Yoonchan Jeong School of Electrical Engineering, Seoul National University

Tel: +82 (0)2 880 1623, Fax: +82 (0)2 873 9953 Email: yoonchan@snu.ac.kr

Electromagnetics: Antenna Patterns and Antenna Parameters

Thin Linear Antennas Antenna Arrays

(11-3, 11-4, 11-5)

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Radiation Pattern or Antenna Pattern

Relative far-zone field strength at a fixed distance from an antenna:

→ E-plane pattern:

→ H-plane pattern:

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Herzian dipole: Far fields:

θβπ

β

φ sin4

=

−

ReIdljH

Rj

θβηπ

β

θ sin4 0

=

−

ReIdljE

Rj

1/2 >>= λπβ RR

θθ sin=→ ENormalized

22 zxxE+

=→ θ

22

2

21

21

=+

±→ zx

E-plane pattern:

H-plane pattern:

2/sin

πθθ θ=

=→ ENormalized

Normalized field strength vs θ for a constant φ

Normalized field strength vs φ for θ = π/2

2 1=

θE

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Antenna Patterns

Typical H-plane radiation patterns:

3

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Antenna Parameters (1)

1. Bandwidth (Width of the main beam):

2. Sidelobe levels:

A typical H-plane radiation pattern:

3. Directivity: avPRU 2=→ Radiation intensity (time-averaged power per unit solid angle):

→ Total time-average power radiated: Ω=⋅= ∫∫ dUdP avr sP

→ Directive gain: av

D UUG ),(),( φθφθ =

→ Directivity: av

max

UUD =

∫ ∫= π π

φθθφθ

π2

0 0

2

2

sin),(

4

ddE

Emax

4

→ Angular width between the half-power or −3-dB points (occasionally, −10-dB points)

→ Typically, −40-dB required for radar applications

πφθ

4/),(

rPU

=π

φθ4/),(

Ω=∫ dU

U

π4/max

rPU

=

Time-average Poynting vector

Antenna Parameters (2)

6. Radiation resistance:

4. Power gain:

π4/max

iP P

UG = lri PPP +=←

Total input power

Radiated power

Power loss (Ohmic, structural, etc.)

5. Radiation efficiency:

i

rr P

P=η

rr RIP 2

21

=

7. Loss resistance: ll RIP 2

21

=

5

DGp=

π4/max

rPU

=av

max

UUD =←

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Current distribution:

)(sin)( zhIzI m −=→ β

Recall: Hertzian dipole

θβπ

β

φ sin4

=

−

ReIdljH

Rj

θβηπ

β

θ sin4 0

=

−

ReIdljE

Rj

)1/2( >>= λπβ RR

θβηπ

ηβ

φθ sin4 00

′

==→′−

ReIdzjdHdE

Rj

( ) 2/122 cos2 θRzzRR −+=′←

∫−− −≅=→

h

h

zjRjm dzezheR

IjHE θββφθ β

πθβηη cos0

0 )(sin4

sin

[ ]∫−− +−=

h

h

Rjm dzzjzzheR

Ij )cossin()coscos()(sin4

sin0 θβθββπ

θβη β

∫ −= − hRjm dzzzheR

Ij0

0 )coscos()(sin2

sin θββπ

θβη β6

Far fields:

Thin Linear Antennas (1)

Elemental contribution

θcoszR −≅

Why?

Why?

Thin Linear Antennas (2)

∫ −≅=→ − hRjm dzzzheR

IjHE0

00 )coscos()(sin

2sin θββ

πθβηη β

φθ

)(60 θβ FeRIj Rjm −=

θβθβθ

sincos)coscos()( hhF −

=←← Pattern function

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

hhλπβ 2

=→

7

Far fields:

Radiation pattern: Zero radiation

Zero radiation

Why?

Half-Wave Dipole (1)

θβθβθ

sincos)coscos()( hhF −

=E-plane Pattern function:

For a half-wave dipole: 2//2 πλπβ ==→ hh2/2 λ=h

[ ]

=→ −

θθπβ

θ sincos)2/(cos60 Rjm e

RIjE

[ ]

=→ −

θθπ

πβ

φ sincos)2/(cos

2Rjm e

RjIH

Far-zone fields:

Time-average Poynting vector: [ ] 2

2

2

sincos)2/(cos15

21

== ∗

θθπ

πφθ RIHEP m

av

Total power radiated:

∫ ∫=π π

φθθ2

0 0

2 sin ddRPP avr

[ ]∫=π

θθ

θπ0

22

sincos)2/(cos30 dIm

254.36 mI≅8

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Radiation resistance: )(1.732

2 Ω==m

rr I

PR

Half-Wave Dipole (2)

Directivity: )2/(2

max πθ == avPRU

π4/max

rPUD =→

Half-power bandwidth: [ ]

= −

θθπβ

θ sincos)2/(cos60 Rjm e

RIjE

[ ]2

1sin

cos)2/(cos=→

θθπ 78≈→ FWHMθ

What about the total impedance?

rrr jXRZ +=For the impedance matching, the dipole length should be slightly shorter than λ/4!

9

[ ] 2

2

2

sincos)2/(cos15

=←

θθπ

πRIP m

av

254.36 mr IP ≅←

215mI

π=

64.154.36

60==

rr RIP 2

21

=←

Quarter-Wave Monopole

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Recall: Image method ← Radiation limited into the upper half-space!

)54.36(21 2

mr IP ×=→

Total power radiated:

Radiation resistance:

)(54.3622 Ω==→m

rr I

PR

Directivity: 64.1

2/===→

πr

max

av

max

PU

UUD

10

For HW DP

Recall: Electromagnetic Waves (RF/MW)

+Q

−Q

z θ

O dl

R

×

E

H aR

A Hertzian dipole

λ/4

λ/4

Half-wave dipole antenna

λ/4

Quarter-wave Monopole antenna

c = fλ

Speed of EM wave

Frequency

Wavelength λ/4 = 3×108 / 2.4×109 / 4 ≈ 0.031 m

Conducting ground

WiFi router (f = 2.4 GHz)

11

Antenna Arrays

Two-element arrays:

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

00

0

),(R

eFEERj

m

β

φθ−

=

11

1

),(ReeFEE

Rjj

m

βξ

φθ−

=

Total electric field:

+=+=

−−

1010

10

),(Ree

ReFEEEE

RjjRj

m

βξβ

φθ φθ cossin01 dRR −≅←

)1(),( cossin

0

0 ξφθββφθ jdjRjm eee

RFEE +=→ −

= −

2cos2),( 2/

0

0ψφθ ψβ jRj

m eeR

FEξφθβψ +=← cossind

2cos),(2

0

ψφθFREE m=→

Element factor

Array factor

12

)1(),(0

0

ψβφθ jRjm ee

RFE += −

Initial phase difference

Why?

Multi-Element Parallel Dipole Arrays

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

For two parallel half-wave dipoles:

2cos),(2

0

ψφθFREE m=

2cos

sin]cos)2/cos[(2

0

ψθ

θπREm=

)cos(21cos

2cos)( ξφβψψ +== dA

→ Normalized array factor:

2/πθ =←

D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

Arrays of equal magnitude and a uniform progressive phase shift: ψψψψ )1(211)( −+⋅⋅⋅+++= Njjj eee

NA

ξφβψ +=← cosd

ψ

ψ

ψ j

jN

ee

NA

−−

=→111)(

)2/sin()2/sin(1

ψψN

N=

2/πθ =←

13

Why zero? θ

βθβθsin

cos)coscos()( hhF −=←

→ Broadside or endfire?

ξφθβψ +=← cossind