# Simulation of the axially symmetrical helical line/Asines simetrijos spiralines sistemos modeliavimas.

Introduction

Helical and meander structures are applied for retardation of electromagnetic waves in the super-wideband electronic devices. Models of helical systems are proposed and their properties are described in [1,2] and other monographs and papers. According to analysis, sometimes we can obtain desirable characteristics of the systems using effects that appear due to their inhomogeneities. This possibility is demonstrated in  where properties of the system consisting of two unshielded helices are described.

Two types of symmetrical systems are possible: systems with plane symmetry and axially symmetrical systems. According to  it is possible to design a simple helical system having good dispersion properties, but it is difficult to ensure low characteristic impedance using the system with plane symmetry.

In the central part of the helical system with the plane symmetry currents flow in opposite directions [1,2]. In the central part of the axially symmetrical helical system, currents flow in the same direction. As a result of this, the magnetic flux, the distributed inductivity and the characteristic impedance of the axially symmetrical system are lower.

In order to reveal properties and possible advantages of the axially symmetrical helical systems we propose its models and present results of simulation in this paper. The multi-conductor line method [1-3] and the CST software package Microwave Studio  are used for simulation and analysis.

Simulation using the method of multi-conductor lines

In order to reveal general properties of the axially symmetrical helical system (Fig. 1), we will use the multi-conductor line method  as in.

The model of the system (Fig. 1) is presented in Fig. 2. It consists of the four-row multi-conductor line and shields. The space between conductors is vacuumed.

[FIGURE 1 OMITTED]

Using the quasi-TEM approximation and taking into account normal modes, we have the following expressions [1,2] for voltages and currents of the conductors in the line:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 2 OMITTED]

where A--coefficients, m--the number of the row of the multi-conductor line, n--the number of the conductor in the row, k = [omega]/c--the wave number, [omega]--the angular frequency, c--the light velocity, [theta]--the phase angle between the voltages on the adjacent conductors of the multi-conductor line; [Y.sub.1E0]= [Y.sub.1E](0,[theta]), [Y.sub.1E[pi]] = [Y.sub.1E]([pi],[theta]), [Y.sub.2E0] = [Y.sub.2E] (0,[theta]) and [Y.sub.2E[pi]]= [Y.sub.2E] ([pi],[theta])--characteristic admittances of the multi-conductor line at the electrical wall in the plane x0z; [Y.sub.1M0] = [Y.sub.1M] (0,[theta]), [Y.sub.1M[pi]] = ([pi],[theta]), [Y.sub.2M0] = [Y.sub.2M] (0,[theta]) and [Y.sub.2M[pi]] =[Y.sub.2M] ([pi],[theta])--characteristic admittances of the multi-conductor line at the magnetic wall in the plane x0z. We can assume that the type of the wall does not have influence on the impedances of the outside rows with m = 1 and m = 4. Then [Y.sub.1E0] = [Y.sub.1M0] = [Y.sub.10] and [Y.sub.1E0] = [Y.sub.10[pi]] = [Y.sub.1[pi]]. The models for calculation of admittances are presented in . The finite difference and the finite element methods were used for calculations.

The section of the multi-conductor line models the axially symmetrical helical system if these symmetry and boundary conditions are satisfied:

[[U.bar.].sub.2n] (x) [[U.bar.].sub.3n]; (5)

[[I.bar.].sub.2n](x) = [[I.bar.].sub.3n] (x); (6)

[U.sub.1n] (0) = [U.sub.2(n+1)] (0); (7)

[U.sub.1n] (0) = [I.sub.2(n+1)(0); (8)

[U.sub.1n] (h) = [U.sub.2n] (h); (9)

[I.sub.1n] (h) = -[U.sub.2n] (h). (10)

Substituting (1) and (2) into (5)-(10), we arrive at a set of algebraic equations. Solving the set, we can derive the following dispersion equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Solving the dispersion equation we can find values of phase angle [theta], corresponding to selected values of the wave number k . After that we can find values of the retardation factor Kr and frequency f :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

f = kc/2[pi], (9)

here [v.sub.ph]--the phase velocity of the traveling-wave and L--the step of the conductors of the helix and multi-conductor line.

The input impedance of the system is dependent on the coordinate x . At x = 0 , according to (1) and (2)

[Z.sub.c](0)= [U.sub.20] (0)/[I.sub.20] (0). (10)

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As an example, a set of characteristics of the axially symmetrical system (retardation factor and input impedance versus frequency) is presented in Fig. 3.

[FIGURE 3 OMITTED]

According to [1, 2] the retardation factor of a single unshielded helix at low frequency is low. It increases with frequency and approaches the constructive retardation factor [K.sub.rc] = 2h L (here 2h corresponds to the turn length and L is the step of helical turns).

The characteristic impedance of the axially symmetrical helical structure periodically changes along the helical wire. Using properties of electrodynamic structures with periodical inhomogeneities [6-9], it is possible to increase the retardation factor of the axially symmetrical line at low frequency, approach it to the constructive retardation, and reduce the dispersion of the slow electromagnetic wave in the line (Fig. 3(a)). According to (11), retardation factor at low frequency becomes equal to the constructive retardation factor if

[Y.sub.1[pi]] = 2[Y.sub.10] =_[Y.sub.2E[pi]] = [Y.sub.2M[pi]] = 2[Y.sub.2E0] = [Y.sub.2Mo] = 0.

