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Chapter 3Chapter 3Radiation IntegralsRadiation Integrals

ECE 5318/6352ECE 5318/6352Antenna EngineeringAntenna Engineering

Dr. Stuart LongDr. Stuart Long

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VECTOR POTENTIALSVECTOR POTENTIALSMathematical tool used to simplify calculation of

radiated fields andE

H

Radiation Fields

,E

H

Vector Potentials

, F

A

Sources

,J

M

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Coordinate system for computing radiating fieldsCoordinate system for computing radiating fields

Fig. 3.2 (b) Coordinate system for computingfields radiated by sources. Source not at origin

source points

field points

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Electric Field Strength [V/m]

Magnetic Field Strength [A/m]

Electric Volume Current Density [A/m²]

Magnetic Volume Current Density [V/m²]

J

HE

M

DEFINITIONSDEFINITIONS

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DEFINITIONSDEFINITIONS (CONT)(CONT)

Electric Vector Potential [Wb/m]

Electrical Scalar Potential [V]

Magnetic Vector Potential [A-sec/m]

Magnetic Scalar Potential [A]

A

F

m

e

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Vector Potential Vector Potential AA for anfor anelectric current sourceelectric current source JJ

Vector Potential Vector Potential FF for a for a magnetic current sourcemagnetic current source MM

1

e j

j

AA

A

A A

B H A

E A

H J E

1

m j

j

F

F

F F

E F

H F

E M H

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Vector Potential Vector Potential AA for anfor anelectric current sourceelectric current source JJ

Vector Potential Vector Potential FF for a for a magnetic current sourcemagnetic current source MM

2 2

'

e

jkR

v

j

k

e dvR

A

A A J

A J4

2 2

'

m

jkR

v

j

k

e dvR

F

F F M

F M4

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The total fields are The total fields are

FHEEE AFA

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j

(solutions derived in Sect. 3.5)(also for surface and linear currents)

FFA EAHHH

j11

[3-29a]

[3-30a]

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FAR FIELD RADIATIONFAR FIELD RADIATION

2

2Dr

r

Fields are essentially TEM to r

A

A

E j A

E j A

F

F

H j F

H j F

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FAR FIELDS RADIATION FAR FIELDS RADIATION (CONT)(CONT)

2

2Dr

r

Fields are essentially TEM to r

F F

F F

E H

E H

AA

AA

EH

EH

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DUALITYDUALITYIdentical equations for fields due to A and F

for J ≠

0M 0

for J 0M ≠

0

EA HA J A k

HF -EF M F k

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RECIPROCITYRECIPROCITY

Assume LINEAR, ISOTROPIC (but not necessarily homogeneous) materials

give sources J1 , M1 E1 , H1

and J2 , M2 E2 , H2

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RECIPROCITYRECIPROCITY (CONT)(CONT)

Lorentz Reciprocity Theorem

(also valid for fields away from finite sources – far fields)

each integral is called a “REACTION”of fields (E, H ) to the sources (J, M )

'1212

'2121 dvdv

vv MHJEMHJE [3-66]

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For two antennas

equivalence of power transferred in both directions for two antennas (p.147)

RECIPROCITYRECIPROCITY (CONT)(CONT)

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equivalence of transmit and receive mode radiation patterns (p.148)

For radiation patterns

RECIPROCITYRECIPROCITY (CONT)(CONT)