Antenna Theory and Design Project

7/30/2019 Antenna Theory and Design Project

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UNIVERSITY OF ARIZONA: ECE 584

Antenna Theory and

Design2013 Take Home ProjectRyan Sessions

4/23/2013

The goal of this assignment is two-fold. The first is the analysis of the performance of antenna arrays,

(Steps 1-5) and the second is to design a low-profile (Planar Inverted F Antenna (PIFA)-like) antenna

using Ansys High Frequency Structure Simulator, HFSS (Step 6). [1]


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Antenna Theory and Design: 2013 Take Home Project

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ContentsI. Introduction ............................................................................................................................................... 3

II. Antenna Arrays ......................................................................................................................................... 4

III. A Simulation of a Uniform Linearly Spaced Array .................................................................................... 7

IV. A Linearly Tapered Antenna Distribution .............................................................................................. 10

V. Electrically Scanned Difference Patterns ................................................................................................ 18

VI. Planar Inverted F Antennas ................................................................................................................... 25

IX. Conclusion .............................................................................................................................................. 36

X. References .............................................................................................................................................. 37

XI. Appendix A: MATLAB Simulation of A Linear Antenna Array ................................................................ 38

Figure 1: A Uniformly Spaced Linear Array ................................................................................................... 4

Figure 2: Simulation Program Flow Chart ..................................................................................................... 6

Figure 3: 6 Element Antenna Pattern Comparison- Approximate Theoretical Result vs. Simulation Result 7

Figure 4: 14 Element Antenna Pattern Comparison- Approximate Theoretical Result vs. Simulation Result

...................................................................................................................................................................... 8

Figure 5: A Comparison Between a 14 Element ULA and a 14 Element Linear Tapered Array (amin = 0.1) 11

Figure 6: Effect of Taper Parameter amin on Maximum Sidelobe Level ...................................................... 12

Figure 7: Effect of Taper Parameter amin on HPBW ..................................................................................... 13

Figure 8: Effect of Taper Parameter amin on Directivity .............................................................................. 14

Figure 9: Comparison Between the Performance of a 14 Element ULA and a 14 Element Tapered Array

(amin = 0.5) ................................................................................................................................................... 15

Figure 10: Grating Lobe for a 14 Element ULA ........................................................................................... 16

Figure 11: Grating Lobe for a 14 Element Tapered Array (amin = 0.5) ......................................................... 17

Figure 12: ULA Difference Array Configuration .......................................................................................... 18

Figure 13: Antenna Excitation Distribution for a 14 Element ULA.............................................................. 20

Figure 14: Difference Pattern for a 14 Element ULA (d = /2).................................................................... 21

Figure 15: Difference Pattern for a 14 Element ULA (d = /2) With Central Null Shifted to 120 .............. 22

Figure 16: Element Distribution for the 14 Element Tapered Difference Pattern ...................................... 23

Figure 17: The 14 Element Tapered Difference Pattern ............................................................................. 23

Figure 18: The 14 Element Tapered Difference Pattern (shifted to 120) .................................................. 24

Figure 19: The Planar Inverted F Antenna (PIFA) ........................................................................................ 25

Figure 20: Multiband PIFA Configurations Studied ..................................................................................... 25

Figure 21: Performance of the No Slot Case ............................................................................................... 26

Figure 22: Performance of the Control Case .............................................................................................. 26

Figure 23: Performance of the Shifted Slot Case ........................................................................................ 27

Figure 24: Performance of the Small Slot Case ........................................................................................... 27

Figure 25: Performance of the Large Slot Case ........................................................................................... 28


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Figure 26: Original PIFA from which all PIFAs in this Paper Are Scaled [3] ................................................. 29

Figure 27: Scale Factor Search Resulting in 780MHz Primary Operating Frequency ................................. 30

Figure 28: Performance of PIFA for SF = 0.8642 ......................................................................................... 30

Figure 29: First Attempt at a Dual Band PIFA Performance Results .......................................................... 31

Figure 30: Simulation Results are Used to Adjust the Frequency of Second Resonance .......................... 32

Figure 31: Performance Characteristics of First Dual Band PIFA Design .................................................... 32

Figure 32: Performance Characteristics of Second Dual Band PIFA Design................................................ 33

Figure 33: Dual Band PIFA Dimension Definition ........................................................................................ 34

Table 1: Comparison Between the Results of the Simulation and Theoretical Predictions ......................... 9

Table 2: Comparison Between the Results of the Simulation and Closed Form Numerical Predictions ...... 9

