Concentrated Loads
5.4 METHOD OF MATHEMATICAL PROGRAMMING
Use of the mathematical programming method [47] plays a major role in solving the inverse problems. The mathematical programming method allows determining optimal parameters of an antenna, in particular its geometric dimensions, magnitudes of the connected in the antenna concentrated and distributed loads, etc. It allows obtaining radiators with the given characteristics or with characteristics so close to the given characteristics as much as possible.
This remark is due to the fact that the variation interval of radiator parameters is bounded, i.e., not every value of antenna electrical characteristic can be realized practically. Different characteristics are optimal for different values of parameters.
Moreover, an antenna should exhibit certain properties not at a single fixed frequency, but in the entire operation range. Therefore, the selected parameters are the result of a compromise, which is reached with the help of the mathematical programming method.
The problem of mathematical programming in the general case is stated as follows:
one has to find vector
x of parameters that minimizes some objective function F( x) under imposed constraints ji(
x Ů &GRGPFKPI QP VJG V[RG QH HWPEVKQPUF( x) and ji(
x), mathematical programming is divided into linear, convex and nonlinear ones. In the case at hand, the problem is solved by nonlinear programming methods, since the type of function F(
x) is unknown.
The objective function (or the general functional) is a sum of several partial functional Fj(
x) with weighting coefficients pj and penalty function Fip: F x p F xj j Fip
i j
( ) =
Â
( ) +Â
. (5.62)The partial functional is an error function for one or the other characteristic. The weighting function allows to take into account an importance of this characteristic and a sensitivity of corresponding functional to results of changing the vector. A penalty function is zero, if the parameters lie within the given intervals, and has a great magnitude, even if only one of the parameters falls outside the interval limits.
Present an example of antenna parameters. Controlled parameters x for an antenna with concentrated loads are magnitudes of the loads, coordinates zn of the points of their placement and the wave impedance W of the cable. The loads are the simple elements or the sets of simple elements (capacitors with capacitances Cn, coils with inductances Ln and resistors with resistances Rn). Values zn, W, Cn,Ln and Rn are to be real, positive and frequency-independent, and zn are to be smaller than antenna length L. These requirements, naturally, limit the interval of parameters change.
Different ways of an error function formation are known. For example, the quasi-Tchebyscheff criterion gives the good results:
F x N
Here, Nf is a number of points of the independent argument (e.g. a number of frequencies in a given range), nf is a point number (e.g. a frequency number), fj(
x) is one of the electrical characteristics of an antenna, fj min(
x) is its minimal value in the considered interval, fj0 is a hypothetical value of the characteristic, which must be reached, S is the index of power, allowing to control the method sensitivity.
A root-mean-square criterion is other error function
F xj N N f x f of frequencies in given range and a number of points on the wire), nf is a frequency number, nl is a point number, fj(x) is one of the electrical characteristic of an antenna (e.g. a current or a voltage), fj0 is a hypothetical value of the characteristic, which must be reached.
The choice of function fj(x) depends on the stated problem. For example, for creation of a wide-band radiator one must use as functions fj(x) the travelling wave ratio (TWR) in the cable and the pattern factor (PF), which is equal to the average level of radiation at predetermined angles: antenna impedance, relative to a wave impedance of a cable, K is a number of angles qk within the limits of angular sector from q1 to qK (e.g. from 90° to 60°), and F(qk) is a magnitude of normalized directional pattern in the vertical plane for an angle qk. If resistors with the resistances Rn are used as the loading elements, it is necessary to supplement the set of fj(x) by the function of antenna efficiency
hA
If it is necessary to obtain a given current distribution J(z), it is expedient to use as functions fj(x) (the electrical characteristics of an antenna) either real and imaginary current components
f1 = Re J(z, f ), f2 = Im J(z, f ), (5.67) or the amplitude and the phase of the current:
f3 = J z f( , ) ,f4=tan-1ÈÎIm( , )/Re( , )z f z f ˘˚. (5.68) In the cases, when analytical expression for objective function F(x) is absent, one can find the minimum of this function by a numerical method, based on searching
the gradient. The gradient method is an iterative procedure, in which we go step by step from one set of parameters xm to another set xm+1 in the direction of the maximal decrease of the function (the steepest descent method):
xm+1=xm-amgradF x( m). (5.69) Here m is the iteration number, a m is the scale coefficient, determined as a result of a linear searching the minimum of the functional in the direction of anti-gradient.
