feeding networks-tuesday 2009 monday [modo de...

Feeding NetworksBasic Concepts, Principal Devicesasic Concepts, incipal evices

Radiation Group Signals, Systems and Radiocommunications Department

Universidad Politécnica de MadridPablo Padilla de la Torre

E-Mail: [email protected]

ANTENNA DESIGN AND MEASUREMENT TECHNIQUES - Madrid (UPM) – March 2009

E Mail: [email protected]

Outline

Introduction

Transmission Lines Architecture

Some Feeding Devices


Printed líne

A printed line Consists of:

Line :

M lli fMetallic surface

Extremely thin surface (thickness: 10 – 50μm ⇒ typical: 18μm y 35μm )

Substrate :

Dielectric layer. Thickness: 0.003λ - 0.05λDielectric Constants within the range: 1 ≤ εr ≤ 12

G d lGround plane.


Transmission Line Theory

TEM Mode2c

z

t

oezuuE γφ −−∇= )(),,(21

φ γ zoezzuuH

−−∇∧=

)(ˆ)(

c1c2

)(·1

zVldec

zo =∇∫ −γφ POTENTIAL WAVE

dlen

dlHndljzI zs oγφ −

∫∇−

=∧∫=∫=ˆ·ˆˆ)(

ηtzuuH =),,(

21dledlHndljzI

ct

cs

cs η∫=∧∫=∫=

222

)(

CURRENT WAVE

Transmission line model:Transmission line model:

g

c

c

z

kn

lde

IzV

Zo

·ˆ

·

)()(

2

1 ηφφ

ηγ

=∇∫∇

==−

Characteristic Impedance

Geometric Constant

Primary Parameters :

gz

c

s ldenzI o·)(

2

η

ηφη

γ∫

∇− −

S

μεωγ =

Secondary Parameters :

kCs

ε= μ

gkLs = δωCtgGs =


gk

Introduction




Basic Models

Microstrip Line StriplineStripline

Covered microstrip line

Coplanar

Suspended microstrip line

Inverted microstrip line

Slot



Features

B i F• Basic Features:

Line width w

Line thickness t

Line losses αc

S b t t thi k hSubstrate thickness h

Substrate dielectric constant εr

Substrate losses: loss tangent tan(δ) Subs e osses: oss ge (δ)

Radiation losses αr

• Main parameters:pCharacteristic impedance Z

Effective dielectric Constant εeff


[1] B. C. Wadell, “Transmission Line Design Handbook”, Artech House, London, 1991

Features

cPhase velocity vp :

L W l th λ

rrp

cv

εμ0=

λλ 0=Layer Wavelength λg :

Non-homogeneous dielectric:

rr

g εμλ =

eff

g ελλ 0=

effgp

cv

ε0=

Non-homogeneous dielectrics: when we have several dielectric layers (multilayer structure) or even when εr, μr change, depending on the dielectric layer positiondielectric layer position.

The effective dielectric layer εeff takes into account the wave propagation inside non-homogeneous dielectric layers.


Limitations

W h t Ad t i l ti⇒ We have to Adopt a compromise solution

Layer thickness hLayer thickness h

Reduce surface wave losses ⇒ h

Increase bandwidth⇒ h

Substrate dielectric constant εr

Tiny structures ⇒ εr

Li idthLine width w

w << λg/2

Decrease undesired line radiation ⇒ w << λg/2Decrease undesired line radiation ⇒ w λg/2


Applications

• Feeding network:

L16 S90-120ETSIT - S.R. MOYANO

g

For antennas

• Printed circuits:

Filters

Dividers

MixersMixers

…


Microstrip Line

w

• The most common transmission structure for planar antennas.

• The main working mode is quasi-TEM ⇒ almost all the field is concentrated inside thesubstrate.

