introduction to microstrip antennas.pdf
TRANSCRIPT
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Introduction to Microstrip Antennas David R. Jackson
Dept. of ECE University of Houston
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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]
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Contact Information
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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).
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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).
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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.
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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
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Notation
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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)
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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
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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.
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Microstrip line feed Coax feed
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Overview of Microstrip Antennas
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Common Shapes
Rectangular Square Circular
Elliptical
Annular ring
Triangular
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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.
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Overview of Microstrip Antennas History
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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).
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Overview of Microstrip Antennas
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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).
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Overview of Microstrip Antennas
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Applications
Satellite communications
Microwave communications
Cell phone antennas
GPS antennas
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Applications include:
Overview of Microstrip Antennas
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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
Overview of Microstrip Antennas
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Overview of Microstrip Antennas
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Arrays
Linear array (1-D corporate feed)
2×2 array
2-D 8X8 corporate-fed array 4 × 8 corporate-fed / series-fed array
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Wraparound Array (conformal)
(Photo courtesy of Dr. Rodney B. Waterhouse)
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Overview of Microstrip Antennas
The substrate is so thin that it can be bent to “conform” to the surface.
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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
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Overview of Microstrip Antennas Rectangular patch
W = 1.5L is typical.
Js
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Circular Patch
x
y
h
a
εr
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Overview of Microstrip Antennas
The location of the feed determines the direction of current flow and hence the polarization of the radiated field.
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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
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Feeding Methods
Some of the more common methods for feeding microstrip antennas are shown.
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The feeding methods are illustrated for a rectangular patch, but the principles apply for circular and other shapes as well.
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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
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x rε h
z
Feeding Methods
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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.)
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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
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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
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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
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Feeding Methods Proximity-coupled Feed
(Electromagnetically-coupled Feed)
Top view Microstrip line
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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
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Feeding Methods Gap-coupled Feed
Top view Microstrip line
Patch
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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
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Feeding Methods Aperture-coupled Patch (ACP)
Top view
Slot
Microstrip line
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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
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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.
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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
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Basic Principles of Operation
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.
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Main Ideas:
(See next slide.)
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Basic Principles of Operation
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).
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A microstrip antenna can radiate well, even with a thin substrate.
x rε hsJ
z 1sJ
h∝
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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≈
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h ( ),zE x y
z
Basic Principles of Operation Thin Substrate Approximation
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( )( )
( )( )
1
1 ˆ ,
1 ˆ ,
z
z
H Ej
zE x yj
z E x yj
ωµ
ωµ
ωµ
= − ∇×
= − ∇×
= − − ×∇
Magnetic field inside patch cavity:
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Basic Principles of Operation
Thin Substrate Approximation
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( ) ( )( )1 ˆ, ,zH x y z E x yjωµ
= ×∇
Note: The magnetic field is purely horizontal. (The mode is TMz.)
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( ),H x y
h ( ),zE x y
z
Basic Principles of Operation Thin Substrate Approximation
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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.)
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( )bot tops sJ J>>assuming
( )ˆbotsJ z H= − ×
Also,
h 0bot
tH ≈
Basic Principles of Operation Magnetic-wall Approximation
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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
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( )ˆ , 0n H x y× =
or
PMC
h 0edge
tH ≈
Actual patch
Basic Principles of Operation Magnetic-wall Approximation
x
y
n
L
W sJ
t
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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× ×∇ = ⋅∇ − ∇ ⋅
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nh
PMC
(Neumann B.C.)
Basic Principles of Operation Magnetic-wall Approximation
x
y
n
L
W sJ
t
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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
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Basic Principles of Operation Resonance Frequencies
We then have
1 0 rk k k ε= =
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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=
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We thus have
x
y
L
W PMC
( ),zE x y
Basic Principles of Operation Resonance Frequencies
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2 2
2mnr
c m nfL Wπ π
π ε = +
Hence mnf f=
(resonance frequency of (m,n) mode)
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Basic Principles of Operation Resonance Frequencies
x
y
L
W PMC
( ),zE x y
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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
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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
= −
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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:
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Basic Principles of Operation
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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.)
