Organization of This Thesis

Leaky modes on two types of substrates are investigated: cylindrical and planar types. The full-wave method, spectral domain approach (SDA) [18], is used to analyze the propagation characteristics of leaky modes on these structures.

Odd-symmetry and even-symmetry current and electric-field basis functions combined with the edge conditions [18] are chosen in the longitudinal and transverse directions according to the symmetry represented by a virtual perfect electric conductor (PEC) or a perfect magnetic conductor (PMC). Leaky modes on both cylindrical and planar structures have wide radiation regions and can be implemented as high-gain antennas.

In Chapter 2, cylindrical microstrip leaky-wave antennas are thoroughly studied.

Propagation constants of different structure parameters are calculated to decide the operating frequency band of cylindrical microstrip leaky-wave antennas. From the numerical results and measurement results, the cylindrical leaky-wave antennas have the similar propagation characteristics and antenna performance like those of planar microstrip leaky-wave antennas [1]-[6].

Most papers discussed the bound mode of coaxial lines [19]-[21] and cylindrical coplanar waveguides (CPW) [22], but few papers mentioned about leaky modes on such structures. In this thesis, the first higher order leaky modes on these two structures are studied in Chapter 3. For the first higher order leaky modes, the longitudinal currents of slotted coaxial lines are even-symmetric while the currents are odd-symmetric of CPWs. Two design examples are also presented to demonstrate the performance of slotted coaxial and CPW leaky-wave antennas.

Chapter 4 includes two broadband planar antennas: slotline leaky-wave antennas and inverted-T leaky-wave antennas. Slotlines and single-conductor strips are complementary structures indeed. From the calculated propagation constants, the first

grounded dielectric slabs. With the broadband microstrip-to-CPW transition [24], a wideband slotline leaky-wave antenna is designed. The slotline leaky-wave antennas are dual-beam antennas with the frequency-scanning feature.

In addition, many leaky-wave antennas require complicated feeding structures and large circuit size. Due to the current symmetry of the first higher order leaky modes, we only need a half of an antenna for space-wave leakage. Half-width leaky-wave antennas have several advantages such as suppression of bound mode [25], smaller size, and simpler feeding structure than full-width leaky-wave antennas. In the section 4.2, we design a novel broadband inverted-T leaky-wave antenna, which is derived from the single-conductor strip leaky-wave antenna.

Chapter 2

Cylindrical Microstrip Leaky-Wave Antennas

This Chapter presents the first higher order leaky mode of cylindrical microstrip lines. The full-wave method is applied to calculate propagation constants under different structural parameters. From the numerical results, we observe the phenomena caused by various parameters and can select the suitable operating frequency band of the antenna. With the aperture-couple feeding structure, a high-gain and wideband cylindrical microstrip leaky-wave antenna is designed and measured. In conclusion, the first higher order leaky modes can exist on cylindrical microstrip lines as those leaky modes exist on planar microstrip lines.

2.1 Spectral Domain Analysis

In this thesis, we use the spectral domain approach (SDA) to analyze the first higher leaky mode of cylindrical microstrip leaky-wave antennas. Fig. 2.1 shows the cross section of the cylindrical microstrip line. Region I is the cylindrical substrate with a permittivity of ε₁ =ε ε_{r 0} and a thickness of h. Region II is free space. The strip with a circumferential width w is placed on top of the substrate. The permeabilities of all materials are µ₀. The thickness of the top strip and the ground conductor are assumed to be zeros. All the conductors and substrates are assumed to be lossless for simplification. First we expand the z-components of fields

(

Ez^, Hz

)

in cylindrical coordinates by inverse Fourier series (suppressing the time-harmonic expression

ej t^ω and z -dependencee^{−jk zz} ) [12] as:

( )

^{1 (1)}

(

¹

)

¹

(

¹

)

β are the attenuation constant and the phase constant, respectively. k_iρ is the ρ-directed propagation constant of each region. The other components of fields

(

^Eρ^{, H}ρ^{, E}φ^{, H}φ

)

creeping waves and surface waves [15] are ignored in this analysis.

The cross-section on the ρφ-plane can be transformed into that of the planar microstrip line with the PEC sidewalls, like shielded microstrip lines. This direct mapping is similar to that used in [26].

φ =0

φL = −π φ_R =π

φR =π φL = −π

ρ =a ρ =b

φ =0

Fig. 2.2 The PEC boundary representation and transformation.

φ

w

φ

_w

φ

Fig. 2.1 The cross-sectional view of the cylindrical microstrip line.

