Introduction to Microstrip Antennas

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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: djackson@uh.edu

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 and lateral waves

5

Notation

6

0
k

wavenumber of free space
1
k

wavenumber of substrate
0


intrinsic impedance of free space
1


intrinsic impedance of substrate
r


relative permtitivity (dielectric consta
nt) of substrate
eff
r


effective relative permtitivity
(accouting for fringing of flux lines a
t edges)
eff
rc


complex effective relative permtitivity
(used in the cavity model to account fo
r all losses)
0


wavelength of free space
c

speed of light in free space
0 0 0 0
2/
k
  
 
1 0
r
k k




0
0
0
376.7303



  
0
/
c f




8
2.99792458 10 m/s
c
 
1 0
/
r
  





0 0
7
0
12
0
2
0
1
4 10 H/m
1
8.854188 10 F/m
c
c

 



 
  
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 and lateral waves

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
D
echamps

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”).


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

Overview of Microstrip Antennas

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

Overview of Microstrip Antennas

Applications


Satellite communications


Microwave communications


Cell phone antennas


GPS antennas

13

Applications include:

Overview of Microstrip Antennas

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

F
ilter

Diplexer

LNA

PD

K
-
connector

DC supply Micro
-
D
connector

M
icrostrip
antenna

Fiber
input with
collimating lens

(Photo courtesy of Dr. Rodney B. Waterhouse)

14

Overview of Microstrip Antennas

Overview of Microstrip Antennas

15

Arrays

Linear array (1
-
D corporate feed)

2

2 array

2
-
D 8X8 corporate
-
fed array

4



8 corporate
-
fed / series
-
fed array

Wraparound Array (conformal)

(
Photo
courtesy of Dr. Rodney B. Waterhouse)

16

Overview of Microstrip Antennas

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.

The
width
W

is usually chosen to be larger than
L

(to get higher bandwidth).
However, usually
W

<

2
L
(to avoid problems with the
(0,2)
mode).


r

17

Overview of Microstrip Antennas

Rectangular patch

W

=

1.5
L

is typical.

Circular Patch

x

y

h

a


r

18

Overview of Microstrip Antennas

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

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 and lateral waves

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
F
eed

A feed along the centerline
is the most
common


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

x

y

L

W

Feed
at (
x
0
,
y
0
)

Surface
current

21

x

r

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


Not easily compatible with arrays

Coaxial
F
eed

2
0
cos
edge
x
R R
L

 

 
 
x

r

h
z

Feeding Methods

x

y

L

W



0 0
,
x y
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

23

Inset Feed

Microstrip line

Feeding Methods

Recent work has shown
that the resonant input
resistance varies as

2
0
2
cos
2
in
x
R A B
L

 
 
 
 
 
 
 
The coefficients
A

and
B

depend on the notch width
S

but (to a good
approximation) not on the line width
W
f

.

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.

24

L

W

W
f

S

x
0

Feeding Methods

Inset
F
eed

Advantages:



Allows for planar feeding


Less line radiation compared to
microstrip feed

Disadvantages:



Requires multilayer fabrication


Alignment is important for input match

Patch

Microstrip
line

25

Feeding Methods

Proximity
-
coupled Feed


(Electromagnetically
-
coupled Feed)

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

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)

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 and lateral waves

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.

ˆ
n
h

PMC

z

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.

30

Main Ideas:

Basic
Principles of Operation



As the substrate gets thinner the patch current radiates less, due to image
cancellation.



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.

31

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

x

r

h
s
J
z

On patch and ground
plane:

0
t
E

ˆ
z
E z E

Inside the patch cavity, because of the thin substrate,

the
electric field vector is approximately independent of
z
.

Hence





ˆ
,,,
z
E x y z z E x y

32

h



,
z
E x y
z

Basic Principles of Operation

Thin Substrate Approximation









1
1
ˆ
,
1
ˆ
,
z
z
H E
j
zE x y
j
z E x y
j



  
  
   
Magnetic field inside patch cavity:

33

Basic Principles of Operation

Thin Substrate Approximation







1
ˆ
,,
z
H x y z E x y
j

 
Note: The magnetic field is purely horizontal.

