Antenna and EM Modeling with MATLAB Antenna Toolbox. Sergey N. Makarov

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Figure 1.1 A generator (its Thévenin equivalent) connected to an antenna.

      The portion depicted in the shaded box is an antenna. The antenna in Figure 1.1 is assumed to be resonant, which means that its equivalent impedance, Z_a, is purely real, i.e.

      (1.1)

      In order words, the resonant antenna is simply modeled by a resistor R_a.

      The antenna resistance R_a includes two parts:

      One thus has

      (1.2)

      Parasitic antenna resistance R_L has the following features:

      Example 1.1

      A small antenna operating at f = 10 MHz uses a thin copper wire with the diameter D of 0.25 mm and with the wire length l of 1 m. Calculate antenna loss resistance R_L.

      Solution: The DC resistance of the wire is given by

      (1.3)

      where σ is the material conductivity and A is the wire cross section. However, we cannot use this formula since most of the high‐frequency current flows in a thin skin layer around the wire. The correct result has the form:

      (1.4)

      where P is wire perimeter and μ₀ is vacuum permeability. A short MATLAB script given below accomplishes the task and gives R_L approximately equal to 1 Ω. This value may be comparable to the radiation resistance of a small antenna, and may even exceed it.

      clear all; mu = 4*pi*1e-7; % Vacuum permeability, H/m sigma = 5.7e7; % Copper conductivity, S/m D = 0.25e-3; % Diameter, m l = 1; % Length, m f = 10e6; % Frequency, Hz RHF = l/(pi*D)*sqrt(mu*2*pi*f/(2*sigma)) % Loss resistance

1.2 MAXIMUM POWER TRANSFER TO (AND FROM) ANTENNA

One question you have to ask yourself is this: for a fixed resistance R_g, can the electric power delivered to the antenna be maximized, and at which value of R_a does the maximum occur? In other words, we would like to know what parameters the antenna should have in order to acquire and radiate maximum electric power from the given RF generator (an RF amplifier). In an electric circuit, the passive load – the antenna – may have only one such parameter – the antenna resistance. All other antenna parameters (geometrical, material, etc.) are implicitly included into antenna's resistance.

      The answer is given by the maximum power transfer theorem and found by solving the circuit in Figure 1.1. We assume that voltages and current are all functions of time and solve the circuit for an arbitrary time moment. First, the current is determined from the given voltage source υ_g(t) and the total resistance using the series equivalent,

      (1.5a)

      This allows us to compute the (instantaneous) power delivered to the antenna based on

(1.5b)СКАЧАТЬ