Applied Science and Innovative Research 

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

Vol. 6, No. 3, 2022 

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1 
 

Original Paper 

About Microstrip Antennas 

Boris Levin1 

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

 

Received: May 23, 2022          Accepted: June 5, 2022         Online Published: June 9, 2022 

doi:10.22158/asir.v6n3p1             URL: http://doi.org/10.22158/asir.v6n3p1 

 

Abstract 

The electrical characteristics of a microstrip antenna are considered. Its design and electrical 

characteristics are compared with design and electrical characteristics of volume self-complementary 

antennas and an antenna located on the faces of a lying rectangular pyramid. The method of a complex 

potential is used for an analysis of the substrate influence on the properties of this antenna. In 

accordance with the results of comparing the characteristics of linear radiators with different current 

distributions, the design of a microstrip antenna with an in-phase distribution was proposed, which 

ensures operation in a wide frequency band. 

Keywords 

Antenna theory, current distribution, extension of frequency band, influence of substrate  

 

1. Introduction 

An explosive wave of theoretical work on antenna technology in the middle of the last century led to 

the creation of rigorous methods for analyzing and then synthesizing antennas. The widespread use of 

new antenna options, one of which was microstrip antenna, was a side effect of this wave. The reason 

for a widespread use of microstrip antennas was the fact that on aircraft, ships and other moving objects, 

where size, weight, price and ease of installation are especially important, low-profile antennas of the 

appropriate shape and design are needed. Such antennas can be mounted flush on metal surfaces, and it 

was need only feed lines, which are usually placed under the ground plane. The designs and electrical 

characteristics of microstrip antennas, as well as examples of their application, are described in books 

(Balanis, 2005; Johnson, 1993) and in articles included in the collection (Microstrip Antennas, 1995). 

The microstrip antenna (Figure 1) presents a thin metal strip 1 (with thickness t) located at a small 

height h above the ground plate 2, at that 𝑡, ℎ ≪ 𝜆, where 𝜆 is the wavelength. The strip and ground 

plate are separated by dielectric plate 3 (substrate) as shown in the figure. The radiating elements and 

the feed lines are usually photoetched on the dielectric substrate. The metallic strip may be any 

configuration, but square, rectangular, and circular are the most common because simpler fabrication 

mailto:levinpaker@gmail.com


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and analysis. The feed line is made or as a conducting strip 4, as a rule, of smaller width (Figure 1a) or 

as coaxial line 5 (Figure 1b), where the inner conductor of the coax is attached to the metal strip 1. 

With help of microstrip antennas one can create linear and circular polarization. Arrays of microstrip 

antennas allow to provide the high directivity. 

 

 

Figure 1. Microstrip Antennas with Feed Lines Made as a Conducting Strip (a) and as Coaxial 

Line (b) 

 

Major operational disadvantages of this antenna are its small efficiency and very narrow frequency 

bandwidth, which is typically only a fraction of a percent or a few percent. The reason of this 

inefficiency is that its thickness is usually very small, and the waves generated within the dielectric 

substrate undergo considerable reflections when they go along the edge of the strip. 

In order to analyze the electrical characteristics of microstrip antenna (patch), several models have been 

proposed. The greatest attention was paid to model in the form of a two-wire long line. This model 

treats the antenna as a segment of a transmission line with length L terminated at both ends by a 

radiation admittance 𝑌𝑇.  It is considered that the transmission line doesn’t radiate because of 

reflections and acts as a transformer. Each radiation admittance is perpendicular to the feed line, and it 

is made in the form of a slot with length b and width h. If the length of the transmission line is 

approximately 𝜆 2,⁄  where 𝜆 is the wavelength in the line, then the fields at the aperture of two slots 

have the same directions, and the slots form a two-element array with a distance 𝜆 2⁄  between the 

elements. The field components parallel to the antenna axis add up in phase and create a maximum 

radiation along the perpendicular to the ground plane. 

As is convincingly stated in (Munson, 1974) the transmission-line model “has the advantage of 

yielding very simple expressions for both the real and imaginary part of 𝑌𝑇, but it has three important 

shortcomings: 

a) The expressions used for 𝑌𝑇 are inaccurate for narrow patches (i.e., for 𝑏 ≤ 𝜆0, where 𝜆0 is 

free-space wavelength). 

b) The mutual coupling between the main radiating slots is neglected. 

c) The influence of the side slots on the radiation conductance is neglected. 

Here the slots with length b are called the main radiating slots and the slots with length L are called the 

side slots. Let’s give the author the right to such terminology, but it is necessary to change the order of 



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the listed deficiencies, since the main deficiency is the weak interest to the side slots. 

It may seem surprising, but the closest relatives of microstrip antennas should be considered volume 

antennas located on a cone, a paraboloid and a pyramid shown in Figure 2. All of the listed antennas 

consist of two symmetrical radiators: metal and slotted, located under a right angle to each other. They 

are excited at the point located on the top of a cone, a paraboloid and a pyramid. 

 

 

Figure 2. Antenna on a Cone (a), a Paraboloid (b) and a Pyramid (c) 

 

Antennas located on a cone and a paraboloid are usually called complementary or self-complementary, 

but it would be more correct to call them self-complementary antennas of limited dimensions, in 

contrast to flat self- complementary antennas that occupy the entire infinite plane. The surface of an 

antenna on the pyramid is not a smooth surface of rotation with axial symmetry and strictly speaking 

such an antenna cannot be called complementary. But its characteristics are similar to those of 

complementary antennas. The upper and lower faces of the pyramid are made metallic and they form 

an electric dipole; the two side faces are slots and they form a magnetic dipole. The pyramid 

cross-section has a rectangular shape. Such structure is easier to realize, than for example conical, and 

on medium waves the lower arm of the antenna is created as a reflection of the upper arm in the 

ground.  

As can be seen from Figure 2с, the pyramid structure is closest to the structure of the microstrip 

antenna. In the case of the microstrip antenna the upper and lower trapezoid faces of the pyramid are 

replaced by rectangular strips, and the metal strips of the antenna and the feed line are joined with each 

other. The slot radiator is also rectangular, although its width is substantially less than the width of the 

metal radiator. Basically, microstrip antenna is no different from the structure of the antenna located on 

the pyramid. The differences are minor. 