(The mode is
TM
z
.)

34



,
H x y
h



,
z
E x y
z

Basic Principles of Operation

Thin Substrate Approximation

On the
edges
of
the patch:

ˆ
0
s
J n
 
ˆ
0
bot
s
J n
 
0
bot
t
H

x

y

ˆ
n
L

W

s
J
ˆ
t
On the bottom
surface of
the patch
conductor, at the edge of the patch,
we have

(
J
s

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

35



bot top
s s
J J



ˆ
bot
s
J z H
  
Also,

h

0
bot
t
H

Basic Principles of Operation

Magnetic
-
wall Approximation

0 ( )
t
H

PMC
Since the magnetic field is approximately
independent of
z
, we have an approximate PMC
condition on the entire vertical edge.

ˆ
n
h

PMC Model

36



ˆ
,0
n H x y
 
or

PMC

h

0
edge
t
H

Actual patch

Basic Principles of Operation

Magnetic
-
wall Approximation

x

y

ˆ
n
L

W

s
J
ˆ
t
Hence,

0
z
E
n









1
ˆ
,,
z
H x y z E x y
j

 


ˆ
,0
n H x y
 




ˆ
ˆ
,0
z
n z E x y
  




ˆ
ˆ
,0
z
z n E x y
 












ˆ ˆ ˆ
ˆ ˆ ˆ
,,,
z z z
n z E x y z n E x y E x y n z
     
37

ˆ
n
h

PMC

(Neumann B.C.)

Basic Principles of Operation

Magnetic
-
wall Approximation

x

y

ˆ
n
L

W

s
J
ˆ
t
2 2
0
z z
E k E
  
cos cos
z
m x n y
E
L W
 
   

   
   
2 2
2
1
0
z
m n
k E
L W
 
 
   
   
 
   
   
 
 
Hence

2 2
2
1
0
m n
k
L W
 
 
   
   
 
   
   
 
 
From
separation of variables
:

(
TM
mn

mode)

x

y

L

W

PMC



,
z
E x y
38

Basic Principles of Operation

Resonance Frequencies

We then have

1 0
r
k k k

 
2 2
2
1
m n
k
L W
 
   
 
   
   
1 0 0 0
r r
k k
  
 
Recall that

2
f
 

Hence

2 2
2
r
c m n
f
L W
 
 
   
 
   
   
0 0
1/
c


39

We thus have

x

y

L

W

PMC



,
z
E x y
Basic Principles of Operation

Resonance Frequencies

2 2
2
mn
r
c m n
f
L W
 
 
   
 
   
   
Hence

mn
f f

(resonance frequency of (
m
,
n
) mode)

40

Basic Principles of Operation

Resonance Frequencies

x

y

L

W

PMC



,
z
E x y
This mode is usually used because
the


radiation pattern has a
broadside beam
.

10
1
2
r
c
f
L

 

 
 
cos
z
x
E
L

 

 
 
0
1
ˆ
sin
s
x
J x
j L L
 

 

   

 
   
   
 
This mode acts as a
wide
dipole
(width
W
)
that has a resonant length of
0.5

guided
wavelengths in the
x

direction.

x

y

L

W

Current

41

Basic Principles of Operation

Dominant (1,0) mode

This is the mode with the lowest resonance frequency.

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
/2
d
r
L



 
1
k L


2 2
2
1
m n
k
L W
 
   
 
   
   
(
1,0
) mode:

42

Basic Principles of Operation

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”
L
e

.

10
1
2
e
r
c
f
L

 

 
 
2
e
L L L
  
y

x

L

L
e


L


L

Note: Some authors use
effective permitt
ivity in this equation.

