1

Antenna for EMC

Prof. Tzong-Lin Wu

EMC Lab.Department of Electrical Engineering National Taiwan University

Introduction

1. Overview2. Steps in evaluation of radiated fields3. Ideal Dipole (Hertzian Dipole)4. Antenna parameters5. Dipole and Monopole6. Small Loop Antenna7. Antenna Arrays8. Examples of Antenna9. Antennas in communication systems

2

How antenna radiate: a single accelerated charged particle

• The static electric field originates at charge and is directed radially away from charge.

• At point A, the charge begins to be accelerated until reachingpoint B.

• The distance between the circles is that distance light would travel in time △t, and △r=rb – ra = △t / c

• Charge moves slowly compared to the speed of the light, ∴ △r >> △z and two circles are concentricconcentric.

• The electric field lines in the △r region are joined together because of required continuity of electrical lines in the absenceof charges.

• This disturbance expands outward and has a transverse component Et , which is the radiated field.

• If charges are accelerated back and forth (i.e., oscillate), a regular disturbance is created and radiation is continuous.

• This disturbance is directly analogous to a transient wave createdby a stone dropped into a calm lake.

How antenna radiate : Evolution of a dipole antenna from an open-circuited transmission line

• Open-circuited transmission line1. The currents are in opposite directions on the two wires and

behaves as a standing wave pattern with a zero current magnitudeat the ends and every half wavelength from the end.

2. The conductors guide the waves and the power resides in the region surrounding the conductors as manifested by the electric and magnetic fields.

3. Electric fields originate from or terminate on charges and perpendicular to the wires.

4. Magnetic fields encircle the wires.

• Bending outward to form a dipole

1. The currents are no longer opposite but are both upwardly directed.2. The bounded fields are exposed to the space.3. The currents on the dipole are approximately sinusoidal.4. The situation on the Fig. is the peak current condition. As time

proceeds and current oscillation occur, the disturbed fields areradiated.

3

How antenna radiate : Time dynamics of the fields for a dipole antenna

• At t = 0, peak charges buildup occurs (positive on the upper half and negative on the lower half). current I = 0.

• At t = T/4, +/- charges are neutralized and I is maximum.

• Because there are no longer charges for the terminationof electric fields, they form closed loop near the dipole.

• At t = T/2, peak charges buildup again, but upper half is negative and lower half is positive. I = 0.

• Notice the definition of λ/2 at t = T/2.

Closed loop

Overview of antenna

4

Overview of antenna

Antenna Concept (I)

5

Antenna Concept (II)

Steps in evaluation of radiation fields : Solution of Maxwell equations for radiation problems)

∇ ⋅ =

∇ × = −

∇ × = +

∇ ⋅ =

H

E j H

H j E J

E

0

ωμ

ωερε

H A= ∇ ×1μ

∇ × + =( )E j Aω 0 E j A+ = −∇ω φ

∇ × ∇ × = ∇ ∇ ⋅ − ∇ = − − ∇ +A A A j j A J( ) ( )2 ωμε ω φ μ

Magnetic vector potential

Electric scalar potential

∇⋅ = −A jωμεφIf we set

∇ + = −2 2A A Jω με μ

(Lorenz condition)

Vector wave equation

A J eR

dvj R

v

=−zzz μ π

β

4'

'

Solution is

6

Steps in evaluation of radiation fields (Far Fields)

1. Find A'

'

ˆ n

4

j rj r r

v

eA Je dvr

ββμ

π

−⋅= ∫∫∫

'

'

ˆ 'ˆ4

j rj r r

zv

eA z J e dvr

ββμ

π

−⋅= ∫∫∫

'

'

' cos 'ˆ ( )4

j rj z

v

eA z I z e dvr

ββ θμ

π

−⋅= ∫∫∫

Z-directed sources

Z-directed line sources

Far field

Steps in evaluation of radiation fields (Far Fields)

2. Find E

ˆ ˆE j A A= − +θ φω( θ φ)

