Introduction to Microstrip Antennas.pdf

Introduction to Microstrip Antennas David R. Jackson

Dept. of ECE University of Houston

1

David R. Jackson

Dept. of ECE N308 Engineering Building 1

University of Houston Houston, TX 77204-4005

Phone: 713-743-4426

Fax: 713-743-4444 Email: [email protected]

2

Contact Information

Purpose of Short Course

Provide an introduction to microstrip antennas.

Provide a physical and mathematical basis for understanding how microstrip antennas work.

Provide a physical understanding of the basic physical properties of microstrip antennas.

Provide an overview of some of the recent advances and trends in the area (but not an exhaustive survey – directed towards understanding the fundamental principles).

3

Additional Resources

Some basic references are provided at the end of these viewgraphs.

You are welcome to visit a website that goes along with a course at the University of Houston on microstrip antennas (PowerPoint viewgraphs from the course may be found there, along with the viewgraphs from this short course).

4

ECE 6345: Microstrip Antennas

http://www.egr.uh.edu/courses/ece/ece6345/web/welcome.html

Note: You are welcome to use anything that you find on this website, as long as you please acknowledge the source.

Outline Overview of microstrip antennas

Feeding methods

Basic principles of operation

General characteristics

CAD Formulas

Radiation pattern

Input Impedance

Circular polarization

Circular patch

Improving bandwidth

Miniaturization

Reducing surface waves and lateral radiation

5

Notation

6

0 0 0 02 /k ω µ ε π λ= =

1 0 rk k ε=

[ ]00

0

376.7303µηε

= ≈ Ω

0 /c fλ =

[ ]82.99792458 10 m/sc = ×

1 0 / rη η ε=

[ ]

[ ]

0 0

70

120 2

0

1

4 10 H/m1 8.854188 10 F/m

c

c

ε µ

µ π

εµ

=

= ×

= ≈ ×

c = speed of light in free space

0λ = wavelength of free space

0k = wavenumber of free space

1k = wavenumber of substrate

0η = intrinsic impedance of free space

1η = intrinsic impedance of substrate

rε = relative permtitivity (dielectric constant) of substrate

effrε = effective relative permtitivity

(accouting for fringing of flux lines at edges)

effrcε = complex effective relative permtitivity

(used in the cavity model to account for all losses)


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


7

Overview of Microstrip Antennas Also called “patch antennas”

One of the most useful antennas at microwave frequencies (f > 1 GHz).

It usually consists of a metal “patch” on top of a grounded dielectric substrate.

The patch may be in a variety of shapes, but rectangular and circular are the most common.

8

Microstrip line feed Coax feed

Overview of Microstrip Antennas

9

Common Shapes

Rectangular Square Circular

Elliptical

Annular ring

Triangular

Invented by Bob Munson in 1972 (but earlier work by Dechamps goes back to1953).

Became popular starting in the 1970s.

G. Deschamps and W. Sichak, “Microstrip Microwave Antennas,” Proc. of Third Symp. on USAF Antenna Research and Development Program, October 18–22, 1953. R. E. Munson, “Microstrip Phased Array Antennas,” Proc. of Twenty-Second Symp. on USAF Antenna Research and Development Program, October 1972. R. E. Munson, “Conformal Microstrip Antennas and Microstrip Phased Arrays,” IEEE Trans. Antennas Propagat., vol. AP-22, no. 1 (January 1974): 74–78.

10

Overview of Microstrip Antennas History

Advantages of Microstrip Antennas

Low profile (can even be “conformal,” i.e. flexible to conform to a surface).

Easy to fabricate (use etching and photolithography).

Easy to feed (coaxial cable, microstrip line, etc.).

Easy to use in an array or incorporate with other microstrip circuit elements.

Patterns are somewhat hemispherical, with a moderate directivity (about 6-8 dB is typical).

11


Disadvantages of Microstrip Antennas

Low bandwidth (but can be improved by a variety of techniques). Bandwidths of a few percent are typical. Bandwidth is roughly proportional to the substrate thickness and inversely proportional to the substrate permittivity.

Efficiency may be lower than with other antennas. Efficiency is limited by conductor and dielectric losses*, and by surface-wave loss**.

Only used at microwave frequencies and above (the substrate becomes too large at lower frequencies).

Cannot handle extremely large amounts of power (dielectric breakdown).

* Conductor and dielectric losses become more severe for thinner substrates.

** Surface-wave losses become more severe for thicker substrates (unless air or foam is used).

12


Applications

Satellite communications

Microwave communications

Cell phone antennas

GPS antennas

13

Applications include:


Microstrip Antenna Integrated into a System: HIC Antenna Base-Station for 28-43 GHz

Filter

Diplexer

LNA PD

K-connector DC supply Micro-D

connector

Microstrip antenna

Fiber input with collimating lens

(Photo courtesy of Dr. Rodney B. Waterhouse) 14



15

Arrays

Linear array (1-D corporate feed)

2×2 array

2-D 8X8 corporate-fed array 4 × 8 corporate-fed / series-fed array

http://www.google.com/url?sa=i&rct=j&q=+microstrip+antenna+integrated+monolithic&source=images&cd=&cad=rja&docid=0_D8RbqqK_a_6M&tbnid=6vLyP73MjOoo6M:&ved=0CAUQjRw&url=http://cegt201.bradley.edu/rfpage/seniorproj/seniorproj.shtml&ei=JPpiUYnECMrE0QGuxoGgCQ&psig=AFQjCNFCqzPd3r6DBMFKoqgtizOqYJQw0A&ust=1365527362189557

http://www.google.com/url?sa=i&rct=j&q=+microstrip+antenna+array&source=images&cd=&cad=rja&docid=YuN3G8qpqwN1UM&tbnid=8lIImwBhdLOPGM:&ved=0CAUQjRw&url=http://www.antenna-theory.com/arrays/main.php&ei=YvpiUZjXDOH20gGhmoEo&psig=AFQjCNHZ-FwGC1Ar36PNUgl7X19rQw6cwQ&ust=1365527519773344

Wraparound Array (conformal)

(Photo courtesy of Dr. Rodney B. Waterhouse)

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The substrate is so thin that it can be bent to “conform” to the surface.

x

y

h L

W

Note: L is the resonant dimension (direction of current flow). The width W is usually chosen to be larger than L (to get higher bandwidth).

However, usually W < 2L (to avoid problems with the (0,2) mode).

εr

17

Overview of Microstrip Antennas Rectangular patch

W = 1.5L is typical.

Js

Circular Patch

x

y

h

a

εr

18


The location of the feed determines the direction of current flow and hence the polarization of the radiated field.


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


19

Feeding Methods

Some of the more common methods for feeding microstrip antennas are shown.

20

The feeding methods are illustrated for a rectangular patch, but the principles apply for circular and other shapes as well.

Coaxial Feed

A feed along the centerline is the most common

(minimizes higher-order modes and cross-pol).

x

y

L

W

Feed at (x0, y0)

Surface current

21

x rε h

z

Feeding Methods

22

Advantages: Simple Directly compatible with coaxial cables Easy to obtain input match by adjusting feed position

Disadvantages: Significant probe (feed) radiation for thicker substrates Significant probe inductance for thicker substrates (limits

bandwidth) Not easily compatible with arrays

Coaxial Feed

2 0cosedgexR R

Lπ =

x rε h

z

Feeding Methods

x

y

L

W ( )0 0,x y

(The resistance varies as the square of the modal field shape.)

Advantages: Simple Allows for planar feeding Easy to use with arrays Easy to obtain input match

Disadvantages: Significant line radiation for thicker substrates For deep notches, patch current and radiation pattern may show distortion

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Inset Feed

Microstrip line

Feeding Methods

Recent work has shown that the resonant input resistance varies as

2 02cos2in

xR A BL

π = −

The coefficients A and B depend on the notch width S but (to a good approximation) not on the line width Wf .

Y. Hu, D. R. Jackson, J. T. Williams, and S. A. Long, “Characterization of the Input Impedance of the Inset-Fed Rectangular Microstrip Antenna,” IEEE Trans. Antennas and Propagation, Vol. 56, No. 10, pp. 3314-3318, Oct. 2008.

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L

W Wf

S

x0

Feeding Methods Inset Feed

Advantages: Allows for planar feeding Less line radiation compared to microstrip feed Can allow for higher bandwidth (no probe inductance, so

substrate can be thicker)

Disadvantages: Requires multilayer fabrication Alignment is important for input match

Patch

Microstrip line

25

Feeding Methods Proximity-coupled Feed

(Electromagnetically-coupled Feed)

Top view Microstrip line

Advantages: Allows for planar feeding Can allow for a match even with high edge impedances, where a notch

might be too large (e.g., when using high permittivity)

Disadvantages: Requires accurate gap fabrication Requires full-wave design

Patch

Microstrip line

Gap

26

Feeding Methods Gap-coupled Feed

Top view Microstrip line

Patch

Advantages:

Allows for planar feeding

Feed-line radiation is isolated from patch radiation

Higher bandwidth is possible since probe inductance is eliminated (allowing for a thick substrate), and also a double-resonance can be created

Allows for use of different substrates to optimize antenna and feed-circuit performance

Disadvantages: Requires multilayer fabrication Alignment is important for input match

Patch

Microstrip line

Slot

27

Feeding Methods Aperture-coupled Patch (ACP)

Top view

Slot

Microstrip line


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


28

Basic Principles of Operation

The basic principles are illustrated here for a rectangular patch, but the principles apply similarly for other patch shapes.

