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49

Chapter 3

Coupling to DRAs

Chapter 2 examined the basic DRAs and presented design equations for predicting

the resonant frequency and radiation Q-factor for the commonly used lower order modes. The models for deriving these equations assumed the DRAs were in

isolation or mounted on an infinite perfect conducting ground plane and did not

account for the feeding mechanisms used to excite the DRAs. The selection of the

feed and that of its location both play an important role in determining which

modes are excited. This, in turn, will determine the input impedance and radiation

characteristics of the DRAs. The coupling mechanism can also have a significant

impact on the resonant frequency and Q-factor, which the previous equations fail

to predict. This chapter begins with a brief review of coupling theory and an

examination of the internal fields within rectangular and cylindrical DRAs. Aknowledge of the internal field configuration is essential for understanding how

the various feeds can excite different modes within the DRA. The more common

feeds are then surveyed and examples provided to highlight practical design

considerations.

3.1 COUPLING COEFFICIENTS

For most practical applications, power must be coupled into or out of the DRAthrough one or more ports. (One notable exception is the DRA reflectarray, which

will be discussed in Chapter 9.) The type of port used and the location of the port

with respect to the DRA will determine which mode will be excited and how much

power will be coupled between the port and the antenna. The mode or modes

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Dielectric Resonator Antenna Handbook 50

generated, the amount of coupling, and the frequency response of the impedance

are all important in determining the performance of the DRA. Although these

quantities are difficult to determine without using numerical methods, a great deal

of insight can be obtained by knowing the approximate field distributions of the

modes of the isolated DRA and by making use of the Lorentz Reciprocity

Theorem and some coupling theory borrowed from resonator circuits [1].

When coupling to a DRA, the source can typically be modeled as either an

electric or magnetic current, and the amount of coupling, χ, between the source

and the fields within the DRA can be determined by applying the reciprocity

theorem with the appropriate boundary conditions. For an electric source J s

χ ∝ E DRA ⋅ J s( )dV

V

(3.1)

and for a magnetic source M s

χ ∝ H DRA ⋅ M s( )dV

V

(3.2)

where V is the volume occupied by the source within which the electric and/or magnetic currents exist, while E DRA and H DRA are the electric and magnetic fields

within the DRA. Equation (3.1) states that in order to achieve strong coupling

using an electric current source (like a probe), then that source should be located in

an area of strong electric fields within the DRA. On the other hand, to achieve

strong coupling using a magnetic current source (like a loop or an aperture) then

from (3.2) the source should be located in an area of strong magnetic fields. It is

thus necessary to have a good understanding of the internal field structures of the

isolated DRA to determine where the feed should be placed to excite the desired

mode. The fields within cylindrical and rectangular DRAs will be examined in thenext section.

In addition to transferring power, the coupling mechanism to the DRA has a

loading effect that will influence the Q-factor of the DRA. An external Q-factor

(Qext ) can be defined in terms of the coupling factor, χ:

Qext =Q

χ (3.3)

and the loaded Q-factor (Q L) of the DRA can then be expressed as:

Q L =1

Q+

1

Qext

−1

=Q

1+ χ (3.4)

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Coupling to DRAs 51

where Q is the unloaded Q-factor. Maximum power is transferred between the

coupling port and the DRA when the coupling factor is 1. This condition is known

as critical coupling. When χ < 1, the DRA is said to be undercoupled, while when

χ > 1, the DRA is overcoupled. The more common coupling methods to DRAs

will be presented later in this chapter.

3.2 FIELDS WITHIN RECTANGULAR AND CYLINDRICAL DRAS

For the rectangular DRA shown in Figure 3.1, with dimensions w > b or d , the

lowest order mode will be TExδ11. Using the dielectric waveguide model, this leads

to the following fields within the DRA [2]:

H x =k y

2 + k z2( )

jωµ ocos k x x( )cos k y y( )cos k z z( ) (3.5)

H y =k yk x( )

jωµ osin k x x( )sin k y y( )cos k z z( ) (3.6)

H z =k zk x( )

jωµ osin k x x( )cos k y y( )sin k z z( ) (3.7)

E x = 0 (3.8)

E y = k z cos k x x( )cos k y y( )sin k z z( ) (3.9)

E z = −k y cos k x x( )sin k y y( )cos k z z( ) (3.10)

where, as was shown in Chapter 2:

k x tan k xd / 2( )= ε r −1( )k o2 − k x

2 (3.11)

and

k x2 + k y

2 + k z2 = ε r k o

2 (3.12)

The e jω t time dependence is suppressed in the above equations. Assuming magnetic

walls along air-dielectric interfaces parallel to the z-axis, then:

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k y =m

wand k z =

nπ

b, where m, n are positive integers.

