Applied Science and Innovative Research 

ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) 

Vol. 6, No. 2, 2022 

www.scholink.org/ojs/index.php/asir 

71 
 

Original Paper 

Method of Electrostatic Analogy 

Boris Levin1 

1 Lod, Israel, E-mail: levinpaker@gmail.com 

 

Received: May 2, 2022         Accepted: May 15, 2022         Online Published: May 21, 2022 

doi:10.22158/asir.v6n2p71           URL: http://doi.org/10.22158/asir.v6n2p71 

 

Abstract 

It is considered the analysis method for electromagnetic fields and directional characteristics of 

antenna arrays consisting of parallel linear radiators based on the calculation of the electrostatic 

fields of charges located on parallel wires. It is demonstrated the advantage of using radiators with 

in-phase current distribution. It is shown that the application of radiators with in-phase current 

distribution allows to decrease the length of antenna array and to provide the high directivity in a wide 

frequency range. It is stated the task of optimization of considered arrays by methods of mathematical 

programming. 

Keywords 

Antenna theory, directional antennas, in-phase current distribution, optimization 

 

1. Introduction 

The purpose of antenna synthesis is creating antennas with high electrical characteristics. In order to 

solve this problem engineers tried for a long time to find a law of current distribution, which provides 

required characteristics. The issue of creating the desired distribution of the current remained open. The 

task of choosing of antenna elements and their dimensions with the aim of optimizing antenna 

characteristics was first staged as the task of mathematical programming in relation to the Yagi-Uda 

antenna (Chaplin, 1969; Chen & Cheng, 1975). The task solution confirmed rightness of chosen 

approach. 

The more general problem is a problem of creating a wideband radiator. It consists in determining a 

type and magnitudes of concentrated loads, which are placed along a linear radiator and provide in a 

given frequency band high electrical characteristics, including matching level, efficiency and required 

directional pattern (Levin, 1998). Its solution was based on understanding advantages of the in-phase 

current distribution and on the hypothesis of Hallen about usefulness of capacitive loads, whose 

magnitudes are changed along a radiator axis in accordance with linear or exponential law. Selected 

approach confirmed the hypothesis of Hallen, demonstrated the rightness of applying capacitive loads 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

72 
Published by SCHOLINK INC. 

and gave numerical results.  

The first of these was determination of loads magnitudes, which provide in a given frequency range the 

required current distribution. The second result is decreasing distortion of directional patterns by 

closely spaced superstructures with help of increasing height of antennas and using loads providing an 

operation in required frequensies band. Finally, the method was used for selecting loads in the wires of 

a symmetrical V-antenna in order to expand significantly the frequencies band, in which this antenna 

has high directivity in the direction of the bisector of its angular aperture. These tasks are consistently 

considered in (Levin, 2017). 

Based on these results, it was decided to consider the possibility of using capacitive loads in directional 

antennas, in particular to use in-phase currents in a director-type antenna. During the work a reasonable 

sequence of solving each problem was defined. At the first stage an approximate method of solution 

must be proposed. Its results are used at the second stage as initial values for numerical solution of the 

problem by the method of mathematical programming. For analyzing the radiator with loads, the 

method of two-wire impedance long line and the method of two-wire metal long line with loads were 

used as approximate methods. Effectiveness of the such methodology is obvious compared with 

helpless method of trial and error. In the case of antennas systems consisting of parallel linear radiators, 

the method of electrostatic analogy can be used as an approximated method of this type.  

The specified method and the procedure of its application are described in Section 2. In Section 3 the 

known variant of directional antenna is considered as an example. This antenna was developed as result 

of the first profound efforts devoted to the optimization of the such antennas. The method of 

electrostatic analogy allow us to consider the issues of expanding the operating range of directional 

antenna and to apply obtained results to a log-periodic antenna. The results of using the approximate 

method to the calculation of a log-periodic antenna are given in Section 4. Based on these results, the 

problem of applying the mathematical programming method to this antenna was stated.  

 

2. Method of Electrostatic Analogy  

Basic directional characteristics of antennas are the two characteristics: directivity D and pattern factor 

PF. The directivity is the ratio of radiation intensity in the direction of maximum to the average value 

of intensity in all directions (to the value of isotropic radiation). In accordance with this formulation, 

the directivity is the coefficient of increasing total radiated power P, if a maximum power is radiated in 

all directions. In this case, a total power is 

𝐷𝑃 = 4𝜋𝑆𝑚, 

where 𝑆𝑚 is a power flux in the direction of maximum radiation, i.e., a total power is equal to a 

product of a power in the direction of a maximum radiation by the area 𝑆 = 4𝜋𝑅2 of a sphere of an 

infinite large radius R. In a cylindrical coordinate system, the radiated power of a typical antenna with 

an axial structure, symmetrical with respect to two planes, passing through coordinates origin, is  



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

73 
Published by SCHOLINK INC. 

𝑃 = 2∫ 𝑑𝜑 ∫ 𝑆𝑚𝐹2(𝜃, 𝜑)𝑑𝜃 = 4𝜋
𝜋 2⁄

0

2𝜋

0
 ∫ 𝑆𝑚𝐹2(𝜃, 𝜑)𝑑𝜃.

