difference being the sign of theexponent. Actually it is the Fourier transform since the signis arbitrary as long as the opposite sign is used in theexponent of the compatible inverse Fourier transform.What this means is that an even distribution of current willresult in a more directional radiation pattern and a tapered(Gaussian) current distribution will result in a broaderradiation pattern with lower sidelobes. See also FourierTransform p30 and Fourier Transform Examples p30.

HALF-WAVE DIPOLE

The half-wave dipole or resonant dipole is the mostcommonly used antenna primarily because of itsimpedance, which is easily matched.

To understand the half-wave dipole, first consider thecurrent on a transmission line. The current on oneconductor is out of phase with current on the other so thatradiation effects are canceled. At -wavelength from theend, where the line will be bent to form the -wave dipole,current is at a maximum.

= 0I

~

Frequency

Source

0I= |I |

As a result of bending, current is now in phase so thatradiation takes place.

Current in the dipole

( ) ( )0 0cos I z I k z =

Impedance: zin = 73+j0 Directivity: 1.64 or 2.15 dB

( )0j 0 2ff0 0 2

0

2 cos cosj sin4 sin

k r Iek

r k

=

E

v

Increasing the thickness of the elements has the effect ofbroadening the antenna bandwidth.

LOG-PERIODIC ANTENNA

The log-periodic dipole array (LPDA) consists of anarray of half-wave dipoles of frequenciesf, rf, r2f, . . .,r

nfwhich, when plotted on a log scale appear equally

spaced. This produces a broadband antenna.

-1

termination

shortcircuit

+1ln

ln

n+1D

l-1n

DnnD

LOG-PERIODIC FREQUENCIES

The relationship between the bandwidth, the numberof elements, and the scaling factor for a log-periodicantenna follows. These are my own observations;textbook information is in the next box.

1log logu

l f

fN

f s=

The crossover points between bands is then:

0 1 2

1 1 1 1, , ,

N

l l l l

f f f f

f f f f

s s s s

L

The frequencies used to determine the length of each half-wavelength element are the center frequencies of each ofthe N bands:

0.5 1.5 2.5 0.5

1 1 1 1, , ,

N

l l l l

f f f f

f f f f s s s s

L

fu = upper bandwidth cutoff frequency [Hz]fl = lower bandwidth cutoff frequency [Hz]N= number of antenna elements

sf= scaling factor [no units]


21/31


LOG-PERIODIC ANTENNA PROPERTIES

Alignment angle tan2

n

n

l

D =

Scaling factor

1 1

n nf

n n

l Ds

l D+ += = , a constant < 1

Spacing parameter 1 cot2 4

fn np

n

sd dSl

= = =

,

where1n n nd D D+= , varies with the element.

and the optimum spacing parameter is4 25.76909996 10 0.0167208176 0.0602945516pS g g

= + +

where g is the gain in dB,and the optimum scaling factor is

3.9866009 0.236230336f ps S= +

Impedance20 1

1 0

88a fZ Z

Z sZ Z

= +

where

{

10

Element lengthto diameterratio

276log 270lZd

=

and

Z0 is the characteristic impedance of the transmissionline. The antenna impedance is matched to the line byadjusting the length to diameter ratio.

Design bandwidth ( )2

1.1 7.7 1 cots w f B B s = +

whereBw is the desired bandwidth as the ratio of highestto lowest frequency. The design bandwidth is greaterthan the desired bandwidth.

Number of elements

( )ln1

ln 1/ s

f

BN

S = +

-WAVE MONOPOLE

A -wave monopole antenna is essentially half of a -wave dipole antenna.

According to image theory,when the monopole ismounted perpendicular to aground plane, it can bemodeled as having a

reflected current opposite theplane with flow in the samedirection as current in themonopole.

0in

0

/ 236.5

VZ

I= =

Reflection

Groundplane

Monopole

Signal

~

I0

APERTURE ANTENNAS

APERTURE THEORY

If a plane wave eminates from a source of finitedimensions, the propagating wave assumes sphericalcharacteristics within its radiation pattern. Thebeamwidth is inversely related to the size (in

wavelengths) of the aperture, that is, a large apertureproduces a more directional beam. However, sidelobelevel is independent of size; it depends on amplitude

taper. For a large aperture, the sine term in theelement pattern disappears since the pattern becomes

highly directional with 90, sin 1.