From said, it follows that the all elements of the so-called transmission-line located along the length L, 

do not simply transfer energy from an excitation point to an aperture at an opposite end, but radiate the 



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signal, i.e., all elements that make up a transmitting strip antenna create the field - just like all elements 

of a two-wire line radiate weak signals, if the distance between the wires is not zero. Ignoring this 

radiation is the main disadvantage of the so-called transmission-line model. Wave attenuation in the 

antenna substrate cannot serve as its justification. 

 

2. Structure of Microstrip Antenna 

Let us compare the microstrip antenna with the antenna on the pyramid, not taking into account the 

permittivity of the substrate. A cross-section of the microstrip antenna is given in Figure 3a. The dotted 

line in the figure shows a mirror image in the metal plane of grounding. The capacitance of the 

symmetric structure between a microstrip antenna and its mirror image is two times less than the 

capacitance of the asymmetric structure. Considering the microstrip antenna as a combination of two 

longitudinal radiators: metal and slot, we determine an input impedance of each one. The capacitance 

of a metal radiator per unit length can be calculated as the capacitance between two identical plates 

with a common plane of symmetry. 

 

 

Figure 3. Cross-section of a Microstrip (a) and a Cylindrical (b) Antenna 

 

A strict expression for its calculation contains complete and incomplete elliptic integrals of the first and 

second kind and requires the solution of an equation consisting of such functions. For calculations 

approximate functions can be used. Here and below, we extensive will use the formulas given in (Iossel, 

Kochanov, & Strunsky, 1981). If the width of the metal plate is equal to b, and the distance between the 

plates is 𝑑, then at 𝑏 𝑑 > 27⁄  the capacitance between the plates per unit length is 𝐶𝑙 ≈ 𝜀 𝑏 𝑑⁄ . 

When  

𝐶𝑙 ≈ 𝜀
𝑏

𝑑
{1 +

𝑑

𝜋𝑏
[1 + ln(2𝜋𝑏 𝑑⁄ )]}. 

If, for example, 𝑑=h=0.7874 mm, b=33.15 mm, then for the capacitance between the metal plates 

(according to the approximate first formula and the more accurate second one) we obtain  

𝐶𝑙1 = 42.1𝜀, 𝐶𝑙2 = 44.2𝜀.  

The wave impedance of a long line formed by two metal strips of width b, located in a free space at a 

distance d from each other, is 𝑊 = 𝑘 (𝜔𝐶)⁄ , where 𝑘 = 2𝜋 𝜆⁄  is propagation constant of the wave in 

air, 𝜔 = 2𝜋𝑓 is circular frequency, 𝜆 is wave length, 𝑓 is frequency, 𝜆𝑓 = 𝑐 = 3 ∙ 108 is speed of 



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light. Since 1 𝑐 = 120𝜋⁄ 𝜀0, where 𝜀0 = 10−9 (36𝜋)⁄  is the absolute permittivity of the free space, 

we get that 𝑊 = 120𝜋𝜀0 𝐶𝑙2⁄ ≅ 2.7𝜋. This magnitude is equal to the wave impedance 𝑊𝑀 of the 

metal radiator included in the structure of the complementary antenna.  

To calculate the wave impedance 𝑊𝑆 of a metal radiator identical to a slot in shape and size, it is need 

to determine the capacitance between the metal strips having dimensions of slots and located on their 

sites. Capacitance between plates with width d, located at a distance b from each other, if 𝑑 𝑏 ≤ 1,⁄  is 

equal to 𝐶𝑙 ≈ 𝜋𝜀0 ln(4 𝑏 𝑑⁄ ),⁄  i.e., 𝐶𝑙 = 0.613𝜀0,  and 𝑊𝑆 = 195.8𝜋. The product of asymmetric 

wave impedances is 𝑊𝐸𝑊𝑠 ≅ (23𝜋)2. For a symmetrical antenna this product is 4 times larger, i.e., it 

is equal to (46𝜋)2. As is known, the product of the wave impedances of self-complementary antennas 

is (60𝜋)2. In the given case, this magnitude is significantly smaller, since the surface on which they 

are located, cannot be called smooth. 

The symmetrical structure consisting of a microstrip antenna and its mirror image is similar to the 

structure of a complementary antenna consisting of two coaxial unclosed shells (Figure 3b) separated 

by angles 2𝜑. The capacitance between them per unit length of shells is equal to (see Iossel, 

Kochanov, & Strunsky, 1981) 𝐶𝑙 = 𝜀 𝐾(√1 − 𝑘2) 𝐾(𝑘),⁄  where 𝐾(𝑘)  is the complete elliptic 

integral of the first kind from the argument 𝑘 = 𝑡𝑎𝑛2 𝜑 2⁄ . Let’s compare this structure with the 

structure of the microstrip antenna. For the given dimensions of this antenna, assuming that the 

structure radius is equal to 𝑅 = 𝑏 2⁄ , and the angle 𝜑 is equal to 𝜑 = ℎ 𝑅⁄ , we find: 𝜑 2⁄ = ℎ 𝑏⁄ =

0.024. In accordance with expression for 𝐶𝑙 we get: 𝑘 = 0.00056, √1 − 𝑘2=0.9999998, 𝐾(𝑘) =

1.5708, 𝐾(√1 − 𝑘2) = 8.87, that is 𝐶𝑙1 = 5.7𝜀. If replace the angle width 2𝜑 of the slot with the 

angular width of the shell (it is equal to 2 ∝= 𝜋 − 2𝜑),  we get: ∝ 2⁄ = 0.7616,  𝑘 = 0.909, 

𝐾(𝑘) = 2.3217, √1 − 𝑘2 = 0.416, 𝐾(√1 − 𝑘2) = 1.6464, that is the capacitance between the slots 

is 𝐶𝑙2 = 0.71𝜀. Wave impedances are respectively 𝑊𝐸 = 21.05𝜋, and 𝑊𝑠 = 169𝜋. The product of 

the wave impedances is 𝑊𝐸𝑊𝑠 = (60𝜋)2  - in full agreement with the properties of the 

self-complementary structures. The difference between the results for the structures shown in Figure 3a 

and Figure 3b is, of course, caused with the difference in the distances between the arms of the metal 

radiators. 

As mentioned above, each of the radiators presented in Figure 1 can be simultaneously viewed as a 

long line and as an antenna. This approach is no different from the usual comparison of a linear antenna 

with an equivalent long line. That greatly simplifies calculating the wave impedance and other 

characteristics of an antenna. The structure of the microstrip antenna and the dimensions of its 

cross-section are constant along its length L, i.e., the equivalent long line is uniform, and its wave 

impedance is constant. 