E j A= − ω

ˆzE j A= ωsinθ θ

Far field

Transverse to the propagation direction r

Z-directed sources

3. Find H

1 ˆH r Eη

= ×

1H Eφ θη=

Far field (Plane wave)

Z-directed source

7

Far-Field Conditions

Parallel ray approximation for far-field calculations

Far Field conditions D is the length of line source

22Dr

r Dr

λ

λ

>

>>>>

'ˆR r r r= − ⋅

Far-Field and Near-field Conditions

8

The ideal dipole (Hertzian dipole) : definition

Ideal dipole: a uniform amplitude current that is electrically small with △z <<λ

∵R～r for small dipole

/ 2'

/ 2

ˆ4

z j R

z

eA z I dzR

β

μπ

Δ −

−Δ

= ∫

ˆ4

j reA I z zr

β

μπ

−

= Δ

The ideal dipole (Hertzian dipole): E and H field

2

2

1 1 ˆ[1 ] sin4 ( )

1 1 ˆ [ ] cos2

j r

j r

I z eE jj r r r

I z ej rr r r

β

β

ωμ θθπ β β

η θπ β

−

−

Δ= + − +

Δ−

1 ˆ(1 ) sin4

j rI z eH jj r r

β

β θφπ β

−Δ= +H A= ∇ ×

1μ

1E Hjωε

= ∇×

Far field condition : βr >> 1

120 377EHθ

φ

ωμ μ η πβ ε

= = = = =

ˆsin4

j rI z eH jr

β

ωμ θφπ

−Δ=

ˆsin4

j rI z eE jr

β

β θθπ

−Δ=

9

The ideal dipole (Hertzian dipole): radiated power

Power flowing density : (Unit : w/m*m)

22

2

1 1 sin ˆ( )2 2 4

I zS E H rrθωμβ

π∗ Δ

= × =

Total radiated power (Unit : W)

2 2 2 2( ) 40 ( )12

zP S ds I z Iωμβ ππ λ

Δ= ⋅ = Δ =∫∫

Antenna Parameters: radiation pattern

The field pattern with its maximum value is 1,max

( , ) EFE

θ

θ

θ φ =

For Hertzian dipole

F(θ)=sinθ

10

Antenna Parameters: power pattern

2( , ) ( , )P Fθ φ θ φ=

( , ) ( , )dB dBP Fθ φ θ φ=

It is worth noting that the field patter and power pattern are the same in decibles.

Power pattern parameters

Antenna Parameters: Directivity

Directivity: The ratio of the radiation intensity in a certain direction to the average radiation intensity.

2

2 2

1 ˆRe( )( , ) ( , ) / 2( , )( , ) ( , ) / / 4ave ave

E H rU U rDU U r P r

θ φ θ φθ φθ φ θ φ π

∗× ⋅= =

Radiation intensity: the power radiated in a given direction per unit solid angle

21 ˆ( , ) Re( )2

U E H r rθ φ ∗= × ⋅

Average radiation intensity:

1( , ) ( , ) / 44aveU U d Pθ φ θ φ ππ

= Ω =∫∫

11

Antenna Parameters: Directivity

When directivity is quoted as a single number without reference to a direction, maximum directivity is usually intended.

24

( , )m

ave

UDU F d

πθ φ

= =Ω∫∫

Directivity of Hertzian dipole2 21( , ) ( ) sin

2 4I zU θ φ ωμβ θπΔ

=

21( , ) / 4 ( )3 4ave

I zU Pθ φ π βωμπΔ

= =

21 ( )2 4m

I zU ωμβπΔ

∴ =

3 10log1.5 1.752

D dB∴ = = =

Antenna Parameters: Gain

Gain : 4π times the ratio of radiation intensity in a given direction toThe net power accepted by the antenna from the connected transmitter

4 ( , )( , )in

UGP

π θ φθ φ =

• If all input power appeared as radiated power (Pin=P), Directivity = Gain.• In reality, some of the power is lost in the antenna absorbed by the

antenna and nearby structures). • Radiation efficiency:

, (0 1)r rin

Pe eP

= ≤ ≤

rG e D∴ =

12

Antenna Parameters: antenna impedance, radiation efficiency

A A AZ R jX= +

RA: Dissipation = Radiation + Ohmic lossXA: Power stored near the antenna

• Input impedance

2 21 12 2in ohmic r A ohmic AP P P R I R I= + = +

• Efficiencyr

rin ohmic A

P P ReP P P R

= = =+

• For Ideal dipole

Rr is very small since △z << λ

2 2 22 2

2 ( ) 80 ( )12r

A

P zR I zIIωμβ ππ λ

Δ= = Δ =

Antenna Parameters: ideal dipole is an ineffective radiator

Example

For radiator △z = 1cm, f0 = 300MHz (λ= 1m) → Rr = 79mΩ

∴ It needs 3.6A for 1W of radiated power.

For radiator △z = 1cm, f0 = 1GHz (λ= 30cm) → Rr = 0.87Ω

∴ It needs 1.5A for 1W of radiated power.

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Antenna Parameters: How to increase the radiation resistance (efficiency) of the short dipole

Practical short dipole with triangular-likecurrent distribution

1. Capacitor-plate antenna: the top-plate supply the charge such that the current on the wire is constant

2 280 ( )rzR πλΔ

=

2 220 ( )rzR πλΔ

=

Antenna Parameters: How to increase the radiation resistance (efficiency) of the short dipole

2. Transmission line loaded antenna

3. Inverted-L antenna or Inverted-F antenna

Looks like uniform distributionif △z<<L

Image theory

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Antenna Parameters: effective aperture

e

R av

he effective aperture of an antenna, A ,is the ratio of power received in its load

impedance, P , to the power density of the incident wave, S ,when the

polarization of the incident wave an

a. T

( )2

d the polarization of receiving antenna

ard matched:

in m

The maximum effective aperture A is the A when the maximum powerem e

transferring to the load

b.

Re

av

PA S=

takes place, which means load impedance is the conjugate

to the antenna impedance.

c. Example: compute the Aem (maximum effective aperture) of a Hertzian dipole antenna.

Ans: To solve Aem, two conditions should be kept in mind:(1)matched polarization for the incident

wave and the antenna.(2)matched load. (ie. )

E θ

θ

LZ radR jX= −

inZ radR jX= +

*

L antZ Z=

Suppose the incident wave is arriving at an

angle

(1)the open-circuit voltage produced at the

terminals of the antenna isˆ = E sin ocV dlθ

θ

θocV LZ r a dR j X= −

i nZ r a dR jX= +

15

2

0

2 2 2 2

2

2

rad

(2)power density in the incident wave

1 2

(3) the load is matched

sin the received power 8 8

1 1 ( I R= R )2 2 2

(4)radiation resistance for H

av

oc

Rrad rad

oc

rad

ES

V E dlP R R

VR

θ

θ

η

θ

=

∴ = =

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∵

∵

( ) ( )

2

2

0

2 220

2

ertzian dipole

80

sin64022

2 00(5) , 1.5 sin ,44

rad

R

dlR

EP

PRA Dem Sav

θ

πλ

λθπ

λλθ φ θ θ φππ

⎛ ⎞⎜ ⎟= ⎜ ⎟⎝ ⎠

∴ =

⎛ ⎞⎜ ⎟∴ = = =⎜ ⎟⎝ ⎠

( ) ( )

( ) ( )

20

20

d. For general antennas, the above relation hold.4

, , for lossless antenna.

4 , , for lossy antenna.

The maximum effective aperture of an antenna used

em

em

D A

G A

πθ φ θ φλ

πθ φ θ φλ

⇒ =

⇒ =

for reception is related to the directive gain in the direction of the incoming wave of

that antenna when it is used for transmission.