We use the cavity model to explain the operation of the patch antenna.

29

Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Experiment on Microstrip Antennas,” IEEE Trans. Antennas Propagat., vol. AP-27, no. 3 (March 1979): 137–145.

nh

PMC

z


The patch acts approximately as a resonant cavity (with short-circuit (PEC) walls on top and bottom, open-circuit (PMC) walls on the edges).

In a cavity, only certain modes are allowed to exist, at different resonance frequencies.

If the antenna is excited at a resonance frequency, a strong field is set up inside the cavity, and a strong current on the (bottom) surface of the patch. This produces significant radiation (a good antenna).

Note: As the substrate thickness gets smaller the patch current radiates less, due to image cancellation. However, the Q of the resonant mode also increases, making the patch currents stronger at resonance. These two effects cancel, allowing the patch to radiate well even for small substrate thicknesses.

30

Main Ideas:

(See next slide.)


As the substrate gets thinner the patch current radiates less, due to image cancellation (current and image are separated by 2h).

However, the Q of the resonant cavity mode also increases, making the patch currents stronger at resonance.

These two effects cancel, allowing the patch to radiate well even for thin substrates (though the bandwidth decreases).

31

A microstrip antenna can radiate well, even with a thin substrate.

x rε hsJ

z 1sJ

h∝

On patch and ground plane: 0tE = ˆ zE z E=

Inside the patch cavity, because of the thin substrate, the electric field vector is approximately independent of z.

Hence ( ) ( )ˆ, , ,zE x y z z E x y≈

32

h ( ),zE x y

z

Basic Principles of Operation Thin Substrate Approximation

( )( )

( )( )

1

1 ˆ ,

1 ˆ ,

z

z

H Ej

zE x yj

z E x yj

ωµ

ωµ

ωµ

= − ∇×

= − ∇×

= − − ×∇

Magnetic field inside patch cavity:

33


Thin Substrate Approximation

( ) ( )( )1 ˆ, ,zH x y z E x yjωµ

= ×∇

Note: The magnetic field is purely horizontal. (The mode is TMz.)

34

( ),H x y

h ( ),zE x y

z

Basic Principles of Operation Thin Substrate Approximation

On the edges of the patch:

ˆ 0sJ n⋅ =

ˆ 0botsJ n⋅ ≈

0bottH ≈

x

y

n

L

W sJ

t

On the bottom surface of the patch conductor, at the edge of the patch, we have:

(Js is the sum of the top and bottom surface currents.)

35

( )bot tops sJ J>>assuming

( )ˆbotsJ z H= − ×

Also,

h 0bot

tH ≈

Basic Principles of Operation Magnetic-wall Approximation

0 ( )tH = PMC

Since the magnetic field is approximately independent of z, we have an approximate PMC

condition on the entire vertical edge.

nh

PMC Model

36

( )ˆ , 0n H x y× =

or

PMC

h 0edge

tH ≈

Actual patch


x

y

n

L

W sJ

t

Hence,

0zEn

∂=

∂

( ) ( )( )1 ˆ, ,zH x y z E x yjωµ

= ×∇

( )ˆ , 0n H x y× =

( )( )ˆ ˆ , 0zn z E x y× ×∇ =

( )( )ˆˆ , 0zz n E x y⋅∇ =( )( ) ( )( ) ( )( )ˆ ˆ ˆˆ ˆ ˆ, , ,z z zn z E x y z n E x y E x y n z× ×∇ = ⋅∇ − ∇ ⋅

37

nh

PMC

(Neumann B.C.)


x

y

n

L

W sJ

t

2 2 0z zE k E∇ + =

cos coszm x n yE

L Wπ π =

2 22

1 0zm n k EL Wπ π − − + =

Hence 2 2

21 0m n k

L Wπ π − − + =

From separation of variables:

(TMmn mode) x

y

L

W PMC

( ),zE x y

38

Basic Principles of Operation Resonance Frequencies

We then have

1 0 rk k k ε= =

2 22

1m nkL Wπ π = +

1 0 0 0r rk k ε ω µ ε ε= =

Recall that

2 fω π=

Hence 2 2

2 r

c m nfL Wπ π

π ε = + 0 01/c µ ε=

39

We thus have

x

y

L

W PMC

( ),zE x y


2 2

2mnr

c m nfL Wπ π

π ε = +

Hence mnf f=

(resonance frequency of (m,n) mode)

40


x

y

L

W PMC

( ),zE x y

This mode is usually used because the radiation pattern has a broadside beam.

101

2 r

cfLε

=

coszxE

Lπ =

0

1ˆ sinsxJ x

j L Lπ π

ωµ − =

The resonant length L is about 0.5 guided wavelengths in the x direction.

x

y

L

W

Current

41

Basic Principles of Operation Dominant (1,0) mode

This structure operates as a “fat planar dipole.”

( ) 1ˆ, sin xH x y yj L L

π πωµ

= −

The resonance frequency is mainly controlled by the patch length L and the substrate permittivity.

Resonance Frequency of Dominant Mode

Note: A higher substrate permittivity allows for a smaller antenna (miniaturization) – but with a lower bandwidth.

Approximately, (assuming PMC walls)

This is equivalent to saying that the length L is one-half of a wavelength in the dielectric.

0 / 2/ 2dr

L λλε

= =1k L π=

2 22

1m nkL Wπ π = +

(1,0) mode:

42


The resonance frequency calculation can be improved by adding a “fringing length extension” ∆L to each edge of the patch to get an

“effective length” Le .

101

2 er

cfLε

=

2eL L L= + ∆

Note: Some authors use effective permittivity in this equation. (This would change the value of Le.)

43

Basic Principles of Operation Resonance Frequency of Dominant Mode

y

x L

Le

∆L ∆L

Hammerstad formula:

( )

( )

0.3 0.264/ 0.412

0.258 0.8

effr

effr

WhL h

Wh

ε

ε

+ + ∆ = − +

1/ 21 1 1 12

2 2eff r rr

hW

ε εε−

+ − = + +

44

Note: Even though the Hammerstad formula involves an effective permittivity, we still use

the actual substrate permittivity in the resonance frequency formula.

10122 r

cfL Lε

= + ∆

Basic Principles of Operation Resonance Frequency of Dominant Mode

Note: 0.5L h∆ ≈

This is a good “rule of thumb” to give a quick estimate.

45


Resonance Frequency of Dominant Mode

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07h / λ0

0.75

0.8

0.85

0.9

0.95

1N

OR

MA

LIZE

D F

REQ

UEN

CY Hammerstad

Measured

W/ L = 1.5

εr = 2.2 The resonance frequency has been normalized by the zero-order value (without fringing):

fN = f / f0

Results: Resonance Frequency

46

0/h λ



Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


47

General Characteristics

The bandwidth is directly proportional to substrate thickness h.

However, if h is greater than about 0.05 λ0 , the probe inductance (for a coaxial feed) becomes large enough so that matching is difficult – the bandwidth will decrease.

The bandwidth is inversely proportional to εr (a foam substrate gives a high bandwidth).

The bandwidth of a rectangular patch is proportional to the patch width W (but we need to keep W < 2L ; see the next slide).

Bandwidth

48

49

2 2

2mnr

c m nfL Wπ π

π ε = +

101

2 r

cfLε

=

022

2 r

cfWε

=

2W L<

Width Restriction for a Rectangular Patch

fc

f10 f01 f02

011

2 r

cfWε

=

W = 1.5 L is typical.

02 011 1

2r

cf fW Lε

− = −


L

W

Some Bandwidth Observations

For a typical substrate thickness (h /λ0 = 0.02), and a typical substrate permittivity (εr = 2.2) the bandwidth is about 3%.

By using a thick foam substrate, bandwidth of about 10% can be achieved.

By using special feeding techniques (aperture coupling) and stacked patches, bandwidths of 100% have been achieved.

50


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

h / λ0

0

5

10

15

20

25

30B

AN

DW

IDTH

(%) ε r

2.2

= 10.8

W/ L = 1.5 εr = 2.2 or 10.8

Results: Bandwidth

The discrete data points are measured values. The solid curves are from a CAD formula (given later).

51

0/h λ

10.8rε =

2.2


The resonant input resistance is fairly independent of the substrate thickness h unless h gets small (the variation is then mainly due to dielectric and conductor loss).

The resonant input resistance is proportional to εr.

The resonant input resistance is directly controlled by the location of the feed point (maximum at edges x = 0 or x = L, zero at center of patch).

Resonant Input Resistance

L

W (x0, y0)

L x

y

52


The patch is usually fed along the centerline (y0 = W / 2) to maintain symmetry and thus minimize excitation of undesirable modes

(which cause cross-pol).

Desired mode: (1,0)

L x

W

Feed: (x0, y0)

y

53

Resonant Input Resistance (cont.)


