yagi uda antenna

Yagi-Uda Antenna

by

Dong Xue

Department of Engineering Mechanics

May 5th, 2002

1

Contents

1 Introduction 4

2 Motivation and objectives 5

3 Problem setup and analysis 63.1 Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 NBS design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Matlab implementation 104.1 Structure of the code . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Highhight of the code . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Simulation results 125.1 E-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 H-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Complex current distributions . . . . . . . . . . . . . . . . . . . 155.4 Characteristic variables . . . . . . . . . . . . . . . . . . . . . . . 30

6 Conclusion 31

A List of Routine 32A.1 yagi.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.2 func.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.3 func2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2

List of Figures

1.1 Geometry of Yagi-Uda array . . . . . . . . . . . . . . . . . . . . . 4

3.1 Configure of 15 element NBS Yagi antenna . . . . . . . . . . . . . . 9

5.1 The E-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 The H-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 The current distribution on the reflector N=14 . . . . . . . . . . . . . 155.4 The current distribution on the feeder N=15 . . . . . . . . . . . . . . 165.5 The current distribution on the director N=1 . . . . . . . . . . . . . . 175.6 The current distribution on the director N=2 . . . . . . . . . . . . . . 185.7 The current distribution on the director N=3 . . . . . . . . . . . . . . 195.8 The current distribution on the director N=4 . . . . . . . . . . . . . . 205.9 The current distribution on the director N=5 . . . . . . . . . . . . . . 215.10 The current distribution on the director N=6 . . . . . . . . . . . . . . 225.11 The current distribution on the director N=7 . . . . . . . . . . . . . . 235.12 The current distribution on the director N=8 . . . . . . . . . . . . . . 245.13 The current distribution on the director N=9 . . . . . . . . . . . . . . 255.14 The current distribution on the director N=10 . . . . . . . . . . . . . 265.15 The current distribution on the director N=11 . . . . . . . . . . . . . 275.16 The current distribution on the director N=12 . . . . . . . . . . . . . 285.17 The current distribution on the director N=13 . . . . . . . . . . . . . 29

3

Chapter 1

Introduction

Yagi-Uda Antenna is a parasitic linear array of parallel dipoles, see Fig 1.1, one ofwhich is energized directly by a feed transmission line while the other act as par-asitic radiator whose currents are induced by mutual coupling. The basic antennais composed of one reflector (in the rear), one driven element, and one or more di-rectors (in the direction of transmission/reception).The Yagi-Uda antenna has re-ceived exhaustive analytical and experimental investigations in the open literatureand else where. The characteristics of a Yagi are affected by all of the geometricparameters of the array. Usually Yagi-Uda arrays have low input impedance andrelatively narrow bandwidth.Improvements in both can achieved at the expense ofothers. Usually a compromise is made, and it depends on the particular design.

s i+1

2a

s i

nl

R(x’, y’, z’)

(x, y, z)

x

y

z

Driven

ElementReflector

21 3 4 5 . . . . . . N

Directors

Figure 1.1: Geometry of Yagi-Uda array

4

Chapter 2

Motivation and objectives

Often one needs to improve reception of a particular radio or television station.One effective way to do this is to build a Yagi-Uda (or Yagi) antenna because oftheir simplicity and relatively high gain.

The goal of the project is to simulate an NBS yagi antenna which covers allthe VHF TV channels. we will calculate and plot

� Far-zone field (both E-plane and H-plane)

� 3-db beamwidths

� Front-to-back ratio

� Directivity

� Complex current distribution on each element

� Input impedance.

We choose frequency��

. Why? Because the VHF TV channelstarts with 54MHz(channel 2) and ends with 216MHz (channel 13). Antennas’gains rise slowly up to the design frequency and fall off sharply thereafter [5]. Itis therefore easier to make the design frequency a little higher than desired.

5

Chapter 3

Problem setup and analysis

In this Chapter, we will introduce formulations and parameters used in my simu-lations.