𝜋 2⁄

0
 

According to these two expressions 

1 𝐷 = 𝑃 (4𝜋𝐸𝑚
2 ) = ∫

𝑆𝑚𝐹2(𝜃)

𝐸𝑚
2

𝑑𝜃,
𝜋 2⁄

0

⁄⁄  

where 𝐸𝑚 is the electric field in the direction of maximum radiation. 

The pattern factor is equal to the average level of radiation in a predetermined range of angles, for 

example, in the range of vertical angles 𝜃 from 60 to 90 degrees:  

𝑃𝐹 =
1

𝐾
∑𝐹(𝜃𝑘),

𝐾

𝑘=1

 

where 𝐹(𝜃𝑘) is the magnitude of normalized vertical directional pattern at an angle 𝜃 within an 

angular sector from 𝜃1 to 𝜃𝐾 . The distance of radio communication depends on the field magnitude 

in the pointed range of angles. Therefore pattern factor is an important characteristic of any antenna. If 

a maximal directivity of an antenna is large, but is located outside the required angular sector, it is 

useless for radio communication. 

As already said, for calculating electrical characteristics of systems consisting of parallel linear 

radiators we will use the method of electrostatic analogy as an approximate method. It is based on the 

similarity of electromagnetic fields created by high-frequency currents of linear radiators with 

electrostatic fields of charges placed on linear conductors. Both fields are directly proportional to the 

magnitude of the current or the magnitude of the charge, and in the far zone they are inversely 

proportional to the distance from the source. This similarity of their fields is a sign of the identical 

mathematical problem and the identical form of equations. This coincidence permits to use the 

principle of correspondence in the solution process.  

When solving mathematical problems, an analogous form of equation often lends essential assistance. 

For example, the first two Maxwell’s equations that became a basis for classical electromagnetic theory 

allowed substantiate the principle of duality (Pistolkors, 1948). Let’s write these equations in the form: 

𝑐𝑢𝑟𝑙�⃗⃗� =𝑗 ⃗⃗  +𝜀0
𝜕�⃗� 

𝜕𝑡
 , 𝑐𝑢𝑟𝑙�⃗� =−𝜇0

𝜕�⃗⃗⃗� 

𝜕𝑡
, 

where �⃗�  and �⃗⃗�  are strengths vectors of electric and magnetic field respectively, 𝑗  is a conduction 

current, 𝜀0 and 𝜇0 
are permittivity and permeability of a surrounding medium. By replacing variables 

in accordance with the equalities �⃗� 1 = √𝜀0�⃗� , �⃗⃗� 1 = √−𝜇0�⃗⃗�   and introducing conventionally a 

magnetic conduction current, we obtain 

𝑐𝑢𝑟𝑙�⃗⃗� 1 = 𝑗 1 +√−𝜀0𝜇0 
𝜕�⃗�  1

𝜕𝑡
, 𝑐𝑢𝑟𝑙�⃗� 1 = +√−𝜀0𝜇0 

𝜕�⃗⃗�  1

𝜕𝑡
, 

where �⃗� 1 and �⃗⃗� 1 are also strength vectors of fields, and 𝑗 1 is a conduction current, but in other units. 

The obtained equations are completely symmetric with respect to �⃗� 1 and �⃗⃗� 1. An introduction of a 

magnetic conduction current makes theirs also symmetric with respect to electric and magnetic 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

74 
Published by SCHOLINK INC. 

conduction currents, i.e., makes possible to treat theirs not only as equations for an electric radiator, but 

also as equations for a magnetic radiator with a magnetic conduction current 𝑗 𝑚1. Expressions for 

fields and characteristics of a magnetic radiator can be recorded by means of the expressions analogous 

to expressions for an electric radiator. For example, an input impedance of a magnetic radiator 𝑍𝑚, 

whose shape and dimensions coincide with the shape and dimensions of an electric radiator, is equal to  

𝑍𝑚 = (120𝜋)2 𝑍𝑒⁄ . 

Here 𝑍𝑒 is an input impedance of an electric radiator. A slot is a real embodiment of a magnetic 

radiator. 

A similar method of solving problems is used in many other cases. For example, there is the known 

analogy between a picture of electrostatic field of charged conducting bodies located in a homogeneous 

and isotropic dielectric and a picture of constant currents in a homogeneous, weakly-conducting 

medium. In this case, bodies placed in the medium must have high conductivity and their shape and 

geometric dimensions must coincide with the shape and dimensions of the conducting bodies located in 

a dielectric. 