Table 3: Comparison Between the Performance of a 14 Element ULA and a 14 Element Tapered Array

(amin = 0.5) ................................................................................................................................................... 15

Table 4: Parametric Study of the Effects of a Slot on the Performance of a PIFA ..................................... 29

Table 5: PIFA Dimensions, Including a PIFA Dimension Calculator ............................................................. 34


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I. IntroductionThe impact of the antenna on the world can hardly be overlooked. One need only examine the

recent surge in wireless enabled devices to note the prevalence of antennas. Indeed, advances in

antenna design have enabled vast improvements in durability and packaging of such devices, partly due

to novel antenna designs such as the Planar Inverted F Antenna (PIFA). Like many recent advances, the

surge in recent developments in antenna design has been facilitated by the availability of quality

simulation tools to design engineers.

This paper will examine several topics utilizing simulation techniques in antenna theory. The

first is that of the antenna array- a configuration in which multiple antennas are combined together in

order to greatly enhance the directivity with which a signal can be transmitted or received. A simulation

of an antenna array is presented and the results compared to theoretical closed form predictions. The

simulation is then used to synthesize an antenna array pattern from requirements. A difference

antenna array pattern is synthesized, then, through the application of phase shifts, the pattern is

scanned angularly.

The second topic that this paper will discuss is adaptation of an existing PIFA design to enable

single and dual band performance in the 4G LTE band 14 frequency range. A parametric study on dual

band antennas is performed and the results are applied to a design problem along with the help of a full

wave simulation tool.


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II. Antenna ArraysIn order to develop a simulation of the linear array, one must first understand the theory behind

an antenna array. An antenna array is composed of N individual radiators. The radiators are assumed to

radiate with equal strength in all directions. A schematic depiction of an antenna array is shown in

figure 1. The antenna elements are assumed to be distributed equally along a straight line with element

spacing d, and relative phase . The superposition of the waves emitted by the antennas is simply the

sum of the individual phasor representation of each radiator. This superposition is known as the array

factor (AF), and represents the total wave from all radiators when seen at a large distance:

Where the variable , which represents the individual phase of a wave from each element, has been

introduced. The variable k is defined to be the wavenumber of the emitted wave.

Figure 1: A Uniformly Spaced Linear Array

For the special case when the array elements all have the same output power, the an coefficients above

may be assumed to be equal to 1. This special case is known as that of the Uniform Linear Array (ULA).

The array factor of the ULA can be reduced through summation of geometric series to:

In cases of sufficiently large arrays, (generally greater than 10), the array factor can be further simplified

by the approximation:


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The resulting patterns can be shown [2] to reach half of the maximum power at the angle:

[ ] One may find the absolute maximum of the array factor at the angle [2]:

Therefore, one can find the angle of the beam which yields at least 50% of the maximum power within

the angle known as the half power beam width, HPBW as [2]:

| |Due to the damped oscillatory nature of the array factor, many times there will be a local maximum at

some angle of power that is less than the maximum power. This is known as a side lobe. The side lobe

angle may be found by [2]:

{ } where s has been defined to serve as an index to distinguish individual side lobes. If one assumes a large

array, then the side lobe levels (SLL) of individual sidelobes can be found by [2]:

| Finally, it is often useful to have a means for quantifying how concentrated the radiation output

of an array is. It is common for arrays to direct most of their energy in one general direction. In order to

measure this, one need only compare the maximum radiation intensity of the array with the average

radiation intensity:

For sufficiently large ULAs, the directivity may be approximated by [2]:


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Using the above relations, a numerical simulation was developed in MATLAB. See Appendix A for source

code. A flow chart of basic operation for the simulation is shown in figure 2 below.

Figure 2: Simulation Program Flow Chart


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III. A Simulation of a Uniform Linearly Spaced ArrayIn order to demonstrate the accuracy of the ULA simulation program, the program results will

be compared to the closed form and approximate results given in section II. The metrics that will be

used are the directivity, the peak sidelobe levels, and half power beam width (HPBW). Two test cases

will be used: a 14 element array and a 6 element array. The simulations were run assuming equal

element excitation amplitudes and no inter-element phase differences.

As can be clearly seen in figures 3 and 4, the results match for results around 90 which

corresponds to =0. As deviates from 90, deviates from 0. Recall the approximation which yielded

the approximate AF formula:

It is clear that this approximation is only valid for small values of . As increases, N/2 becomes

larger than sin(N /2). This means that the approximate AF will get smaller than the actual AF as the

difference increases, i.e. as deviates away from 0, which is exactly what is shown in figures 3 and 4.



