microstrip antennas rectangular patch - dr. montoya's...
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Microstrip Antennas- Rectangular Patch Chapter 14 in Antenna Theory, Analysis and Design (4th Edition) by Balanis
Cavity model
Microstrip antennas resemble dielectric-loaded cavities that are
bounded by conductors on the top & bottom (i.e., tangential electric
fields are zero) and magnetic walls (i.e., tangential magnetic fields
are zero) on the sides, simulate open circuits.
A pure cavity model does not take into account that part of the field
is radiated. Radiation loss is worked into the model by introducing
an effective loss tangent δeff =
1/Q where Q is the antenna quality
factor.
Since h << , the electric field is nearly normal to the patch (neglect
fringing) inside the cavity. This leads us to consider only the
transverse magnetic (TMx) field configurations or modes.
y
z
x
W
L
h
Rectangular microstrip patch geometry.
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TMx field configurations or modes
The wave equation that will be solved, for the dielectric cavity is
2 2 0x xA k A
where Ax is the x-component of the vector magnetic potential and k is the
wave number. The general solution for Ax is
1 1 2 2
3 3
cos( ) sin( ) cos( ) sin( )
cos( ) sin( )
x x x y y
z z
A A k x B k x A k y B k y
A k z B k z
where kx, ky, and kz are the wave numbers in the indicated directions.
The applicable boundary conditions are
Top and Bottom of cavity (conductors)
Ey(x’ = 0, 0 ≤ y’ ≤ L, 0 ≤ z’ ≤ W ) = Ey(x’ = h, 0 ≤ y’ ≤ L, 0 ≤ z’ ≤ W ) = 0
Sides of cavity (magnetic walls)
Hy(0 ≤ x’ ≤ h, 0 ≤ y’ ≤ L, z’ = 0 ) = Hy(0 ≤ x’ ≤ h, 0 ≤ y’ ≤ L, z’ = W ) = 0
Hz(0 ≤ x’ ≤ h, y’ = 0, 0 ≤ z’ ≤ W ) = Hz (0 ≤ x’ ≤ h, y’ = L, 0 ≤ z’ ≤ W ) = 0
where the primed coordinates represent the inside of the cavity.
Applying these boundary conditions leads to
cos( ')cos( ')cos( ')x mnp x y zA A k x k y k z
where Amnp is the product of the amplitude coefficients and the wave
numbers are
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0,1,2,
00,1,2,
(can't all be zero)
0,1,2,
x
y
z
mk m
h
m n pnk n
L
pk p
W
and
2 2 2 2 2
x y z r rk k k k .
The subscript r refers to the resonant frequency. Therefore, the resonant
frequency is
2 2 2
1
2r mnp
m n pf
h L W
.
After solving for Ax, the electric and magnetic fields can be found from
the vector magnetic potential using
2
2
2
2
1
1
1
xx x
xy
xz
AE j k A
x
AE j
x y
AE j
x z
and
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0 (transverse magnetic)
1
1
x
xy
xz
H
AH
z
AH
y
which yields
22
cos( ')cos( ')cos( ')
sin( ')sin( ')cos( ')
sin( ')cos( ')sin( ')
xx mnp x y z
x y
y mnp x y z
x zz mnp x y z
k kE j A k x k y k z
k kE j A k x k y k z
k kE j A k x k y k z
and
0
cos( ')cos( ')sin( ')
cos( ')sin( ')cos( ')
x
zy mnp x y z
y
z mnp x y z
H
kH A k x k y k z
kH A k x k y k z
The electric field configurations for the lowest few cavity modes are
shown in Figure 14.13.
The dominant mode (i.e., the mode with the lowest resonant frequency)
depends on the dimensions of the cavity (patch). Since the cavity height
h << L and h << W, the length L and width W of the patch will control the
dominant mode.
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Figure 14.16 Field configurations (modes) for rectangular microstrip
patch. [From Balanis, Antenna Theory, Analysis and Design (Fourth Edition)]
If L > W > h, the dominant mode (i.e., the desired mode) is the TMx
010
where the resonant frequency is
010
1
2 2r
r
cf
L L .
Further, if L > W > L/2 > h, the next highest mode (after TMx010) is the
TMx001
001
1
2 2r
r
cf
W W .
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However, if W > L > h, the dominant mode is the TMx001 whose resonant
frequency has already been given. Fortunately, if a centered microstrip
feed is used, the TMx010 mode can be excited, even if it is not the
dominant mode (see Fig 14.13a).
If L > L/2 >W > h, the dominant mode would be the TMx020
020
1r
r
cf
L L .
If W > W/2 > L > h, the next highest mode (after TMx001) is the TM
x002
002
1
2 2r
r
cf
W W .
