**2.1. Integral and matrix equations to determine current in arbitrary thin conductor**

Wire antennas are simulated numerically by the method of integral equations. In the present chapter, we have used Pockington's integral equation [Mittra, 1973]. The integral equation is solved by Galerkin method (method of the moments) [Harrington, 1968; Mittra, 1973; Fletcher, 1984]. To integrate the integral equation to a system of linear algebraic equations, impulse functions have been used further as basis and weight functions. In this case, the basic part of each matrix coefficient of this system is defined by an exact analytical expression. That ensures the enhancement of accuracy of solution of the integral equation and reduces this problem time.

According to [Mittra, 1973], Pockington's integral equation looks like:

$$Z\_s(l\_p) \cdot I(l\_p) + Z\_l(l\_p) \cdot I(l\_p) + \prod\_{q=1}^{L} I(l\_q) Z(l\_p, l\_q) dl\_q = \mathcal{U}(l\_p) \,, \tag{1}$$

where

( ) *<sup>q</sup> I l* is the sought current that flows in a conductor under the influence of an extraneous field, which excites the conductor;

( ) *Z l R iX sp s s* is the surface resistance of a conductor;

( ) *Z l l p* is the lumped complex impedance connected to the rupture of an antenna conductor – a load;

*L* is the conductor length.

The kernel of the integral equation Z(l , ) <sup>p</sup> *<sup>q</sup> l* is defined by the expression:

$$\mathbf{Z}(l\_p, l\_q) = -i\Im 0k \left[ \mathbf{G}(l\_p, l\_q) \cdot \overrightarrow{(l\_o, \overset{\rightarrow}{S}\_o)} - \frac{1}{\mathbf{k}^2} \frac{d^2 \mathbf{G}(l\_p, l\_q)}{dl\_p dl\_q} \right]. \tag{2}$$

The right side depends on distribution of the extraneous field that excites the conductor:

$$\begin{array}{c} \stackrel{\rightarrow}{L}\mathcal{U}(l\_{\mathbf{p}}) = -\stackrel{\rightarrow}{(E\_{\alpha\circ}(l\_{\mathbf{p}}), l\_{\circ})} , \end{array} \tag{3}$$

where *i* 1 ; <sup>2</sup> *<sup>k</sup>* is the wave number of free space, is the wavelength of the field

that excites the conductor; 

158 Numerical Simulation – From Theory to Industry

attributed to complex antennas;

**conductor** 

where

– a load;

and reduces this problem time.

field, which excites the conductor;

The kernel of the integral equation Z(l , ) <sup>p</sup> *<sup>q</sup>*

*L* is the conductor length.

( ) *Z l R iX sp s s* is the surface resistance of a conductor;


The above specified questions are the contents of the present chapter.

According to [Mittra, 1973], Pockington's integral equation looks like:

**2. Method of analysis and mathematical model of wire antenna** 

**2.1. Integral and matrix equations to determine current in arbitrary thin** 

Wire antennas are simulated numerically by the method of integral equations. In the present chapter, we have used Pockington's integral equation [Mittra, 1973]. The integral equation is solved by Galerkin method (method of the moments) [Harrington, 1968; Mittra, 1973; Fletcher, 1984]. To integrate the integral equation to a system of linear algebraic equations, impulse functions have been used further as basis and weight functions. In this case, the basic part of each matrix coefficient of this system is defined by an exact analytical expression. That ensures the enhancement of accuracy of solution of the integral equation

L

( ) *<sup>q</sup> I l* is the sought current that flows in a conductor under the influence of an extraneous

( ) *Z l l p* is the lumped complex impedance connected to the rupture of an antenna conductor

<sup>o</sup> <sup>p</sup> <sup>2</sup> <sup>1</sup> (,) Z(l , ) 30 ( , ) ( , ) <sup>k</sup> *p q <sup>o</sup> <sup>q</sup> p q*

The right side depends on distribution of the extraneous field that excites the conductor:

*l i k Gl l l S*

q 0

( ) ( ) ( ) ( ) I(l ) ( , ) ( ) *Z l I l Z l I l Z l l dl U l sp p lp p pq q p* , (1)

*l* is defined by the expression:

2

*dGl l*

*dl dl*

*p q*

. (2)



> *Eos* is the vector of the extraneous electric field that excites the conductor.

Green's function is defined by the expression:

$$\mathbf{G}(\mathbf{l}\_{\mathbf{p}}, l\_q) = \frac{e^{i\mathbf{k}\cdot\mathbf{R}}}{R} \,. \tag{4}$$

The remaining quantities included in (1) – (4) are shown in fig. 1.