43

Basic Principles of Operation

Resonance
Frequency of Dominant Mode

Hammerstad formula:





0.3 0.264
/0.412
0.258 0.8
eff
r
eff
r
W
h
L h
W
h


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1/2
1 1
1 12
2 2
eff
r r
r
h
W
 


 
 
 
 
  
 
 
 
 
 
 
44

Note:
Even though the
Hammerstad

formula
involves an effective permittivity, we still use
the
actual substrate permittivity
in the
resonance frequency formula.

10
1
2
2
r
c
f
L L

 

 
 
 
Basic Principles of Operation

Resonance
Frequency of Dominant Mode

Note:

0.5
L h
 
This is a good “rule of
thumb” to give a quick estimate.

45

Basic Principles of Operation

Resonance
Frequency of Dominant Mode

0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
h /

0
0.75
0.8
0.85
0.9
0.95
1
NORMALIZED FREQUENCY
Hammerstad
Measured
W
/
L

= 1.5


r

= 2.2

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

f
N

=
f
/
f
0

Results: Resonance
Frequency

46

0
/
h

Basic Principles of Operation

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 and lateral waves

47

General Characteristics


The bandwidth is directly proportional to substrate thickness
h
.


However, if
h

is greater than about
0.05


0

, the probe inductance
(for a
coaxial feed) becomes
large enough so that matching is difficult.


The bandwidth is inversely proportional to

r

(a foam substrate gives a high
bandwidth
).


The bandwidth of a rectangular patch is proportional to the patch width
W
(but remember, we need to keep
W <
2
L
).

Bandwidth

48

49

2 2
2
mn
r
c m n
f
L W
 
 
   
 
   
   
10
1
2
r
c
f
L

 

 
 
02
2
2
r
c
f
W

 

 
 
2
W L

Width Restriction for a Rectangular Patch

f
c

f
10

f
01

f
02

01
1
2
r
c
f
W

 

 
 
W

=

1.5

L

is typical.

02 01
1 1
2
r
c
f f
W L

 
  
 
 
General Characteristics

Some Bandwidth Observations


For a typical substrate thickness (
h

/

0


= 0.02
), and a typical substrate
permittivity (

r

= 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

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


0
5
10
15
20
25
30
BANDWIDTH (%)

r
2.2
= 10.8
W
/
L

= 1.5


r

= 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

10.8
r


2.2
General Characteristics


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

r
.


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

(
x
0
,
y
0
)

L

x

y

52

General Characteristics

The
patch is usually fed along the
centerline

(
y
0

=

W

/ 2
)


to maintain symmetry and thus minimize excitation of undesirable
modes

(
which cause cross
-
pol
).

Desired mode: (1,0)

L

x

W

Feed
:

(
x
0
,
y
0
)

y

53

Resonant Input Resistance

General Characteristics

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.



2
0 0
,
in z
R E x y

For (
1
,
0
) mode:

2
0
cos
in
x
R
L

 

 
 
L

x

W

(
x
0
,
y
0
)

y

54

Resonant Input Resistance

General Characteristics

This will be seen from the cavity
-
model
eigenfunction

analysis later.

Hence, for (1,
0
) mode:

2
0
cos
in edge
x
R R
L

 

 
 
The value of
R
edge

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

Resonant Input Resistance

General Characteristics

L

x

W

(
x
0
,
y
0
)

y

0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
h /


0
50
100
150
200
INPUT RESISTANCE (

2.2
r
= 10.8


r

= 2.2
or

10.8

W
/
L

= 1.5

x
0

= L/
4

Results: Resonant
Input
R
esistance

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

L

x

W

(
x
0
,
y
0
)

y

y
0

=
W
/2

56

0
/
h

10.8
r


2.2
General Characteristics

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.

r
r
tot
P
e
P

57

General Characteristics

58

Radiation Efficiency

General Characteristics

surface wave

TM
0

cos (

) pattern

x

y

J
s



r r
r
tot r c d sw
P P
e
P P P P P
 
  
P
r

=

radiated power

P
tot

=

total input power

P
c

=

power dissipated by conductors

P
d

=

power dissipated by dielectric

P
sw

=

power launched into surface wave

Hence,

59

Radiation Efficiency

General Characteristics


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.


