Dependence of the Input Resistance of Vibrator and Microstrip Antennas on the Primary Field
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TRODYNAMICS AND WAVE PROPAGATION
Dependence of the Input Resistance of Vibrator and Microstrip Antennas on the Primary Field S. I. Eminov* Yaroslav Mudryi Novgorod State University, Veliky Novgorod, 173003 Russia *e-mail: [email protected] Received March 13, 2020; revised March 13, 2020; accepted May 7, 2020
Abstract—The dependence of input impedances of vibrator and microstrip antennas on the profile of the primary field is investigated using the analytical inversion of the main hypersingular operator and the explicit form of the inverse integral operator. General patterns in the behavior of input resistances for arbitrary primary fields were found. Numerical calculations were performed and agreement with theoretical results has been obtained. DOI: 10.1134/S1064226920110054
INTRODUCTION: FORMULATION OF THE PROBLEM
acteristics of the antennas change if we take another function, also localized in the interval [−ε, ε] ?
Electrodynamic analysis of dipole antennas is based on solving a hypersingular equation of form [1, 2]
Suppose function f ( τ) is zero outside the interval [−1,1] , continuous on [−1,1] , is nonnegative, and 1
1
1 ∂ u t ∂ ln 1 dt () π ∂τ −1 ∂t τ − t
1
+
f (t ) dt = 1.
(1)
K ( τ, t ) u (t ) dt = f ( τ) , − 1 ≤ τ ≤ 1.
Based on f ( τ) we construct function
()
1 f τ , τ ≤ ε, fε ( τ) = ε ε 0, τ > ε.
−1
Equation (1) also describes microstrip [3, 4] and slot antennas [5]. Among the unexplored problems is the study of the dependence of solution (1) on the profile of the primary field, i.e., from f ( τ) . For active antennas, the primary field is localized in a small area in comparison with the antenna length and with the wavelength. Therefore, when developing approximate calculation methods, the primary field was represented in form [6–8]
f (τ) = δ ( τ) ,
(3)
−1
(2)
where δ ( τ) is the Dirac delta function. However, as early as in [9] it was found that the exact solution of Eq. (1) at zero tends to infinity. Therefore, model (2) is not applicable. In this regard, in the theory of active antennas, it is often assumed that function f ( τ) is nonzero in small area [−ε, ε] (ε much less than one), and on this interval it is constant and equal to 1 2ε . We are interested in the question: how will the solution of Eq. (1) and the char-
(4)
Function fε ( τ) is localized on small interval [−ε, ε] and satisfies relation (3), i.e., the integral is 1. Further, fε ( 0) = 1 f ( 0) and as it decreases ε the value of the ε function increases at zero fε ( 0) . As shown in [10, p. 97], function fε ( τ) approximates δ ( τ) in the integral sense, i.e., for arbitrary smooth function ϕ ( τ) ε
f
−ε
1 ε
(t ) ϕ (t ) dt → δ (t ) ϕ (t ) dt −1
= ϕ ( 0) at ε → 0.
Therefore, we replace delta function δ ( τ) by approximating function fε ( τ) , which is continuous and, as a consequence, belongs to the space of square-summable functions L2[−1,1].
1263
1264
EMINOV
The aim of this study is to analyze the influence of approximating function fε ( τ) on antenna characteristics at small values of ε,
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