If a picture of an electric field of linear charges is known, then, using the correspondence principle, one 

can construct a picture of a magnetic field of constant linear currents, provided the currents and charges 

are distributed in space identically. The difference between these images is only that lines of equal 

magnetic potential are located on places of lines of electric field strength, and lines of magnetic field 

strength are located on places of lines of equal electric potential (Neyman & Kalantarov, 1959). 

Generalizing the principle of correspondence, it is advisable to compare electromagnetic fields created 

by high-frequency currents of linear radiators with electrostatic fields of charges, which are placed on 

linear conductors. This approach is based on an analogy between two structures consisting of 

high-frequency currents and constant charges. It is assumed that shapes and dimensions of radiators 

coincide with shapes and dimensions of conductors. In the case of several radiators a ratio of emf in 

their centers is equal to a ratio of charges placed on the conductors. The positive charge, equal to 𝑄0, is 

located on the conductor 0, which corresponds to the active radiator. The negative charges (their 

number is equal to N) are located on the conductors i, corresponding to passive radiators. They are 

equal to −𝑄𝑖  and their sum is ∑ (−𝑄𝑖) = −𝑄0
𝑁
𝑖=1 , i.e., the sum of all charges is zero and the 

conductors form an electrically neutral system. In this system 

𝑄𝑖 𝑄0 = 𝐶0𝑖 ∑𝐶0𝑖 ,

(𝑖)

⁄⁄  

where 𝐶0𝑖 is the partial capacitance between conductors 0 and i. As it follows from this  

equality, the charges of the conductors i are directly proportional to the partial capacitances 𝐶0𝑖 

between these conductors and the conductor 0. See, for example, (Iossel, Kochanov, & Strunsky, 1981). 

Equivalent replacement of a complex structure of high-frequency radiators by a structure with constant 

charges placed on conductors sharply simplifies the problem, reducing it to the electrostatic problem. In 

accordance with what has been said, it is natural to call the proposed method by the method of 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

75 
Published by SCHOLINK INC. 

electrostatic analogy. 

The considered method allows analyzing the problem in a general view, for example, to study and to 

compare different laws of current distribution along the individual radiators. This is an undoubted 

advantage of the method. Characteristics of complex antennas are usually calculated using complex 

programs based on the moments method. For discussed problems, such a method is in essence a 

trial-and-error method. The moments method does not permit comparison of antennas with different 

distribution of currents in a common view. Therefore, this method is not applicable here. The 

approximate method does not give exact results. But if this method is correct, i.e. corresponds to the 

physical essence of the problem, its accuracy is the same for different distributions of the current, and it 

allows us to choose the best option. 

 

3. Director Antenna  

It is expedient to consider the procedure of applying the electrostatic analogy method on a concrete 

example. As an example, the director antenna (Yagi-Uda antenna) described in (Chaplin, Buchazky, & 

Mihailov, 1983). was adopted. This article was one of the first and most profound works devoted to 

optimization of directional antennas. 

The antenna circuit is given in Figure 1. The antenna consists of four metal radiators, their numbers are 

given in parentheses: active radiator 0, reflector 1, directors 2 and 3. The antenna dimensions are shown 

in figure in metres. They were defined by solving the optimization problem in a rigorous formulation. 

Let’s start with the capacitance between the wires. If the radii of the wires i and 0 are the same and 

equal to a = 0.001 m and the lengths 𝑙0 and 𝑙𝑖 of these wires are slightly different from each other, 

then the partial capacitance 𝐶0𝑖 between these wires in the first approximation is equal to 

 

 

Figure 1. Dimensions of Optimized Directional Antenna from Metal Radiators 

 

𝐶0𝑖 = 𝜋𝜀0𝑙𝑖 𝑙𝑛(𝑏𝑖 𝑎⁄ )⁄ ,
 

where 𝜀0 is the medium permittivity and 𝑏𝑖 is the distance between the wires.  

If to divide the wire 0 of the antenna into three conductors and to denote these conductors by indices 0i, 

then the circuit is divided into three circuits. Each circuit consists of two conductors: of conductor i and 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

76 
Published by SCHOLINK INC. 

conductor 0i (Figure 2). The generator is consists of three generators located in the centers of 

conductors 0i. Their emfs are defined as 𝑒𝑖 = 𝑒 𝑄𝑖 𝑄0,⁄  where e is the emf of the active radiator 0.
 