For a given mode, it can be shown that the resonant input resistance is proportional to the square of the cavity-mode field at the feed point.

( )20 0,in zR E x y∝

For (1,0) mode:

2 0cosinxRL

π ∝ L

x

W (x0, y0)

y

54


This is seen from the cavity-model eigenfunction analysis (please see the Appendix).


Hence, for (1,0) mode:

2 0cosin edgexR RL

π =

The value of Redge depends strongly on the substrate permittivity (it is proportional to the permittivity).

For a typical patch, it is often in the range of 100-200 Ohms.

55


L x

W (x0, y0)

y


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08h / λ0

0

50

100

150

200

INPU

T R

ESIS

TAN

CE

( Ω

)

2.2

r = 10.8ε

εr = 2.2 or 10.8 W/L = 1.5

x0 = L/4

Results: Resonant Input Resistance

The discrete data points are from a CAD formula (given later.)

L x

W

(x0, y0)

y

y0 = W/2

56

0/h λ

10.8rε =

2.2


Region where loss is important

Radiation Efficiency

The radiation efficiency is less than 100% due to

Conductor loss

Dielectric loss

Surface-wave excitation

Radiation efficiency is the ratio of power radiated into space, to the total input power.

rr

tot

PeP

=

57


58

Radiation Efficiency (cont.)


surface wave

TM0

cos (φ) pattern

x

y

Js

( )r r

rtot r c d sw

P PeP P P P P

= =+ + +

Pr = radiated power

Ptot = total input power

Pc = power dissipated by conductors

Pd = power dissipated by dielectric

Psw = power launched into surface wave

Hence,

59

General Characteristics Radiation Efficiency (cont.)

Conductor and dielectric loss is more important for thinner substrates (the Q of the cavity is higher, and thus more seriously affected by loss).

Conductor loss increases with frequency (proportional to f 1/2) due to the skin effect. It can be very serious at millimeter-wave frequencies.

Conductor loss is usually more important than dielectric loss for typical substrate thicknesses and loss tangents.

1 2sR δ

σδ ωµσ= =

Rs is the surface resistance of the metal. The skin depth of the metal is δ.

60

0

2sR fωµσ

= ∝

Some observations:

General Characteristics Radiation Efficiency (cont.)

Surface-wave power is more important for thicker substrates or for higher-substrate permittivities. (The surface-wave power can be minimized by using a thin substrate or a foam substrate.)

61

For a foam substrate, a high radiation efficiency is obtained by making the substrate thicker (minimizing the conductor and dielectric losses). There is no surface-wave power to worry about.

For a typical substrate such as εr = 2.2, the radiation efficiency is maximum for h / λ0 ≈ 0.02.


Radiation Efficiency (cont.)

εr = 2.2 or 10.8 W/L = 1.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1h / λ0

0

20

40

60

80

100EF

FIC

IEN

CY

(%)

exactCAD

Results: Efficiency (Conductor and dielectric losses are neglected.)

2.2

10.8

Note: CAD plot uses Pozar formula (given later). 62

10.8rε =

2.2

0/h λ


0 0.02 0.04 0.06 0.08 0.1h / λ0

0

20

40

60

80

100EF

FIC

IEN

CY

(%)

ε = 10.8

2.2

exactCAD

r

εr = 2.2 or 10.8 W/L = 1.5 63

7

tan 0.0013.0 10 [S/m]

δ

σ

=

= ×

Results: Efficiency (All losses are accounted for.)

0/h λ

10.8rε =

2.2


Note: CAD plot uses Pozar formula (given later).

64

General Characteristics Radiation Pattern

E-plane: co-pol is Eθ

H-plane: co-pol is Eφ

Note: For radiation patterns, it is usually more convenient to place the origin at the middle of the patch

(this keeps the formulas as simple as possible).

x

y

L

W E plane

H plane

Probe

Js

Comments on radiation patterns:

The E-plane pattern is typically broader than the H-plane pattern.

The truncation of the ground plane will cause edge diffraction, which tends to degrade the pattern by introducing:

Rippling in the forward direction

Back-radiation

65

Pattern distortion is more severe in the E-plane, due to the angle dependence of the vertical polarization Eθ on the ground plane.

(It varies as cos (φ)).

General Characteristics Radiation Patterns (cont.)

66

x

y

L

W E plane

H plane

Edge diffraction is the most serious in the E plane.

General Characteristics Radiation Patterns

Space wave

cosEθ φvaries as

Js

-90

-60

-30

0

30

60

90

120

150

180

210

240

-40

-30

-30

-20

-20

-10

-10

E-plane pattern

Red: infinite substrate and ground plane Blue: 1 meter ground plane

Note: The E-plane pattern “tucks in” and tends to zero at the horizon due to the presence of the infinite substrate.

67


Red: infinite substrate and ground plane Blue: 1 meter ground plane

-90

-45

0

45

90

135

180

225

-40

-30

-30

-20

-20

-10

-10

68

H-plane pattern


Directivity

The directivity is fairly insensitive to the substrate thickness.

The directivity is higher for lower permittivity, because the patch is larger.

69


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1h / λ0

0

2

4

6

8

10D

IREC

TIVI

TY (

dB)

exact

CAD

= 2.2

10.8

ε r

εr = 2.2 or 10.8 W/ L = 1.5

Results: Directivity (relative to isotropic)

70

0/h λ

2.2rε =

10.8



Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


71

CAD Formulas CAD formulas for the important properties of the

rectangular microstrip antenna will be shown.

72

D. R. Jackson, “Microstrip Antennas,” Chapter 7 of Antenna Engineering Handbook, J. L. Volakis, Editor, McGraw Hill, 2007.

D. R. Jackson, S. A. Long, J. T. Williams, and V. B. Davis, “Computer-Aided Design of Rectangular Microstrip Antennas,” Ch. 5 of Advances in Microstrip and Printed Antennas, K. F. Lee and W. Chen, Eds., John Wiley, 1997.

D. R. Jackson and N. G. Alexopoulos, “Simple Approximate Formulas for Input Resistance, Bandwidth, and Efficiency of a Resonant Rectangular Patch,” IEEE Trans. Antennas and Propagation, Vol. 39, pp. 407-410, March 1991.

Radiation efficiency Bandwidth (Q) Resonant input resistance Directivity

0 0 1 0

1 3 11/ 16 /

hedr

r avehed s rr d

eeR Le

h p c W hε

π η λ λ

= + +

where

tand δ= = loss tangent of substrate

12sR ωµ

σδ σ= = =surface resistance of metal

73

Note: “hed” refers to a unit-amplitude horizontal electric dipole.

CAD Formulas Radiation Efficiency

( ) / 2ave patch grounds s sR R R= +

Comment: The efficiency becomes small as the substrate gets thin.

where

Note: “hed” refers to a unit-amplitude horizontal electric dipole.

( ) ( )2 20 12

0

1 80hedspP k h cπ

λ=

1

1

hedsphed

r hedhed hedswsp swhed

sp

Pe

PP PP

= =+ +

( )3

3 302

0

1 160 1hedsw

r

P k h πλ ε

= −

74

CAD Formulas Radiation Efficiency (cont.)

Note: When we say “unit amplitude” here, we assume peak (not RMS) values.

( )3

01

1

3 1 11 14

hedr

r

e

k hc

πε

=

+ −

Hence, we have

Physically, this term is the radiation efficiency of a horizontal electric dipole (hed) on top of the substrate.

75


1 2

1 2 / 51r r

cε ε

= − +

( ) ( ) ( ) ( )

( ) ( )

2 4 2220 2 4 0 2 0

2 22 2 0 0

3 11 210 560 5

170

ap k W a a k W c k L

a c k W k L

= + + + +

+

The constants are defined as

2 0.16605a = −

4 0.00761a =

2 0.0914153c = −

76


Improved formula for HED surface-wave power (due to Pozar)

( )( ) ( )

3/22200 0

2 21 0 0 1

18 1 ( ) 1 1

rhedsw

r r

xkPx k h x x

εη

ε ε

−=

+ + − +

20

1 20

1

r

xxxε

−=

−

2 2 20 1 0 1 0

0 2 21

21 r r r

r

xε α α ε ε α α α

ε α− + + − +

= +−

77

D. M. Pozar, “Rigorous Closed-Form Expressions for the Surface-Wave Loss of Printed Antennas,” Electronics Letters, vol. 26, pp. 954-956, June 1990.

( )0 0tans k h sα = ( ) ( )( )0

1 0 20

1 tancos

k h sk h s

s k h sα

= − +

1rs ε= −

CAD Formulas

Note: The above formula for the surface-wave power is different from that given in Pozar’s paper by a factor of 2, since Pozar used RMS instead of peak values.

1

0 0 0

1 1 16 1/ 32

aves

d hedr r

R p c h WBWh L eπ η λ ε λ

= + +

BW is defined from the frequency limits f1 and f2 at which SWR = 2.0.

2 1

0

f fBWf−

= (multiply by 100 if you want to get %)

78

12

QBW

=

CAD Formulas Bandwidth

Comments: For a lossless patch, the bandwidth is approximately proportional to the patch width and

to the substrate thickness. It is inversely proportional to the substrate permittivity.