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

1 2
s
R

 
 
R
s

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

.

60

0
2
s
R f


 
Some observations:

Radiation Efficiency

General Characteristics


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

r

= 2.2
, the radiation efficiency is maximum for
h

/

0



0.02
.

Radiation Efficiency

General Characteristics


r

= 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
0.1
h /

0
0
20
40
60
80
100
EFFICIENCY (%)
exact
CAD
Results:
Efficiency
(Conductor
and dielectric losses are
neglected.)

2.2

10.8

Note: CAD plot uses
Pozar

formula (given later).

62

10.8
r


2.2
0
/
h

General Characteristics

0
0.02
0.04
0.06
0.08
0.1
h /

0
0
20
40
60
80
100
EFFICIENCY (%)

= 10.8
2.2
exact
CAD
r

r

= 2.2
or

10.8

W
/
L

= 1.5

63

7
tan 0.001
3.0 10 [S/m]



 
Results:
Efficiency
(All losses are accounted for.)

0
/
h

10.8
r


2.2
General Characteristics

Note: CAD plot uses
Pozar

formula (given later).

64

General Characteristics

Radiation Pattern

E
-
plane: co
-
pol

is
E


H
-
plane: co
-
pol

is
E


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

J
s

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
E



on the ground plane, as well as
the surface
-
wave pattern
.


(Both
vary as
cos

(

))
.

General Characteristics

Radiation Patterns

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

cos
E


varies as
J
s

-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

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

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

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

0
0
2
4
6
8
10
DIRECTIVITY (dB)
exact
CAD
= 2.2
10.8

r

r

= 2.2
or

10.8

W
/
L

= 1.5

Results: Directivity

70

0
/
h

2.2
r


10.8
General Characteristics

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 and lateral waves

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 1
1
/16/
hed
r
r
ave
hed
s
r
r d
e
e
R
L
e
h pc W h

  

   
    
 
   
 
   
    
 
   
   
 
    
   
where

tan
d

 
loss tangent of substrate
1
2
s
R

 
  
surface resistance of metal
73

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

CAD
Formulas

Radiation Efficiency



/2
ave patch ground
s s s
R R R
 
where

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





2
2
0 1
2
0
1
80
hed
sp
P k h c



1
1
hed
sp
hed
r
hed
hed hed
sw
sp sw
hed
sp
P
e
P
P P
P
 




3
3
3
0
2
0
1 1
60 1
hed
sw
r
P k h

 
 
 
 
 
 
 
 
 
74

CAD
Formulas

Radiation Efficiency

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



3
0
1
1
3 1 1
1 1
4
hed
r
r
e
k h
c



  
 
  
  
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

1
2
1 2/5
1
r r
c
 
  












2 4 2
2
2
0 2 4 0 2 0
2 2
2 2 0 0
3 1
1 2
10 560 5
1
70
a
p k W a a k W c k L
a c k W k L
   
    
   
   
 

 
 
The constants are defined as

2
0.16605
a
 
4
0.00761
a

2
0.0914153
c

76

CAD
Formulas

Radiation Efficiency

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







3/2
2
2
0
0 0
2 2
1 0 0 1
1
8
1 ( ) 1 1
r
hed
sw
r r
x
k
P
x k h x x


 


   
2
0
1
2
0
1
r
x
x
x




2 2 2
0 1 0 1 0
0
2 2
1
2
1
r r r
r
x
     
 
    
 

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 0
tan
s k h s

 

 






0
1 0
2
0
1
tan
cos
k h s
k h s
s
k h s

 
 
  
 
 
 
 
 
 
1
r
s

 
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
/3
2
ave
s
d
hed
r r
R
pc
h W
BW
h L e
   
 
    
   
   
  
 
    
   
   
   
 
   
    
 
BW

is defined from the frequency limits
f
1

and
f
2

at which
SWR

= 2.0
.