 

As follows from the said, the calculation of the electromagnetic field of the presented antenna is reduced 

to a calculation of the fields created by the separate generators and to the summation of the obtained 

results. Since all elements of the system are known, calculations can be performed using a program based 

on the method of moments. As another variant of the calculation, the procedure based on the described in 

(King & Harrison, 1970) theory of folded antennas can be used. This procedure makes possible to receive 

results in the form of analytical expression. In this case the structure of the directional antenna is reduced 

to three circuits, one of them is shown in Figure 2. Using approximate expressions for the active and 

reactive components of the dipole input impedance and for the reactance of a long line, calculating the 

currents in the circuit elements and summing the currents in the wires, it is possible to determine the 

total field at the observation point, taking in the account distance from each wire to this point. 

Electrostatic analogy between emfs in the radiator centers and the charges of conductors is approximate 

in nature. But if this approach is correct, i.e., corresponds to the physical essence of the problem, then its 

accuracy is the same for different laws of current distribution and allows us to choose the best option. 

 

 

Figure 2. Circuit from Two Parallel Vertical Conductors 

 

As is shown in the theory of folded antennas, the circuit of two parallel vertical wires with identical 

dimensions located at a distance 𝑏𝑖 from each other can be divided into a dipole and a long line, open 

at both ends. The current at the center of each dipole is equal to  

𝐽𝑖𝑑 = 𝑒𝑖 (4𝑍𝑖𝑑)⁄ . 

The reactive component of its input impedance is 𝑋𝑖𝑑 = −120 𝑙𝑛(2𝐿0 𝑎𝑒𝑖⁄ ) 𝑐𝑜𝑡 𝑘𝐿0,  where 

𝐿0 = 𝑙0 2 ⁄ is the arm length of the active radiator, 𝑎𝑒𝑖 = √𝑎𝑏𝑖  is its equivalent radius, k is the 

propagation constant. The current at the center of each wire of the long line is equal to 

𝐽𝑖𝑙 = 𝑒𝑖 (2𝑗𝑋𝑖𝑙)⁄ , 

where 𝑗𝑋𝑖𝑙 = −𝑗120 𝑙𝑛(𝑏𝑖 𝑎⁄ ) 𝑐𝑜𝑡 𝑘 𝐿𝑖 is the reactive input impedance of the long line with length 

𝐿𝑖 = 𝑙𝑖 2⁄ . The currents of the active (0i) and passive (i) radiators are equal to the sum and difference of 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

77 
Published by SCHOLINK INC. 

the dipole wire current 𝐽𝑖𝑑 and the current 𝐽𝑖𝑙 of the long line, i.e., the total current 𝐽0 of the active 

radiator and the total current 𝐽𝑖 of each passive radiator are  

𝐽0 = ∑(𝐽𝑖𝑑 + 𝐽𝑖𝑙),   𝐽𝑖 = 𝐽𝑖𝑑

3

𝑖=1

− 𝐽𝑖𝑙 . 

The amplitude and phase of the fields created by each radiator depend on its structure. The radiator 

structure determines the law of current distribution along it. If the radiator arm is a straight metallic 

conductor, its current is distributed according to the sinusoidal law 𝐽(𝑧) = 𝐽(0) 𝑠𝑖𝑛 𝑘 (𝐿 − |𝑧|). 

In this case the far field of the radiator is equal to 

𝐸𝜃 =
𝑗60𝐽(0)

𝑆𝑖𝑛𝜃
∙
𝑒𝑥𝑝(−𝑗𝑘𝑅)

𝜀𝑟𝑅
[𝑐𝑜𝑠(𝑘𝐿 𝑐𝑜𝑠 𝜃) − 𝑐𝑜𝑠 𝑘 𝐿], 

where R is the distance to the observation point. If the capacitive loads realize the in-phase current, 

distributed along the radiator in accordance with a linear law 𝐽(𝑧) = 𝐽(0)(1 −
|𝑧|

𝐿
), then 

 

From Figure 1 it is clear that the maximum radiation of the considered antenna is directed to the right, 

that is, towards the radiator 3. Since the radiator 1 is located on the left from the active radiator 0 at a 

distance 𝑏1, its field lags behind the field of the active radiator, firstly, per phase corresponding to a 

time of signal propagation from radiator 0 to radiator 1 and, secondly, per phase corresponding to the 

propagation time of the signal in the opposite direction - from the radiator 1 to the radiator 0 (the signal 

of radiator 1 must come to the radiator 0 an angle 𝜃, i.e., the path length between the wires is 

𝑏1/ 𝑠𝑖𝑛 𝜃). The total phase difference is equal to 𝜓1 = −𝑘𝑏1 (1 + 𝑠𝑖𝑛 𝜃) 𝑠𝑖𝑛 𝜃⁄ . Similarly, in the case 

of radiators 2 and 3, this phase difference is equal to 𝜓2 = 𝑘𝑏2 (𝑠𝑖𝑛 𝜃 − 1) 𝑠𝑖𝑛 𝜃⁄  and to 𝜓3 =

−𝑘𝑏3 (𝑠𝑖𝑛 𝜃 − 1) 𝑠𝑖𝑛 𝜃⁄ , respectively. 

