CAD OF RECTANGULAR MICROSTRIP ANTENNAS

David R. Jackson, Stuart A. Long, Jeffery T. Williams, and Vickie B. Davis

Department of Electrical and Computer Engineering University of Houston

Houston, TX 77204-4005 Publication information: Chapter 5 of Advances in Microstrip and Printed Antennas, Kai Fong Lee and Wei Chen, Eds., John Wiley, 1997.

1

5.1 INTRODUCTION This chapter develops simple CAD formulas for the rectangular microstrip patch antenna. The CAD formulas are closed-form approximate expressions that describe the basic properties of the patch antenna. CAD formulas are presented for the resonance frequency, input resistance at resonance, radiation efficiency, bandwidth, and directivity. With the exception of the formulas for resonance frequency, all of the CAD formulas are derived from accurate analytical approximations of exact formulas, and are therefore not simply empirical in nature. The CAD formulas account for radiation into space, surface-wave radiation, dielectric loss, and conductor loss. The formulas in all cases become more accurate as the substrate thickness decreases. With the exception of input resistance, the CAD formulas are independent of the specific feeding mechanism. The formula for resonant input resistance applies for the specific case of a probe feed. A CAD formula for the probe reactance is also given. Although derived for the rectangular patch, the CAD formulas for bandwidth, radiation efficiency, and directivity may be used in an approximate fashion for the circular patch as well, by applying the formulas to an equivalent square patch having the same area as the original circular patch. In addition, a simple CAD model of the patch antenna is introduced for the calculation of input impedance. This model consists of a parallel RLC circuit (modeling the patch resonance) in series with a reactance (the probe reactance). This simple circuit, which follows directly from cavity model theory, can be used to calculate the input impedance at any frequency once the resonant input resistance, resonance frequency, and bandwidth of the patch are known. Hence, this simple circuit can be used directly with the above mentioned CAD formulas. Alternatively, the CAD formulas for radiated and dissipated power can be used to calculate an effective loss tangent of the substrate, which can be used directly in a cavity-model analysis of the patch. Results from the CAD formulas are compared with rigorous results from a spectral-domain analysis. Results show that the formulas for bandwidth, radiation efficiency, and directivity are very accurate, and even suffice for final design equations provided the substrate thickness is small enough so that ε λr h / 0 ≤ 0.10. The CAD formula for resonant input resistance loses accuracy sooner as the substrate thickness increases, but is accurate for substrate thicknesses in the range ε λr h / 0 ≤ 0.03. This chapter also discusses CAD formulas for the far-field radiation pattern of the rectangular patch on an infinite substrate. A comparison is made between formulas obtained from an electric current model and a magnetic current model. Both models are derived from different applications of the equivalence principle. In the electric current model the horizontal patch current is integrated to find the far-field pattern. In the magnetic current model the equivalent magnetic current at the boundary of the patch is used as the radiating source. It is demonstrated that both models yield the same result provided that the electric and magnetic currents in the two models correspond to the same cavity mode, and that the frequency of radiation is the resonance frequency of the cavity mode. Formulas obtained from both models are also presented for the far-field radiation pattern of a patch with a substrate that is truncated at the edges, and results are presented to show how these patterns differ from those for an infinite substrate.

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5.2 CAD MODEL FOR RECTANGULAR PATCH ANTENNA A probe-fed rectangular patch is shown in Fig. 1. In the cavity model [1]-[2], a perfect magnetic-wall boundary is placed on the edges of the patch to form an ideal closed cavity. In order to account for fringing, the effective length of the patch is taken as Le = L + 2 ΔL, where ΔL is an edge extension that is chosen to produce the correct resonance frequency for the dominant cavity mode of the patch. CAD formulas for ΔL are discussed in the next section. The resonance frequency of the dominant cavity mode, f0 , is related to the effective patch length by

f cLr r e

0 2=

ε μ, (5.1)

where c is the speed of light, 2.9979254 × 108 m/s, and ε r and μ r are the relative permittivity and permeability of the substrate, respectively. The effective width of the patch, We, is chosen as We = W + 2 ΔW, where the fringing width is chosen as [3]

ΔW h≈ FHGIKJ

ln4π

. (5.2)

The fringing width is much less important than the fringing length, since it is the fringing length that determines the resonant frequency of the patch. The ideal magnetic wall allows for a simple modal expansion of the fields in terms of an eigenfunction expansion. The electric field Ez inside the cavity, as well as the eigenfunctions, are independent of z, provided the assumption is made that the probe current Jz is constant (this is one restriction that limits the validity of the model to substrates that are thin compared to a wavelength). The eigenfunctions ϕmn x y( , ) satisfy the eigenvalue equation ∇ + =2 2 0ϕ ϕmn mn mnx y k x y( , ) ( , ) . (5.3) The eigenfunctions are cavity modes that can exist inside the magnetic-wall cavity, and the eigenvalues kmn are the corresponding wavenumbers of the resonant cavity modes. Because of the ideal cavity approximation, the eigenfunctions are complete and orthogonal, and the total field excited by the feed may be expanded in terms of these functions. Furthermore, the eigenvalues are all real numbers, independent of the substrate loss tangent (they are actually the cutoff wavenumbers of a corresponding rectangular waveguide with magnetic walls). The eigenfunction expansion of the electric field inside the cavity is [1]

E x y A x yz mnnm

mn( , ) ( , )==

∞

=

∞

∑∑00

ϕ , (5.4)

3

where

A j Jk kmn

z mn

mn mn e mn

=−

ω μ ϕϕ ϕ( , )

( , )1

2 2 . (5.5)

The inner product notation (f, g) denotes integration of the product f•g over the area of the patch, and the effective complex wavenumber ke is defined by the relation k k je e

212 1= −( ) , where

k k r r12

02= ε μ and e is the effective loss tangent of the substrate. This effective loss tangent

accounts for all loss mechanisms, including radiation into space, surface waves, dielectric loss, and conductor loss. Note that the wavenumber k1 is real-valued even though the actual substrate may have a nonzero loss tangent - this loss is included in the effective loss tangent. For the rectangular patch, the eigenfunctions are

ϕ π πmn

e e

x y m xL

n yW

( , ) cos cos=FHGIKJFHGIKJ (5.6)

with eigenvalues

k mL

nWmn

e e

22 2

=FHGIKJ +FHGIKJ

π π . (5.7)

The effective loss tangent is related to the Q of the patch by the relation

e =1Q

. (5.8)

The total quality factor Q may be expressed in terms of the Q factors associated with radiation into space (Qsp), radiation into surface waves (Qsw), dielectric loss (Qd ), and conductor loss (Qc ), through the relation 1 1 1 1 1Q Q Q Q Qsp sw d c

= + + + . (5.9)

The dominant rectangular patch resonance occurs at the frequency for which k1 ≈ k10 . Assuming the probe current I0 is one Amp, Eq. (5.5) predicts that the dominant (1,0) cavity mode is excited at resonance with a maximum amplitude of

Ak

x LW Lr

e

e

e e10 0

12

012

= −FHGIKJω μ μ

πcos //

b g . (5.10)

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This equation is used later to derive the CAD formula for input resistance at resonance. The input impedance is calculated from an average voltage, obtained by integrating over the probe, as

Z VI

hI

E x y hI

J Ein

ave

zave

z z= = − = −0 0

0 002( , ) ( , ) , (5.11)

which yields

Z j hI

Jk kin r

nm

z mn

mn mn e mn

= −−=

∞

=

∞

∑∑ω μ μ ϕϕ ϕ0

02

00

2

2 2

1( , )( , )

. (5.12)

For the rectangular patch, it is most convenient to model the cylindrical probe of radius a, centered at (x0, y0), as a flat strip of width wp, parallel to the y-z plane axis and centered at (x0, y0). One choice for the current distribution on the strip is a “Maxwell function” that includes the edge singularity. Assuming a probe current I0, the current density would be

J I

wy y

y yw

sz

p

p=FHGIKJ − −

− <1

2

20

2

02

0πb g

, | | . (5.13)

For this choice, the strip width that best models the original probe would be [4] w ap =4 . (5.14) Alternatively, a simpler choice for the probe current is a uniform current density,

J Iw

y yw

szp

p= − <00 2

, | | . (5.15)

For this choice, the correct strip width is [4] w e ap =

3 2/ . (5.16) The advantage of using the uniform current is that the formula for the reaction between the probe current and the eigenmodes (the numerator term in Eq. (5.12)) is simpler, involving only the sin(x) function instead of the J0 (x) Bessel function. In particular, for the uniform probe current density

5

J m xL

J y n yW

dyz mne

szy w

y w

ep

p, cos ( )cos/

/ϕ π πb g= F

HGIKJ

FHGIKJ−

+z02

2

0

0

=FHGIKJFHGIKJFHGIKJI m x

Ln y

Wn w

We e

p

e0

0 0

2cos cosπ π π

sinc , (5.17)

where

sinc x xx

a f a f≡

sin . (5.18)

For the rectangular patch, Eq. (5.12) then becomes (setting I0 = 1 Amp)

Z j h

m xL

n yW

W L

n wW

k mL

nW

in rnm

e e

e em n

p

e

ee e

= −

FHGIKJFHGIKJ

FHIK + +

FHGIKJ

−FHGIKJ −FHGIKJ

=

∞

=

∞

∑∑ω μ μ

π π

δ δ

π

π π0

00

2 0 2 0

0 0 22 2

41 1

2cos cos

b gb g

sinc2

, (5.19)

where the notation δ m0 denotes the Kronecker delta (1 if m = 0, and 0 otherwise). From equation (5.12) it follows that the input impedance may be written in the form

Z Zin inm n

nm

==

∞

=

∞

∑∑ ( , )

00

, (5.20)

where

Z j ak kin

m n mn

e mn

( , ) = −−

ω 2 2 . (5.21)

In this equation the amplitude terms

a hI

Jmn r

z mn

mn mn

= μ μ ϕϕ ϕ0

02

2( , )( ),

(5.22)

are frequency-independent. The Zin

m n( , ) terms in Eq. (5.20) represent impedances connected in series to yield the total input impedance. Taking the reciprocal of these impedance terms yields the admittance terms

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Y ja

k kinm n

mne mn

( , ) ( )= −1 2 2

ω

= − −ja

k j kmn

e mn1 11

2 2

ωb g

=FHGIKJ +FHGIKJFHGIKJ −

LNMM

OQPP

ka

j ka

e

mn

mn

mn mn

12 2 2

1ω ω

ωω

. (5.23)

