## 3 Effect of Error Sources on Antenna Gain
### The Influence of Feed Offset Error on Antenna Gain  
The amplitude distribution of the circular symmetric aperture field of the parabolic antenna can be defined as [22]:  
\[A\left(r,\phi^{\prime}\right)=B+C\left(1-\frac{r^{2}}{a^{2}}\right)^{p} \tag{11}\]  
where \(B+C=1\), edge taper = 20 log \(B\), \(1\leq P\leq 2\).  
The antenna is simulated and analyzed using Fast Fourier Transform (FFT) with the following input parameters: antenna diameter \(d\) = 12 m, wavelength \(\lambda\) = 5 cm, focal length \(f\) = 4.8 m, feed axial offset \(\delta_{a}\) = 0.5\(\lambda\), feed lateral offset \(\delta_{l}\) = 0.5\(\lambda\), edge taper = \(-\)12 dB.  
Assuming that the antenna is an ideal parabolic surface, the Equations (4), (5), and (11) are substituted into Equation (2). The far-field pattern changes caused by the feed offset are shown in Figure 6. It can be seen from the figure that the axial offset of the feed causes the main lobe of the far-field pattern to decrease, the side lobe to rise, and the main lobe to fuse. The lateral offset of the feed causes the main lobe of the far-field pattern to shift and the side lobe to decrease, showing left-right asymmetry.  
The antenna gain can be expressed as follows [26]:  
\[G=\frac{4\pi}{\lambda^{2}}\frac{\left|\int_{0}^{2\pi}\int_{0}^{1}A(r,\phi^{ \prime})e^{i\left(r,\phi^{\prime}\right)}rdrd\phi^{\prime}\right|^{2}}{\int_{0}^ {2\pi}\int_{0}^{1}A^{2}(r,\phi^{\prime})rdrd\phi^{\prime}} \tag{12}\]  
The gain changes due to the feed offset error are calculated by substituting Equations (4), (5), and (11) into Equation (12). When calculating the feed lateral offset error, a best-fit linear phase plane needs to be subtracted from the whole aperture first to eliminate the phase error introduced due to system alignment deviation. The relationship curves of the gain loss increasing with increasing feed offset error for different values of _f/d_ are plotted in Figure 7. In the presence of 0.5\(\lambda\) the feed offset error, the antenna _f/d_ decreases from 0.5 to 0.3, the antenna gain loss caused by the feed axial offset error increases by a factor of about 3, and the gain loss caused by the feed lateral offset error increases by a factor of about 5. Obviously, the larger _f/d_ is, the smaller the effect of the feed offset error on antenna gain loss.  
By fitting the feed offset error curve in Figure 7, the relationship equation between the feed offset error and antenna gain loss is obtained as:  
\[\eta=a_{gl}\left(\frac{\delta}{\lambda}\right)^{2} \tag{13}\]  
where \(a_{gl}\) is the feed offset error gain loss coefficient with the value shown in Table 1, and \(\delta\) is the feed offset error.  
Figure 6: The far-field pattern caused by the feed offset. The blue curve is the ideal far-field pattern; the yellow curve is the far-field pattern caused by the axial offset of 0.5\(\lambda\); and the orange curve is the far-field pattern caused by the lateral offset of 0.5\(\lambda\).  
Figure 7: The relationships between the gain loss of different antenna focal diameter ratios and the feed offset error: (**a**) the influence of the axial offset error of the feed with different focal diameter ratios on the gain loss of the antenna; (**b**) the influence of the lateral offset error of the feed with different focal diameter ratios on the gain loss of the antenna.  
From Equation (13), it can be seen that the antenna gain loss is proportional to the square of the feed offset error, by which the antenna gain loss caused by the feed offset error can be quickly calculated.