The value δ can be defined as the fraction of a half-cycle of the field variation in

the z-direction and is given by:

δ =k x

/ d (3.13)

For the lowest order mode (m = n = 1), a sketch of the field configuration is shownin Figure 3.1. The Hx component of the magnetic field is dominant along the

center of the DRA, while the E-fields (predominantly Ey and Ez) circulate aroundthe Hx component. These fields are similar to those produced by a short magneticdipole. A plot of the relative amplitudes of the electric and magnetic fields in the

x-y plane of the DRA is shown in Figure 3.2. A knowledge of the relative

amplitudes of these fields as a function of location within the DRA is important

for determining where to place the feed mechanism to efficiently excite the DRA.

Figure 3.1 Sketch of the fields for the TExδ1 mode of the rectangular DRA.

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Coupling to DRAs 53

-1

0

1

EzHx

R e l a t i v e F i e l d S t r e n g t h

y (at z = 0)

-w/2 +w/20

Figure 3.2 Relative field strength of the TE

x

δ11 mode within the rectangular DRA.

Higher-order modes of rectangular DRAs can also be excited for certain

aspect ratios. Figure 3.3 shows sketches of the electric fields of some of these

modes. The TEx

δ31 and TEx

δ13 modes will produce radiation patterns similar to the

TEx

δ11 mode, having a peak in the broadside direction (along the z-axis), while the

TEx

δ21mode will have a null at broadside. (Note that the TE

x

δ12mode cannot exist

for the case of the DRA mounted on the ground plane, due to the boundary

condition that forces the tangential E-field to zero at z = 0, since the TEx

δ12 would

require the E-field to be maximum at that location.) By properly combining one or

more of the higher-order modes with the fundamental mode, a wider bandwidth or

dual-band operation can be achieved [3].

The fields of a cylindrical DRA operating in the TE01δ mode can be

approximated by [4-6]:

H z ∝ J o β r ( )cos π 2h

z

(3.14)

H r ∝ J 1 β r ( )sinπ

2h z

(3.15)

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Figure 3.3 Sketches of the E-fields for selected higher-order modes within the rectangular DRA.

E φ ∝ J 1 β r ( )cos2h

z (3.16)

E z = E r = H φ = 0 (3.17)

where J o( β ) and J 1( β ) are Bessel functions of the first kind, and β is the

solution to J o( β a) = 0.

The TM01δ fields are similar to those of the TE01δ with the magnetic and

electric field components interchanged. For the HE11δ mode of the cylindricalDRA, the field components can be expressed as:

E z ∝ J 1 α r ( )cosπ

2h z

cos

sinφ

(3.18)

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Coupling to DRAs 55

E r ∝∂ J 1 r ( )

∂ α r ( )

sinπ

2h

z cosφ

sinφ

(3.19)

E φ ∝ J 1 α r ( )sinπ

2h z

sin

cosφ

(3.20)

H r ∝ J 1 α r ( )cosπ

2h z

sinφ

cosφ

(3.21)

H φ ∝J 1 r ( )∂ α r ( )

cosπ

2h z

cosφ

sinφ

(3.22)

H z ≈ 0 (3.23)

where α is the solution to J1(αa) = 0. The choice of cosφ or sinφ depends on the

location of the feed.

The fields for these three modes are sketched in Figure 3.4. Examples of the

relative field strengths for these modes are shown in Figures 3.5 and 3.6. The

actual values will depend on the dielectric constant and radius of the DRAs;

however, these figures are useful for visualizing the locations of high electric andmagnetic fields within the DRA for the different modes, to assist in determining

what type of feed is best suited and where the feed should be located to optimize

the DRA excitation. The following sections will examine the more conventional

coupling mechanisms in closer detail.

3.3 APERTURE COUPLING

One common method of exciting a DRA is through an aperture in the ground plane upon which the DRA is placed. Figure 3.7 shows some of the aperture

shapes that have been used for exciting DRAs. The small rectangular slot is

probably the most widely used aperture [7-14]. By keeping the slot dimensions

electrically small, the amount of radiation spilling beneath the ground plane can be

minimized. Annular slots have also been used for exciting cylindrical DRAs [15],

while cross-shaped and C-shaped slots are used to excite circular polarization [16-

18]. The aperture can itself be fed by a transmission line (either microstrip or

coaxial) or a waveguide [19, 20], as shown in Figure 3.7. Aperture coupling offers

the advantage of having the feed network located below the ground plane,isolating the radiating aperture from any unwanted coupling or spurious radiation

from the feed.

