Theoretical model of VICTS antenna

This section gives the main structure and the principles of the VICTS antenna. As shown in Fig. 2, the radiation plate is composed of several slot in series, which match the active impedance of the array through a single matching section [Reference Potelon, Ettorre, Coq, Bateman, Francey, Lelaidier, Seguenot and Sauleau20]. In order to make the aperture field distribution relatively uniform, two types of radiation slots with the same period d and different widths w ₁ and w ₂ are selected.

Fig. 2. Partial side view of VICTS antenna.

The radiation efficiency of the slot is controlled by multiple parameters, which can be calculated by the full-wave simulation software CST [Reference Hao, Cheng and Wu9]. The radiation efficiency of the slot increases as the height h decreases, and as the width w _i(i = 1, 2) increases. When w _i and h are determined, the radiation efficiency of the slot is also determined and recorded as η_0i(i = 1, 2).

In addition, the radiation efficiency of the slot is related to the operating frequency of the antenna. It is necessary to obtain the radiation capability of the slot through full-wave simulation while scaling to a different frequency.

Mathematical model of radiation pattern of VICTS

The theoretical model of the VICTS antenna is shown in Fig. 3 when there is no relative rotation between the radiation plate and feeding plate. Although the radiation plate keeps circular, the energy is mainly distributed in the rectangle which is slightly wider than the width of the parallel plate waveguide. Therefore, only the contribution in the rectangular region is considered in order to simplify the analysis. The length L _x and width L _y of the rectangle correspond to the narrow beamwidth axis and the wide beamwidth axis of the radiation pattern, respectively.

Fig. 3. Schematic diagram of the equivalent array of VICTS when the plate is not rotating.

In the main radiation region, the number of narrow slots is N ₁. The number of wide slots is N ₂, thus

(1)$$L_{x} = d( N_{1} + N_{2}) $$

Since each slot is parallel to the feed port, the electric field at different positions of the same slot does not change with the change of the Y coordinate. ${ {\dot I}}_{n}$ can be expressed as follows:

(2)$${\dot{I}}_n = I_n\ e^{\,jk\varepsilon_{e}dn}$$

where k = 2πf/c, f is the operating frequency, c is the speed of light in vacuum, ɛ _e is the equivalent dielectric constant of the slow-wave structure [Reference Verbitskii21, Reference You, Lu, You, Wang, Huang and Lancaster22], I _n is the unit amplitude, which can be expressed as

(3)$$\eqalign{I_{n} & = \sqrt{\prod_{i = 1}^{n}{( 1-\eta_{i-1}) \eta_{i}}}\ \ \ \ n = 1\ldots N_{1} + N_{2}\cr & \quad \quad\quad\quad\quad \eta_{i} = \left\{\matrix{ 0,\; \hfill & i = 0 \hfill \cr \eta_{01},\; \hfill & 0< i \leq N_{1} \hfill \cr \eta_{02},\; \hfill & N_{1}< i \leq N_{1} + N_{2} \hfill }\right.}$$

In this case, one-dimensional linear array theory can be used to calculate the antenna's radiation pattern F(θ, φ).

(4)$$F\left(\theta,\; \varphi\right) = \sum_{n = 1}^{N_1 + N_2}{C{\dot{I}}_ne^{\,jkdn\sin{\theta}\cos{\varphi}}}$$

where C is the cell pattern.

As shown in Fig. 4, when the two plates of the VICTS antenna rotate relatively, the angle between the radiation slot and the equal phase plane is denoted as γ. The amplitude and phase of different positions on the same slot are no longer the same, so the radiation pattern cannot be simply calculated as before.

Fig. 4. Schematic diagram of the equivalent array of VICTS antenna when the plate rotates γ.

The VICTS antenna is regarded as a two-dimensional array by dispersing the different positions on the radiation slot at equal distance. The element spacing in X-axis direction and Y-axis direction is d _x and d _y, respectively [Reference Porter19].

(5)$$d_x = {d}/{\cos{\gamma}}\quad d_y = {d}/{\sin{\gamma}} $$

There are $M = a\lfloor {L_y}/{d_y}\rfloor$ rows in the Y-axis direction where a is a coefficient. The larger a is, the more points there are, and the more accurate the calculation results are, but the larger the amount of calculation is. a should be increased to ensure the accuracy when the rotation angle γ is larger. $\lfloor \bullet \rfloor$ is a down rounding function.

The number of elements in each row in X-axis direction is $N = \lfloor {L_x}/{d_x}\rfloor$, of which the small radiation capacity is N ₀. Therefore, with the decrease of the Y coordinate, the number of narrow slots is N ₀ increases step by step.

(6)$$\eqalign{N_0 & = \left\lfloor{L_x-\tan{\gamma}\left(L_y-2{m-1 \over a}d_y\right)-d_x \over 2d_x} + 1\right\rfloor \cr & \quad \quad \quad \quad \quad \quad m = 1\ldots M}$$

The coordinates of each element can be expressed as follows:

(7)$$\eqalign{X_{mn} = & \left(n + {m\over a}-\left\lfloor{m\over a}\right\rfloor-1\right)d_x\cr Y_{mn} = & {m\over a}d_y}$$

When the radiation plate and the feeding plate rotate relatively, the electric field of each point is different because the radiation plate is composed of a variety of branches with different radiation efficiency. After determining the position of each element and the number distribution of slots parallel to the X-axis, the electric field of each element can be expressed as:

(8)$$\eqalign{\dot{I}_{mn} = \sqrt{\prod_{i = 1}^{n}{\left(1-\eta_{i-1}\right)\eta_i}}e^{\,j\left(k\varepsilon_eX_{mn}\right)}\ \ \ \ \quad \matrix{m = 1\ldots M \cr n = 1\ldots N}\cr \eta_i = \left\{\matrix{ 0,\; \hfill & i = 0 \hfill \cr \eta_{01},\; \hfill & 0< i\leq N_{0} \hfill \cr \eta_{02},\; \hfill & N_{0}< i\leq N \hfill }\right. }$$

Therefore, the pattern of the antenna can be expressed as:

(9)$$F\left(\theta,\; \varphi\right) = \sum_{m = 1}^{M}\sum_{n = 1}^{N}{C{\dot{I}}_{mn}e^{\,jk( X_{mn}\cos{\varphi + Y_{mn}\sin{\varphi}}) \sin{\theta}}}$$

The antenna pattern is converted to U − V coordinate by the following transformation:

(10)$$\eqalign{ u = & \sin{\theta}\cos{\varphi}\cr v = & \sin{\theta}\sin{\varphi}}$$

The short axis pattern of the antenna can be obtained by taking the section parallel to the U-axis and passing through the maximum value of the pattern.