Planewave

Sphericalwave

Aperture

The far field radiation is proportional to the Fouriertransform of the current distribution.

Array Factor:

Azimuth:( )sin

AF xx

x

ka

ka

=

Elevation:( )sin

AFy

y

y

ka

ka

=

k= 2/, phase constant of the carrier wave [m-1

]2ax = width of aperture [m]2ay = height of aperture [m] = azimuth angle [radians] = elevation angle [radians]

TOTAL FIELD for apertures and arrays

The total radiated field is a product of the elementpattern and the array factor.

( ) ( )Element ArrayTotal Field pattern factor= The far field radiation is proportional to the Fouriertransform of the current distribution.


22/31


CIRCULAR APERTURE ANTENNA

The circular aperture antenna is parabolic in shape sothat the wave emitted from the horn is reflected inphase as a plane wave.

In phaseHornParabolicreflector

For uniform amplitude across the beam:

1 radian 60Beamwidth

/ /D D

=

VERY IMPORTANT

( )2

4 AreaDirectivity

=

For a tapered aperture (amplitude varies across theaperture:

( )2

4 AreaDirectivity a

=

edge

center

Edge taper 20logJ

J=

a = aperature efficiency, a value less than or equal to 1D = aperature diameter [m]J= current density [A/m2]

= wavelength [meters]Sidelobe level is dependent on amplitude taper, not size:

Sidelobe Levels

Edge Taper

[dB]Beamwidth [] Sidelobe [dB]

Aperture

Efficiency

-8

-10-12

-14-16

-18

-20

65.3/(D/)67.0/(D/)68.8/(D/)70.5/(D/)72.2/(D/)73.9/(D/)75.6/(D/)

-24.7

-27.0-29.5

-31.7-33.5

-34.5

-34.7

0.918

0.8770.834

0.7920.754

0.719

0.690

ANTENNA ARRAYSWhen two or more antennas are used together, thecombination is called an array. A planar arrangement ofclosely spaced antenna elements essentially produces anaperture of equivalent area.

AF ARRAY FACTOR

The Array Factor is the far-field radiation intensity ofthe elements of an array, assuming the elements to beisotropic radiators. That is, it contains no informationabout the type of element used in the array.

AF ARRAY FACTORfor a linear series of elements

d

( )

( )0

sin / 2AF , sin sin

sin / 2

Nuu k d

u= = +

N= number of array elements [no units]d= spacing between adjacent elements [m]

= angle of elevation. This can be taken as 90for thex-yplane [radians]

= phase angle between adjacent elements (The directionof the antenna array can be electronically steered bydriving the elements at different phases, i.e.progressively altering the phase of each element by the

angle with respect to the previous element.) [radians]

|AF| is periodic with period 2. There are N-2 sidelobes inthe AF between 0 and 2.

Array Factors for N=2, 3, 5, & 10, plotted versus u


23/31


0 ARRAY STEERING [radians]

The direction of the main lobe of the antenna array can beelectronically steered away from the perpendicular by an

angle of by driving the elements at different phases, thatis, progressively altering the phase of each element by the

angle with respect to the previous element.

100

sin k d =

= phase angle of the driving signal between adjacentelements [radians]

k0= phase constant in free space [m-1]

d= the distance between adjacent elements [m]

GRATING LOBES

Grating lobes are secondary lobes in the radiationpattern having the same magnitude as the main lobe.When the spacing between array elements becomes

too large, grating lobes appear. Grating lobes are aserious detriment to the directivity of an antennasystem. To prevent the occurance of grating lobes,the element spacing should be less than onewavelength.

