# Fundamental parameters of antennas

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```					Chapter 2 : Fundamental parameters of antennas
– Radiation intensity – Beamwidth – Directivity – Antenna efficiency – Gain – Polarization

Chapter 2 : Topics (2)
• Antenna effective length and effective area • Friis transmission equation • Radar range equation

• From Circuit viewpoint
– Input Impedance

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• Once the electromagnetic (EM) energy leaves the antenna, the radiation pattern tells us how the energy propagates away from the antenna. • Definition : Mathematical function or a graphical representation of the radiation properties of an antenna as a function of space coordinates
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• Can be classified as:
– Isotropic, directional and omnidirectional

• Principal patterns (or planes):
– E-plane : the plane containing the electric field vector and the direction of maximum radiation – H-plane : the plane containing the magnetic field vector and the direction of maximum radiation

• Isotropic: Hypothetical antenna having equal radiation in all directions • Directional: having the property or receiving EM energy more effectively in some directions other than others • Omnidirectional: having an essentially nondirectional pattern in a given plane and a directional pattern in any orthogonal plane
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1

Eθ Hφ

Field Regions

Omnidirectional

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Reactive Near Field Region
• Region surrounding the antenna, wherein the reactive field predominates
For D > λ : R < 0.62 D3

• Region between reactive near-field and farfield regions (Fresnel zone)
For D > λ : 2D 2

λ λ 2π

λ

> R > 0.62

D3

λ λ
2π D3

For D < λ (small antenna) : R < ⇒ R < max[ D3 λ ,0.62 ] 2π λ

For D < λ (small antenna) : 3λ > R > ⇒ max[3λ , 2D 2 ] > R > max[

λ
2π

λ

,0.62

λ

]

Angular field distribution depends on distance from antenna
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Radiation fields predominate but angular field distribution still depends on distance from antenna

Far Field Region
• Region where angular field distribution is essentially independent of the distance from antenna (Fraunhofer zone)

Change of antenna amplitude pattern shape

For D > λ : R >

2D 2

λ
2D 2 ]

For D < λ (small antenna) : R > 3λ ⇒ R > max[3λ ,

λ

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2

Spherical coordinate and Solid Angle : Steradian
• Measure of solid angle: 1 steradian = solid angle with its vertex at the center of a sphere of radius r that is subtended by a surface of area r2

• Poynting vector = Power density r r r W =E × H
r W : instantaneous Poynting vector [W/m2 ] r E : instantaneous electric field density [V/m] r H : instantaneous magnetic field density [A/m]

• Total power:

r r ˆ P = ∫∫W ⋅ ds = ∫∫W ⋅ nda
S S

dA = r 2 sin θdθdφ = r 2 dΩ; dΩ = sin θdθdφ = solid angle
Quiz: What’s the solid angle subtended by a sphere?
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P : instantaneous total power [W] ˆ n : unit vector normal to the surface da : infinitesimal area of the closed surface [m2 ]

• For time-harmonic EMrfields r
E ( x, y, z; t ) = Re[E( x, y , z )e jωt ] r r H ( x, y, z; t ) = Re[ H( x, y, z )e jωt ]

• Poynting vector r r r 1 r r r r 1 W =E × H = Re[E × H * ] + Re[E × He j 2ωt ] 2 2 • Time average Poynting vector (average power density or radiation density) r r r r 1 Wav ( x, y, z ) = [ ( x, y, z; t )]av = Re[E × H * ] W 2
1 2
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r r r r 1 ˆ Prad = Pav = ∫∫ Wav ⋅ ds = ∫ Wav ⋅ nda = ∫∫ Re[E × H * ] ⋅ ds 2 S S S
Example 2.1: The average power density is given by r sin θ ˆ ˆ Wav = rWr = rA0 2 [ W / m 2 ] r The total radiated power becomes r ˆ Prad = ∫∫ Wav ⋅ nda
S

