Vector Potentials and Antenna Fundamentals

Vector Potentials 1

Vector Potentials and Antennas

Vector Potentials 2

∇×E = − jωµ

H ∇ i ε

E = ρ

∇×H = jωε

E +J ∇ i µ

H = 0

Maxwell’s Equations – Time-Harmonic Electromagnetic Fields – Homogeneous Medium

Vector Potentials 3

∇ iB = 0

Since

Now for any vector field

∇ i∇×A = 0

Define

B = µ

H = ∇×

A

or

H = 1

µ∇×A

The Vector Potential

Vector Potentials 4

Substituting

H = 1

µ∇×A

∇×E = − jωµ

H

∇×E = − jωµ 1

µ∇×A

∇×E + jω

A( ) = 0

Note that for any scalar function

∇×∇Φ ≡ 0∇×

E + jω

A( ) = 0 = ∇× −∇Φ( )

E + jω

A = −∇Φ

Φ

Vector Potentials 5

The Wave Equation for A – Time-Harmonic Fields So what?

B = ∇×

A

∇×B = µ∇×

H = µ jωε

E +J( ) = ∇×∇×

A = ∇∇ i

A−∇2

A

µ jωε −∇Φ− jωA( ) + J⎡⎣ ⎤⎦ = ∇∇ i

A−∇2

A

− jωµε∇Φ+ω 2µεA+ µ

J = ∇∇ i

A−∇2

A

∇ ∇ iA( )−∇2

A = µ

J − jωµε∇Φ+ω 2µε

A

Vector Potentials 6

Helmholtz Theorem (the divergence and the curl are independent): Any vector field may be written as: Proof:

C = −∇Φ+∇×

A

Ci = −∇Φ irrotational - zero curl( )Cs = ∇×

A soleniodal - zero divergence( )

∇×C = −∇×∇Φ

≡ 0 +∇×∇×

A

∇ iC = −∇ i∇Φ+∇ i∇×

A

≡ 0

Vector Potentials 7

The Wave Equation for A – Time-Harmonic Fields So what? We specified the curl of as to satisfy We are also free to specify its divergence. (This is known as the Lorentz condition)

∇ ∇ iA( )−∇2

A = µ

J − jωµε∇Φ+ω 2µε

A

⇒∇ ∇ iA( ) = − jωµε∇Φ

⇒−∇2A = µ

J +ω 2µε

A

H = 1

µ∇×A

∇ iA = − jωµεΦ

Vector Potentials 8

The Wave Equation for A – Time-Harmonic Fields Look how nice this is:

E + jω

A = −∇Φ

∇ iE + jω∇ i

A = −∇ i∇Φ

ρε+ jω∇ i

A = −∇2Φ⇒ jω∇ i

A = −∇2Φ− ρ

ε∇ ∇ i

A( ) = − jωµε∇Φ⇒ jω∇ i

A =ω 2µεΦ

⇒ jω∇ iA = −∇2Φ− ρ

ε⇒ω 2µεΦ = −∇2Φ− ρ

ε

⇒∇2Φ+ω 2µεΦ = − ρε

Vector Potentials 9

The Wave Equation for A – Time-Harmonic Fields Look how nice this is:

⇒∇2A+ω 2µε

A = −µ

J

⇒∇2Φ+ω 2µεΦ = − ρε

Vector Potentials 10

Solution of the Inhomogeneous Vector Potential Wave Equation

Once A is known:

∇2A+ β 2

A = −µ

J

H = 1

µ∇×A

E = −∇Φ− jω

A

Φ = jωµε

∇ iA

⎫

⎬⎪

⎭⎪

E = − j

ωµε∇∇ i

A− jω

A

or

∇×H = jωε

E +J ⇒

E = 1

jωε∇×H


a2d 2 ydt2

+ a1dydt

+ a0 y = f t( ) = δ t( ) = 0, t > 0( )Recall:

Poejωtδ r( ) = 0, r > 0( )

Point Source at origin

Region of Solution (Source Free)


x

y

z

( ), ,′ ′ ′x y z

ϕ

θ ( ), ,x y z r

Consider as a source the limiting case of a point source current density (z-directed) located at the origin.