Below are radiation patterns of a 5-element dipole array with

spacings d=0.8, d=0.9, and d=. The main lobes are thethin vertical lobes and the grating lobes are the largehorizontally opposed lobes.

d= 0.8 d= 0.9 d=

VISIBLE REGION

The Array Factor is an unbounded function with a period of

2. However, only a finite region of the function plays a partin the radiation pattern of the array. This is called thevisible region and is defined by:

0 0k d u k d + + +

= phase angle between adjacent [radians]

k0= phase constant in free space [m-1]

d= the distance between adjacent elements [m]

GRAPHICAL METHOD

The Radiation Pattern can be determined using theinformation covered in the preceeding three sections. Apolar plot is created below the plot of the array factorversus u. The diameter of the polar plot of the array factor

versus is fitted into the visible range. Lines extendingdownward from the nodes to the perimeter of the polar plot

and then to its center determine where the lobes are drawn.For the example below:

N = 5, i.e. a 5-element arrayd = /2, elements are spaced /2 apart0 = 40, the steering angle= -2.019 or -0.643, the phase angle between elements-1.64 to 0.36 is the visible range

The Matlab code for creating these two plots is inGraphicalMethod.m and requires a second file polar3.m.


24/31


FRIIS FREE SPACE TRANSMISSIONFORMULA

The Friis transmission formula is used in measuringthe gain of antennas and in determining the properaperture size for an application. The formulaexpresses the relationship between received andtransmitted power, also called the power transfer ratio:

( ) ( )2

DIR DIR4

R

T R

T

P

P r

=

To determine the size of the antennas needed, it isnecessary to know the minimum amount of power requiredat the receiving antenna:

( )minimumminimum S/NR NP P=

whereN sP kT B=

Ts = equivalent system noise "temperature" [K]k= Boltzmann constant 1.38065810-23 [J/K]

PT= transmitted power [W]PR = power received [W]r= distance between antennas [m](DIR)T= directivity of the transmitting antenna [no units](DIR)R = directivity of the receiving antenna [no units]

RADAR

Maximum detectable range:

( )

( ) ( ) ( )

12 42

max 3

min

DIR

4 S/NT

s

Pr

k T

=

[m]

Radar cross-section:

( ) ( )

2

2 2back2

incident

4 4s

i

Er r

E = =

S

S[m2]

Ambient noise: N SP kT B= [W]Incident power at the target:

( )2incident DIR4T

T

P

r=

S [W/m2]

Receiver bandwidth:1

B = [Hz]

Ts = equivalent system noise "temperature" [K]k= Boltzmann constant 1.38065810-23 [J/K] = duration of transmitted pulse [s]PT= transmitted power [W]

TL TRANSMISSION LOSS [dB]

The transmission loss between a transmitting andreceiving antenna depends on the antenna gains, thedistance and the frequency:

( ) ( )4

DIR DIR 20logT R

rTL

= +

[dB]

(DIR)T= directivity of the transmitting antenna [no units](DIR)R = directivity of the receiving antenna [no units]r= distance between antennas [m] = wavelength [meters]see also PLANE EARTH TRANSMISSION LOSSp25.

NUMERICAL METHODS

COURANT STABILITY CONDITION

The FDTD numerical method for calculating anelectromagnetic field requires that the propagation

speed of the calculations be equal to or greater thanthe propagation speed of the wave:

1

2t

c

t= increment of time between calculations [s] = increment of distance between discrete values, e.g. the

distance between (i,j) and (i+1,j) [s]c = speed of light, (2.998108 m/s in free space)


25/31


FDTD FINITE DIFFERENCE TIME DOMAIN

In this class, we did a 2D FDTD computer simulation.Refer to the document FDTD-homework10.pdf. TheFDTD is the simplest numerical method. It is basedon the Yee Grid..

The 2-dimensional YeeGrid at right is acoordinate system withelectric field componentslocated discretely atinteger intercepts in the

xy-plane and the magneticcomponents locatedintermediately as shown.

2

3

H (0,)

E (0,0)

H (,0)y

z

x

0 1 2 3 Hx Values ofHx depend on the past value of Hx

at the same position and values ofEz aboveand below.

( ) ( )

( ) ( )

1 12 21 1

2 2

0

, ,

, 1 ,

n n

x x

n n

z z

H i j H i j

tE i j E i j

+ + = +

+

Hy Values ofHy depend on the past value ofHy at the same position and values ofEzto the left and right.

