ELEE 5318/6352

Special Problems-Part I

Hamed Rahmani

Problem 1

1 Introduction

Dipole antenna is the simplest and widely used antenna in telecommunication applications whichconsists of two identical conductive metal wires. Dipole antenna performance is dependent on manyparameters such as metal conductivity, length, radius of wire, surrounding medium, feeding pointand etc. but, In this article, we are studying the effect of length and radius on the performance ofthese kind of antennas. In the following sections, we describe principles underlying dipole antennaoperation and introduce the most important performance metrics used for characterizing it. Inputpower to the antenna can be calculated knowing the antenna input impedance and source outputimpedance. In this article, we study the effect of length on input impedance of a dipole antenna.

2 Dipole Antenna Overview

Antenna is a electric device that converts electrical power to radio waves and vice versa. It radiateselectromagnetic waves generated by the source into free-space. For dipole antenna, the radiationmechanism is easy to understand. The center-fed dipole antenna is derived by a Alternative Current(AC) source which established an AC current in two cords. Traveling Electric field lines establishedbetween two cords of a dipole antenna create a close loop with the new electric fields which haveopposite direction in the next cycle of AC source and detach from the antenna. Therefore, the mainmechanism for dipole antenna radiation is movement of the free electrons on the conductors , asin all other types of antenna. It is worthwhile to mention that antennas are completely reciprocaldevices. It means that presence of same electromagnetic field between two cords of dipole antennaestablishes same voltage difference and AC current between dipole antenna terminals. Because ofantenna reciprocity, we only study dipole antenna in transmit mode.

2.1 Current Distribution a Thin Dipole Antenna

Movement of electrons on the conductors creates a current distribution on a linear dipole. Subse-quently, it leads to radiation of electromagnetic waves into-free space. The current distribution on

1

a linear antenna can be calculated by knowing the source frequency and dipole length. Knowingthe current distribution, we can calculate electric and magnetic field at any point in the environ-ment except the points located on the z axis. Following the same terminology as in the textbook,we categorized dipole antennas into three different types based on the length (l) compared to thewavelength showed as λ.For infinitesimal dipole (l < λ/50), the current is constant over dipole length as shown in Figure 1.

Figure 1: Current distribution of an infinitesimal dipole antenna over length.

Dipole antenna with length of l (λ/10>l>λ/50 ) are categorized as small dipoles and currentdistribution over length is a triangular distribution with maximum value at the center. Finite lengthdipoles length is grater than λ/10 and the current distribution over length is a sinusoidal functionof horizontal location as shown in equation 1.

Ie(x′ = 0, y′ = 0, z′) = azI0sin([k( l2− | z′ |])(1)

Where k is wavenumber of the wave coming form the source and is calculated as 2π/l. Thecurrent distribution of multiple dipoles with different lengths is shown in figure 2.

2.2 Performance Parameters

Generally, antennas are used to transmit and receive radio waves. In order to design most optimumantenna for a particular applications, we need to describe antenna performance using different per-formance parameters and optimize them regarding to the requirements. In the following sections, we

2

Figure 2: Current distribution of multiple dipole antennas with different length.

study some of the most importance performance parameters of a dipole antenna assuming far-fieldregion operation.

2.2.1 Radiation Pattern

As describe in the introduction part, a transmitter antenna converts electrical energy to radio wavesand send it into free-space. The radiated energy is confined in traveling electromagnetic fields anddecays as the distance from the antenna increases. Radiation pattern is defined as a mathematicalor graphical representation of the radiation properties of the antenna as a function of space coordi-nates. Dipole antenna is an omnidirectional antenna which means the radiated power is constant inazimuth plane but it changes with elevation angle. Radiation intensity is defined as radiated powerform an antenna per unit solid angle and it is expressed in equation 2 for a dipole antenna.

U = η |I0|2

8π2 [cos( kl

2 cos(θ))−cos(kl2 )

sin(θ) ]2for0<φ<2π, 0<θ<π (2)

According to equation 2, the radiation intensity for a dipole antenna is a function of antennalength and elevation angle and it is independent of azimuth angle which is consistent with the fact

3

that dipole is a omnidirectional antenna. Radiation pattern for dipole antenna with four differentlength is plotted in figure 3. Plotting the radiation pattern shows that it is composed of multiple

Figure 3: Radiation Intensity pattern for four different lengths of dipole antenna.

parts known as radiation lobes. The largest lobe is called major lobe and the other lobes are knownas minor lobes.Radiation pattern is a symmetric function with respect to θ and and the number oflobes is a even number.Depending on the length of the antenna, radiation pattern may contains twoor more than two major lobes and it may no minor lobes. Usually, minor lobes are undesired fordesigners because some amount of power is radiated in an unintended direction. Side lobe level is agood parameter to obtain valuable insight about how much power is confined in the major lobes. Itis defined as the amount of largest minor lobe over the value of major lobe and is usually reported indB. Half Power Beam width (HPBW) is another important metric in antenna design and is defined asthe difference in the angles in which the radiation pattern value is half of its maximum. Consideringfigure 3, HPBW decreases when the ratio of length over wave length increases. the angles in which theradiation pattern has zero value are called nulls angles in which radiation pattern has its maximumvalue are known as maxima. No need to mention that the number and value of maximas and nullsand value of HPBW and SLR changes with variation in dipole length. Figure 4 shows HPBW versusl/λ for the ratio from 0.02 to 20 with 0.02 increment in ratio. A fitting curve that is close enough tothe actual data is obtained by MATLAB basic fitting data tools and is demonstrated in the figure.