Equation (5.23) provides the key for a physical interpretation of the result in Eq. (5.20). Each term Yin

m n( , ) has the form

YR

jLin = +FHGIKJFHGIKJ −

LNMM

OQPP

1 1 10

2

ωωω

, (5.24)

which is the admittance for a parallel RLC circuit with a resonance frequency

ω π0 02 1= =f

LC. (5.25)

Comparing Eqs. (5.23) and (5.24), it is seen that the two equations have the same mathematical form except for the fact that the constant 1/R term in Eq. (5.24) corresponds to a frequency-dependent term in Eq. (5.23). However, near the resonance frequency of the RLC circuit ω is approximately constant, and thus the first term in the last line of Eq. (5.23) is approximately constant. Hence, there is an approximate equivalence between the models. Therefore, the equivalent circuit shown in Fig. 2 is an accurate model for the input impedance of the probe-fed patch. In this model the RLC circuit corresponding to the (0,0) mode is missing the inductance term. This mode corresponds to a uniform electric field in the patch cavity, with no magnetic field. The resonance frequency of this mode is zero, and it corresponds to a static DC eigenmode (which can exist independently, with no source, only at DC). The resistance R in each of the circuits except the dominant (1,0) circuit can be neglected to a first approximation, since most of these modes radiate much less than the dominant mode, and because these resistances are effectively short-circuited by the inductors in the circuits (capacitor for the (0,0) mode). If a narrow frequency region around ω10 , the dominant rectangular patch mode resonance, is considered, the impedances for all other RLC circuits can be lumped into a single inductance, which models the stored energy in these other modes. The simple CAD model shown in Fig. 3 is then obtained. The inductance Lf is called the feed inductance, since most of the energy stored in the magnetic field of the higher-order modes exists near the feed (the total magnetic field inside the patch cavity is strongest at the surface of the probe, and would be infinite on the axis of the probe if it were a filamentary wire). The Q of the RLC circuit in Fig. 3 is

7

Q=R

Lω 0

, (5.26)

where the resonance frequency of the circuit is ω ω0 10

1= =

LC. (5.27)

A simple circuit analysis allows the input impedance of the CAD model in Fig. 3 to be written near resonance as

Z jX Rj Q fin f

r

≈ ++ −1 2 1( )

, (5.28)

where Xf = ω Lf and the frequency ratio fr is defined as fr = f / f0 . (5.29) The bandwidth (BW) of the patch may be calculated by assuming the patch is matched to an incoming transmission line with characteristic impedance Z0 = R at the resonance frequency (neglecting the feed inductance), and calculating the lower and upper frequency limits f1 and f2 at which the standing wave ratio (SWR) is some specified value. The bandwidth is then defined as

BW =−

= −f f

ff fr r

2 1

02 1 . (5.30)

(This bandwidth is unitless - it is multiplied by 100 to put the result in percent.) A simple calculation yields the result

BW SWRQ SWR

=−1 . (5.31)

In this chapter the specific choice SWR = 2.0 will be adopted, so that

BWQ

=12

. (5.32)

The radiation efficiency, er, of the patch is defined as the power radiated into space divided by the total power, which includes the power radiated into space and surface waves, the power dissipated in the dielectric, and the power dissipated in the patch and ground plane. In terms of the Q factors,

8

ersp

=Q

Q. (5.33)

The radiation efficiency and bandwidth of the patch are determined once the Q factors are known. In section III, CAD formulas will be presented for each of the Q factors in Eq. (5.9). To determine input impedance from the CAD formula (5.28), three additional unknown parameters must first be determined, in addition to the Q of the patch: the feed reactance (Xf ), the input resistance at resonance (R), and the resonance frequency (f0) (which is needed to determine fr for any given frequency). In the following sections CAD formulas will be presented for each of these quantities. These CAD formulas are then used along with Eq. (5.28) to determine the input impedance. 5.3 CAD FORMULAS FOR RESONANCE FREQUENCY Of primary concern in many applications is the value of the resonance frequency for a microstrip patch antenna. Since the bandwidth of these radiators is often times quite small, the accurate prediction of their resonance frequency is most important. The simplest estimation is that the resonant dimension of the patch L is just equal to one-half wavelength in the dielectric material. This zero-order result (denoted as f0

0 ) is given by Eq. (5.1) with Le = L, so that

f cL r

00

2=

ε. (5.34)

(In this formula, and all the following formulas in this section, the substrate is assumed to be nonmagnetic, so that μ r = 1.) This approximation does not take into account the finite thickness of the substrate or the non-resonant width of the patch. Thus, it is only approximately correct for very thin substrates. An equation for the fringing length proposed by Hammerstad [5] can be used to obtain a more accurate expression for the resonance frequency. Alternatively, a formula proposed by James, Hall, and Wood [6, pp. 99-100] determines the resonance frequency directly. Both formulas are commonly used for more accurate predictions for rectangular patches when the substrate is not necessarily extremely thin. Both equations share the concept of an effective dielectric constant, ε eff , given by [7]

ε ζ εε

ζeff b g

b g=

++

− +FHG

IKJ−

rr

h1

2

1 1 10

2

1 2/

, (5.35)

9

where ζ is a dimension variable that can be either L or W in the formulas presented later. Using the concept of fringing length, the resonance frequency is given by Eq. (5.1), with the effective length Le obtained by adding the fringing length ΔL to each end of the patch. The formula is

f cL L r

0 2 2=

+ Δb g ε. (5.36)

The Hammerstad formula for the fringing length [5] is

ΔLh W W

h

W Wh

=+ +FH

IK

− +FHIK

0 0 3 0 264

0 258 0 8

.412( ) . .

. .

ε

ε

eff

eff

a fb ga fb g

. (5.37)

There is one important point that should be mentioned in connection with Eq. (5.36). This formula is often used in the literature with ε eff instead of ε r . The original reference by Hammerstad [5] does not make it clear which is correct, since no formula for resonance frequency is actually given there. However, in our comparisons with measurements and the results from James (discussed below), we have obtained more accurate results by using ε r when Eq. (5.37) is used for the fringing length. Therefore, Eq. (5.36) is used in the form shown for all of the results presented here. James, et al. obtain a formula for the resonance frequency by directly modifying the zero-order result in Eq. (5.34). The formula is

f fW L

r0

00

1=

+ε

ε ε δeff effa f a f a f , (5.38)

where

δεε

ε

π ε= +

−LNM

OQP+

+ + +FHGIKJ

FHG

IKJ

L

N

MMMM

O

Q

PPPP

L

N

MMMM

O

Q

PPPPhL

Lhr

r

r

r

0 8820164 1

1 0 758 188

2..

. ln .b g b g. (5.39)

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Figure 4 shows a comparison of CAD formulas (5.36) and (5.37) with formulas (5.38) and (5.39). The normalized resonance frequencies f f0 0

0/ predicted by the formulas are plotted versus the electrical thickness of the substrate in wavelengths (λ0) for two different values of substrate relative permittivity. Figure 4a shows results for a low-permittivity substrate (ε r = 2.2), while Fig. 4b shows results for a high-permittivity substrate (ε r = 10.8). Measured results for the resonance frequency of a probe-fed rectangular patch are also included in the figures. The measured results were obtained by using a variety of different substrate thicknesses and patch sizes (with W / L = 1.5 for all patches). As can be seen, both CAD formulas predict a reduction of the resonance frequency as the substrate thickness increases. The Hammerstad formula agrees better with the measured results for the low-permittivity substrate, while the measured results are spread between the Hammerstad and James results for the high-permittivity substrates. 5.4 CAD FORMULAS FOR THE Q FACTORS In this section CAD formulas for the quality factors are presented. In the following sections these formulas are used together with the formulas introduced in Section II to obtain CAD formulas for bandwidth, radiation efficiency, and input resistance at resonance. 5.4.1 Dielectric and Conductor Q Factors CAD formulas for Qc and Qd are well known [6, pp. 76-77]. Assuming the magnetic-wall approximation, a relatively straightforward analysis yields

Qd =1

d

(5.40)

and

Qc =FHGIKJ

12 0

0η μ rs

k hR

, (5.41)

where d is the actual loss tangent of the substrate, μr is the relative permeability of the substrate, and Rs is the surface resistance of the patch and ground plane metal. (If the patch and ground plane have different surface resistances, then Rs is the average of the two resistances.) The surface resistance is calculated from the usual formula [8]

Rs =ω μσ

0

2, (5.42)

11

where ω = 2 π f and σ is the effective metal conductivity. Due to surface roughness, the effective conductivity for copper patches and ground planes is usually taken to be approximately 3.0 × 107 mhos/meter rather than that of a pure bulk material (5.8 × 107 mhos/meter). Comparing Eqs. (5.40) and (5.41), it is seen that the dielectric loss Q is independent of substrate thickness, while the conductor loss Q is proportional to the electrical thickness of the substrate (k0 h). Hence, for thin substrates, the conductor loss (which is inversely related to Qc) will always dominate over the dielectric loss.

5.4.2 Relation Between Surface-Wave and Space-Wave Q Factors The remaining two Q factors, Qsp and Qsw , determine the amount of power radiated into space and surface waves. To relate these two Q factors, a radiation efficiency er

0 is defined, which is the radiation efficiency assuming no dielectric loss ( d = 0 ) and no conductor loss (σ = ∞). This efficiency accounts only for power loss due to the excitation of surface-waves. In terms of the Q factors,

err

sp

0 =QQ

, (5.43)

where the radiation quality factor Qr is defined as

1 1 1Q Q Qr sp sw

= + . (5.44)

To a good approximation, the efficiency er

0 may be approximated by the radiation efficiency er

hed of a horizontal electric dipole on top of the lossless substrate (the loss tangent d of the substrate is set to zero). In general, the power radiated either into space or into surface waves is not approximated to a high degree of accuracy by the corresponding power radiated by a dipole, since the radiation from the patch comes from a distributed electric current and not a single point. There is, in effect, an array-factor due to the distribution of the current over the surface of the patch. There are array-factor terms for both the power radiated into space and the power radiated into surface waves. It turns out that these two array factors are similar, so that the ratio of power radiated into space divided by power radiated into surface waves is very similar for both the patch and the dipole on the lossless substrate (the accuracy of this approximation will be demonstrated later when numerical results for the radiation efficiency are presented). A simple algebraic manipulation of Eqs. (5.43) and (5.44) yields

Q Qsw spr

r

ee

=−FHGIKJ

0

01. (5.45)

Hence, to a good approximation,

12

Q Qsw sprhed

rhed

ee

=−FHG

IKJ1

. (5.46)

The surface-wave quality factor is thus known from the space-wave quality factor, provided the radiation efficiency of the dipole is known. The radiation efficiency for a horizontal electric dipole may be written as

eP

P Prhed sp

hed

sphed

swhed=

+, (5.47)

where Psp

hed is the power radiated into space by a unit-strength horizontal electric dipole on the

lossless substrate, and Pswhed is the power radiated into surface waves by the dipole. In [9], a CAD

formula for Psphed is derived. The result is

P k h csphed

r=1 80

02 0

2 2 21λ

π μb g c h , (5.48)

where c1 is a constant that depends only on the substrate material,

cn n1

12

141 1 2 5

= − +/ , (5.49)

and n1 is the index of refraction of the substrate, n r r1 = ε μ . (5.50) Equation (5.48) is derived from an asymptotic approximation of the exact space-wave power formula, assuming that the substrate is thin, so that k h0 << 1. This formula, therefore, becomes increasingly accurate as the substrate thickness decreases. Two CAD formulas for the surface-wave power Psw

hed have been previously derived. The first one, derived in [9], is

P k hnsw

hedr= −FHGIKJ

LNMM

OQPP

1 60 1 1

02 0

3 3 3

12

3

λπ μb g . (5.51)