3.1 Formulations� Pocklington’s Integral Equation

Pocklington’s Integral Equation is used on finite diameter wires. We havethe field equation [1]: �

��

(3.1.1)

where

� �� is the total electric field.

� �� is the scattered electric field

produced by the induced current current density � .

� � �� is the incident

electric field.

We get derive Pocklington’s integral equation as followings,�� "!#�%$ �$ � � �'& � ��( ��)+*#,-/.10 ��243 �� '5/6%798 �� (3.1.2)

with0 �;: ��<>=?< � � � �@��AB=CA � � � �D�E�F=C� � � � .

� Fourier expansion of the current

GIH �E��J� � KLMON P G M HRQTS�UWV � ��X = � � . � �Y H[Z6

whereG M H

represents the complex current coefficient of modeX

on element� andY H

represents the corresponding length of the � element. Pocklinton’sintegral (0.02)reduces to� KM N P G M H�� += � � M � P � ��X = � � .Y H � �� <�� < ��9A� A � � Y H

�� V & � = � ��X = � � � . �Y �H Z � ��

� � �� <�� < � �9A� A �� H � QTS/U>V � ��X = � � . � �HY H Z 3 � �H�� 5 -/. 6%7 �#8�� (3.1.3)

where

� �� <�� < ��9A�� A �� #��H � � ( ��) *#,�� ( ��) *#,�� (3.1.4)

and �� : � <W=?< � � � �D��AB=CA � � � �� D��'� � � � (3.1.5)

where � � � � � �! ��#"#"#"$�!%and

% �total number of elements.

0 �is the

distance from the center of each wire radius to the center of any other wire,as shown in Fig. 1.1

� Method of Moments We use Method of Moments to obtain the complexcurrent coeffient

G M H.

1. On driven element

– The matching is done on the surface.

–8 �� '&

at� = �

points.

– The�)(+*

equation on the feed element isKLM N P GTH M �E� � �,& �.- H N�/ � �

2. On all other element

– The matching is done at the center of the wire

7

–8 �� E� � � � � � &

� Far-Field Pattern

Once the current distribution is found, the far-zone field generated by sum-ming the contributions from each.8 � �

/L H N P 8 � H � = 5R6��

� � �/L H N P � � H � =�� ( ��) *��-/.� U� � � /L H N P � ( ) *�� H �� H � � H �

� KM N P GTH M V U� � � � � �� U� � � � � �

� � Z � Y H�

with

� � � V � � X = � � .Y H � & QTS�U � Z Y H� (3.1.6)

� � � V � ��X = � � .Y H = & QTS�U � Z Y H� (3.1.7)

3.2 NBS design

A government document has been published which provides extensive data ofexperimental in investigations carried out by National Bureau of Standards [6].Wecan obtain the desired data from the government document, see Fig3.1.

� Element Number% � ��

.

� Radius of each element� � & " & &! "�

� Directors lengthY P � Y � � & " - � - � Y$# � & " - � & � Y&% � & " - &"' � Y&( � & " - & � Y&) �& " !* � Y,+ � & " !* - � Y = Y � �'& " !* &

� Reflector lengthY P % � & " - '!�

� Feeder lengthY P ( �'& " -

� Space between directors is 0.308.

8

14

0.475

x

y

Driven

Element

1 2 43 5 7615 8 10 119 12 13

Director

.390.390.390.390.390.390.394.398.424 .424

.466.420 .407 .403

Reflector

z

0.2.308 .308

2a = 0.017

Figure 3.1: Configure of 15 element NBS Yagi antenna

� Space between feeder and reflector is 0.2

The overall antenna length would be �� - " ��

. Parameters (element lengths andspacings) are given in terms of wavelength.

9

Chapter 4

Matlab implementation

In this project, I did not use any existing fancy code. Instead, I write some codeof my own, using matlab.

