Russian Physics Journal, Vol. 49, No. 9, 2006
NUMERICAL SIMULATION OF AXIALLY SYMMETRIC ULTRAWIDEBAND RADIATORS V. I. Koshelev, A. A. Petkun, and S. Liu
UDC 621.373
A code based on the finite-differences time-domain method has been developed to simulate axially symmetric ultrawideband radiators. The results obtained with the use of this code are compared with the now available data. Approaches to the determitation of the edges of the radiation zones for a short electric monopole are discussed.
INTRODUCTION Now interest in radiators of ultrawideband (UWB) electromagnetic pulses has been quickened [1]. This is related to the extended application of UWB radiation pulses in such fields as radiolocation [2], communications [3], electromagnetic compatibility [4], and exposure of objects and media [5], biological ones included [6] to this type of radiation. The main problem here is to extend the bandwidth of the radiator [7]. For the development of methods for extending the bandwidth it is necessary to have knowledge about the physical processes of conversion of the energy of the electric pulse driving the antenna into the energy of electromagnetic radiation. A powerful tool for obtaining necessary information is a numerical simulation with the use of the finite-difference time-domain method [8]. A primary problem in analyzing UWB radiators is selection of zones: near, intermediate, and far. In doing this, it is desirable that in developing criteria for this selection the geometric dimensions of the antenna and the parameters of the drive pulse be taken into account. This problem has been solved for the aperture antenna [9]. It has been shown that the zone edges depend on the diameter of the aperture and on the risetime of the drive pulse. To determine the edge of the far zone for UWB arrays [10], the criterion Ep R = const , where Ep is the peak electric field strength at a distance R from the radiator, is used. The quantity Ep R is called an effective potential. It was of interest to consider the zone edges for a short linear radiator, whose characteristics are rather well understood, with the use of numerical simulation methods [11].
MODEL AND TESTING For the simulation of axisymmetric radiators a code has been developed which is based on the finite-difference time-domain method. The geometry of the radiator with the network is given in Fig. 1. The finite-difference analogs of two-dimensional Maxwell's equations in cylindrical coordinates ( r , z ) with axial symmetry, ∂ / ∂φ = 0 , have the following form: n+ 1
Hφ
2
n− 1
[ri , z j ] = H φ
2
[ri , z j ] −
Δt Δt ( Ern [ri , z j + 1 ] − Ern [ri , z j − 1 ]) + ( Ezn [ri + 1 , z j ] − Ezn [ri − 1 , z j ]) , μ 0 Δz μ0 Δr 2 2 2 2
Ern +1 [ri , z j + 1 ] = Ern [ri , z j + 1 ] − 2
2
n+ 1 n+ 1 Δt ( H φ 2 [ri , z j +1 ] − H φ 2 [ri , z j ]) , ε0 Δz
Institute of High Current Electronics of the Siberian Branch of the Russian Academy of Sciences, e-mail:
[email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 63–67, September, 2006. 970
1064-8887/06/4909-0970 ©2006 Springer Science+Business Media, Inc.
r
b
a
h
z
Fig. 1. The calculation domain with the network.
n+ 1 r Δt ri +1 n + 12 H φ [ri +1 , z j ] − i +1 H φ 2 [ri , z j ]) . ( ri + 1 ε0 Δz ri + 1
Ezn +1 [ri + 1 , z j ] = Ern [ri + 1 , z j ] + 2
2
2
2
All components of the field are calculated at the nodes of the space networks separated from each other by steps (Δr , Δz ) ; the magnetic field is separated in time (superscript) by a half-step Δt relative to the electric field [12]. Here, ε0 and μ0 are the absolute dielectric and magnetic constants. The boundary conditions at z = 0 inside the coaxial line are of the waveguide type, i.e. such that the TEM wave propagating to the left leaves the line without reflection. At the other exterior boundaries an absorbing layer of thickness 7– 10 meshes is used with a finite conductivity adjusted by an empirical polynomial formula [13]. For the absorbing layer the finite-difference equation for Er with a finite conductivity σ takes the form Ern +1 [ri , z j + 1 ] = aij Ern [ri , z j + 1 ] − bij 2
2
n+ 1 n+ 1 Δt ( H φ 2 [ri , z j +1 ] − H φ 2 [ri , z j ]) , ε0 Δz
where for aij and bij only one subscript is used at a time: i on the boundary in r , and j on the boundary in z. Here, aij =
2ε0 − Δt σij 2ε0 + Δt σij
bij =
,
2Δt , 2ε0 + Δt σij
σij → σ( x) = ( x ) m σopt , d
σopt ≅
0.8 ( m + 1) ε0 , Δ μ0