r r appears because E, H fields represent peak values

=∫

2π

0

∫

π

0

ˆ (rA0

sin θ ˆ ) ⋅ rr 2 sin θdθdφ = π 2 A0 [ W ] r2

• For an isotropic antenna
Prad =∫ r r = ∫∫ W0 ⋅ d s
S

• Definition : The power radiated from an antenna per unit solid angle

2π

0

∫

π

0

ˆ ˆ [rW0 ( r )] ⋅ rr sin θdθdφ = 4πr W0 [ W ]
2 2

The power density is then given by

r P ˆ ˆ W0 = rW0 = r rad2 [ W / m 2 ] 4πr

Total power can be given by

Ω

2π

0

∫

π

0

U sin θdθdφ

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• Radiation intensity is related to the far-zone electric field of antenna
U (θ , φ ) = r r r | E( r , θ , φ ) | 2 ≅ [| Eθ ( r ,θ , φ ) |2 + | Eφ ( r , θ , φ ) |2 ] 2η 2η 1 ≅ [| Eθo (θ , φ ) |2 + | Eφo (θ , φ ) |2 ] 2η

Example 2.2: The radiation intensity is given by

U = r 2Wrad = A0 sin θ

Prad = ∫ = A0 ∫
2π 0

2π

0

∫
π

π

0

U sin θdθdφ

∫

0

sin 2 θdθdφ = π 2 A0 [ W]

r r e − jkr E(r ,θ , φ ) : far - zone electric - field intensity of the antenna = E o (θ , φ ) r Eθ , Eφ : far - zone electric - field components of the antenna η : intrinsic impedance of the medium (≈ 377 Ω in free space)
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For an isotropic antenna

Prad = ∫∫U 0 dΩ = U 0 ∫∫ dΩ = 4πU 0
Ω Ω

⇒ U0 =

Beamwidth
• Beamwidth is the angular separation between two identical points on opposite site of the pattern maximum • Half-power beamwidth (HPBW): in a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the beam • First-Null beamwidth (FNBW): angular separation between the first nulls of the pattern
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Beamwidth (2)

Beamwidth (3)
Example 2.3: The normalized radiation intensity of an antenna is represented by

Directivity
• Ratio of radiation intensity in a given direction from the antenna to the average radiation intensity

U (θ ) = cos 2 θ (0 ≤ θ ≤ π ,0 ≤ φ ≤ 2π )
The angle θh at which the function equal to half of its maximum can be found by

U (θ ) |θ =θ h = cos 2 θ = 0.5 ⇒ cos θ h = 0.707 4 Since the pattern is symmetric with respect to the maximum, HPBW = 2 θh = π/2
Likewise, FNBW = 2θn = π since U (θ ) |θ =θ = 0 ⇒ θ n = cos −1 (0) =
n

D(θ , φ ) =

U (θ , φ ) 4πU = (dimension - less) U0 Prad

θ h = cos −1 (0.5) =

π
Note that the average radiation intensity equals to the radiation intensity of an isotropic source.
π
2
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Directivity (2)
Since
Prad = ∫∫U (θ ' , φ ' )dΩ'
Ω'

Directivity (3)
• If the direction is not specified, it implies the directivity of maximum radiation intensity (maximum directivity) expressed as
U max 4πU max = (dimension - less) U0 Prad

D(θ , φ ) = 4π

U (θ , φ )

∫∫U (θ ' , φ ' )dΩ'
Ω'

Dmax = 4π

U max

∫∫U (θ ' , φ ' )dΩ'
Ω'

=

4π ΩA

Dmax: maximum directivity

ΩA is called beam solid angle, and is defined as “solid angle through which all the power of the antenna would flow if its radiation intensity were constant and equal to Umax for all angles within ΩA
U (θ ' , φ ' ) Ω A = ∫∫ dΩ' ⇒ Prad =Ω AU max U max Ω'
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Dmax = D0 =

Directivity (4)
Example 2.4: The radial component of the radiated power density of an infinitesimal linear dipole is given by

Antenna Efficiency
• The overall antenna efficiency take into the following losses:
– Reflections because of the mismatch between the transmission line and the antenna – Conduction and dielectric losses

sin 2 θ [ W/m 2 ] r2 where A0 is the peak value of the power density. The radiation density is given by ˆ ˆ Wav = rWr = rA0

