# Dynamic Correction of the Influence of Long Measuring Path Irregularity in Antenna Tests
## 4 Errors in Measuring the Electrical Length
### Influence of Microwave Discontinuities a Long Transmission Path on the Node Shift of the Mixed Wave
The mean value on the plot is 0.5; \(M1[\arcsin\rho]=18.33^{\circ}\) at 1 GHz; \(M2[\arcsin\rho]=4.68^{\circ}\) at 5 GHz; \(M3[\arcsin\rho]=2.32^{\circ}\) at 9 GHz. The standard deviation is found as the difference between \(M\) and its value at the level of 0.159 along the cumulative frequency (upper x-axis) or \(-1\) along the theoretical quantile (lower x-axis). \(\sigma_{1}=0.085^{\circ}\) at 1 GHz; \(\sigma_{2}=0.031^{\circ}\) at 5 GHz; \(\sigma_{3}=0.013^{\circ}\) at 9 GHz. Thus, the distribution of the electrical length fluctuations, caused by the influence of microwave discontinuities, is close to normal. Accordingly, this makes it possible to correct the accuracy of the electrical length control by the studied method of unmodulated reflection from the load. Figure 5: Quantile-quantile plots for node shift of the mixed wave: (**a**) 1 GHz, (**b**) 5 GHz, (**c**) 9 GHz. Attenuation Influence on Accuracy of the Electrical Length Control of a Inhomogeneous Microwave Path  
A statistical study of random microwave discontinuities along the lossless transmission path was carried out in a number of works [31; 32]. A key finding of the previous studies is that in a lossless path with irregularities the input wave is determined by equivalent reflection coefficients. As a first approximation, they are the sum of partial reflection coefficients \(\rho_{i}\), translated to the input of the path. In real microwave paths with attenuation losses the magnitude of the incident wave along the path does not remain constant. The further the irregularity is from the beginning of the transmission path, the less influence it makes, which reduces the probable value of the equivalent reflection coefficient. Let us carry out a statistical analysis of the influence of \(n\) random irregularities along the real transmission path with attenuation losses on the equivalent input reflection coefficient, taking into account the conclusions obtained. Let us recalculate the partial reflection coefficients into the common section at the path beginning, taking into account the attenuation losses:  
\[ \begin{array}{c}\dot{\rho_{i}}^{\prime}=\rho_{i}e^{-2\beta L_{i}}e^{-2\alpha L _{i}}e^{j\varphi_{i}}=\rho_{i}e^{-2\alpha L_{i}}e^{-j(2\beta L_{i}-\varphi_{i} )}=\\ =\rho_{i}e^{-2\alpha L_{i}}\cos(2\beta L_{i}-\varphi_{i})+j\rho_{i}e^{-2\alpha L _{i}}\sin(2\beta L_{i}-\varphi_{i})=\rho_{xi}+j\rho_{yi},\end{array} \]  
where \(\rho_{i}\) and \(\varphi_{i}\) are the magnitude and phase of the partial reflection coefficient, respectively, \(\alpha\) is attenuation of transmission path, \(\beta=2\pi/\lambda\) is phase constant, \(L_{i}\) is the distance to the \(i\)-th irregularity from the beginning of the path. Phase shifts \((2\beta L_{i}-\varphi_{i})\) are random variables uniformly distributed over the \([0;2\pi]\) interval, thus the mean values \(\rho_{xi}\) and \(\rho_{yi}\) are equal to zero. The input reflection coefficient of the transmission path is defined as the sum of the partial reflection coefficients \(\dot{\rho}_{i}^{\prime}\), translated to the input section of the path. \[\dot{\rho}=\sum\limits_{i=1}^{n}\Bigl{[}\rho_{i}e^{-2\alpha L_{i}}\cos(2\beta L _{i}-\varphi_{i})\Bigr{]}+j\sum\limits_{i=1}^{n}\Bigl{[}\rho_{i}e^{-2\alpha L _{i}}\sin(2\beta L_{i}-\varphi_{i})\Bigr{]}=\rho_{x}+j\rho_{y}\]  
\(\rho_{x}\) and \(\rho_{y}\) are the sum of normally distributed quantities and, by the central limit theorem, have a normal distribution with probability density \(W(\rho_{x})\) and \(W\Bigl{(}\rho_{y}\Bigr{)}\). Assuming \(\rho_{x}\) and \(\rho_{y}\) to be statistically independent, we find the joint probability as the integral of the product of \(W(\rho_{x})\) and \(W\Bigl{(}\rho_{y}\Bigr{)}\) in polar coordinates \(\rho\), \(\varphi\).