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A comparison between the results of the simulation program and the theoretical predictions are

shown in table 1 below. When the numerical calculation that performs the summation of the phasors is

replaced by the closed form solution:

The results in table 2 are obtained. The reader can clearly see that the closed form solution provides

excellent support for the results obtained from the simulation.

14

Element

SLL

Theoretical Simulation

() SLL (dB) Angle SLL (dB)

1 77.6 -13.5 78.2 -13.1

2 69.1 -17.9 69.4 -17.4

3 60.0 -20.8 60.2 -19.9

4 50.0 -23.0 50.2 -21.5


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5 38.2 -24.8 38.3 -22.4

6 21.8 -26.2 21.8 -22.9

HPBW 7.3 7.3

Directivity 14.0 14.0

6

Element

SLLTheoretical Simulation


1 60.0 -13.5 61.2 -12.4

2 33.6 -17.9 34.1 -15.3

HPBW 16.9 17.2

Directivity 6 6.0

Table 1: Comparison Between the Results of the Simulation and Theoretical Predictions

14

Element

SLL



1 78.2 -13.1 78.2 -13.1

2 69.4 -17.4 69.4 -17.4

3 60.2 -19.9 60.2 -19.9

4 50.2 -21.5 50.2 -21.5

5 38.3 -22.4 38.3 -22.4

6 21.8 -22.9 21.8 -22.9

HPBW 7.3 7.3


6

Element

SLL



1 61.2 12.4 61.2 -12.4

2 34.1 15.3 34.1 -15.3

HPBW 17.2 17.2

Directivity 6 6.0

Table 2: Comparison Between the Results of the Simulation and Closed Form Numerical Predictions


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IV. A Linearly Tapered Antenna DistributionThe performance of a linear array may be improved by adding a taper to the distribution. This

has the effect of reducing the sidelobe levels. Unfortunately, this comes at the price of a broader main

beam. Many methods exist to allow a designer to allocate the necessary excitation to each array

element. Perhaps the simplest however, is that of the linear taper. The excitation is decreased linearly

from the centermost elements of the array to the outermost elements of the array, where it takes the

value amin. This method is straightforward enough that a designer with simulation tools (like the afore

presented simulation program) may easily determine the necessary parameters of the taper by trial and

error, provided of course, that the desired sidelobe levels are sufficiently above levels where only the

most sophisticated methods will provide a good design.

To illustrate this, an example design will be synthesized so that the 1st

sidelobe level is below -

18dB. A parametric study will be made of the taper parameters and the resulting sidelobe levels. The

taper is assumed to have the form:

{ ( ) A 14 element, d = /2 example of this distribution is plotted in figure 5. A variety of values between

0


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Figure 5: A Comparison Between a 14 Element ULA and a 14 Element Linear Tapered Array (a min = 0.1)


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Figure 6: Effect of Taper Parameter amin on Maximum Sidelobe Level


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Figure 7: Effect of Taper Parameter amin on HPBW


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Figure 8: Effect of Taper Parameter amin on Directivity

The effect of the taper parameter on the half power beam width and the directivity was also

studied and is shown in figure 7. The HPBW was found to decrease with increasing amin. This stands to

reason since the case of amin = 1 corresponds to a ULA. The directivity was found to increase as the taper

parameter amin increase to 1 (see figure 8). Based on a design requirement of -20dB maximum SLL, the

value amin = 0.5 was selected. The tapered distribution parameters are compared to an analogous ULA in

table 3. The directivity is found to suffer some from the taper, as does the HPBW, but not severely. All

sidelobe levels were found to be decreased by 3-7dB when a taper was implemented.

14

Element

SLL

ULA Tapered (amin = 0.5)


1 78.2 -13.1 77.2 -20.1

2 69.4 -17.4 69.1 -20.5

3 60.2 -19.9 59.9 -25.0

4 50.2 -21.5 50.0 -25.3

5 38.3 -22.4 38.2 -27.0

6 21.8 -22.9 21.8 -27.1


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HPBW 7.3 8.1


Table 3: Comparison Between the Performance of a 14 Element ULA and a 14 Element Tapered Array (a min = 0.5)

Figure 9: Comparison Between the Performance of a 14 Element ULA and a 14 Element Tapered Array (a min = 0.5)

Finally, a parametric study was made of the impact of increasing the value of d from /2 to

slightly greater than . It was discovered that grating lobes, a case where sidelobes occur with the same

amplitude as the main beam, were introduced as soon as d became greater than or equal to . This is

illustrated in figures 10 and 11 for both the ULA and the tapered array.


