Numerical analysis of Rotman lens multi-beam forming network

Figure 1 Microstrip line and its equivalent slab waveguide model

In order for this equivalent model to reflect the effects of higher-order modes, it is applicable in the entire frequency band, we consider the dispersion effect of equivalent parameters, and the dynamic equivalent dielectric constant εe (f) and equivalent width we (f) vary with frequency The relationship is [2]:

(1)
(2)

Where εeff and weff are the static equivalent dielectric constant and equivalent width, respectively, and fT is the cutoff frequency of the lowest order mode, defined as:

(3)

Where c is the speed of light in free space, and Δw represents the fringe field effect at the edge of the microstrip. For different aspect ratios w / h, the expressions of Z0 and εeff are also different. This paper uses Gupta's empirical formula [3] to calculate Z0, εeff and Δw. The static equivalent width weff can be determined by the following formula:

(4)

After calculating fT, εeff and weff, εe (f) and we (f) at different frequencies can be obtained according to equations (1) and (2), thus the microstrip line can be equivalent to the corresponding planar waveguide Model. For the microstrip Rotman lens shown in Figure 2, the equivalent slab waveguide model should be the similar shape of the original lens. When the shape of the lens is determined, the width of all its ports can be known, using the aforementioned method can be obtained The equivalent width of each port, and then use these equivalent port widths as constraints, and keep the center of the lens unchanged and each section of the equivalent profile is parallel to the corresponding section of the original lens. Find the similar shape of the original lens. Obtain the outline of the equivalent slab waveguide model that satisfies the above frequency dispersion effect. The following analysis is carried out for this equivalent model.

Figure 2 Rotman lens schematic

3. Multi-mode scattering parameters and processing of high-order modes are shown in Figure 2. The outline of the Rotman lens is composed of three parts: input port (beam port), output port (array port) and virtual port. Since the thickness of the medium is very small, it can be It is approximated that the field in the z direction perpendicular to the surface of the conductor does not change, so only the three components of Ez, Hx and Hy are not zero. The electric field distribution derived from the two-dimensional Hemholtz equation at the edge of the lens cavity can be described by the following contour integral equation [ 4, 5]:

(5)

Where H (2) 0 and H (2) 1 represent the second-order zero-order and first-order Hankel functions, s and s ′ denote the field point and source point on the contour c, respectively, r = r Is a vector from point s to s ′, Is the direction of the outer normal at s ′, and kd is the wavenumber in the medium.
Due to the arbitrariness of the lens shape, the surrounding line integral in equation (5) must be discretized. The lens outline is divided into enough N segments, and each segment is approximated as a straight line. Let the field point sm be on the mth segment , The source point s′n is located on the nth segment, according to equation (5), the field at sm can be written as:

(6)

Because only the three components of Ez, Hx, and Hy are not zero in the lens cavity, only TEp0 (p = 0,1,2, ...) modes exist in the planar waveguide connected to them, which are expressed by these orthogonal mode functions Field distribution on the port (ie M patterns):

(7)

Where Akp and Bkp represent the p-order modes incident and reflected at the k-th port, respectively, Ykp and γkp are their mode admittance and propagation constant, wk is the width of the k-th port, and εp is the Newman constant, when p = 0 (corresponding to TEM mode), εp = 1, p ≠ 0, εp = 2. Using the local coordinate system to analyze each port, the following equation can be obtained from equation (7):

(8)

Write the expansion of Ez (sm) and Ez (s′n) according to equation (7), and substitute equation (8) into equation (6) to get:

(9)

Using Galerkin method, weight function , And take the inner product on both sides of the above formula and simplify it to get:

(10)

among them:

(11)
(12)

When m ≠ n, that is, the field point and the source point are in different regions, the double integral of the above two equations can be conveniently obtained by numerical integration method, when m = n, that is, the field point and the source point are located in the same region, Gmnij = 0 , And Fmnij appears singularity, which can be solved by variable substitution combined with singular point removal method [6].
In order to facilitate the processing of higher-order modes, the port numbers need to be rearranged. Let the ports corresponding to the main mode be 1 ～ N, and the ports corresponding to the higher-order modes are numbered from N + 1 to N × M according to the mode from low to high, and the Sum the exchange order and write a matrix equation, we can get:

CB = DA (13)

The elements of matrix C and D are:

Ckl = (Ynj) -1 (γnjFmnij-kdGmnij),
Ckk = (Ynj) -1 (γnjFmnij-kdGmnij) + j2 / Ymi (14)
Dkl = (Ynj) -1 (γnjFmnij + kdGmnij),
Dkk = -j2 / Ymi + (Ynj) -1 (γnjFmnij + kdGmnij),
k = 1,2,…, N × M, l = 1,2,…, N × M (15)

A and B respectively represent the following column vectors:

A = [A10A20 ... AN0 A11 ... AN1 ... A1M-1 ... ANM-1] T (16)
B = [B10B20 ... BN0 B11 ... BN1 ... B1M-1 ... BNM-1] T (17)

Where Akp and Bkp represent the incident and reflected waves of the p-th mode of the k-th port, respectively. Solving the matrix equation of (13) gives the multi-mode scattering matrix of the Rotman lens:

S = C-1D (18)

Normally, the lens port is connected to the transmission line that only transmits the main mode through the gradient line. The energy of the higher-order mode excited by the port is continuously reflected back to the lens cavity within the gradient line, and finally only the main mode propagates. We put these withered higher-order modes It is regarded as the loading of the multi-port network of lenses. The characteristic impedance Zm0i of the i-th mode of the m-th port can be calculated by the following formula [7]:

(19)

Then the load reflection coefficient corresponding to the i-th mode of the m-th port is:

Γ (i-1) N + m = (Zm0i-Zm0) / (Zm0i + Zm0) (20)

Where Zm0 is the characteristic impedance of the equivalent planar waveguide of the corresponding port. Set in all N × M ports, the 1st to Nth ports corresponding to the main mode are zone I, and the remaining (M-1) × N higher order modes The corresponding port N + 1 ～ N × M is the second zone, then the single-mode scattering matrix [Sm] of the lens loaded by the higher-order mode is excited by the main mode. Out [8]:

[Sm] = [SⅡ] + [SⅠ Ⅱ] {[Γ] -1- [SⅡ Ⅱ]}-1 [SⅡ Ⅰ] (21)

Where [Γ] is a diagonal matrix composed of the load reflection coefficients of high-order mode ports N + 1 ～ N × M.

4. Calculation example and discussion Using the above method, the 32 × 32 yuan Rotman lens shown in Figure 3 is calculated and compared with the experimental results. It has 32 input ports, 32 output ports, and 16 virtual load ports. Angle α = 35 °, scanning angle β = 45 °, dielectric constant εr = 2.2, substrate thickness d = 1mm. Because there are many lens ports, we only use the most representative edge beam opening 1 and center beam opening 16 As an example, the lens is given the amplitude and phase distribution of the output port when the two ports are excited at an intermediate frequency f = 12GHz. These calculation results are obtained by taking four expansion mode functions.

Figure 3 32 × 32 yuan Rotman lens used in calculation and experiment

The box line in Figure 4 (a) represents the amplitude distribution of the output port measured by the experiment when the beam port 1 is excited. The circle line is the theoretical calculation result, and Figure 4 (b) is the corresponding phase distribution. Figure 5 (a) and 5 (b) is the amplitude and phase distribution of the output port when the central beam port 16 is excited. It can be seen from the figure that the amplitude fluctuation of the edge port is relatively large, and the amplitude of the center port is relatively stable, regardless of the edge port or the center port, The experimental results and the calculation results are in good agreement. Except for individual points, the amplitude error is usually about 0.5dB, and the phase error is generally below ten degrees. The selection is more appropriate, so although there are many input ports, the coupling between them is small and the isolation is good. The isolation between the beam ports 1 and 16 and other beam ports given in Table 1 can be seen a little.

Figure 4 (a) Amplitude distribution of output port when input port 1 is excited; (b) Phase distribution of output port when input port 1 is excited

Figure 5 (a) Amplitude distribution of output port when input port 16 is excited; (b) Phase distribution of output port when input port 16 is excited

Table 1 Isolation between input ports