GENERALIZED INDUCED EMF METHOD

An analytical solving problem of antenna radiation has been obtained for a small number of the simplest variants of radiators. As a rule, the small-scale radiators situated in free space were considered. This is explained by the difficulty of the problem. In this connection, numerical methods allowing reducing the problem to solution of a set of linear algebraic equations became frequent practice in solving integral equations for the antenna current. These methods permit to find characteristics of complex antennas of great dimensions (in comparison with a wavelength), and also to take into account the influence of nearby antennas and metal bodies.

Integral equation reduces to a set of algebraic equations with the help of Moment method. In the general case the integral equation for the current in a wire antenna has the form

where J(V) is the sought function (the current distribution along a wire), K(z, V) is the kernel of the equation, which depends on coordinate Z of the observation point and on coordinate V of the integration point. F(V) is a known function, it is determined by extraneous sources of the field. The terms proportional to the current may enter into this function, for example in the case of antenna with loads. Here this is of no great importance. The integral is taken over an all wire length. It is easy to verify that the equations considered earlier are particular cases of the equation (2.77).

Unknown current J(V) is expressed in the form of a sum of linearly independent function f_n(V), which are called by the basis functions:

J I f_{n n}

where I_n are unknown coefficients, which in the general case are complex. Substituting (2.78) into (2.77), we obtain:

I_n f_n K z d F z

Often the second system of linearly independent functions jp(z) is introduced. They are called by the weight functions. If to multiply both parts of equation (2.79) by jp(z) and to integrate over entire wire length and then to repeat the operation at different p, we shall obtain the set of equations:

I_{n p} z f_n K z d dz _p z F z dz p N

Obviously number N of equation (2.70) must coincide with the number N of unknown magnitudes. The integration result of each expression is its moment. From this the method’s name comes.

If the system of weight functions coincides with the system of basis functions, such a variant of the Moment method is known as Galerkin’s method. In this case

I_n f z_p f_n K z d dz f z F z dz_p p N

One can rewrite this set of equations as

I Z_{n np} U_p p N

Equation (2.82) is true also for the set of equations (2.80), if one replaces f_p(z) with j_p(z) in formulas for Z_np and U_p.

Expression (2.82) is the set of linearly independent algebraic equations with N unknown I_n, having the dimensionality of the current. Coefficients Z_np and U_p have the dimensionalities of the impedance and voltage; they can be calculated, e.g. by means of numerical integration. Accordingly, one can interpret the expression (2.82) as Kirchhoff’s equation for the contour p with current I_p and emf U_p, which enters into the system of N coupled contours. Here Z_pp is the own impedance of the contour element, and Z_np is the mutual impedance of the contours n and p.

The set of equations (2.82) can be solved on the computer with the help of standard software. If to write down the set in a matrix form:

[I][Z] = [U], (2.83)

where [Z] is the impedance matrix, [I] and [U] are a current and a voltage vectors, then one can say that the solution is obtained by means of the standard method of matrix inversion:

[I] = [Z]^–1[U]. (2.84)

Substitution of values In into (2.78) allows to calculate current distribution J(V), and afterwards all electrical characteristics of the radiator.

In practice the calculation of matrix elements Znp may prove to be difficult, since it is connected with the double numerical integration. To alleviate the difficulties, one can use d-functions in the capacity of weight functions: jp(z) = d(z – zp). Then, the double integral in the calculation of Znp becomes a simple integral, the calculation of Up requires no integration, and the expression (2.80) takes the form

I_n f_n K z_p d F z_p p N

n l N

( ) , ( ), ,

( )

V

( )

Â

Ú

One can obtain this equation directly from (2.77) and (2.78), if the left and right parts of the equation (2.77) are equated to each other at isolated points. Their number N corresponds to that of the obtained equations. For this reason, the variant of the Moment method is known as the point-matching technique or the collocation method (see, e.g., [31]).