J = az Jz


∇2A+ β 2

A = −µ

J

J = az Jz ⇒ Ax = Ay = 0

⇒∇2 Az + β2 Az = −µJz

Since a point source must be independent of angle, Az = Az r( )⇒∇2Az + β 2Az = 0

⇒ 1r2

∂∂rr2

∂Az r( )∂r

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ β 2Az = 0

⇒d 2Az r( )dr2

+ 2rdAz r( )dr

+ β 2Az = 0

For all points except the origin: 2 2 0β∇ + =z zA A


Solutions are known:

( ) ( )

( )

22

2

1 2

2 0

β β

β

− +

+ + =

= +

z zz

j r j r

z

d A r dA rA

dr r dr

e eA r C Cr r

Since Represents an inward traveling radial wave, the source a the origin requires that

β+ j re

2 0≡C


( ) 1

β−

=j r

zeA r Cr

For the static case: 0 0ω β= ⇒ =

( ) 1=zCA rr

Which is the solution to Poisson’s Equation (the wave equation with no time variation):

2 µ∇ = −z zA J


For this static case, it is reminiscent of a charge density at the origin, satisfying (from electrostatics): With solution Thus

∇2Φ = − ρ

ε

Φ = 1

4περrV

∫∫∫ d ′v

2

4µµπ

′∇ = − ⇒ = ∫∫∫ zz z z

V

JA J A dvr


By analogy to the static case, for

2 2

4

βµβ µπ

−

′∇ + = − ⇒ = ∫∫∫j r

z z z z zV

eA A J A J dvr

0ω ≠

r =r


If the current density had x and y components as well then:

4

4

4

β

β

β

µπµπµπ

−

−

−

′=

′=

′=

∫∫∫

∫∫∫

∫∫∫

j r

x xV

j r

y yV

j r

z zV

eA J dvr

eA J dvr

eA J dvr

A = µ

4πJ e− jβr

rV∫∫∫ d ′v


x

y

z

( ), ,′ ′ ′x y z

ϕ′

θ ′ ( ), ,x y z ′r

r

R

ϕ

θ

If the source is moved from the origin and place at coordinates : ( ), ,′ ′ ′x y z

A = µ

4πJ ′x , ′y , ′z( ) e− jβR

RV∫∫∫ d ′v

R =R = r − ′r


For filamentary currents:

A = µ

4πIe ′x , ′y , ′z( ) e− jβR

RC∫ d ′


Example: A very short linear current element. (An Infinitesimal Dipole)

Ie ′x , ′y , ′z( ) = az Ie

Ie a constant

λ

x

y

z

ϕ

θ ( ), ,x y zr


An Infinitesimal Dipole

A = µ

4πIe ′x , ′y , ′z( ) e− jβR

RC∫ d ′

since → 0, R ≈ r ′r = 0( )

= az

µ4π

Ie

− 2

2

∫e− jβR

Rd ′z = az

µIe

4πe− jβr

rd ′z

− 2

2

∫

Ie ′x , ′y , ′z( ) = az Ie

Ie a constant



A = az

µIe

4πe− jβr

rd ′z

− 2

2

∫

= az

µIe

4πe− jβr

rd ′z

− 2

2

∫

= az

µIe4πr

e− jβr


Rectangular to Spherical Conversion

sin cos sin sin coscos cos cos sin sinsin cos 0

sin cos cos cos sinsin sin cos sin coscos sin 0

θ

ϕ

θ

ϕ

θ ϕ θ ϕ θθ ϕ θ ϕ θ

ϕ ϕ

θ ϕ θ ϕ ϕθ ϕ θ ϕ ϕ

θ θ

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟= −⎜ ⎟ ⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞−⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠

r x

y

z

x r

y

z

A AA AA A

A AA AA A


Ar

Aθ

Aϕ

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=sinθ cosϕ sinθ sinϕ cosθcosθ cosϕ cosθ sinϕ −sinθ−sinϕ cosϕ 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