When solving an integral equation by the method of moments, a conductor of arbitrary shape is divided into M rectilinear segments of length *L* . Coordinates of the beginning ( 1 , 1 , 1 *XYZ mmm* ) and the end ( 2 , 2 , 2 *XYZ mmm* ) of each segment are calculated (m is the segment number, 1 *m M* ).

The use of impulse basis and weight functions leads to the following system of linear algebraic equations:

$$\mathbf{Z}\_{sn} + \mathbf{Z}\_{\text{lk}} + \sum\_{\mathbf{m}=1}^{\text{M}} \mathbf{I}\_{\text{m}} \mathbf{K}\_{mn} = \mathbf{U}\_n \tag{5}$$

where

M is the number of segments, into which the whole conductor of length L is divided;

m is the number of a segment, in which point Q (source point) is located (fig. 1), 1 *m M* ; n is the number of a segment, in which point P (point of observation) is located (fig. 1), 1 *n M* ;

*Zsn* is the surface resistance in a segment with number n;

*Zlk* is the series load resistance of a conductor, *k* is the number of a segment, to which the resistance is connected;

*mI* is the sought current in a segment with number m.

The matrix coefficients are determined by the expression:

$$\mathbf{K}\_{\mathrm{mn}} = \int\_{\Delta l\_{\mathrm{n}}} \int\_{\Delta l\_{\mathrm{m}}} \left[ Z(l\_{p'}, l\_{q}) (\overset{\rightarrow}{l}\_{\mathrm{om}}, \overset{\rightarrow}{S}\_{\mathrm{om}}) \right] dl\_{q} dl\_{p} = A\_{\mathrm{mn}} + B\_{\mathrm{mn}} \,\, \, \, \tag{6}$$

where

, *n m L L* are the lengths of segments with numbers n and m;

*l S on om* , are unit vectors, tangent to a conductor in the centre of segments with numbers n and m;

**Figure 1.** Conductor model

$$\mathbf{A}\_{\mathrm{mn}} = -i \, 30k \int\_{Z\_{\mathrm{nl}}}^{Z\_{\mathrm{p}}} \int \mathbf{G}(z\_{p'} z\_q) dz\_q dz\_{p'} \,\tag{7}$$

11 1 1 1 1 1 1 ( )( )( )a *R X X YY ZZ n m nm n m o* . (14)

. (15)

, (16)

Un *U* , (17)

*inp* , (18)

<sup>2</sup> <sup>2</sup> 2 2

The right side of the system of equations (5) according to (3) turns out to be equal to:

*L*

*n*

*L*

number n. Therefore, it follows from (15) that:

where

where

where

number n;

**2.2. Radiation problem** 

U is the excitation voltage;

space. In this case, it follows from (16) that:

*<sup>n</sup> E*

U ( ,l ) ( ) <sup>n</sup> *n*

The weight function is distinct from zero and is equal to unity only within a segment with

U ( ) exp( ) <sup>n</sup>

is the component of an extraneous field, tangent to a conductor in a segment with

*Un* , *n* are the amplitude and the phase of an exciting voltage in a segment with number n.

When solving a radiation problem, the extraneous field is considered to be distinct from zero only in the space of excitation and constant as for its amplitude and phase within the

0 *Un* only in those segments, which are within the space of excitation. It means that, in the system of equations (5), the right side is given in accordance with (17) in the equations, the numbers of which correspond to the numbers of segments within the space of excitation.

As a result of numerical solution of the system of linear algebraic equations (5), the current in each segment *mI* is found. The amplitude and the initial phase of the current within each segment is considered to be constant. According to the current distribution in a conductor

and the given voltage U, the input resistance Z R iX inp inp inp is determined,

inp

Z

inp

U

I

The further transformation into (16) is different for radiation and scattering problems.

*E l dl U*

*os on np p*

*np p n n*

*E f l dl*

$$B\_{mn} = i\frac{\Im 0}{k} [G\_{22} - G\_{12} - G\_{21} + G\_{11}],\tag{8}$$

$$\mathbf{G}\_{22} = \frac{e^{ikR\_{22}}}{R\_{22}}; \quad \mathbf{G}\_{12} = \frac{e^{ikR\_{12}}}{R\_{12}}; \tag{9}$$

$$\mathbf{G}\_{21} = \frac{e^{ikR\_{21}}}{R\_{21}}; \quad \mathbf{G}\_{11} = \frac{e^{ikR\_{11}}}{R\_{11}}; \tag{10}$$

$$R\_{22} = \sqrt{(X\_{n2} - X\_{m2})^2 + (Y\_{n2} - Y\_{m2})^2 + (Z\_{n2} - Z\_{m2})^2 + \mathbf{a}\_o^2} \tag{11}$$

$$R\_{12} = \sqrt{(X\_{n1} - X\_{m2})^2 + (Y\_{n1} - Y\_{m2})^2 + (Z\_{n1} - Z\_{m2})^2 + a\_o^2} \tag{12}$$

$$R\_{21} = \sqrt{\left(X\_{n2} - X\_{m1}\right)^2 + \left(Y\_{n2} - Y\_{m1}\right)^2 + \left(Z\_{n2} - Z\_{m1}\right)^2 + \mathbf{a}\_o^2} \tag{13}$$

$$R\_{11} = \sqrt{\left(X\_{n1} - X\_{m1}\right)^2 + \left(Y\_{n1} - Y\_{m1}\right)^2 + \left(Z\_{n1} - Z\_{m1}\right)^2 + \mathbf{a}\_o^2} \,. \tag{14}$$