16

Antenna Factor

a. Antenna factor is a common way to characterize the reception properties of EMC antenna.

b. Antenna factor is defined as the ratio of the incident electric field at the surface of the measurement antenna to the received voltage at the antenna terminals:

recZocV

radR jX+

Antenna Spetrum

analyzer

recV

recZ

recV

incEReceived

dBExpressed as dB AF

inc

rec

EAF

VVdB dB Vmμ μ

=

⇒ = −

(incident field) (received voltage)

d. Ex: please find the antenna factor at 100MHz for following measurement data.(1)the field intensity of antenna surface

is 60dBuV/m(2)The specturm Analyzer measures

40dBuV(3)coaxial loss=4.5dB/100ft

sol:

.The antenna factor is usually furnished by the manufacture of the antenna.c

Specturm

Analyzer30fta n tV

recV

dB3AF 60 (40 4.5 )

10 =18.65

VdB dB V dBm

dB

μ μ= − + ⋅

17

2

0

2 20

22 20 0

20

.

4

4 . .4

f av

out av e

rav r

L

rr

L r L r

ES

P S A

VS GZ

E EVG A FZ V Z G

η

λπ

λ πηη π λ

=

= ⋅

= ⋅ ⋅ =

∴ ⋅ ⋅ = ⇒ = =

rV

o u tP

ZL =50Ω

E

The Friss transmission equation

a. The coupling of two antennas.

T Tθ ,φ

PtPR

d

R Rθ ,φ

( )( )

T T T

eT T T

G θ ,

A θ ,

φ

φ( )( )

R R R

eR R R

G θ ,

A θ ,

φ

φ

18

2

R

(1) The power density at the receiving antenna of distance d.

( , )4

(2) and by the definition of effective asperture, the received power P

b. Friss transmission equation:

Tav T T T

PS Gd

θ φπ

=

( )( )

( ) ( )

( )

2

2

20

,

( , ) ,

4 (3) and we know

4 , ,

( , ) , ( )4

(4) In terms of dB

10 log

R av eR R R

T T T eR R RR

T

R R R eR R R

RT T T R R R

T

R

T

P S A

G APP d

G A

P G GP d

PP

θ φ

θ φ θ φπ

πθ φ θ φλ

λθ φ θ φπ

=

∴ =

=

∴ = ⋅ ⋅

⎛, , 20 log 20log 147.6T dB R dBG G f d

⎞= + − − +⎜ ⎟

⎝ ⎠

c. : Friss transmission equation is valid under two assumptions: (1) the receiving antenna must be matched to its load impedance and the polarization of the inco

Note

2

ming wave. (2) two antenna should be in the far field . . d > 20 / ( for surface antennas ) d > 3 ( for wire antennas )

o

o

i e λλ

19

Half-wave Dipole Antenna:

( ) sin[ ( )]4mI z I zλβ= −

'' cos '

2

sin ( )4

2 cos[( / 2)cos ] sinsin

j rj z

j rm

eE j I z e dzr

I ejr

ββ θ

θ

β

ωμ θπ

π θωμ θβ θ

−

−

=

=

∫

Input impedance = 73+ j42.5Ω

If the length is slightly reduced to achieve resonance, the input impedance is about 73 + 0Ω

cos[( / 2)cos ]( )sin

F π θθθ

=

Half-wave Dipole Antenna: performance comparison between some dipoles

20

Dipole Antenna: length v.s. directivity

TypeShort dipoleA λ/2 dipoleA λ dipolelonger

Directivity1.51.642.41↓

HP width78°47°32°

Out-of-phase and Cancellation in the far field

In-phase and enhancement in the far-field

Directivity is dropped

Dipole Antenna: length v.s. radiation resistance

Input resistance is near infinity becausethe input current is zero at the feeding point.

21

Monopole

,,

,, ,

1122

A dipoleA mono

A A dipoleA mono A dipole

VVZ Z

I I= = =

1. Radiation resistance is half2. Directivity is double

2mono dipoleD D=

EMI due to Common-mode current can be explained byThe theory of monopole antenna.