Note: These calculations ignore the effects of fringing and assume that
the dielectric substrate is only under the patch.
Radiation (TM
x010 mode)
Assuming the active or dominant mode is the TMx010, the fields in the
cavity are
0 0cos ' and sin '
0
x z
y z x y
E E y H H yL L
E E H H
where n =
1, m
=
p
=
0, E0
= -jA010, and H0
= (π/µL) A010. See Figure
14.13a for pictures of the electric field distribution. Radiation occurs
from the two end slots (located at y =
0 and y
=
L). The rectangular slots
have dimensions of W h, and are separated by about /2 at resonance.
The side slots (located at z =
0 and z = W ) are non-radiating because the
radiation from the fields along the sides cancel each other in the far-field
(note that along half the side slots the electric field points up and on the
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other half it points down).
The far-field radiated electric fields radiated by each slot (see Chapter
12) are
0
0 0
0
sin( ) sin( )sin
2
r
jk r
E E
k hW E e X ZE j
r X Z
where and are the standard spherical coordinate angles, and
0
0
sin cos2
cos2
k hX
k WZ
.
If k0 h << 1, then E reduces to
0 00 sin cos
2sin
cos
jk r k WhE e
E jr
Note, the voltage across the slot is V0=h E0.
Modeling the radiating slots as a two-element array (see Chapter 6) of
rectangular aperture antennas leads to
0tot 0 0 0 eff
0
sin( ) sin( )sin cos sin sin
2
r
jk r
E E
k hW E e k LX ZE j
r X Z
Again, if k0 h << 1, this reduces to
0 0tot 0 0 eff2 sin cos
cos sin sin2sin 2
cos
jk r k WhE e k L
E jr
Note, the voltage across the slot is V0 = h E0 .
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The radiated electrical field in the two principal planes is
0
0
tot 0 0 0 eff
0
sin cos2 cos sin
2cos
2
jk r
k h
k WhE e k LE j
k hr
in the E-plane (x-y plane above the ground, = 90), and
0
0 0
tot 0 0
0 0
sin sin sin cos2 2
sin
sin cos2 2
jk r
k h k W
k WhE eE j
k h k Wr
in the H-plane (x-z plane above the ground, = 0 or 180).
Figure 14.18 shows examples of typical E-plane and H-plane radiation
patterns. Note that the experimental, theoretical, and MoM results agree
well in the H-plane. However, there are some differences at low angles
(near the dielectric substrate) between the methods in the E-plane. This
primarily because the cavity theory assumed the dielectric substrate was
truncated at the edges of the cavity, which does not happen in reality.
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Figure 14.18 Predicted and measured E- and H-plane patterns of
rectangular microstrip patch (L=0.906cm, W=1.186cm, y0=0.3126cm,
εr=2.2, f0=10GHz). [From Balanis, Antenna Theory, Analysis & Design (Second Edition)]
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Directivity
Knowing the fields allows the directivity of the rectangular patch to be
calculated. In particular, we are interested in the maximum directivity
max maxmax 0
0 rad
4U UD D
U P
.
For the typical case that k0h << 1, the maximum radiation intensity and
the power radiated by a single rectangular slot are
22
0
max 2
0 02
V WU
and 2
20
0 3
rad 2
0 0
sin cossin2
2cos
k WV
P d
.
The maximum directivity of a single rectangular slot is then
2
max 0
0 1
2 1WD D
I
where I1 is 2
03
1
0
00 0 0
0
sin cossin2
cos
sin( )2 cos( ) ( )i
k W
I d
k Wk W k W S k W
k W
The maximum directivity of a single slot is shown in Figure 14.19.
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The maximum directivity of a rectangular patch (2 radiating slots) is
2
tot tot
max 0
0 2 rad 0
2 2
15
W WD D
I G
where Grad is the radiation conductance and
2
03 2 0 eff
2
0 0
sin cossin cos sin sin2
2cos
k Wk L
I d d
.
A slightly simpler expression for the maximum directivity is
tot tot
max 0 0
12 1
2
1 /D D D
G G
.
The directivities for two slots (i.e., rectangular patch) are shown in
Figures 14.19 and 14.20 as functions of slot width W and substrate
height h.
The HPBWs in the E-plane and H-planes are (very) approximately
21 0
2 2 2
eff
7.032sin (14-58)
4 3E
L h
and
1
0
12sin (14-59)
2H
k W
.
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Figure 14.22 Computed directivity of one and two slots as a function of
the slot width. [From Balanis, Antenna Theory, Analysis and Design (Fourth Edition)]
Figure 14.23 Directivity variations as a function of substrate height for a
square microstrip patch antenna (courtesy of D. M. Pozar). [From Balanis,
Antenna Theory, Analysis and Design (Fourth Edition)]