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Figure 3.4 Sketch of the cylindrical DRA field configurations.

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Coupling to DRAs 57

0

0.2

0.4

0.6

0.8

1

Hz (Ez)

Eφ (Hφ)


DRA Radius (r )0 a

Figure 3.5 Example of relative field strength of the TE01δ (TM01δ) mode in the cylindrical DRA.

0

0.2

0.4

0.6

0.8

1

Hr

Ez


DRA Radius (r )0 a

Figure 3.6 Example of the relative field strength of the HE11δ mode in the cylindrical DRA.

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Figure 3.7 Various slot apertures.

The electric fields within a rectangular slot are sketched in Figure 3.8. For

coupling purposes, the slot can be considered as an equivalent magnetic current

whose direction is parallel to the slot length. To achieve strong coupling to the

DRA, the aperture should be located in a region of strong magnetic fields, as

indicated by (3.2). Figure 3.9 shows a rectangular slot feeding a rectangular DRA.

The orientation of the slot will excite the TE

x

δ11 mode of the DRA [8]. Centering theDRA over the slot will ensure strong coupling to the internal magnetic fields. Some

degree of impedance matching can be achieved by offsetting the DRA from the slot

center. A rectangular slot can also be used to excite the HE11δ mode of a cylindrical

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Coupling to DRAs 59

DRA [12] or the TE01δ mode of a split-cylinder DRA, as shown in Figures 3.10 and

3.11 [9, 12].

Feeding the aperture with a microstrip transmission line is the most common

approach, since printed technology is easy to fabricate. Microstrip lines also offer a

Figure 3.8 Equivalent magnetic current for slot apertures.

Figure 3.9 Slot aperture coupling to a rectangular DRA.

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degree of impedance matching not available with coaxial lines or waveguides. As

shown in Figure 3.12, the microstrip line can be extended by a distance s beyond

the slot. This extension behaves like an open-circuit stub, whose admittance is in parallel with the admittance of the slot. By adjusting the length s of the stub, the

reactive component of the slot admittance can be reduced (and, in theory,

completely cancelled at the design frequency), resulting in an improved impedance

match to the microstrip line. The techniques for aperture coupling to DRAs are

similar to those of a microstrip patch antenna, and as a rule of thumb, the stub

length is chosen to be s = λ g /4, where λ g is the guided wavelength of the microstrip

line. The slot length l s and width w s will control the amount of coupling from the

microstrip line to the DRA. The area of the slot should, in general, be kept as small

as possible to avoid excessive radiation beneath the ground plane. Also, if theaperture is too large, it will significantly load the DRA, and the resonant frequency

will shift compared to the theoretical value obtained using the models in Chapter 2.The aperture introduces an air gap beneath the DRA, which will then no longer see

a continuous ground plane. If the aperture is too large, the previous assumptions

using image theory to double the height of the DRA become less accurate,resulting in larger errors in the predicted resonant frequency and Q-factor.

Figure 3.10 Slot aperture coupling to a cylindrical DRA.

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Coupling to DRAs 61

Figure 3.11 Slot aperture coupling to a split-cylindrical DRA.

Computational electromagnetics methods, such as the finite element method,

the method of moments, or the finite-difference time domain method are typically

used to determine the input impedance of slot-fed DRAs. Several commercial

software packages are available for analyzing three-dimensional electromagnetic

problems, which can be used to predict the input impedance with a reasonably

high degree of accuracy. These software tools are better suited for analysis thandesign, since the computational time can be lengthy, especially for high values of

the dielectric constant. Although there are no simple equations for designing the

slot dimensions given the various antenna parameters, the following guidelines can

be used as a starting point for rectangular slots:

1) The slot length l s is chosen large enough so that sufficient coupling

exists between the DRA and the feed line but small enough so that it

does not resonate within the band of operation, which usually leads

to a significant radiated back lobe. A good starting value is [10, 21]:

ls =0.4λ o

ε e(3.24)

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Figure 3.12 Microstrip-fed rectangular slot.

where ε e =ε r +ε s

2(3.25)

and ε r and ε s are the dielectric constants of the DRA and substrate,

respectively.