where x is the direction toward the boundary ( r or z ); d is the thickness of the layer; m is the power of the polynomial, which is equal to three, and Δ is the space step of the network corresponding to x. On the metal surfaces of the feeder, screen, and radiator the tangential component of the electric field strength is equal to zero. The excitation of short pulses in the coaxial line is described by the initial conditions for Maxwell's equations. For −1
t = 0 the values of Er0 (ri , z j ) and H φ 2 (ri , z j ) are set that correspond to the principal TEM mode of the waveguide with a given z-dependence of the field which propagates to the right. For long pulses, the excitation is realized by means of current J r in a fixed cross-section of the waveguide with a given dependence on time. In this case, a finite-difference equation with a right member (source) is used: Ern +1 [ri , z j + 1 ] = Ern [ri , z j + 1 ] − 2
2
n+ 1 n+ 1 Δt Δt n + 12 Jr . ( H φ 2 [ri , z j +1 ] − H φ 2 [ri , z j ]) + ε0 Δz ε0
971
The simulation used a Gaussian pulse V (t ) = V0 exp(−0.5(t τp ) 2 ) , the differentiated Gaussian pulse V (t ) = −V0 (t τp )exp(−0.5(t τp )2 − 0.5) , and a harmonic signal V (t ) = V0 a(t )sin(ωt ) . Here, V0 is the pulse amplitude, a (t ) is the profile function varying in the range from zero to unity, and τp is the characteristic duration of the Gaussian pulse. In this case, the full duration of the Gaussian pulse is τu = 8τp . The step of the network was chosen in view of the Courant criterion c Δt ≤
Δr 2 Δz 2
( c being the velocity of light in the free space). Δr 2 + Δz 2 The code allows one to simulate electromagnetic fields in the calculation domain and also to calculate the characteristics of an antenna, such as the pulse reflected from the antenna inlet and, hence, the energy efficiency of the radiator [14], the voltage standing-wave ratio (VSWR), and the complex impedance of the antenna or its complex admittance. To calculate the complex impedance of the antenna, the following well-known formula was used: Za = Z0
1 − jKC tan( γd min ) . KC − j tan( γd min )
Thus, the admittance was determined by the relation
Ya = G + jB =
where Γ=
τ = tan
ϕ = tan( γd min ) , 2
KC − j τ 1 ⎡ KC (1 + τ2 ) ⎤ 1 ⎡ τ( KC 2 − 1) ⎤ j = + ⎢ ⎥ ⎢ ⎥, Z 0 (1 − jKC τ) Z 0 ⎢⎣ 1 + KC 2 τ2 ⎦⎥ Z 0 ⎢⎣ 1 + KC 2 τ2 ⎦⎥
γ = 2π / λ ,
KC = U max / U min ,
G=
KC (1 + τ2 ) , Z 0 1 + KC 2 τ2
B=
2 τ ( KC − 1) , Z 0 1 + KC 2 τ2
μ0 μ 1 Eref K −1 b = C , Z0 = ln , ε and μ are the relative dielectric penetrability and magnetic permeability of the Einit KC + 1 ε 0 ε 2π a
space between the conductors of the line, b and a are the outer and inner radii of the line conductors (Fig. 1). The calculations were performed for an air-filled coaxial line ( μ = ε = 1). Here, KC is the VSWR; d min is the distance from the antenna inlet to the point of a minimum voltage across the line, ϕ is the phase of the coefficient of reflection Γ for the excitation of the antenna by a harmonic signal with wavelength λ . For testing the code the results of the numerical simulation performed by Maloney and co-workers [11] were used. Figure 2 presents the pulses U ref reflected from the antenna inlet that were obtained in our calculations (curve 1) and by the authors of [11] (curve 2) for the amplitude of the drive voltage pulse V0 = 1 V . To estimate quantitatively the differences in the results of the calculations, the root-mean-square deviation was used which was calculated by the formula N
∑ ( xi − yi )
σ=
i =1
N
∑
i =1
2
⋅100% . xi2
Here, N is the length of the series that represents the function under investigation in the time or frequency domain; xi is the function obtained in our calculations, and yi is the function obtained in [11]. For the data presented in Fig. 2, σ is equal to about 7% at a network step of 0.25 mm. The data were obtained for the following parameters: b / a = 2.3 , h / a = 32.8 , and a Gaussian pulse with τ p / τa = cτ p / h = 1.61 ⋅ 10−1 .
972
0.4
25
1
20
0.2
G, B, mS
Uref, V
0.3 2
0.1 0.0
4
10 5
3
0
-0.1 -0.2 0
15
G 1 2
B
-5
2
4
6
8
10
12 t/τa
Fig. 2
-10 0.10
0.15
0.20
0.25
0.30
0.35 h/λ
Fig. 3
Fig. 2. The reflected pulse (curve 1: our calculation; curve 2: data of [11]). Fig. 3. The admittance of the antenna (curves 1, 3: our calculation; curves 2, 4: data of [11]).
Figure 3 presents the complex admittance Y = G + jB calculated with the following parameters: b / a = 3.0 , a / λ = 7.02 ⋅ 10−3 , 0.1 ≤ h / λ ≤ 0.4 . These results were obtained at a network step of 0.25 mm. For the curves G and B the value of σ was equal to 4.6 and 3.4%, respectively.