The described procedure permits to determine the total field of the director antenna. In accordance with 

expression for 𝑒𝑖  emfs of the different radiators are equal to 𝑒1 = 0.388𝑒 , 𝑒2 = 0.335𝑒 , 𝑒3 =

0.277𝑒. The total field of this antenna with radiators in the form of straight metal wires is 

𝐸𝜃 =
𝑗60𝐽(0)

sin 𝜃
∑𝑒𝑖 ∙

3

𝑖=1

𝑒𝑥𝑝(−𝑗𝑘𝑅)

𝜀𝑟𝑅
∙ 

{(
1

4𝑍𝑖𝑑
+

1

2𝑍𝑖𝑙
) [cos(𝑘𝐿0 cos 𝜃) − cos 𝑘 𝐿0] + (

1

4𝑍𝑖𝑑
−

1

2𝑍𝑖𝑙
) 𝑒𝑥𝑝(𝑗𝜓𝑖)[cos(𝑘𝐿𝑖 cos 𝜃) − cos 𝑘𝐿𝑖]}. 

The inclusion of concentrated capacitive loads along the linear radiators, the magnitudes of which vary 

in accordance with the linear law, permits to create radiators with in-phase current. While retaining the 

dimensions of the radiators and the distances between them, we get the director antenna shown in 

Figure 3. The total field of such an antenna is calculated by the formula 

𝐸𝜃 =
𝑗60𝐽(0)

𝑘𝐿sin 𝜃𝑐𝑜𝑠2𝜃
∑𝑒𝑖 ∙

3

𝑖=1

𝑒𝑥𝑝(−𝑗𝑘𝑅)

𝜀𝑟𝑅
∙ 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

78 
Published by SCHOLINK INC. 

{(
1

4𝑍𝑖𝑑
+

1

2𝑍𝑖𝑙
) [1 − cos(𝑘𝐿0 cos 𝜃)] + (

1

4𝑍𝑖𝑑
−

1

2𝑍𝑖𝑙
) 𝑒𝑥𝑝(𝑗𝜓𝑖)[1 − cos(𝑘𝐿𝑖 cos 𝜃)]}. 

 

 

Figure 3. Directional Antenna with In-phase Straight Radiators 

 

The results of calculating the directivity and the pattern factor of a director antenna with straight metal 

wires are given in Figure 4 (curves 1). Since here an approximate calculation procedure was used, these 

results are not identical to the results presented in (Chaplin, Buchazky, & Mihailov, 1983), but are similar 

to them. Practically the antenna operates at the same frequency. The results of calculating the same 

characteristics for the antenna with in-phase currents are also given in Figure 4 (curves 2). They speak 

for themselves. This antenna operates over a wide frequency range and its directivity steadily and 

smoothly increases with increasing frequency, that is, the quality factor of this antenna is small. Of 

course, we must bear in mind that these characteristics are valid only if the current is in-phase. But the 

frequency ratio of antennas with capacitive loads with a high level of matching is a value of the order 10. 

 

 

Figure 4. Directional Characteristics of Director Antennas with Sinusoidal (1) and In-phase (2) 

Currents 

 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

79 
Published by SCHOLINK INC. 

As has already been said, the method of electrostatic analogy is based on the resemblance of a structure 

with high-frequency currents and a structure with constant charges. Comparison of electromagnetic 

fields created by high-frequency alternating currents of linear radiators with electrostatic fields of 

charges placed on linear conductors of electrically neutral system shows similarity of the mathematical 

structures of both fields. 

This method allows us to propose a simple and effective procedure for calculating the directional 

characteristics of the director antennas consisting from linear radiators. The procedure uses knowledge 

about the basic antenna dimensions and current distributions along the radiators. The detailed 

information on the types and magnitudes of concentrated loads does not require. As calculations were 

shown, the director antennas consisting from the linear radiators with in-phase currents provide a high 

directivity and smooth variation of characteristics in a wide frequency range. The results obtained with 

help of this method can be used to solve the problem of optimizing various director antennas by 

mathematical programming methods.  

The electrostatic analogy method was proposed in (Levin, 2017). But the need for such a method was 

understood long ago. A.F. Chaplin with his pupils stubbornly paved a way to its creation. A.F. Yakovlev 

persistently tried to improve the electrical characteristics of a log-periodic antenna (Yakovlev & 

Pyatnenkov, 2007). Their results will be discussed in Section 4. 

 

4. Log-periodic Antenna 

In Section 3 it is shown that using in-phase currents in director antennas provide high directivity in a 

wide frequency range. The result was obtained for the antenna array, consisting from the radiators with 

loads, by a method of electrostatic analogy. This method allows us to propose a new approach for 

solving the problem of reducing dimensions of the log-periodic antenna. Its structure does not 

withstand rough intervention, i.e., attempts to decrease longitudinal dimensions of an antenna by 

violating geometric progression relationships is causing a sharp deterioration of electrical 

characteristics and gives an insignificant decrease in overall dimensions. 