For very thin substrates the bandwidth will increase, but as the expense of efficiency.


79

CAD Formulas Quality Factor Q

1 1 1 1 1

d c sp swQ Q Q Q Q= + + +

0sUQ

Pω≡

0

1

s

PQ Uω

=

sU = energy stored in patch cavity

P = power that is radiated and dissipated by patch

d c sp swP P P P P= + + +

80

CAD Formulas Q Components

1 / tandQ δ=

0 0( )2c ave

s

k hQR

η =

1 0

3 116 /

rsp

LQpc W hε

λ ≈

1

hedr

sw sp hedr

eQ Qe

= − ( )

3

01

1

3 1 11 14

hedr

r

e

k hc

πε

=

+ −

The constants p and c1 were defined previously.


2 0cosmaxin edge

xR R RL

π = =

Probe-feed Patch

0

0

1

0 0 0

4

1 16 1/ 3

edges

d hedr r

L hW

RR p c W h

h L e

ηπ λ

π η λ ε λ

=

+ +

81

CAD Formulas Resonant Input Resistance

Comments: For a lossless patch, the resonant resistance is approximately independent of the substrate thickness. For a lossy patch it tends to zero as the

substrate gets very thin. For a lossless patch it is inversely proportional to the square of the patch width and it is proportional to the substrate permittivity.

( ) ( )tanc tan /x x x≡

where

( ) ( )( )212

1 1

3 tanctan

r

r

D k hpc k h

εε

= +

82

CAD Formulas Directivity

1 0 rk k ε=

The constants p and c1 were defined previously.

1

3Dp c

≈

For thin substrates:

(The directivity is essentially independent of the substrate thickness.)

83

CAD Formulas Directivity (cont.)


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


84

Radiation Pattern

85

h rεx

Patch

Probe

Coax feed

z

Note: The origin is placed at the center of the patch, at the top of the substrate, for the pattern calculations.

There are two models often used for calculating the radiation pattern:

Electric current model Magnetic current model

86

h rεx

Patch

Probe

Coax feed

Electric current model: We keep the physical currents flowing on the patch (and feed).

Radiation Pattern

patch top bots s sJ J J= +

h rεx

patchsJ

probesJ

87

h rεx

Patch

Probe

Coax feed

Magnetic current model: We apply the equivalence principle and invoke the (approximate) PMC condition at the edges.

ˆesM n E= − ×

Radiation Pattern

Equivalence surface

ˆ

ˆ

es

es

J n H

M n E

= ×

= − ×

h rεx

esM e

sM

The equivalent surface current is

approximately zero on the top surface (weak fields) and the sides (PMC).

We can ignore it on the ground plane (it does not radiate).

88

Theorem

The electric and magnetic models yield identical patterns at the resonance frequency of the cavity mode.

Assumption:

The electric and magnetic current models are based on the fields of a single cavity mode, corresponding to an ideal cavity with PMC walls.

D. R. Jackson and J. T. Williams, “A Comparison of CAD Models for Radiation from Rectangular Microstrip Patches,” Intl. Journal of Microwave and Millimeter-Wave Computer Aided Design, vol. 1, no. 2, pp. 236-248, April 1991.

Radiation Pattern

89

Comments on the Substrate Effects

The substrate can be neglected to simplify the far-field calculation.

When considering the substrate, it is most convenient to assume an infinite substrate (in order to obtain a closed-form solution).

Reciprocity can be used to calculate the far-field pattern of electric or magnetic current sources inside of an infinite layered structure.

When an infinite substrate is assumed, the far-field pattern always goes to zero at the horizon.


Radiation Pattern

90

Comments on the Two Models For the rectangular patch, the electric current model is the simplest since

there is only one electric surface current (as opposed to four edges).

For the rectangular patch, the magnetic current model allows us to classify the “radiating” and “nonradiating” edges.

For the circular patch, the magnetic current model is the simplest since there is only one edge (but the electric surface current is described by Bessel functions).

Radiation Pattern

On the nonradiating edges, the magnetic currents are in opposite

directions across the centerline (x = 0).

sinzxE

Lπ = −

ˆesM n E= − ×

L

x W

y “Radiating edges”

“Nonradiating edges”

esM

sJ

10ˆ cossxJ x A

Lπ =

(The formula is based on the electric current model.)

The origin is at the center of the patch.

L

rεh

Infinite ground plane and substrate

x

The probe is on the x axis. coss

πxˆJ xL

y

L

W E-plane

H-plane

x (1,0) mode

91

Radiation Pattern Rectangular Patch Pattern Formula

( ) 22

sin cos2 2( , , ) , ,2

2 2 2

y x

hexi i

y x

k W k LWLE r E r k W k L

πθ φ θ φπ

= −

0 sin cosxk k θ φ=

0 sin sinyk k θ φ=

The “hex” pattern is for a horizontal electric dipole in the x direction,

sitting on top of the substrate.

ori θ φ=

The far-field pattern can be determined by reciprocity.

92

x

y

L

W sJ

Radiation Pattern


( ) ( )( ) ( )

0

0

, , sin

, , cos

hex

hex

E r E F

E r E Gφ

θ

θ φ φ θ

θ φ φ θ

= −

=

where

( ) ( ) ( )( )( )( ) ( )

0

0

2 tan1

tan secTE k h N

Fk h N j N

θθ θ

θ θ θ= + Γ =

−

( ) ( )( ) ( )( )( )( ) ( )

0

0

2 tan coscos 1

tan cos

TM

r

k h NG

k h N jN

θ θθ θ θ εθ θ

θ

= + Γ =−

( ) ( )2sinrN θ ε θ= −

000 4

jk rjE er

ω µπ

− −=

93

Radiation Pattern

Note: To account for lossy substrate, use ( )1 tanr rc r jε ε ε δ→ = −


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


94

Input Impedance

95

Various models have been proposed over the years for calculating the input impedance of a microstrip patch antenna.

Transmission line model The first model introduced Very simple

Cavity model (eigenfunction expansion) Simple yet accurate for thin substrates Gives physical insight into operation

CAD circuit model Extremely simple and almost as accurate as the cavity model

Spectral-domain method More challenging to implement Accounts rigorously for both radiation and surface-wave excitation

Commercial software Very accurate Can be time consuming

96

Results for a typical patch show that the first three methods agree very well, provided the

correct Q is used and the probe inductance is accounted for.

2.2tan 0.001

1.524 mm

r

h

εδ

==

= 7

6.255 cm/ 1.5

3.0 10 S/m

LW Lσ

==

= ×

0

0

6.255 cm0

0.635mm

xya

==

=

Input Impedance Comparison of the Three Simplest Models

Circuit model of patch Transmission line model of patch

Cavity model (eigenfunction expansion) of patch

CAD Circuit Model for Input Impedance

97

The circuit model discussed assumes a probe feed. Other circuit models exist for other types of feeds.

Note: The mathematical justification of the CAD circuit model comes from the cavity-model eigenfunction analysis (see Appendix).

Input Impedance

The transmission-line model and the cavity model are discussed in the Appendix.

Near the resonance frequency, the patch cavity can be approximately modeled as a resonant RLC circuit.

The resistance R accounts for radiation and losses.

A probe inductance Lp is added in series, to account for the “probe inductance” of a probe feed.

Lp R C

L

Zin

Probe Patch cavity

98

Probe-fed Patch

Input Impedance

0

0

1in p

RZ j Lf fjQf f

ω≈ +

+ −

0

RQLω

= 12

BWQ

= BW is defined here by SWR < 2.0 when the RLC circuit is fed by a matched line (Z0 = R).

0 012 fLC

ω π= =

99

Lp R C

L

Zin

Input Impedance

in in inZ R jX= +

0

maxin in f f

R R R=

= =

R is the input resistance at the resonance of the patch cavity (the frequency that maximizes Rin).

100

2

0

0

1in

RRf fQf f

=

+ −

Input Impedance

Lp

R

C L

0f f=

maxinR

0f f=

(resonance of RLC circuit)

The input resistance is determined once we know four parameters:

101

f0: the resonance frequency of the patch cavity

R: the input resistance at the cavity resonance frequency f0

Q: the quality factor of the patch cavity

Lp: the probe inductance

(R, f0, Q)

Zin

CAD formulas for the first three

parameters have been given

earlier.

0

0

1in p

RZ j Lf fjQf f

ω≈ +

+ −

Input Impedance

Lp

R

C L

4 4.5 5 5.5 6FREQUENCY (GHz)

0

10

20

30

40

50

60

70

80R

in (

Ω )

CADexact

Results: Input Resistance vs. Frequency

εr = 2.2 W/L = 1.5 L = 3.0 cm

Frequency where the input resistance

is maximum (f0): Rin = R

102

Rectangular patch

Input Impedance

Note: “exact” means the cavity model will all infinite modes.