The right side of the system of equations (5) according to (3) turns out to be equal to:

$$\mathbf{U}\_n = -\int\_{\Delta l\_n} (\stackrel{\rightarrow}{E}\_{os} \stackrel{\rightarrow}{l}\_{on}) f\_n(l\_p) dl\_p \,. \tag{15}$$

The weight function is distinct from zero and is equal to unity only within a segment with number n. Therefore, it follows from (15) that:

$$\mathbf{U}\_n = -\int\_{\Delta l\_n} E\_{\tau n}(l\_p) dl\_p = \left| \mathbf{U}\_n \right| \exp(\Psi\_n) \, \prime \tag{16}$$

where

160 Numerical Simulation – From Theory to Industry

**Figure 1.** Conductor model

2 2

*Z Z*

R

Q

1 1 A 30 ( , ) mn *n m*

30[ ] *mn B iGGGG*

*pq qp*

Electric current

22 12 21 11

22 12

22 12 G ;G ; *ikR ikR e e*

21 11

21 11 G ;G ; *ikR ikR e e*

2 2 2 2 22 2 2 2 2 2 2 ( )( )( )a *R X X YY ZZ n m nm n m o* ; (11)

2 2 2 2 21 2 1 2 1 2 1 ( )( )( )a *R X X YY ZZ n m nm n m o* ; (13)

<sup>2</sup> <sup>2</sup> 2 2 *R (X X ) (Y Y ) (Z Z ) a* 12 1 2 1 2 1 2 *n m nm n m o* ; (12)

*i k G z z dz dz* ; (7)

P

; (8)

*R R* (9)

*R R* (10)

*n m*

*Z Z*

22 12

21 11

*k*

*<sup>n</sup> E* is the component of an extraneous field, tangent to a conductor in a segment with number n;

*Un* , *n* are the amplitude and the phase of an exciting voltage in a segment with number n. The further transformation into (16) is different for radiation and scattering problems.

#### **2.2. Radiation problem**

When solving a radiation problem, the extraneous field is considered to be distinct from zero only in the space of excitation and constant as for its amplitude and phase within the space. In this case, it follows from (16) that:

$$\mathbf{U}\_{\mathbf{n}} = -\mathbf{U}\_{\mathbf{n}} \tag{17}$$

where

U is the excitation voltage;

0 *Un* only in those segments, which are within the space of excitation. It means that, in the system of equations (5), the right side is given in accordance with (17) in the equations, the numbers of which correspond to the numbers of segments within the space of excitation.

As a result of numerical solution of the system of linear algebraic equations (5), the current in each segment *mI* is found. The amplitude and the initial phase of the current within each segment is considered to be constant. According to the current distribution in a conductor and the given voltage U, the input resistance Z R iX inp inp inp is determined,

where

$$\mathbf{Z}\_{\rm imp} = \frac{\mathbf{U}\_{\rm imp}}{\mathbf{I}\_{\rm imp}} \; \tag{18}$$

U ,I inp inp are the input voltage and the input current. The input voltage is the given voltage U in the space of excitation. The input current is the current at the ends of the segment (segments) that fills (fill) the space of excitation in a conductor model. The situation is explicated by fig. 2 that shows a part of a conductor of an antenna and some segments.

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 163

is extraneous, and *E E os i*

in the expression (3) have three orthogonal components:

of power flux density of an incident wave. The direction

, (20)

components;

Q

 

, (21)

 *Hi* 

. In general, the vector *Ei*

in relation to the X, Y,

of a

The figure shows a scattering conductor in the X, Y, Z coordinate system, vectors *Ei*

of the incident wave propagation is set by an angle θ<sup>i</sup> that is the angle between the Z axis

*<sup>i</sup> o o ix iy iz <sup>o</sup> E ExEyEz* 

*o o* cos cos cos *<sup>o</sup> xyz <sup>o</sup> lx y z*

*<sup>z</sup>* are the direction cosines of the vector *l <sup>o</sup>*

( ) ( ) cos ( ) cos ( ) cos *Ul E l E l E l <sup>p</sup> ix p x iy p y iz p z*

If the antenna is irradiated by a plane electromagnetic wave, the amplitude of the electric field is the same at all the points of the antenna and equal to *io E* . In this case, in the

 

Y

. (22)

are the unit vectors of the coordinate system of X, Y, Z.