However, it is necessary to note a significant difference between a microstrip antenna and a long line 

(and antennas associated with it). Known antennas are often divided into symmetrical (e.g., the dipole) 

and asymmetric (e.g., the monopole). In this case, a structure is called asymmetric, if it consists of a 

real antenna and its mirror image in the ground or grounding. Asymmetry essentially means a change in 



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the sign or direction of the current. The asymmetry of the metal plate of a microstrip antenna is caused 

by the difference in the media adjacent to it from above and below and the large width of the plate 

compared to the thickness of the substrate. This results in a different magnitude of currents along the 

top and bottom surfaces of the plate. In this case, the EMF on both surfaces at the point of connection 

of the generator is the same. The voltages are also the same on both sides of the plate along all its 

length. 

Thus, the structure of a microstrip antenna is more complex than that of a common long line. To 

simplify this structure, let’s replace the metal plate with width b by a metal cylinder 1 with a radius 

𝑎 = 𝑏 𝜋⁄  (with the same surface area). The cross-section of such a structure is shown in Figure 4. 

 

 

Figure 4. Approximate Equivalent Structure of the Microstrip Antenna: 1 - radiator, 2 - substrate, 

3 – grounding, 4 – surface of zero potential 

 

As it can be seen from the figure, in the vicinity of the radiator there are two surfaces with the same 

(zero) potential. This is, firstly, the metal surface 3, on which the antenna is located (grounding). This is, 

secondly, the surface of zero potential 4, which by definition is located at a distance from the radiator 

equal to the total length 2L of the antenna. The substrate is noted by the number 2. 

The wave impedance of a long line located between the lower surface of the plate and the surface 3 of 

zero potential is equal to 𝑊1 = 𝑘1 (𝜔𝐶1),⁄  where 𝑘1 = 𝑘𝑛 is a wave propagation constant along the 

line (here 𝑛 = √𝜀1 is a wave slowing, 𝜀1 is a relative substrate permittivity). Capacitance 𝐶1 in the 

first approximation, as previously shown, is equal to 𝐶1 = 𝜀1𝜀0𝑏 ℎ.⁄  As a result, we get 

𝑊1 = 120
𝜋ℎ

𝑏√𝜀1
 .                               (1) 

This value is usually specified in microstrip antenna manuals. 

If the structure used in this calculation is replaced by the new structure shown in Figure 4, then the 

result gives some error. But the new structure permits to take into account the influence of the zero 

potential surface, indicated by the number 4. This surface is usually not taken into account, and that 



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leads to a significant error. From the solution of the integral equation for the current in a symmetrical 

linear radiator with a length 2L it follows that its wave impedance is approximately equal to 𝑊𝐴 =

120(ln 2𝐿 𝑎⁄ − 1), where a is the wire radius. The capacitance of a single wire per unit length is twice 

as large as the capacitance of the long line, and the wave impedance of a single wire is half as much as 

the wave impedance of the long line, i.e., it is equal to 

𝑊2 = 60(ln 2𝐿 𝑎⁄ − 1).                           (2) 

As follows from the above, the replacement of the calculated structure by the structure shown in Figure 

4 leads to an approximate choice of the value a in the expression for the antenna wave impedance. But 

this is a standard substitution used in the derivation and solution of integral equations for current, and 

the error caused by it is minimal. 

Hence, the total wave impedance of the antenna shown in Figure 1a is 

𝑊 =
𝑊1𝑊2

𝑊1+𝑊2
.                                 (3) 

To determine the reactive component of the input impedance, it is necessary to clarify the value of the 

propagation constant. 

As can be seen from Figure 1, microstrip antennas are made in two versions: asymmetric and 

symmetrical. In the first version, the antenna equivalent is a segment of a long line, excited at one end 

and open at the other end (analogous to a monopole). Wave impedance W was obtained for the first 

option. The symmetrical radiator (dipole) is the equivalent of the second option. It consists of two 

identical line segments, excited in the center of the structure and open at the ends remote from each 

other. If the dimensions of the arms coincide with the dimensions of the first option, then the wave 

impedance of the second option is twice as large. 

 

3. Substrate 

One of the main elements of a microstrip antenna is the substrate. A substrate changes all electric 

characteristics of antenna: capacitance per unit length, wave length, propagation constant, electric field 

and complicates antenna calculation. If the antenna is located in a homogeneous medium with 𝜀𝑟 ≠ 1, 

the capacitance С1 per unit length of the antenna increases 𝜀𝑟 times. At the same time in an antenna 

with a substrate, cut from a material with the same dielectric constant, the capacitance increases much 

less, only 𝜀𝑟е times, and this increase depends on the antenna design and substrate dimensions. In this 

case the Leontovich’s equation for the current in a mode of own oscillations has the form 

𝑑2𝐽

𝑑𝑧2
+ 𝑘1

2𝐽 = 0, 

where the wave propagation constant 𝑘1 = 𝑘√𝜀𝑟е is increased by √𝜀𝑟е times. Assuming that the 

radius 𝑎 of the antenna conductor is small compared to the antenna length 2L, i.e., the surface of zero 

potential is located at a distance 2L from the antenna surface, we obtain for the self-capacitance of a 

conductor per unit its length С10 = 2𝜋𝜀𝑟е (ln 2𝐿 𝑎⁄ − 1),⁄  and for the capacitance per unit length 



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between the antenna arms С1 = С10 2.⁄  An increase of the propagation constant leads to a 

corresponding increase of the antenna wave impedance and the reactive component of its input 

impedance. 

From the said it follows that in order to determine the electrical characteristics of a radiator located on a 

substrate, it is necessary to find out to what extent the substrate changes a capacitance per unit antenna 

length. This means that the analysis of the substrate effect is reduced to an electrostatic problem. The 

electrostatic problem considers a system of charged conducting bodies located in a dielectric medium, 

in which volume charges are absent. If the charge of each body and the charge distribution over the 

surface of the body are known, then in a homogeneous medium with a dielectric constant ε the 

potential U, determined by the entire set of charges located in the volume V, can be found using the 

expression. Here 𝜌𝑣 is the charge density of the conducting body, r is the distance from the point of 

charge placement (integration point) to the observation point. The presented integral is a solution of the 

Poisson equation. 