Results: Input Reactance vs. Frequency

εr = 2.2 W/L = 1.5

4 4.5 5 5.5 6FREQUENCY (GHz)

-40

-20

0

20

40

60

80X i

n ( Ω

)

CADexact

L = 3.0 cm

Frequency where the input resistance is maximum (f0)

Frequency where the input impedance is real

Shift due to probe reactance

103

Rectangular patch

Input Impedance

Xp

Note: “exact” means the cavity model will all infinite modes.

0.577216γ

( )( )

00

0

2ln2p

r

X k hk a

η γπ ε

= − +

(Euler’s constant)

Approximate CAD formula for probe (feed) reactance (in Ohms)

p pX Lω=

0 0 0/ 376.7303η µ ε= = Ω

a = probe radius h = probe height

This is based on an infinite parallel-plate model.

104

rε h2a

Input Impedance

( )( )

00

0

2ln2p

r

X k hk a

η γπ ε

= − +

Feed (probe) reactance increases proportionally with substrate thickness h.

Feed reactance increases for smaller probe radius.

105

If the substrate gets too thick, the probe reactance will make it difficult to get an input match, and the bandwidth will suffer.

(Compensating techniques will be discussed later.)

Important point:

Input Impedance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Xr

0

5

10

15

20

25

30

35

40

X f (

Ω)

CADexact

Results: Probe Reactance (Xf =Xp= ωLp)

xr = 2 ( x0 / L) - 1 The normalized feed location ratio xr is zero at the center of the patch (x = L/2), and is 1.0 at the patch edge (x = L).

εr = 2.2

W/L = 1.5

h = 0.0254 λ0

a = 0.5 mm

106

rxCenter Edge

Rectangular patch

Input Impedance

L

W (x0, y0)

L x

y

Note: “exact” means the cavity model with all infinite modes.

107

Design a probe-fed rectangular patch antenna on a substrate having a relative permittivity of 2.33 and a thickness of 62 mils (0.1575 cm). (Rogers RT Duroid 5870). Choose an aspect ratio of W / L = 1.5. The patch should resonate at the operating frequency of 1.575 GHz (the GPS L1 frequency). Ignore the probe inductance in your design, but account for fringing at the patch edges when you determine the dimensions. At the operating frequency the input impedance should be 50 Ω (ignoring the probe inductance). Assume an SMA connector is used to feed the patch along the centerline (at y = W / 2), and that the inner conductor of the SMA connector has a radius of 0.635 mm. The copper patch and ground plane have a conductivity of σ = 3.0 ×107 S/m and the dielectric substrate has a loss tangent of tanδ = 0.001. 1) Calculate the following:

The final patch dimensions L and W (in cm) The feed location x0 (distance of the feed from the closest patch edge, in cm) The bandwidth of the antenna (SWR < 2 definition, expressed in percent) The radiation efficiency of the antenna (accounting for conductor, dielectric, and surface-

wave loss, and expressed in percent) The probe reactance Xp at the operating frequency (in Ω) The expected complex input impedance (in Ω) at the operating frequency, accounting for the

probe inductance Directivity Gain

2) Plot the input impedance vs. frequency.

Design Example

108

Design Example

1) L = 6.07 cm, W = 9.11 cm 2) x0 = 1.82 cm 3) BW = 1.24% 4) er = 81.9% 5) Xp = 11.1 Ω 6) Zin = 50.0 + j(11.1) Ω 7) D = 5.85 (7.67 dB) 8) G = (D)(er) = 4.80 (6.81 dB)

Results from CAD formulas

x

y

L

W

Feed at (x0, y0)

Y0 = W/2

1.5 1.525 1.55 1.575 1.6 1.625 1.6520−

10−

0

10

20

30

40

50

6050.255

13.937−

Rin fghz( )

Xin fghz( )

1.651.5 fghz

109

f0 = 1.575 ×109 Hz R = 50 Ω Q = 56.8 Xp = 11.1 Ω

0

0

1in p

RZ jXf fjQf f

≈ +

+ −

Results from CAD formulas

Rin

Xin

f (GHz)

Ω

Design Example


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


110

Circular Polarization

Three main techniques:

1) Single feed with “nearly degenerate” eigenmodes (compact but small CP bandwidth).

2) Dual feed with delay line or 90o hybrid phase shifter (broader CP bandwidth but uses more space).

3) Synchronous subarray technique (produces high-quality CP due to cancellation effect, but requires even more space).

111

The techniques will be illustrated with a rectangular patch.

L

W

Basic principle: The two dominant modes (1,0) and (0,1) are excited with equal amplitude, but with a ±45o phase.

The feed is on the diagonal. The patch is nearly

(but not exactly) square.

L W≈

112

Circular Polarization Single Feed Method

(1,0)

(0,1)

Design equations:

2x y

CP

f ff

+=

112x CPf fQ

=

112y CPf fQ

= ±

Top sign for LHCP, bottom sign for RHCP.

in in x yZ R R R= = =

The resonant input resistance of the CP patch is the same as what a linearly-polarized patch fed at the same position would be.

L

W

x

y

12

BWQ

=

(SWR < 2 )

The optimum CP frequency is the average of the x and y resonance frequencies.

0 0x y=

The frequency f0 is also the resonance frequency:

113


(1,0)

(0,1)

Other Variations

Patch with slot Patch with truncated corners

Note: Diagonal modes are used as degenerate modes

L

L

x

y

L

L

x

y

114


115

1 (SWR 2)2

LPSWRBW

Q= <

2 (SWR 2)CPSWRBW

Q= < ( )0.348 AR 2 (3dB)CP

ARBWQ

= <

Here we compare bandwidths (impedance and axial-ratio):

Linearly-polarized (LP) patch:

Circularly-polarized (CP) single-feed patch:

The axial-ratio bandwidth is small when using the single-feed method.

W. L. Langston and D. R. Jackson, “Impedance, Axial-Ratio, and Receive-Power Bandwidths of Microstrip Antennas,” IEEE Trans. Antennas and Propagation, vol. 52, pp. 2769-2773, Oct. 2004.


Phase shift realized with delay line:

116

Circular Polarization Dual-Feed Method

L

L

x

y

P

P+λg/4

RHCP

Phase shift realized with 90o quadrature hybrid (branchline coupler)

117


This gives us a higher bandwidth than the simple power divider, but requires a load resistor.

λg/4

RHCP

λg/4

0Z0 / 2Z

Feed

50 Ohm load

0Z

0Z

Multiple elements are rotated in space and fed with phase shifts.

0o

-90o

-180o

-270o

Because of symmetry, radiation from higher-order modes (or probes) tends to be reduced, resulting in good cross-pol.

118

Circular Polarization Synchronous Rotation


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


119

Circular Patch

x

y

h

a

εr

120

a PMC

From separation of variables:

( ) ( )1cosz mE m J kφ ρ=

Jm = Bessel function of first kind, order m.

0z

a

E

ρρ =

∂=

∂( )1 0mJ k a′ =

121

Circular Patch Resonance Frequency

1 0 rk k ε=

1 mnk a x′=

a PMC

(nth root of Jm′ Bessel function)

2mn mnr

cf xπ ε

′=

122

This gives us

( )1 0mJ k a′ =


0 0

1cµ ε

=

123

n / m 0 1 2 3 4 5 1 3.832 1.841 3.054 4.201 5.317 5.416 2 7.016 5.331 6.706 8.015 9.282 10.520 3 10.173 8.536 9.969 11.346 12.682 13.987

Dominant mode: TM11

11 112 r

cf xaπ ε

′=11 1.841x′ ≈

Table of values for mnx′


124

Dominant mode: TM11 y

x

a

Circular patch Square patch

The circular patch is somewhat similar to a square patch.

W = L

x

y

Circular Patch

Fringing extension

11 112 e r

cf xaπ ε

′=

“Long/Shen Formula”:

a PMC

a + ∆a

ln 1.77262r

h aah

ππε

∆ ≈ + 21 ln 1.7726

2er

h aa aa h

ππ ε

= + + or

125

L. C. Shen, S. A. Long, M. Allerding, and M. Walton, "Resonant Frequency of a Circular Disk Printed-Circuit Antenna," IEEE Trans. Antennas and Propagation, vol. 25, pp. 595-596, July 1977.

Circular Patch

ea a a= + ∆

(The patterns are based on the magnetic current model.)

( ) ( )( )

1 1

1 1

1, cosz

J kE

J k a hρ

ρ φ φ =

(The edge voltage has a maximum of one volt.)

a

y x

E-plane

H-plane

In patch cavity:

The probe is on the x axis.

2a

rεh

Infinite GP and substrate

x

The origin is at the center of the patch.