4.1 Structure of the code

There are three subroutines of my code. They are The data we should design are:yagi.m, func.m and func2.m,See Appendix

In the main subroutine yagi.m, the input variables are,

� The total number of elements of the array (%

).

� The corresponding lengths of each elements (Y �� ).

� The spacings between elements (U � * � � ).

The output the the following,

� Far-zone field (both E-plane and H-plane)

� 3-db beamwidths

� Front-to-back ratio

� Directivity

� Complex current distribution on each element

� Input impedance.

10

4.2 Highhight of the code

In my matlab code, the following points must be emphasized:

� My code is based on pocklington’s integral equation. The entire domaincosinusoidal (fourier) basis modes are used for each of the antenna ele-ments..

� All the elements are along the y-axis, with the driven element at the origin.

� If the effect of a mode is found on the element that it is located then distanceis the radius

�of the element, otherwise the distance is found using the

formula

3�;: ��

.

� When we calculate the radiated fields,

– E-plane� � � * & � � �� & � � �� & ��

� ' & � � �� & � � & � �– H-plane

� � * & � � �� & � �! & � �� When we calculate the directivity

� � * & � � � * & � "

11

Chapter 5

Simulation results

All the following performance figures are theoretical calculations, see Chapter 3.That means, for instance, that the actual gain will be slightly less than that given.

5.1 E-plane

We can plot far-zone Electric Field both in polar coordinate and cartesian coordi-nate. see Fig.5.1.The corresponding beamwidth and font-to-back ratio are shownin red.

12

50

100

30

210

60

240

90

270

120

300

150

330

180 0

0 50 100 150 200 250 300 3500

10

20

30

40

50

60

| E(θ

, φ=

π/2

or

3π/2

|

Observation angle θ

The E−plane

3−db beamwidth = 28.8770 degreesfront−to−back ratio = 12.0779 db

zoom in

Figure 5.1: The E-plane

13

50

100

30

210

60

240

90

270

120

300

150

330

180 0

0 50 100 150 200 250 300 3500

10

20

30

40

50

60

| H(θ

= π

/2, φ

|

Observation angle φ

The H−plane

3−db beamwidth = 30.5268 degreesfront−to−back ratio = 12.0811 db

zoom in

Figure 5.2: The H-plane

5.2 H-plane

We can plot far-zone Magnetic Field both in polar coordinate and cartesian coordi-nate. see Fig.5.2.The corresponding beamwidth and font-to-back ratio are shownin red.

14

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25166.8

166.9

167

167.1

167.2

The

pha

se o

f cur

rent

Element length zm

The current distribution on the reflector N= 14

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The

mag

nitu

de o

f cur

rent

Element length zm

Figure 5.3: The current distribution on the reflector N=14

5.3 Complex current distributions

We can get the complex current distributions(both phase and magnitude) on eachof the 15 elements, see the following figures.

