# 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
\[W(\rho)=\int\limits_{0}^{2\pi}W(\rho_{x})W\Bigl{(}\rho_{y}\Bigr{)}\rho d\varphi =\frac{\rho}{2\pi\sigma^{2}}\int\limits_{0}^{2\pi}e^{\frac{-\rho_{y}^{2}-2 \alpha L_{i}}{2\sigma^{2}}}d\varphi=\frac{\rho}{\sigma^{2}}e^{-\frac{\rho^{2}} {2\sigma^{2}}}e^{-2\alpha L_{i}}\]  
where  
\[\rho_{1}=\sum\limits_{i=1}^{n}\rho_{i},\quad L_{s}=\sum\limits_{i=1}^{n}L_{i}\]  
and \(W(\rho_{x})=\frac{1}{\sqrt{2\pi\sigma_{x}}}e^{-\rho_{x}^{2}/2\sigma_{x}^{2}}\), \(W\Bigl{(}\rho_{y}\Bigr{)}=\frac{1}{2\pi\sigma_{y}}e^{-\rho_{y}^{2}/2\sigma_{y}^ {2}}\) are probability density distributions with variances \(\sigma_{x}^{2}=\sigma_{y}^{2}=\sigma^{2}=\frac{1}{2}\sum\limits_{i=1}^{n}\sigma_ {i}^{2}\). Thus, the magnitude distribution of the input reflection coefficient is subject to the law of  
\[W^{\prime}(\rho)=\frac{\rho}{\sigma^{2}}e^{-\frac{\sigma^{2}}{2\sigma^{2}}e^{- 2\alpha L}} \tag{19}\]  
Let us show that this is the Rayleigh distribution law. Changing of the parameters \(\sigma^{2}=A_{1}\), \(e^{-2\alpha L}=A_{2}\), we equate (19) to some Rayleigh law with parameter \(\sigma^{\prime}\) and variable \(\rho^{\prime}\):\[\frac{\rho}{A_{1}}e^{-\frac{\rho^{2}}{2C_{1}}A_{2}}=\frac{\rho^{\prime}}{\sigma^{ \prime 2}}e^{-\frac{\rho^{\prime 2}}{2\sigma^{\prime 2}}}\]  
From this equality follows a system of equations:  
\[\left\{ \begin{array}{c}\frac{\rho}{A_{1}}=\frac{\rho^{\prime}}{\sigma^{ \prime 2}}\\ \frac{\rho^{2}}{2A_{1}}A_{2}=\frac{\rho^{2}}{2\sigma^{\prime 2}}\end{array} \right. \tag{20}\]  
Solving the system (20), we obtain:  
\[\left\{ \begin{array}{c}\rho^{\prime}=\rho e^{-2\alpha L}\\ \sigma^{\prime 2}=\sigma^{2}e^{-2\alpha L}.\end{array} \right.\]  
Thus, the desired magnitude distribution of the input reflection coefficient is subject to the law of Raleigh. \[W(\rho^{\prime})=\frac{\rho^{\prime}}{\sigma^{\prime 2}}e^{-\frac{\rho^{ \prime 2}}{2\sigma^{\prime 2}}}\]  
Its parameter is the most probable value of the input reflection coefficient \(\rho^{\prime}_{m}=\sigma^{\prime}=\sigma\sqrt{e^{-2\alpha L}}\), corresponding to the top of distribution. That is, the input reflection coefficient variance \(\sigma^{\prime 2}\) of the transmission path with attenuation losses is \(e^{2\alpha L}\) times less than in the case of a lossless path \(\sigma^{2}\), by reducing the additional node shift \(\Delta l\) of the wave. Let us find the difference between node shifts for a transmission path with and without attenuation losses. The magnitude of the input reflection coefficient is assumed to be \(\rho^{\prime}_{0.9}=2.14\rho^{\prime}_{m}=2.14\sigma\sqrt{e^{-2\alpha L}}\), where \(\rho^{\prime}_{0.9}\) is reflection coefficient with level of probability \(P(\rho<\rho^{\prime}_{0.9})=0.9\). Relative node shift of the wave:  
\[\Delta l^{\prime}/\lambda=\frac{1}{180^{\circ}}\arcsin\rho^{\prime}_{0.9}= \frac{1}{180^{\circ}}\arcsin\left(\rho_{0.9}\sqrt{e^{-2\alpha L}}\right)\]  
where \(\rho_{0.9}=2.14\sigma\). The difference in relative shifts in terms of the electrical length for a transmission path with and without attenuation losses:  
\[\delta\beta l=\Delta\beta l-\Delta\beta l^{\prime}=2\left(\arcsin\rho_{0.9}- \arcsin(\rho_{0.9}\sqrt{e^{-2\alpha L}})\right)\]  
Figure 6 depicts the dependence \(\delta\beta l(2\alpha L)\). As the attenuation over transmission path increases, the node shift of the wave decreases, and consequently the shift difference grows. As an example, for \(2\alpha L=4.6\) Np, and \(\rho_{0.9}=0.3\), the node shift in the lossless path is equal to \(35^{\circ}\), whereas in the path with attenuation it is \(3.4^{\circ}\). The standing wave arises due to reflection from microwave discontinuities along the transmission path. Thus, an increase in the attenuation losses of the path reduces the probable node shift of the mentioned standing  
Figure 6: Note shift difference of the mixed standing and travelling wave. wave, and, therefore, reduces their influence on the result of antenna measurements.