J. Eur. Opt. Society-Rapid Publ. 21, 2( 2025) 13
The“ phase difference within the array” between the Nth antenna unit and the 1st antenna unit is
u ¼ 2p ðN � 1Þd sin h m ð5Þ k 0
Figure 1. Schematic diagram of phased array.
angle at which the beam pointing power is maximum is h 0, the phase difference between adjacent antenna elements emitting microwave signals at h 0 satisfies the following formula
u ¼
2p k d sin h 0
In formula( 1), d is the spacing between adjacent elements, and k is the wavelength of the transmitted microwave signal.
The phase difference between the elements at both ends is
u ¼ 2p k ðN � 1Þd sin h 0 ¼ 2p
L sin h 0 ð2Þ k ð1Þ
In formula( 2), L is the total length of the linear array antenna, which is the distance between the two end elements. According to the principles of antennas and vector field theory, the directional diagram of a one-dimensional linear structure array antenna can be represented as
FðhÞ ¼ XN�1 j2p it a i e i¼0 ð Þf ð sin h�sin h 0Þ ð3Þ
Among them, h is the antenna scanning angle, a i is the power value of the radiated signal of each antenna unit, 2pfDt is the phase difference between adjacent antenna units, Dt is the signal transmission time between adjacent antenna units.
According to formula( 3), the directional pattern of the antenna is not only determined by the beam pointing angle, but also by the frequency of the radar signal.
2.2 Beam squint phenomenon
Assuming the center frequency of the signal is f 0 and the maximum scanning angle of the antenna array beam is h m [ 22 ]. The“ spatial phase difference” between adjacent elements at the maximum scanning angle should be
u 0 ¼ 2p d sin h m ð4Þ k 0
Different from the“ spatial phase difference”, this phase difference is the phase difference that needs to be provided by the phase shifter. Assuming that the distance L between the Nth antenna and the 1st antenna is( N�1) d, this distance is called the linear array aperture. Since the wavelength of the signal k is equal to the speed of light c divided by the center frequency of the signal f 0, the phase difference can be expressed as
Where
T 0 u ¼ 2pf 0 T 0
T 0 ¼ L sin h m = c
is called the“ aperture transition time” of an array antenna, which is the time difference between the signals radiated by the antenna elements at both ends of the array antenna reaching the target located in the direction of the maximum beam. If the antenna is in the receiving state, T 0 reflects the time difference between the two antenna units receiving the target signal from the maximum scanning angle h m direction.
The phase difference within the array u increases with the increase of the antenna array, and its value can be m times 2p, wherem is any integer. If a phase shifter is used to control the antenna pattern, due to the limitation of the phase shift value provided by the phase shifter, the phase shift value generated by the Nth antenna element is:
u 0 ¼ u � 2mp ð6Þ
ð7Þ
ð8Þ
The phase shift value provided by a phase shifter usually does not vary with frequency. When the signal frequency increases from f 0 to( f 0 + Df), for the target located in the direction h m, the phase difference between the echo signal between the Nth unit and the 1st unit will become
u s ¼ 2pLðf 0 þ f Þsin h m = c ð9Þ
u s ¼ u s0 þ u s u s ¼ 2pfT 0 ð10Þ ð11Þ
It can be considered that the direction of the antenna beam depends on the balance between the“ spatial phase difference u s” of the antenna array and the“ intra array phase difference u 0” provided by the phase shifter, that is, it needs to meet the condition u s = u 0. When the signal frequency changes from f 0 to( f 0 + Df), then u s > u s0, breaking the balance relationship will cause the beam direction to deviate.
From formula( 6) and formula( 7), sinh m can be calculated as
sin h m ¼ u c ð12Þ 2pf 0 L