ZONE EDGES FOR A SHORT RADIATOR To check the possibility of using the relation Ep R = f ( R) for the estimation of the zone edges for a short radiator (see Fig. 1), calculations have been performed with the following parameters: a = 1 mm, b = 2.5 mm, and h = 10 mm. The calculation domain in ( r , z ) was within an area of 300×450 mm. The radiator was driven by a Gaussian pulse of duration τp = 0.222–0.5 ns. The dependence Ep R ( R ) was calculated for angles θ = 45 and 90º. The angle θ was counted from the z-axis. The value of R was determined by the formula R = r 2 + z 2 . In the calculations with a step of 0.25 mm the magnetic field component H φ was found. The electric field was determined by the relation E = W0 H φ , where W0 is the resistance of the free space, equal to 120π Ω. The peak field strength at each point was found. For the chosen angles, the dependences Ep R ( R ) are similar to each other. The results of the calculations for θ = 90 are given in Fig. 4 (curves 1–
4). The maxima of the plots for θ = 45º are closer to the axis of the radiator. As the duration of the drive pulse is decreased, the curves become different in character and the falling section disappears (curve 1). Calculations with the chosen parameters have shown that the far zone criterion ( Ep R = const) is fulfilled at a distance R = 2τp с . At R = τp с the value of Ep R decreases by about 1%, and the root-mean-square deviation of the shape of the radiated pulse makes about 3% relative to that in the far zone. It is proposed to use as a criterion of the boundary between the near and intermediate zones the abrupt (by two orders of magnitude or greater) decrease of the electric field radial component Er ( R ) near the radiator (curve 5). The region of abrupt inflection of the curve can be taken for the boundary. Calculations have shown that the distance to the abrupt inflection region weakly depends on pulse duration and approximately corresponds to the longitudinal dimension of the radiator. It should also be noted that the basic changes in the dependence Ep R ( R ) occur at distances corresponding to the region of abrupt fall in Er ( R ) , and this is an argument in favor of the proposed approach to the determination of the edge of the near zone. Downstream of this edge, in the intermediate zone, a traveling wave propagates whose amplitude drops as 1 / R in the far zone. Figure 5 shows the time dependence for a Gaussian drive pulse of duration τp = 0.333 ns (curve 1) and the related electric field strengths in the
973
1 2 3 U, E, rel. units
EpR, Er, rel. units
0.8 0.6 4
0.4 0.2 0.0
1
0.75
2
0.50
3
0.25 0.00 -0.25 -0.50
5 0
10
20
Fig. 4
30
40 R, mm
-0.75 0.0
0.2
0.4
0.6
0.8
1.0
t, ns
Fig. 5
Fig. 4. The effective potential (τp = 0.222 (curve 1), 0.267 (curve 2), 0.333 (curve 3), and 0.50 ns (curve 4)) and the radial component of the electric field (curve 5) versus distance from the radiator axis. Fig. 5. Voltage (curve 1) and field strength waveforms for R = 10 (curve 2) and 200 mm (curve 3).
inflection section of the dependence Er ( R ) (curve 2) and in the far zone (curve 3) for the principal direction of the diagram θ = 90º. It can be seen that the duration of the radiated pulse in the far zone is greater than the duration of the drive pulse, and this is related to the finite bandwidth of the radiator under investigation.
CONCLUSIONS A code has been developed and tested for numerical simulation of the axially symmetric UWB radiators drived by pulses of different waveform propagating along a coaxial feeder. For a short cylindrical radiator, the possibility of determination of the far and near zones with the use of the dependences Ep R ( R ) and Er ( R ) , respectively, has been demonstrated. This work was supported in part by the Russian Foundation for Basic Research (grant No. 06-08-00295).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
974
H. Schants, The Art and Science of Ultrawideband Antennas, Artech House, Boston (2005). J. D. Taylor, ed., Ultra-Wideband Radar Technology, CRC Press, NY (2000). M. Ghavami, L. B. Michael, and R. Kohno, Ultra Wideband Signals and Systems in Communication Engineering, Wiley, London (2004). D. V. Giri, High-Power Electromagnetic Radiators. Nonlethal Weapons and Other Applications, Harvard University Press, Cambridge, Mass. (2004). A. B. Shvartsburg, Usp. Fiz. Nauk, 175, 833–861 (2005). K. H. Schoenbach, S. Katsuki, R. H. Stark, et al., IEEE Trans. Plasma Sci., 30, 293–300 (2002). V. P. Belichenko, Yu. I. Buyanov, V. I. Koshelev, and V. V. Plisko, Radiotekh. Elektron., 44, No. 2, 178–184 (1999). A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, Artech House, Boston, Ma, (2000). O. V. Mikheev, S. A. Podosenov, K. Yu. Sakharov, et al., IEEE Trans. Electromagn. Compat., 39, 48–54 (1997).
10. 11. 12. 13. 14.
V. P. Gubanov, A. M. Efremov, V. I. Koshelev, et al., Prib. Tekh. Eksper., 3, 46–54 (2005). J. G. Maloney, G. S. Smith, W. R. Scott, Jr., IEEE Trans. Antennas Propagat., 38, 1059–1068 (1990). K. S. Yee, IEEE Trans. Antennas Propagat., 14, 302–307 (1966). J.-P. Berenger, Ibid., 45, 466–473, 1997. Yu. A. Andreev, Yu. I. Buyanov, and V. I. Koshelev, Radiotekh. Elektron., 50, 585–594 (2005).
975