00Az

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Ar = cosθ Az = cosθµIe4πr

e− jβr

Aθ = −sinθ Az = −sinθµIe4πr

e− jβr

Aϕ = 0



Ar = cosθ Az = cosθ

µIe4πr

e− jβr , Aθ = −sinθ Az = −sinθµIe4πr

e− jβr


H = 1

µ∇×A

E = 1

jωε∇×H

∇×A =

ar

r sinθ∂ Aϕ sinθ( )

∂θ−∂Aθ

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

aθ

r1

sinθ∂Ar

∂ϕ−∂ rAϕ( )∂r

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+aϕ

r∂ rAθ( )∂r

−∂Ar

∂θ⎡

⎣⎢⎢

⎤

⎦⎥⎥



H = 1

µ∇×A =

aϕ

r1µ

∂ rAθ( )∂r

−∂Ar

∂θ⎡

⎣⎢⎢

⎤

⎦⎥⎥

=aϕ

r1µ

∂ −r sinθµIe4πr

e− jβr⎛⎝⎜

⎞⎠⎟

∂r−∂ cosθ

µIe4πr

e− jβr⎛⎝⎜

⎞⎠⎟

∂θ

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= −aϕ

rsinθ

Ie4π

∂ e− jβr( )∂r

+Ie

4πr∂ cosθ( )

∂θe− jβr

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −aϕ

r− jβ sinθ

Ie4π

e− jβr −Ie

4πrsinθe− jβr⎡

⎣⎢

⎤

⎦⎥

=Ie4π

sinθaϕ

re− jβr 1

r+ jβ⎡

⎣⎢

⎤

⎦⎥



E = 1

jωε∇×H = 1

jωεar

r sinθ∂ Hϕ sinθ( )

∂θ−

aθ

r

∂ rHϕ( )∂r

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= 1jωε

ar

r∂Hϕ

∂θ+ ar

cosθr sinθ

Hϕ − aθ

∂Hϕ

∂r− aθ

Hϕ

r⎡

⎣⎢⎢

⎤

⎦⎥⎥

= 1jωε

ar

r∂Hϕ

∂θ+ ar

cosθr sinθ

Hϕ − aθ

∂Hϕ

∂r− aθ

Hϕ

r⎡

⎣⎢⎢

⎤

⎦⎥⎥

= 1jωε

ar

r∂Hϕ

∂θ+ ar

cosθr sinθ

Hϕ − aθ

∂Hϕ

∂r− aθ

Hϕ

r⎡

⎣⎢⎢

⎤

⎦⎥⎥

Hϕ = jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr

Hr = Hθ = 0



E = 1

jωεar

r∂Hϕ

∂θ+ ar

cosθr sinθ

Hϕ − aθ

∂Hϕ

∂r− aθ

Hϕ

r⎡

⎣⎢⎢

⎤

⎦⎥⎥

Hϕ = jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr



E = 1

jωε

ar

r∂∂θ

jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr⎛

⎝⎜⎞⎠⎟

+arcosθr sinθ

jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr

−aθ∂∂r

jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr⎛

⎝⎜⎞⎠⎟

−aθ1r

jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr + 0 i aϕ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥



