Electromagnetism: an introduction

MAXWELL’s EQUATIONS

0B

D

JDt

H

Bt

E

e

e

GIVEN

• E, Electric field [V/m]

• H, Magnetic field [A/m]

• B, Magnetic Flux density, [T = Wb/m2]

• D, Electric Flux density [C/m2]

• Je, Current density [A/m2]

• e, Electric Charge density [C/m3]

Moreover, B and D fields depend on E and H fields through

constitutive relations of the media where they propagate.

In vacuum or in an isotropic media: D= ·E, B= ·H where [F/m]

is the permittivity while [H/m] is the permeability.

When frequency = 0 (stationary fields), E field

and H field are uncoupled

CV

Only E field

inside capacitor induced by the Voltage V

V

I=V/R

Only B field

Induced by the

current loop

I

ee JJDt

H

0Bt

E

I

P

FIELD SOURCES ARE ELECTRIC CURRENTS

Following Maxwell’s equations, a non stationary current I inside any wire

generates in P a Magnetic field H and an Electric field E

E, H

eJD

tH

Bt

E

RADIATION PROPERTIES OF A SHORT

DIPOLE ANTENNA

Short dipole antenna = Short dl wire with a current I.

More complex wired antennas can be seen as a set of elementary dipole antennas

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r

rrrjsineE

rrcose2E

3

5

2

43

3

2

2

1

rj2

0

2j

21

21

21

21

e2

Z4

Idle 0

Y

Z

X

Er

Er

dl

For a given frequency, the structure of the Electric field radiated from a short dipole

antenna strictly depends on r/ value

(r/ )-1

(r/ )-2

(r/ )-3

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32

21

r

rrrjsineE

rrcose2E

(r/ )-1

(r/ )-2

(r/ )-3

35

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32

21

r

rrrjsineE

rrcose2E

(r/ )-1

(r/ )-2

(r/ )-3

35

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32

21

r

rrrjsineE

rrcose2E

For a given frequency, the structure of the Electric field radiated from a short dipole

antenna strictly depends on r/ value

NEAR FIELD ZONE (r/ <<1) STATIONARY or NON RADIATIVE FIELD

Electrotecnics formulations to solve such problems… (think about to 50 Hz

electric sources: f = 50 Hz = 6000 km… the radiated field does not vary

inside typical circuits length!)

Electric Field and Magnetic field are independent

With reference to the small dipole antenna, both Er and E Electric field

components coexist

RADIATIVE FIELD (r/ > 1)

Raileigh zone (r<< D2/(2 ) )

Fresnel zone (D2/(2 ) <r< 2D2/ )

Fraunhofer or FAR FIELD zone (2D2/ << r)

With reference to the small dipole antenna, Er Electric

field component vanishes for r/ >> 1

DI1

I2

I3

P

r

FAR FIELD or Fraunhofer zone

The following IMPEDANCE RELATION (planar wave) can be written

Conditions

r>>

r>2D2/

Ex. Microwave communications: f=1GHz, =30 cm

HrZE 0

E

B

r

120 impedance vacuumZ0

DI1

I2

I3

P

r

RADIATED FIELD ZONES

for the feeder of a parabolic antenna

(HORN ANTENNA with d /2)

Zona di Rayleigh

Zona di Fresnel

Zona di Fraunhofer

d2/(2 ) 3 cm

2d2/ 15 cm f=1GHz

=30 cm

d

RADIATED FIELD ZONES

for the PARABOLIC ANTENNA

Attention: POLARIZATION ASPECTS

The radiated Electromagnetic field is a VECTORIAL FIELD.

Vertical linear

polarization

Clockwise

circular

polarization

E

B

B

E

E E

B B

Horizontal

linear

polarization

Anti

Clockwise

circular

polarization