Z

 , *Hi* 

where , , *ix iy iz EEE* are the complex amplitudes of the vector *Ei*

field of an incident wave, vector *П<sup>i</sup>*

. The wave field *Ei*

and vector *П<sup>i</sup>*

where cos

*xo* , *<sup>o</sup> y* , *z <sup>o</sup>* 

and the unit vector *l <sup>o</sup>*

 *<sup>x</sup>* cos *<sup>y</sup>* cos

**Figure 3.** Scatterer conductor in space

X

Scatter

It follows from (3), (20), (21) that:

expression (20):

Z coordinate axes;

**Figure 2.** Conductor and its model

If there is one segment in the space of excitation, o inp n I I is the current in a segment of excitation, *<sup>o</sup> n* is the number of a segment of excitation. The current in each segment is the complex quantity Re( ) Im( ) *inp inp inp I I iI* . Therefore, *Z R iX inp inp inp* , where *Rinp* and *Xinp* are the active and reactive parts of input resistance of the antenna:

$$R\_{imp} = \text{LI} \frac{\text{Re}(I\_{imp})}{\left|I\_{imp}\right|^2}; \quad \mathcal{X}\_{\text{imp}} = -\text{LI} \frac{\text{Im}(\mathbf{I}\_{\text{imp}})}{\left|\mathbf{I}\_{\text{imp}}\right|^2} \tag{19}$$

According to the found current distribution in the conductor, the geometry of the conductor, it is easy to determine numerically the conductor field in space in any zone, and further, the basic characteristics and parametres of a wire antenna: its directional pattern, phase diagram, polarization pattern, directivity factor. These questions compose the contents of an exterior problem of the theory, it will be studied in more detail below.

#### **2.3. Scattering problem**

The mathematical model of a wire antenna in the scattering mode includes the above relationships that determine current in scattering conductors, and the expression that defines the right side of the matrix equation (3). We will study (3) for the case of excitation of a conductor by a plane wave falling from the given direction. The problem is explicated by fig. 3.

The figure shows a scattering conductor in the X, Y, Z coordinate system, vectors *Ei Hi* of a field of an incident wave, vector *П<sup>i</sup>* of power flux density of an incident wave. The direction of the incident wave propagation is set by an angle θ<sup>i</sup> that is the angle between the Z axis and vector *П<sup>i</sup>* . The wave field *Ei* , *Hi* is extraneous, and *E E os i* . In general, the vector *Ei* and the unit vector *l <sup>o</sup>* in the expression (3) have three orthogonal components:

$$
\stackrel{\rightarrow}{E}\stackrel{\rightarrow}{E}\_i = E\_{ix}\stackrel{\rightarrow}{\cdot x}\_o + E\_{iy}\stackrel{\rightarrow}{\cdot y}\_o + E\_{iz}\stackrel{\rightarrow}{\cdot z}\_o \tag{20}
$$

where , , *ix iy iz EEE* are the complex amplitudes of the vector *Ei* components;

$$\stackrel{\rightarrow}{l}\_{o} = \stackrel{\rightarrow}{x\_{o}} \cos \alpha\_{x} + \stackrel{\rightarrow}{y\_{o}} \cos \alpha\_{y} + \stackrel{\rightarrow}{z\_{o}} \cos \alpha\_{z\_{o}} \tag{21}$$

where cos *<sup>x</sup>* cos *<sup>y</sup>* cos *<sup>z</sup>* are the direction cosines of the vector *l <sup>o</sup>* in relation to the X, Y, Z coordinate axes;

*xo* , *<sup>o</sup> y* , *z <sup>o</sup>* are the unit vectors of the coordinate system of X, Y, Z.

**Figure 3.** Scatterer conductor in space

162 Numerical Simulation – From Theory to Industry

Space of excitation

**Figure 2.** Conductor and its model

Segments

**2.3. Scattering problem** 

by fig. 3.

segments.