U = r 2Wr = A0 sin 2 θ

e0 = ecd er = ecd (1− | Γ |2 )
e0 : total efficiency er : reflection (mismatch) efficiency ecd : antenna radiation efficiency = ec ed ec , ed : conduction, dielectric efficiencies Γ : voltage reflection coefficient at the input terminal
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The maximum radiation is directed along θ = π/2 and Umax = A0. The total radiated power is given by
Prad = ∫∫ UdΩ = A0 ∫
2π Ω 0

∫

π

0

sin 2 θ sin θdθdφ = A0

8π 3

Thus,
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D0 =

4πU max 3 = and D = D0 sin 2 θ = 1.5 sin 2 θ Prad 2

Gain
• It takes into account the efficiency of the antenna as well as its directional properties. (Directivity only measures
directional properties.)

Gain (2)
• Using ecd, Prad= ecdPin and

G (θ , φ ) = 4π

U (θ , φ ) Pin

G (θ , φ ) = 4π

U (θ , φ ) ecd = ecd D(θ , φ ) Prad

Pin : Power input to antenna; Pin = (1− | Γ |2 ) Po = Prad + Ploss ( Ploss : ohmic and dielectric power loss)
Gain : ratio of radiation intensity in a given direction to the average radiation intensity that would be obtained if all the power input to the antenna were radiated isotropically
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Relative gain: ratio of power gain in a given direction to the power gain of a reference antenna in the same direction. The power input must be the same for both antennas. If the reference antenna is a lossless isotropic source, then

G (θ , φ ) = 4π

U (θ , φ ) Pin (lossless isotropic source)

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Gain (3)
• Absolute gain takes into account impedance mismatch losses at the input terminals in addition to losses within antenna

Polarization
• Property of an EM wave describing the time varying direction and relative magnitude of the electric field. The figure traced as a function of time by the tip of the electric field and the sense in which is traced, as observed along direction of propagation.
Wave propagating in –z direction

Gabs (θ , φ ) = 4π = 4π

U (θ , φ ) U (θ , φ ) = 4π (1− | Γ |2 ) Po Pin U (θ , φ ) (1− | Γ |2 )ecd = e0 D (θ , φ ) Prad

r ˆ ˆ E ( z; t ) = xEx ( z; t ) + yEy ( z; t )

Let E x ( z ) = E xo e jφ x e jkz , E y ( z ) = E yo e y e jkz where E xo , E yo ≥ 0
ˆ z
jφ

e jωt time dependence
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Ex ( z; t ) = E xo cos(ωt + kz + φ x ) Ey ( z; t ) = E yo cos(ωt + kz + φ y )

Polarization (2)
A. Linear Polarization

Polarization (3)
B. Circular Polarization

(i) E xo = 0 or E yo = 0
or (ii)δ = φ y − φ x = nπ
Example
y

(i) E xo = E yo = Eo ⇒ γ = tan −1 (1) = π / 4
and
2 2 E xo + E yo

where n = 0,±1,±2,...

γ = tan −1  

 E yo  π , 0 ≤ γ ≤ E xo  2  

=0 = /4
ˆ z

45°

x

 1  2n + 2 π ; CW/RCP   (ii)δ = φ y − φ x =  −  2n + 1 π ; CCW/LCP     2  where n = 0,1,2,...
Note that the sense of rotation is observed along the direction of propagation.
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δ, γ determine polarization state
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Polarization (4)
Example: RCP

γ = π / 4, δ = π / 2
ˆ z
Ex ( z; t ) = Eo cos(ωt + kz + φ x ) Ey ( z; t ) = Eo cos(ωt + kz + φ x + π / 2) = − Eo sin(ωt + kz + φ x )

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Polarization (5)
C. Elliptic Polarization
A wave is elliptically polarized if it is not linearly or circularly polarized. Linear and circular polarization are special cases of elliptic polarization. To have elliptic polarization: 1. Field must have two orthogonal linear components. 2. The two components can be of the same or different magnitude.