• To avoid surface waves electrically thin substrate (0.003λ < h <0.05 λ)

• Dielectric constant εr : 2.2 < εr < 12.r r


Microstrip Line

•Features:

Characteristic line impedance Z : ⇒ 0 05 ≤ w/h ≤ 100 ε ≤ 16 ⇒ 0 2% errorCharacteristic line impedance Z : ⇒ 0.05 ≤ w/h ≤ 100, εr ≤ 16 ⇒ 0.2% error

⎟⎟⎞

⎜⎜⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ⋅

−2

666.300

4)62(6

7528.0

hhew

h

πεμ

η

Layer Impedance

Geometric constant

Eff i di l i ≤ 16 0 05 ≤ /h ≤ 20 1%

⎟⎟⎟

⎠⎜⎜⎜

⎝

⋅++

⋅⋅−+==

⎠⎝

20 4

1)62(6

ln2

0

w

h

w

heKZ

eff

g

eff

πεπε

εη

Effective dielectric constant εeff : ⇒ εr ≤ 16, 0.05 ≤ w/h ≤ 20 ⇒ 1% error

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ⋅+

−+

+=

− 22

1

104.012

12

1

2

1

h

w

w

hrreff

εεε w/h < 1⎥⎦

⎢⎣

⎠⎝⎠⎝22 hw

2

1

121

2

1

2

1−

⎟⎠⎞

⎜⎝⎛ ⋅+

−+

+=

w

hrreff

εεε w/h ≥ 1

< 2% error considering εr > 16

< 2% error considering 05.0<h

w


h

[1] B. C. Wadell, “Transmission Line Design Handbook”, Artech House, London, 1991.

Microstrip Line

• Typical dielectric substrates used to perform microstrip lines are:

RT-Duroid 5880 ε = 2 2RT Duroid 5880 εr 2.2

Alumina (ceramic, Al2O4 (97%)) εr = 9.8• Real prototype width/thickness values within the range 101.0 ≤≤

hw

• Real prototype characteristic impedance in the range 10 ≤ Z ≤ 200 ohmsh



• Considering the same dimensions as in a conventional microstrip the covered• Considering the same dimensions as in a conventional microstrip, the covered microstrip line has a lower impedance, due to the existence of a new ground plane.

F th i d Z ( i th ti l i t i ) th li idth i• For the same impedance Z (as in the conventional microstrip) ⇒ the line width is smaller.

• The effective dielectric constant εeff for the same dimensions (as in conventional microstrip) is lower.

• When h2 ⇒ microstrip line

• When h ⇒ Z y ε


• When h2 ⇒ Z y εeff



•Features :

Characteristic impedance Z :

⎞⎛ ⎞⎛ 75280

Layer impedance

geometric constant

eff

w

h

eff

g

eff

Z

w

h

w

heKZ

επ

επεμ

εη Δ

−

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

⋅++

⋅⋅−+==

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ⋅

−

2

2

666.30

0

0

41

)62(6ln

2

7528.0

0

eff

Z

effeff

striplíneamicro

εεπε⎟⎠

⎜⎝

⎞⎛ 32

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

⎟⎞

⎜⎛ +

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

−⋅

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−−+−=Δ −

2

2

32

1

2

2

1

027.0177.0012.0tanh0109.1

1

389.11706.0192.1tanh1270

h

hw

hw

hw

hhh

hZ

⎟⎠

⎜⎝

⎟⎠

⎜⎝+⎟

⎠⎜⎝ ⎥⎦⎢⎣

⎟⎠

⎜⎝ 1

hh


⇒ ±0.1% error, 0.1 ≤ w/h ≤ 6; ⇒ ± 0.2% error, 6 ≤ w/h ≤ 10[1] B. C. Wadell, “Transmission Line Design Handbook”, Artech House, London, 1991.