If the distribution of charges is unknown, then in accordance with the uniqueness theorem, one should 

to establish the necessary and sufficient conditions, which uniquely determine the field. With allowance 

for these conditions, the field on the surface of the conducting body must satisfy the Laplace’s equation, 

and these surfaces must be surfaces of equal potential, i.e., on each of them 𝑈 = 𝑈𝑚 = const. The 

total charge 𝑞𝑚 of each body m should be equal 𝑞𝑚 = −∫ 𝜀
(𝑆𝑚)

𝜕𝑈

𝜕𝑛
𝑑𝑆. Here 𝑆𝑚 is the surface of a 

body m, n is the normal to its surface. The structure of an electrostatic field is determined by curves of 

equal potential, along which U = const, and by curves of field strength (by force lines) V = const. Each 

of these curves is a collection of points, at which this or that equality is true. The curves corresponding 

to these two equalities are two families of curves intersecting at right angles. 

The problem of calculating the electric field of charged bodies is greatly simplified, if the field depends 

on only two coordinates. Such a field is called by a plane-parallel field. This, for example, is the field 

of a system consisting of several infinitely long cylindrical wires parallel to each other and to z axis of 

a rectangular coordinate system. In this case the wires charges are uniformly distributed along their 

length, the field structure is the same in all transverse planes z = const, the lines of field strength and 

equal potential in these planes depend only on two coordinates, x and y. 

Let dl is an element of the line of field strength, and ds is an element of the line of equal potential. As 

already mentioned, they are mutually perpendicular. The potential U increases in the direction opposite 

to the vector �⃗� , i.e., in the direction of decreasing l. It is generally accepted that the flux 𝛹𝐸 of the 

electric field vector grows in the direction of increasing s. In accordance with these conditions, the 

electric field strength may be related with magnitudes U and V by dependences 

𝐸 = −𝜕𝑈 𝜕𝑙 = 𝜕𝑉 𝜕𝑠.⁄⁄  In a rectangular coordinate system, we obtain 𝐸𝑥 = −𝜕𝑈 𝜕𝑥 = 𝜕𝑉 𝜕𝑦,⁄⁄  

 𝐸𝑦 = −𝜕𝑈 𝜕𝑦 = −𝜕𝑉 𝜕𝑥.⁄⁄  Repeated differentiation shows that both functions (U and V) satisfy 



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Laplace’s equation, which in the rectangular coordinate system has the form (e.g., for U) 
𝜕2𝑈

𝑑𝑥2 +
𝜕2𝑈

𝑑𝑦2 +

𝜕2𝑈

𝑑𝑧2 = 0, and consequently, they satisfy the first requirement of the uniqueness theorem. 

If to consider the plane of the cross-section of a plane-parallel field as the plane of a complex variable 

𝑧 = 𝑥 +  𝑗y, then for the regular analytic function 𝜍(𝑧) = 𝜉(𝑥, 𝑦) + 𝑗𝜂(𝑥, 𝑦) of the variable z we can 

write in accordance with the Cauchy-Riemann conditions 𝜕𝜂 𝜕𝑥 = −𝜕𝜉 𝜕𝑦,⁄⁄  𝜕𝜂 𝜕𝑦 = −𝜕𝜉 𝜕𝑥.⁄⁄  It 

is easy to be convinced that these equalities coincide with the expressions for 𝐸𝑥 and 𝐸𝑦, i.e., we can 

adopt that ξ = V, η = U and use the regular analytic function 

ζ (z) = V (x, y) + j U (x, y), 

which is called by the complex potential of the field. In each specific case, the problem of calculating 

the field will be completely solved if the function ζ (z) is found that satisfies the boundary conditions 

on the surfaces of conducting bodies. 

The method of complex potential and examples of its use for solving problems devoted to conducting 

bodies located in a homogeneous medium are described in the monograph (Mirolubov, Kostenko, 

Levinstein, & Tichodeev, 1963). In particular, there it is shown that for a solitary circular wire with a 

charge τ per unit length, located in a homogeneous medium with a dielectric constant ε, the complex 

potential has the form 

𝜁(𝑧) = −𝑗
𝜏

2𝜋𝜀
𝑙𝑛𝑧 + 𝐶.                             (4)  

Here 𝐶 = 𝐶1 + 𝑗𝐶2 is a constant, and factors 𝐶1 and 𝐶2 depend on the choice of the initial line of 

field strength and the line of zero potential. Introducing the notation 𝑧 = 𝜌𝑒𝑗𝜑, we find 

𝑉(𝑧) =
𝜏𝜑

2𝜋𝜀
+ 𝐶1, 𝑈(𝑧) = −

𝜏

2𝜋𝜀
𝑙𝑛𝜌 + 𝐶2,                       (5)  

whence for the line of field strength and the line of equal potential we obtain, respectively 

𝜑 = 𝑐𝑜𝑛𝑠𝑡, 𝜌 = 𝑐𝑜𝑛𝑠𝑡.                            (6) 

The constant coefficients included in expression (5) are determined based on the condition that when 

going around a closed contour around the wire section, the angle φ increases by 2𝜋, and the function 

𝑉(𝑧) receives an increment equal to Ψ𝐸 𝑙,⁄  where Ψ𝐸  is the flux of the electric field vector �⃗�  

through a cylindrical surface covering the wire segment with a length l. 

The method of complex potential allows us to determine all the electrical characteristics of a metal 

radiator located in the homogeneous medium, in particular, the self-capacitance of a conductor per unit 

of its length and the capacitance between the antenna arms.  

If the medium is inhomogeneous, for example, piecewise homogeneous, the problem is significantly 

complicated. Such a problem, in particular, arises if the wire is located at the boundary of two media, 

air and dielectric. In this case it is advisable to go from the flux Ψ𝐸 of the vector of electric field �⃗�  to 

the flux Ψ𝐷 of the electric displacement vector �⃗⃗� , taking into account the different density of this 

flux in different media.  