Polarization (6)
C. Elliptic Polarization
 1  2n + 2 π ; CW/REP   (i)if δ = φ y − φx =  −  2n + 1 π ; CCW/LEP     2  where n = 0,1,2,... AND E xo ≠ E yo

(ii)if δ = φ y − φ x ≠ ±

nπ 2

> 0; CW/REP  < 0; CCW/LEP

where n = 0,1,2,... FOR ∀E xo , ∀E yo
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Polarization (7)
Axial Ratio (AR) = Major Axis ; 1 ≤ AR ≤ ∞ Minor Axis

Polarization Loss Factor (PLF)
• Electric field of incoming wave r ˆ E w = ρ w Ei • Electric field of receiving antenna r ˆ E a = ρ a Ea
where ˆ ρ w : unit vector of the wave ˆ ρ a : polarization vector

τ=

π

 2E E  1 − tan −1  2 xo yo cos δ  2 E −E  2 2 yo  xo 

ˆ ρw ˆ ρa

ˆ ˆ PLF =| ρ w ⋅ ρ a |2 =| cosψ p |2 (dimensionless)

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PLF example
LCP wave: δ=-π/2,φx=0 φy=-π/2

Input Impedance
Impedance presented by the antenna at its terminal
Transmitting case

E x = Eo e e ; E y = Eo e e r ˆ ˆ ˆ ˆ ˆ E = ( xEo + yEo e − jπ / 2 )e jkz = ( x − jy ) Eo e jkz = ρ w 2 Eo e jkz ˆ ˆ x − jy ˆ where ρ w = 2 *
jkz jkz

jφ x

jφ y

Z A (ω ) = R A (ω ) + jX A (ω )
RA = Rr + Rloss Rr : Radiation resistance

[Ω ]

Rloss : Loss resistance (ohmic, dielectric) Rloss = Rohmic + Rd

If the antenna is also LCP,

ˆ ˆ ρa = ρw
2

ˆ ˆ ˆ ˆ x − jy x + jy PLF = ⋅ = 1 = 0 dB 2 2 ˆ − jy ˆ x ˆ If the antenna is RCP, ρ a = PLF =
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Z g = Rg + jX g ,Vg = I g ( Z g + Z A ) Vg , I g are peak values
Maximum power delivered to the antenna occurs when conjugate matched: R = R , X = −X
A g A g
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ˆ ˆ ˆ ˆ x − jy x − jy ⋅ =0 2 2

2

2

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Input Impedance (2)
When conjugate matched:

Input Impedance (3)
Power supplied by generator when conjugate matched:

Ig =

Vg 2 RA

|V | 1 Rr | I g |2 Rr = g 2 8 ( Rr + Rloss ) 2
2

2

Vg* | Vg |2 | Vg |2 1 1 * Ps = Re[ Vg I g ] = Vg = = 2 2 2 RA 4 R A 4 Rg

Ploss =
Pg =

|V | 1 Rloss | I g |2 Rloss = g 2 8 ( Rr + Rloss ) 2 Power loss to heat
2

|V | 1 1 | I g |2 R g = g 2 8 Rr + Rloss

Power loss in Rg

1 Ps 2 1 Pin = Prad + Ploss = Ps 2 Prad Rr ecd = = Pin Rr + Rloss Pg =

Pg = Pin

NOTE:
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If Rloss = 0 ⇒ Ploss = 0, Prad = Pin , ecd = 1

Receiving Antenna
Impedance presented by the antenna at its terminal

Receiving Antenna (2)
Power supplied by generator when conjugate matched:

Z A = RA + jX A , Z T = RT + jX T
Receiving case

Ploss =

under conjugate matched condition RT = RA = Rr + Rloss , X T = − X A

Rloss 1 | V |2 | I T |2 Rloss = T 2 8 ( Rr + Rloss ) 2 Power lost to heat PT = Pscatt + Ploss
captured/collected power

VT 1 | V |2 ⇒ PT = | I T |2 RT = T 2 RT 2 8 RT Power delivered to VT , I T are peak value s load IT =