Er =cosθωε

1+ 1jβr

⎛⎝⎜

⎞⎠⎟β Ie4πr 2 e− jβr

= cosθωε

1+ 1jβr

⎛⎝⎜

⎞⎠⎟ω µε Ie

4πr 2 e− jβr

= µε

cosθ 1+ 1jβr

⎛⎝⎜

⎞⎠⎟

Ie4πr 2 e− jβr

=ηcosθ 1+ 1jβr

⎛⎝⎜

⎞⎠⎟

Ie4πr 2 e− jβr



Eθ = − 1ωε

β Ie4π

sinθ ∂∂r

1r+ 1

jβr 2

⎡

⎣⎢

⎤

⎦⎥e− jβr⎛

⎝⎜⎞

⎠⎟+ 1

r 2 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= − 1ωε

β Ie4π

sinθ

1r+ 1

jβr 2

⎛⎝⎜

⎞⎠⎟∂∂r

e− jβr( ) + e− jβr ∂∂r

1r+ 1

jβr 2

⎛⎝⎜

⎞⎠⎟

+ 1r 2 1+ 1

jβr⎛⎝⎜

⎞⎠⎟

e− jβr

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= −ηIe4π

sinθ− jβ 1

r+ 1

jβr 2

⎛⎝⎜

⎞⎠⎟− 1

r 2 +2

jβr3

⎛⎝⎜

⎞⎠⎟

+ 1r 2 1+ 1

jβr⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

e− jβr



Eθ = −ηIe4π

sinθ− jβ 1

r+ 1

jβr 2

⎛⎝⎜

⎞⎠⎟− 1

r 2 +2

jβr3

⎛⎝⎜

⎞⎠⎟

+ 1r 2 1+ 1

jβr⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

e− jβr

= −ηIe4π

sinθ − jβ 1r+ 1

jβr 2

⎛⎝⎜

⎞⎠⎟− 1

jβr3

⎡

⎣⎢

⎤

⎦⎥e− jβr

= jβηIe4π

sinθ 1r+ 1

jβr 2 −1

β 2r3

⎡

⎣⎢

⎤

⎦⎥e− jβr

= jβηIe4πr

sinθ 1+ 1jβr

− 1β 2r 2

⎡

⎣⎢

⎤

⎦⎥e− jβr



Er =ηcosθ 1+ 1jβr

⎛⎝⎜

⎞⎠⎟

Ie4πr 2 e− jβr

Eθ = jβηIe4πr

sinθ 1+ 1jβr

− 1β 2r 2

⎡

⎣⎢

⎤

⎦⎥e− jβr

Eϕ = 0

Hϕ = jβ Ie4πr

sinθ 1+ 1jβr

⎡

⎣⎢

⎤

⎦⎥e− jβr

Hr = Hθ = 0

Valid everywhere except on the current element.


What have we learned? The fields produced by finite antennas are spherical waves, i.e., they contain the radial factor Also, the fields decay by a factor of The general form for the potential in spherical coordinates is

β− j re

1r n , n = 1, 2, 3,…

A = ar Ar r,θ ,ϕ( ) + aθ Aθ r,θ ,ϕ( ) + aϕ Aϕ r,θ ,ϕ( )


What have we learned? The potential may be written as

A = ar Ar r,θ ,ϕ( ) + aθ Aθ r,θ ,ϕ( ) + aϕ Aϕ r,θ ,ϕ( )= 1

re− jβr ar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

+ 1r 2 e− jβr ar Ar

′′ θ ,ϕ( ) + aθ Aθ′′ θ ,ϕ( ) + aϕ Aϕ

′′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

+ 1r3 e− jβr ar Ar

′′′ θ ,ϕ( ) + aθ Aθ′′′ θ ,ϕ( ) + aϕ Aϕ

′′′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

+This type of r-separation will occur in all we do.


The Far Field

As r →∞A 1

re− jβr ar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

E = − jω

A− j 1

ωµε∇ ∇ i

A( )