2) A fairly narrow slot width is usually chosen to avoid a large backlobe

component. A reasonable choice is:

ws = 0.2ls (3.26)

At high frequencies, (3.26) might result in a very narrow slot that

may be difficult to fabricate due to etching limitations. At these

frequencies, a wider slot width can be used.

3) The stub extension s is selected so that its reactance cancels out that

of the slot aperture. It is generally initially chosen to be:

s =λ g

4(3.27)

where λ g is the guided wave in the substrate.

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Coupling to DRAs 63

The amount of coupling actually achieved using the above guidelines is not always

as high as desired. Oftentimes the coupling can be significantly improved simply

by slightly offsetting the DRA with respect to the slot. This solution requiresneither a second design iteration nor the fabrication of a new circuit and is an

attractive method for obtaining a good impedance match [2].

An example of a slot feed for a rectangular DRA design at 5.5 GHz with an

8% fractional impedance bandwidth is now illustrated. To design the DRA, the

procedure outlined in Section 2.3.2 can be followed. Using a material with a

dielectric constant of ε r = 10, the following DRA dimensions were chosen:

w = 14 mm, and d = h = 8 mm, as shown in Figure 3.13. Substituting these values

into (2.19) the theoretical resonant frequency is 5.6 GHz and from (2.24), the

radiation Q-factor is Qrad = 7.9, which translates to a fractional bandwidth of 8.9%.The microstrip line was printed on a substrate have a dielectric constant of

ε s = 3.38 and thickness of 0.5 mm. Using (3.25) the effective dielectric constant is

ε e = 6.7, from (3.24) the slot length is l s = 8.4 mm, and from (3.26) the slot width

is w s = 1.7 mm. The stub length extension, from (3.27) is s = 8.3 mm, where a

guided wavelength λ g = 33.2 mm was calculated for the microstrip line shown in

Figure 3.13. Based on these initial calculations, several feed circuits were

fabricated with different combinations of slot length (l s) and stub length ( s). For

these prototype circuits, a somewhat narrower dimension was chosen for the slot

width (w s = 1.0 mm) to help minimize the potential for high backlobes.

Figure 3.13 Design example of a rectangular DRA fed by a rectangular slot.

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Figure 3.14 shows the effects of varying the slot length for a fixed stub length of

s = 5 mm. Increasing the slot length from 8 to 12 mm results in an increase in the

10-dB return loss bandwidth from 6.3% to 13.3% as well as an upward shift in thefrequency response. The increase in bandwidth might come at the expense of

higher backlobes, if there is increased radiation from the longer slots.

The effect of varying the stub length from s = 3 mm to 10 mm is seen in

Figure 3.15, where the slot length was kept constant at l s = 10 mm. A significant

amount of frequency tuning can be achieved by adjusting the stub length.

-25

-20

-15

-10

-5

0

4.0 4.5 5.0 5.5 6.0 6.5 7.0

l s = 8 mm

l s = 10 mm

l s

= 12 mm | S 1 1 | ( d B )

Frequency (GHz)

Figure 3.14 Effects of slot length the DRA return loss ( s = 5 mm).

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Coupling to DRAs 65

As the stub length is reduced from 10 mm down to 3 mm, there is an upward shift

in the frequency response and an increase in the 10-dB return loss bandwidth.

Trimming the stub length could thus be a practical method for tuning the returnloss response of the DRA. Instead of fabricating several microstrip circuits with

different stub lengths, a single circuit could be fabricated with a stub length

somewhat larger than the value obtained using (3.27). If the desired impedance

response is not obtained, the stub could then be trimmed to improve the results.

-25

-20

-15

-10

-5

0

4.0 4.5 5.0 5.5 6.0 6.5 7.0

s = 3 mm

s = 5 mm

s = 7 mm

s = 10 mm

| S 1 1 | ( d B

)

Frequency (GHz)

Figure 3.15 Effects of stub length on the DRA return loss (l s = 10 mm).

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A second method for tuning the impedance response is to offset the location of the

DRA with respect to the slot. Figure 3.16 shows the change in the impedance

response that occurs by offsetting the DRA a few millimeters from the center of the slot. The response is shifted down in frequency by a significant amount,

without an appreciable narrowing of the impedance bandwidth. The results of

these measurements help confirm that the procedure outlined above for

determining the slot and stub dimensions is a useful design tool for slot-coupled

feeds for DRAs.