 



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The corresponding generalization of the complex potential method was accomplished in the article 

(Levin, 1997). In this case the increment of the function V(z) in going along closed contour is 

Δ𝑉 =
1

𝑙
∑ Ψ𝐷𝑚 𝜀𝑚⁄(𝑚)  (m is a number of a medium). Integrating with respect to the volume v of the 

conductor the left and right parts of the Maxwell equation 𝑑𝑖𝑣�⃗⃗� = 𝜌𝑣 (𝜌𝑣 is the charge density on the 

conductor):  

∫ 𝑑𝑖𝑣�⃗⃗� 𝑑𝑣 = ∑ ∮ �⃗⃗� 
𝑆𝑚

(𝑚)𝑣
𝑑𝑆  , ∫ 𝜌𝑣𝑣

𝑑𝑣 = ∑ ∫ 𝜎𝑚𝑑𝑆
𝑆𝑚

𝑚  

(here S is the conductor surface, 𝑆𝑚 is the part of this surface located beside to the medium m, 𝜎𝑚 is 

the charge density of this surface part) and equating these results, we find that ∑ Ψ𝐷𝑚 = ∑ 𝑞𝑚,(𝑚)(𝑚)  

i.e., the flux Ψ𝐷𝑚 of the displacement vector through the conductor surface to medium m is equal to 

the charge 𝑞𝑚 of the surface part located beside to this medium. Respectively 

Δ𝑉 =
1

𝑙
∑ 𝑞𝑚 𝜀𝑚⁄ .(𝑚)                               (7) 

Let’s introduce the magnitude 𝜀𝑒, satisfying the equality ∑ 𝑞𝑚 𝜀𝑚⁄ = 𝑞 𝜀𝑒 ,⁄(𝑚)  where q is the total 

charge of the conductor. It can be seen that this magnitude has the meaning of the equivalent 

permittivity of an inhomogeneous medium. If Δ𝜑𝑚 is the circumference arc beside to medium m, and 

∑ 𝑞𝑚 = 𝑞,(𝑚)  then it is easy to show that 

𝜀е =
1

2𝜋
∑ 𝜀𝑚Δ𝜑𝑚(𝑚) .                               (8) 

If N media with the same angular width are located beside to the wire, then 𝜀е =
1

𝑁
∑ 𝜀𝑚.𝑁

𝑚=1     

It is important to mark that in plane-parallel field the potential on the surface of each antenna wire (for 

example, on each dipole arm) is constant. The potential at an infinitely distant point is not equal to zero, 

otherwise at finite distances from the radiator it will turn out to be infinitely large. At the same time, it 

is obvious that when moving in the plane 𝑧 = 𝑐onst away from the antenna axis, the potential is the 

same in all directions. Therefore, it should be assumed that lines of equal potential are circumferences 

with centers at the origin, and surfaces of equal potential are circular cylinder. 

As it follows from expression (7), in the case of a homogeneous medium, the magnitude of the electric 

field flux between two wires depends on the angle φ between the lines of equal potential passing 

through the axes of the wires. As shown in Figure 5a, in the case of a symmetrical two-wire long line 

located in free space, the lines 1 of equal potential are arcs of a circumference, and the angle between 

them near each wire 2 is close to 𝜋. This means that half of the field flux passes through the volume 

bounded by the dotted line. Half of the energy between the wires of the long line passes through the 

same volume. In the case of a single-wire long line located above a perfectly conducting ground 3, 

contrary to popular opinion, only half of the energy passes through the space bounded by a semicircle. 

The second half passes through the surrounding space. These simple facts are given in (Levin, 2021). If 

the wire is located at the boundary of two media with the same form and volume (Figure 5c), then half 

of the energy passes through each medium, Δ𝜑1 = Δ𝜑2 = 𝜋, and as can be seen from (8), 𝜀е =

0.5(𝜀0 + 𝜀1). 

 



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Figure 5. Cross-sections of Two-wire (a) and Single-wire (b) Long Lines and a Wire between Two 

Media (c) 

 

In the case of a microstrip antenna, an approximate equivalent of its simpler version (monopole) is 

shown in Figure 4. In this case, the field structure is plane-parallel, since it is formed by two parallel 

horizontal surfaces of zero potential. As can be seen from the figure, the flux of the electric 

displacement is divided into two parts by a horizontal line coinciding with the axis of the antenna metal 

strip. This means that in this case, as in the case of Figure 5c, Δ𝜑1 = Δ𝜑2 = 𝜋, i.e., the fluxes through 

the air and the substrate are the same and the equivalent permittivity 𝜀𝑒 of the medium surrounding the 

radiator is equal to half sum of the permittivity of air and a substrate. Respectively the propagation 

constant of the current wave along the upper and lower plate surfaces is the same and equal to 

𝑘𝑒 = 𝑘√𝜀𝑒 , i.e., reactive and active components of the input impedance of the microstrip antenna 

shown in Figure 1a, are equal to 

𝑋𝐴1 = −𝑊 cot 𝑘𝑒𝐿 , 𝑅𝐴 = 80 (
𝑘

𝑘𝑒
tan

𝑘𝑒𝐿

2
)
2

.                      (9) 

In order to experimentally verify the validity of the proposed method of calculating the equivalent 

permittivity of a medium next to a microstrip antenna, we will use the results of the development of a 

microstrip antenna with circular polarization (Barbakadze, Tabatadze, Petoev, & Zaridze, 2018). As 

known, a circular polarization compared to linear polarization has a number of advantages in the 

transmission and reception of radio signals. In this paper it was considered a square microstrip antenna 

excited by a coaxial cable (Figure 6). It has become significant step forward in expanding the 

employment of microstrip antennas. 

 



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Figure 6. General View of the Antenna: 1 - Metal Plate, 2 - Substrate, 3 - Grounding, 4 –Cable,  

5 - Cut Corners 

 

The main option of the antenna was designed to operate at the frequency 2.49 GHz. The dimensions of 

the antenna model (in meters) are given in the figure. The thickness of the metal plate (strip) is 

approximately 0.01 mm. Two corners of the conductive strip were cut off in order to obtain circular 

polarization. The circular polarization is created by the rotating current and depends on the depth of the 

cut off. This depth is selected experimentally and is equal to 0.0028 m., The thickness of the substrate 

is 3.48 mm, its relative permittivity is 𝜀𝑟 = 6.8. To optimize the model performance at the specified 

frequency, the cable connection point is shifted from the center and located at a distance of 0.0242 and 

0.2095 m from the left and bottom edges of the grounding. The cable radius is 1.6 mm.  

The paper confirms the possibility of creating a circularly polarized antenna with sufficiently high 

electrical characteristics: the antenna SWR at the fundamental frequency is 1.5, the 6 dB bandwidth is 

equal to 0.13 GHz, and the axial ratio is 1.27 db. 