The collocation method ensures an exact equality of the left and right parts of the equation (2.77), at N points at least. In the intervals between the points the difference between the two parts of the equation may increase sharply. When using the Moment method with weight functions of other type, the equality may not take place in all points of the interval of z changing. But equating of both moments of function (integration with some weight) minimizes the difference between the left and right parts at whole interval of z changing. This property in the final analysis is almost always more important than the exact equality at isolated points. Therefore, Galerkin’s method allows providing, as a rule, an essentially more accurate solution than the collocation method. Yet, sometimes the collocation method is useful too.

The choice of basis functions is of great importance for using the Moment method, since the successful selection of the system permits to decrease the amount of calculation under given accuracy or increases the accuracy under the same calculation time. For that end, as a rule, the basis functions must correspond to the physical sense of the problem, i.e. must coincide, in the first approximation, with the actual distribution of the current along a radiator or its elements.

Basis functions are subdivided into two types: entire domain functions, which are other than zero along the entire radiator length, and functions of sub-domains, which are other than zero along segments of radiator. In the capacity of basis functions of the first type, one can use, for example, terms of Fourier series and polynomials of Tchebyscheff or Legendre. Their field of application is limited mainly by solitary radiators of a simple shape. Basis functions of sub-domains are typically employed for an antenna of a complex shape. In particularly, such approach is expedient, if the antenna consists of arbitrarily situated segments of straight wires partially connected with each other. A straight radiator may also consist of physically isolated segments, if concentrated loads are located in the conductor of the radiator at given distances from each other. Piecewise-constant (impulse) functions (Figure 2.4a), piecewise-linear functions (Figure 2.4b), and piecewise-parabolic functions (Figure 2.4c) are shown at Figure 2.4 for illustration of basis functions of sub-domains. These basis functions are special cases of a wider class of basis functions – of polynomials. A simplest variant of approximation with the help of a polynomial is proposed in [32]:

J I_{nm n} ^m

Here, Mn is the selected degree of the polynomial on the segment n, and Inm are unknown coefficients. Comparing this expression with (2.78), we obtain:

I I f I

Figure 2.4 The curve line as the sum of pulsed (a), linear (b) and piecewise-parabolic (c) basis functions of sub-domains.

One can use terms of Fourier series as basis functions of sub-domains. A particular case of such functions are piecewise-sinusoidal functions:

J I k I k

Comparing this expression with (2.78) and choosing a simpler variant, one can write:

f

Application of expression (2.78) with the basis functions in the form of (2.86) is equivalent to dividing of wire onto short dipoles with overlapped arms and with centers at points V_p, wherein I_p is the current at the center of dipole p. In this sense, expressions (2.78) and (2.86) are the generalization of expression (1.8). When lengths of short dipoles are decreased, piecewise-sinusoidal basis functions are converted to piecewise-linear functions. Figure 2.4bpermits to visualize how the basis functions of sub-domains form the curve line corresponding to distribution of the current along an antenna.

In [33] it is proposed to use the functions in the form (2.86) as the basis and weight functions. Such variant of the Moment Method has two advantages. First, a rapid convergence of results is ensured, i.e. dimension of the matrix [Z] is small in comparison with dimensions of the matrixes when using other basis and weight functions. This means that application of piecewise-sinusoidal functions as the basis and weight functions corresponds to the physical content of the problem. Second, expressions containing sine integrals and cosine integrals can be used to calculate many matrix elements.

If to substitute the current distribution (2.78) with weight functions (2.86) into the equation (2.74) for the complicated wire radiator and to multiply in accordance with Galerkin’s method, both parts of the equation to weight function f_S(z) and after that to integrate along the entire wire length, then we obtain, repeating this operation for different s, a set of p equations of type (2.82) with p unknown magnitudes I_p and with the coefficients

Comparing (2.87) with expression (1.50), where magnitude Eps is taken from (2.73), it is easy to verify that the formula for Zps corresponds to the mutual impedance between dipoles p and n, calculated by the induced emf method. As seen from (2.87), the dipoles are considered as isolated, i.e. the current of each dipole follows the sinusoidal law.

Substituting extraneous field Ks(z) into (2.87), we see that magnitude Us is the emf of the generator connected at the center of the dipole s. Therefore, the set of equations (2.82) with coefficients Zps and Us is the set of Kirchhoff equations for the set of dipoles constituting the wire antenna.