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Figure 10: Grating Lobe for a 14 Element ULA


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Figure 11: Grating Lobe for a 14 Element Tapered Array (amin = 0.5)

Figure 12: Antenna Excitation Distribution for Lobe for a 14 Element Tapered Array (amin = 0.5)


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V. Electrically Scanned Difference PatternsThe antenna array patterns that have been discussed previously are referred to as sum patterns

because they act to transmit and receive in a central window. Sometimes, it is advantageous to attempt

the converse- to not transmit and receive in a central window. The antenna pattern which has this

feature is referred to as a difference pattern. It is achieved by using a uniform linear array with a 180

phase shift applied to one entire half of the array (figure 12). Please note that in the following

treatment, the array is assumed to have 2N elements.

Figure 13: ULA Difference Array Configuration

In order to describe this, the contributions of the array are handled separately, as in above in

Section II. The Array factor is now the sum of two array factors:


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This can be simplified to:

Where S1 and S2 are given by:

Both of the above terms have the form of a geometric series. This means that their sum has the form:

( )

Consequently, it can be shown that:

Therefore, the array factor can be shown to be:

In order to electrically scan the central minimum of the array pattern, an appropriate inter

element phase shift to be determined. In order to define this, let the case of a minimum be considered.

In order for a minimum to exist, the numerator of AF must be zero:


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Furthermore, this implies that the argument of sin() must be an integer multiple of :

From the above, it can be shown that the angle where the central null is formed is given by:

Which also implies:

In order to test this result, the above results are integrated into a simulation. First the

simulations ability to create the necessary excitation distribution (figure 13) and difference pattern

(figure 14) is tested. Next, the simulations ability to scan the main null is tested by scanning the null to

120, as shown in figure 15.

Figure 14: Antenna Excitation Distribution for a 14 Element ULA


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Figure 15: Difference Pattern for a 14 Element ULA (d = /2)


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Figure 16: Difference Pattern for a 14 Element ULA (d = /2) With Central Null Shifted to 120

Finally, the above results were also simulated for a tapered difference array. Figures 16 and 17

show the distribution and the unshifted pattern, respectively. Notice that no null except the central null

in the pattern drops below -21.56dB. Figure 18 shows that the same method used to scan the null of

the ULA difference pattern will also shift the null of the tapered distribution.


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Figure 17: Element Distribution for the 14 Element Tapered Difference Pattern

Figure 18: The 14 Element Tapered Difference Pattern


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Figure 19: The 14 Element Tapered Difference Pattern (shifted to 120)


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VI. Planar Inverted F AntennasThe final section of this paper shall consider the Planar Inverted F Antenna (PIFA). As will be

shown, the PIFA allows monopole antenna-like performance, but in a much less obtrusive package. The

PIFA consists of a ground plane that is shorted to a parallel plate. The parallel plate is then fed coaxially

through a hole in the ground plane. For an example, see figure 19. When multi-band performance is

required, an L shaped slot is often cut in the top plate of the PIFA. In order to study this in greater

detail, multiple configurations were studied (see figure 20).

Figure 20: The Planar Inverted F Antenna (PIFA)

Figure 21: Multiband PIFA Configurations Studied


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The return loss, impedance, and far field antennas of each of multiple configurations was

studied in order to determine the effect of increasing/decreasing slot width and moving the right angle

bend in the slot further from the feed location. The results of these experiments is tabulated in table 4

and figures 21 through 25.

Figure 22: Performance of the No Slot Case

Figure 23: Performance of the Control Case


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Figure 24: Performance of the Shifted Slot Case

Figure 25: Performance of the Small Slot Case


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Figure 26: Performance of the Large Slot Case

As is visible in figures 21-25 above, the addition of a slot adds a second resonance to the PIFA.

The width of the slot appears to control how strongly coupled the second antenna is. When the slot is

wide, the second resonance has a wide bandwidth. The shift in the slot and also the large slot also

appear to increase the frequency of the second resonance. The wide slot also appears to increase the

frequency of the secondary resonance. This is likely caused by the length of the tab created by the L

shaped cut. The length of a PIFAs top plate will determine its resonant frequency- with longer plates

causing lower resonant frequencies [3].