Let us apply to this antenna the described method of calculating characteristics. The equivalent 

permittivity of the antenna substrate is equal to 𝜀е = (6.8 + 1) 2⁄ =3.9. Since frequency is 𝑓 = 2,49 ∙

109, the wave length is 𝜆 = 𝑐 𝑓⁄ = 0.12 m, and slowing is 𝑛 = √𝜀е =1.97, then the resonant length 

of the antenna is 𝐿 = 𝜆 (2√𝜀𝑒)⁄ = 0.03 m. Its geometric length, i.e., the diagonal of the square, 

taking into account the cuts, is somewhat less - 0.026 m. Therefore, it had to slightly increase this 

length by shifting the excitation point. This result serves as a clear confirmation of the validity of the 

described technique and shows that the substrate even of a small thickness significantly increases the 

equivalent permittivity of the environment. 

 

 

 



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4. Dependance of Antenna Characteristics on Current Distribution  

In the article it was already said about the importance of expanding the frequency range and improving 

the characteristics of microstrip antennas. The solution of these problems primarily depends on the 

nature of the current distribution along a particular antenna. This Section is devoted to comparing the 

results of applying known distributions. 

We will calculate the radiation resistance of the antennas by means of the method of the Poynting 

vector (Poynting, 1884). In accordance with this method the radiation resistance 𝑅Σ is determined as 

the result of dividing the radiated power 𝑃  by the square of the antenna input current 𝐽𝐴(0): 

𝑅Σ = 𝑃 𝐽𝐴
2(0).⁄  At that the power is defined as the sum of fields fluxes through the surface of a sphere 

of infinitely large radius 𝑅0, the center of which coincides with the antenna center. Such a radius 

allows us to assume that the field passing through the sphere is flat, i.e., the ratio of the electric and 

magnetic fields is equal to 𝐸 𝐻⁄ = 𝑍0, where 𝑍0 = 120𝜋 is the wave impedance of free space. This 

means that the power flux 𝑃 through the unit area of the sphere S is equal to 𝑃/𝑆 = 𝐸𝜃
2/𝑍0. Here 𝐸𝜃 

is the projection of the electric field strength onto the sphere surface. 

In order to determine the power flux through the surface of the sphere, it is necessary to calculate the 

integral of the value 𝐸𝜃
2 over the angles θ and φ. Integration over φ is reduced to a multiplication by 

2π, since 𝐸𝜃 does not depend on the angle φ. Integration over the angle θ is a laborious problem, 

which is solved, as a result of multiple integration by parts.  

The used further method of calculation is based on two simplifications. Firstly, the expression for the 

electric field of a vertical radiator at an arbitrary angle θ is complex, while the analogous formula for 

the field in a direction perpendicular to the radiator is much simpler. Secondly, the antenna dimensions 

are small compared to the infinite radius of the spherical surface, i.e., the distance from any point of the 

antenna to any point of this surface is the same. Therefore, the field created by the antenna on this 

surface at an angle θ is equal to 𝐸𝜃 = 𝐸𝜃(𝜋 2⁄ ) sin 𝜃, and the power flux in the indicated direction is 

proportional to 𝐸𝜃
2. That’s why the integral of 𝐸𝜃

2 over the entire spherical surface in the spherical 

coordinate system is 

∫ 𝐸𝜃
2 sin 𝜃 𝑑

𝜋 2⁄

0

𝜃 = ∫ 𝐸𝜃
2 (𝜋 2⁄ )sin3 𝜃𝑑𝜃.

𝜋 2⁄

0

 

This result is easy to verify and confirm by a well-known example. In accordance with the above 

expressions, the radiation resistance of the antenna is 

𝑅Σ = 4𝜋𝑅0
2 ∫

𝐸𝜃
2

𝑍0𝐽𝐴
2(0)

𝜋 2⁄

0
sin𝜃𝑑𝜃 = 𝑅0

2 ∫
𝐸𝜃

2(𝜋 2⁄ )

30𝐽𝐴
2(0)

sin3𝜃𝑑𝜃.
𝜋 2⁄

0
            (10) 

The effective length of a dipole with an arm length 𝐿 = 𝜆 4⁄  is equal to 𝑙𝑒 = 2 𝑘,⁄  i.e., in the 

horizontal direction 𝐸𝜃(𝜋 2⁄ )𝑅0 = 60, and we obtain the well-known value of 𝑅Σ: 

𝑅Σ = 120∫ sin3𝜃𝑑𝜃 = 120 ∙
2

3
= 80

𝜋 2⁄

0
 0hm. 

In accordance with this technique, we will consider different options of current distribution along the 

radiator. Let’s start with the usual sinusoidal distribution. In this case the electric field of a symmetrical 



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linear radiator is  

𝐸𝜃 = 𝑗
60𝐽𝐴(0)

sin𝑘𝐿

cos(𝑘𝐿 cos𝜃)−𝑐𝑜𝑠𝑘𝐿

sin𝜃

𝑒𝑥𝑝(−𝑗𝑘𝑅0 )

𝜀𝑟𝑅0 
.                     (11) 

In the horizontal direction, the field is equal to 

𝐸𝜃(𝜋 2⁄ ) = 𝑗60𝐽𝐴(0)
1−𝑐𝑜𝑠𝑘𝐿

sin 𝑘𝐿

𝑒𝑥𝑝(−𝑗𝑘𝑅0 )

𝜀𝑟𝑅0 
=

𝑗30𝐽𝐴(0)𝑘𝑙𝑒
𝜀𝑟𝑅0 

. 

In this case the value 𝑙𝑒 , which proceeding from the analogy with the value 2L in expression for the 

field of an elementary dipole is called the equivalent effective length, is 𝑙𝑒 =
2

𝑘
tan

𝑘𝐿

2
. And in 

accordance with (10) and (11) we get 

𝑅Σ = 20𝑘2𝑙𝑒
2.                                  (12) 

When 𝐿 > 𝜆 4,⁄  the value 𝑅Σ  grows with increasing L, but the expression (12) becomes very 

approximate. Along with the growth of 𝑅Σ the reactive component of the input impedance increases, 

and the input current of the antenna decreases. At 𝐿 > 𝜆 2⁄  𝑅Σ begins to decrease, since segments with 

anti-phase current appear along the antenna. At 𝐿 >
𝜆

4
 the directivity D starts to grow. But an increase of 

D can lead to a decrease in a communication range, if the signal in a horizontal direction decreases. The 

pattern factor PF was introduced as an indicator of such a result. It is equal to the average level of 

radiation under angles θ from 60° to 90°. When 𝐿 < 𝜆 4,⁄  𝑃𝐹1 = ∫ sin 𝜃𝑑𝜃 = 0.5.
𝜋 2⁄

𝜋 3⁄
 With growth of 

L this factor begins to decrease, sharply limiting the operation range of frequencies. 