The input impedance of the axially symmetrical system (Fig. 3(b)) decreases if the distance between helices decreases and it can be sufficiently lower than the input impedance of the helical line with the plane symmetry.

Simulation using the package Microwave Studio

The multi-conductor line method allows us relatively easily to find the way to obtain desired properties of the periodic electrodynamic structures. On the other hand, it is used to model infinitively long systems. Besides, changes of characteristic impedances of vertical parts of the helical wires (Fig. 1) are not taken into account when the simplest model (Fig. 2) is used. In order to verify the conclusions based on the multi-conductor method, the CST software package Microwave Studio was used for simulation of the system presented in Fig. 1.

[FIGURE 4 OMITTED]

The model of the symmetrical helical systems with axial symmetry, developed using the CST Microwave Studio graphical editor, is presented in Fig. 4. The calculation methodology of characteristics of slow-wave structures using the package Microwave Studio is described in .

The set of curves illustrating dispersion properties of the helical system with axial symmetry (Fig. 4) is presented in Fig. 5. Characteristics confirm that there are possibilities to increase the retardation factor at low frequency and reduce dispersion. According to Fig. 5 the dispersion of retardation is less than (3-4) %.

The values of the system characteristic impedances in the low frequency region are 212, 110 and 63 [OMEGA]. according to Fig. 3(b), and 173, 98 and 59 [OMEGA] as a result of calculations using CST Microwave Studio package. So the values of the characteristic impedance in the low frequency region determined using the CST Microwave Studio package are lower (by 6-18 %) with respect to the values when the multi-conductor line method was used.

The differences of retardation factor frequency dependences and characteristic impedances can be explained taking into account that the analyzed system investigated by the Microwave Studio package was not infinitively long (Fig. 2) but has the real length and input and output ports (Fig. 4).

[FIGURE 5 OMITTED]

The transfer characteristics of the axially symmetrical helical delay line (length-40 mm) are presented in Fig. 6. Oscillations of characteristics are caused by reflections from the ports of the system . They increase with frequency because the characteristic impedance of the system decreases with frequency (Fig. 3(b)). The period of the ripples (in the frequency domain) is related to the delay time in the system ([DELTA]f [congruent to] 1/2[t.sub.d], where [DELTA]f is the period of the ripples and [t.sub.d] is the delay time).

[FIGURE 6 OMITTED]

According to Fig. 6, the amplitude frequency distortions are less than 3 dB in the frequency range from 0 to 4 GHz. The comparison of these results with those obtained in  shows that at properly selected dimensions the axially symmetrical helical systems ensure the wider pass-band with respect to the plane symmetry systems.

Conclusions

Properties of the helical line with axial symmetry are revealed using the multi-conductor line method and the CST software package Microwave Studio.

There are possibilities to design simple symmetrical helical lines with axial symmetry (Fig. 1) having good dispersion properties and relatively low characteristic impedance.

Simulation made using the CST Microwave Studio package confirmed the properties of the systems obtained using the multi-conductor line method.

We recommend to use the synergy of multi-conductor method and a commercial package for analysis and design of the wide-band helical and meander structures. The multi-conductor line method allows relatively easily to find the way to obtain desired properties of a structure. A modern commercial package allows to model the structure with the real length and ports and can be used for the final analysis and design.

References

[1.] [TEXT NOT REPRODUCIBLE IN ASCII.] 1986.-266 c.

[2.] [TEXT NOT REPRODUCIBLE IN ASCII.], 1993.-360 c.

[3.] Daskevicius V., Skudutis J., Staras S. Simulation of Symmetrical and Asymmetrically Shielded Helical Lines // Electronics and Electrical Engineering.--Kaunas: Technologija, 2008.--No. 3(83).--P. 3-6.

[4.] CSTMicrowave Studio: http://www.cst.com

[5.] Kleiza A., Staras S. Daugialaidziu liniju banginiu varzu skaiciavimas // Elektronika ir elektrotechnika.--Kaunas: Technologija, 1999.--Nr. 4(22).--P. 41-44.

[6.] Staras S., Burokas T. Nevienalyciu spiraliniu sistemu savybes ll Elektronika ir elektrotechnika.--Kaunas: Technologija, 2003.--Nr. 1(43).--P. 17-20.

[7.] Staras S. Simulation and Properties of the Twined Helical Deflecting Structure // IEEE Trans. on ED.-2002.--Vol. 49, No 10.-P. 1826-1830.

[8.] Aloisio M., Waller P. Analysis of helical slow-wave structures for space TWTs using 3-D electromagnetic simulators //IEEE Trans. Electron Devices.-2005.-Vol. 52, No. 5.-P. 749-754.

[9.] Burokas T., Daskevicius V., Skudutis J., Staras S. Simulation of the Wide-Band Slow-Wave Structures using Numerical Methods Eurocon 2005-The International Conference on Computer as a Tool.--2005.--P. 848-851.

[10.] Staras S., Burokas T. Simulation and properties of transitions to traveling-wave deflection systems IEEE Trans. on ED.-2004.--Vol. 51, No 7.--P. 1049-1051.

Department of Computer Engineering, Vilnius Gediminas Technical University, Naugarduko str. 41, LT-03227, phone: +370 5 2 744767; e-mail: julius.skudutis@el.vgtu.lt

S. Staras

Department of Electronic Systems, Vilnius Gediminas Technical University, Naugarduko str. 41, LT-03227, phone: +370 5 2 744755; e-mail: stanislovas.staras@el.vgtu.lt
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