∇ iA = 1

r 2

∂∂r

r 2 Ar( ) + 1r sinθ

∂∂θ

sinθ Aθ( ) + 1r sinθ

∂Aϕ

∂ϕ

∇ψ = ar∂ψ∂r

+ aθ1r∂ψ∂θ

+ aϕ1

r sinθ∂ψ∂ϕ


∇ iA = 1

r 2

∂∂r

r 2 Ar( ) + 1r sinθ

∂∂θ

sinθ Aθ( ) + 1r sinθ

∂Aϕ

∂ϕA 1

re− jβr ar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

∇ iA = 1

r 2

∂∂r

r 2 1r

e− jβr Ar′ θ ,ϕ( )⎡

⎣⎢

⎤

⎦⎥ +

1r sinθ

∂∂θ

sinθ 1r

e− jβr Aθ′ θ ,ϕ( )⎡

⎣⎢

⎤

⎦⎥

+ 1r sinθ

∂∂ϕ

1r

e− jβr Aϕ′ θ ,ϕ( )⎡

⎣⎢

⎤

⎦⎥

= 1r 2 Ar

′ θ ,ϕ( ) ∂∂rre− jβr⎡⎣ ⎤⎦ +

e− jβr

r 2 sinθ∂∂θ

sinθ Aθ′ θ ,ϕ( )⎡

⎣⎢⎤⎦⎥

+ e− jβr

r 2 sinθ∂∂ϕ

Aϕ′ θ ,ϕ( )⎡

⎣⎢⎤⎦⎥


∇ iA = 1

r 2 Ar′ θ ,ϕ( ) ∂∂r

re− jβr⎡⎣ ⎤⎦ +e− jβr

r 2 sinθ∂∂θ

sinθ Aθ′ θ ,ϕ( )⎡

⎣⎢⎤⎦⎥

+ e− jβr

r 2 sinθ∂∂ϕ

Aϕ′ θ ,ϕ( )⎡

⎣⎢⎤⎦⎥

= 1− jβrr 2 Ar

′ θ ,ϕ( )e− jβr + e− jβr

r 2 sinθsinθ

∂Aθ′ θ ,ϕ( )∂θ

+ cosθ Aθ′ θ ,ϕ( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+ e− jβr

r 2 sinθ∂Aϕ

′ θ ,ϕ( )∂ϕ


∇ iA = 1− jβr

r 2 Ar′ θ ,ϕ( )e− jβr + e− jβr

r 2 sinθsinθ

∂Aθ′ θ ,ϕ( )∂θ


⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ e− jβr

r 2 sinθ∂Aϕ

′ θ ,ϕ( )∂ϕ

∇ ∇ iA( ) = ar

∂∂r

+ aθ1r∂∂θ

+ aϕ1

r sinθ∂∂ϕ

⎛⎝⎜

⎞⎠⎟∇ iA

= ar∂∂r

1− jβrr 2 Ar


r 2 sinθsinθ

∂Aθ′ θ ,ϕ( )∂θ


⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ e− jβr

r 2 sinθ∂Aϕ

′ θ ,ϕ( )∂ϕ

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

+aθ1r∂∂θ

1− jβrr 2 Ar


r 2 sinθsinθ

∂Aθ′ θ ,ϕ( )∂θ


⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ e− jβr

r 2 sinθ∂Aϕ

′ θ ,ϕ( )∂ϕ

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

+aϕ1

r sinθ∂∂ϕ

1− jβrr 2 Ar


r 2 sinθsinθ

∂Aθ′ θ ,ϕ( )∂θ


⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ e− jβr

r 2 sinθ∂Aϕ

′ θ ,ϕ( )∂ϕ

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪


∇ ∇ iA( ) = ar Ar

′ ∂∂r

1− jβrr 2 e− jβr⎡

⎣⎢

⎤

⎦⎥ + sinθ

∂Aθ′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥∂∂r

e− jβr

r 2

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aθ1r

1− jβrr 2 e− jβr ∂Ar

′

∂θ+ e− jβr

r 2

∂∂θ

1sinθ

sinθ∂Aθ

′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aϕ1

r sinθ1− jβr

r 2

∂Ar′

∂ϕe− jβr + e− jβr

r 2 sinθsinθ

∂2 Aθ′

∂ϕ ∂θ+ cosθ

∂Aθ′

∂ϕ+∂2 Aϕ

′

∂ϕ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪


∇ ∇ iA( ) = are

− jβr Ar′ − jβ 1− jβr

r 2 + − jβr 2 −1+ j2βr 2

r 4

⎡

⎣⎢

⎤

⎦⎥ − sinθ

∂Aθ′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

jβr 2 + 2rr 4

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aθ1r


′

∂θ+ e− jβr

r 2

∂∂θ

1sinθ

sinθ∂Aθ

′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aϕ1

r sinθ1− jβr

r 2

∂Ar′


r 2 sinθsinθ

∂2 Aθ′

∂ϕ ∂θ+ cosθ

∂Aθ′

∂ϕ+∂2 Aϕ

′

∂ϕ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

∇ ∇ iA( ) = are


r 2 − 1− jβr 2

r 4

⎡

⎣⎢