In a linear antenna with a constant surface impedance of an inductive nature, the current distribution 

remains sinusoidal, but the wave slows down, and the propagation constant 𝑘1 increases. Calculations 

show that for the same geometric length, the radiation resistance of an antenna with a constant inductive 

impedance is somewhat greater than the radiation resistance of a metal antenna, and for the same 

electrical length, it is much less.  

In-phase current distributions created by capacitive loads connected at regular intervals along the entire 

length of the antenna are of particular interest. In the case of a linear distribution, the current amplitude 

decreases uniformly from the generator to the free ends of the radiator. The electric field of a vertical 

antenna with allowance for the mutual compensation of components whose signs are opposite in 

different arms of a symmetrical radiator is calculated as 

𝐸𝜃 = 𝑗30𝑘 𝐽𝐴(0) sin 𝜃 𝑒𝑥𝑝(−𝑗𝑘𝑅0 ) (𝜀𝑟𝑅0 )⁄ ∙ ∫ 𝑓1(|𝑧|)𝑒𝑥𝑝(𝑗𝑘𝑧 cos 𝜃)𝑑𝑧.
𝐿

−𝐿

 

Here 𝑓1(|𝑧|) = 1 − |𝑧| 𝐿⁄  is a symmetric function of current distribution along the radiator. 

Respectively 

𝐸𝜃 = 2𝐴∫ (1 − 𝑧 𝐿⁄ ) cos(𝑘𝑧 cos 𝜃)𝑑𝑧,
𝐿

0

 

where 𝐴 =  𝑗30𝑘 𝐽𝐴(0) sin 𝜃 𝑒𝑥𝑝(−𝑗𝑘𝑅0 ) (𝜀𝑟𝑅0 )⁄ . As a result of integration, 

𝐸𝜃 =
2𝐴

𝑘2𝐿𝑐𝑜𝑠2𝜃
[1 − cos(𝑘𝐿 cos 𝜃)]. 



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If θ→0, and 
1−cos(𝑘𝐿 cos𝜃)

𝑐𝑜𝑠2𝜃
→ 0.5𝑘2𝐿2,  we get 𝐸𝜃(𝜋 2⁄ ) = 𝑗30𝑘𝐿𝐽𝐴(0) 𝑒𝑥𝑝(−𝑗𝑘𝑅0 ) (𝜀𝑟𝑅0 )⁄ ,  i.e., 

𝑅Σ = 20𝑘2𝐿2. In this case, in contrast to (12), the value 𝑅Σ grows continuously with increasing L. At 

that the values D and PF do not change, i.e., with an increase in the radiator length the field and the 

radiation resistance grow within the frequency band, in which the in-phase distribution is preserved. 

In the case of an exponential in-phase current distribution along the radiator, when the distribution 

function is equal to 𝑓2(|𝑧|) =
𝑒−∝𝑧−𝑒−∝𝐿

1−𝑒−∝𝐿 , the field is 

𝐸𝜃 =
2𝐴

1 − 𝑒−∝𝐿
∫ [𝑒−∝𝑧 − 𝑒−∝𝐿]

𝐿

0

cos(𝑘𝑧 cos 𝜃)𝑑𝑧. 

As a result of integration, we find 

𝐸𝜃 = 
2𝐴 ∝ 𝑘 cos 𝜃

(1 − 𝑒−∝𝐿)(∝2+ 𝑘2𝑐𝑜𝑠2𝜃)𝑘 cos 𝜃
∙ {1 − 𝑒−∝𝐿[cos(𝑘𝐿 cos 𝜃)+∝ sin(𝑘𝐿 cos 𝜃)/(𝑘 cos 𝜃)]}. 

If ∝ is small, 𝐸𝜃 = [2𝐴 (𝑘2𝐿)𝑐𝑜𝑠2𝜃⁄ ][1 − cos(𝑘𝐿 cos 𝜃)].  

As θ→0, 𝐸𝜃(𝜋 2⁄ ) → 𝑗30𝑘𝐿𝐽𝐴(0)
𝑒𝑥𝑝(−𝑗𝑘𝑅0 )

𝜀𝑟𝑅0 
, i.e., as expected, the value 𝑅Σ coincides with (11). If 

∝ is not small, then, taking into account that cos (𝑘𝐿 cos
𝜋

2
) = 1, we obtain: 

𝐸𝜃(𝜋 2⁄ ) =  𝑗60
𝑘

∝
𝐽𝐴(0)

𝑒𝑥𝑝(−𝑗𝑘𝑅0 )

𝜀𝑟𝑅0 
, 

and therefore  

𝑅Σ = 80𝑘2 ∝2⁄ .                                (13) 

Figures 7 and 8 show the results of characteristics calculating for asymmetrical linear metal radiators 

(monopoles) with the same dimensions depending on the current distributions: sinusoidal (1), in-phase 

with a linear distribution (2), in-phase with an exponential distribution at ∝= 1 𝐿⁄  (3) and ∝= 4 𝐿⁄  

(4). The radiators with an arm length 1 m and a radius 0.01 m are excited by the cable with a wave 

impedance 60 Ohm. The fields and the radiation resistances of radiators are given correspondingly on 

Figures 7a and 7b, the reactive components of input impedances for different current distributions are 

shown on Figure 8a. As is known, an operation frequency band depends on a level of travelling wave 

ratio (TWR). The magnitudes of TWR are presented in Figure 8b. 



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Figure 7. Fields (a) and Radiation Resistances (b) of Antennas with Different Current 

Distributions 

 

 

Figure 8. Input Reactance (a) and TWR (b) of Antennas with Different Current Distributions 

 

The wave impedance of this radiator with sinusoidal current distribution is 𝑊1 = 120 (ln 2𝐿 𝑎⁄ − 1) =

516 Ohm. 

A reactive component of its input impedance is equal to 𝑋𝐴1 =- 𝑊1 cot 𝑘𝐿.  The inclusion of 

capacitive loads leads to a sharp decrease in the inductance per unit length of the wire and the 

propagation constant 𝑘1 along this wire. As a result, the wave impedance 𝑊2 of an antenna with 

capacitive loads is decreased by √𝑘1/𝑘 times. In the author’s book (Levin, 2019) it is erroneously 

stated that this impedance coincides with 𝑊1 .  Therefore, the reactive component of an input 

impedance of a radiator with a linear in-phase current distribution is 𝑋𝐴2 =- 𝑊2 (𝑘𝐿)⁄ . 