Effective dielectric constant εeff :

11

–It depends on the superior metallic plane distance, and the line 2

12

1 −+

+= rr

eff

εαεε

width:

( )⎟⎟⎟⎞

⎜⎜⎜⎛

⎟⎟⎟⎞

⎜⎜⎜⎛

−⎟⎞

⎜⎛ +⎟⎟

⎞⎜⎜⎛

−+=thhh

ab

α 2ln2101

164.112100431tanh 2

where:

⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝

−⎟⎠

⎜⎝+⎟⎟

⎠⎜⎜⎝

−+=

hw

hwhh π

α 1121.0043.1tanh2

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛++

⎞⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

+=3

4

222

1181ln

718152

1

ln491

1h

w

w

hw

hw

a053.0

39.0

564.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= rb

ε

⎟⎟

⎠⎜⎜

⎝

⎟⎠

⎜⎝ ⎠⎝

+⎟⎠⎞

⎜⎝⎛ 1.187.18

432.049 h

hw 3 ⎟

⎠⎜⎝ +rε


⇒ ±0.2% error , 0.01 ≤ w/h ≤ 100 y 1 ≤ εr ≤ 100 ; [1] B. C. Wadell, “Transmission Line Design Handbook”, Artech House, London, 1991.


εr

εair

h

h2

w

The s spended microstrip line is sed hen lo losses are needed• The suspended microstrip line is used when low losses are needed.

• The effective dielectric constant εeff is reduced. For the same width (as inmicrostrip line) ⇒ Z is higher.

• For the same impedance (as in microstrip line) ⇒ the line is wider.

• When h2 ⇒ microstrip lineWhen h2 ⇒ microstrip line

• When h2 ⇒ Z y εeff




• Features:

Ch t i ti i d Z ± 0 2% ≤ 20Characteristic impedance Z : ⇒ ± 0.2% error, εr ≤ 20

⎞⎛ ⎞⎛ 75280

Layer impedance

Geometricconstant

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

++⋅−+

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

2

666.30

0

0

41

)62(6ln

2

7528.0

0

uu

eKZ

u

gπ

επεμ

εη

2hh

wu

+=

Eff ti di l t i t t ± 0 2% ≤ 20

⎟⎟

⎠⎜⎜

⎝2 uueffeff επε 2

Effective dielectric constant εeff : ⇒ ± 0.2% error, εr ≤ 20

2

1

⎤⎡ ⎞⎛⎞⎛ ⎞⎛=effε

4

2

' ln1251.08621.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

hh

2

'''

2

11

ln1⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

r

ff

hw

hhhh

ε

2 ⎠⎝ ⎠⎝4

2

'' ln1397.04986.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

hh


⎦⎣ 2 ⎠⎝ ⎠⎝



εr

εair

h

h2 w

• The inverted microstrip line is also used when low losses are desired• The inverted microstrip line is also used when low losses are desired.• Effective dielectric constant εeff is reduced. For the same width (as in

conventional microstrip line) ⇒ Z is higher.• For the same impedance Z (as in conventional microstrip) ⇒ the line is• For the same impedance Z (as in conventional microstrip) ⇒ the line is

wider.• When h2 ⇒ microstrip line

Wh h ⇒ Z ( 1 Ai )• When h2 ⇒ Z y εeff (εeff ≈ 1, Air )



Inverted microstrip linep

• Features:

Characteristic impedance Z : ⇒ ± 1% error, εr ≤ 20

⎞⎛ ⎞⎛ ⎞⎛75280μ

Layer impedance

Geometricconstant

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

++⋅−+

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

2

666.30

0

0

41

)62(6ln

2

7528.0

0

uu

eKZ

u

ff

g

ff

πεπεμ

εη

h

wu =

p

Effective dielectric constant εeff : ⇒ < 0.6% error, εr ≤ 20; 0.5 ≤ w/h2 ≤ 10 y 0 06 ≤ h/h ≤ 1 5

⎟⎠

⎜⎝

2 uueffeff επε2h

0.06 ≤ h/h2 ≤ 1.5

( )2⎤⎡

⎟⎞

⎜⎛

⎟⎞

⎜⎛ wh

2

' ln1515.05173.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

hh

( )2

'''

2

1ln1⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+= reff h

whh

h

h εε 2⎟⎠

⎜⎝ ⎠⎝ h

2

'' ln1047.03092.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

hh


2⎟⎠

⎜⎝

⎟⎠

⎜⎝ h


Stripline line

εairwh εr

h/2

t

• For the same width ε and h/2 = h as in conventional microstrip line• For the same width, εr and h/2 = hmicrostrip as in conventional microstrip line ⇒ Z is lower.

