### I. Introduction

### II. APE Method for Fast Calculation

*D*and that of the central dish is

*d*. The angle error of the panel misalignment is

*α*degrees, and the surface error is

*ɛ*meters. If the feeder is located at the focal point of the parabola, the phase of the reflected wave is uniform at the aperture plane, as the path length in traveling from the focal point to the reflector surface and to the aperture plane is constant for any point in the aperture. The panel misalignment results in the surface error

*ɛ*, which results in the phase error at the aperture plane.

*C*(≤1) is the edge illumination relative to that of the center [2].

*P*has the value between 1 and 2 and is determined by comparing the aperture distributions from (2) and the radiation pattern of the feed horn. In our calculation,

*C*is assumed to be 0.25, which corresponds to the edge tapering of −12 dB and

*P*= 2.

### 1. Uniform Panel Misalignment

##### (3)

*δ*changes only in the radial direction (

*ρ̂*) but does not change in

*φ̂*because the misalignment is uniform for every panel. Performing the integration in

*φ̂*using a closed-form solution [2], the resulting expression in the radiated field becomes

*α*= 0°, 0.5°, and 0.8°. The number of panels is 30 in the GRASP simulation. The calculation by the APE method has been verified to provide reasonable results. In our calculation, the angle error

*α*is limited to be less than the maximum angle error of 0.5°.

*α*increases from 0° to 0.5°. For the uniform misalignment, the direction of the main lobe is preserved to be aligned with the antenna axis as the error is symmetric in the

*φ*–direction. The beam width grows and the SLL increases with the increase in

*α*. The first side lobe is merged into the main lobe for

*α*= 0.1°, 0.2°, and 0.3°, and the first and second side lobes are merged for 0.4° and 0.5°.

*ρ̂*direction because of the panel misalignment, the collimation of the rays is not perfect. The calculated radiation pattern shows the increase in the beam width by the panel misalignment, and the gain decreases. The loss of the gain is 5.86 dB for an angle error of 0.5°.

### 2. Uniform Misalignment in the Quarter Zones

##### (5)

*δ*

*(*

_{i}*ρ̂*) represents the surface error due to the angle error corresponding to the

*i*

*quarter. The analytic solution of the integrals in (5) is obtained in*

^{th}*φ̂*using [9]

*I*

_{0}and

*L*

_{0}are the modified Bessel and the modified Struve functions, respectively. (6) is verified by comparing the numerical calculations of the integrals with the results from the formula on the right-hand side. The resulting expression for the radiated field is obtained for

*φ*as

##### (7)

*ψ*=

*kρ̂*sin

*θ*. The radiation pattern is obtained for the observation points at

*n*.

*φ*= 0° plane is shown when the misalignment errors are uniform for the panels in the first and fourth quarter zones with angle errors of 0°–0.5°. The main lobe is tilted to the opposite direction from the side of the misaligned panel because of the geometry of the misalignment. The SLL is higher on the opposite side of the incompletely deployed panels. The first side lobe is merged into the main lobe for

*α*= 0.3° and 0.5°, and the corresponding beam widths slightly increases.