Small loop antenna:

2

2

ˆsin4

ˆsin4

1 ˆˆ sin4

j r

j r

j r

IeA jrIeE S

rIeH r E S

r

β

β

β

μ θφπ

ηβ θφπ

ηβ θθη π

−

−

−

=

=

= × = −

The fields depend only on the magnetic moment (IS) and not the loop shape.

2 2 2 222 / 20( ) 31200( )r

SR P I Sβλ

= = ≅ Ω

The radiation resistance can be increased by using multiple turns (N)

2231200( )r

NSRλ

= Ω

S is the area of the loop

22

Small loop antenna:

• The small loop antenna is very popular in receiving antenna.

• Single turn small loop antennas are used in pagers.

• Multi-turn small loops are popular in AM broadcast receiver.

• Small loop antennas are also used in direction-finding receivers and for EMI field strength probes.

Yagi-Uda Antenna

Array Antenna can be used to increase directivity.Array feed networks are considerably simplified if only a few elements are fed directly. Such array is referred to as a parasitic array.A parasitic linear array of parallel dipole is called a Yagi-Udaantenna.Uda antennas are very popular because of their simplicity and relatively high gain.

23

Yagi-Uda Antenna: Principles of operation

incident driverE E=

If a parasitic element is positioned very close to the driver element,it is excited with roughly equal amplitude.

Because the tangential E field on a good conductor should be zero.

From array theory, we know that the closely spaced, equal amplitude,opposite phase elements will have end-fire patterns.

parasitic incident driverE E E= − = −

Why null here?

Yagi-Uda Antenna: Principles of operation

The simplistic beauty of Yagi is revealed by lengthening the parasiteThe dual end-fire beam is changed to a more desirable single end-fire beam.

The H-plane pattern is obtained by the numerical method.

Such a parasite is called a reflector because it appears to reflect radiationfrom the driver.

0.49λ 0.4781λ

24

Yagi-Uda Antenna: Principles of operation

0.45λ0.4781λ

Spacing = 0.04λ

If the parasite is shorter than the driver, but now placed on the other side of the driver, the patter effect is similar to that when using a reflector in the sensethat main beam enhancement is in the same direction.

This parasite is then referred to as a director since it appears to direct radiation in the direction fromthe driver toward the director.

Yagi-Uda Antenna: Principles of operation

The single endfire beam can be further enhanced with a reflector and a director on oppositeSides of a driver.

The maximum directivity obtained from three-element Yagi is about 9dBi or 7dBd.

25

Yagi-Uda Antenna: Principles of operation

Optimum reflector spacing SR (for maximum directivity) is between 0.15 and 0.25 wavelengths.

Director-to-director spacings are typically 0.2 to 0.35 wavelengths, with the larger spacingsbeing more common for long arrays and closer spacings for shorter arrays.

The director length is typically 10 to 20% shorter than their resonant length.

The addition of directors up to 5 or 6 provides a significant increase in gain.satuation

Yagi-Uda Antenna: Principles of operation

The main effect of the reflector is on the driving point impedance at the feeding point and on the back lobe of The array.

Pattern shape, and therefore gain, are mostly controlled byThe director elements.

26

Broadband Measurement Antennas

a.FCC prefers to use tuned half-wave dipoles. For measurement of radiated emission from 30MHz~1GH, it is time consuming because its length must be physically adjusted to provide /2 at each fλ requency.

b. Broadband measurement antennas for EMC. 1.Input / output impedence is fairly constant over the frequency band. 2.The pattern is fairly constant over the frequency band. two commonly used antenna 1.biconical antenna -- 30MHz~200MHz 2.log-periodic antenna -- 200MHz~1GHz

The Biconical Antenna

^Eθ

^Hφ

z

hθθ

+

-

h

^ ^ ^

^ ^ ^

a. Infinite biconical antenna 1. half angle . 2.feed point with small gap.b. In the free space , the symmetry

suggests:

H H a

E E aθ

φ φ

θ

θ

=

= , 0

(1/ sin ) / (sin ) 0 .....(1) (-1/ ) / ( ) .....(2)

r

H j E J for J

r H j Er r rH j E

φ

φ θ

ωε

θ θ θ ωεωε

× = + =

∂ ∂ = =

∂ ∂ =

▽

27

c. by the Faraday’s and Ampere’s law

0

0

^0

^0 0 0

^ ^0

( /sin ) / from (1) => sin / 0

and ( / sin ) / from (2) and TEM mode type

=> so , we may uniquely define voltage between two point

j r

j r

H H e rH

E H e r

E H

βφ

φ

βθ

θ φ

θθ θ

β ωε θ

η

−

−

=∂ ∂ =

=

=s on the

cores as was the case in tx lines.

0

0 0

0

^

0 0

0 0 0 0

2^ ^-

00

d. ( ) - - ( / sin )*

- / sin 2 ln (cot( / 2))

by Ampere's law => ( ) sin 2

e

h h

h h

h

h

j r

j r j rh

j r

V r E dl H e r rd

H e d H e

I r H r d H e

θ θβ

θ π θ θ π θ

θβ β

π θ

πβ

η θ θ

η θ θ η θ

θ φ π

−

= − = −

− −

−

Φ

Φ=

= =

= =

= ⋅ ⋅ =

∫ ∫

∫

∫

^ ^ ^

00^ ^

. Input impedence : ( at 0)

( ) / ( ) ( / )ln(cot( / 2)) 120ln(cot( / 2))

Usually , the half angle is choosen such that

in h hr

in C

r

Z V r I r

Z Z

η π θ θ=

=

= = =

=

28

^

rad2-2 ^ 2

00

-22

0 0 0 0

f. for infinite biconnical antenna

R = Z

proof => /2 sin

* /sin 2 *ln(cot( /2))

h

h

h

h

in

avrad S

h

P S ds E r d d

H d H

π θπ

θθ θ

π θ

θ θ

η θ θ φ

πη θ θ πη θ

=

=

⋅= =

= =

∫ ∫ ∫

∫2

0 0^ ^ ^2 2

(1/2)(2 ) ( / *ln(cot( /2)))

(1/2)( ) (1/2)( )

h

inrad

H

I R I Z

π η π θ=

= =

^

^'

in

f. note : (1) Z is independent of frequency ( flat response) for infinite core (2) but , practice antenna is finite length , so

Z = R + jX some

in

→ storage energy will apearg. EMC biconical antenna

- +

With wire element

- +

29

Log-Periodic Dipole Array (LPDA) Antenna

1 1n

n

RR

τ += <

Scale factor

Spacing factor

2n

n

dL

σ =

Why is the LPDA constructed as below?

Log-Periodic Dipole Array (LPDA) Antenna

It is convenient to view the LPDA operation as being similar to that of a Yagi-Uda antenna.

The longer dipole behind the most active dipole behaves as a reflector and the adjacent shorter dipole in front acts as a director.

The radiation is then off of the apex.

30

Antenna Arrays: • Hertzian dipole, small loop, half dipole, and monopole are all omnidirectionalbecause of symmetry of the structure.

• Two or more omni-directional antennascan change the pattern (create null ormaximum). Better both for communicationand EMI considerations.

• This can be achieved by phasing the currentsto the antennas and separated them sufficientlysuch that the fields will add constructively anddestructively to result in the null/maximum patterns.

Antenna arrays

a. By using two or more omni-directional antennas to produce maxima and/or nulls in the resulting pattern.

b. The far fields at point P due to each antenna are of the form:

ψx

y

z

d

θ1 θ2θ

rr1r2

-d/2 d/2

P

0

0

11

22

ˆˆ1

ˆ 0ˆ2

j r

j r

MIE er

MIE er

βθ

βθ

α −

−

∠=

∠=

31

1 2

note: 1. 0 and , we assume that the currents of the two antennas are equal in magnitude but with phase difference .

ˆ 2. depends on the type of antennas used.

