Propagation of current waves along randomly located multiconductor transmission lines inside a rectangular resonator is investigated. Two different line configurations are considered: parallel wires and perpendicular lines. The analytical solutions for both types of lines are obtained, using the method of symmetrical lines inside the resonator. Computer realization of this method allows it to obtain a fast solution. Therefore, it becomes possible to investigate statistical properties of induced currents. It could be shown that the mutual coupling of the currents in the wires is strong when the frequency is near resonances in the system “cavity and wires” and relatively small far from these resonances. Stochastization leads to a broadening of the resonance curves.
Transmission lines of different kinds are widely used in different branches of modern technique and industry, especially in power engineering and electronics. They can serve as a channel for the propagation of useful signals and energy, as well as a receiver of different kind of intentional and unintentional electromagnetic interferences. Thus, the study of signal propagation along transmission lines and the external electromagnetic field coupling with transmission lines and wiring systems is an important task of electromagnetic compatibility. This problem is complex itself, especially for the high frequency case, when the transversal dimension of the transmission line is comparable with the wavelength, and also for the treatment of multiconductor lines. Moreover, this task is additionally complicated by the fact that transmission lines are often placed inside resonator-like objects: car bodies, aircraft fuselages, computer cases, shielded rooms, etc. Due to the resonator behavior there is a strong coupling of the penetrated field with the wires inside and vice versa; this can greatly affect the signal propagation on conductors. In addition, often the exact positions of the lines and the corresponding electrical parameters per-unit length are unknown and have randomly distributed values. This leads to the necessity to study stochastic transmission lines in the resonator, including the more complex multiconductor variant.
In literature, there are a number of papers which deal with stochastic cavities, i.e., with cavities whose parameters are only known stochastically. This activity is mainly connected with the practical necessity to study Mode Stirred Chambers (MSC) and to clarify the correspondence between measurements in a MSC and in free space. For those studies papers of S. Hemmadi and T.M. Antonsen and their groups [
In contrast, we are interested in stochastic lines inside the resonator with deterministic parameters. This view is linked to the need for a statistical analysis of the vulnerability of electronic devices to intentional and nonintentional interferences, mutual coupling interferences, etc. We only know one paper which deals with a single-wire line of stochastic geometry inside a resonator [
Recently, we proposed an exact solution method for a loaded line in a resonator, which can be excited by an arbitrary electromagnetic field [
Note that our method is applicable to a line with geometric symmetry of the resonator: the line connects two opposite rectangular walls and it is parallel to the four other walls (see Figure
Two-wire symmetrically loaded line in a resonator.
In [
In the present paper we extend the method [
We consider N parallel wires inside a rectangular resonator with sides a, b, and h. The wires keep the symmetry of the resonator: they connect two opposite walls of the resonator and they are parallel to all other walls and to the z axis (see Figure
Thus, one can write the function
Then, in the definition of the function
In Figures
Deterministic parallel lines in a resonator. Response at the end of the first line calculated by TL approximation and the exact solution with and without accounting for the second line.
Deterministic parallel lines in resonator. Response at the end of the first line calculated by the exact analytical solution with CST code.
Deterministic parallel lines in resonator. Response at the end of the second line calculated by the exact analytical solution with and CST code.
(For the numerical simulation we used the values
Consider now the excitation of the transmission line by lumped sources which have the same coordinate z1, but different amplitudes:
The same trick can be done for the lumped loads at the right terminal z2=h-Δ (Δ→0):
Then, the column vector for total exciting field for the loaded multiconductor line with lumped sources at the left terminal is
Deterministic parallel lines in resonator. Response at the end of the first, loaded, line calculated by the exact analytical solution and by the CST code.
Deterministic parallel lines in resonator. Response at the end of the second, short-circuited, line calculated by the exact analytical solution and by the CST code.
As in a previous example, the comparison has shown a good agreement between developed theory and numerical simulation. Far from the resonances current in the first wire is approximately constant, i.e., has TL structure (see [
In this section to realize stochastization of multiconductor TL, we consider random positions of the parallel wires. We consider a similar resonator with dimensions a=1.5 m, b=1.2 m, and h=0.9 m. As in the previous case, the first line has a left lumped source; both lines have zero loads at both terminals. The radiuses of the wires are 1 mm. The transversal positions of the both wires in the plane (x,y) are defined by
The simulation was carried out in the frequency band 150-550 MHz with 500 frequency points. For each frequency point we made 100 statistical events. All calculation required about 4 hours on the computer HP ENVY 17 Notebook PC with processor Intel(R) Core(TM) i7-5500U@ 2.40 GHz. Code was realized on the Fortran programming language.