The variables n of the inverse Fourier series (2.1)-(2.4) are integers on the real axis (x-axis) of the n-plane. In solving planar microstrip leaky-wave structures [27]-[30], the variables used in integrations over the real axis are continuous; however, variables are discrete in solving cylindrical structures. Since the surface-wave leakage phenomenon in planar transmission lines will not occur in cylindrical substrates, no consideration of the surface wave leakage is needed in this analysis. The branch points in solving planar microstrip leaky-wave structures are not needed in solving cylindrical structures because the variables n are irrelevant tok_iρ.

The sign of ρ-direction wavenumber in free space k_2ρ = k₂²−k_z² in the series (2.1)-(2.4) should be chosen properly to satisfy the exponential growth condition of the space-wave leakage in the positive ρ-direction, and this square root is at the improper Riemann sheet. The similar choice of branch is defined in [31].

From (2.1)-(2.4) and boundary conditions between Region I and II with some algebraic operations, the unknowns (A_1n,B_1n,C_1n,D_1n,A_2n,C_2n) are solved and we obtain the relationships between Fourier transforms of currents and electric fields, J

and E
:

with unknown coefficients

(

^{up, vq}

)

.The basis functions used in the computations are:

For numerical computation, we take the inner products of the above two equations with the other sets of basis functions Jzd

( )

φ and J_φg

( )

φ as testing functions (Galerkin’s method) over the top strip. The numbers of testing functionsJ_zpandJ_φq, arep_max and

qmax, respectively. After the inner product operations, we can obtain a matrix form:

zz z

coefficients. The nontrivial solution of the complex propagation constant k_{z exists} only if

2.2 Numerical Results

2.2.1 Convergence Test

Each element of the Q-matrix is an infinite series, e.g.

( _{z zz zz}) _zz( ). show the relative deviation (from n=120 case) of real and imaginary parts in Fig. 2.3 to check the convergence of the series. Smaller outer radius causes the series to converge a little bit faster. Good convergences are obtained with the number of the summation terms n > 100 and with more than six basis functions used. The relative deviation of propagation constants for the number of basis functions is displayed in Fig. 2.4.

Number of terms (n)

20 40 60 80 100

Relative error (%) of JzzGzzJzz

-100

2.2.2 Comparison of the Results from the SDA to the Result from the Leaky-Mode Scattering Parameter Extraction

To check the validity of the numerical results, we introduce the leaky-mode scattering parameter extraction technique [32] to extract the propagation constants from scattering parameters.

We choose two sections of leaky-wave microstrip lines, with the same design except their longitudinal lengths. Then we place two identical leaky-mode excitation circuits at the both ports of each of the two leaky-wave microstrip lines. After we obtain the two sets of scattering parameters of these leaky-wave microstrip lines from the full-wave simulation software, we can extract the propagation constants with some simple calculations based on the transmission line theory [33]. Numerical results obtained by this method are checked with those obtained by SDA. They show a good agreement. An example is presented in Fig. 2.5.

Number of basis functions

Fig. 2.4 Convergence of normalized phase constant and normalized attenuation constant of cylindrical microstrip leaky-wave antenna. w = 10 mm, h = 0.508 mm, εr = 2.2, b = 10 and 20 mm at 9.5GHz.

2.2.3 Effect of Different Outer Radii

First we use angular width R of the line width w to the outer radius b of the substrate,

/ .

R=w b (2.14) This definition represents the curvature of the cylindrical structure. If the radius becomes larger with a fixed line width, the angular width R decreases, which indicates that the structure is flatter. For example, the substrate thickness h is 0.508 mm and the antenna width is 10 mm. We choose three different outer radii, b = 10, 20, and 30 mm with three angular widths: 1, 0.5, and 1/3. It is reasonable that when the radius increases, the propagation constants gradually approach those of the planar microstrip leaky-wave modes.

Fig. 2.6 shows the computed normalized phase constant β / k₀and Fig. 2.7 shows the computed normalized attenuation constants α/ k₀, where k_z =β− jα .We find

Frequency (GHz)

Fig. 2.7 Normalized attenuation constants of the first higher order mode (w = 10 mm, h = 0.508 mm, εr = 2.2, b = 10, 20, 30 mm and planar ).

can be used as the lower frequency limit of effective space-wave leakage [3]. The lower frequency limit occur at about 9.65, 9.45, and 9.25GHz, for b= 10, 20, and 30 mm, respectively.