2 1
0
f f
BW
f


(multiply by
100

if you want to get %)

78

1
2
Q
BW

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.



/2
ave patch ground
s s s
R R R
 
79

CAD
Formulas

Q

Components

1 1 1 1 1
d c sp sw
Q Q Q Q Q
   
1/tan
d
Q


0 0
( )
2
c
ave
s
k h
Q
R

 
 

 
 
 
 
1 0
3 1
16/
r
sp
L
Q
pc W h


 
 
 

 
 
 
 
 
 
1
hed
r
sw sp
hed
r
e
Q Q
e
 

 

 


3
0
1
1
3 1 1
1 1
4
hed
r
r
e
k h
c



  
 
  
  
The constants
p

and
c
1

were defined previously.



/2
ave patch ground
s s s
R R R
 
2
0
cos
max
in edge
x
R R R
L

 
 
 
 
Probe
-
feed Patch



0
0
1
0 0 0
4
1 16 1
/3
edge
s
d
hed
r r
L h
W
R
R
pc
W h
h L e

 
   
 
   
 
   
   
 

    
   
   
 
    
   
   
   
   
    
80

CAD
Formulas

Resonant Input Resistance

Comments: For a lossless patch, the resonant resistance is approximately
independent of the substrate thickness. It is inversely proportional to the square
of the patch width. It is proportional to the substrate permittivity.





tanc tan/
x x x

where







2
1
2
1 1
3
tanc
tan
r
r
D k h
pc k h


 
 

 
 

 
 
81

CAD
Formulas

Directivity

1 0
r
k k


1
3
D
pc

For thin substrates:

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

82

CAD
Formulas

Directivity

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 and lateral waves

83

Radiation Pattern

84

h
r

x
P
atch

P
robe

C
oax 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

85

h
r

x
P
atch

P
robe

C
oax feed

patch top bot
s s s
J J J
 
h
r

x
patch
s
J
probe
s
J
Electric current model:

We keep the physical currents flowing on the patch (and feed).

Radiation Pattern

86

h
r

x
P
atch

P
robe

C
oax feed

Magnetic current model:

We apply the equivalence principle and invoke the (approximate) PMC condition

at the edges.

ˆ
e
s
M n E
 
Radiation Pattern

Equivalence surface

ˆ
ˆ
e
s
e
s
J n H
M n E
 
  
h
r

x
e
s
M
e
s
M
87

Theorem

The electric and magnetic models yield identical patterns

at the
resonance frequency
of the cavity mode.

Assumptions:


1)
The electric and magnetic current models are based on the fields
of a single cavity mode, corresponding to an ideal cavity with
PMC walls.

2)
The probe current is neglected in the electric current model.

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

88

Comments on the models


For the electric current model, 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.


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


When an infinite substrate is assumed in 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

89

Comments on the 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 more than one component of electric surface current,
described by Bessel functions).

Radiation Pattern

On the nonradiating edges, the
magnetic currents are in opposite
directions across the centerline (
x

= 0
).

sin
z
x
E
L

 
 
 
 
ˆ
e
s
M n E
 
L

x

W

y

“Radiating edges”

“Nonradiating edges”

e
s
M
s
J
10
ˆ
cos
s
x
J x A
L

 

 
 
(The formulas are based
on
the electric
current
model.)

The origin is at the
center of the patch.