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Basic Principles of Operation Resonance Frequency of Dominant Mode
y
x L
Le
∆L ∆L
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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−
+ − = + +
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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
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Note: 0.5L h∆ ≈
This is a good “rule of thumb” to give a quick estimate.
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Basic Principles of Operation
Resonance Frequency of Dominant Mode
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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
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0/h λ
Basic Principles of Operation
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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
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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
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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
− = −
General Characteristics
L
W
![Page 50: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/50.jpg)
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
General Characteristics
![Page 51: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/51.jpg)
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
General Characteristics
![Page 52: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/52.jpg)
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
General Characteristics
![Page 53: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/53.jpg)
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.)
General Characteristics
![Page 54: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/54.jpg)
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
General Characteristics
This is seen from the cavity-model eigenfunction analysis (please see the Appendix).
Resonant Input Resistance (cont.)
![Page 55: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/55.jpg)
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
General Characteristics
L x
W (x0, y0)
y
Resonant Input Resistance (cont.)
![Page 56: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/56.jpg)
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
General Characteristics
Region where loss is important
![Page 57: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/57.jpg)
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
General Characteristics
![Page 58: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/58.jpg)
58
Radiation Efficiency (cont.)
General Characteristics
surface wave
TM0
cos (φ) pattern
x
y
Js
![Page 59: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/59.jpg)
( )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.)
![Page 60: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/60.jpg)
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.)
![Page 61: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/61.jpg)
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.
General Characteristics
Radiation Efficiency (cont.)
![Page 62: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/62.jpg)
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
General Characteristics
![Page 63: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/63.jpg)
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
General Characteristics
Note: CAD plot uses Pozar formula (given later).
![Page 64: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/64.jpg)
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
![Page 65: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/65.jpg)
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.)
![Page 66: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/66.jpg)
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
![Page 67: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/67.jpg)
-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
General Characteristics Radiation Patterns
![Page 68: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/68.jpg)
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
General Characteristics Radiation Patterns
![Page 69: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/69.jpg)
Directivity
The directivity is fairly insensitive to the substrate thickness.
The directivity is higher for lower permittivity, because the patch is larger.
69
General Characteristics
![Page 70: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/70.jpg)
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
General Characteristics
![Page 71: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/71.jpg)
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
71
![Page 72: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/72.jpg)
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
![Page 73: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/73.jpg)
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.
![Page 74: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/74.jpg)
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.
![Page 75: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/75.jpg)
( )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
CAD Formulas Radiation Efficiency (cont.)
![Page 76: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/76.jpg)
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
CAD Formulas Radiation Efficiency (cont.)
![Page 77: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/77.jpg)
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.
![Page 78: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/78.jpg)
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.
( ) / 2ave patch grounds s sR R R= +
![Page 79: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/79.jpg)
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= + + +
![Page 80: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/80.jpg)
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.
( ) / 2ave patch grounds s sR R R= +
![Page 81: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/81.jpg)
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.
![Page 82: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/82.jpg)
( ) ( )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.
![Page 83: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/83.jpg)
1
3Dp c
≈
For thin substrates:
(The directivity is essentially independent of the substrate thickness.)
83
CAD Formulas Directivity (cont.)
![Page 84: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/84.jpg)
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
84
![Page 85: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/85.jpg)
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
![Page 86: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/86.jpg)
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
![Page 87: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/87.jpg)
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).
![Page 88: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/88.jpg)
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
![Page 89: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/89.jpg)
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.
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
![Page 90: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/90.jpg)
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 =
![Page 91: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/91.jpg)
(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
![Page 92: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/92.jpg)
( ) 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
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.
![Page 93: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/93.jpg)
( ) ( )( ) ( )
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→ = −
![Page 94: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/94.jpg)
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
94
![Page 95: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/95.jpg)
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
![Page 96: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/96.jpg)
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
![Page 97: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/97.jpg)
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.
![Page 98: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/98.jpg)
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
![Page 99: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/99.jpg)
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= +
![Page 100: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/100.jpg)
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)
![Page 101: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/101.jpg)
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
![Page 102: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/102.jpg)
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.
