Mode Distinction - Physical Properties of Leaky Waves

Chapter 2 Physical Properties of Leaky Waves

2.2 Mode Distinction

After our previous discussions, we will follow the classification by Lin in [10], [11] to divide the frequency range into the following four regions:

1. , (large ) ---reactive cutoff region.

2. , (small ) ---surface wave and space wave leakage region.

3. , (small ) ---surface wave leakage region.

4. , ---bound mode region.

In the radiation region ( ) with a larger attenuation constant is reactive and below cutoff, and has a different mode nature from that with a smaller attenuation constant. Therefore, this radiation-frequency region can further be divided into two regions with decreasing frequency:

the antenna-mode region ( , ) where most of the guided power leaks away in the b

Fig. 2.10 Radiation leakage in a microstrip line at the air-substrate interface.

forms of space wave and surface wave, and the reactive cutoff region, where most of the power is reflected back to the feed line, however, the propagation constant is still a complex number with a small real part, , and a large imaginary part, , indicating that the mode is not strictly cutoff, but a very small portion of energy still propagates down the transmission line. In design of leaky-wave antennas, this distinction of mode nature in the radiation region is essential since the antenna efficiency is low in the reactive-mode region. We can simply define the lower frequency edge ( ) and the upper frequency edge ( ) of the usable frequency range for the leaky wave antenna:

(2-5) (2-6) Fig. 2.11 shows the normalized propagation constant for the first higher-order mode of microstrip line as an example. For this case ( , , ), the usable frequency range for the leaky wave antenna is approximately 5.1 to 6.1 GHz.

Fig. 2.11 Behavior of the normalized phase constant and the attenuation constant as a function of frequency for the first higher-order mode of the microstrip line.

Chapter 3 Analysis and Numerical Results

Planar transmission line analysis in the Fourier transform domain (or spectral domain) is superior to many numerical methods in the spatial domain. The spectral domain approach (SDA) was presented by Itoh and Mittra [12], this method is basically a modification of Galerkin’s approach adapted for application in the Fourier transform domain, or spectral domain (In SDA, Galerkin’s method is used to yield a homogeneous system of equations to determine the propagation constant). The Fourier transform is taken along the direction parallel to the substrate and perpendicular to the strip. The main reason the SDA is numerically efficient is that it requires a significant analytical preprocessing. This feature in turn imposes a certain restriction on the applicability of the method. One of the limitations is that SDA requires infinitesimal thickness for the strip conductor. It is also difficult to treat the structure with a strip having finite conductivity. No discontinuity in the substrate in the sideward direction is allowed. In spite of these limitations, however, SDA is one of the most popular and widely used numerical techniques.

In this chapter, the general approach (field approach) is described [13].

3.1 SPACE DOMAIN TO SPECTRAL DOMAIN

Before the detailed formulation process is presented, let us compare the types of equations obtained by the SDA and those obtained by a typical space domain formulation [13].

Fig. 3.1 shows a shielded microstrip line with its cross-sectional view as an example. In conventional space domain analysis, the structure can be analyzed by first formulating the following coupled homogeneous integral equations.

The equations will then be solved for the unknown propagation constant .



Zzz⁽xx y J x^{', )} z^{( ')}Zzx⁽xx y J x^{', )} x^{( ')}



dx^'E xz^{( )} sides of the equations are therefore required to be zero on the strip. These equations can be solved if Z_zz, Z_zx, Z_xz and Z_xx are given. As we will see shortly, the following algebraic equations, instead of the coupled integral equations, are obtained in the spectral domain formulation. These equations are Fourier transforms of the coupled integral equations.

( , ) ( ) ( , ) ( ) ( , ) Where quantities with tildes (^~) are Fourier transforms of corresponding quantities.

Ground Plane

Fig. 3.1 Corss-sectional view of a shielded microstrip line.

Ground Plane

The Fourier and inverse Fourier transform are defined as ( )k_x ^ ( ) x e^{jk x}^x dx integration over all x, not only over the strip. The equations contain four unknowns J_z, J_x, Ez, E_x with unknown . E_z and E_x, however, will be eliminated in the solution process based on the Galerkin procedure.

3.2 SDA ON SINGLE-CONDUCTOR STRIP STRUCTURE

3.2.1 Field Approach

In this section, the Green’s impedance functions Z_zz, Z_zx, Z_xz, Z_xx will be derived for the single-conductor strip structure in Fig. 3.2. Only one perfect conducting and infinitely thin strip is located at the interface between a semi-infinite air layer and an isotropic lossless substrate, with a dielectric constant of and a thickness of h. This structure is assumed to be uniform and infinite in both x- and z-directions. First, the hybrid fields are expressed in terms of Superposition of TE-to-y and TM-to-y expressions [14] with scalar potentials ^e and ^h as follows:

Fig. 3.2(a) Single-conductor strip structure. Fig. 3.2(b) Cross-sectional view of a single-conductor strip structure.