-25

-20

-15

-10

-5

0

4.0 4.5 5.0 5.5 6.0 6.5 7.0

Centered

Offset

| S 1 1 | ( d B )

Frequency (GHz)

Figure 3.16 Effects of offsetting the DRA on return loss.

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Coupling to DRAs 67

3.4 PROBE COUPLING

Another common method for coupling to DRAs is with a probe, as shown in

Figure 3.17 [22-28]. The probe usually consists of the center pin of a coaxial

transmission line that extends through the ground plane. The center pin can also be

soldered to a flat metal strip, that is placed adjacent to the DRA [29], whose length

and width can be adjusted to improve the impedance match. Instead of a coaxial

line, the flat metal strip can also be connected to a microstrip line [30-32]. For

coupling purposes, the probe can be considered as a vertical electric current, as

shown in Figure 3.17 and, from (3.1), it should be located in a region of the DRA

having high electric fields to achieve strong coupling. The probe can either be

Figure 3.17 Vertical probe sources.

located adjacent to the DRA or can be embedded within it. The amount of coupling can be optimized by adjusting the probe height and the DRA location.

Also, depending on the location of the probe, various modes can be excited.

A probe located adjacent to (or slightly inset into) a rectangular DRA, as

shown in Figure 3.18, will excite the TExδ11 mode. Similarly, the HE11δ mode of a

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cylindrical DRA or the TE01δ mode of the split cylinder can be excited with a

probe located adjacent to (or slightly inset into) the DRA, as in Figures 3.19 and

3.20. For a probe located in the center of a cylindrical (or ring) DRA, the TM01δ

mode is excited, as shown in Figure 3.21. One advantage of coaxial probe

excitation is the direct coupling into a 50-Ω system without the need for a

matching network. Probes are useful at lower frequencies where aperture-coupling

may not be practical due to the large size of the slot required.

The probe length is generally chosen to be less than the height of the DRA, to

avoid probe radiation. (A notable exception is the hybrid monopole-DRA,

described in Section 4.5.2, where the probe is actually designed to act as both a

feed for the DRA and a monopole radiator.) Rigorous analyses for probe-fed

hemispherical and cylindrical DRAs have been carried out, showing the effects of

both the probe position and length on the input impedance and resonant frequency

Figure 3.18 Probe coupling to a rectangular DRA.

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Coupling to DRAs 69

of the DRA [33-35]; however, there are no simple equations to design the required

probe height for a given set of DRA dimensions and dielectric constant. In terms

of practicality, locating the probe feed adjacent to the DRA is preferred since it

does not require drilling into the DRA. (This, of course, cannot be avoided for the

TM01δ mode of cylindrical DRAs where the probe must be at the center of the

DRA.) If the center conductor of a coaxial cable is used as the probe, one approach

is to begin with a probe height slightly taller than that of the DRA, then trimming

the height until the desired match is achieved. Similarly, if a flat metallic strip is

used, it is recommended to start with a taller, wider strip which can be then

trimmed for impedance tuning.

Figure 3.19 Probe coupling to the HE11δ mode of the cylindrical DRA.

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Figure 3.20 Probe coupling to split-cylindrical DRAs.

Figure 3.21 Probe coupling to the TM01δ mode of the cylindrical DRA.

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Coupling to DRAs 71

3.5 MICROSTRIP LINE COUPLING

A common method for coupling to dielectric resonators in microwave circuits is

by proximity coupling to microstrip lines. Figure 3.22 shows this feeding

technique applied to DRAs [36]. Microstrip coupling can be used to excite the

TExδ11 mode of the rectangular DRA or the HE11δ mode of the cylindrical DRA, as

shown in Figure 3.23. This sketches the magnetic fields in the DRA and the

equivalent short horizontal magnetic dipole mode.

Figure 3.22 Microstrip line coupling to DRAs.

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The amount of coupling from the microstrip line to the DRA can be controlled

to a certain degree by adjusting s in Figure 3.22, which represents the spacing

between the DRA and the line for the side-coupled case or the length of the line

underneath the DRA for the direct-coupled case. A more dominant parameter

affecting the degree of coupling is the dielectric constant of the DRA. For higher

values (ε r > 20), strong coupling is achieved; however, the maximum amount of

coupling is significantly reduced if the dielectric constant of the DRA is lowered.

This can be problematic if low dielectric constant values are required for obtaining

wideband operation. For series-fed linear arrays of DRAs (discussed in Chapter 9),

the lower level of coupling may not be an impediment, since each DRA element

usually only requires coupling a small amount from the microstrip feed line [37-

38].