L

r
ε
h

I
nfinite ground plane
and substrate

x

The probe is on the
x

axis.

cos
s
πx
ˆ
J x
L
æ ö
÷
ç
=
÷
ç
÷
ç
è ø
y

L

W

E
-
plane

H
-
plane

x

(1,0) mode

90

Radiation Pattern

Rectangular Patch



2
2
sin
cos
2
2
(,,),,
2
2
2 2
y
x
hex
i i
y
x
k W
k L
WL
E r E r
k W
k L

 

 
 
 
 
 
 
 
 
 
   
 
 

 
 
 
 
 
 

 
 
 
 
 
 
 
 
0
sin cos
x
k k
 

0
sin sin
y
k k
 

The “hex” pattern is for a

horizontal
electric dipole in the
x

direction
,


sitting on top of the substrate.

or
i
 

The far
-
field pattern can be determined by reciprocity.

91

x

y

L

W

s
J
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
.









0
0
,,sin
,,cos
hex
hex
E r E F
E r E G


  
  


where















0
0
2tan
1
tan sec
TE
k h N
F
k h N j N

 
  
  

















0
0
2tan cos
cos 1
tan cos
TM
r
k h N
G
k h N j
N
 
  

 

  





2
sin
r
N
  
 
0
0
0
4
jk r
j
E e
r



 


 
 
92

Radiation Pattern

Note: To account for lossy substrate, use



1 tan
r rc r
j
   
  
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 and lateral waves

93

Input Impedance

94

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

95



1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
-20
-10
0
10
20
30
40
50
Frequency (GHz)
Z
i
n
(

)
RLC Circuit model of a Microstrip antenna


1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
-20
-10
0
10
20
30
40
50
60
Frequency (GHz)
Z
i
n
(

)
Transmission Line model of a Microstrip antenna
l
eff
includes all loses

1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
-20
-10
0
10
20
30
40
50
60
Frequency (GHz)
Z
i
n
(

) (Eigenfunction expansion)
Cavity model of a Microstrip antenna
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.

2.2
tan 0.001
1.524 mm
r
h





7
6.255 cm
/ 1.5
3.0 10 S/m
L
W L



 
0
0
6.255 cm
0
0.635mm
x
y
a



Input Impedance

Comparison of Simplest Models

Circuit model of patch

Transmission line model of patch

Cavity model (
eigenfunction

expansion) of patch

Transmission Line Model
for
Input Impedance

96



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 L
y y W
 
 
0
eff
e rc
k k


Denote



1
eff
rc r eff
j l
 

 
1 1 1 1 1
eff
d c sp sw
l
Q Q Q Q Q
    
Note:



L

is from
Hammerstad’s

formula



W

is from Wheeler’s formula


Physical patch dimensions (
W



L
)


97

A CAD formula for
Q

has been given earlier.

Input Impedance

L
e

W
e

x

y

0 0
(,)
e e
x y
PMC

s
J
eff
rc


98

ln4
W h

 
 
 
 
0.262
0.300
0.412
0.258
0.813
eff
r
eff
r
W
h
L h
W
h


 

 
 

 
 
 

 
 

 
(Hammerstad formula)

1 1
1
2 2
1 12
eff
r r
r
h
W
 

 
   
 
   
   
 

 
 
(Wheeler formula)

Commonly used fringing formulas

Input Impedance

99

where

TL
in in p
Z Z jX
 
0.57722
( )

Euler's constant






0 0
0 0
1/tan
tan
TL TL eff eff e
in in c c
eff eff e
c c e
Y Z jY k x
jY k L x
 
 
0
,
eff eff
c c
Z k
0
x

e
x L

p
L
0
e
x x

p p
X L


Z
in

0 0
1/
eff eff
c c
Y Z





0
0
0
2
ln
2
p
r
X k h
k a




 
 
  
 
 
 
 
 
 
Input Impedance

(from a parallel
-
plate model of probe inductance)

0 0
1
eff eff
c c
eff
e e
rc
h h
Z
W W
 

 
CAD Circuit Model for Input Impedance

100

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 analysis, discussed later.

Input Impedance


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
L
p

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

L
p

R

C

L

Z
in



P
robe

P
atch
cavity

101

Probe
-
fed Patch

Input Impedance

0
0
1
in p
R
Z j L
f f
jQ
f f

 
 
 
 
 
0
R
Q
L


1
2
BW
Q

BW

is defined here by
SWR

<

2.0
.