126

Patterns Circular Patch

1 0 rk k ε=

( ) ( ) ( ) ( )01 1 0

0

, , 2 tanc cos sinRz

EE r a k h J k a Qθ θ φ π φ θ θη

′=

( ) ( ) ( ) ( )1 001

0 0

sin, , 2 tanc sin

sinR

z

J k aEE r a k h Pk aϕ

θθ φ π φ θ

η θ

= −

( ) ( )( ) ( )( )( ) ( )0

2cos 1 cos

tan secTE jN

Pk hN jN

θθ θ θ θ

θ θ θ

−= − Γ =

−

( ) ( ) ( )

( )( ) ( )0

2 cos1

tan cos

r

TM

r

jN

Qk h N j

N

ε θθ

θ θ εθ θθ

−

= − Γ =−

tanc tanx x / xwhere

( ) ( )2sinrN θ ε θ= −

127

Patterns

Circular Patch

Note: To account for lossy substrate, use ( )1 tanr r jε ε δ′= −

000 4

jk rjE er

ω µπ

− −=

( )( )

21 1 0

21 1

in edge

J kR R

J k aρ

≈

a

0ρ

128

Input Resistance

Circular Patch

1 0 rk k ε=

12edge r

sp

R eP

=

( ) ( )( )

( ) ( ) ( ) ( )

/ 22 2

0 00 0

2 22 21 0 0

tanc8

sin sin sin

sp

inc

P k a k hN

Q J k a P J k a d

ππ θη

θ θ θ θ θ θ

=

′⋅ +

∫

( ) ( )1 /incJ x J x x=

where

Psp = power radiated into space by circular patch with maximum edge voltage of one volt.

er = radiation efficiency

129

Input Resistance (cont.)

Circular Patch

20

0

( )8sp cP k a Iπη

=

CAD Formula:

43c cI p= ( )

62

0 20

kc k

kp k a e

=

= ∑

0

2

43

64

85

107

12

10.400000

0.0785710

7.27509 10

3.81786 10

1.09839 10

1.47731 10

eeeee

ee

−

−

−

−

== −=

= − ×

= ×

= − ×

= ×

130

Circular Patch Input Resistance (cont.)


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


131

Improving Bandwidth

Some of the techniques that have been successfully developed are illustrated here.

132

The literature may be consulted for additional designs and variations.

L-shaped probe:

Capacitive “top hat” on probe:

Top view

133

Improving Bandwidth Probe Compensation

As the substrate thickness increases the probe inductance limits the bandwidth – so we

compensate for it.

SSFIP: Strip Slot Foam Inverted Patch (a version of the ACP).

Microstrip substrate

Patch

Microstrip line Slot

Foam

Patch substrate

Bandwidths greater than 25% have been achieved.

Increased bandwidth is due to the thick foam substrate and also a dual-tuned resonance (patch+slot).

134

Improving Bandwidth

Note: There is no probe inductance to worry about here.

J.-F. Zürcher and F. E. Gardiol, Broadband Patch Antennas, Artech House, Norwood, MA, 1995.

Bandwidth increase is due to thick low-permittivity antenna substrates and a dual or triple-tuned resonance.

Bandwidths of 25% have been achieved using a probe feed.

Bandwidths of 100% have been achieved using an ACP feed.

Microstrip substrate

Coupling patch

Microstrip line Slot

Patch substrates

Top Patch

135

Improving Bandwidth Stacked Patches

Bandwidth (S11 = -10 dB) is about 100%

Stacked patch with ACP feed 3 4 5 6 7 8 9 10 11 12

Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

Ret

urn

Loss

(dB

)

MeasuredComputed

136

Improving Bandwidth Stacked Patches


Stacked patch with ACP feed

0

0.2

0.5 1 2 5 1018

017

016

015

014

0

130120

110 100 90 80 7060

50

4030

2010

0-10

-20-30

-40

-50-60

-70-80-90-100-110-120-130

-140

-150

-160

-170

4 GHz

13 GHz

Two extra loops are observed on the Smith chart.

137

Stacked Patches Improving Bandwidth


Radiating Edges Gap Coupled Microstrip Antennas

(REGCOMA).

Non-Radiating Edges Gap Coupled Microstrip Antennas

(NEGCOMA)

Four-Edges Gap Coupled Microstrip Antennas

(FEGCOMA)

Bandwidth improvement factor: REGCOMA: 3.0, NEGCOMA: 3.0, FEGCOMA: 5.0?

Mush of this work was pioneered by

K. C. Gupta.

138

Improving Bandwidth Parasitic Patches

Radiating Edges Direct Coupled Microstrip Antennas

(REDCOMA).

Non-Radiating Edges Direct Coupled Microstrip Antennas

(NEDCOMA)

Four-Edges Direct Coupled Microstrip Antennas

(FEDCOMA)

Bandwidth improvement factor: REDCOMA: 5.0, NEDCOMA: 5.0, FEDCOMA: 7.0

139

Improving Bandwidth Direct-Coupled Patches

The introduction of a U-shaped slot can give a significant bandwidth (10%-40%).

(This is due to a double resonance effect, with two different modes.)

“Single Layer Single Patch Wideband Microstrip Antenna,” T. Huynh and K. F. Lee, Electronics Letters, Vol. 31, No. 16, pp. 1310-1312, 1986.

140

Improving Bandwidth U-Shaped Slot

A 44% bandwidth was achieved.

Y. X. Guo, K. M. Luk, and Y. L. Chow, “Double U-Slot Rectangular Patch Antenna,” Electronics Letters, Vol. 34, No. 19, pp. 1805-1806, 1998.

141

Improving Bandwidth Double U-Slot

A modification of the U-slot patch.

A bandwidth of 34% was achieved (40% using a capacitive “washer” to compensate for the probe inductance).

B. L. Ooi and Q. Shen, “A Novel E-shaped Broadband Microstrip Patch Antenna,” Microwave and Optical Technology Letters, vol. 27, No. 5, pp. 348-352, 2000.

142

Improving Bandwidth E Patch

Multi-Band Antennas

General Principle:

Introduce multiple resonance paths into the antenna.

A multi-band antenna is sometimes more desirable than a broadband antenna, if multiple narrow-band channels are to be covered.

143

Dual-band E patch

High-band

Low-band

Low-band

Feed

Dual-band patch with parasitic strip

Low-band

High-band

Feed

144

Multi-Band Antennas


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


145

Miniaturization

• High Permittivity • Quarter-Wave Patch • PIFA • Capacitive Loading • Slots • Meandering

Note: Miniaturization usually comes at a price of reduced bandwidth!

146

Usually, bandwidth is proportional to the volume of the patch cavity, as we will see in the examples.

The smaller patch has about one-fourth the bandwidth of the original patch.

L

W

1rε =

(Bandwidth is inversely proportional to the permittivity.)

147

4rε =

/ 2L L′ =

/ 2W W′ =

Size reduction

Miniaturization High Permittivity

(Same aspect ratio)

L

W

Ez = 0

The new patch has about one-half the bandwidth of the original patch.

0s

r

UQP

ω=Neglecting losses: / 2s sU U′ =

/ 4r rP P′ =2Q Q′ =

Short-circuit vias

148

W

/ 2L L′ =

Note: 1/2 of the radiating magnetic current

Miniaturization Quarter-Wave patch

The new patch has about one-half the bandwidth of the original quarter-wave patch, and hence one-fourth the bandwidth of the regular patch.

(Bandwidth is proportional to the patch width.)

149

/ 2W W′ =

A quarter-wave patch with the same aspect ratio W/L as the original patch

Width reduction

L

W

Ez = 0 Short-circuit vias

/ 2L L′ =

W ′W

/ 2L L′ =

Miniaturization Smaller Quarter-Wave patch

Fewer vias actually gives more miniaturization!

(The edge has a larger inductive impedance: explained on the next slide.)

150

L L′′ ′<

W

L′′

W

/ 2L L′ =

Use fewer vias

Miniaturization Quarter-Wave Patch with Fewer Vias

151

Short Open

Inductance

The Smith chart provides a simple explanation for the length reduction.

Miniaturization Quarter-Wave Patch with Fewer Vias

A single shorting strip or via is used.

This antenna can be viewed as a limiting case of the via-loaded patch, or as an LC resonator.

Feed Shorting strip or via Top view

152

Miniaturization Planar Inverted F (PIFA)

The capacitive loading allows for the length of the PIFA to be reduced.

Feed Shorting plate Top view

153

01LC

ω =

Miniaturization PIFA with Capacitive Loading

The patch has a monopole-like pattern

Feed c

b

Patch Metal vias

2a

(Hao Xu Ph.D. dissertation, University of Houston, 2006)

The patch operates in the (0,0) mode, as an LC resonator

154

Miniaturization Circular Patch Loaded with Vias

Example: Circular Patch Loaded with 2 Vias

Unloaded: resonance frequency = 5.32 GHz.

0.5 1 1.5 2 2.5

Frequency [GHz]

-20

-15

-10

-5

0

S11[

db]

0

45

90

135

180

225

270

315

-40

-30

-30

-20

-20

-10

-10

E-thetaE-phi

(Miniaturization factor = 4.8) 155

Miniaturization Circular Patch Loaded with Vias

The slot forces the current to flow through a longer path, increasing the effective dimensions of the patch.

Top view

Linear CP

0o ±90o

156

Miniaturization Slotted Patch

Meandering forces the current to flow through a longer path, increasing the effective dimensions of the patch.

Meandering also increases the capacitance of the PIFA line.