8 9 10 11 12 13

2.2.4 Effect of Different Substrate Thicknesses

Fig. 2.8 plots the computed normalized attenuation constants

α

/k₀ and the

normalized phase constants

β / k

₀ for two different substrate thicknesses h = 0.508 and 1.570 mm, with the same outer radius b = 10 mm. The antenna width w is 10 mm. When the substrate is thicker, the radiation region moves to a lower

frequency band. In planar microstrip leaky-wave antennas, similar effects can be observed [5]. The attenuation constant α/k₀also becomes smaller, which means less attenuation. The lower frequency limit of effective space-wave leakage (α =β) is 8.75 and 9.65GHz for h = 1.570 and 0.508 mm, respectively.

8 9 10 11 12 13

2.2.5 Effect of Different Line Widths

Fig. 2.9 shows the effects of the line width on the propagation constants of the cylindrical microstrip leaky-wave antennas. From Fig. 2.9, a smaller microstrip line width results in a higher frequency band for space-wave leakage. In planar microstrip leaky-wave antennas, similar effects can be observed [1]. The lower frequency limits of effective space-wave leakage appear at about 8.55, 9.30, and 10.15GHz, corresponding to widths w = 11, 10, and 9 mm, with almost the same values α =β =0.2. As the line width reduces, the lower frequency limit of the effective space-wave leakage increases.

Fig. 2.10 The feeding and the antenna structure.

2.3 Design Example and Measurement

Since the first higher leaky mode of cylindrical microstrip lines has a wide radiation region, a broadband cylindrical type leaky-wave antenna can be implemented.

We design aperture-coupling feeding structures with enough bandwidth to excite the first higher leaky mode and deliver power into antennas. The antenna system configuration in Fig. 2.10 consists of two layers of substrates. The upper substrate is utilized for the antenna, and lower substrate is applied for the feeding microstrip line, respectively. Both substrates share the same ground with the coupling slot. The top view of the antenna system in Fig. 2.11 shows the positions of the coupling slot and the feeding microstrip line.

Both of these substrates have a dielectric constant of ε_r= 2.2 and a thickness of h = 0.508 mm. On the other hand, substrates thicker than 0.508 mm are too firm to bend into the cylindrical shape. The energy goes through the coupling slot to the leaky-wave microstrip and excites the first higher leaky mode [4].

The dimensions used are the following: the feeding line width w_f = 1.88 mm, the distance from the feed line end to the antenna center lm = 3.9 mm , the slot width w _s = 0.5 mm, the distance from the slot end toward the antenna to the antenna starting point lc = 3 mm, the slot length (lc + lo) with lo = 9 mm, the antenna

width w = 10 mm, and the antenna length L =150 mm. All of these structural parameters are selected to obtain a good impedance matching with enough bandwidth.

Two sets of antennas with two different outer radii b = 20 and 30 mm are designed, which are specified as Antenna I and Antenna II, respectively. From the measured scattering parameters shown in Fig. 2.12, the bandwidth (with S11 < -10dB) of Antenna I is 9.59~12.71 GHz; and those for Antenna II is 9.77~12.89GHz. They both have 3.12GHz bandwidth. It is noted that this kind of antenna has the potential to be a good wideband antenna.

From the results in Fig. 2.12, we can find that a larger radius (b=30 mm) causes a lower operating frequency. There are two dips near 11GHz and 12.5GHz for each cases (b=30 for the solid line and b=20 mm for the dashed line). The dips are mainly due to the original resonant modes of the feeding micrsotrip lines, the feeding substrates, and the ground plane.

Fig. 2.11 The top view of feeding and the antenna structure.

Frequency (GHz)

8 9 10 11 12 13 14

Return Loss (dB)

-40 -30 -20 -10 0

b=30mm b=20mm

Fig. 2.12 Measured return loss of the cylindrical microstrip antenna with two angular widths (w = 10 mm, h = 0.508 mm, εr = 2.2, b = 20 and 30 mm).

-15 -5 5 15

Fig. 2.13 Measured copolarization radiation patterns of the cylindrical microstrip antenna with an outer radius b = 20 mm.

Copolarization radiation patterns of Antenna I and Antenna II in xz-plane are plotted in Fig. 2.13 and Fig. 2.14. Antenna I have the peak gains 11.79, 14.86, and 11.76 dBi for 10, 11, and 12GHz, respectively, in Fig. 2.13. The corresponding mainbeam directions starting from the z-direction (endfire direction) are 67°, 46°, and 33°. This is the well-known frequency-scanning feature of leaky-wave antennas.