13 corresponding quantity in the space domain (see Appendix A). The Fourier-transformed Helmholtz equation is expressed as (see Appendix A)

2 The solution for this homogeneous differential equation is described in the form of

2 2 2 2

coefficients. is the propagation constant in the y-direction and may be written as . Solution of (3-7) (3-9) into (3-4) yields the field expressions in the three regions:

1 1 boundary conditions in the spectral domain are obtained as the Fourier transforms of those in the space domain. In space domain, the boundary conditions are written as follows,

15 Notice these quantities need to be introduced so that the boundary conditions are specified for the entire range of x. Otherwise; it is not possible to take Fourier transforms. In the spectral domain, the boundary conditions are now given by the following equations.

At :

Finally, the algebraic equations are derived in matrix from as follows:

The derivation of these formulations is detailed in Appendix B. It should be noted that there is one more set of boundary conditions not used up to this stage. In the space domain, it is

0 for 2 at

z x

E E  x w yh (3-20) This set of conditions is incorporated in the solution process as we will see below.

and J_x have known forms, and the only unknowns in their representation are the amplitude coefficients c_m and d_m. The basis functions must be chosen to correspond to the odd or even symmetry of the currents for the mode of interest. The current is nonzero only on the strip. Therefore, the basis functions J_zm( )k_x and J_xm( )k_x must also be chosen such that

their inverse Fourier transforms are nonzero only on the strip x w 2. The accuracy of the numerical results can be increased by selecting higher values of M and N, but it is relatively low efficiency because of taking more time during numerical computation. It means that the accuracy and efficiency depend on the numbers of basis functions. As shown in Fig. 3.3, due to the structural symmetry, a virtual perfect electric wall is placed at the center of the single

PEC

Fig. 3.3 Cross-sectional view of a single-conductor strip structure. A virtual perfect electric wall is placed at the center of this structure for the first higher-order mode.

Note that the definitions given above are only over the strip x w 2 and the functions are zero elsewhere. The functions in (3-22) incorporate the correct edge singularity. The shapes of the first three functions are shown in Fig. 3.4. The Fourier transforms of (3-22) and (3-23) are (see Appendix C for derivations)

where J₀ denotes the zero-order Bessel function of the first kind.

Fig. 3.4 Shapes of basis functions

zm( )

J x J_xm( )x

3.2.3 Method of Solution

In this section, an efficient method for solving (3-13) is presented. It is noted that the two equations in (3-13) contain four unknowns J_z, J_x, E_z and E_x. The latter two unknowns

J k , respectively, for different values of k and l. This process yields the matrix equation

1 1

The right hand sides of (3-26) are zero by virtue of Parseval’s theorem, because the currents

zk( )

J x , J_xl( )x and the field components E x h_z( , ) , E x h_x( , ) are nonzero in the complementary regions of x. For instance, if the inner product of E_z on the left-hand side of (3-13) and J_zk( )k_x is taken, one obtains

Equations (3-26) will be expressed in matrix form as follows

(1,1) (1,2) process, the true value of k_z is obtained at each frequency.

21 depending upon the choice of the integration contour. The conventional path, which lies along the real axis in the complex k_x-plane (contour C₀ of Fig. 3.5), yields the solution for the proper (bound) mode. The other two paths C₁ and C₂ in Fig. 3.5 are used to obtain leaky-mode solutions [15]. The absence of a ground plane allows the TM0 and the TE0

surface-wave modes to exist in the single-conductor strip structure [6]. Because both of the TM0 and the TE0 surface-wave modes exist for a dielectric slab structure (see Appendix D), so two proper surface-wave poles of vertical wavenumber k _y [16]. This path is used to obtain the solution for a leaky mode that has leakage into only the TM0 and the TE0 surface waves [8]. In order to yield the solution for a leaky mode that energy leaks into both the space wave and the surface wave, the path C₂ is utilized. This path lies partly on the improper sheet of the k_x plane in the region between the branch cuts and includes both proper surface-wave poles for the TM₀ and the TE₀ modes [6].

For the first higher-order leaky mode of a single conductor strip structure:

1) propagation constant k_z   j (3-33) 2) k_y₀   k₀² k_x²k_z² , (proper sheet "", improper sheet "+") (3-34)

3) branch points (k_y₀ 0): k_xb   k₀² k_z² (3-35) These branch points define a two-sheeted Riemann surface in the k_x plane. Using the Sommerfeld choice for defining the corresponding branch cuts,

: integral path

Fig. 3.5 Integral paths of the inverse Fourier transform on the k_x plane.

: branch cut : branch point

: proper surface wave poles (TM₀, TE₀)

23 derivation is detailed in Appendix E.

