CHAPTER 1 INTRODUCTION
2.3 Mathematical Analysis of the Leaky Wave Antenna
2.3.4 Transmission Line Network Representation
B
y the electromagnetic boundary conditions, the tangential electric and magnetic field components are continuous at the interfaces between air and dielectric slab, those are the interface planes at ⎟⎠ view of this, it is convenient to cascade these three sections of finite-length transmissions lines, as mentioned in the above section, to form a transmission line network. As for the terminations of the transmission line network, we can apply the electromagnetic boundary conditions to determine the terminations of the transmission line network. Moreover, one must make some efforts to treat the discontinuity problem, in particular for the case of waveguide opening immediate to a half space.
We look at the left-hand side of the waveguide first. Based on the electromagnetic boundary condition, the tangential electric field vanishes on the surface of PEC (perfect electric conductor) wall (x = -a). Since the tangential electric field is related to the voltage wave of the transmission line, as depicted in equation (14), it means that the voltage at the left-hand side must be zero, that is, v(-a) = 0, this shows a short-circuit termination at x = -a.
On the other hand, at the right hand side, the slit is an immediate opening to the half space.
It is a problem of continuous spectra rather than the discrete modes in a closed waveguide.
By the approximation formulation in Waveguide Handbook [27], we could obtain the lumped circuit elements, which contain a conductance (G) and a susceptance (B) to model the radiation and storage energy, respectively, in the vicinity of the slit discontinuity. The detailed mathematical derivation can be referred to [27] (Waveguide Handbook). With the equivalent lumped circuit at the right-hand side and shorted circuit at the left-hand side, one can obtain a complete transmission line network, where the values of conductance G and susceptance B can be referred to [26-27] and given as:
a
, where e = 2.718 and γ = 1.781.
2.3.5 Transverse Resonance Equation
W
ith the transmission line network representation shown in the above section, the complete network can be drawn in figure 3(h). Because the wave is propagating along a uniform waveguide along the z-direction, we are able to apply the resonance condition in the transverse plane, which is the vanishing sum of input impedances seen when looking into left and right of this reference plane (x=-c, where c is smaller than the waveguide width a). This transverse resonance equation defines the dispersion relation of the waveguide. The reference plane to perform the resonance condition can be chosen at any position within the network. However, according to our experience, to obtain an accurate dispersion root, one could choose the transverse resonance reference plane at the plane where the field strength is relatively strong. Here, both the input impedance seen when looking into left or right are functions of propagation constant κ, and the dispersion relation can be expressed as,0 right, respectively, from some arbitrary reference plane. If we choose the reference plane
as ⎟
x , then the input impedance of a section of short-circuited transmission line
with length ⎟
Zo(l)
and kx(l)
represent the characteristic wave impedances and the wave propagation constants along x direction. As for the wave impedance seen when looking to the right of the reference plane
Γ
, and the conductance G and the susceptance B have been given in equation (23)
(28)
Thus, the analytical dispersion relation is derived for solving kz =βz − jαzby substituting equations (25)-(29) into equation (24). To obtain the dispersion roots in equation (24), we will use the conventional root-searching algorithm such as Newton’s method. To accelerate the root-searching process, the analytical solution of the leaky-wave waveguide free of dielectric slab, which is listed in [26], can be used as an initial point for further iteration process. The propagation constant κz in general is a complex number. Imaginary part of κz represents the attenuation of wave propagating along the longitudinal direction of the waveguide and characterizes the power leakage from the slit into the open half plane. This shows the basic physical mechanism of the leaky-wave waveguide antenna.
2.3.6 Far Field Radiation Pattern
T
o compare the theoretical and the experimental results, we have to calculate the radiation pattern of the leaky waveguide antenna with the propagation constant which has been already found from the above derivations. Assuming that the antenna slit width is relatively small compared with the waveguide height. This makes the electric field across the antenna slit uniform in the y-direction, and propagates in the z-direction with its amplitude proportional toexp(-jβzz)exp(-αzz). With the field distribution known inside the cavity, it is thus convenient to use the equivalence principle to find the far field radiation pattern of the antenna concerned.
The equivalence principle states that two sets of sources producing the same field within a region of space are said to be equivalent within that region. If we are interested in the fields in a specific region, we are free to choose an equivalent set of source such that they produce the same field distribution within that region. Actual information about the original sources is not really required, equivalent sources will serve as well [38]. The equivalence principle is shown in figure 4(a) and figure 4(b). Figure 4(a) represents the sources internal to a closed surface S and free space external to S, E and H are the original electromagnetic fields all over the space. To find an equivalent source to produce equivalent electromagnetic fields outside the specific surface, it is allowed to choose the fields exist outside surface S the original field, and the field inside S the null field. To support this field, electric and magnetic surface currents, Js and Ms, must exist on S according to
(30) H
Js = nˆ×
s = E×nˆ
M (31)
, where is the unit vector directed outward of surface S, and E and H are the original fields over S. Since the currents are acting into the unbounded free space, we are thus able to determine the fields from the electric vector potential
nˆ
F for an equivalent magnetic source, and the magnetic vector potential A for an equivalent electric source, they can be represented as
where r is the observation position vector and r′ is the source position vector that is located within S.