A method for calculating the radiation field of a satellite-borne antenna array, which is applied to underwater communication and navigation, wherein the antenna array consists of N parallel linear antennas of the same size and operates at a frequency range of very-low-frequency (VLF), and the method comprises:
- step (1) solving the radiation field excited by an electric dipole in an ionospheric environment, comprising:
- calculating a kernel function of a antenna, namely the radiation field excited by the electric dipole in the ionospheric environment according to a relative position relationship between the electric dipole and an observation point, comprising:
- calculating a relative transverse propagation distance Δρ and a relative longitudinal propagation distance ΔZ between the observation point and the electric dipole, and
- performing a three-dimensional Fourier transform on Maxwell's equations, to obtain the radiation field generated by the electric dipole at any position, wherein the radiation field generated by the electric dipole is equivalent to the radiation field generated by a current element on any antenna of the antenna array, that is, the expression of the radiation field of the electric dipole is the kernel function G(Δρ, Δz) for each antenna of the antenna array in the ionospheric environment;
- step (2) establishing and solving N current integral equations satisfied by the antenna array in the ionospheric environment using the kernel function obtained in step (1) according to a boundary condition to obtain a current distribution Ii of each antenna of the antenna array, comprising:
- calculating the current distribution of each antenna of the antenna array according to the following current integral equation:
- η
- 4
-
- π
-
- k
- 0
-
- ε
- 3
-
- ∫
- -
- h
- h
-
- ∑
- i
- =
- 1
- N
- G
-
- (
- b
- ki
- ,
- z
- -
- z
- ′
- )
-
- I
- i
- (
- z
- ′
- )
-
- dz
- ′
- =
- -
- V
- k
-
- δ
-
- (
- z
- )
- wherein according to the boundary condition, an electric field tangential component on a surface of each antenna equals to zero everywhere and is equal to a feed voltage at an antenna feed point;
- and
- wherein η represents a wave impedance in free space, k0 represents a wave number in free space, ε3 represents a diagonal component of a permittivity tensor of the ionosphere, Ii represents a current distribution of an ith antenna of the antenna array, Vk represents a feed voltage of a kth antenna of the antenna array, h represents a half-length of each antenna, bki represents a transverse distance between the kth antenna and the ith antenna, z represents a longitudinal height of the observation point, z′ represents a longitudinal height of the current element on the ith antenna, δ(z) represents a Dirac function and G represents the kernel function;
- and
- step (3) calculating the radiation field of each antenna of the antenna array according to the kernel function obtained in step (1) and the current distribution obtained in step (2), and calculating the total radiation field of the antenna array by summing the radiation field of all the antennas of the antenna array, the radiation field of each antenna is obtained by multiplying the current distribution by the kernel function and performing integration along the antenna length, and an expression for the total radiation field of the antenna array is:
- E
- array
- =
- η
- 4
-
- π
-
- k
- 0
-
- ε
- 3
-
- ∫
- -
- h
- h
-
- ∑
- i
- =
- 1
- N
- G
-
- (
- Δρ
- i
- ,
- Δ
-
- z
- i
- )
-
- I
- i
- (
- z
- ′
- )
-
- dz
- ′
- where Δρi and Δzi are a relative transverse propagation distance and a relative longitudinal propagation distance between the current element, equivalent to an electric dipole, at location z′ on the ith antenna and the observation point, respectively, Earray represents the total radiation field of the antenna array, and a component of the radiation field corresponds to a component of the kernel function G.