4) TM0 poles: poles along the k_x integral path. These poles are encountered when the denominator of Z_e

or Z_h is equal to zero. The zeros of the denominators of Z_e and Z_h correspond to the odd sense. This means that the integrands of Eqs. (3-28)-(3-31) involve four residual contributions which correspond physically to excitation of the TM₀ and TE₀ surface waves; therefore we

can rearrange each of them as the quotient form as follows:

( ) ( ) ( ) ( ) Thus, the spectral integrals need to be evaluated along C₂ and can be simplified to

, TM0 , TM0

Fig. 3.6 Integral paths of the inverse Fourier transform on the k_x plane.

3.2.5 Numerical Results

Fig. 3.7 plots the normalized phase constant  k₀ and the normalized attention constant  k₀ against the frequency of the first higher order mode of the single conductor strip structure with and for different strip widths. Because of the resonance of the transverse currents, the leaky region obviously shifts to a higher frequency for a narrow strip. Also, a narrow strip increases the attention constant.

Fig. 3.7 Behavior of the normalized phase constants and the normalized attenuation constants as a function of frequencies for the first higher order mode of the single conductor strip structure with and for different strip widths.

Fig. 3.8 plots the normalized phase constant  k₀ and the normalized attention constant  k₀ against the frequency of the first higher order mode of the single conductor strip structure with and for different dielectric constants. A substrate with a high dielectric constant will rapidly increase the normalized phase constant.

Increasing the dielectric constant of the substrate also increase the attenuation constant.

Fig. 3.8 Behavior of the normalized phase constants and the normalized attenuation constants as a function of frequencies for the first higher order mode of the single conductor strip structure with and for different dielectric constants.

Fig. 3.9 plots the normalized phase constant  k₀ and the normalized attention constant  k₀ against the frequency of the first higher order mode of the single conductor strip structure with and for different substrate thicknesses. When the thickness of the substrate increases, the normalized phase constant increases, since the effective strip width is reduced by the fringing effect. The attenuation constant of a thin substrate is less than that of a thick substrate. Most of the energy is focused under the strip when the dielectric constant is high or the substrate is thick. The energy leaks out to the air more easily if the thickness or the dielectric constant of the substrate is lower or smaller.

Fig. 3.9 Behavior of the normalized phase constants and the normalized attenuation constants as a function of frequencies for the first higher order mode of the single conductor strip structure with and for different substrate thicknesses.

Chapter 4 Antenna Design, Simulation and Measurement

Broadband, high gain, and frequency-scanning are the main features of leaky wave antennas [9]. In some applications, such as point-to-point communication, the frequency-scanning characteristic is undesired. In this chapter, we use a single-conductor strip leaky wave antenna, which is proposed in [6]. Unlike other leaky-wave antennas, whose main beam changes according to frequency, the single-conductor strip leaky-wave antenna has a fixed main beam in the end-fire direction over a broadband region. Compared to the resonant antennas such as microstrip patch antenna and dipole antenna, whose radiation patterns are in the broadside direction, the pattern feature of single-conductor strip leaky-wave antenna is useful in some applications that require a main beam in the end-fire direction. Finally, to alleviate the problem of a large back lobe [17], the feeding structure of the single-conductor strip leaky-wave antenna is modified with two broadband planar baluns [18]. The simulated and measured results are also presented in this chapter.

4.1 DESIGN OF SINGLE-CONDUCTOR STRIP LEAKY-WAVE ANTENNA

4.1.1 Broadband Planar Feeding Structure

As described above, the single-conductor strip structure has only a single-conductor strip on a substrate without a practical ground plane. For the first higher-order leaky mode in this structure, an infinite virtual PEC boundary is assumed at the center of the strip, in which the longitudinal currents are odd-symmetric and the transverse currents are even-symmetric with respect to the center [6]. As shown in Fig. 4.1, these current distributions help us design an appropriate feeding structure which generates two out-of-phase currents to feed this

single-conductor strip structure and thus excites the first higher order leaky mode. As shown in Fig. 4.2, a broadband planar feeding structure which consists of a conventional microstrip line、a microstrip-to-balanced-microstrip-line transition、a balanced-microstrip-line T-junction power divider and one set of the balanced microstrip lines is changed to form the inverted balanced microstrip lines is developed for the first higher-order leaky mode of the single-conductor strip line [6]. As shown in Fig. 4.3, gradually tapering the ground plane to a width equal to the strip width w makes conventional microstrip line a balanced microstrip line, with a strip of positive voltage on the upper side of the substrate and a strip of negative voltage on the lower side of the substrate.

y

z

x

PEC symmetry plane

To excite the first higher-order leaky mode w

h

No ground

Fig. 4.1 An appropriate feeding structure which generates two out-of-phase currents is used to feed the single-conductor strip structure and thus excites the first higher order leaky mode.