This formula comes from an asymptotic approximation of the exact surface-wave power, assuming that the substrate is thin, so that k h0 << 1. For thin substrates, only the fundamental

13

TM0 surface wave mode is above cutoff, and Eq. (5.51) gives the power radiated into this surface wave. An improved formula for Psw

hed that is more accurate than Eq. (5.51) has been derived by Pozar [10], by keeping higher-order terms in the asymptotic approximation. The result, derived in [10] for the case of a nonmagnetic substrate (μr = 1), is

( )[ ] ( )

32 2200 0

2 21 0 0 1

1

4 1 1 1rhed

sw

r r

xkP

x k h x x

εη

ε ε

−=

⎡ ⎤+ + − +⎣ ⎦, (5.52)

where

x xxr

102

02

1=

−−ε

and

x r r r

r0

20 1

20 1 0

2

212

12

= +− + + − +

−

ε α α ε ε α α αε αc h ,

with α 0 0= s k h stan b g ,

α 1 00

20

1= − +LNMM

OQPPs

k h sk h s

k h stan

cosb g b g

b g ,

and s r= −ε 1 . Although Eq. (5.52) is more accurate than Eq. (5.51), Eq. (5.51) has the advantage of providing more physical insight into the behavior of the surface-wave power. It is seen from Eq. (5.51) that the surface-wave power varies with substrate thickness as k h0

3b g , while the space-wave power

varies as k h02b g , from Eq. (5.48). Hence, the radiation efficiency er

0 approaches 1.0 as the substrate becomes thinner. Therefore, for thin substrates, the surface-wave loss becomes negligible.

14

Figure 5 shows the accuracy of CAD formulas (5.51) and (5.52), for the radiation efficiency of a horizontal electric dipole on a lossless substrate (where d has been set to zero). In this figure these two CAD formulas are compared with an exact result, obtained from a spectral-domain calculation [11] (where the dipole was approximated by a patch having dimensions much smaller than a wavelength). Results are shown for two different substrate permittivities, corresponding to a typical duroid substrate and a high-permittivity substrate, in Figs. 5a and 5b, respectively. It is seen that both Eqs. (5.51) and (5.52) are fairly accurate up to approximately h / λ0 ≈ 0.10 for the low-permittivity substrate. For the high-permittivity substrate, Eq. (5.51) is accurate up to h / λ0 ≈ 0.02 while Eq. (5.52) is accurate up to h / λ0 ≈ 0.04. In Fig. 5b, the results from Eq. (5.52) are plotted only up to h / λ0 ≈ 0.078. At this point the equation breaks down since the argument of the square-root in the calculation of the term x0

becomes negative. The exact curve in Fig. 5b exhibits a slope discontinuity at h / λ0 ≈ 0.08, corresponding to the cutoff of the TE1 surface-wave mode on the grounded dielectric slab. 5.4.3 Space-Wave Quality Factor Equation (5.46) along with Eqs. (5.47), (5.48), and either (5.51) or (5.52), determine the surface-wave quality factor once the space-wave quality factor is known. To determine the space-wave quality factor, the power radiated by the patch antenna must be calculated, as well as the energy stored in the patch cavity. The space-wave Q factor is then determined from the standard formula for Q, Q sp

s

sp

UP

=ω 0 , (5.53)

where Us is the energy stored inside the cavity and Psp is the power radiated into space by the patch current. Both of these terms are calculated assuming a dominant mode current on the patch, corresponding to the (1,0) mode of the cavity. The current density is taken as

J xs =FHGIKJcos 'π x

Le

, (5.54)

where x x Le' /= − 2 (the primed coordinate system x y' , 'b g is measured from the center of the patch). This current density corresponds to a y -directed magnetic field inside the patch cavity, with Hy = Jsx . The length of the patch in Eq. (5.54) has been taken as the effective length, even though the actual surface current exists only over the physical dimensions of the patch. The actual surface current Jsx on the patch exhibits an edge effect, vanishing near the patch edges as

s , where s is the distance from the patch edge at x = 0 or x = L. The dominant cavity-mode current in Eq. (5.54) vanishes linearly (proportional to s) near the patch edges. The edge effect causes an additional amount of radiating current to be present on the patch, and this is approximately accounted for by choosing the patch length to be Le in Eq. (5.54). The effective length Le in this and all subsequent formulas is obtained from Le = L + 2ΔL, where Eq. (5.37) is

15

used for ΔL. For the same reasons, the effective width W W We = +2Δ is used in all subsequent CAD formulas, with ΔW calculated from Eq. (5.2).

To calculate the stored energy inside the cavity, it is noted that the energy stored in the electric and magnetic fields are equal at resonance (this is true for any cavity resonator), so that U Us

EsH= . Hence, the total stored energy is given by

U U h H x y dss s

H

sr y= = z2 2 1

4 0

2μ μ ,a f

=FHGIKJ−− zzμ μ π

02

2

2

2

2

2re

L

L

W

Wh xL

dx dye

e

e

e cos ' ' '/

/

/

/,

which results in

U h W Ls r e e=4 0μ μ . (5.55)

To calculate the power radiated into space, the patch is first replaced by an equivalent dipole that has the same dipole moment meq , defined from

m J x y dx dyeq L

L

sxW

W

e

e

e

e=

−− zz /

/

/

/' , ' ' '

2

2

2

2 b g

= 2π

W Le eb g . (5.56)

A horizontal electric dipole of amplitude I meq= will radiate approximately the same power into space as the patch (ignoring the array factor due to the distribution of the patch current over the surface of the patch). Therefore, to a first approximation, P m Psp eq sp

hed≈ 2 , (5.57) where Psp

hed is given by Eq. (5.48). Equation (5.57) becomes more accurate as the size of the patch decreases, since this approximation ignores the patch array factor. It is possible to modify Eq. (5.57) to account for the array factor, and thus improve the accuracy. To do this, a power ratio p is defined as the ratio of the actual power radiated into space by the patch, to the power radiated into space by the equivalent dipole with moment meq ,

16

pP

m Psp

eq sphed≡ 2 . (5.58)

The factor p approaches 1.0 as the size of the patch becomes small compared to a wavelength (which in turn implies that the substrate permittivity becomes large). An approximate formula for the p factor may be obtained from some algebraic manipulations. Equation (5.58) is first written in terms of the far-field power patterns as

pS r r d d

m S r r d d

r

eq rhex

= zzzz

, , sin

, , sin

/

/

θ φ θ θ φ

θ φ θ θ φ

ππ

ππ

b gb g

2

0

2

0

2

2 2

0

2

0

2 , (5.59)

where Sr is the power density (radial component of the Poynting vector) in the far-field from the patch current, and Sr

hex is the power density from an x-directed unit-strength dipole. To evaluate Eq. (5.59), the power density radiated by the patch is expressed as S r m S r Ar eq r

hex, , , , ,θ φ θ φ θ φb g b g b g= 2 2, (5.60)

where A θ φ,b g is the patch array factor, defined as

Am

J x y e dx dyeq

sxs

j k x k yx yθ φ, ' , ' ' 'b g b g d i= z +1

=FHGIKJ−−

+zzπ π2 2

2

2

2

W Lx

Le dx dy

e e eL

L

W

W j k x k y

e

e

e

e x ycos ' ' '/

/

/

/ ' 'd i , (5.61)

where k kx = 0 sin cosθ φ , k ky = 0 sin sinθ φ . Evaluating the integrals in Eq. (5.61) gives the result A T Tθ φ θ φ θ φ, , ,b g b g b g= 1 2 , (5.62)

where

17

T

k W

k W

y e

y e1

2

2

θ φ,sin

b g =FHGIKJ

(5.63)

and

T

k L

k L

x e

x e2 2 2

2

1 22

θ φ

π

,cos

b g =FHGIKJ

− FHGIKJFHGIKJ

. (5.64)

Substituting Eqs. (5.63) and (5.64) into Eq. (5.62), and then inserting Eq. (5.62) into Eq. (5.60), allows the power density radiated by the patch to be written in terms of the power density radiated by the unit-strength dipole. Equation (5.60) is then used in Eq. (5.59) to evaluate the factor p. A closed-form approximate evaluation of Eq. (5.59) may be performed, if the assumption is made that the substrate is thin compared to a wavelength and the patch dimensions are not too large compared to a wavelength [9]. This evaluation is performed in Appendix A. The result is

pa

k W a a k W c k L a c k W k Le e e e e= + + + FHGIKJ + FHG

IKJ + F

HGIKJ1

102 3

56015

170

20

222

4 04

2 02

2 2 02

02b g c h b g b g b g b g ,

(5.65) where a2 016605= − . , a4 0 00761= . , c2 0 0914153= − . . It should be pointed out that in reference [9] an error was made in the derivation of the factor p, so the above expression is different from the one in [9]. Figure 6 shows a plot of the factor p versus the electrical patch length, for different aspect ratios We / Le. The results from Eq. (5.65) are compared with the exact p factor, calculated by numerically evaluating Eq. (5.59). Figure 6 shows that CAD formula (5.65) is quite accurate for resonant patches, since the maximum patch length (corresponding to an air substrate) would be Le / λ 0 ≈ 0.5. Equation (5.58) allows the space-wave power radiated by the patch to be written as P p m Psp eq sp

hed= 2 . (5.66)

18

Substituting Eqs. (5.48) and (5.56) into Eq. (5.66) yields the final CAD formula for the space-wave power radiated by the patch. Equations (5.66) and (5.55) may then be substituted into Eq. (5.53) to obtain a final CAD formula for the space-wave quality factor. After some simplifications, the final result is

Qspr e

epcLW h

=FHGIKJFHGIKJFHGIKJ

316

1

1 0

ελ/

. (5.67)

From this expression it is seen that the space-wave quality factor varies inversely with the electrical thickness of the substrate (under the assumption that this thickness is small, which is one of the assumptions in the derivation). 5.5 CAD FORMULA FOR BANDWIDTH Using the CAD formulas for the quality factors derived in Section IV, a final CAD formula for the bandwidth may be obtained, using the formulas presented in Section II. 5.5.1 CAD Formula The bandwidth of the rectangular patch antenna is found from Eq. (5.32), using Eq. (5.9) along with Eqs. (5.40), (5.41), (5.46), and (5.67). The result is