U ,I inp inp are the input voltage and the input current. The input voltage is the given voltage U in the space of excitation. The input current is the current at the ends of the segment (segments) that fills (fill) the space of excitation in a conductor model. The situation is explicated by fig. 2 that shows a part of a conductor of an antenna and some

U Сonductor

If there is one segment in the space of excitation, o inp n I I is the current in a segment of excitation, *<sup>o</sup> n* is the number of a segment of excitation. The current in each segment is the complex quantity Re( ) Im( ) *inp inp inp I I iI* . Therefore, *Z R iX inp inp inp* , where *Rinp* and

According to the found current distribution in the conductor, the geometry of the conductor, it is easy to determine numerically the conductor field in space in any zone, and further, the basic characteristics and parametres of a wire antenna: its directional pattern, phase diagram, polarization pattern, directivity factor. These questions compose the contents of an

The mathematical model of a wire antenna in the scattering mode includes the above relationships that determine current in scattering conductors, and the expression that defines the right side of the matrix equation (3). We will study (3) for the case of excitation of a conductor by a plane wave falling from the given direction. The problem is explicated

2 2 inp

Re( ) Im(I ) : X

inp

(19)

Model of conductor

I

*Xinp* are the active and reactive parts of input resistance of the antenna:

exterior problem of the theory, it will be studied in more detail below.

*inp*

*inp*

*I*

*I R U U*

inp

*inp*

It follows from (3), (20), (21) that:

$$LI(l\_p) = E\_{ix}(l\_p) \cdot \cos \alpha\_x + E\_{iy}(l\_p) \cdot \cos \alpha\_y + E\_{iz}(l\_p) \cdot \cos \alpha\_z \,. \tag{22}$$

If the antenna is irradiated by a plane electromagnetic wave, the amplitude of the electric field is the same at all the points of the antenna and equal to *io E* . In this case, in the expression (20):

$$\left| E\_{ix}(l\_p) \right| = E\_{i\flat} \cos \theta\_i \sin \varphi\_{i\prime} \cdot \left| E\_{iy}(l\_p) \right| = E\_{i\flat} \cos \theta\_i \cos \varphi\_{i\prime} \cdot \left| E\_{iz}(l\_p) \right| = E\_{i\flat} \sin \theta\_{i\prime} \tag{23}$$

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 165

considered to be constant within each segment with number *m* . Therefore, the integral (25)

*<sup>m</sup> M kR o m om m m*

*R*

; *Rm* is the distance from a segment with number m to point Р:

are represented in the form of the sum of projections on the X, Y, Z axes:

can be expressed by the projections , , *xyz EEE* :

 

 

is determined by a field of the antenna in the radiation

is determined by the field in the scattering mode. The

, (31)

is different, as the current distribution is different in

22 2 ( )( )( ) *R X X YY ZZ m m p mp m p* , (27)

. (26)

. (29)

, (28)

(30)

. (32)

1

*m m mm m m* 2 1 21 2 1 *om <sup>o</sup> <sup>o</sup> <sup>o</sup> X X YY ZZ S xy z LLL* 

> *o o xyz <sup>o</sup> E Ex Ey Ez*

( , ) sin ( cos sin )cos ,

 

 

*z xy*

fields in the radiation and scattering modes are described by the same expressions (30). Therefore, the directional pattern and the scattering pattern of the antenna are determined

max ( , ) ( , )/ ( , ) *<sup>s</sup> F E EF*

The directivity factor ( *D* ) is determined by a directional pattern [Balanis; Aisenberg,

4

( , )sin

*F dd*

 

2

 

 

is the unit vector, tangent to a conductor in a segment with number m.

( , ) sin cos .

*E EE*

 

> 

where max *E* is the maximum value of a field on a sphere *R const* .

2

 

 

*G*

0 0

 and ( , ) *<sup>s</sup> F* 

antenna conductors in the radiation and scattering modes.

*E E EE*

 

*x y*

 

*<sup>e</sup> E E IS*

is transformed into the sum of fields of M segments:

 and *E*

 

**2.5. Directional pattern, directivity factor and gain** 

 

where 60 / *<sup>o</sup> Ei L*

*Som* 

Vectors *Som* 

and *E* 

The cross components *E*

The directional pattern *F*(,)

by the same expression:

The shape of *F*(,)

mode, the scattering pattern ( , ) *<sup>s</sup> F*

 

Jampolsky and Terjoshin, 1977]:

where *<sup>i</sup> i* are the angular coordinates of a transmitter of an irradiating electromagnetic wave located at point *Q* (see fig. 3, angle *<sup>i</sup>* is calculated from the Х axis to the Y axis).

The phase ( ) *U lp* in a segment with number m depends on the segment position in space, and is determined by the expression:

$$\Psi\_m = -k(X\_m \cos \theta\_i \cos \phi\_i + Y\_m \sin \theta\_i \sin \phi\_i + Z\_m \cos \theta\_i) \tag{24}$$

where , , *XYZ mm m* are the coordinates of the centre of the segment with number m.