In accordance with a well-known method of calculating log-periodic antenna (Carrel, 1961) we 

consider an active region of this antenna array, consisting of three radiators (Figure 5), and determine 

fields of these radiators when emf e is located in the center of a middle radiator. Let the arm length of 

the middle radiator is 𝐿0 4⁄  ( is the wavelength), the arm length of the left (longer) radiator is 

𝐿1 /4 , and the arm length of the right (shorter) radiator is 𝐿2 /4, where  is a denominator of a 

geometric progression, according to which the radiators’ dimensions are changed. A magnitude  is 

another parameter, it is equal to   0.251cot, where  is an angle between an array axis and a line 

passing through radiators ends. The value  is the distance measured in the wavelengths between the 

half-wave radiator and the smaller neighboring radiator: 𝑏2   , and respectively the distance 

measured in the wavelengths between the half-wave radiator and the larger neighboring radiator is equal 

to 𝑏1 = σλ τ⁄ . As said in (Yakovlev & Pyatnenkov, 2007), the generalization of technic information 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

80 
Published by SCHOLINK INC. 

leads to the conclusion that the minimal changes in the electrical characteristics of the log-periodic 

antenna with metal radiators within the frequency range from f to f occur when /  0.19. If to assume 

that this relation is also valid for the considered array and for definiteness to take that  is equal to 0.9, 

then  = 0.146, 𝑏1=0.19 𝜆, 𝑏2 = 0.171𝜆, 𝐿1 = 𝐿0 0.9 = 0.278𝜆, 𝐿2 = 𝐿0⁄ ∙ 0.9 = 0.225𝜆,  

 

 

Figure 5. Directional Antenna of Three Radiators 

 

As it was said in Section 3, since all elements of the system are known, calculations can be performed in 

accordance with the known methods. Let us apply the calculation method used in the analysis of the 

antenna array presented in Figure 1. At that we will consider that radii of all radiators are the same. The 

radiators’ lengths, strictly speaking, are not the same, but they differ slightly. In the written below 

formulas, the difference between them is not fixed in order to simplify the written expressions. But the 

calculation with allowance for difference in their lengths shows that this difference has practically no 

effect on the result.  

In order to use this procedure, we divide the active radiator (conductor 0) into two parallel conductors 

(with numbers 01 and 02), and afterwards the obtained system of the four conductors is divided into 

two circuits (conductors with numbers 01 and 1 in the first circuit, conductors with numbers 02 and 2 

in the second circuit). In accordance with the method of the electrostatic analogy for two arrays, that is, 

accordingly to the physical content of the problem, one must assume that the ratio of emf in the centers 

of wires 01 and 02 is equal to the ratio of the partial capacitances 𝐶01 and 𝐶02, i.e., 𝑒1 = 0.52e, 

𝑒2 = 0.48e. A system of two parallel conductors located at the distance 𝑏1 can be divided into a dipole 

and a long line, opened on both ends. The current in the center of each dipole is equal to 𝐽𝑖𝑑 =

𝑒𝑖 (4𝑍𝑖𝑑)⁄ ; the current in the center of each conductor of a long line is equal to 𝐽𝑖𝑙 = 𝑒𝑖 (2𝑍𝑖𝑙).⁄  

Accordingly, the current in the active radiator center is the sum of the currents of the dipole and the 

long line, and the current in the passive radiator center is the difference between these magnitudes, i.e., 

𝐽0 = ∑ (𝐽𝑖𝑑 + 𝐽𝑖𝑙)
2
𝑖=1 , 𝐽𝑖 = 𝐽𝑖𝑑 − 𝐽𝑖𝑙. In the case of metal wires in the first approximation  

𝑍𝑖𝑑 = 𝑅𝑑 + 𝑗𝑋𝑖𝑑, 𝑅𝑑 = 80tan2(𝑘𝐿0 2⁄ ), 

𝑋𝑖𝑑 = 120𝑙𝑛(2𝐿0 𝑎𝑒𝑖⁄ ) cot 𝑘𝐿0,   𝑗𝑋𝑖𝑙(𝐿𝑖 , 𝑏𝑖) = −𝑗120 𝑙𝑛(𝑏𝑖 𝑎⁄ ) 𝑐𝑜𝑡 𝑘 𝐿𝑖 , 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

81 
Published by SCHOLINK INC. 

where 𝑎𝑒𝑖 = √𝑎𝑏𝑖 is the equivalent radius of dipole.  

The field amplitude and phase of each radiator depends on its structure and location in an antenna. 