( ) ( )

( ) ( )

1 12 21 1

2 2

0

, ,

1, ,

n n

y x

n n

z z

H i j H i j

tE i j E i j

+ + = +

+ +

Ez Values ofEz depend on the past value ofEz at the same position and values ofHxabove and below, and values of Hy to theleft and right.

( ) ( ) ( )

( ) ( )

( ) ( )

1

2

1 12 2

1 12 2

1 120

1 12 2

12

0

, , ,

, ,

, ,

nn n z z y

n n

y x

n n

x z

tE i j E i j H i j

H i j H i j

t H i j J i j

++

+ +

+ +

= + +

+

+

Calculations proceed in the order shown,Hx,Hy, thenEz.Special treatment is given to the boundaries.

n is used to describe the relative order in time, e.g. n+occurs after n.

t= increment of time between calculations [s] = increment of distance between discrete values, e.g. the

distance between (i,j) and (i+1,j) [s]

GEOMETRICAL OPTICS

GEOMETRICAL OPTICS APPROXIMATION

Observationpoint

Reflectedray0

Image

Specularpoint

2 , ,

, 0 0

Direct ray

Antenna

1

( )( )

( )

( )00 j 02 21j 01 101 02 21

kkA Ae y e y

+ = + +

Er

A() = far field radiation pattern01 = distance from point 0 to point 1= reflection coefficient, see REFLECTION COEFFICIENT

p14.

PLANE EARTH TRANSMISSION LOSS

This is an approximation based on geometrical optics,not valid for tall antennas.

Receivingantenna

Transmittingantenna r

h

( ) ( ) 410log 10log DIR DIR 40logRT R

T

PTL h r

P = =

TL = transmission loss [dB](DIR) = directivity, unitless here (not dB)h = antenna height [m]r= distance between antennas [m]see also TRANSMISSION LOSSp24.


26/31


MATHEMATICS

x +jy COMPLEX NUMBERS

0

y

m

A

Rex

jj cos j sin x y A Ae A A+ = = = +

{ } cos x y x A+ = = Re

{ } sin x y y A+ = = Im

{ } 2 2Magnitude j x y A x y+ = = +

{ } 1Phase j tany

x y

x

+ = =

452j

j j e e

= =

Expressing a complex number in terms of the naturalnumber e. Note that when using a calculator, the exponentof e must be in radians.

( )+j+jj cba b a e a e + = = , where /180b c=

Taking the square root of a complex number:

( )1 1

-j45 -j22.52 41 j 2 2 1.10 j0.455e e = = =

With frequency information:

{ } ( )j j cosz tae e a t z = +Re and

{ } ( )j jj sinz tae e a t z = +Re

COMPLEX CONJUGATES

The complex conjugate of a number is simply thatnumber with the sign changed on the imaginary part.This applies to both rectangular and polar notation.When conjugates are multiplied, the result is a scalar.

22))(( bajbajba+=+

2))(( ABABA = Other properties of conjugates:

*)*****()*( FEDCBAFDEABC ++=++ jBjB ee + =)*(

TRIG IDENTITIES

j j 2cose e+ + = j j 2sine e+ = j cos j sine =

SINC( ) FUNCTION

( )( )sin

sincv

vv

=

The sinc function may be involved when finding the -power beamwidth using a field expression. The solutionis

( )sin 1, 1.391558 radians

2

vv

v= =

WORKING WITH LINES, SURFACES, AND

VOLUMESl(r') means "the charge density along line l as a

function of r'." This might be a value in C/m or it

could be a function. Similarly, s(r') would be thecharge density of a surface and v(r') is thecharge density of a volume.

For example, a disk of radius a having a uniform

charge density of C/m2, would have a totalcharge of a2, but to find its influence on pointsalong the central axis we might considerincremental rings of the charged surface as

s(r') dr'= s2r'dr'.

If dl' refers to an incremental distance along a circularcontour C, the expression is r'd, where r' is theradius and d is the incremental angle.