![Page 103: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/103.jpg)
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.
![Page 104: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/104.jpg)
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
![Page 105: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/105.jpg)
( )( )
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
![Page 106: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/106.jpg)
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.
![Page 107: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/107.jpg)
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
![Page 108: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/108.jpg)
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
![Page 109: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/109.jpg)
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
![Page 110: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/110.jpg)
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
110
![Page 111: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/111.jpg)
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.
![Page 112: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/112.jpg)
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)
![Page 113: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/113.jpg)
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
Circular Polarization
(1,0)
(0,1)
![Page 114: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/114.jpg)
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
Circular Polarization
![Page 115: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/115.jpg)
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.
Circular Polarization
![Page 116: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/116.jpg)
Phase shift realized with delay line:
116
Circular Polarization Dual-Feed Method
L
L
x
y
P
P+λg/4
RHCP
![Page 117: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/117.jpg)
Phase shift realized with 90o quadrature hybrid (branchline coupler)
117
Circular Polarization
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
![Page 118: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/118.jpg)
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
![Page 119: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/119.jpg)
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
119
![Page 120: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/120.jpg)
Circular Patch
x
y
h
a
εr
120
![Page 121: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/121.jpg)
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=
![Page 122: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/122.jpg)
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′ =
Circular Patch Resonance Frequency
0 0
1cµ ε
=
![Page 123: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/123.jpg)
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′
Circular Patch Resonance Frequency
![Page 124: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/124.jpg)
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
![Page 125: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/125.jpg)
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= + ∆
![Page 126: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/126.jpg)
(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=
![Page 127: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/127.jpg)
( ) ( ) ( ) ( )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
− −=
![Page 128: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/128.jpg)
( )( )
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=
![Page 129: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/129.jpg)
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
![Page 130: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/130.jpg)
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.)
![Page 131: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/131.jpg)
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
131
![Page 132: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/132.jpg)
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.
![Page 133: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/133.jpg)
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.
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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.
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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
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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
(Photo courtesy of Dr. Rodney B. Waterhouse)
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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
(Photo courtesy of Dr. Rodney B. Waterhouse)
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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
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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
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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
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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
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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
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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
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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
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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
145
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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.
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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)
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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
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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
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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
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151
Short Open
Inductance
The Smith chart provides a simple explanation for the length reduction.
Miniaturization Quarter-Wave Patch with Fewer Vias
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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)
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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
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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
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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
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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
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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
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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
158
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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
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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
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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= =
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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
Reducing Surface and Lateral Waves
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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
Reducing Surface and Lateral Waves
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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
Reducing Surface and Lateral Waves
SIDE VIEW
z
b
h x
Shorted annular ring
Ground plane Feed
a ρ0
TOP VIEW
a
ρ o
b
Feed
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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 =
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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
Reducing Surface and Lateral Waves
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!).
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-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 and Lateral Waves
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Reducing surface-wave excitation and lateral radiation reduces mutual coupling.
Space-wave radiation
Lateral radiation
Surface waves
168
Reducing Surface and Lateral Waves
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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
Reducing Surface and Lateral Waves
1/r
1/r3
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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
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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
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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
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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
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174
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175
Transmission line model
Cavity model (eigenfunction expansion)
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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).
![Page 177: Introduction to Microstrip Antennas.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563dbaf0550346aa9aa8e6f1/html5/thumbnails/177.jpg)
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
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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
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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
= =
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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.
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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 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
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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
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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)
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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
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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
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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
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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
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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
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*
*
*
,
*
,
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
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*
,
*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
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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
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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
=
= +
=
( )1effrc r effj lε ε ′= −
2 2
0 0
, cos cose eL W
mn mne e
m x n ydx dyL Wπ πψ ψ
=
∫ ∫
192
Input Impedance
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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
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( )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
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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
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( ) ( )
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
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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
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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
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199
Summary (cont.)
( )1effrc r effj lε ε ′= −
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
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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
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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
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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
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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
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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
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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