Pc =

1 | V |2 * Re[VT I T ] = T 2 4 RT

Pscatt =
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1 | V |2 Rr | I T |2 Rr = T 2 8 ( Rr + Rloss ) 2
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1 PT = Pc 2

Pscatt + Ploss =

1 Pc 2

Pc = Pscatt + Ploss + PT

Antenna equivalent Area
• Used to describe the power capturing characteristics of an antenna when a wave impinges on it
Ae (θ , φ ) = effective area (aperture) in direction (θ , φ ) = Power available at terminals of receiving antenna Incident power flux density from direction (θ , φ )

Antenna Equivalent Area (2)
Ae = | VT |2 2Wi   RT  2 2  ( Rr + Rloss + RT ) + ( X A + X T )  Under conjugate matched condition:

Aem =

| VT |2 | VT |2 1 = 8Wi RT 8Wi Rr + Rloss

Maximum effective area

P | I |2 RT Ae = T = T Wi 2 Wi

[m ]

2

As =

| VT |2 Rr 8Wi ( Rr + Rloss ) 2

Ae : effective area, which when multiplied by the incident

PT : power delivered to load RT Wi : power density of incident wave
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power density, is equal to power delivered to load RT As : scattering area, which when multiplied by indent power density, is equal to the scattered or re - radiated power

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Antenna Equivalent Area (3)
Under conjugate matched condition:

Antenna Equivalent Area (4)
Example 2.5: a uniform plane wave is incident upon very short dipole, whose radiation resistance is Rr=80(πl/λ)2. Assume that Rloss = 0, the maximum effective area reduces to | V |2 Aem = T 8Wi Rr Since the dipole is very short, the induced current can be assumed to be constant and of uniform phase. The induced voltage is

Aloss = Ac =

| VT |2 Rloss 8Wi ( Rr + Rloss ) 2

Ae = As + Aloss

| VT |2 | VT |2 Rr + Rloss + RT = 4Wi RT 8Wi ( Rr + Rloss ) 2

Ac = Ae + As + Aloss

VT = El

Aloss : loss area, which when multiplied by the incident power density, is equal to power delivered to load Rloss Ac : scattering area, which when multiplied by indent power density, is equal to the total power captured by antenna
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For a uniform plane wave, the incident power density is given by E2 Wi = 2η ( El ) 2 3λ2 Aem = = = 0.119λ2 thus 2 2 2 2 8( E / 2η )(80π l / λ ) 8π

Vector Effective Length
• Vector effective length (or height) is a quantity used to determine the voltage induced on the open-circuit terminals of an antenna when a wave impinges on it. It is r a far-field quantity.

Effective area & Directivity
If antenna #1 were isotropic, its radiated power density at a distance R would be P W0 = t 2 4πR where Pt is the total radiated power. Because of the directivity, the actual power density becomes

ˆ ˆ le (θ , φ ) = θlθ (θ , φ ) + φlφ (θ , φ ) [m] ri r Voc = E ⋅ le

Wt = W0 Dt =

Pt Dt 4πR 2
Wr = Wt Ar = or Dt Ar = Pt Dt Ar 4πR 2

r E i : incident electric field VT = Voc : open - circuit voltage

r r kI l ˆ ˆ E a = θEθ + φE φ = − jη in e − jkr le 4πr

The power collected by the antenna would be

Example 2.6 : The electric field of a short dipole is given by
r ˆ kI le E a = θjη in 8πr
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− jkr

sin θ

r ˆl le = −θ sin θ 2
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Pr 4πR 2 (1) Pt

Effective area & Directivity(2)
If antenna #2 is used as a transmitter, 1 as a receiver, and the medium is linear, passive and isotropic, one obtains P Dr At = r 4πR 2 (2) Pt Dt Dr From (1) and (2), = At Ar Increasing the directivity of an antenna increases its effective area: D0 t D0 r = Atm Arm If antenna #1 is isotropic,

Effective area & Directivity(3)
For example, if antenna #2 is a short dipole, whose effective area is 3λ2/8π and directivity is 1.5, one obtains A 3λ2 2 λ2 Atm = rm = = D0 r 8π 3 4π λ2 and A =D A =D
rm 0r tm 0r