Thus, the variant of Galerkin’s method, which was proposed by Richmond for calculating the current distribution in a complicated antenna, is equivalent to dividing of the radiator onto isolated dipoles. Their self- and mutual impedances are calculated by the induced emf method. For this reason, Richmond’s method can be named by the generalized induced emf method.

It is expedient to divide the antenna wire with connected in it concentrated loads onto short dipoles so that to place each load in the center of a dipole. Then, in accordance with (2.59), one can generalize the set of equations (2.82) and write it in the form:

I Z_{p ps} U_s I Z s_{s s} N

The accuracy of the induced emf method for calculating a dipole as is known decreases when the dipole length increases. The accuracy of calculation is acceptable at dipole arm length L ŭ l. The advantage of the generalized induced emf method consists in the fact that one can divide the long dipole onto several short dipoles, e.g., with the arm length no greater than 0.2l. That allows ensuring the required exactness.

Calculation of the coefficients Zps requires the double numerical integration. But the problem is simplified essentially, if the method described in [8] is used for calculating the mutual impedance of two arbitrarily situated dipoles. Here, the double integrals are reduced to ordinary integrals, and each integral is a sum of alternating series. The components of series are calculated by means of recurrence formulas, almost as quickly as the components of the power series.

From all the above it follows that the induced emf method is a constant companion and satellite of the integral equation method. Also it is inseparable from the concept of an equivalent long line open at the end with the the sinusoidal current distribution coinciding with the current distribution along the symmetrical dipole. In the case of the

usual line of metal wires, the propagation constant of a wave along the line is equal to the propagation constant of a wave in the air.

The generalized induced emf method in substance is the basis of all programs of calculation used in modern computers. This allows us to stop talking about the strict theory of thin linear radiators. But before that, we should say a few words on the cross-sectional shape of the radiators.

The radiator’s models considered in the first two chapters, are shaped like a straight circular cylinder. But a cross-section of the dipoles may have an arbitrary shape. In practice, the circular (see Figure 1.4) and rectangular cross-sections are encountered most often. The circular cross-section is the usual cross-section of metal radiator. Slot radiators in a metal sheet have a rectangular cross-section. Appearance of printed circuits caused an interest in thin dipoles of rectangular cross-section. They are printed on dielectric substrates and excited by strip lines. In order to provide a distinction between dipoles of circular and rectangular cross-section the latter dipoles often are called strip dipoles.

Calculations as a rule are limited by the variants, in which the dipoles consist of conductors with perfect conductivity. Maximal linear dimension of the cross-section is u  L. The currents flow along conductors’ surface. This implies that a potential difference, created by a generator between edges of a gap, is not a function of a coordinate along the perimeter of the cross-section. But the field of antenna depends on the shape and dimensions of its cross-section.

This question is considered in [16]. The author studies the dependence of the vector-potential of the field on the antenna shape. The solution is based on the first expression of (1.28), in which the integral is taken along the perimeter of the antenna cross-section, particularly along the perimeters of the gap and adjacent segments. Using the mean value theorem in order to simplify the problem, it is possible to calculate the vector potential for the cross-section of an elliptical cylinder, and compare it with the vector potential of a circular cylinder. They are equal to each other, if the radius of the circular cross-section is equal to a_e = 0.5(a_e + b_e). Here 2a_e and 2b_e are large and small axes of the ellipse.

The similar result is given in [13]. Here it is assumed that the equivalent radius of the flat antenna with a width b is equal to a_e = b/2. That permits to find the self capacitance of the flat antenna with this equivalent radius. It is equal to C₀₁ = 2pe/ln(2L/a_e) = 2pe/

ln(4L/b). The other self-capacitance of a rectangular plate with the length 2L and the width b per unit of its length is given in [34]. When 2L/b Ů  C₀₂ = 2pe/ln (2.4L/b).

The formulas obtained for C₀₁ and C₀₂ lie within the limits of accepted accuracy.

3.1 ELECTRICALLY RELATED LONG LINES PARALLEL TO