BW Rh

pc h WL ed

s

r r

e

e rhed= +

FHG

IKJFHGIKJ +FHIKFHGIKJFHGIKJFHGIKJFHGIKJ

LNMM

OQPP

12

1 163

1

0 0

1

0π η μ λ ε λ/. (5.68)

Equations (5.47) and (5.48) are used along with either Eq. (5.51) or (5.52) to calculate the er

hed in this equation (Eq. (5.52) will give more accurate results). Equations (5.49) and (5.65) are used to calculate the terms c1 and p, respectively. If dielectric and conductor losses are neglected (valid for substrates of moderate thickness), the bandwidth formula simplifies to

BW pc h WL er

e

e rhed

0 1

0

12

163

1= FHIKFHGIKJFHGIKJFHGIKJFHGIKJ

LNM

OQPε λ. (5.69)

This CAD formula shows that the bandwidth is proportional to the electrical substrate thickness h / λ 0 (although the electrical substrate thickness must still be relatively small in order for this formula to be accurate). For thicker substrates, the radiation efficiency er

hed begins to decrease due to surface-wave excitation, which causes the bandwidth to increase even faster than linearly. However, this is normally undesirable, since bandwidth is being increased at the expense of lost power. For thin substrates, the bandwidth will be determined by the dielectric and conductor Q factors, so that the first two terms inside the bracket of Eq. (5.68) dominate. Ultimately, for very thin substrates, it is the conductor Q that dominates. In this region the bandwidth increase is

19

inversely proportional to the electrical substrate thickness. Here again, however, bandwidth is being increased at the expense of lost power, which is normally undesirable. It is also noted from Eq. (5.69) that the bandwidth is inversely proportional to the substrate permittivity. Hence, higher bandwidths are obtained by using thicker substrates with a low permittivity. The bandwidth is also increased by increasing the aspect ratio W / L. For very thin substrates, where dielectric and conductor losses dominate, the bandwidth is seen from Eq. (5.68) to be essentially independent of the patch dimensions (this is an undesirable region of operation). 5.5.2 Results

Figure 7a shows a comparison between CAD formula (5.68) (which includes the dielectric and conductor losses) and the exact bandwidth, obtained from the cavity model, for a patch with W / L = 1.5. The same effective width and length are used in the cavity-model calculation as in the CAD formula. Results are shown for a resonant patch on a low-permittivity substrate (ε r = 2.2) and a high-permittivity substrate (ε r = 10.8). In these figures a loss tangent of d = 0.001 is assumed for both substrates, and a conductivity of σ = 3.0 × 107 mhos / meter is assumed for the patch and ground plane. A frequency of 5.0 GHz is assumed so that the factor Qc may be calculated. The results are then displayed versus electrical thickness h / λ 0 . Equation (5.52) has been used to calculate er

hed in Eq. (5.68). In the cavity-model calculation, the frequencies for which SWR = 2.0 are determined numerically, assuming a perfect match at the center frequency. The feed position is at x0 = L / 4, y0 = W / 2 for the cavity-model calculation (although bandwidth is not sensitive to feed location). When the electrical thickness of the substrate exceeds h / λ 0 ≈ 0.05 for the low-permittivity substrate, the patch can no longer be made resonant by changing the length of the patch. In this region the probe inductance is sufficient to make the input impedance inductive for all patch lengths. The patch resonance is then defined from the patch length that minimizes the input reactance. The value of the reactance at the minimum point is subtracted from the input impedance at each frequency, and the SWR is then calculated by assuming a perfect match at resonance. The hollow dots for the low-permittivity case in Fig. 7a indicate those points where this modified calculation has been used.

There is excellent agreement between the CAD formula and the “exact” cavity-model result for thin substrates. The CAD formula is accurate up to h / λ 0 ≈ 0.06 for the low-permittivity substrate, and up to h / λ 0 ≈ 0.03 for the high-permittivity substrate. In both cases, the CAD formula is therefore accurate up to h d/ λ ≈ 0.1, where λ d is the wavelength in the dielectric. The results from the CAD formula are only plotted up to h / λ 0 = 0.078 for the high-permittivity substrate, for the same reason discussed previously in connection with Fig. 5b. Figure 7a shows that the bandwidth increases almost monotonically with increasing substrate thickness. This is true until the substrate thickness becomes very small, at which point the conductor loss begins to dominate.

Figure 7b shows a comparison between the same CAD results shown in Fig. 7a and recently obtained measured results. For the measured results, a slightly different definition of resonance

20

frequency was used, namely the frequency for which the real part of the input impedance was maximum. However, this causes a negligible difference in the bandwidth calculation, provided the electrical thickness of the substrate is relatively small. As for Fig. 4, the measured results were obtained by using a variety of substrate thicknesses and patch sizes (with W / L = 1.5 for all patches). The agreement is seen to be quite good for both the low- and high-permittivity substrates, for the ranges plotted. 5.6 CAD FORMULA FOR RADIATION EFFICIENCY Using the CAD formulas for the quality factors derived in Section IV, a final CAD formula for the radiation efficiency may be obtained, using the formulas presented in Section II. 5.6.1 CAD Formula Equation (5.33), along with Eqs. (5.9) and (5.40), (5.41), (5.46), and (5.67), give the final CAD formula for radiation efficiency. The result is

e e

e Rh p c

LW h

rrhed

rhed

ds

r

r e

e

=

+ +FHG

IKJFHGIKJ

LNM

OQPFHGIKJFHGIKJFHGIKJFHGIKJ

LNM

OQP

1 1 316

10 0 1 0π η μ λ

ελ/ /

. (5.70)

As for the bandwidth formula, Eqs. (5.47) and (5.48) are used along with either Eq. (5.51) or (5.52) to calculate er

hed in this equation (Eq. (5.52) being more accurate). Equations (5.49) and (5.65) are used to calculate c1 and p. Equation (5.70) indicates that the radiation efficiency of the patch approaches the radiation efficiency of a horizontal electric dipole on a corresponding lossless substrate as the electrical thickness of the substrate increases. For thicker substrates the radiation efficiency will therefore decrease due to increased surface-wave excitation. For electrically thin substrates the dielectric and conductor losses dominate the surface-wave loss. In this region the efficiency increases as the substrate thickness increases. For very thin substrates the conductor loss is the dominant loss mechanism. In this region the radiation efficiency is proportional to h / λ 0

2b g and is very poor. Equation (5.70) predicts that the efficiency increases quadratically for very thin substrates, but eventually reaches a maximum for a particular substrate thickness. The efficiency then decreases as the substrate thickness is increased beyond this point, due to surface-wave loss. The optimum substrate thickness will depend on the permittivity of the substrate (it will increase as the permittivity decreases). For thicker substrates the dimensions of the patch do not significantly affect the radiation efficiency, since the efficiency is essentially the same as that of the horizontal electric dipole. However, for thin substrates the efficiency is improved by making the aspect ratio W / L larger.

21

5.6.2 Results Figure 8a shows a comparison between CAD formula (5.70) and the exact radiation efficiency obtained from a spectral-domain calculation [11], for a resonant patch on a low-permittivity substrate and a high-permittivity substrate, with W / L = 1.5. A loss tangent of d = 0.001 is assumed for the substrate, and a conductivity of σ = 3.0 × 107 mhos / meter is assumed for the patch and ground plane. A frequency of 5.0 GHz is again assumed so that conductive losses may be calculated. Equation (5.52) has been used to calculate er

hed in Eq. (5.70). The spectral-domain calculation does not directly account for conductor loss, since only space-wave and surface-wave powers are calculated. But conductor loss may be calculated separately by making use of the approximate formula (5.41) for Qc , along with Eq. (5.55) for the stored energy and the definition of Qc [6]. Because Eq. (5.41) is used in both the CAD and spectral-domain calculations, this comparison does not reflect the accuracy of Eq. (5.41). However, when the substrate is thin enough so that conductor loss is important in the calculation of efficiency, Eq. (5.41) is expected to be quite accurate. Figure 8a shows that CAD formula (5.70) is accurate for substrate thicknesses up to h / λ0 ≈ 0.1 for the low-permittivity substrate, and up to h / λ0 ≈ 0.03 for the high-permittivity substrate. Figure 8a also shows that the efficiency is higher for the low-permittivity substrate, for the same electrical thickness h / λ0. The optimum substrate thickness is h / λ0 ≈ 0.02 for the low-permittivity substrate, and h / λ0 ≈ 0.015 for the high-permittivity substrate. The CAD formula is accurate for predicting the optimum substrate thickness. Figure 8b shows the radiation efficiency for the same patch and substrate as in Fig. 8a, but with the dielectric and conductor losses neglected. That is, the loss tangent of the substrate has been set to zero, and the patch and ground plane are assumed to be perfectly conducting. The radiation efficiency therefore only accounts for surface-wave loss. In this case it is seen that the efficiency closely resembles that of a horizontal electric dipole on the same lossless substrate, shown in Figs. 5a and 5b. When conductor and dielectric losses are neglected the efficiency decreases monotonically as the substrate thickness increases, at least until the substrate becomes quite thick.

22

5.7 CAD FORMULA FOR INPUT RESISTANCE The input impedance of the patch may be calculated from the cavity model formula (5.19), using Eqs. (5.8) and (5.9) along with the CAD formulas for the Q factors that were given in Section II. At resonance, the main contribution to the input impedance (which is purely resistive) comes from the dominant (1,0) mode. A simple CAD formula for the input resistance at resonance may be derived by including only the (1,0) contribution to the sum in Eq. (5.19). This resistance is the value of the resistor R that appears in the CAD circuit of Fig. 3. Assuming that only the dominant (1,0) mode is significant, the electric field inside the cavity is found from the (1,0) term of Eq. (5.4), with A10 given by Eq. (5.10). The input resistance is then determined from Eq. (5.11) as

R h A xLin

e

≈−FHGIKJ10

0cos π

=FHGIKJFHGIKJFHGIKJω μ μ π

0 012

2 02 1r

e e e e

hW L k

xL

cos . (5.71)

Using the relation k10 Le = π, valid at resonance, this simplifies to

R LW

h xLin r

e

e

e e

=FHGIKJFHGIKJFHGIKJFHGIKJ

4 10

0

2 0

πμ η

λπb g cos . (5.72)

Using Eqs. (5.8) and (5.9) along with (5.40), (5.41), (5.46), and (5.67) to evaluate the effective loss tangent e , Eq. (5.72) becomes

R

LW

h xL

Rh

pc WL

he

in

re

e e

ds

r r

e

e rhed

=

FHGIKJ

FHGIKJFHGIKJFHGIKJ

+FHG

IKJFHGIKJ +FHGIKJFHGIKJFHGIKJFHGIKJFHGIKJ

4

1 163

1

00

2 0

0 0

1

0

πμ η

λπ

π η μ λ ε λ

b g cos

/

. (5.73)

