Formulario

Parameter FormulaEquationNumber

Innitesimal areaof sphere

dA = r2 sin d d (2-1)

Elemental solidangle of sphere

d2 = sin d d (2-2)

Average powerdensity

Wav = 12 Re[E H] (2-8)

Radiatedpower/averageradiated power

Prad = Pav =#S

Wav ds = 12#S

Re[E H] ds (2-9)

Radiation densityof isotropicradiator

W0 = Prad4r2 (2-11)

Radiation intensity(far eld)

U = r2Wrad = B0F(, ) r2

2

[|E(r, , )|2 + |E(r, , )|2](2-12),

(2-12a)

DirectivityD(, )

D = UU0

= 4UPrad

= 42A

(2-16),(2-23)

Beam solid angle2A

2A = 2

0

0Fn(, ) sin d d

Fn(, ) = F(, )|F(, )|max

(2-24)

(2-25)

(continued overleaf )

TABLE 2.3 Summary of Important Parameters and Associated Formulas and EquationNumbers

110 FUNDAMENTAL PARAMETERS OF ANTENNAS

TABLE 2.3 (continued )


Maximumdirectivity D0

Dmax = D0 = UmaxU0

= 4UmaxPrad

(2-16a)

Partial directivitiesD,D

D0 = D +D

D = 4UPrad

= 4U(Prad ) + (Prad )

D = 4UPrad

= 4U(Prad ) + (Prad )

(2-17)

(2-17a)

(2-17b)

Approximatemaximumdirectivity (onemain lobe pattern)

D0 441r42r

= 41,25341d42d

(Kraus)

D0 32 ln 2421r +422r

= 22.181421r +422r

= 72,815421d +422d

(Tai-Pereira)

(2-26),

(2-27)

(2-30),

(2-30a),

(2-30b)

Approximatemaximumdirectivity(omnidirectionalpattern)

D0 101HPBW(degrees) 0.0027[HPBW(degrees)]2(McDonald)

D0 172.4 + 191

0.818 + 1HPBW(degrees)

(Pozar)

(2-33a)

(2-33b)

Gain G(, )G = 4U(, )

Pin= ecd

[4U(, )

Prad

]= ecdD(, )

Prad = ecdPin

(2-46),

(2-47),

(2-49)

Antenna radiationefciency ecd

ecd = RrRr + RL (2-90)

Loss resistance RL(straightwire/uniformcurrent)

RL = Rhf = lP

0

2(2-90b)

Loss resistance RL(straight wire//2dipole)

RL = l2P0

2



Maximum gain G0 G0 = ecdDmax = ecdD0 (2-49a)

Partial gainsG,G

G0 = G +G

G = 4UPin

, G = 4UPin

(2-50)

(2-50a),

(2-50b)

Absolute gainGabs

Gabs = erG(, ) = erecdD(, ) = (1 |?|2)ecdD(, )= e0D(, )

(2-49a)

(2-49b)

Total antennaefciency e0

e0 = ereced = erecd = (1 |?|2)ecd (2-52)

Reectionefciency er

er = (1 |?|2) (2-45)

Beam efciencyBE BE =

20

10

U(, ) sin d d 20

0U(, ) sin d d

(2-54)

Polarization lossfactor (PLF)

PLF = |w a |2 (2-71)

Vector effectivelength e(, )

e(, ) = a l (, )+ al(, ) (2-91)

Polarizationefciency pe

pe = |e Einc |2

|e|2|Einc |2 (2-71a)

Antennaimpedance ZA

ZA = RA + jXA = (Rr + RL)+ jXA (2-72),(2-73)

Maximumeffective area Aem

Aem = |VT |2

8Wi

[1

Rr + RL]= ecd

(2

4

)D0|w a |2

=(2

4

)G0|w a |2

(2-96),

(2-111),

(2-112)

Apertureefciency ap

ap = AemAp

= maximum effective areaphysical area

(2-100)


112 FUNDAMENTAL PARAMETERS OF ANTENNAS



Friis transmissionequation

Pr

Pt=(

4R

)2G0tG0r |t r |2

(2-118),

(2-119)

Radar rangeequation

Pr

Pt= G0tG0r

4

[

4R1R2

]2|w r |2

(2-125),(2-126)

Radar crosssection (RCS)

= limR

[4R2

Ws

Wi

]= lim

R

[4R2

|Es |2|Ei |2

]

= limR

[4R2

|Hs |2|Hi |2

] (2-120a)

BrightnesstemperatureTB(, )