Expressions for the currents distributions and the far fields of metal radiators and radiators with 

capacitive loads decreasing to antenna ends in accordance with the linear law are given in Section 3. 

Consider an impact of the radiator location on the far field of the antenna by way of a specific example 

of the directional antenna, shown in Figure 5. The arm length of the middle (active) radiator is equal to 

0.3 м, i.e., the wave length of the first (series) resonance is equal to 1.2 м. Radii of all conductors are 

the same and are equal to 0.001 m. The magnitude 𝜏 is 0.9. It is obvious that the maximal radiation of 

the antenna should be directed to the right, toward the radiator 2. Since the radiator 1 is located from 

the left of the active radiator 0, at a distance 𝑏1 from it, its field lags behind the active radiator field, 

firstly by 𝑘𝑏1 in phase, i.e., on the propagation time of the signal from the active radiator to the 

passive radiator 1, and, secondly, by 𝑘𝑏1 sin 𝜃⁄  in phase, i.e., on the time of the signal propagation in 

the opposite direction, from the radiator 1 to the active radiator (signal of the wire 1, radiated at an 

angle 𝜃, must come to the active radiator at the same angle 𝜃, i.e., it travels the distance 𝑏1 sin 𝜃⁄  , 

and not the distance 𝑏1. The total change in phase is equal to 𝜓1 = −𝑘𝑏1 ∙
1+sin𝜃

sin𝜃
. Similarly, in the 

case of radiator 2, this phase change is equal to 𝜓2 = −𝑘𝑏2 ∙
sin𝜃−1

sin𝜃
. 

The total field of the antenna structure shown in Figure 5 with in-phase current distribution at angle 𝜃 

on the base of aforesaid may be written in the form 

𝐸 =
𝐴𝐽(0) 𝑠𝑖𝑛 𝜃

𝑐𝑜𝑠2 𝜃
∑𝑒𝑖 {(

1

4𝑍𝑖𝑑
+

1

2𝑍𝑖𝑙
) [1 − 𝑐𝑜𝑠(𝑘𝐿0 𝑐𝑜𝑠⟨𝜃 − 𝜃0⟩)] +

2

𝑖=1

 

+[1 (4𝑍𝑖𝑑)⁄ − 1 (2𝑍𝑖𝑖)⁄ ]𝑒𝑥𝑝(𝑗𝜓𝑖)[1 − cos(𝑘𝐿𝑙 cos 𝜃)] }. 

The directivity magnitude is determined by the expression  

 

where Δ is the interval between neighboring values 𝜃𝑛, N is the number of these intervals between  

𝜃 = 0 and 𝜃 = 𝜋 2⁄ . 

The results of calculating directivity of a structure, shown in Figure 5, are given in Figure 6 depending 

on an electrical length 𝑘𝐿0 of the active radiator arm. The curve 1 demonstrates the 

directivity of the structure with in-phase currents in each element. The directivity of the structure with 

sinusoidal currents is given for comparison by the curve 2. Radiators with 

concentrated capacitances distributed in accordance with linear or exponential law along each arm 

allow to ensure a high level of matching with a cable in the range with a frequency ratio of the order 10. 

Therefore, graphics are made so that to show the structure directivity in the range from 𝑘𝐿0 = 𝜋 to 

𝑘𝐿0 = 10𝜋. 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

82 
Published by SCHOLINK INC. 

 

Figure 6. Directivity of Antennas of Three Radiators 

 

Radiation of neighboring elements of log-periodic antenna with the arm length 0.27 m and 0.333 m 

respectively requires calculating fields in the structures presented in Figures 7a and 7b. Results of these 

calculations are given in Figure 6: curve 3 corresponds to Figure 7a, curve 4 - to Figure 7b. The 

directivity magnitudes in Figure 6 for specific values 𝑘𝐿 correspond to the same frequencies, i.e., the 

same values 𝑘𝐿 correspond to the elements of equal length in the all three circuits (for example, to the 

elements with the arm length 0.3 m). The figure shows that the directivity magnitudes at the same 𝑘𝐿 

are close to each other. This is natural, since the directivity in each scheme increases slowly with 

increasing frequency. Small increasing the active radiator length causes at the same 𝑘𝐿 the small 

increase of the resonant wavelength, i.e., the small decrease of the resonance frequency and the 

directivity. 

 

 

Figure 7. Structures of Smaller (a) and Greater (b) Neighboring Radiators 

 

The obtained results show that each active radiator with in-phase current, included in the structure of 

the log-periodic antenna, provides high radiation directivity in a wide frequency range. Neighboring 

radiators have similar directivity magnitude. The direction of the radiation is the same. A signal 

propagates along the distribution line from short elements to the long dipoles (in Figures 5 and 7 to the 

left), the radiated signal propagates in the opposite direction. The total path difference is quite large. 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

83 
Published by SCHOLINK INC. 

For example, for the signal radiated by a half-wave neighboring radiator it is equal to 0.38𝜆, i.e., it is 

close to a half wavelength. Crossing wires of the distribution line in an interval between the elements 

permits to reduce dramatically this path difference. 