If dielectric and conductor losses are neglected, Eq. (5.73) further simplifies to

Rpc

e LW

xLin

r rrhed e

e e

=FHG

IKJFHGIKJFHGIKJ

34

0

1

22 0

πε μ η πcos . (5.74)

Equation (5.74) indicates that the resonant input resistance is approximately independent of the substrate thickness, provided the electrical substrate thickness is large enough so that dielectric and conductor losses may be neglected (the only term in Eq. (5.74) that varies with substrate

23

thickness is erhed , which is approximately equal to 1.0 for relatively thin substrates). For thin

substrates, Eq. (5.73) shows that the input resistance tends to zero as h / λ 02b g . The input

resistance is independent of the probe position along the nonresonant dimension (y axis), according to these formulas. These formulas predict that the resistance is maximum along the resonant edges of the patch (x0 = 0, L), and zero along the centerline (x0 = L / 2). Equation (5.74) also shows that the input resistance decreases as the aspect ratio W / L increases, and that the input resistance increases as the substrate permittivity increases. When implementing Eq. (5.73), x0 is taken to be the x-axis distance from the probe to the effective edge of the patch, after the fringing length ΔL has been added to both edges of the patch. Therefore, the original distance x0 (measured from the physical edge of the patch) is replaced by x L0 + Δ in Eq. (5.73). Figure 9a shows a plot of resonant input resistance versus electrical substrate thickness for a patch with W / L = 1.5, for both a low-permittivity and a high-permittivity substrate. Again, a frequency of 5.0 GHz is assumed so that conductive losses may be calculated. The feed position in this case is at a fixed position, halfway between the center of the patch and the edge at x0 = L / 4, y0 = W / 2, and the probe radius is 0.05 cm. The results from Eq. (5.73) are compared with results from the cavity model. This figure shows that the higher-order modes become more important as the substrate thickness increases, since there is more deviation between the results from Eq. (5.73) and the cavity model. The CAD formula is accurate for substrate thicknesses up to h / λ 0 ≈ 0.04 for the low-permittivity substrate, and h / λ 0 ≈ 0.05 for the high-permittivity substrate. The dashed part of the CAD curves indicates that the patch reactance never went to zero. In this case, as in Fig. 7a, the patch resonance was defined from the patch length that minimized the reactance. The hollow dots and squares for the low-permittivity and high permittivity cavity-model curves in Fig. 9a indicate that the patch was never resonant in this model, and resonance was therefore defined in the same way as for the CAD results. Both models predict that the patch ceases to become resonant at about the same substrate thickness, h / λ 0 ≈ 0.04 for the low-permittivity substrate, and h / λ 0 ≈ 0.05 for the high-permittivity substrate. Figure 9b shows a comparison between the same CAD results shown in Fig. 9a and the results from a spectral-domain calculation [12], which assumes a filamentary probe feed and one full-domain basis function in both the x and y directions. It should be commented that the filamentary probe model used to obtain the spectral-domain results is known to lose accuracy for thicker substrates - the accuracy is probably questionable for h / λ 0 > 0.02. Because the probe inductance is not accounted for in the spectral-domain calculation, the resonance frequency will be shifted from that predicted by the CAD formula and the cavity model. This results in a significant difference in the predicted input resistance at resonance, since the input resistance is a function of frequency. Neglecting the probe inductance implies that the input resistance will be a maximum at the resonance frequency. With a nonzero probe inductance, the input resistance at resonance will be less than the maximum input resistance.

24

A more accurate calculation of input impedance can be obtained by using more basis functions together with an attachment mode on the patch [13]. Figure 9c shows a comparison between the CAD results in Fig. 9a with recently obtained measured results for the resonant input resistance. As mentioned previously in connection with Fig. 7b, the definition of resonance frequency for the measurements was slightly different than the one used for the calculations. The measured results were obtained by using a variety of different substrate thicknesses and patch sizes, with all patches having an aspect ratio of W / L = 1.5. The agreement between the measured results and the CAD results is quite good, agreeing fairly well for substrate thickness in the range h / λ 0 < 0.02 for the low-permittivity substrate, and h / λ 0 < 0.015 for the high-permittivity substrate. The spectral-domain results shown in Fig. 9b do not appear to agree as well with the measured results as the CAD and cavity model. Due to the assumption of a filamentary probe feed and only one full-domain basis function in the x and y directions, the spectral-domain results are probably not very accurate beyond h / λ 0 ≈ 0.02. 5.8 CAD FORMULA FOR PROBE REACTANCE The last remaining element of the CAD model in Fig. 3 that needs to be determined is the probe reactance Xf = j ω Lf . All of the terms needed in Eq. (5.28) will then be known, and this equation may be used to calculate input impedance at any frequency. The probe is assumed to be a circular wire of radius a, centered at (x0, y0). Comparing Figs. 2 and 3, the probe reactance is that part of the input impedance due to all other modes except the dominant (1,0) mode. Therefore, from Eq. (5.12),

X hI

Jk kf r

z mn

mn mnm n mn

= −−≠

∑ω μ μ ϕϕ ϕ0

02

2

1 0 12 2

1( , ),( , ) ( , ) b g . (5.75)

In the preceding equation the effective wavenumber ke has been taken as the real-valued wavenumber k k r r1

202= ε μ , which is a sufficiently accurate approximation for all modes except

the dominant (1,0) mode. Specializing to the case of the rectangular patch, using Eq. (5.19), the formula for probe reactance becomes

X h

m xL

n yW

W L

n w

k mL

nW

f re e

e em n

m n

p

e e

= −

FHGIKJFHGIKJ

FHGIKJ + +

FHGIKJ

−FHGIKJ −FHGIKJ

≠∑ω μ μ

π π

δ δ

π

π π0

2 0 2 0

0 01 0

12

2 2

41 1

2cos cos

( , ) ( , ) b gb g

sinc2

. (5.76)

Equation (5.76) represents the exact probe reactance, defined so as to be consistent with Fig. 3. Although it is not difficult to numerically evaluate Eq. (5.76), it is convenient to approximate this equation with a simple CAD formula. It is especially useful to have such a simple CAD

25

formula for patches on thin substrates, since the probe reactance will not be very large in this case, so that even a fairly approximate formula should suffice. One simple approximate formula for Xf can be obtained by ignoring the cavity boundaries and calculating the reactance of the same probe in an infinite parallel-plate waveguide [14, p. 228]. The resulting expression for the probe reactance is

X k hk af r

r r

= − +FHG

IKJ

LNMM

OQPP

ηπμ γ

ε μ0

002

2b g b gln , (5.77)

where γ = 0.577216 is Euler’s constant. This formula neglects the variation of the probe reactance with position, and becomes more accurate as the probe radius becomes smaller. (As the probe radius becomes smaller, the energy stored in the magnetic field is concentrated more strongly near the probe, and thus the feed position or the shape of the patch becomes less important.) Because of the logarithm term, the probe reactance will increase very slowly as the probe radius decreases. If the probe radius is not too small, and the substrate is electrically thin, the probe reactance will be small and can usually be neglected. However, for thicker substrates the probe reactance may be appreciable. Figure 10 shows a comparison between Eq. (5.77) and the exact probe reactance, Eq. (5.76). The probe reactance in Eq. (5.76) was actually calculated by finding the input reactance when the patch length is varied slightly to maximize the input resistance. At this point the RLC circuit in Fig. 3 is resonant, so the input reactance is the probe reactance. This is equivalent to calculating the probe reactance with Eq. (5.76). The reactance is plotted versus feed position at 5.0 GHz, for a probe radius of 0.05 cm. The probe position is varied in the x direction, keeping y0 = W / 2. The patch is of resonant length on a low-permittivity substrate, with an aspect ratio W / L = 1.5. This figure shows that the CAD formula underestimates the probe reactance. The actual probe reactance is a function of feed position, increasing as the probe approaches the edge of the patch. Equation (5.77) is an asymptotic formula, becoming more accurate as the probe radius decreases. Equation (5.77) is seen to be fairly accurate when the probe is not too close to an edge of the patch. 5.9 RESULTS FOR INPUT IMPEDANCE Equation (5.28) is the final CAD formula for input impedance. Equation (5.77) is used to determine Xf . Equation (5.9) along with Eqs. (5.40), (5.41), (5.46), (5.47), (5.48) and (5.52) are used to determine Q. Equation (5.73) is used to determine R. Equation (5.36) is used to determine the resonance frequency f0, in order to calculate the frequency ratio fr in Eq. (5.29). Figures 11a and 11b show a comparison between input impedance (real and imaginary parts, respectively) calculated by Eq. (5.28) and by the cavity model (Eq. (5.19)). The input impedance is plotted versus frequency for a patch of dimensions L = 2.0 cm, W = 3.0 cm, on a low-permittivity substrate of moderate thickness (h = 0.1524 cm, corresponding to h / λ 0 =

26

0.0254 at 5.0 GHz), with an aspect ratio of W / L = 1.5. A loss tangent of d = 0.001 is assumed for the substrate, and a conductivity of σ = 3.0 × 107 mhos / meter is assumed for the patch and ground plane. This comparison shows that the simple CAD model in Fig. 3 is quite accurate over the bandwidth of the patch. 5.10 RADIATION PATTERNS The calculation of far-field radiation patterns from patch antennas is well-known. In this section the two basic models that are commonly used, the electric-current model and the magnetic-current model, are reviewed. To obtain simple formulas, the radiation patterns are based on the currents associated with the dominant cavity mode only. In this case it is shown that the two models give exactly the same results, provided the patch is at resonance. Final formulas are first presented for the rectangular patch on an infinite substrate and ground plane (Fig. 1). Formulas are also presented for the radiation pattern of a patch antenna on a truncated substrate, where the substrate terminates at the edges of the patch (while the ground plane is still infinite). For this case both an electric- and a magnetic-current model may also be used, with both methods again agreeing at resonance. The presentation here follows that in [15]. 5.10.1 Infinite Substrate 5.10.1.1 Electric-current model The electric-current model for the rectangular patch antenna in Fig. 1 uses the patch surface current in Eq. (5.54) to determine the far field. The radiation comes from the actual patch current on the substrate, as shown in Fig. 12a. The far-field components Ei (i = θ or φ) are calculated from E r E r J x y e dx dyi i

hexsxL

L

W

W j k x k y

e

e

e

e x y( , , ) , , ' , ' ' '/

/

/

/ ' 'θ φ θ φ=−−

+zza f a f d i2

2

2

2, (5.78)

where Ei

hex is the corresponding far-field pattern of a unit-strength horizontal electric dipole in the x direction on top of the infinite substrate, and k kx = 0 sin cosθ φ , k ky = 0 sin sinθ φ . The dipole pattern Ei

hex is given in Appendix B. Performing the integration, the expression for the far-field pattern becomes