• For the same impedance as in conventional microstrip line (same εr and h/2 h ) th li i= hmicrostrip ) ⇒ the line is narrower.


Symmetric stripline

• Features:

Characteristic impedance Z : ⇒ 0.5% error, εr ≤ 20 y w/h > 0.1

μLayer

impedanceGeometricconstant

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛++== 27.6

8885.01ln

2

2

'''0

0

0

w

h

w

h

w

hKZ

r

g

eff πππεπεμ

εη

⎠⎝ ⎦⎣reff

ttw

wwΔ

+='

23

5ln ⎟

⎠⎞

⎜⎝⎛

=Δ th

w

Effective dielectric impedance εeff :

t 2.3t

reff εε =



Asymmetric stripline

εairh

h

air

εr

h2 wt

• For the same dimensions (as symmetric stripline) ⇒ Z is lower.• For the same impedance ⇒ the width is lower.



Asymmetric stripliney p

• Features:

– Characteristic impedance Z : ⇒ 0.5% error, εr ≤ 20 y w’/h > 0.1

⎟⎞

⎜⎛ μ

Layer impedance

Geometric constant

⎟⎟⎟⎟

⎜⎜⎜⎜

Δ−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛++== Z

w

h

w

h

w

hKZ g

2

'''0

0

27.6888

5.01ln2

10

πππεπεμ

εεη

( )⎟⎟

⎠⎜⎜

⎝

⎟⎠

⎜⎝ ⎥⎦⎢⎣ ⎠⎝ www

thwZ

rreff

rsymmtric ,,,

2

ε

πππεπεε

22

9.2

2

2.2

2

22'' 25.0826.0

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++

+−=Δ

thhwt

thh

th

ZZπ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=

),,,(),,,(

),,,(),,,(2

2

2''

thwZthwZ

thwZthwZZ

rsymmetricrsymmetric

rsymmetricrsymmetric

εεεε

– Effective dielectric constant εeff :

⎟⎠

⎜⎝

⎦⎣

ff εε =


reff εε[1] B. C. Wadell, “Transmission Line Design Handbook”, Artech House, London, 1991.

Coplanar

b

εrh

a

•There is no ground plane.

d i d i b λ•In order to prevent superior mode propagation: b < λg

•Practical example: couplers, dipole feeding…



Coplanar line

• Features:

Ch t i ti i d Z 2%– Characteristic impedance Z : 2% error,

⎟⎞

⎜⎛0 aaμLayer

impedanceGeometric constant

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

++==

4

0

0

41

412ln

2

10

aaba

ba

KZeff

g

eff πεεμ

εη

Eff ti di l t t t 1%

⎟⎠

⎜⎝

−+ 4 41ba

baeffeff εε

– Effective dielectrc constant εeff : 1% error

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

++++4

4 41411aa

kk⎟⎠⎞

⎜⎝⎛

ha

k4

sinhπ

⎟⎟⎟

⎠⎜⎜⎜

⎝⎟⎟⎟

⎠⎜⎜⎜

⎝−+

−−+++−

+=4

411

411

4141

412ln

21

1

ba

ba

bbkk

kkr

eff

εε⎟⎠⎞

⎜⎝⎛

⎠⎝=

hbh

k

4sinh

41 π


⎠⎝ ⎠⎝ bb


Coplanar line

13εr = 13


Slot line

εrh

w

• The slot line is a TE structure not a TEM oneThe slot line is a TE structure, not a TEM one.

• This structure needs a substrate with a high dielectric constant value εr (εr ≥ 10)⇒ on behalf to concentrate propagating fields and decrease radiation.

