# Dynamic Correction of the Influence of Long Measuring Path Irregularity in Antenna Tests
## 3 Analysis of the Electrical Length Control Method  
Long transmission path stabilization is determined by two factors: the accuracy of the electrical length control and the total error tracking. The accuracy of phase error tracking is determined by the parameters of stabilization system. The accuracy of the electrical length control is determined by the chosen measurement method. A number of works have been devoted to methods for measuring the electrical length of long paths [20; 21; 22; 23]. The viability of using a particular method depends on the specific situation. In our work, a measurement method using load reflections was investigated. The method of unmodulated reflection from the load has found application mainly in the impedance and reflection coefficient measurements [22], as well as in power supply systems of linear accelerators [19]. It is proposed to use this method for measuring the electrical length of a long microwave path as follows. When oscillations are excited in the lossless line terminated with a mismatched load, a part of the incident wave, reflected from the load propagates in the opposite direction. Due to the interference of the incident and reflected waves, the mixed standing and travelling wave mode is established in the line. When changing the line length, the waves shift along it, by an amount equal to the electrical length deviation from the nominal value. By registering the standing wave shifting, for example, on the displacement of the wave nodes, it is possible to control the electrical length of the microwave path. The accuracy of electrical length control with the presented method is influenced by several factors: reflections from microwave discontinuities within the path, the cable attenuation, the load reflection, as well as the signal source reflection. With a theory of sensitivity, we will estimate the degree of influence of these factors on the oscillation signal in case of the electrical length control [24]. Sensitivity analysis is associated with the study of the effect of parameters variation on the behavior of dynamic systems. The goal is to find the sensitivity matrix:  
\[\mathbf{U}(p)=\left[\frac{\partial\mathbf{x}}{\partial\mathbf{p}}\right]=\left[\frac{ \partial\mathbf{x}_{n}}{\partial p_{m}}\right], \tag{11}\]  
where \(\mathbf{x}\) is the \(N\)-dimensional vector of the system state; \(\mathbf{p}\) is the \(M\)-dimensional vector containing system parameters. In practice, the sensitivity matrix is most often determined with the mathematical model of the system [25]. To design and analyze the model, we will use the well-known theory of directed graphs [26; 27]. A measuring system for determining the electrical length of a long transmission line using the unmodulated reflection method can be considered as a cascade connection of several blocks. The first of them is the two-terminal device, which represents a signal source, then there is the four-terminal device, which stands for a transmission path with microwave discontinuities and attenuation, and finally the two-terminal device, which acts as the load (Figure 1). To obtain the magnitude and phase of the input reflection coefficient of the path, it is useful to represent (13) in trigonometric form:  
\[\dot{\Gamma}_{in}=\Gamma_{in}\cos\varphi_{in}+j\Gamma_{in}\sin \varphi_{in}\] \[\dot{S}_{11}=S_{11}\cos\varphi_{11}+jS_{11}\sin\varphi_{11}\] \[\dot{S}_{12}=S_{12}\cos\varphi_{12}+jS_{12}\sin\varphi_{12}\] \[\dot{S}_{21}=S_{21}\cos\varphi_{21}+jS_{21}\sin\varphi_{21}\] \[\dot{S}_{22}=S_{22}\cos\varphi_{22}+jS_{22}\sin\varphi_{22}\] \[\dot{\Gamma}_{G}=\Gamma_{G}\cos\varphi_{G}+j\Gamma_{G}\sin\varphi_{G}\] \[\dot{\Gamma}_{Z}=\Gamma_{Z}\cos\varphi_{Z}+j\Gamma_{Z}\sin\varphi_{Z}\]  
The elements of the sensitivity matrix of the investigated system (11) are determined from (13).