In the case of an exponential in-phase distribution the reactive component of the input impedance is 

𝑋𝐴 =- 𝑊3,4 (𝑘𝐿)⁄ , where the wave impedance 𝑊3,4 is given depending on ∝ by the expression 

𝑊3,4 = 0.5 ∝ 𝐿𝑊2[1 + cot(∝ 𝐿 2⁄ )]. 

 



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The formulas for calculating the radiation resistance of the compared radiators and their electric fields 

along the perpendicular to the antenna axis are given above. 

The obtained results show that antennas with in-phase current distribution permit in the transmit mode 

to provide the high electrical characteristics in the wide frequency range, including the range, which 

used for operation with sinusoidal distribution. They confirm advantages of applicating the in-phase 

current distribution. Other options of loads allow to operate in different ranges using short waves, 

which length is much less than the length of an antenna arm, i.e., at frequencies where a sinusoidal 

distribution is not applicable. 

At the same time, with increasing frequency, the signal level of the vertical radiator at angles close to 

the earth’s surface, grows with the general growth of the field. This circumstance is very important for 

ensuring reliable radio communications over long distances. These properties are equally valid for both 

electric and magnetic radiators. 

Antennas with switchable loads make it possible to operate in all need ranges. 

The self-complementary antennas consisting of magnetic and electric radiators located side by side on a 

common surface and filling this entire surface deserve special consideration. In this case, the input 

impedance of each radiator is purely active, does not depend on frequency in the operating range and is 

equal to 60π. These characteristics, as is usually stated, are partially preserved at smaller sizes of the 

structure. Attentive analysis shows that this situation is valid at high frequencies, more precisely, when 

the wavelength of the radiated signal is half the dimension of the occupied surface. 

Typical input characteristics of symmetrical flat complementary antennas with an arm length L 

depending on the wavelength 𝜆, taken from the book (Levin, 2019) are shown in Figure 9. Curves 1 

are given for an antenna consisting of one metal and one slot radiator with the identical angular width 

𝜋 2⁄ , curves 2 are given for two antennas with the same angular width, which are connected to the 

generator poles in parallel to each other. It can be seen from the figure that as the frequency increases 

(wavelength decreases), the curves slowly converge to known limit values: the active resistance of each 

radiator converges to 60π, the reactive impedance - to zero. This limit is realized when 𝜆 𝐿 ≈ 1,⁄  i.e., 

at a wavelength equal to half the length of the radiator. 

As follows from the foregoing, radiators with in-phase current distribution included in the 

self-complementary structure differs from single radiators with a similar current distribution by a purely 

active input impedance. 

 



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Figure 9. Active (a) and Reactive (b) Components of the Input Impedances of Flat 

Self-complementary Antennas 

 

5. Expansion of Frequency Range 

Obtained results allow us to analyze the performances of microstrip antennas and suggest methods for 

improving their characteristics. Their main disadvantage is low efficiency and narrow frequency range. 

The small thickness of the substrate leads to the losses in the substrate and reduces efficiency. 

Despite the individual structural elements that distinguish microstrip antennas from other radiators, and 

features of electrical characteristics caused by these elements, a sober look at the operation of these 

antennas shows that in principle they little differ from ordinary radiators. And therefore, to improve 

their performance, one must use the usual methods. The current in a microstrip antenna is distributed 

along a sinusoid, as in a conventional linear radiator.  

A linear antenna with a sinusoidal current distribution is equivalent to a resonant circuit operating in a 

narrow frequency range. To expand the frequency range of such a structure, it is necessary in particular 

to lower its Q factor, i.e., increase the resistance of radiation and losses. In the case of a microstrip 

antenna, this method can be implemented by using a two-story structure, in which the upper antenna 

(second floor) connected to the main (first floor), provides additional radiation and partially extends the 

operating range (see, for example, Croq & Pozar, 1991). But a significant expansion of the operating 

range is possible only by creating in a linear radiator the in-phase current distribution with the help of 

capacitive loads. 

Capacitive loads, located along the axis of a linear electrical radiator, make it possible to provide over 

the entire its length a current close to the in-phase in a wide frequency range. Amplitude of in-phase 

current decrease toward antenna ends in a linear or exponential law. As a result, constant loads allow 

several times to expand a frequency range with a required matching level. Changing of loads by means 

of their switching greatly improve this result. In accordance with the duality principle the inductive 

loads connected along the slot provide similar result in a magnetic radiator. 

This design can be applied in a microstrip antenna. Figure 10 shows three options of such an antenna 

with capacitive and inductive loads located along its axis. The capacitive loads are installed on a metal 



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plate 1 and are made in the form of transverse slots 5 with metal selvages 6 (see Figure 10a) or with 

capacitors 7 (Figure 10b). The inductive loads are made in the form of metal tapes 8 and are placed on 

the substrates’ sides 4. The third option (Figure 10c) makes it possible to switch capacitive loads and 

change operating ranges. 

 

 

Figure 10. Designs of Microstrip Antenna: First Option (a), Second Option (b), Third Option (c) 1 

– metal strip, 2 – ground plane, 3 – substrate, 4 – side of substrate, 5 – transverse slot, 6 – metal 

selvage, 7 – capacitor, 8 – transverse strip 

 

The use of capacitive loads in an electric radiator of the microstrip antenna makes it possible to expand 

the operating range and significantly improve the characteristics of the antenna. Calculating the wave 

impedance of a metal radiator, identical to the slot in shape and size, performed in Section 2, shows that 

magnetic radiator of the microstrip antenna at the currently used sizes has little effect on the magnitude 

of the radiated signal. To improve the characteristics of the magnetic radiator, it is necessary to increase 

the thickness of the microstrip antenna. 

 

6. Conclusion  

Microstrip antennas are widely used in various objects. Sufficiently simple changes in their design can 

significantly improve their characteristics and expand the possibilities of application. 

 



www.scholink.org/ojs/index.php/asir             Applied Science and Innovative Research                  Vol. 6, No. 3, 2022 

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Published by SCHOLINK INC. 

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