0 0
1
2
f
LC
 
 
102

L
p

R

C

L

Z
in



Input Impedance

in in in
Z R jX
 
0
max
in in
f f
R R R

 
R

is
the input resistance at the resonance of the patch
cavity


(the frequency that maximizes
R
in
).

L
p

R

C

L

103

0
f f

max
in
R


2
0
0
1
in
R
R
f f
Q
f f

 
 
 
 
 
 
 
Input Impedance

0
f f

(resonance of
RLC

circuit)

The
input resistance
is determined once we know four parameters:

104



f
0
: the resonance frequency of the patch cavity



R
: the input resistance at the cavity resonance frequency
f
0



Q
: the quality factor of the patch cavity



L
p
: the probe inductance

L
p

R

(
R
,
f
0
,

Q
)

L

C



Z
in

CAD formulas
for these four
parameters
have been
given earlier.

0
0
1
in p
R
Z j L
f f
jQ
f f

 
 
 
 
 
Input Impedance

4
4.5
5
5.5
6
FREQUENCY (GHz)
0
10
20
30
40
50
60
70
80
R
in

(

)
CAD
exact
Results:
Input
Resistance
vs.
Frequency


r

= 2.2

W
/
L

= 1.5

L

= 3.0 cm

Frequency
where
the input resistance
is maximum (
f
0
)

105

Rectangular patch

Input Impedance

Results: Input
Reactance
vs.
Frequency


r

= 2.2

W
/
L

= 1.5

4
4.5
5
5.5
6
FREQUENCY (GHz)
-40
-20
0
20
40
60
80
X
in

(

)
CAD
exact
L

= 3.0 cm

Frequency
where the input
resistance is maximum (
f
0
)

Frequency
where the
input impedance is real

Shift
due to probe reactance

106

Rectangular patch

Input Impedance

0.577216





0
0
0
2
ln
2
p
r
X k h
k a




 
 
  
 
 
 
 
 
 
(Euler’s constant)

Approximate CAD formula for
probe

(feed)
reactance (in Ohms)

p p
X L


0 0 0
/376.7303
  
  
a

=

probe radius

h

=

probe height

This is based on an infinite parallel
-
plate model.

107

r

h
2
a
Input Impedance





0
0
0
2
ln
2
p
r
X k h
k a




 
 
  
 
 
 
 
 
 

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


Feed reactance increases for smaller probe radius.

108

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
1
X
r
0
5
10
15
20
25
30
35
40
X
f
(

)
CAD
exact
Results: Probe
Reactance
(
X
f

=
X
p
=

L
p
)

x
r

= 2 (
x
0

/
L
)
-

1

The normalized feed location ratio
x
r

is
zero at the center
of the
patch (
x

=
L
/2
),
and is
1.0

at
the patch edge (
x

=
L
).


r

= 2.2

W
/
L

= 1.5

h
= 0.0254

0


a

= 0.5 m
m


109

r
x
Center

Edge

Rectangular patch

Input Impedance

L

W

(
x
0
,
y
0
)

L

x

y

Cavity Model


110



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 L
y y W
 
 
0
eff
e rc
k k


Denote



1
eff
rc r eff
j l
 

 
1 1 1 1
eff
d c sp sw
l
Q Q Q Q
   
Note:



L

is from
Hammerstad’s

formula



W

is from Wheeler’s formula


L
e

W
e

x

y

0 0
(,)
e e
x y
PMC

Physical patch dimensions (
W



L
)


111

Input Impedance

A CAD formula for
Q

has been given earlier.

eff
rc


112

ln4
W h

 
 
 
 
0.262
0.300
0.412
0.258
0.813
eff
r
eff
r
W
h
L h
W
h


 

 
 

 
 
 

 
 

 
(Hammerstad formula)

1 1
1
2 2
1 12
eff
r r
r
h
W
 

 
   
 
   
   
 

 
 
(Wheeler formula)

Commonly used fringing formulas

Input Impedance

Next, we derive the Helmholtz equation for
E
z
.