The results of simulations for the right terminal of the first (active) and the second (receiving) wires are represented at Figures
Stochastic parallel lines in resonator. Averaged response function at the end of the first, active, line compared with the deterministic result.
Stochastic parallel lines in resonator. Variance of the response function of the first, active line.
Stochastic parallel lines in resonator. Averaged response function at the end of the second, receiving, line compared with the deterministic result.
Stochastic parallel lines in resonator. Variance of the response function of the second, receiving, line.
In the present section, we consider another configuration of multiconductor lines with symmetrical geometry in rectangular resonator, which can be interesting for the practical applications on the one hand and allows the analytical consideration on the other hand. Namely, we consider two perpendicular lines in resonator (see Figure
Perpendicular wires with symmetrical geometry in resonator.
As in the previous case, we expand the exciting tangential electrical fields and induced currents into the Fourier series, but now the basic cosines functions are different for the wires of different directions:
Then one can write zero boundary conditions for tangential components of the total electric field on the boundary of the first and second wire by a standard way (of course, it is again assumed that the thin-wire approximation is valid):
Note that, unlike the case of two parallel wires, considered in the previous section, the amplitudes of current modes for different wire couple with each other; i.e., the system of equations (
The first method, analytical, is an iteration approach. Due to the lack of space in this paper, we will only briefly describe this method. On the first step, the nondiagonal terms in (
The second method, more numerical, reduces the equation to the linear system for current amplitudes. Restricting the summation by some index
The numerical results for the deterministic lines are presented in Figures
Current in the first, active wire at the point z=0 with and without the second, receiving wire.
Comparison of the currents in the perpendicular wires at the points z=0 and x=0, correspondingly.
The investigation of the perpendicular wires in resonator allows making the following conclusions:
Using the developed method one can investigate (within a reasonable time) the statistical properties of induced currents and fields for the case of stochastically arranged symmetrical lines in rectangular resonators. The results of such investigations are presented in Figures
Stochastic perpendicular lines in resonator. Averaged response function at the end of the first, active, line compared with the deterministic result.
Stochastic perpendicular lines in resonator. Variance of the response function of the first, active line.
Stochastic perpendicular lines in resonator. Averaged response function at the end of the second, receiving, line compared with the deterministic result.
Stochastic perpendicular lines in resonator. Variance of the response function of the second, receiving, line.
where 0<r <1 is uniformly distributed stochastic variable. Again, for each frequency point we made 100 statistical events.
Analysis of the curves has shown the next characteristic features. As in the case of parallel wires, the stochastization changes the positions of resonances near the cavity ones, but not the positions of transmission line resonances. For the receiving wire the influence of stochastization is more strong (see Figures
In this work the propagation of current waves along stochastic transmission lines inside a resonator was investigated. Two wire configurations were considered: parallel straight wires and perpendicular straight wires. These configurations keep the symmetry of the resonator: the wires connect two opposite walls of the resonator and are parallel to the other four walls. The stochastic line was created by randomizing the positions of the wires. To find the current, an earlier developed method for symmetrical wires inside the resonator was applied for the multiconductor problem. This method gives the general solver for arbitrary excitations, including the considered case of excitation of one wire by a lumped voltage source.
For both geometries it was shown that the influence of the receiving line on the active line is small, when the frequency is far from cavity resonances, but it is essential near these resonances. These effects are kept, for both deterministic and stochastic lines.
The stochastization changes the positions of resonances of the system “wires in resonator” near the cavity ones, but not the positions of transmission line resonances. For the receiving wire the influence of stochastization on the resonant picture is stronger. Unlike the mean value, the statistical variance for the active and receiving wire has the same order of magnitude. Qualitatively the frequency dependency of the variance corresponds to those of paper [
The obtained results have several fields of application. They can be used to evaluate the statistical properties of parasitic mutual coupling for the propagation of the signal along a TL in resonators; one can calculate the response of the lines to the external excitation by the EM field penetrating into the resonator through slits and apertures and can evaluate the corresponding statistical properties. Also the results can be used to investigate the damping of the penetrated field by the scattering of the loaded lines (see [
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.