With the same three frequencies selected above, Antenna II has 12.76, 14.82, and 11.96 dBi peak gains, as shown in Fig. 2.14. Compared to the Antenna I, larger outer radius of the antenna causes smaller mainbeam directions from the endfire, due to its larger phase constants. This phenomenon can be deduced from the data in Fig. 2.6.

-15 -5 5 15 -180

-150

-120

-90

-60

-30 0 30 60

90 120

150

10GHz, b=30mm 11GHz, b=30mm 12GHz, b=30mm

Fig. 2.14 Measured copolarization radiation patterns of the cylindrical microstrip antenna with an outer radius b = 30 mm.

2.4 Summary

This Chapter proposes a cylindrical microstrip antenna structure with a full-wave analysis and its implementation. We investigate how the angular widths and the substrate thicknesses affect the propagation constants and radiation patterns of this cylindrical structure. Measured scattering parameters and radiation patterns of these realized antennas circuits are presented. Finally, we conclude that cylindrical leaky-wave antennas have the high-gain and wideband features, similar to those of the planar leaky-wave antennas.

Chapter 3

Leaky Modes on Cylindrical Substrates

In this Chapter, the first higher order leaky modes on the other two types of cylindrical structures are demonstrated: slotted coaxial lines and cylindrical coplanar waveguides. They have different field symmetries which are represented with virtual PMC or PEC about the center. With the same method used in Chapter 2, spectral domain approach, propagation characteristics of the first higher order leaky modes on both structures are investigated. From the measurement results of two design examples, these two types of antennas can also be high-gain and wideband. Briefly, we have shown that such two structures can support space-wave leaky modes.

3.1 Leaky Modes of Slotted Coaxial Lines

3.1.1 Spectral Domain Analysis of Slotted Coaxial Lines

This section discusses the propagation characteristics of slotted coaxial lines. Fig.

3.1(a) shows the cross section of slotted coaxial lines. The slot with width w is on top of the substrate. The inner ground conductor and the outer conductor shell are located at circumferences of radii a and b, respectively. The permeabilities of all materials are µ₀. The thickness of the outer shell and the ground conductor are assumed to be zero.

In the analysis of the first high order leaky mode of slotted coaxial lines, we place a PMC boundary at the bottom (φ=π ) in Fig. 3.1(b) to represent the appropriate field symmetry about the center.

The z-direction fields represented as inverse Fourier series are the same as in (2.1)-(2.4). Similarly, the time-harmonic expression e^{j tω} and z-dependence e^{−jk zz} are assumed. The longitudinal (z-direction) electric field on the slot is

φwφ_w

φ φ =⁰

φL = −π φ_R =π

(a) (b)

Fig. 3.1 (a) The cross-sectional view of the slotted coaxial line. (b) The PMC boundary representation the slotted coaxial line.

even-symmetric about φ = , whereas the transverse electric field (0 φ-direction ) is odd-symmetric about φ = . For slotted coaxial lines, the longitudinal electric field 0 distributions are like the transverse current distributions of cylindrical microstrip lines, and the transverse electric field distributions are like the longitudinal current distributions of cylindrical microstrip lines. If the electric fields of the first higher leaky mode in slotted coaxial lines are equivalent to magnetic currents (M_z =E_φ×ρ, M_φ =E_z×ρ), these magnetic currents will be similar to electric currents of the first higher leaky mode in cylindrical microstrip lines.

We solve the coefficients by applying boundary conditions. Therefore we obtain substrate thickness h, are investigated. The dielectric constant of the substrate εr is 2.2.

In Fig. 3.2, the radius b is 100 mm, the thickness h is 10 mm, and the widths w are 10, 15 and 20 mm. As the slot width w increases, the curve of the propagation constant moves to a lower frequency region. Fig. 3.3 shows how the radius b affects the curves.

The slot width w is 10 mm, the thickness h is 10 mm, and the outer radii b are 20, 40, and 80mm.The larger radius also changes the propagation constant to a lower frequency band.

Fig. 3.4 plots the curves for three different substrate thicknesses h, which are 6, 8, and 10mm. The slot width w is 10 mm and the outer radius b is 50 mm. The lower

Frequency (GHz)

2 4 6 8 10 12

β /k

0

, α /k

0

0.0 0.5 1.0 1.5 2.0 2.5

β/k₀, w=10mm α/k₀, w=10mm β/k₀, w=15mm α/k₀, w=15mm β/k₀, w=20mm α/k₀, w=20mm

Fig. 3.2 The normalized propagation constants of different slot widths ( b = 100 mm, h = 10 mm, εr = 2.2, and w = 10, 15 and 20 mm).

bounds

(

^α=β

)

of the radiation regions

(

α≤β≤k0

)

occur at 9.675, 8.750, and 6.125GHz, corresponding to h = 6, 8, and 10 mm. It is notably that these frequencies are approximately proportional to the reciprocal of substrate thicknesses. We conclude that the outer radius and slot widths do not affect propagation constants very much, whereas the substrate thickness may dominate the operating frequency.