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

i eff
c
H J j E
E j H


  
  






2
2 2
2 2
1
i eff
c
i
e
i
e
i
e
E J j E
j
E j J k E
E E j J k E
E k E j J





    
    
     
  

113

Input Impedance

(Ampere’s law)

(Faraday’s law)

Hence

Denote

2 2
i
z e z z
E k E j J

  
(,) (,)
z
x y E x y


where

(,) (,)
i
z
f x y j J x y


Then


114

2 2
(,)
e
k f x y
 
  
(scalar Helmholtz equation)

Input Impedance

Introduce “
eigenfunctions


mn

(
x
,
y
)
:

For rectangular patch we have, from separation of variables,

2 2
(,) (,)
mn mn mn
x y x y
 
  
0
mn
C
n




2 2
2
(,) cos cos
mn
e e
mn
e e
m x n y
x y
L W
m n
L W
 

 

   

   
   
 
   
 
 
   
 
   
 

115

Input Impedance

Assume an “
eigenfunction

expansion”:

Hence

,
(,) (,)
mn mn
m n
x y A x y
 


2 2
(,)
e
k f x y
 
  
2 2
,,
(,)
mn mn e mn mn
m n m n
A k A f x y
 
  
 
Using the properties of the eigenfunctions, we have



2 2
,
(,) (,)
mn e mn mn
m n
A k x y f x y
 
 

This must satisfy


116

Input Impedance

Note that the eigenfunctions are orthogonal, so that

Denote

Next, we multiply by and integrate.

(,)
m n
x y

 
(,) (,) 0,(,) (,)
mn mn
S
x y x y dS m n m n
 
 
 
 

2
,(,)
mn mn mn
S
x y dS
  
  



2 2
,,
mn e mn mn mn mn
A k f
   
     
We then have


117

Input Impedance

Hence, we have

For the patch problem we then have

2 2
,
1
,
mn
mn
mn mn e mn
f
A
k

  
 
 

 
  
 
2 2
,
1
,
i
z mn
mn
mn mn e mn
J
A j
k


  
  
 

  
  
  
The field inside the patch cavity is then given by

,
(,) (,)
z mn mn
m n
E x y A x y




118

Input Impedance

*
*
*
,
*
,
1
(,)
2
1
(,)
2
1
2
1
,
2
i
in z z
V
i
z z
S
i
mn mn z
m n
S
i
mn mn z
m n
P E x y J dV
h E x y J dS
h A J dS
h A J


 
 
 
   





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


119

Input Impedance

*
,
*
2 2
,
2
2 2
,
1
,
2
1,1
,
2,
,
1 1
2,
i
in mn mn z
m n
i
i
mn z
mn z
m n
mn mn e mn
i
mn z
m n
mn mn e mn
P h A J
J
h j J
k
J
h j
k


 
  


  
   
 
  
 
   
 
  
  
  
 
 
 
 
 
 
 
    
 
 



Hence

Also,

2
1
2
in in in
P Z I

so

2
2
in
in
in
P
Z
I


120

Input Impedance

Hence we have

where

2
2
2 2
,
,
1 1
,
i
mn z
in
m n
mn mn e mn
in
J
Z j h
k
I


  
 
 
 
 
 
 
 
  
 
 

,0 0
m n m n
 
 

  

121

Input Impedance

Rectangular patch:

where

2 2
2
0
cos cos
mn
e e
mn
e e
eff
e rc
m x n y
L W
m n
L W
k k
 

 


   

   
   
   
 
   
   



1
eff
rc r eff
j l
 

 
2 2
0 0
,cos cos
e e
L W
mn mn
e e
m x n y
dx dy
L W
 
 
   

   
   
 

122

Input Impedance

so

To calculate , assume a strip model as shown below