27/31


NABLA, DEL OR GRAD OPERATOR[+ m-1]

The del operator operates on a scalar to produce a

vector. Compare the operation to taking the timederivative. Where /tmeans to take the derivativewith respect to time and introduces a s-1 component to

the units of the result, the operation means to takethe derivative with respect to distance (in 3dimensions) and introduces a m-1 component to theunits of the result. terms may be called spacederivativesand an equation which contains the operator may be called a vector differential

equation. In other words A is how fastA changesas you move through space.

in rectangularcoordinates:

A A A

A x y z x y z

= + +

in cylindricalcoordinates:

1 A A A

A r zr r z

= + +

in sphericalcoordinates: 1 1 sin A A AA r r r r = + +

2 THE LAPLACIAN [+ m-2]The divergence of a gradient

Laplacian of a scalar inrectangular coordinates:

2 2 22

2 2 2

A A AA

x y z

= + +

Laplacian of avector in rectan-gular coordinates:

22 22

2 2 2 yx z

AA A A x y z

x y z

= + +

v

In spherical and

cylindricalcoordinates:

( )( ) ( )

2

grad div curl curl

A A A

A A =

v v v

v

, div D DIVERGENCE [+ m-1]

The divergence operation is performed on a vectorand produces a scalar. The del operator followed bythe dot product operator is read as"the divergence of"and is an operation performed on a vector. In

rectangular coordinates, means the sum of thepartial derivatives of the magnitudes in the x,y, andzdirections with respect to thex,y, andz variables. Theresult is a scalar, and a factor of m-1 is contributed tothe units of the result.

In the electrostatic context, divergence is the totaloutward flux per unit volume due to a source charge.For example, in this form of Gauss' law, where D is a

density per unit area, D becomes a density per unitvolume.

div yx zDD D

D D x y z

= = + + =

v v

Dv = electric flux density vector D E= v v [C/m2] = source charge density[C/m3]

in rectangularcoordinates:

yx zDD D

D x y z

= + +

v

in cylindricalcoordinates:

( )1 1

zrD D

D rDr r r z

= + +

v

in spherical coordinates:

( ) ( )22

sin1 1 1

sin sin

rr D DDD

r r r r

= + +

v


28/31


CURL curl B B= v v

[+ m-1]

The circulation around an enclosed area. The curloperator acts on a vector field to produce anothervector field. The curl of vector B isin rectangular coordinates:

curl

y yx xz z

B B

B BB BB B x y z

y z z x x y

= =

+ +

v v

This can be written in determinant form and may beeasier to remember in this form.

curl

x y z

x y z

B B x y z

B B B

= =

v v

in cylindrical coordinates:

( )

curl

1 1 z r z r

B B

rBB B B B Br z

r z z r r r

= = + +

v v

in spherical coordinates:

( )

( ) ( )

sin1curl

sin

1 1 1 sin

r r

B B B B r

r

rB rBB B

r r r r

= = +

+

v v

The divergence of a curl is always zero:

( ) 0H =v

DOT PRODUCT [= units2]The dot product is a scalar value.

( ) ( )zzyyxxzyxzyx BABABABBBAAA ++=++++= zyxzyxBA

ABcos = BABA

0 =yx , 1 =xx

( ) yzyx BBBB =++= yzyxyB

B

A

AB

Projection of Balong :

( )aaB

B

B

The dot product of 90 vectors is zero.The dot product is commutative and distributive:

ABBA = ( ) CABACBA +=+

CROSS PRODUCT

The cross product is an operation performed on twovectors resulting in a third vector perpendicular to theplane in which the first two lie.

( ) ( )( ) ( ) ( )xyyxzxxzyzzy

zyxzyx

BABABABABABA

BBBAAA

++=

++++=

zyx

zyxzyxBA

ABsin = BAnBA

where n is the unit vector normal toboth A and B (thumb of right-hand rule).