TB(, ) = (, )Tm = (1 |?|2)Tm (2-144)

Antennatemperature TA TA =

20

0TB(, )G(, ) sin d d 2

0

0G(, ) sin d d

(2-145)

COMPUTER CODES 215

TABLE 4.2 Summary of Important Parameters and Associated Formulas and EquationNumbers for a Dipole in the Far Field

Parameter Formula Equation Number

Innitesimal Dipole(l /50)

Normalized powerpattern

U = (En)2 = C0 sin2 (4-29)

Radiation resistanceRr

Rr = (

23

)(l

)2= 802

(l

)2 (4-19)Input resistanceRin

Rin = Rr = (

23

)(l

)2= 802

(l

)2 (4-19)Wave impedanceZw

Zw = EH

= 377 ohms

Directivity D0 D0 = 32 = 1.761 dB (4-31)

Maximum effectivearea Aem

Aem = 32

8(4-32)

Vector effectivelength e

e = a l sin (2-91),|e|max = Example 4.2

Half-powerbeamwidth

HPBW = 90 (4-65)

Loss resistance RL RL = lP

0

2= l

2b

0

2(2-90b)

Small Dipole

(/50 < G /10)


U = (En)2 = C1 sin2 (4-36a)


Rr = 202(l

)2(4-37)

Input resistance Rin Rin = Rr = 202(l

)2(4-37)


216 LINEAR WIRE ANTENNAS



Wave impedanceZw

Zw = EH

= 377 ohms (4-36a), (4-36c)

Directivity D0 D0 = 32 = 1.761 dB


Aem = 32

8


e = a l2 sin (2-91),

|e|max = l2 (4-36a)

Half-powerbeamwidth

HPBW = 90 (4-65)

Half Wavelength Dipole(l = /2)


U = (En)2 = C2

cos

(2

cos )

sin

2

C2 sin3 (4-87)


Rr = 4 Cin (2) 73 ohms (4-93)

Input resistance Rin Rin = Rr = 4 Cin(2) 73 ohms (4-79), (4-93)

Input impedanceZin

Zin = 73 + j42.5 (4-93a)

Wave impedanceZw

Zw = EH

= 377 ohms

Directivity D0 D0 = 4Cin(2)

1.643 = 2.156 dB (4-91)


e = a

cos(

2cos

)sin

(2-91),

|e|max = = 0.3183 (4-84)

Half-powerbeamwidth

HPBW = 78 (4-65)

MULTIMEDIA 217



Loss resistance RL RL = l2P0

2= l

4b

0

2Example (2-13)

Quarter-Wavelength Monopole(l = /4)


U = (En)2 = C2

cos

(2

cos )

sin

2

C2 sin3 (4-87)


Rr = 8 Cin (2) 36.5 ohms (4-106)

Input resistance Rin Rin = Rr = 8 Cin (2) 36.5 ohms (4-106)

Input impedanceZin

Zin = 36.5+ j21.25 (4-106)

Wave impedanceZw

Zw = EH

= 377 ohms

Directivity D0 D0 = 3.286 = 5.167 dB


e = a

cos(

2cos

)(2-91)

|e|max = = 0.3183 (4-84)

MULTIMEDIA 271

TABLE 5.1 Summary of Important Parameters, and Associated Formulas andEquation Numbers for Loop in Far Field


Small Circular Loop (a < /6,C < /3)

(Uniform Current)


U = |En|2 = C0 sin2 (5-27b)

Wave impedanceZw

Zw = EH

= 377 Ohms (5-28)

Directivity D0 D0 = 32 = 1.761 dB (5-31)


Aem = 32

8(5-32)

Radiation resistanceRr (one turn)

Rr = 202(C

)4(5-24)

Radiation resistanceRr (N turns)

Rr = 202(C

)4N2 (5-24a)

Input resistance Rin Rin = Rr = 202(C

)4(5-24)

Loss resistance RL(one turn)

RL = lP

0

2= C

2b

0

2(2-90b)

Loss resistance RL(N turns)

RL = NabRs

(Rp

R0+ 1

)(5-25)

Loop externalinductance LA

LA = 0a[

ln(

8ab

) 2

](5-37a)

Loop internalinductance Li

Li = ab

0

2(5-38)


e = ajk0a2 cosi sin i (5-40)

Half-powerbeamwidth

HPBW = 90 (4-65)


272 LOOP ANTENNAS



Large Circular Loop (a /2, C 3.14)(Uniform Current)