#### **2.4. Radiation (scattered) field of thin conductor**

The field at an arbitrary point of space P is determined according to the found current distribution ( ) *<sup>q</sup> I l* in a conductor. The positon of the point Р is set by spherical coordinates R, , - fig. 4. To calculate the electric field at the point Р, the vector-potential method has been used [Stratton, 1941; Volakis, 2007]:

$$\stackrel{\rightarrow}{E} \approx -i \frac{60\pi}{\lambda} \int\_{L} I(l\_q) \cdot \stackrel{\rightarrow}{S}\_o \frac{e^{-kR\_{pq}}}{R\_{pq}} dl\_{q\ \prime} \tag{25}$$

where *Rpq* is the distance from point of observation Р to point *Q* on the antenna (see fig. 1).

**Figure 4.** Point of observation in space

When calculating the current ( ) *<sup>q</sup> I l* , all the antenna conductors are divided into rectilinear segments of length *L* . The current amplitude *mI* and the current phase *m* are considered to be constant within each segment with number *m* . Therefore, the integral (25) is transformed into the sum of fields of M segments:

$$\stackrel{\rightarrow}{E} \approx E\_o \sum\_{m=1}^{M} I\_m \stackrel{\rightarrow}{S}\_{om} \frac{e^{-kR\_m}}{R\_m} \cdot \tag{26}$$

where 60 / *<sup>o</sup> Ei L* ; *Rm* is the distance from a segment with number m to point Р:

$$R\_m = \sqrt{(X\_m - X\_p)^2 + (Y\_m - Y\_p)^2 + (Z\_m - Z\_p)^2} \tag{27}$$

*Som* is the unit vector, tangent to a conductor in a segment with number m. Vectors *Som* and *E* are represented in the form of the sum of projections on the X, Y, Z axes:

$$\stackrel{\rightarrow}{S}\_{\text{our}} = \frac{X\_{m2} - X\_{m1}}{\Delta L} \stackrel{\rightarrow}{x}\_o + \frac{Y\_{m2} - Y\_{m1}}{\Delta L} \stackrel{\rightarrow}{y}\_o + \frac{Z\_{m2} - Z\_{m1}}{\Delta L} \stackrel{\rightarrow}{z}\_{o\text{ \textquotedblleft}}\tag{28}$$

$$
\stackrel{\rightarrow}{E} = E\_x \stackrel{\rightarrow}{x}\_o + E\_y \stackrel{\rightarrow}{y}\_o + E\_z \stackrel{\rightarrow}{z}\_o \cdot \tag{29}
$$

The cross components *E* and *E*can be expressed by the projections , , *xyz EEE* :

$$\begin{aligned} E\_{\theta}(\theta,\varphi) &= -E\_z \sin\theta + (E\_x \cos\varphi + E\_y \sin\varphi)\cos\theta, \\ E\_{\phi}(\theta,\varphi) &= -E\_x \sin\varphi + E\_y \cos\varphi. \end{aligned} \tag{30}$$

#### **2.5. Directional pattern, directivity factor and gain**

164 Numerical Simulation – From Theory to Industry

wave located at point *Q* (see fig. 3, angle *<sup>i</sup>*

and is determined by the expression:

been used [Stratton, 1941; Volakis, 2007]:

**Figure 4.** Point of observation in space

segments of length *L*

where *<sup>i</sup> i* 

 ,  **2.4. Radiation (scattered) field of thin conductor** 

where , , *XYZ mm m* are the coordinates of the centre of the segment with number m.

( ) cos sin ; ( ) cos cos ; ( ) sin , *ix p io i i iy p io i i iz p io i El E El E El E*

The phase ( ) *U lp* in a segment with number m depends on the segment position in space,

( cos cos sin sin cos ) *m m i i m i im i k X*

The field at an arbitrary point of space P is determined according to the found current distribution ( ) *<sup>q</sup> I l* in a conductor. The positon of the point Р is set by spherical coordinates R,

<sup>60</sup> ( )

*<sup>e</sup> E i I l S dl*

where *Rpq* is the distance from point of observation Р to point *Q* on the antenna (see fig. 1).

When calculating the current ( ) *<sup>q</sup> I l* , all the antenna conductors are divided into rectilinear

. The current amplitude *mI* and the current phase *m* are


*pq kR o q q L pq*

, (25)

*R*

are the angular coordinates of a transmitter of an irradiating electromagnetic

 

 

is calculated from the Х axis to the Y axis).

 *Y Z* , (24)

(23)

The directional pattern *F*(,) is determined by a field of the antenna in the radiation mode, the scattering pattern ( , ) *<sup>s</sup> F* is determined by the field in the scattering mode. The fields in the radiation and scattering modes are described by the same expressions (30). Therefore, the directional pattern and the scattering pattern of the antenna are determined by the same expression:

$$F(\theta,\varphi) = E(\theta,\varphi) / \,\, E\_{\text{max}} = F\_s(\theta,\varphi) \,\,\,\,\,\tag{31}$$

where max *E* is the maximum value of a field on a sphere *R const* .