Figure 3.23 Fields and equivalent radiation models of microstrip line-coupled DRAs.

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Coupling to DRAs 73

3.5.1 The Multisegment DRA

As mentioned in Section 3.5, to achieve strong coupling between the microstrip

line feed and the DRA, the dielectric constant of the DRA needs to be relatively

high (usually ε r > 20). Since the radiation Q-factor is proportional to the dielectric

constant, the bandwidth of these DRAs is typically narrow. For wider-band

applications, DRAs with lower values of dielectric constant are required, but only

a small amount of coupling is achievable between the microstrip line and the

DRA, resulting in poor radiation efficiency. One solution to overcoming the weak

coupling of DRAs with lower dielectric constants is the multisegment DRA [39-

40]. The multisegment dielectric resonator antenna (MSDRA) consists of a

rectangular DRA of low permittivity under which one or more thin segments of

different dielectric constant substrates are inserted, as shown in the exploded view

of Figure 3.24. The inserts serve to transform the impedance of the DRA to that of

the microstrip line by concentrating the fields underneath the DRA; this

significantly improves the coupling performance. In general, more than one insert

can be added to obtain the required impedance match, but to reduce the

complexity of the fabrication process and ultimately the cost, it is desirable to use

only a single insert, as shown in Figure 3.25.

Figure 3.24 Exploded view of the multisegment DRA.

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Figure 3.25 MSDRA with a single insert.

The MSDRA can be designed using the equations for the rectangular DRA in

Section 2.4, with some modifications. To account for the effect of the insert and of

the microstrip substrate on the resonant frequency of the MSDRA, the dielectric

waveguide model equations are modified by including an effective permittivity

(ε eff ) and effective height ( H eff ). Adopting a simple static capacitance model, theeffective permittivity of the MSDRA is calculated using:

ε eff = H eff

h / ε r + t / ε i + s / ε s(3.28)

where ε r , ε i, and ε s are the dielectric constants of the DRA, insert, and substrate,

respectively. The effective height ( H eff ) is simply the sum of the DRA height (h),

insert thickness (t ), and substrate thickness ( s):

H eff = h + t + s (3.29)

Equations (3.28) and (3.29) are substituted into (2.19), with ε eff replacing ε r and

2 H eff replacing b.

The selection of the insert thickness and dielectric constant should be based

on two considerations. The first is to ensure that the insert itself does not radiate.

The dielectric waveguide model equations can be used with the insert parameters

to ensure that the resonant frequency of the insert is well above the desiredfrequency of operation. The second consideration relates to the impedance

bandwidth obtained by the various inserts. In general, the higher the dielectric

constant of the insert, the narrower the impedance bandwidth, for a given insert

thickness. MSDRAs with higher dielectric constant inserts also show a greater

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Coupling to DRAs 75

sensitivity to their position with respect to the open end of the microstrip line.

Since one of the objectives of using the MSDRA is to obtain a wide impedance

bandwidth, the insert dielectric constant should be chosen accordingly. This places

an upper bound on permittivity. Based on the investigations carried out in [39], a

set of guidelines was proposed for designing the MSDRA.

(1) Determine the dimensions (w, h, d ) of the DRA, using the dielectric

waveguide model equations for the desired resonant frequency and

radiation Q-factor. The dielectric constant of the DRA should be

chosen within the range of approximately 10 ≤ ε r ≤ 12 for wideband

operation.

(2) Choose an insert dielectric constant in the range of 20 ≤ ε i ≤ 40 and

an insert thickness (t ) such that 0.1 < T < 0.3 (where T = t /(t+h)).

Ensure that the resonant frequency of the isolated insert is well above

the desired frequency of operation by substituting the insert

parameters into the dielectric waveguide model equations.

(3) Estimate the resonant frequency of this MSDRA structure by using

(2.19) with the effective dielectric constant (ε eff ) and effective height

( H eff ), based on the insert parameters chosen in (2). The insert

parameters or the DRA dimensions might require some adjustment if

there is a significant shift in the desired resonant frequency.

(4) Once fabricated, some experimental optimization may be required to

maximize the coupling. The simplest form of optimization is done by

adjusting the position of the MSDRA with respect to the open end of

the microstrip line. If this is not sufficient, a second iteration of the

MSDRA parameters may be required.