Frequency (GHz)

Fig. 3.3 The normalized propagation constants of different slot widths ( b = 100 mm, h = 10 mm, εr = 2.2, and w = 10, 15 and 20 mm).

Fig. 3.4 The normalized propagation constants of different substrate thicknesses ( b = 50 mm, h = 6, 8, and 10 mm, εr = 2.2, and w = 10 mm).

Fig. 3.5 The proposed conductor-backed slotline leaky-wave antenna.

3.1.3 Design Example: A Conductor-backed Slotline Leaky-Wave Antenna

From the numerical results in Section 3.1.2, the radius of the coaxial line is too large (6 mm or more) in fabrication for operating frequencies less than 14 GHz.

Therefore we design a planar type of slotted coaxial lines, which are equivalent to the conductor-backed slotlines to demonstrate that such a leaky mode can propagate in this structure. Fig. 3.5 illustrates the proposed conductor-backed slotline leaky-wave antenna, with a simple microstrip line and a gap resonator feeding. The substrate is chosen as air to reduce the electrical length of the substrate thickness, and the operating frequency is lowered. The substrate thickness h is 10 mm, the slot width w₁ is 10 mm, the width of two side top conductors w2 is 20mm, and the length of top conductors is 150 mm.

Fig. 3.6 The normalized phase constants and attenuation constants.

Fig. 3.6 displays the normalized phase constants and attenuation constants. The radiation region starts at 12 GHz. As shown Fig. 3.7, the bandwidth of this antenna is about 2.95 GHz, which begins from 15.95 to 18.90 GHz. At 13, 14, 15 GHz, the antenna gains plotted in Fig. 3.8 are 9.75, 12.16, 14.01 dBi, respectively. At 16, 17, 18 GHz, the antenna gains plotted in Fig. 3.9 are 15.02, 14.95, 16.52 dBi, respectively.

This conductor-backed slotline leaky-wave antenna has the similar frequency-scanning property as a microstrip leaky-wave antenna does.

Fig. 3.7 The return loss of the conductor-backed slotline antenna.

Fig. 3.8 The copolarization radiation patterns at 13, 14 15 GHz in the xz-plane.

Fig. 3.9 The copolarization radiation patterns at 16, 17, 18 GHz in the xz-plane.

φwφ_w φ

φ =0

φL = −π

φR =π φs φ_s

(a) (b)

Fig. 3.10 (a) The cross-sectional view of the CPW on the cylindrical substrate.

(b) The PEC boundary representation.

3.2 Leaky Modes of

Coplanar Waveguides on Cylindrical Substrates

3.2.1 Spectral Domain Analysis of Coplanar Waveguides on Cylindrical Substrates

The proposed leaky CPW on cylindrical substrate is plotted in Fig. 3.10(a).

Region I and III are both free space. Region II is the cylindrical substrate with a permittivity of ε2=εrεo and a thickness of h. The center strip (with the width w) and the slots (with the widths s) are on the top of the substrate. The center strip, slots, and ground conductor are all located at circumferences of radius b. As the same symmetry shown in Chapter 1, a PEC boundary is placed at the top and the bottom (φ =0, π ) in Fig. 3.10(b). The longitudinal currents (z-direction) on the center strip and the ground conductor are odd-symmetric about φ =0, π . The transverse currents (φ-direction) are even-symmetric about φ =0, π. This CPW leaky mode is similar to the coupled slotline mode [34]-[35], and the main difference is that the currents of CPW leaky

Region I and III are both free space. Region II is the cylindrical substrate with a permittivity of ε2=εrεo and a thickness of h. The center strip (with the width w) and the slots (with the widths s) are on the top of the substrate. The center strip, slots, and ground conductor are all located at circumferences of radius b. As the same symmetry shown in Chapter 1, a PEC boundary is placed at the top and the bottom (φ =0, π ) in Fig. 3.10(b). The longitudinal currents (z-direction) on the center strip and the ground conductor are odd-symmetric about φ =0, π . The transverse currents (φ-direction) are even-symmetric about φ =0, π. This CPW leaky mode is similar to the coupled slotline mode [34]-[35], and the main difference is that the currents of CPW leaky

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