Feed

Via

Meandered quarter-wave patch

Feed

Via

Meandered PIFA

157

Miniaturization Meandering


Feeding methods



CAD Formulas

Radiation pattern

Input Impedance


Circular patch

Improving bandwidth

Miniaturization


158

Reducing Surface and Lateral Waves Reduced Surface Wave (RSW) Antenna

SIDE VIEW

z

b

h x

Shorted annular ring

Ground plane Feed

a ρ0

TOP VIEW

a

ρ o

b

Feed

D. R. Jackson, J. T. Williams, A. K. Bhattacharyya, R. Smith, S. J. Buchheit, and S. A. Long, “Microstrip Patch Designs that do Not Excite Surface Waves,” IEEE Trans. Antennas Propagat., vol. 41, No 8, pp. 1026-1037, August 1993.

159

Reducing surface-wave excitation and lateral radiation reduces edge diffraction.

Space-wave radiation (desired)

Lateral radiation (undesired)

Surface waves (undesired)

Diffracted field at edge

160

Reducing Surface and Lateral Waves

0 coszVEh

φ= −

0 cossVMhφ φ= −

( )ˆˆ ˆs zM n E zEρ= − × = − ×

TM11 mode:

( ) ( ) ( )0 1 11 1

1, coszE V J khJ k a

ρ φ φ ρ −

=

At edge:

( ) ( ),s zM E aφ φ φ=

a x

y

sM φ

161

Reducing Surface and Lateral Waves Principle of Operation

1 0 rk k k ε= =

0 cossVMhφ φ= −

Surface-Wave Excitation:

( ) ( )0 0

0 0

21cos zTM jk z

z TM TME A H eφ β ρ −=

( )0 01TM TMA AJ aβ′=(z > h)

( )01 0TMJ aβ′ =Set

162

a x

y

sM φ

Substrate


0 1TM na xβ ′=

11 1.841x′ ≈For TM11 mode:

01.841TM aβ =

Patch resonance: 1 1.841k a =

Note: 0 1TM kβ < The RSW patch is too big to be resonant.

163

Hence

a x

y

sM φ

Substrate


01.841TM bβ =

The radius a is chosen to make the patch resonant:

( )( )

0

0

1 111

1 1

1 1 1 111

TM

TM

k xJJ k aY k a k xY

β

β

′′ = ′′

164


SIDE VIEW

z

b

h x

Shorted annular ring

Ground plane Feed

a ρ0

TOP VIEW

a

ρ o

b

Feed

0 cossVMhφ φ= −

Space-Wave Field: 01cos jkSP

z SPE A e ρφρ

− =

( )1 0SPA CJ k a′=

(z = h)

( )1 0 0J k a′ =Set

Assume no substrate outside of patch (or very thin substrate):

165

a x

y

sM φ

Ground plane

Reducing Surface and Lateral Waves Reducing the Lateral Radiation

0 1.841k a =

For a thin substrate:

0 0TM kβ ≈

The same design reduces both surface-wave fields and

lateral-radiation fields.

166

a x

y

sM φ

Substrate


Note: The diameter of the RSW antenna is found from

0 1.841k a =

0

2 0.586aλ

=Note: The size is approximately independent of the

permittivity (the patch cannot be miniaturized by choosing a higher permittivity!).

-90

-60

-30

0

30

60

90

120

150

180

210

240

-40

-30

-30

-20

-20

-10

-10-90

-60

-30

0

30

60

90

120

150

180

210

240

-40

-30

-30

-20

-20

-10

-10

Conventional RSW

Measurements were taken on a 1 m diameter circular ground plane at 1.575 GHz. E-plane Radiation Patterns

Measurement Theory (infinite GP)

167


Reducing surface-wave excitation and lateral radiation reduces mutual coupling.

Space-wave radiation

Lateral radiation

Surface waves

168


Reducing surface-wave excitation and lateral radiation reduces mutual coupling.

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 1 2 3 4 5 6 7 8 9 10

Separation [Wavelengths]

S 12 [

dB]

RSW - MeasuredRSW - TheoryConv - MeasuredConv - Theory

“Mutual Coupling Between Reduced Surface-Wave Microstrip Antennas,” M. A. Khayat, J. T. Williams, D. R. Jackson, and S. A. Long, IEEE Trans. Antennas and Propagation, Vol. 48, pp. 1581-1593, Oct. 2000.

E-plane

169


1/r

1/r3

References General references about microstrip antennas:

Microstrip Antenna Design Handbook, R. Garg, P. Bhartia, I. J. Bahl, and A. Ittipiboon, Editors, Artech House, 2001.

Microstrip Patch Antennas: A Designer’s Guide, R. B. Waterhouse, Kluwer Academic Publishers, 2003.

Advances in Microstrip and Printed Antennas, K. F. Lee, Editor, John Wiley, 1997.

Microstrip Patch Antennas, K. F. Fong Lee and K. M. Luk, Imperial College Press, 2011. Microstrip and Patch Antennas Design, 2nd Ed., R. Bancroft, Scitech Publishing, 2009.

170

References (cont.)

General references about microstrip antennas (cont.):

Millimeter-Wave Microstrip and Printed Circuit Antennas, P. Bhartia, Artech House, 1991.

The Handbook of Microstrip Antennas (two volume set), J. R. James and P. S. Hall, INSPEC, 1989.

Microstrip Antenna Theory and Design, J. R. James, P. S. Hall, and C. Wood, INSPEC/IEE, 1981.

Microstrip Antennas: The Analysis and Design of Microstrip Antennas and Arrays, D. M. Pozar and D. H. Schaubert, Editors, Wiley/IEEE Press, 1995.

CAD of Microstrip Antennas for Wireless Applications, R. A. Sainati, Artech House, 1996.

171

Computer-Aided Design of Rectangular Microstrip Antennas, D. R. Jackson, S. A. Long, J. T. Williams, and V. B. Davis, Ch. 5 of Advances in Microstrip and Printed Antennas, K. F. Lee, Editor, John Wiley, 1997.

More information about the CAD formulas presented here for the rectangular patch may be found in:

References (cont.)

Microstrip Antennas, D. R. Jackson, Ch. 7 of Antenna Engineering Handbook, J. L. Volakis, Editor, McGraw Hill, 2007.

172

References devoted to broadband microstrip antennas:

Compact and Broadband Microstrip Antennas, K.-L. Wong, John Wiley, 2003.

Broadband Microstrip Antennas, G. Kumar and K. P. Ray, Artech House, 2002.

Broadband Patch Antennas, J.-F. Zürcher and F. E. Gardiol, Artech House, 1995.

References (cont.)

173

174

175

Transmission line model

Cavity model (eigenfunction expansion)

Transmission Line Model for Input Impedance

176

The model accounts for the probe feed to improve accuracy.

The model assumes a rectangular patch (it is difficult to extend to other shapes).

Input Impedance

We think of the patch as a wide transmission line resonator (length L).

0 0

0 0

e

e

x x Ly y W

= + ∆

= + ∆

0eff

e rck k ε=

Denote

( )1effrc r effj lε ε ′= −

1 1 1 1 1eff

d c sp sw

lQ Q Q Q Q

= = + + +

Note: ∆L is from Hammerstad’s formula ∆W is from Wheeler’s formula

Physical patch dimensions (W × L)

177

A CAD formula for Q has been given earlier.

Input Impedance

Le

We

x

y

0 0( , )e ex yPMC

sJ

effrcε

178

ln 4W hπ

∆ =

0.2620.3000.4120.258 0.813

effreffr

WhL h Wh

εε

+ +∆ = − +

(Hammerstad formula)

1 1 12 2

1 12

eff r rr

hW

ε εε + − = + +

(Wheeler formula)

Commonly used fringing formulas

Input Impedance

179

where

TLin in pZ Z jX= +

0.57722( )γ Euler's constant

( )( )( )

0 0

0 0

1 / tan

tan

TL TL eff eff ein in c c

eff eff ec c e

Y Z jY k x

jY k L x

= =

+ −

0 ,eff effc cZ k

0x = ex L=

pL

0ex x=

p pX Lω=

Zin

0 01 /eff effc cY Z=

( )( )

00

0

2ln2p

r

X k hk a

η γπ ε

= − +

Input Impedance

(from a parallel-plate model of probe inductance)

0 01eff eff

c c effe erc

h hZW W

η ηε

= =

Cavity Model

180

It is a very efficient method for calculating the input impedance.

It gives a lot of physical insight into the operation of the patch.

The method is extendable to other patch shapes.

Input Impedance

Here we use the cavity model to solve for the input impedance of the rectangular patch antenna.

Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Experiment on Microstrip Antennas,” IEEE Trans. Antennas Propagat., vol. AP-27, no. 3, pp. 137-145, March 1979.

0 0

0 0

e

e

x x Ly y W

= + ∆

= + ∆

0eff

e rck k ε=

Denote


1 1 1 1eff

d c sp sw

lQ Q Q Q

= + + +

Note: ∆L is from Hammerstad’s formula ∆W is from Wheeler’s formula

Le

We

x

y

0 0( , )e ex y PMC

Physical patch dimensions (W × L)

181

Input Impedance

A CAD formula for Q has been given earlier.

effrcε

182

ln 4W hπ

∆ =

0.2620.3000.4120.258 0.813

effreffr

WhL h Wh

εε

+ +∆ = − +

(Hammerstad formula)

1 1 12 2

1 12

eff r rr

hW

ε εε + − = + +

(Wheeler formula)

Commonly used fringing formulas

Input Impedance

Next, we derive the Helmholtz equation for Ez.