Summarize the analysis of the log-periodic antenna. Known log-periodic antenna with sinusoidal 

current distribution along the radiators have a property of automatic currents “cut-off”, i.e., a separate 

antenna segment (active region) radiates a signal in a narrow frequency band. Outside this band and 

outside the borders of the active region the signal decays rapidly. Wide frequency range is provided by 

a large antenna length, which is equal to the sum of the lengths of the active regions. Attempts to 

reduce the antenna length by disturbing a geometric progression and increasing a radiators number lead 

to a small decrease of dimensions and a sharp deterioration of electrical characteristics. The more 

effective methods are firstly a two-fold use of each active region by an application of linear-spiral 

radiators and secondly an employment of an asymmetrical log-periodic antenna with coaxial 

distribution line (Yakovlev & Pyatnenkov, 2007). This allows to decrease the antenna length by 

25-30%. 

Replacement of the straight metal radiators by the radiators with concentrated capacitive loads provides 

reusable active area and allows us to obtain a high directivity in a wide frequency range, using a simple 

structure with three radiators. Results obtained with the help of the method of electrostatic analogy may 

be used for solving optimization problem by methods of mathematical programming. Increasing the 

radiators number in the structure may allow dramatically increasing its directivity. 

 

5. Conclusions 

In this work the following results are obtained. 

A simple and effective procedure, based on the method of electrostatic analogy, in accord with which it 

is assumed that the ratio of emf in the radiator centers is equal to the ratio of the charges placed on the 

conductors, is used for calculating directional characteristics of log-periodic antennas. 

With the help of this procedure the directional characteristics of log-periodic antennas with sinusoidal 

and in-phase current distribution in linear radiators are compared. It is shown that replacement of the 

straight metal radiators by the radiators with concentrated capacitive loads allows to provide, when 

only three radiators are used, the high directivity in a wide frequency range, i.e., decrease sharply the 

antenna length. Increasing the number of radiators allows to increase substantially the antenna 

directivity.  

Results obtained with the help of the method of electrostatic analogy may be used for solving 

optimization problem by methods of mathematical programming. 

 

 

 

 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 2, 2022 

84 
Published by SCHOLINK INC. 

References 

Carrel, R. L. (1961). The design of log-periodic dipole antennas. IRE Int. Convention Record, Part 1, 

61-75. https://doi.org/10.1109/IRECON.1961.1151016 

Chaplin, A. F. (1969). Synthesis of arrays of passive radiators. Izvestiya Vuzov of USSR 

-Radioelectronics, 6, 559-562. (In Russian). 

Chaplin, A. F., Buchazky, M. D., & Mihailov, M. Yu. (1983). Optimization of director antennas. 

Radiotechnics, 7, 79-82. (In Russian). 

Chen, C. A., & Cheng, D. K. (1975). Optimum element length for Yagi-Uda arrays. IEEE Trans. 

Antennas Propagation, 1, 8-15. https://doi.org/10.1109/TAP.1975.1141001 

Iossel, Yu. Ya., Kochanov, E. S., & Strunsky, M. G. (1981). Calculation of Electrical Capacitance. 

Energoizdat: Leningrad. (In Russian). 

King, R. W. P., & Harrison Ch. (1970). Antennas and Waves, a Modern Approach. MIT Press. 

Levin, B. M. (1998). Monopole and Dipole Antennas for Ship Radio Communications. Abris: 

St.-Petersburg. (In Russian). 

Levin, B. M. (2017). Antenna Engineering: Theory and Problems. CRC Press: London, New York. 

Levin, B. M. (2017). Directional antennas with in-phase currents. Proc. of 11 Int. Conference on 

Antenna Theory and Techniques ICATT’17 (Kyiv, Ukraine), 110-113. (In Russian). 

Neyman, L. R., & Kalantarov, P. L. (1959). Theoretical Background of Electro Engineering, Part 3. 

Gosenergoizdat: Moscow, Leningrad. (In Russian). 

Pistolkors, A. A. (1948). Theory of the circular diffraction antenna. Proc. of IRE, 1, 56-60. 

https://doi.org/10.1109/JRPROC.1948.230915 

Yakovlev, A. F., &Pyatnenkov, A. E. (2007). Wide-band Directional Antennas Arrays of Dipoles. 

St.-Petersburg. (In Russian). 

 

 

http://ieeexplore.ieee.org.ezproxy.hit.ac.il/xpl/articleDetails.jsp?tp=&arnumber=1697503&contentType=Journals+%26+Magazines&searchField%3DSearch_All%26queryText%3DPistolkors