The minimum of the functional and the values of parameters, which are correspond to this minimum, are determined for each iteration. Each iteration is, in essence, a search for the minimum of a function of one variable – a. The method with increasing the step (e.g. with doubling it) and subsequent interpolation function in the considered interval by a polynomial of the given power is the most rational. It is convenient to apply the cubic interpolation, since the number of interpolation nodes is great enough (four), and the root of the derivative (the value of a, causing the derivative to vanish) is found analytically. If the first step results in an increase, rather than decrease, of the objective function, the step should be reduced by a factor of 10p, where p = 1, 2…, whereupon the linear search goes on again with doubling of a step.
A modification of the steepest descent method is the method of the conjugate gradients. In this case the iteration 1, (Q – 1), (2Q + 1) and so on are calculated according to anti-gradient (here Q is a number of parameters) and the rest steps correspond to the expression
xm+1=xm-amGm, (5.70)
where
Gm =gradF x( m)+ gradF x( m)/gradF x( m-1)2Gm-1.
The calculation is over, when the decrease of the objective function from iteration to iteration becomes smaller than a preset value, or the magnitude of iterations exceeds certain limit (m Ů M).
The mathematical programming method (synthesis) presupposes frequentative computations of the antenna electrical characteristics at different initial parameters (analysis). Performing such calculations requires incorporation of a special program into the synthesis software. This program allows to determine at given loads and exciting emf’s all electrical characteristics of an antenna, i.e. calculating functions fj(x) for known vector x of initial parameters.
The most laborious in the calculation is computation of the self- and the mutual impedances between the antenna segments (between so-called short dipoles). Therefore, in order to speed up the calculations, it is expedient to fixate, for example, points of placing concentrated loads, in order to the coordinates of short dipoles and their mutual impedances do not might change from iteration to iteration. If there are enough many loads, i.e. the distances between them are small in comparison with the wave length, this restriction will have no effect on the synthesis results.
As the initial values of the antenna parameters, one must use the magnitudes, found by the approximate method, according to the physical content of the problem. The results of calculations show that the computational process in this case is accelerated, and most importantly, the error probability decreases, since the process of optimization at the arbitrary choice of the initial parameters may lead to a local, rather than true extremum of the objective function. Examples of the approximate physical methods are presented in the following sections.
The synthesis program, based on the mathematical programming method, permits to bring the problem solution to an end. Other methods of solving often stop and do not reach the goal. For example, earlier the synthesis of the antennas with given electrical characteristics was broken up into two stages: at the first stage the distribution of current was computed. The parameters of the antenna, providing such distribution, must be determined at the second stage. The first question has been investigated sufficiently.
It covers a wide class of the tasks (the task of creating a wide-band antenna is one of possible variants). Far less attention has been paid to the second stage of synthesis.
In principle, if the required current distribution along a wire antenna is known, one can split the wire of an antenna onto short dipoles and define currents at the centers of these dipoles. The amplitudes of piecewise-sinusoidal basis functions are equal to the magnitudes of the currents at the corresponding antenna points. It is easy to calculate the magnitudes of loads, which one must connect at these points to obtain the desired currents.
But the impedances of loads, calculated by this method, consist of active and reactive components, which are changed with frequency. The calculated active component of load impedance may be obtained negative, and this is an evidence of impossibility to create such distribution with the help of passive elements. As to the reactive component, it is necessary to solve still the problem of its implementation in the given frequency range with the help of a set of simple elements. Therefore, it is necessary to solve the problem of creating an antenna with the chosen type of loads in order to ensure in the desired range not the given current distribution, but the current distribution close to the desired distribution as much as possible. This problem, as the problem of creating a wide-band radiator, is solved by the mathematical programming method.
The method of mathematical programming offers wide scope for solution of various problems of synthesis.