Substituting Faradays law into Ampere’s law, we have:

i effcH J j E

E j Hωε

ωµ∇× = +∇× = −

( )

( )( )

2

2 2

2 2

1 i effc

ie

ie

ie

E J j Ej

E j J k E

E E j J k E

E k E j J

ωεωµ

ωµ

ωµ

ωµ

− ∇× ∇× = +

∇× ∇× = − +

∇ ∇ ⋅ − ∇ = − +

∇ + =

183

Input Impedance

(Ampere’s law)

(Faraday’s law)

Hence

Denote

2 2 iz e z zE k E j Jωµ∇ + =

( , ) ( , )zx y E x yψ =

where

( , ) ( , )izf x y j J x yωµ=

Then

184

2 2 ( , )ek f x yψ ψ∇ + =

(scalar Helmholtz equation)

Input Impedance

Introduce “eigenfunctions” ψmn (x,y):

For rectangular patch we have, from separation of variables,

2 2( , ) ( , )mn mn mnx y x yψ λ ψ∇ = −

0mnCn

ψ∂=

∂

2 22

( , ) cos cosmne e

mne e

m x n yx yL W

m nL W

π πψ

π πλ

=

= +

185

Input Impedance

Assume an “eigenfunction expansion”:

Hence

,( , ) ( , )mn mn

m nx y A x yψ ψ= ∑

2 2 ( , )ek f x yψ ψ∇ + =

2 2

, ,( , )mn mn e mn mn

m n m nA k A f x yψ ψ∇ + =∑ ∑

Using the properties of the eigenfunctions, we have

( )2 2

,( , ) ( , )mn e mn mn

m nA k x y f x yλ ψ− =∑

This must satisfy

186

Input Impedance

Note that the eigenfunctions are orthogonal, so that

Denote

Next, we multiply by and integrate. ( , )m n x yψ ′ ′

( , ) ( , ) 0, ( , ) ( , )mn m nS

x y x y dS m n m nψ ψ ′ ′ ′ ′= ≠∫

2, ( , )mn mn mnS

x y dSψ ψ ψ< > = ∫

( )2 2 , ,mn e mn mn mn mnA k fλ ψ ψ ψ− < > = < >

We then have

187

Input Impedance

Hence, we have

For the patch problem we then have

2 2

, 1,

mnmn

mn mn e mn

fAk

ψψ ψ λ

< >= < > −

2 2

, 1,

iz mn

mnmn mn e mn

JA jk

ψωµψ ψ λ

< >= < > −

The field inside the patch cavity is then given by

,( , ) ( , )z mn mn

m nE x y A x yψ= ∑

188

Input Impedance

*

*

*

,

*

,

1 ( , )21 ( , )2121 ,2

iin z z

V

iz z

S

imn mn z

m nS

imn mn z

m n

P E x y J dV

h E x y J dS

h A J dS

h A J

ψ

ψ

= −

= −

= −

= − < >

∫

∫

∑∫

∑

To calculate the input impedance, we first calculate the complex power going into the patch as

189

Input Impedance

*

,

*2 2

,

2

2 2,

1 ,2

1 , 1 ,2 ,

,1 12 ,

iin mn mn z

m n

iimn z

mn zm n mn mn e mn

imn z

m n mn mn e mn

P h A J

Jh j Jk

Jh j

k

ψ

ψωµ ψψ ψ λ

ψωµ

ψ ψ λ

= − < >

< >= − < > < > −

< > = − < > −

∑

∑

∑

Hence

Also, 21

2in in inP Z I=

so 2

2 inin

in

PZI

=

190

Input Impedance

Hence we have

where

2

2 2 2,

,1 1,

imn z

inm n mn mn e mnin

JZ j h

kI

ψωµ

ψ ψ λ

< > = − < > − ∑

, 0 0m n m n

∞ ∞

= =

=∑ ∑ ∑

191

Input Impedance

Rectangular patch:

where

2 22

0

cos cosmne e

mne e

effe rc

m x n yL W

m nL W

k k

π πψ

π πλ

ε

=

= +

=


2 2

0 0

, cos cose eL W

mn mne e

m x n ydx dyL Wπ πψ ψ

=

∫ ∫

192

Input Impedance

so

To calculate , assume a strip model as shown below.

( )( )0 0, 1 12 2

e emn mn m n

W Lψ ψ δ δ = + +

0

1, 00, 0m

mm

δ=

= ≠

, imn zJψ

pa

0 0( , )e ex y

193

0 0( , )e ex ypW

Actual probe Strip model

Input Impedance

( )0 02

2

0

, ,2 2

2

4

p pe einsz

p e

p p

W WIJ y y yW

y y

W a

π

= ∈ − +

− −

=

For a “Maxwell” strip current assumption, we have:

Note: The total probe current is Iin.

194

0 0( , )e ex ypW

Input Impedance

0 0

32

, ,2 2

4.482

p pe einsz

p

p p p

W WIJ y y yW

W a e a

= ∈ − +

=

For a uniform strip current assumption, we have:

Note: The total probe current is Iin.

195

0 0( , )e ex ypW

D. M. Pozar, “Improved Computational Efficiency for the Moment Method Solution of Printed Dipoles and Patches,” Electromagnetics, Vol. 3, pp. 299-309, July-Dec. 1983.

Input Impedance

( ) ( )

0

0

20

2

20

0

2

20 0 0

2

1, cos cos

cos cos '

cos cos cos ' sin sin

pe

pe

p

p

p

p

Wy

ei

mn z inW e e p

y

We

ein

Wp e e

We e e

in

Wp e e e

m x n yJ I dyL W W

I m x n y y dyW L W

I m x n y n yy y dW L W W

π πψ

π π

π π π

+

−

+

−

+

−

=

′ = +

′ ′= −

∫

∫

∫

0 0cos cos sinc2

e epin

pp e e e

y

n WI m x n y WW L W W

ππ π =

Assume a uniform strip current model (the two models gives almost identical results):

0ey y y′= +

Use

Integrates to zero

196

Input Impedance

Hence

0 0, cos cos sinc2

e epi

mn z ine e e

n Wm x n yJ IL W W

ππ πψ

=

sinc2

p

e

n WW

π

Note: It is the term that causes the series for Zin to converge.

We cannot assume a prove of zero radius, or else the series will not converge – the input reactance will be infinite!

197

Input Impedance

2

2 2 2,

,1 1,

imn z

inm n mn mn e mnin

JZ j h

kI

ψωµ

ψ ψ λ

< > = − < > − ∑

198

Summary of Cavity Model

0 0, cos cos sinc2

e epi

mn z ine e e

n Wm x n yJ IL W W

ππ πψ

=

( )( )0 0, 1 12 2

e emn mn m n

W Lψ ψ δ δ = + +

2 22mn

e e

m nL Wπ πλ

= +

0eff

e rck k ε=

Input Impedance

199

Summary (cont.)


1 1 1 1 1eff

d c sp sw

lQ Q Q Q Q

= = + + +

A CAD formula for Q has already been given.

Input Impedance

Hence

Note that

(1,0) = term that corresponds to dominant patch mode (which corresponds to the RLC circuit).

2

2 2 2( , )

(1,0)

,1 1,

imn z

pm n mn mn e mnin

JjX j h

kI

ψωµ

ψ ψ λ≠

= − −

∑

2

2 2 2,

,1 1,

imn z

inm n mn mn e mnin

JZ j h

kI

ψωµ

ψ ψ λ

< > = − < > − ∑

200

Input Impedance Probe Inductance

where

,

,

m nin in

m nZ Z= ∑

The input impedance is in the form

201

where , 0 0m n m n

∞ ∞

= =

=∑ ∑ ∑

2

,2 2 2

,1 1,

imn zm n

inmn mn e mnin

JZ j h

kI

ψωµ

ψ ψ λ

< > = − < > −

After some algebra, we can write this as follows:

Input Impedance RLC Model

202

,

1m n mnin

rmn rmnrmn

RZf jQ f

f

=

+ −

2

1

eff

rmnmn

mnmn mn

mn eff

Ql

fff

PRk l

ω

=

=

=

where

2

2

,1,

imn z

mnmn mnin

JP h

I

ψµ

ψ ψ=

and

Input Impedance

2 1rmnf ≈For , we have

,

11

m n mnin

rmnrmn

RZjQ f

f

≈

+ −

This is the mathematical form of the input impedance of an RLC circuit.

203

Input Impedance

Circuit model

Note: This circuit model is accurate as long as we are near the resonance of any particular RLC circuit.

204

Note: The (0,0) mode has a uniform electric field, and hence no magnetic field. It also has negligible radiation (compared to the (1,0) mode).

(0,0)(1,0) (0,1)

Zin

Input Impedance

Approximately, 0 ( , ) (1,0)mnR m n≈ ≠

205

(1,0)

pjX

Zin

Hence, an approximate circuit model for operation near the resonance frequency of the (1,0) mode is

Input Impedance RLC Model

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Introduction to Microstrip Antennas.pdf