The shape of *F*(,) and ( , ) *<sup>s</sup> F* is different, as the current distribution is different in antenna conductors in the radiation and scattering modes.

The directivity factor ( *D* ) is determined by a directional pattern [Balanis; Aisenberg, Jampolsky and Terjoshin, 1977]:

$$G = \frac{4\pi}{\int\_{\rho=0}^{2\pi} \left[ \int\_{\theta=0}^{\pi} F^2(\theta, \rho) \sin \theta \cdot d\theta \right] d\rho} \,\tag{32}$$

The antenna gain ( *G* ) characterises the antenna directional properties and power heat loss in antenna elements [Mittra, 1973]:

$$\mathbf{G} = \mathbf{R}^2 \frac{E\_{\text{max}}^2}{30 U\_{imp} \cdot I\_{imp}}.\tag{33}$$

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 167

Shkolnikov, 1974]. The irradiating wave creates conduction currents in the antenna conductors. These currents radiate the field that is a structural component of the scattered field. The RCS corresponding to this field is *<sup>s</sup> S* . At the same time, the antenna currents excite the transmission line connected to the antenna. The wave in the transmission line is reflected from a load and is radiated by the antenna. This scattered field is the antenna component. The RCS corresponding to this field is *<sup>a</sup> S* . If the load is matched to the transmission line, *<sup>a</sup> S* =0. The antenna components and the structural components of the scattered field differ from each other not only by their scattering pattern, but by their phase diagram as well. Therefore, after having determined separately the components *<sup>a</sup> S* , *<sup>s</sup> S* and the scattered fields corresponding to these components, it is impossible to found the total *RCS* as their arithmetical sum [Crispin and Maffett, 1965]. The antenna component of the RCS is related to an antenna gain and a VSWR in the transmission line. When calculating the *RCS* by the formula (34), it is necessary to connect the matched load to the antenna port. Such an oppotunity is provided in the integral equation (1). The number of the segment *k* , to which the load *Zlk* is connected, shall coincide with the number of a segment of

The scattering pattern differs from the directional pattern (DP) of an antenna in the transmission (reception) mode. It is related to the fact that a structural component predominates over an antenna component in a scattered field. The structural component of the scattered field is the result of radiation of the currents created by an incident EMW in the antenna. Distribution of these currents can differ essentially from current distribution in the

An integral equation and the subsequent part of a mathematical model of a wire antenna allow modelling wire antennas with an arbitrary geometry, located in the nearfield region. If several antennas are located in the near-field region, the length *L* is the total length of conductors of all the antennas. If, in the radiation mode, only one antenna from group functions (an active antenna), and entries of all the remaining antennas are loaded with matched loads (passive antennas), it is possible to determine an isolation ratio between the active antenna and every other passive antenna. The isolation ratio of a field is calculated as the ratio of complex amplitudes of currents at the entry of the active antenna ( ) *inp a I* and a passive antenna ( ) *inp p I* . The power isolation ratio *Kp* is

> 2 ( ) ( ) *inp a*

*<sup>I</sup>* . (36)

*inp p*

*I*

The described mathematical model has been used in the program, with the help of which

*p*

*K*

excitation.

equal to:

antenna in the transmission mode.

the results below have been obtained.

**2.7. Interaction of antennas located in near-field region** 

### **2.6. Radar Cross Section**

The radar cross section (RCS) of antennas can exceed considerably the effective area of an object, on which the antennas are installed. That is why it is necessary to take into consideration the scattering properties of the antenna, when developping countermeasures to radar reconnaissance. In accordance with [Crispin and Maffett, 1965; Scolnic, 1970;]

$$S = 4\pi R^2 \left| \frac{E}{E\_i} \right|^2,\tag{34}$$

where

S is the radar cross section;

*R* is the distance from an antenna to a point of observation in space;

*E* is the amplitude of a scattered field in the point of observation;

*<sup>i</sup> E* is the amplitude of the electric field of an irradiating wave in the place of antenna location.

The scattering pattern and RCS depend on the antenna structure, frequency, polarization, and direction of falling of an electromagnetic wave on the antenna. RCS is described in general by the polarization scattering matrix:

$$S = \begin{vmatrix} \mathbb{S}\_{\partial \partial} & \mathbb{S}\_{\partial \varphi} \\ \mathbb{S}\_{\varphi \partial} & \mathbb{S}\_{\varphi \partial \varphi} \end{vmatrix}' \tag{35}$$

where

*S* is the RCS calculated according to a scattered field component *E* , when the object is irradiated by a field *<sup>i</sup> E*;

*S* is the RCS calculated according to a scattered field component *E* , when the object is irradiated by a field *<sup>i</sup> E*;

*S* is the RCS calculated according to a scattered field component *E* , when the object is irradiated by a field *<sup>i</sup> E*;

*S* is the RCS calculated according to a scattered field component *E* , when the object is irradiated by a field *<sup>i</sup> E*.