4π

In general, maximum effective area of any antenna is related to its maximum directivity by

A Atm = rm D0 r
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Aem =
If there’re conductiondielectric loss, polarization loss and mismatch:

λ2 D0 4π
Aem =

λ2
4π

ˆ ˆ D0 ecd (1− | Γ |2 ) | ρ w ⋅ ρ a |2

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Friis Transmission Equation
• The Friis transmission equation relates the power received to the power transmitted between two antennas separated by a distance R > 2D2/λ. Power density at distance R from the transmitting antenna: P G (θ , φ ) e P D (θ , φ ) Wt = t t t 2 t = t t t 2t t 4πR 4πR
(θ r , φr )

Friis Transmission Equation(2)
The ratio of the received to the input power:

Pr λ2 Dt (θ t , φt ) Dr (θ r , φr ) ˆ ˆ = et er | ρt ⋅ ρ r | Pt ( 4πR ) 2
or λ2 Dt (θ t , φt ) Dr (θ r , φr ) Pr ˆ ˆ = ecdt ecdr (1− | Γt |2 )(1− | Γr |2 ) | ρt ⋅ ρ r | Pt ( 4πR ) 2 For reflection and polarization-matched antennas aligned for maximum directional radiation and reception:

(θ t , φt )

The effective area of the receiving antenna:

ˆ ( Pt , Gt , Dt , ecdt , Γt , ρt )

ˆ ( Pr , Gr , Dr , ecdr , Γr , ρ r )

The amount of power collected by the receiving antenna: λ2 Dt (θ t , φt ) Dr (θ r , φr ) Pt λ2 ˆ ˆ | ρt ⋅ ρ r | Pr = er Dr (θ r , φr ) Wt = et er 4π ( 4πR ) 2
55 56

λ2 Ar = er Dr (θ r , φr ) 4π

Pr  λ  =  G0t G0 r Pt  4πR 

2

• Radar cross section or echo area (σ) of a target is defined as the the area intercepting that amount of power which, when scattered isotropically, produces at the receiver a density which is equal to that scattered by the actual target. r r    W  | E s |2  | H s |2  σ = lim 4πR 2 s  = lim 4πR 2 r i 2  = lim 4πR 2 r i 2  R→∞ Wi  R →∞  | E |  R →∞  |H |  
σ : radar cross section [m 2 ]
R : observation distance from targe [m] Wi , Ws : incident, scattered power density [W/m 2 ] r r E i , E s : incident, scattered electric field [V/m] r r H i , H s : incident, scattered magnetic field [A/m]
57 58

• The radar range equation relates the power delivered to the receiver load to the power transmitted by an antenna, after it has been scattered by a target with a radar cross section of σ. The amount of captured power at the target with the distance R1 from the transmitting antenna:
ˆ ρw
ˆ ( Pt , Gt , Dt , ecdt , Γt , ρt )

The amount of captured power at the target with the distance R1 from the transmitting antenna: 2 P D (θ , φ ) D (θ , φ )  λ  Pr = ArWs = et erσ t t t t r r r   4πR R   4π 1 2   Thus, 2
Pr D (θ , φ ) D (θ , φ )  λ    = ArWs = et erσ t t t r r r   Pt 4π  4πR1 R2 

With polarization loss:
Pr D (θ , φ ) D (θ , φ )  λ    ˆ ˆ 2 = ArWs = ecdt ecdr (1− | Γt |2 )(1− | Γr |2 )σ t t t r r r   | ρw ⋅ ρr | Pt 4π  4πR1R2 
2

P G (θ , φ ) e P D (θ , φ ) Pc = σWt = σ t t t 2 t = t t t 2t t 4πR1 4πR1

The scattered power density:
ˆ ( Pr , Gr , Dr , ecdr , Γr , ρ r )

W= s

P PD(θ ,φ ) c =etσ t t t t2 2 4πR2 (4πRR2) 1
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For reflection and polarization-matched antennas aligned for maximum directional radiation and reception:
Pr G G = σ 0t 0 r Pt 4π  λ    4πR R    1 2 
2

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