27

E r E r W Lk W

k W

k L

k Li ihex e e

y e

y e

x e

x e

( , , ) , ,sin cos

θ φ θ φ ππ

= FHIK

FHGIKJ

L

N

MMMM

O

Q

PPPP

FHIK

FHIK −FHIK

L

N

MMMM

O

Q

PPPPa f

22

2

2

2 2

2 2 . (5.79)

5.10.1.2 Magnetic-current model The magnetic-current model is based on applying the equivalence principle to the patch cavity to find equivalent sources at the boundary [14, pp. 106-110]. Assuming an idealized patch cavity with magnetic-wall boundary conditions, only equivalent magnetic currents will exist on the sides on the patch, since the tangential magnetic field is zero on the side walls. The electric field inside the cavity that is used to find the equivalent magnetic currents is based on the assumption of a patch current given by Eq. (5.54). From Maxwell’s equations, the corresponding electric field is

E x y j L xLz r e

e

' , ' sin 'b g b g=FHGIKJπ

ω μ μ π0 . (5.80)

(The primed coordinate system is defined with the origin at the center of the patch surface, on top of the substrate.) The equivalent magnetic currents are found from M n Es =− × , with n being the outward normal and E the electric field at the edges of the patch. The edges at x Le' = ± / 2 are termed the “radiating edge” (RE), while the edges at y We' /= ± 2 are termed the “nonradiating edges” (NRE) [16, pp. 122-123]. The expressions for the equivalent magnetic currents at the edges are M ys M= 0 x Le' /= ± 2 (RE) (5.81)

M xse

M xL

=FHGIKJ∓ sin '

0π y We' /= ± 2 (NRE), (5.82)

where

M j Lr e0 0=π

ω μ μb g . (5.83)

In the magnetic-current model most of the radiation comes from the “radiating” edges, since the currents are uniform along these edges, and in the same direction on both edges. The magnetic currents are assumed to reside inside the substrate, as shown in Fig. 12b. The radiation pattern is found from integrating over these currents. The total pattern comes from the RE and NRE contributions,

28

E r E r E ri i

REiNRE, , , , , ,θ φ θ φ θ φb g b g b g= + , (5.84)

where

E r k L M E r z e dy dziRE

x e W

W

h ihmy jk y

e

e y, , cos / , , ; ' ' '/

/ 'θ φ θ φb g b g b g=−− zz2 2 02

20 (5.85)

and

E r j k W M xL

E r z e dx dziNRE

y ee

L

L

h ihmx jk x

e

ex, , sin / sin ' , , ; ' ' '

/

/ 'θ φ π θ φb g d i b g= −FHGIKJ−− zz2 2 02

20. (5.86)

In these equations E r zi

hmx , , ; 'θ φb g and E r zihmy , , ; 'θ φb g are the far-field components of a unit-

strength horizontal magnetic dipole at x' = 0 , y' = 0 , and arbitrary z' (height h z− | ' | above the ground plane), in the x or y direction, respectively. For completeness, formulas for these terms are given in Appendix B. Performing the integrations in Eqs. (5.85) and (5.86) yields,

E r hW M E r k L k Wk hi

REe i

hmy x e y ez, , , , ; cosθ φ θ φb g b g b g b g= F

HGIKJFHGIKJ2 0

2 20 1sinc tanc (5.87)

and

E r L h M E rk W

k h

k L k L

k LiNRE

e ihmx y e

z

x e x e

x e

, , , , ; sincos

θ φ θ φπ

b g b g b g b g=FHGIKJ

FHGIKJFHGIKJ

FHGIKJ −FHGIKJ

L

N

MMMM

O

Q

PPPP2 0

22 2

2 2

0 1 2 2tanc , (5.88)

where k k nz1 0 1

2 2= − sin θ (5.89) and n1 is the index of refraction of the substrate (Eq. (5.50)). The sinc function appearing in Eq. (5.87) is defined in Eq. (5.18), while the tanc function is defined as

tanc xx

xb g b g

=tan

. (5.90)

It is proven in [15] that the electric-current model (Eq. (5.79)), gives the exact same result as the magnetic-current model (Eq. (5.84) with Eqs. (5.87) and (5.88)), provided the dimensions of the patch are such that the dominant cavity mode is at resonance, so that

29

k Le1 =π . This insures that the cavity field used to determine the patch surface current (electric-current model) and the magnetic currents at the edges (magnetic current model) is a valid field that satisfies Maxwell’s equations. Figure 13 shows the E- and H-plane radiation patterns for a typical patch on a low-permittivity substrate, where the patch length has been chosen to make the patch resonant. Only one pattern is shown in each plane, since both models give the same result. The E-plane is broader than the H-plane, and the pattern in both planes tends to zero at the horizon ( )θ = 90o . 5.10.2 Truncated Substrate The radiation pattern from a patch on a truncated substrate can also be obtained using either an electric-current model or a magnetic-current model. The geometry is the same as in Fig. 1, except that the substrate terminates at the edges of the patch. The ground plane is assumed to remain infinite, however. In this case the derivations make use of the patterns of dipoles in free-space over a ground plane, rather than dipoles on top of an infinite substrate. 5.10.2.1 Electric-current model The electric-current model uses the same patch current given in Eq. (5.54), but replaces the substrate with equivalent polarization currents J d and M d . A z-directed electric polarization current exists in the substrate. A y-directed magnetic polarization current will also exist if the substrate is magnetic (μ r ≠ 1). Although this is not usually the case in practice, this possibility will be included for completeness. The radiation model is shown in Fig. 12c. The far-field pattern is obtained from summing the contributions from the patch surface current, the electric polarization current, and the magnetic polarization current so that E r E r E r E ri i

sie

im, , , , , , , ,θ φ θ φ θ φ θ φb g b g b g b g= + + , (5.91)

where the superscripts s, e, and m denote patch surface current, electric polarization current, and magnetic polarization current, respectively. The contribution from the patch surface current is exactly the same as that in Eq. (5.79) for the infinite substrate case, provided that ε r = 1 and μ r = 1 are used when calculating the dipole pattern Ei

hex (which will then be denoted Eihex0 ).

Therefore, using Eq. (5.79),

E r E r W Lk W

k W

k L

k Lis

ihex e e

y e

y e

x e

x e

( , , ) , ,sin cos

θ φ θ φ ππ

= FHIK

FHGIKJ

L

N

MMMM

O

Q

PPPP

FHIK

FHIK −FHIK

L

N

MMMM

O

Q

PPPP0

2 222

2

2

2 2

a f , (5.92)

where

30

E r j E k h ehex jk hθ

θθ φ θ φ θ00 02 0, , cos cos sin cos cosb g b g= − (5.93)

and E r j E k h ehex jk hφ

θθ φ φ θ00 02 0, , sin sin cos cosb g b g=− − (5.94)

with

E jr

e jk ro0

0

4=− −ω μ

π. (5.95)

The distance r in this last equation is measured from the origin of the primed coordinate system (at the middle of the patch surface) to the far-field observation point. The far-field contribution from the electric polarization current J j Ez

dr z= −ω ε ε0 1b g (5.96)

is polarized in the θ direction only. This contribution is found from E r E r k h

J x y k h z e dx dy dz

e hez

zd

L

L

W

W

h

j k x k y

e

e

e

e x y

θ θθ φ θ φ θ

θ

, , , , ; sec cos

' , ' cos ( ' ) cos ' ' '/

/

/

/ ( ' ' )

a f a f b ga f b g

=

⋅ +−−−

+zzz0

0

2

2

2

20

0

0

, (5.97)

where E rhezθ θ φ0 0, , ;b g is the far-field of a unit-strength vertical electric dipole at x' = 0 , y' = 0 ,

z' = 0 (height h above the ground plane), given as E r E k h ehez jk hθ

θθ φ θ θ00 00 2 0, , ; sin cos cos cosb g b g= − − . (5.98)

Substituting Eq. (5.96) and (5.80) into Eq. (5.97) and performing the integrations, the result is E r E W L h M e

k Wk h

k L k L

k L

ie

e e rjk h

y e

x e x e

x e

, ,

sin coscos

cosθ φ ω ε ε

θ θπ

θb g b gb gb g

b g

= −

⋅FHGIKJ

FHGIKJFHGIKJ

FHGIKJ −FHGIKJ

L

N

MMMM

O

Q

PPPP

−2 1

22 2

2 2

0 0 0

0 2 2

0

sinc sinc . (5.99)

If the substrate is magnetic, a magnetic polarization current

31

M j H j Jyd

r y r sx= − = −ω μ μ ω μ μ0 01 1b g b g (5.100) will exist. The far-field contribution from this current will be E r E r k h

M x y k h z e dx dy dz

im

ihmy

yd

L

L

W

W

h

j k x k y

e

e

e

e x y

, , , , ; sec cos

' , ' cos ( ' ) cos ' ' '/

/

/

/ ' '

θ φ θ φ θ

θ

a f a f b ga f b g d i

=

⋅ +−

−

−−

+zzz0

0

2

2

2

20

0

0

, (5.101)

where E ri

hmy0 0, , ;θ φb g are the far field components from a unit-strength horizontal magnetic dipole in the y direction at x' = 0 , y' = 0 , z' = 0 . These components are given by

( ) ( ) 0 cos00 0

0

2, , ;0 cos cos cos jk hhmyE r E k h e θθ θ φ φ θ

η−=− . (5.102)

( ) ( ) 0 cos00 0

0

2, , ;0 cos sin cos cos jk hhmyE r E k h e θφ θ φ θ φ θ

η−= . (5.103)

Inserting Eqs. (5.100) and (5.54) into Eq. (5.101) and performing the integrations, the result is E r j W L h E r e

k W k h

k L

k L

im

r e e ihmy jk h

y e

x e

x e

, , , , ;

/ coscos

cosθ φ π ω μ μ θ φ

θπ

θb g b gb gb g b g

d i b g

= −

⋅

FHGIKJ

FHGIKJ −FHGIKJ

L

N

MMMM

O

Q

PPPP

−

21 0

2 2

2 2

00

0 2 2

0

sinc sinc . (5.104)

5.10.2.2 Magnetic-current model The magnetic-current model for the patch on a truncated substrate is the same as for the magnetic-current model on the infinite substrate, with the substrate replaced by free-space. The radiation model is shown in Fig. 12d. The radiation pattern is given by Eq. (5.84) along with Eqs. (5.87) and (5.88), with the terms Ei

hmx and Eihmy calculated by assuming that ε r = 1 and

μ r = 1. In this case (using a 0 superscript again to denote the absence of the substrate), the far-field components Ei

hmy0 are given by Eqs. (5.102) and (5.103). The far-field components Eihmx0

are given by

( ) ( ) 0 cos00 0

0

2, , ;0 sin cos cos jk hhmxE r E k h e θθ θ φ φ θ

η−= (5.105)