15

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25166.8

166.9

167

167.1

167.2

The

pha

se o

f cur

rent

Element length zm

The current distribution on the reflector N= 14

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The

mag

nitu

de o

f cur

rent

Element length zm

Figure 5.4: The current distribution on the feeder N=15

16

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−177.84

−177.82

−177.8

−177.78

−177.76

−177.74

Element length Zm

The current distribution on the director N = 1

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

1.2

The

mag

nitu

de o

f cur

rent

Element length zm

The

pha

se o

f cur

rent

Figure 5.5: The current distribution on the director N=1

17

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.2516.225

16.226

16.227

16.228

16.229

Element length zm

The current distributions on the director N =2

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

1.2

The

mag

nitu

de o

f cur

rent

Element length zm

The

pha

se o

f cur

rent


18

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−138.45

−138.4

The

pha

se o

f cur

rent

Element length Zm

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

The

mag

nitu

de o

f cur

rent

Element length zm



19

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.2594.7

94.75

94.8

94.85

The

pha

se o

f cur

rent


−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

The

mag

nitu

de o

f cur

rent

Element length zm

Element length zm


20

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−50

0

50

100

150

200

Element length Zm


−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

The

mag

nitu

de o

f cur

rent

Element length zm

The

pha

se o

f cur

rent


21

−0.3 −0.2 −0.1 0 0.1 0.2 0.3−200

−150

−100

−50

0

50

Element length zm

The current distributions on the director N = 6

−0.3 −0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

Element length zm

The

pha

se o

f cur

rent


22

−0.2 0 0.278.138

78.139

78.14

78.141

78.142

78.143

78.144

78.145

Element length zm


−0.2 0 0.20

0.1

0.2

0.3

0.4

Element length zm

The

mag

nitu

de o

f cur

rent


23

−0.2 0 0.2−22.738

−22.737

−22.736

−22.735

−22.734

−22.733

Element length zm


−0.2 0 0.20

0.1

0.2

0.3

The

pha

se o

f cur

rent

Element length zm


24

−0.2 0 0.2−150.68

−150.66

−150.64

Element length zm


−0.2 0 0.20

0.1

0.2

0.3

0.4

0.5

Element length zm

The

mag

nitu

de o

f cur

rent


25

−0.4 −0.2 0 0.2 0.4−150.68

−150.66

−150.64

The length of Element zm

The current distributions on the director N = 10

−0.4 −0.2 0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

The length of element zm


26

−0.4 −0.2 0 0.2 0.4

Element length zm

−0.4 −0.2 0 0.2 0.40

0.1

0.2

0.3

Element length zm


27

−0.4 −0.2 0 0.2 0.4−150.345

−150.34

−150.335

−150.33

−150.325

−150.32

−150.315

−150.31

Element length zm

The current distribution on the director N=12

−0.4 −0.2 0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

The

mag

nitu

de o

f cur

rent

Element length zm

The

pha

se o

f cur

rent


28

−0.4 −0.2 0 0.2 0.469.85

69.9

69.95

70

70.05

70.1

70.15

zm


−0.4 −0.2 0 0.2 0.40

0.1

0.2

0.3

0.4

zm


29

5.4 Characteristic variables

The characteristic variables of the designed NBS antenna can be calculated andlisted in the following table.

Directivity 14.2106Front-to-back ratio in e-plane 12.0779Front-to-back ratio in h-plane 12.08113-db beamwidth in the e-plane 28.87703-db beamwidth in the h-plane 30.5268Imput impedance 27.46051ohms

Table 5.1: The characteristics of this antenna design

30

Chapter 6

Conclusion

From National Bereau of Standards, we know 15-element yagi antenna has maxi-mum directivity(gain) =

� - " � 3��. In our simulation result we obtain the directivity=� - " ��#&

is almost the same as NBS design. What’s more, we got the resonableE(H)-field plot,complex current distributions, front-to-back ratio, 3db-beamwidthand input-impedance.

31

Appendix A

List of Routine

A.1 yagi.m

A.2 func.m

A.3 func2.m

32

Bibliography

[1] Balanis and Contantine A. Advanced Engineering electromagnetics. JohnWiley– Sons, 1989.

[2] G.A.Thiele. Yagi-uda type antennas. IEEE Trans. Antennas Propagat, 17:21–31, 1969.

[3] Harrington. Field computation by Moment Methods. MacMillan, 1968.

[4] H.Schwetlick. Numerische losung nichtlinearer gleichungen. In DeutscherVerlag der Wissenschaften, 1979.

[5] N.K.Takla and L.C.Shen. Bandwidth of a yagi array with optimum directivity.IEEE Trans. Antennas Propagat, 25:913–914, 1977.

[6] P.P.Viezbicke. Yagi antenna design. NBS Technical Note 688., 1976.

[7] Gordon W.J. and C.A. Hall. Transfinite element methods: Blending functioninterpolation over arbitary curved element domains. Numer. Math., 21:109–129, 1973.

33

yagi uda antenna

Documents

pocklingtons

element length

element input

db beamwidths

complex current

element zm

zone eld

driven element