(I) (II) (III) (IV)

Fig. 4.2(a) Diagram of the broadband balun structure used to excite the first higher order leaky mode of the single-conductor strip line. The strips of the balun on the lower side of the substrate are connected to each other and returned back to the original ground plane of the microstrip line. This feeding structure consists of (I) a conventional microstrip line; (II) a microstrip-to-balanced-microstrip-line transition;

(III) a balanced-microstrip-line T-junction power divider and (IV) one set of the balanced microstrip lines is changed to form the inverted balanced microstrip lines.

Fig. 4.2(b) The strip of the balun on the upper side of the substrate.

Inverter

Fig. 4.2(c) The strip of the balun on the lower side of the substrate.

Circumfluence fabrication

w

Fig. 4.3(a) Gradually tapering the ground plane to a width equal to the strip width w makes conventional microstrip line a balanced microstrip line.

(a) (b) (c)

Microstrip line Transition Balanced microstrip line w Top layer strip

Bottom layer strip

Fig. 4.3(b) Cross-sectional view of a balanced microstrip line, with a strip of positive voltage on the upper side of the substrate and a strip of negative voltage on the lower side of the substrate.

The position of the positive strip on the upper side of the substrate of the inverted balanced microstrip line structure is exchanged with that of the negative strip on the lower side of the substrate after a microstrip phase inverter illustrated in Fig. 4.4(a). As shown in Fig. 4.4(b), each of these positive and negative strips (strip1 and strip2) is terminated with a chamfered right-angled bend in an opposite direction, and each of the subsequent strips (strip3 and strip4) is also headed with a chamfered right-angle bend but in the other direction. A slanted gap separates two strips on the same side of the substrate. The bent stubs , and on the upper and lower sides of the substrate, respectively, can be used to compensate for the reactance induced by the via holes and the slanted gaps, and may have different lengths. The positive strip on the upper substrate is connected vertically through a cylindrical via to the subsequent strip on the lower side of the substrate. The negative strip on the lower side is similarly connected to the subsequent strip on the upper side. Hence, the positions of the positive and negative strips alternate as shown in Fig. 4.4(c). The details of the design method for such a microstrip phase inverter structure are presented in [19].

via

Fig. 4.4(a) Top view of the inverted balanced microstrip line.

Top layer strips Bottom layer strips

Strip1 Strip3 Strip2 Strip4

Fig. 4.4(b) The width of the bent stubs: ; the length of the bent stubs: and ; the gap width: ; the diameter of vias: ; the slanted angle is 45 degrees.

Top layer signal flow Bottom layer signal flow

Fig. 4.4(c) Signal flow graph of the inverted balanced microstrip line.

Using a balanced-microstrip-line T-junction power divider with inverted balanced microstrip lines substituted for balanced microstrip lines in one of the two output ports, two pairs of broadband planar baluns can be formed as shown in Fig 4.5.

The balun on the lower side of the substrate produces a disturbed radiation because no background metal plane is located beneath the single-conductor strip structure. Fig. 4.2(a) depicts a method for preventing such disturbed radiation pattern. The strips of the balun on the lower side of the substrate are connected to each other and returned back to the original ground plane of the microstrip line. Hence, only the balun on the upper side of the substrate feeds the single-conductor strip and a closed metal loop is formed on the lower side of the substrate. As shown in Fig. 4.2(c), on this closed metal loop, two semicircles beneath the two feeding strips are etched out with diameters equal to the widths of the feeding strips, respectively, to enhance the electrical field transition from feeding points to the single-conductor strip.

Fig. 4.5 Two pairs of the broadband planar baluns on the upper and lower substrate sides, respectively.

4.1.2 Performance of Single-Conductor Strip Leaky-Wave Antenna

Fig. 4.6 shows the geometry of the single-conductor strip leaky-wave antenna with a strip width of w32.5mm and a strip length of L100mm, on a substrate with a dielectric constant of _r 3.55 and a thickness of h0.508mm. The substrate used to design this antenna is Rogers 4003. The geometry of the broadband feeding structure of this antenna is shown in Fig. 4.7, and the S parameter magnitudes and phase difference of this broadband feeding structure are plotted in Fig. 4.8(a) and Fig. 4.8(b), respectively. The return loss is always below -10 dB from 3 GHz to 8 GHz and insertion losses are between -3.4 dB and -3.8 dB over the frequency range of 5 GHz to 6 GHz. The phase imbalance is calculated as 180^ 

 

S21  

 

S31 , and is shown to be less than 10^ from 3 GHz to 8 GHz. Within the frequency from 5 GHz to 6 GHz, the phase imbalance is less than 2^. Careful attention is

 

S21  

 

S31 , and is shown to be less than 10^ from 3 GHz to 8 GHz. Within the frequency from 5 GHz to 6 GHz, the phase imbalance is less than 2^. Careful attention is

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