BAAB =

zyx = zxy = 0= xx =z r = r z

The cross product is distributive:

( ) CABACBA +=+ Also, we have:

( ) ( ) ( ) = A B C A C B A B C

n

AB

A

B

Cylindrical Coordinates:

r z z r z r = = =

Spherical Coordinates:

r r r = = =

GEOMETRY

SPHERE

Area 24 rA =

Volume3

3

4rV =

ELLIPSEArea ABA = Circumference

22

22 baL

+

STOKES' THEOREM

S is any unbroken surface (doesn't have to be flat). Lis is the closed path (line) around its border. Stokes'theorem says that the line integral of a vector fieldaround the pathL is related to the surface integral ofthe curl on that vector field.

( ) L S

V dl V dS= v vv v


29/31


COORDINATE SYSTEMS

Cartesian or Rectangular Coordinates:

( , , ) x y z xx yy zz= + +r x is a unit vector222 zyx ++=r

Spherical Coordinates:

),,( rP ris distance from center is angle from vertical is the CCW angle from thex-axis

r, , and are unit vectores and are functions ofpositiontheir orientation depends on where theyare located.

Cylindrical Coordinates:

),,( zr C ris distance from the vertical (z) axis is the CCW angle from thex-axisz is the vertical distance from origin

COORDINATE TRANSFORMATIONS

Rectangular to Cylindrical:

To obtain: ( , , ) r zr z rA A zA = + +A

22 yxAr += cos sinr x y= +

xy1tan = sin cosx y = +

zz = z z= Cylindrical to Rectangular:

To obtain: ( , , ) x y z xx yy zz= + +r

= cosrx cos cosx r=

= sinry sin cosr y = + zz = z z=

Rectangular to Spherical:

To obtain: ( , , ) rr rA A A

= + + A

222 zyxAr ++= sin cos sin sin cosr x y z= + +

222

1cos

zyx

z

++=

cos cos cos sin sin x y z = +

x

y1tan = sin cosx y = +

Spherical to Rectangular:

To obtain: ( , , ) x y z xx yy zz= + +r = cossinrx

sin cos cos cos sinx r=

= sinsinry

sin sin

cos sin

cos

y r=

+

+

= cosrz

cos sinz r=

r

x

y

z

Point


30/31


VECTOR IDENTITIES

( ) + = +

( ) A B A B + = + v vv v

( ) A B A B + = + v vv v

( ) = +

( ) A A A = + v v v

( ) A B B A A B = v v vv v v

( ) A A A = + v v v

( ) ( ) ( ) A B A B B A B A A B = + v v v v vv v v v v

2 =

0A =v

0 =

( ) 2 A A A = v v v

( )

( ) ( ) ( ) ( )

A B

A B B A A B B A

=

+ + +

v v

v v v vv v v v

A B C B C A C A B = = v v v v v vv v v

( ) ( ) ( ) A B C B A C C A B = v v v v v vv v v

FOURIER TRANSFORM

The Fourier transform converts a function of time to afunction of frequency.

( ) ( ) j tF f t e dt +

=

Inverse Fourier transform:

( ) ( ) j1

2t

f t F e d + +

=

Note that the signs in the exponent for these two functions

are shown as they are by convention but they could bereversed as long as one is positive and the other isnegative.

FOURIER TRANSFORM EXAMPLES

A gaussian with a sharp peak (heavy line) becomes agaussian with a "softer" peak after performing theFourier transform.

A gaussian with a more rounded peak (heavy linebelow) becomes a gaussian with a sharper peakafter performing the Fourier transform.

Taking this to the extreme, the Fourier transform of aconstant (horizontal line) becomes a spike and thetransform of a spike is a horizontal line. All this is relevantto antennas because in the farfield expression for a linecurrent is a space factor, which is actually the Fouriertransform of the current in the antenna element.

( )0

0

j+j cosff

0 0

"space factor""element pattern"dependent on currentthe field due to andistribution and lengthinfinitesimal dipole

j sin4

k rh

k z

h

ek I z e dz

r

+

=

Ev

144424443144424443


31/31

GLOSSARY

anisotropic materials materials in which the electricpolarization vector is not in the same direction as the electric

field. The values of , , and are dependent on the fielddirection. Examples are crystal structures and ionizedgases.

anechoic chamber an enclosed space that absorbes

radiation so that reflections do not interf

Antennas and Wireless Propagation

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