4

Figure 4: HPBW Vs. l/λ for a dipole antenna and 6th order fitting curve.

SLR variation with length is depicted in figure 5. Radiation pattern nulls and maximas canbe calculated from equation 2. Derivation with respect to θ and equating to zero leads to findingmaxima points and equating the equation to zero gives us the nulls. The total number of nulls andmaximas in radiation pattern and the locations change with l / λ ratio as depicted in figure 6.

Fig 6 can be formulated as below.

#Nulls = #Maximas =

2 + 2( lλ − 1)for lλ ∈ N

2 + 4[ lλ − 1]for lλ /∈ N(3).

The exact value of the radiation pattern maxima and null points for a dipole antenna withlλ ∈ N is shown in table 1. Based on the values in table 1, we introduce an approximation methodto estimate these numbers. Table 1 contains maxima and nulls in half plane (0 ≤ θ ≤ π) becausedipole antenna radiation pattern is symmetric with respect to θ = π.

Assume that lλ = M,M ∈ N, The half-plane nulls and maximas can be calculated form equation

4 and 5. which is an approximation of actual values in table 1. θn and θm stand for representing

5

lλ half-plane maximas ◦ half-plane minimas ◦

1 90 0 , 1802 57.43,122.56 0,90,1803 45.8,90,134.18 0,70.52,109.47,1804 39.25,75.31,104.68,140.74 0,60,90,120,1805 34.9,66.17,90,113.82,145.09 0,53.13,78.46,101.53,126.86,1806 31.73,59.74,80.35,99.65,120.25,148.26 0,48.18,70.52,90,109.47,131.81,180

Table 1: Radiation Pattern maximas and nulls for a dipole antenna which its l/λ is an integerbetween 1 and 6.

nulls and maximas respectively.

θ◦m(K) = 180KM+1 ,∀K = 1, ...,M (4)

θ◦n(K) = 90(2K+1)M+1 ,∀K = 2, ...,M − 1 (5)

Dipole antenna radiation pattern is always equal to zero for θ = 0 , θ = π and equation 5 resultsin finding other half-plane nulls. The accuracy of this equation decreases as the ratio increases andit can not be used for dipole antennas have high l/λ ratio. Figure 7 and 8 show the variation ofmaximas and nulls location in a dipole radaition pattern for different values of l

λ .

Figure 5: Minus SLR vs length for a dipole antenna.

2.2.2 Directivity

Directive is another fundamental parameters of an antenna and it measures how directional an an-tenna’s radiation pattern is. It is defined as the ratio of the radiation intensity in a given directionfrom antenna to the average over all directions.

6

Figure 6: Total Number of Nulls and Maximas in Radiation Pattern Vs. l/λ for a Dipole Antenna .

D = UU0

= 4πUP rad

(6)

Usually, the direction is not specified and directivity term implies on maximum dirctivity of theantenna which is defined in equation 7.

Dmax = D0 = U |maxU0

= 4πUmaxP rad

(7)

Directivity is reported as unit less number or in dB which is 10 times of directivity value’slogarithm. It is evident that directivity is also dependent on the the length of antenna because it isderiven from radaition pattern. Figure 9 shows maximum directivity variation with respect to l/λratio. In order to have a better understanding of directivity and radiation pattern, Directivity offour dipole antennas which have same length as in figure 3 is plotted in figure 10.

According to figure 10, dipole antenna directivity increases as its length increases. It is consistentwith the result obtained from HPBW plot because length increment causes narrower HPBW whichis equivalent to high directional antennas. Three dimensional illustration of radiation pattern forthe length values in figure 3 is shown in figure 11.

2.3 Conclusion

In this article, we reviewed dipole antenna operation principles and studied the effect of lengthon current distribution in dipole and consequence effects on radiation pattern and directive. Weintroduced a new formula for estimating number of lobes and location of maximas and nulls inradiation pattern. The effect of changing dipole antenna length on current distribution, SLR, numberof lobes, HPBW, nulls and maximas was discussed and showed in different figures in correspondingsection.

7

Figure 7: Radaiation Pattern Maximas for a Dipole antenna with different length

8

Figure 8: Radaiation Pattern nulls for a Dipole antenna with different length

9

Figure 9: Maximum Directivity vs l/λ in a Dipole antenna

Figure 10: Directivity Pattern of dipole antenna with four different lengths.

10

Figure 11: Three Dimensional Radiation Pattern of a Dipole antenna with four different lengths.

11