• The slot line is combined with microstrip in designing directional couplers• The slot line is combined with microstrip in designing directional couplers,branchlines...



Slot line

• Features:Characteristic impedance Z : ⇒ 2% error, 0.02 ≤ w/h ≤ 1

1.002.050⎞⎛

⎟⎞

⎜⎛ −⎟⎞

⎜⎛ −

ww

( ) ( )( )

( ) ( )( ) ( )2

ln976813411441ln46382110ln73680

ln0849.4528.44100ln3026.21.002.050

ln028.8162.72

⎟⎟⎞

⎜⎜⎛

−−⋅⎥⎤

⎢⎡ ++−−

−⎟⎠⎞

⎜⎝⎛+

⎟⎠

⎜⎝⎟⎠

⎜⎝+−= rr

hw

hw

hw

hhZ

εεε

εε

( ) ( )( ) ( )100

ln9768.134.1144.1ln4638.211.0ln7368.0 ⎟⎟⎠

⎜⎜⎝

−−⋅⎥⎦⎢⎣++−−

grrr h λεεε

Effective dielectric constant εeff : ⇒ 2% error, 0.02 ≤ w/h ≤ 1

2−⎤⎡ ⎞⎛( )

2

0100ln1082.06678.02.0ln0316.1923.0

−

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +−+−=

λεε h

hw

hw

reff



Typical losses for different linesdifferent lines

Typical losses with those structures when applying high frequency(10 GHz)(10 GHz)

Network Losses (dB/m)

Waveguide 0 2Waveguide 0.2

Suspended line 1.8 – 3.0

Stripline 2.7 – 5.6

Mi t i li 4 6Microstrip line 4 - 6


SubstratesPrinted lines at high frequency need quality materials⇒ loss tangent : tan(δ) < 0.002

F f b f i d li 10 GHFeatures of substrates for printed lines at 10 GHz

Dielectric constant: εr losses: tan(δ)

Epoxy fiberglass FR-4 4.4 0.01

Laminex 4.8 0.03

Taconic 2.33 0.0009

Kapton 3.5 0.002p

CuClad 2.17 0.0009

RT Duroid 5880

(teflon + glass fiber)

2.2 0.0009

(teflon + glass fiber)

Alumina

(Ceramic form of Al2O4)

9.9 0.0003

RT D roid 6010 10 5 0 002RT Duroid 6010

(PTFE1 ceramic)

10.5 0.002

GaAs (Gallium Arsenide)

(S i d di l i )

12.8 0.0006


(Semiconductor dielectric)

1 Polytetrafluoroethylene (Teflon)

Substrate selection

Substrate thickness εr

Reduction in line radiation Quite little high

Small dimensions little high

Low Losses little low

Reduction in losses due to surface currents little low

Increase in band width big low

Thin substrates with high εr are used in microwave circuitry, as feedingThin substrates with high εr are used in microwave circuitry, as feeding network:

Advantages:Lower line width.Reduction in radiation and coupling effects, although they should not be

neglected.Disadvantaged:

Higher losses


Higher losses.Lower efficiency. Lower bandwidth.

Introduction




λ/4 Adaptator

Z (d) Z (0)

λ/4

Z

(d) (0)

Transformation (considering no losses):

Z 02

Z 01 Z Adap

Transformation (considering no losses):

( )djsenZdZ

djsenZdZZdZ

d

adapadap ββ

ββ)0(cos

cos)0(

+

+=

djsenZdZadap ββ )0(cos +

λ/4 case

( ))0(

2)0(

2cos

22cos)0(

Z

ZZ

jsenZZ

jsenZZZdZ adap

adap

adap

adap

adap =+

+=

ππ

ππ

)()0( dZZZ adap =02)0( ZZ =

0101)( ZZdZ == ∗


Power Divider

Zλ/4

Z 02

λ/4Z 02

Z 02

Zadap Zadap

Bended Corner: To prevent reflection

02 02

A B C

02)( ZAZ =Point A:

)( ZCZ =Point C: 02)( ZCZ =Point C:

Point B: )()·(

)(//)()( derizq BZBZBZBZBZ

2

)()(Z

ZBZBZ adap

izqizq ==

02)( ZBZ =

)()(

)()()(//)()(

derizq

derizqderizq BZBZ

BZBZBZ+

== 02Z

02

2

2202

2

02

2

02 2

·

Z

Z

ZZ

Z

Z

Z

Z

Z adap

adapadap

adapadap

=

+

=022ZZ adap =


0202 ZZ

Directional Coupler

Z02

34

Z 02

Z 02

• Gate 1 : input

G 2 di12

• Gate 2 : direct output

• Gate 3 : isolated output

• Gate 4 : coupled output

1

A portion of the injected power (input) yields coupling in the alongside line(coupled output), the rest flows through the coupled output.

Power relation between direct and coupled output is a designer choice.

Direct and coupled outputs have a signal phase delay (90º).


p p g p y ( )

Ideally, there is no power at the isolated gate.

Hibrid Coupler 90º

• Gate 1 : input34Z 2

Gate 1 : input

• Gate 2 : -90º output

• Gate 3 : -180º output

G 4 i l d

34

λ/4 λ/4Z Z 1 1λ/4

Z 2

• Gate 4 : isolated port1 2

λ/4 λ/41 1λ/4

⎤⎡01 ZZ =

2/02 ZZ =3 dB coupler

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−−

−−

=001

100

010

2

1

j

j

j

S

A portion of the injected power (1) reaches 3 with -180º phase delay), therest flows through the -90º output (2)

⎥⎦

⎢⎣ −− 010 j

rest flows through the 90 output (2).

The output signals at 2 and 3 have a 90º phase delay, refered one to theother.


Ideally, there is no power at the isolated gate (4).

Circular Coupler (Rat Race)

• Gate 1 : input2 3λ/4 • Gate 2 : output -90º

• Gate 3 : isolated port

• Gate 4 : output -270º

2 3

λ/4

λ/4

λ/4 Gate 4 : output 270

1 4⎟⎟⎞

⎜⎜⎛ −

00

00

1 jj

jj

λ/4

⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝ −−−

−−=

00

00

00

2

1

jj

jj

jjS

3λ/4The injected power at 1 reaches 2 and 4 with a phase delay of -/+90º

respectively.

3λ/4

The output signals at 2 and 4 have a 180º phase delay, referred one tothe other.


Ideally, there is no power at the isolated gate (3).

Referencesf

[1] B C Wadell “Transmission Line Design Handbook” Ed Artech House[1] B. C. Wadell, Transmission Line Design Handbook , Ed. Artech House, London, 1991.

[2] E. Hammerstad, O.Jensen, “Accurate Models for Microstrip Computer-Aided Design”, IEEE MTT-S Symposium Digest, pp.407-409, June 1980.

[3] S. J. Orfanidis, “Electromagnetic Waves & Antennas”, 2004.

[4] H. Howe, “Stripline Circuit Design”, Ed. Artech House, London, 1982.

[5] D.M. Pozar, “Microwave Engineering”, Ed. John Wiley & Sons, 1980.

[6] K.C. Gupta, R. Garg, I. Bahl and P. Bhartia, “Microstrip Lines and Slotlines”, Artech House, London, 1996.

[7] P. Barthia & al., “Milimeter wave microstrip and printed circuit antennas”,[7] P. Barthia & al., Milimeter wave microstrip and printed circuit antennas , Norwood, Mass, Ed. Artech House,1991.

[8] Ansoft Ensemble, “Ansoft Ensemble: Getting Started and Tutorials”, V ió 8 0 C i ht 1984 1999 A ft C ti 1999


Versión 8.0 Copyright 1984-1999, Ansoft Coporation, 1999.

feeding networks-tuesday 2009 monday [modo de...

Documents