I I I I

M

αα

= ∠ = ∠

( )

0 0

0

1 2

ˆ for Hertzian dipole: sin4

ˆ for long dipole : .c. The total field at point P

0 1 0 22 2 2ˆ ˆ ˆ ˆ 1 2

using for field a

dlM j

M j F

j j jj r j re eE E E MIe e e

r rθ θ θ

η β θπ

ε θ

α α αβ β

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

⎛ ⎞− −⎜ ⎟

= + = +⎜ ⎟⎜ ⎟⎝ ⎠

1 2

1

2

pproximation. (1) for distance term.

(2) cos sin sin2 2

cos sin sin ,where cos sin sin2 2 r y

r r rd dr r r

d dr r r a a

γ θ φ

γ θ φ γ θ φ

≅ ≅

≅ + ≅ +

≅ − ≅ − = ⋅ =

sin sin sin sinˆ 0 02 2 2 202ˆ

02 ˆ 2 cos sin sin20

pattern of individual F

d dj j jj rMIE e e e er

j j rIe de M

r

α αα β θ φ β θ φβθ

α βπ αθ φλ

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎢ ⎥∴ = +⎢ ⎥⎢ ⎥⎣ ⎦

−⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟= × −⎢ ⎥ ⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

( )

( )array

array , elementsd. The principle of Pattern Multiplication: The resultant field is the product of the individual antenna element and the array factor F , .

e. Some example

θ φ

θ φ

s of the patternsof two-element array

32

d=λ/2,α=0(1)

I∠α I∠0

ψX

Y0

arraysinF ( =90 , )=cos2

π θθ φ ⎛ ⎞⎜ ⎟⎝ ⎠

(2) d=λ/2,α=π

I∠α I∠0

d

d

x

y0

arraysinF ( =90 , )=cos2 2

π θ πθ φ ⎛ ⎞−⎜ ⎟⎝ ⎠

(3) d=λ/4,α=π/2

d

ψ

x

y

0array

sinF ( =90 , )=cos4 4

π θ πθ φ ⎛ ⎞−⎜ ⎟⎝ ⎠

33

Antenna Arrays:

• Two arbitrary omni-directional antennas

• Separate a distance d

• currents on two antennas are equal in amplitude and phase delay α

0/ 2

0

2 [ ] cos( sin sin )2

j rj Ie dE e M

r

βα

θπ αθ φλ

−

= × −

Pattern of individual antenna

Array pattern

2 212r

m nfa bμε

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

port1(5cm,4cm)

port2(4cm,8cm)port3(1cm,9cm)

port4(6cm,6cm)

10cm

10cm

1.6mmX

Y

(0,0)

Microstrip Antenna and Resonance frequency

34

Antenna Examples: Printed Dipole (at 2.12 GHz)

Source excitation

Current distribution

2D

3D

35

Antenna Examples: Printed Dipole

Radiation Patterns Omni-directional

Gain: 2.1dBEfficiency : 99.1%

Antenna Examples: Microstrip Antenna with edge coupled (at 2.1GHz)

3cm x 4cm

ε=2.2

Current distribution

2D

3D

36

Antenna Examples: Microstrip Antenna with edge coupled (at 2.1GHz)

Gain: 6.6dBEfficiency: 71.1%

Antenna Examples: Microstrip Antenna with inserted feed and edge coupled (at 2.1GHz)

Current distribution

2D

3D

37

Antenna Examples: Microstrip Antenna with inserted feed and edge coupled (at 2.1GHz)

Gain: 6.4dBEfficiency: 49.1%

Antenna Examples: Small Array (at 9.76 GHz)

Current distribution

2D

3D

X = 7.5mmY = 3.75mm

38

Antenna Examples: Small Array (at 9.76 GHz)

Gain: 10.5dB

Antenna Examples: slot coupled microstrip antenna (at 2.213 GHz)

Current distribution

2D

3D

39

Antenna Examples: slot coupled microstrip antenna (at 2.213 GHz)

Gain: 7.1dBEfficiency: 55.3%

Edge coupled patch with matching feed network

Antenna Examples: many others

Crossed slot coupled patch for dual polarization