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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 electrical 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.

A pure cavity model does not account for the part of the field that 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 << λ, the electric field is nearly normal to the patch (neglect fringing) inside the cavity, i.e., the electric field is in the x-direction. This leads us to consider only the transverse magnetic (TMx) field configurations or modes.

y

z

x

W

L

hE-plane

( = 90 , 0)θ ° ≥ x

H-plane( = 0 , 0)φ ° ≥ x θφ

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:

a) Top and Bottom of cavity (electrical conductors)

Ey (x’ = 0, 0 ≤ y’ ≤ L, 0 ≤ z’ ≤ W ) = Ey (x’ = h, 0 ≤ y’ ≤ L, 0 ≤ z’ ≤ W ) = 0

b) 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 mh

m n pnk nLpk pW

π

π

π

= = = = ≠= = = =

and 2 2 2 2 2

x y z r rk k k k ω µ ε+ + = = .

The subscript r refers to the resonant frequency. The resonant frequency is

( )2 2 21

2r mnp

m n pfh L Wπ π π

π µ ε = + +

.

After solving for Ax, the electric and magnetic fields can be found from the vector magnetic potential using

22

2

2

1

1

1

xx x

xy

xz

AE j k Ax

AE jx y

AE jx z

ωµε

ωµε

ωµε

∂= − + ∂

∂= −

∂ ∂

∂= −

∂ ∂

and

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0 (transverse magnetic)1

1

x

xy

xz

HAHz

AHy

µ

µ

=∂

=∂∂

=∂

which yields 2 2

cos( ')cos( ')cos( ')

sin( ')sin( ')cos( ')

sin( ')cos( ')sin( ')

xx mnp x y z

x yy 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

ωµε

ωµε

ωµε

−= −

= −

= −

and

0

cos( ')cos( ')sin( ')

cos( ')sin( ')cos( ')

x

zy mnp x y z

yz mnp x y z

HkH A k x k y k z

kH A k x k y k z

µ

µ

=

= −

=

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 (substrate thickness) is much smaller than the length and width of the patch, i.e., 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 (and desired mode) is the TMx010 where

the resonant frequency is

( )010

12 2r

r

cfL Lµε ε

= = .

Further, if L > W > L/2 > h, the next highest mode (after TMx010) is the

TMx001

( )001

12 2r

r

cfW Wµε ε

= = .

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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 TMx

010 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 TMx

020

( )020

1r

r

cfL Lµε ε

= = .

If W > W/2 > L > h, the next highest mode (after TMx001) is the TMx

002

( )002

12 2r

r

cfW Wµε ε

= = .

Note: These calculations ignore the effects of fringing and assume that the dielectric substrate is only under the patch.

Radiation (TMx010 mode) Assuming the active or dominant mode is the TMx

010, 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

π π = =

= = = =

where n = 1, m = p = 0, E0 = -jωA010, 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 λ/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 other half it points down).

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The far-field radiated electric fields radiated by each slot by itself (see Chapter 12) are

00 0

0

sin( ) sin( )sin2

rjk r

E E

k hW E e X ZE jr X Z

θ

φ θπ

−

≈ ≈

=

where θ and φ are the standard spherical coordinate angles, and

0

0

sin cos2

cos2

k hX

k WZ

θ φ

θ

=

=.

If k0 h << 1, then Eφ reduces to

0 00 sin cos

2sincos

jk r k WhE eE jrφ

θθπ

θ

− =

Note, the voltage across the slot is V0 = h E0. Modeling the two 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 sin2

rjk r

E E

k hW E e k LX ZE jr X Z

θ

φ θ θ φπ

−

≈ ≈

=

.

Again, if k0 h << 1, this reduces to

0 0tot 0 0 eff2 sin cos cos sin sin2sin 2

cos

jk r k Wh E e k LE jrφ

θ θ φθπ

θ

− =

.

Note, the voltage across the slot is V0 = h E0 .

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The radiated electrical field in the two principal planes are

00

tot 0 0 0 eff

0

sin cos2 cos sin

2cos2

jk rk h

k WhE e k LE jk hrφ

φφ

π φ

−

=

on the E-plane (x-y plane above the ground, 0 ≤ φ ≤ 90° and 270° ≤ φ < 360° and θ = 90°), and

00 0

tot 0 0

0 0

sin sin sin cos2 2sinsin cos

2 2

jk rk h k W

k WhE eE jk h k Wrφ

θ θθ

π θ θ

−

=

on the H-plane (x-z plane above the ground, φ = 0 and 0 ≤ θ ≤ 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 on the H-plane. However, there are some differences in the results at low angles (near the dielectric substrate) between the different methods for 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 = 1.186 cm, y0 = 0.3126 cm, 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 DU P

π= = = .

For the typical case that k0 h << 1, the maximum radiation intensity and the power radiated by a single rectangular slot are

220

max 20 02

V WU πη π λ

=

and 2

2 00 3

rad 20 0

sin cos sin22cos

k WVP d

π

θ

θ θ θη π

θ=

=

∫ .

The maximum directivity of a single rectangular slot is then 2

max 00 1

2 1WD DI

πλ

= =

where I1 is 2

03

10

00 0 0

0

sin cos sin2cos

sin( )2 cos( ) ( )i

k WI d

k Wk W k W S k Wk W

π

θ

θ θ θθ=

=

= − + + +

∫

The maximum directivity of a single slot is shown in Figure 14.22.

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The maximum directivity of a rectangular patch (2 radiating slots) is 2

tot totmax 0

0 2 rad 0

2 215

W WD DI G

π πλ λ

= = =

where Grad is the radiation conductance and the parameter 2

03 2 0 eff

20 0

sin cos sin cos sin sin2 2cos

k W k LI d dπ π

φ θ

θ θ θ φ θ φθ= =

=

∫ ∫ .

An alternate expression for calculating the maximum directivity of a rectangular patch (2 radiating slots) is

tot totmax 0 0

12 1

21 /

D D DG G

= = +

where D0 is the directivity of a single slot, G1 is the slot conductance, and G12 is the mutual conductance between the slots. The directivities for two slots (i.e., rectangular patch) are shown in Figures 14.22 and 14.23 as functions of slot width W and substrate height h. The HPBWs in the E-plane and H-planes are (very) approximately

( )2

1 02 2 2

eff

7.032sin (14-58)4 3E L h

λπ

−Θ+

and 1

0

12sin (14-59)2H k W

−Θ+

.

To get accurate HPBWs, use the expressions for totEφ in the E- and H-planes to find where tot tot tot

,max ,max/ 2 0.707E E Eφ φ φ= = with respect to θ or φ as applicable.

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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)]