32

( ) ( ) 0 cos00 0

0

2, , ;0 cos cos cos cos jk hhmxE r E k h e θφ θ φ θ φ θ

η−= . (5.106)

A proof in [15] (similar to the one for the infinite substrate) shows that the electric- and magnetic-current models give the same result for the truncated patch, provided the dominant cavity mode is resonant ( k Le1 =π ). Figure 14 shows the E- and H-plane patterns for the same resonant patch as in Fig. 13, but with a truncated substrate. In this figure only one curve is shown for each plane as before, since the two models give the same result. The patterns of the truncated-substrate patch are very similar to those of the infinite substrate patch shown in Fig. 13. The main difference is near the horizon, where the E-plane pattern remains nonzero as θ approaches 90 o in the truncated case. 5.11 CAD FORMULA FOR DIRECTIVITY In this section a CAD formula for the directivity of a rectangular patch antenna is derived, using the results of the previous section for the radiation pattern along with the results from Section IV for the space-wave power radiated by the patch. The directivity is calculated at broadside, which will always be the maximum directivity (at least for reasonably thin substrates). The CAD formula is derived for the case of a patch on an infinite substrate, although the formula should be fairly accurate for the case of a truncated substrate as well, since the patterns are nearly the same for these two cases except near the horizon. Because the directivity formula is derived from the CAD formula for space-wave power, it has the same property of being more accurate for thinner substrates. The directivity at θ = 0, relative to an isotropic radiator, is defined as

D rS

Pr

sp

= 40 02π

,b g , (5.107)

where Sr 0 0,b g is the radial component of the Poynting vector Sr θ φ,b g in the far field at broadside, and Psp is the space-wave power, given by Eq. (5.66). The Poynting vector at broadside is calculated from

S Er 0 0 12

0 00

2, ,b g b g=

η θ , (5.108)

where the far-field component Eθ is related to the far-field of the unit-strength horizontal electric dipole as E m Eeq

hexθ θ0 0 0 0, ,b g b g= , (5.109)

33

with the far-field component E hexθ θ φ,b g given in Appendix B, and the equivalent dipole moment

meq given in Eq. (5.56). At broadside,

E Ek hn

k hn jn

hex

rθ ε0 0 2 0

0 1

0 11

,tan

tanb g b g

b g=

−

L

N

MMMM

O

Q

PPPP, (5.110)

where E0 is defined by Eq. (5.95), and the substrate index of refraction is defined in Eq. (5.50). Substituting Eqs. (5.110), (5.56) into Eq. (5.109), and then using Eqs. (5.108) and (5.57) in Eq. (5.107) allows the final CAD formula to be obtained directly. The final result is

Dpc

k h n

k h nr

r

=FHGIKJFHGIKJ

+FHGIKJ

L

N

MMMMM

O

Q

PPPPP

ηπ μ

ε

0

1

0 1

20 1

401

1

tanc2 b gb gtan

, (5.111)

where the tanc(x) function appearing in the numerator is defined in Eq. (5.90), and the terms p and c1 are defined in Eqs. (5.65) and (5.49), respectively. Equation (5.111) is the final CAD formula for directivity at broadside, relative to an isotropic radiator. If the substrate is electrically thin, the term inside the square brackets in Eq. (5.111) is nearly unity, and can be taken as such. In this case the CAD formula becomes even simpler,

Dp c

=FHGIKJFHGIKJ

ηπ0

1401 . (5.112)

A further simplification can be made, if it is assumed that the patch is small enough so that the p factor can be taken to be unity, and if it is further assumed that the substrate index of refraction is large enough so that the c1 factor can be approximated as unity. In this case Eq. (5.112) becomes

D ≈ ≈ηπ0

4030. . (5.113)

Equation (5.113) indicates that the directivity of a rectangular patch antenna on a thin substrate with a high permittivity will be close to the value of 3.0 (4.77 dB) relative to an isotropic radiator. For a typical patch antenna on a low-permittivity duroid substrate, Eq. (5.112) is used with ε r = 2 2. , We / Le = 1.5, and k Le1 =π . Evaluating the p and c1 factors, the result for the directivity from Eq. (5.112) is D = 6 09. (7.85 dB). The directivity thus decreases with increasing permittivity.

34

Figure 15 shows a plot of directivity versus electrical substrate thickness for a resonant patch with an aspect ratio of W / L = 1.5. Results are shown for a patch on a low-permittivity substrate and a high-permittivity substrate. The results from CAD formula (5.111) are compared with the exact directivity, obtained by numerically integrating the radiation power pattern to find the exact space-wave power. For thin substrates, the CAD formula is quite accurate, for both the low- and high-permittivity substrates. The CAD formula is accurate for substrate thicknesses up to h / λ 0 ≈ 0.07 for the low-permittivity substrate, and h / λ 0 ≈ 0.03 for the high-permittivity substrate. 5.12 CONCLUSIONS In this chapter CAD formulas for a rectangular patch antenna have been presented. CAD formulas for the basic characteristics of the patch were presented, including resonance frequency, bandwidth, radiation efficiency, and directivity. These quantities are essentially independent of the specific type of feed. A CAD model for input impedance was also presented, for the specific case of a probe-fed patch. CAD formulas for the resonant input resistance of the probe-fed patch and the probe reactance were given. These formulas, along with the CAD model, provide a simple means of calculating the input impedance at any frequency near the dominant-mode resonance. With the exception of the CAD formulas for resonance frequency, the CAD formulas were derived from asymptotic approximations of the corresponding exact quantities, assuming that the substrate is thin compared to a wavelength. These approximations therefore become increasingly accurate as the substrate thickness decreases. Numerical results obtained from a spectral-domain method were presented for the various quantities, to determine the accuracy of the CAD formulas. In all cases, the accuracy improves as the substrate thickness decreases. The CAD formulas for bandwidth, radiation efficiency, and directivity are the most accurate. They are sufficiently accurate that they may be used as final design equations for substrate thickness up to ε λr h / 0 ≈ 0.10. The CAD formula for resonant input resistance loses accuracy sooner, being accurate only up to ε λr h / 0 ≈ 0.03. In addition to the basic patch properties mentioned above, a comparison of CAD formulas for the radiation pattern of a rectangular patch was presented. Formulas for the radiation pattern based on two different models, the electric-current model and the magnetic-current model, were given. Formulas were given for a patch on an infinite substrate, and a truncated substrate, where the ground plane is infinite but the substrate exists only under the patch. In either case, the electric- and magnetic-current models give exactly the same results, provided the patch dimensions are such that the dominant cavity mode of the patch is in resonance. The pattern for the truncated substrate case is very similar to that for the infinite substrate, except near the horizon.

35

APPENDIX A: DERIVATION OF THE p FACTOR In this appendix the p factor in Eq. (5.65) is derived. From the far-field components for a horizontal electric dipole on top of a grounded substrate (see Appendix B), it follows that the far-field power pattern is of the form S cr

hed = +22 2 2sin cos cosφ θ φc h , (5.114)

where c2 is a constant of proportionality. Using the identity

0

2 2 2 2

0

2 43

π πφ θ φ θ θ φ πz z + =sin cos cos sin

/ c h d d , (5.115)

Eq. (5.65) becomes

p T T d d= +z z30

2 2 2 2

0

2

12

22

πφ θ φ θ φ θ φ θ θ φ

π π/ /sin cos cos , , sinc h b g b g . (5.116)

From [17], the following approximations are used for the trigonometric functions appearing in the functions T1 and T2 , sin x

xa x a xb g

≈ + +1 2 44 (5.117)

cos x b x b xb g ≈ + +1 2 4

4 , (5.118) where a2 016605=− . a4 0 00761= . b2 0 49670= − . b4 0 03705= . . Using the above approximations, along with a power series expansion for (1-x)-1 , the following approximation is obtained,

cos x

xc x c xb g

1 212

22

24

4

− FHGIKJ

≈ + +

π

, (5.119)

36

where c2 0 0914153= − . c4 0 0000008= . . The term c4 may be neglected. Using the above approximations, the expression for the p factor becomes

p

a k W a k W c k L d de e e

= +

⋅ + FHGIKJ +FHG

IKJ +FHG

IKJ

LNMM

OQPP

zz31

2 2 2

2 2 2

0

2

0

2

20

2

40

4

20

2

πφ θ φ θ

θ φ θ φ θ φ θ φ

ππsin cos cos sin

sin sin sin sin sin cos

// c h

. (5.120)

The evaluation of the integrals in Eq. (5.120) is straightforward, but somewhat tedious. The following identities are used in the evaluation.

sin/ 2

0

2

4φ φ ππ

d =z (5.121)

cos/ 2

0

2

4φ φ ππ

d =z (5.122)

sin/ 4

0

2 316

φ φ ππd =z (5.123)

cos/ 4

0

2 316

φ φ ππd =z (5.124)

sin/ 6

0

2 532

φ φ ππd =z (5.125)

sin cos/ 2 2

0

2

16φ φ φ ππ

d =z (5.126)

sin cos/ 2 4

0

2

32φ φ φ ππ

d =z (5.127)

sin cos/ 4 2

0

2

32φ φ φ ππ

d =z (5.128)

37

sin/

θ θπ

d =z0 21 (5.129)

sin/ 3

0

2 23

θ θπ

d =z (5.130)

sin/ 5

0

2 815

θ θπ

d =z (5.131)

cos sin/ 2

0

2 13

θ θ θπ

d =z (5.132)

cos sin/ 2 3

0

2 215

θ θ θπ

d =z (5.133)

cos sin/ 2 5

0

2 8105

θ θ θπ

d =z . (5.134)

Using Eqs. (5.121) - (5.134), Eq. (5.120) reduces to the final result

pa

k W a a k W c k L a c k W k Le e e e e= + + + FHGIKJ + FHG

IKJ + F

HGIKJ1

102 3

56015

170

20

222

4 04

2 02

2 2 02

02b g c h b g b g b g b g ,

which is the same result shown in Eq. (5.65).

38

APPENDIX B: RADIATION FORMULAS FOR HED AND HMD In this appendix the final expressions for the far-field of horizontal electric dipole (HED) and a horizontal magnetic dipole (HMD) are given. The HED is assumed to be on the top of the grounded substrate. The HMD is assumed to be embedded at z' , corresponding to a height h z+ ' above the ground plane (the origin is at the center of the patch, on top of the substrate). The HED is assumed to be x-directed, while formulas are given for both the x-directed and y-directed HMD. These formulas are needed in the electric and magnetic current radiation models. A. x-directed HED For an x-directed horizontal electric dipole, E r E Ghexθ θ φ φ θ, , cosb g b g= 0 (5.135)

E r E Fhexφ θ φ φ θ, , sinb g b g= − 0 , (5.136)

where

Fk h N

k h N jN

r

θθ

θθ

μθ

b g b gc hb gc h b g=

−

2 0

0

tan

tan sec (5.137)

Gk h N

k h N jN

rθ

θ θ

θ εθ

θb g b gc h

b gc h b g=

−

2 0

0

tan cos

tan cos (5.138)

and

E jr

e jk r0

0

40=

−FHG

IKJ

−ω μπ

N nθ θb g b g= −1

2 2sin n r r1 = ε μ .