⎤

⎦⎥ − sinθ

∂Aθ′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

jβr 2 + 2rr 4

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aθ1r


′

∂θ+ e− jβr

r 2

∂∂θ

1sinθ

sinθ∂Aθ

′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aϕ1

r sinθ1− jβr

r 2

∂Ar′


r 2 sinθsinθ

∂2 Aθ′

∂ϕ ∂θ+ cosθ

∂Aθ′

∂ϕ+∂2 Aϕ

′

∂ϕ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪


∇ ∇ iA( ) = are


r 2 − 1− jβr 2

r 4

⎡

⎣⎢

⎤

⎦⎥ − sinθ

∂Aθ′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

jβr 2 + 2rr 4

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aθ1r


′

∂θ+ e− jβr

r 2

∂∂θ

1sinθ

sinθ∂Aθ

′

∂θ+ cosθ Aθ

′ +∂Aϕ

′

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+aϕ1

r sinθ1− jβr

r 2

∂Ar′


r 2 sinθsinθ

∂2 Aθ′

∂ϕ ∂θ+ cosθ

∂Aθ′

∂ϕ+∂2 Aϕ

′

∂ϕ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

As r →∞

∇ ∇ iA( ) = −ar

β 2

re− jβr Ar

′


The Far Field

As r →∞A 1

re− jβr ar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

E = − jω

A− j 1

ωµε∇ ∇ i

A( ), ∇ ∇ i

A( ) = −ar

β 2

re− jβr Ar

′

E = − jω

re− jβr ar Ar

′ + aθ Aθ′ + aϕ Aϕ

′⎡⎣⎢

⎤⎦⎥ + j 1

ωµεar

β 2

re− jβr Ar

′

E = − jω

re− jβr ar Ar

′ + aθ Aθ′ + aϕ Aϕ

′⎡⎣⎢

⎤⎦⎥ + jar

ωr

e− jβr Ar′

E = − jω

re− jβr ar Ar

′ − Ar′( ) + aθ Aθ

′ + aϕ Aϕ′⎡

⎣⎢⎤⎦⎥

E = − jω

re− jβr aθ Aθ

′ + aϕ Aϕ′⎡

⎣⎢⎤⎦⎥


The Far Field

H = 1

µ∇×A

A 1

re− jβr ar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

∇ ×A =

ar

r sinθ∂ Aϕ sinθ( )

∂θ−∂Aθ

∂ϕ

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

aθ

r1

sinθ∂Ar

∂ϕ−∂ rAϕ( )∂r

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+aϕ

r∂ rAθ( )∂r

−∂Ar

∂θ⎡

⎣⎢⎢

⎤

⎦⎥⎥


The Far Field

∇×A =

are− jβr

r 2 sinθ

∂ sinθ Aϕ′( )

∂θ−∂Aθ

′

∂ϕ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+

aθ

r1

sinθe− jβr

r∂Ar

′

∂ϕ− Aϕ

′∂ e− jβr( )

∂r

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+aϕ

rAθ′ ∂ e− jβr( )∂r

− e− jβr

r∂Ar

′

∂θ

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

r →∞

∇×A = −

aθ

rAϕ′∂ e− jβr( )

∂r+

aϕ

rAθ′ ∂ e− jβr( )∂r

= − jβ e− jβr

raθ Aϕ

′ − aϕ Aθ′( )


The Far Field

r →∞H = 1

µ∇×A

= − j βµ

e− jβr

raθ Aϕ

′ − aϕ Aθ′( )

= − jωη

e− jβr

raθ Aϕ

′ − aϕ Aθ′( )

β =ω µε , η = µε


The Far Field

r →∞A e− jβr

rar Ar

′ θ ,ϕ( ) + aθ Aθ′ θ ,ϕ( ) + aϕ Aϕ

′ θ ,ϕ( )⎡⎣⎢

⎤⎦⎥

E = − jω e− jβr

raθ Aθ

′ + aϕ Aϕ′( )

H = − jω

ηe− jβr

raθ Aϕ

′ − aϕ Aθ′( )

Er 0 Hr 0

Eθ − jω Aθ Hθ − jωη

Aϕ = −Eϕ

η

Eϕ − jω Aϕ Hϕ + jωη

Aθ = +Eθ

η

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

A pair of plane wavestraveling in the r direction


For the Far Field

E = − jω

A

H = 1

ηar ×E = − jω

ηar ×A

Vector Potentials and Antenna Fundamentals

Documents