Using these guidelines, several MSDRA designs have been carried out at various

frequency bands, achieving impedance bandwidths of up to about 20% [39]. For

the single insert case, an empirical study has led to design guidelines for the

optimum values of the dielectric constant and thickness [41]:

ε i =η o ε r

Z o(3.30)

t =c

4 f o ε i(3.31)

where η o is the intrinsic impedance of free space, Z o is the characteristic

impedance of the transmission line, c is the speed of light and f o is the center

frequency of the DRA. Arrays of MSDRAs will be investigated in Chapter 9.

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The effects of changing the position of the MSDRA with respect to the open

circuit of the microstrip transmission line can be seen in the example shown in

Figure 3.26. The frequency response of the return loss was measured for various positions of the MSDRA, such that the stub extension (l s) ranged from +3 mm to

-3 mm, where the positive values indicate that the stub extends past the MSDRA

while the negative values indicate that the MSDRA extends past the end of the

open circuit. The measured return loss curves for the various positions are overlaid

in Figure 3.27.

Figure 3.26 Effects of the microstrip stub length on the MSDRA return loss.

-25

-20

-15

-10

-5

0

6 7 8 9 10 11 12

s = -3 mms = -2 mms = -1 mms = 0 mms = +1 mm

s = +2 mms = +3 mm

| S 1 1 | ( d B )

Frequency (GHz)

Figure 3.27 Return loss response as a function of stub length for the MSDRA.

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Coupling to DRAs 77

By simply moving the position of the MSDRA with respect to the open, the return

loss can be tuned for a broadband response. For this particular case, the position

with (l s = 0) offered the widest frequency response. In general, either a positive or a negative value for l s could result in the widest response. The length of the stub

can also be used to slightly shift the frequency response for fine tuning.

3.6 COPLANAR COUPLING

Coupling to DRAs can also be achieved by using coplanar feeds with some

examples shown in Figure 3.28 [42-48]. Open-circuit coplanar waveguides can be

used to directly feed DRAs similar to the open-circuit microstrip lines examined

earlier. Additional control for impedance matching can be achieved by addingstubs or loops at the end of the line. Figure 3.29 shows a cylindrical DRA coupled

to a coplanar loop. The coupling level can be adjusted by positioning the DRA

over the loop. The coupling behavior of the coplanar loop is similar to that of the

coaxial probe, but the loop offers the advantage of being nonobtrusive. By moving

the loop from the edge of the DRA to the center, one can couple into either the

HE11δ mode or the TE01δ mode of the cylindrical DRA [42]. As with aperture

coupling, the dimensions of the coplanar feed should be chosen large enough to

ensure proper coupling, but small enough to avoid excessive radiation in the

backlobe.

Figure 3.28 Various coplanar feeds for coupling to DRAs.

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Figure 3.29 Coplanar loop coupling to a cylindrical DRA.

3.7 DIELECTRIC IMAGE GUIDE COUPLING

The final method of coupling to DRAs presented in this chapter is by way of adielectric image guide, as shown in Figure 3.30. Dielectric image guides offer

advantages over microstrips at millimeter-wave frequencies, since they do not

suffer as severely from conductor losses. As with microstrip lines, the amount of

coupling to the DRA is generally quite small, especially for DRAs with lower

dielectric constants, although it may be possible to increase the coupling by

operating the guide closer to the cut-off frequency. The dielectric image guide is

thus best utilized as a series feed to a linear array of DRAs. It will be examined in

Chapter 9 [38, 49, 50].

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Coupling to DRAs 79

Figure 3.30 Dielectric image guide feed for DRAs.

3.8 SURVEY OF ANALYTICAL METHODS

One important parameter in designing a feed to excite the DRA is the input

impedance. Knowledge of the input impedance as a function of frequency isneeded to determine the bandwidth of operation and for matching the antenna to

the circuit. In this chapter, guidelines were given for the various coupling

mechanisms to help obtain a good impedance match that did not rely on a

knowledge of the DRA input impedance. These guidelines offer a good starting

point for the design, but do not allow for precise designs if specific impedances are

required. Unfortunately, there are no simple closed-form expressions for

predicting the input impedance of the DRA when excited by a particular feed and

rigorous analytical or numerical techniques are required. This section provides a

brief survey of some of the techniques that have been used to predict the inputimpedance for DRAs excited by the various feeds.