U = |En|2 = C1J 21 (ka sin ) (5-57)

Wave impedanceZw

Zw = EH

= 377 Ohms (5-28)

Directivity D0(a > /2)

D0 = 0.677(C

)(5-63b)

Maximum effectivearea Aem (a > /2)

Aem = 2

4

[0.677

(C

)](5-63c)

Radiation resistance(a > /2),(one turn)

Rr = 602(C

)(5-63a)

Input resistance(a > /2),(one turn)

Rin = Rr = 602(C

)(5-63a)

Loss resistance RL(one turn)

RL = lP

0

2= C

2b

0

2(2-90b)

Loss resistance RL(N turns)

RL = NabRs

(Rp

R0+ 1

)(5-25)

External inductanceLA

LA = 0a[

ln(

8ab

) 2

](5-37a)

Internal inductanceLi

Li = ab

0

2(5-38)


e = ajk0a2 cosi sin i (5-40)

Small Square Loop (Figure 5.17)(Uniform Current, a on Each Side

Normalized powerpattern (principalplane)

U = |En|2 = C2 sin2 (5-70)

REFERENCES 273



Wave impedanceZw

Zw = EH

= 377 Ohms (5-28)


Rr = 20(

2a

)4= 20

(C

)4

Input resistanceRin

Rin = Rr = 20(

4a

)4= 20

(P

)4

Loss resistance RL RL = 4aP

0

2= 4a

2b

0

2(2-90b)

External inductanceLA

LA = 20 a

[ln(ab

) 0.774

](5-37b)

Internal inductanceLi

Li = 4aP

0

2= 4a

2b

0

2(5-38)

Ferrite Circular Loop (a < /6, C < /3)

(uniform current)

Radiation resistanceRf (one turn)

Rf = 202(C

)42cer

(5-73)

cer = fr1+D(f r 1) (5-75)

Radiation resistanceRf (N turns)

Rf = 202(C

)42cerN

2 (5-74)

Ellipsoid: D =(al

)2 [ln(

2la

) 1

]

Demagnetizingfactor D

l a (5-75a)

Sphere: D = 13

312 ARRAYS: LINEAR, PLANAR, AND CIRCULAR

TABLE 6.5 Nulls, Maxima, Half-Power Points, andMinor Lobe Maxima for Uniform AmplitudeHansen-Woodyard End-Fire Arrays

NULLS n = cos1[

1 + (1 2n) 2dN

]n = 1, 2, 3, . . .n = N, 2N, 3N, . . .

MAXIMA m = cos1{

1 + [1 (2m+ 1)] 2Nd

}m = 1, 2, 3, . . .d/ 1

HALF-POWERPOINTS

h = cos1(

1 0.1398 Nd

)d/ 1N large

MINOR LOBEMAXIMA

s = cos1(

1 sNd

)s = 1, 2, 3, . . .d/ 1

TABLE 6.6 Beamwidths for Uniform AmplitudeHansen-Woodyard End-Fire Arrays

FIRST-NULLBEAMWIDTH(FNBW)

4n = 2 cos1(

1 2dN

)

HALF-POWERBEAMWIDTH(HPBW)

4h = 2 cos1(

1 0.1398 Nd

)d/ 1N large

FIRST SIDE LOBEBEAMWIDTH(FSLBW)

4s = 2 cos1(

1 Nd

)d/ 1

N-ELEMENT LINEAR ARRAY: DIRECTIVITY 313

TABLE 6.7 Maximum Element Spacing dmax to Maintain Either One or Two AmplitudeMaxima of a Linear Array

Array Distribution TypeDirection

of MaximumElementSpacing

Linear Uniform Broadside 0 = 90 only dmax < 0 = 0, 90, 180simultaneously

d =

Linear Uniform Ordinaryend-re

0 = 0 only dmax < /20 = 180 only dmax < /20 = 0, 90, 180simultaneously

d =

Linear Uniform Hansen-Woodyardend-re

0 = 0 only d /40 = 180 only d /4

Linear Uniform Scanning 0 = max dmax < 0 < 0 < 180

Linear Nonuniform Binomial 0 = 90 only dmax < 0 = 0, 90, 180simultaneously

d =

Linear Nonuniform Dolph-Tschebyscheff

0 = 90 only dmax

cos1( 1zo

)0 = 0, 90, 180simultaneously

d =

Planar Uniform Planar 0 = 0 only dmax < 0 = 0, 90 and 180;0 = 0, 90, 180, 270simultaneously

d =

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