39

B. y-directed HMD For a y-directed horizontal magnetic dipole,

E r z E Ck h z N

k h Nhmyθ θ φ

ηφ θ

θθ

, , ; ' coscos ( ' )

cosb g b g b gc h

b gc h=+L

NMM

OQPP

0

0

0

0

(5.139)

E r z E Dk h z N

k h Nhmyφ θ φ

ηθ φ θ

θθ

, , ; ' cos sincos ( ' )

cosb g b g b gc h

b gc h= −+L

NMM

OQPP

0

0

0

0

, (5.140)

where

Cj

N

k h N jN

r

rθ

εθ

θ

θ εθ

θb g b g

b gc h b g=

−

−

2

0

cos

tan cos (5.141)

Dj

N

k h N jN

r

r

θ

θμ

θ

θθ

μθ

b gb g

b gc h b g=−

−

2

0

sec

tan sec. (5.142)

C. x-directed HMD For an x-directed horizontal magnetic dipole,

E r z E Ck h z N

k h Nhmxθ θ φ

ηφ θ

θθ

, , ; ' sincos ( ' )

cosb g b g b gc h

b gc h= −+L

NMM

OQPP

0

0

0

0

(5.143)

E r z E Dk h z N

k h Nhmxφ θ φ

ηθ φ θ

θθ

, , ; ' cos coscos ( ' )

cosb g b g b gc h

b gc h= −+L

NMM

OQPP

0

0

0

0

. (5.144)

40

REFERENCES [1] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Experiment on Microstrip

Antennas,” IEEE Trans. Antennas and Propagation, Vol. AP-27, pp. 137-145, March 1979.

[2] W. F. Richards, Y. T. Lo, and D. Harrison, “An Improved Theory for Microstrip Antennas and Applications,” IEEE Trans. Antennas and Propagation, Vol. AP-29, pp. 38-46, Jan. 1981.

[3] H. A. Wheeler, “Transmission-Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. Microwave Theory and Techniques, Vol. MTT-13, pp. 172-185, March 1965.

[4] 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.

[5] E. O. Hammerstad, “Equations for Microstrip Circuit Design,” Proc. 5th European Microwave Conf., pp. 268-272, Hamburg, Sept. 1975.

[6] J. R. James, P. S. Hall, and C. Wood, Microstrip Antennas - Theory and Design, Stevenage, U.K.: Peter Peregrinus Ltd., 1981.

[7] M. V. Schneider, “Microstrip Dispersion,” Proc. IEEE, pp. 144-146, Jan. 1972.

[8] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., p. 154, John Wiley & Sons, New York, 1993.

[9] 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. AP-39, pp. 407-410, March 1991.

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

[11] D. M. Pozar, “Considerations for Millimeter Wave Printed Antennas,” IEEE Trans. Antennas and Propagation, Vol. AP-31, pp. 740-747, Sept. 1983.

[12] D. M. Pozar, “Input Impedance and Mutual Coupling of Rectangular Microstrip Antennas,” IEEE Trans. Antennas and Propagation, Vol. AP-30, pp. 1191-1196, Nov. 1982.

[13] J. T. Aberle and D. M. Pozar, “Analysis of Infinite Arrays of One- and Two-Probe-Fed Circular Patches,” IEEE Trans. Antennas and Propagation, Vol. AP-38, pp. 421-432, April 1990.

[14] R. F. Harrington, Time-Harmonic Electromagnetic Fields, New York: McGraw-Hill, 1961.

[15] 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 Engineering, Vol. 1, no. 2, pp. 236-248, April 1991.

41

[16] K. F. Lee and J. S. Dahele, “Characteristics of Microstrip Patch Antennas and Some Methods of Improving Frequency Agility and Bandwidth,” Ch. 3 of Handbook of Microstrip Antennas, Vol. 1, IEE Electromagnetic Wave Series Vol. 28, London, U.K.: Peter Peregrinus, 1989.

[17] M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions, Nat. Bur. Stand., AMS 55, Eqs. 4.3.96 and 4.3.98, p. 76, Dec. 1972.

42

LIST OF FIGURES

Fig. 1. The probe-fed rectangular microstrip patch antenna. The feeding probe is a circular wire of radius a, located at (x0, y0). Fig. 2. Equivalent circuit of the probe-fed patch antenna. Fig. 3. CAD model of the probe-fed patch antenna, obtained from the circuit of Fig. 2. This simple model is valid near the resonance frequency. Fig. 4. Normalized resonance frequency versus the electrical thickness of the substrate, for a rectangular patch with W / L = 1.5. The normalized resonance frequency is the resonance frequency f0 divided by the zero-order value f0

0 , which neglects fringing. λ 0 is the free-space wavelength at frequency f0

0 . A comparison is shown between results from the CAD formula of Hammerstad, the CAD formula of James et al., and measured resonance frequencies. The solid dots denote the measured values. (a) ε r = 2.2. (b) ε r = 10.8. Fig. 5. Radiation efficiency of a horizontal electric dipole on a lossless grounded substrate versus the electrical thickness of the substrate, for two different values of substrate permittivity. The CAD formulas are based on Eq. (5.47). One formula uses Eq. (5.51) to calculate the surface-wave power, while the other uses the improved formula (5.52). An exact result, obtained from a spectral-domain analysis, is also shown. (a) ε r = 2.2. (b) ε r = 10.8. Fig. 6. Plot of the factor p defined in Eq. (5.58). The results from CAD formula (5.65) are compared with an exact calculation, obtained by numerically evaluating Eq. (5.59). Fig. 7a. The percent bandwidth of a rectangular patch versus the electrical thickness of the substrate, at a frequency of 5.0 GHz. The patch is resonant for all substrate thicknesses (input impedance is pure real), with W L/ = 1.5. Results for two different substrate permittivities are shown, ε r= 2.2 and 10.8. A loss tangent d = 0.001 is assumed in both cases. The conductivity of the patch and ground plane are both taken as σ = 3.0 × 107 mhos / meter. The CAD formula is compared with results from a cavity-model calculation, which numerically finds the frequencies at which the SWR = 2.0 for a patch that is matched to the incoming feed line at the resonance frequency. The cavity-model calculation assumes that the feed is located at x L0 4= / , y W0 2= / (the CAD formula does not account for the feed location). The results from the CAD formula are shown with solid lines. The results from the cavity-model analysis are shown with dots and squares, for the low- and high-permittivity substrates, respectively. The hollow dots indicate that no frequency can be found for which the input reactance is zero. In this case the resonance frequency is taken as the frequency for which the input reactance is a minimum.

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Fig. 7b. The same CAD results in Fig. 7a are compared with measured results. The dots and squares correspond to the measured bandwidths for the low- and high-permittivity substrates, respectively. Fig. 8. The radiation efficiency of a rectangular patch versus the electrical thickness of the substrate, at a frequency of 5.0 GHz. The patch is resonant for all substrate thicknesses, with W L/ = 1.5. Results for two different substrate permittivities are shown, ε r = 2.2 and 10.8. The CAD formula is compared with results from a spectral-domain calculation. (a) A loss tangent d = 0.001 is assumed for both substrate permittivities. The conductivity of the patch and ground plane are both taken as σ = 3.0 × 107 mhos / meter. (b) The loss tangent is assumed to be zero, and the patch and ground plane are assumed to be perfectly conducting. Fig. 9a. The resonant input resistance of a probe-fed rectangular patch versus the electrical thickness of the substrate, at a frequency of 5.0 GHz. Resonance is defined from the patch length that gives a pure real input impedance. The patch is resonant for all substrate thicknesses, with W L/ = 1.5. The feed probe is located at x L0 4= / , y W0 2= / , and has a radius of 0.05 cm. Results for two different substrate permittivities are shown, ε r = 2.2 and 10.8. A loss tangent d = 0.001 is assumed for both substrate permittivities. The conductivity of the patch and ground plane are both taken as σ = 3.0 × 107 mhos / meter. The CAD formula is compared with results from a cavity-model analysis, shown with dots and squares for the low- and high-permittivity substrates, respectively. The dashed part of the CAD curve indicates that no frequency can be found for which the input reactance is zero in the CAD model. Similarly, the hollow dots and squares indicate that no frequency can be found for which the input reactance is zero in the cavity model. In this case the resonance frequency is taken as the frequency for which the input reactance is a minimum. Fig. 9b. The same CAD results for resonant input resistance shown in Fig. 9a, compared with the results from a spectral-domain calculation of resonant input resistance. The spectral-domain calculation assumes a filamentary probe feed, and thus neglects the probe reactance. Fig. 9c. The same CAD results for resonant input resistance shown in Fig. 9a, compared with measurements. The dots and squares correspond to the measured resonant input resistances for the low- and high-permittivity substrates, respectively. Fig. 10. The probe reactance versus normalized feed position x x Lr = −1 2 0 / . The substrate has a permittivity of ε r = 2.2 and an electrical thickness h / λ 0 = 0.02. The patch is resonant at 5.0 GHz with W L/ = 1.5. The radius of the cylindrical probe is 0.05 cm. The exact result is compared with the CAD formula based on a parallel-plate waveguide model. Fig. 11. Comparison of input impedance versus frequency for a rectangular patch with L = 2.0 cm and W L/ = 1.5. The feed probe is located at x L0 4= / , y W0 2= / , and has a radius of 0.05 cm. Results from the CAD model of Fig. 3 are compared with results from the cavity model. The substrate has a permittivity of ε r = 2.2 and a thickness of 0.1524 cm. (a) Real part of input impedance. (b) Imaginary part of input impedance.

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Fig. 12. Radiation models for a rectangular patch antenna. (a) Electric current model for an infinite substrate. (b) Magnetic current model for an infinite substrate. (c) Electric current model for a truncated substrate. (d) Magnetic current model for a truncated substrate. Fig. 13. Radiation patterns (E- and H-plane) for a rectangular patch antenna on an infinite substrate of permittivity ε r = 2.2 and thickness h / λ 0 = 0.02. The patch is resonant with W L/ = 1.5. Fig. 14. Radiation patterns (E- and H-plane) for a rectangular patch antenna on a truncated substrate of permittivity ε r = 2.2 and thickness h / λ 0 = 0.02. The patch is resonant with W L/ = 1.5. Fig. 15. Directivity of a rectangular patch on an infinite substrate versus the electrical thickness of the substrate. The patch is resonant for all substrate thicknesses, with W L/ = 1.5. Results for two different substrate permittivities are shown, ε r = 2.2 and 10.8. The CAD formula is compared with results from an exact calculation.