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3.8.1 Green's Function Analysis

For a probe-fed DRA, the input impedance ( Z in) can be determined using the

following equation:

Z in =−1

I o2

E ⋅ J s( )dS so

(3.32)

where E represents the electric fields of the DRA, J s is the applied source current

density on the probe, I o is the magnitude of the current on the probe, and the

integral is carried out over the surface S o of the probe. The electric fields of the

DRA will, in turn, depend on the source excitation, and are determined using:

E = G ⋅ J s( )dS so

(3.33)

where G represents Green's function for the DRA. Thus the determination of the

input impedance requires a knowledge of Green's function of the DRA. Using

some simplifying assumptions about a single-mode operation and the currents on

the probe, the Green's function for a hemispherical DRA was first derived in [51]

and was then used to predict the input impedance of the probe-fed DRA operating

in the TE111 mode. Moderately good agreement was achieved between the

predicted and measured input impedance. More rigorous derivations soon

followed [52-53] which led to more accurate predictions. This technique was also

applied to a probe-fed hemispherical DRA operating in the TM101 mode [54], as

well as other variations of hemispherical DRA geometries [55-57]. The input

impedance of conformal strip feeds [58] and aperture feeds [59-62] can also be

analyzed using Green's function approach. The advantage to this technique is the

relatively fast computation time required to obtain the input impedance. It is,

therefore, a useful method for analyzing the effects of altering probe dimensions

and probe location and can be used for optimizing the input impedance. The main

drawback is its limitation to hemispherical DRA geometries, due to the fact that

Green's function has only been determined for DRAs of hemispherical shape. For

other DRA shapes, different analytical techniques are required.

3.8.2 Frequency Domain Analysis

Two common frequency domain techniques that have been used to analyze DRAs

are the method of moments (MOM) and the finite element method (FEM). The

MOM involves discretizing the antenna into a number of small segments and

solving for a set of unknown coefficients, each coefficient representing the current

on one segment due to a known incident field [63]. Once the currents are

determined, the input impedance of the antenna can then be calculated. The MOM

was first developed for wire or metal antennas of arbitrary shape, but can be

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Coupling to DRAs 81

extended to include dielectric materials by introducing equivalent currents. By

using the MOM, analysis of DRAs is not limited to a hemispherical shape, and the

technique can be used to also analyze simple cylindrical and rectangular DRA

shapes [63-73]. Determining the DRA input impedance using the MOM technique

will require more computer memory and time than applying Green's function,

since the DRA must usually be finely segmented to obtain an accurate solution

(especially when they are made of high dielectric constants). The MOM technique

is therefore not as convenient a tool for optimizing the DRA performance.

The FEM is a second frequency domain technique and can be used to analyze

DRAs of arbitrary shape. Similar to the MOM, it involves a discretization of the

geometry (usually using small tetrahedrons), but whereas in the MOM only the

DRA and the ground plane require segmentation, in the FEM technique, the entirevolume surrounding the DRA must also be discretized, thereby increasing the

computational size of the problem. The advantage of the FEM is that it does not

require the formulation of equivalent currents and can thus be readily applied toarbitrary shapes. Another advantage of the FEM is its availability as commercial

software where graphical user interfaces are provided to simplify the geometrical

definition of the problem. Examples of the use of the FEM to analyze DRAs can be found in [74-76]. In Chapter 6, the FEM is used to determine the effects of a

finite ground plane on the radiation patterns of a DRA.

3.8.3 Time Domain Analysis Techniques

Two time domain techniques that have been applied to analyzing DRAs are the

finite difference time domain (FDTD) method and the transmission line method

(TLM). Just as with the FEM, these time domain techniques require the entirevolume around the DRA to be discretized and thus can be memory and time

intensive. For the FDTD and TLM methods, small cubes are used for

discretization, instead of tetrahedrons, and care must be taken to properly modelcurved geometries, due to the stair-stepping effect. Time domain techniques use a

wideband pulse to excite the DRA, and by transforming the solution into thefrequency domain, the input impedance can be determined over a wide frequency

range. For the frequency domain techniques, the problem would have to be

resimulated at every frequency of interest and obtaining the impedance response

over a broad frequency range could be very time consuming. Commercial software

has also been developed for the FDTD and TLM techniques, eliminating the

necessity for designers to develop their own codes. Examples of DRAs analyzed

using time domain methods can be found in [77-86]. Again, as with the frequency

domain methods, the time domain methods are good tools for analyzing the

performance of a given DRA geometry, but are less useful for optimizing the performance of DRAs. However, with the continual increase in the speed and

memory of computers, it may not be long before these methods can also serve as

optimization tools, providing solutions within reasonable times.

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