The scattered field and RCS of the antenna can be represented as the sum of two components: antenna component *<sup>a</sup> S* and structural component *<sup>s</sup> S* [Sazonov and Shkolnikov, 1974]. The irradiating wave creates conduction currents in the antenna conductors. These currents radiate the field that is a structural component of the scattered field. The RCS corresponding to this field is *<sup>s</sup> S* . At the same time, the antenna currents excite the transmission line connected to the antenna. The wave in the transmission line is reflected from a load and is radiated by the antenna. This scattered field is the antenna component. The RCS corresponding to this field is *<sup>a</sup> S* . If the load is matched to the transmission line, *<sup>a</sup> S* =0. The antenna components and the structural components of the scattered field differ from each other not only by their scattering pattern, but by their phase diagram as well. Therefore, after having determined separately the components *<sup>a</sup> S* , *<sup>s</sup> S* and the scattered fields corresponding to these components, it is impossible to found the total *RCS* as their arithmetical sum [Crispin and Maffett, 1965]. The antenna component of the RCS is related to an antenna gain and a VSWR in the transmission line. When calculating the *RCS* by the formula (34), it is necessary to connect the matched load to the antenna port. Such an oppotunity is provided in the integral equation (1). The number of the segment *k* , to which the load *Zlk* is connected, shall coincide with the number of a segment of excitation.

The scattering pattern differs from the directional pattern (DP) of an antenna in the transmission (reception) mode. It is related to the fact that a structural component predominates over an antenna component in a scattered field. The structural component of the scattered field is the result of radiation of the currents created by an incident EMW in the antenna. Distribution of these currents can differ essentially from current distribution in the antenna in the transmission mode.

#### **2.7. Interaction of antennas located in near-field region**

166 Numerical Simulation – From Theory to Industry

in antenna elements [Mittra, 1973]:

**2.6. Radar Cross Section** 

S is the radar cross section;

general by the polarization scattering matrix:

;

> ;

;

.

where

location.

where

irradiated by a field *<sup>i</sup> E*

irradiated by a field *<sup>i</sup> E*

irradiated by a field *<sup>i</sup> E*

irradiated by a field *<sup>i</sup> E*

*S*

*S*

*S*

*S*

The antenna gain ( *G* ) characterises the antenna directional properties and power heat loss

The radar cross section (RCS) of antennas can exceed considerably the effective area of an object, on which the antennas are installed. That is why it is necessary to take into consideration the scattering properties of the antenna, when developping countermeasures to radar reconnaissance. In accordance with [Crispin and Maffett, 1965; Scolnic, 1970;]

<sup>2</sup> 4

*<sup>i</sup> E* is the amplitude of the electric field of an irradiating wave in the place of antenna

The scattering pattern and RCS depend on the antenna structure, frequency, polarization, and direction of falling of an electromagnetic wave on the antenna. RCS is described in

*S S*

*S S* 

The scattered field and RCS of the antenna can be represented as the sum of two components: antenna component *<sup>a</sup> S* and structural component *<sup>s</sup> S* [Sazonov and

 

*S*

is the RCS calculated according to a scattered field component *E*

is the RCS calculated according to a scattered field component *E*

is the RCS calculated according to a scattered field component *E*

is the RCS calculated according to a scattered field component *E*

*<sup>E</sup> S R <sup>E</sup>* 

*R* is the distance from an antenna to a point of observation in space; *E* is the amplitude of a scattered field in the point of observation;

2

*i*

*<sup>E</sup> G R*

2 2 max 30 *inp inp*

*U I* . (33)

, (34)

, (35)

, when the object is

, when the object is

, when the object is

, when the object is

An integral equation and the subsequent part of a mathematical model of a wire antenna allow modelling wire antennas with an arbitrary geometry, located in the nearfield region. If several antennas are located in the near-field region, the length *L* is the total length of conductors of all the antennas. If, in the radiation mode, only one antenna from group functions (an active antenna), and entries of all the remaining antennas are loaded with matched loads (passive antennas), it is possible to determine an isolation ratio between the active antenna and every other passive antenna. The isolation ratio of a field is calculated as the ratio of complex amplitudes of currents at the entry of the active antenna ( ) *inp a I* and a passive antenna ( ) *inp p I* . The power isolation ratio *Kp* is equal to:

$$\mathcal{K}\_p = \left| \frac{(I\_{imp})\_a}{(I\_{imp})\_p} \right|^2. \tag{36}$$

The described mathematical model has been used in the program, with the help of which the results below have been obtained.