In essence, a PIFA with a slot in it acts like two coupled antennas. The shorter antenna is the

shorter tab which corresponds to a higher frequency resonance, while the L shaped longer tab

corresponds to the lower frequency resonance. In order to create a dual band PIFA, the overall size of

the PIFA is scaled to realize an appropriate lower frequency. Then, the length of the small tab is

adjusted to provide an appropriate higher frequency resonance. In order to achieve a higher frequency

resonance, make the tab shorter. In any case, the change in slot configuration has a relatively large

impact on the frequency of the secondary resonance- around 30% variation among the cases studied

here.


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Case

F1

(GHz)

F2

(GHz)

RL @ F1

(dB)

R @F1

()

X @F1

() Effect (relative to control)

Control 1.5 2.08 -26 52.3 4.5 N/A

Shift slot 1.56 2.17 -50 50.1 -0.3

Higher 2nd resonant frequency.

Primary resonance is relatively

unaffected.

Large Slot 1.54 2.55 -35 51.5 1

Higher 2nd resonant frequency.


unaffected.

Small Slot 1.53 2.08 -31 50.5 -2.8

Tighter coupling of secondary

resonance. Primary resonance is

relatively unaffected.

No Slot 1.59 N/A -28 48.8 -3.7


unaffected with/without the slot.

Table 4: Parametric Study of the Effects of a Slot on the Performance of a PIFA

In order to demonstrate how one may design a dual band PIFA with an appropriate primary and

secondary resonance, this paper will design single and dual band PIFAs with primary resonance in the 4G

LTE Band 14 frequency band of 758-798MHz. The design goal will be specifically to place the primary

resonance in the middle of the band- around 778MHz. Then, a dual band antenna will be designed that

implements the same primary resonant behavior but also:

The highest possible secondary resonant frequency (below 2GHz) The lowest possible secondary resonant frequency

The dimensions of the PIFA will be scaled from a PIFA presented in a paper by Virga and Rahmat-Samii

[3]. The overall width and height of the PIFA will be divided by a scale factor denoted in figures as SF.

Figure 27: Original PIFA from which all PIFAs in this Paper Are Scaled [3]


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For the purpose of the following discussion, the modifications to the above PIFA will be referred

to by a scale factor SF, which corresponds to the amount each of the above dimensions is divided down,

and a scale factor SFt, which corresponds to the location and length of the slot. For more details, refer

to Appendix B.

The first stage in the design is the adjustment of the overall size of the PIFA to match it to778MHz. A full wave simulation tool, HFSS, was used to simulate various scale factors in order to

determine the optimal value of the overall scale factor SF. The results of these simulations indicate that

a value of 0.8642 is appropriate to get a primary resonance of 778MHz (see figure 27). The performance

of the PIFA is shown in figure 28.

Figure 28: Scale Factor Search Resulting in 780MHz Primary Operating Frequency

Figure 29: Performance of PIFA for SF = 0.8642


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The radiation patterns of the PIFA are a fair match for the patterns provided in [3] and are quite

broad. The primary resonance is 780MHz and has a good match to 50 with an impedance of

49.3+j0.9. The same scale factor is applied to a PIFA with a 3mm wide slot and SFt = 1.5. The primary

resonance was slightly affected so the PIFA was rescaled to have SF=0.91 and SFt=1.5. The performance

characteristics of the resulting device are summarized in figure 29. The resonant frequencies of the

antenna are 770MHz and 1.32GHz. The antenna has a good match to the feed with and impedance of

59+j4.5 at 770MHz and an impedance of 43.2+j1.7 at 1.32GHz.

Figure 30: First Attempt at a Dual Band PIFA Performance Results

Two different antennas were obtained from this configuration by changing the size of the tab.

The size of the tab was decreased in order to increase the frequency of the secondary resonance. The

size of the tab was increased in order to decrease the frequency of the secondary resonance. Like the

scale factor SF, the appropriate value of the scale factor SFt was selected by a manual search of values

based on simulation results. Figure 30 illustrates how the simulation results are used to adjust tab size.


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Figure 31: Simulation Results are Used to Adjust the Frequency of Second Resonance

The design which maximizes the secondary frequency of resonance utilizes a slot of width 3mm,

SF = 0.91, and SFt = 2. This design achieves a primary resonance frequency of 770MHz and a secondary

resonance of 1.68GHz. The impedances at these frequencies are 55.4+j4.5 and 49+j2.9, respectively.

The radiation patterns are broad, although the pattern at the secondary resonance is less well behaved

than the pattern at primary resonance.

Figure 32: Performance Characteristics of First Dual Band PIFA Design


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The design which minimizes the secondary frequency of resonance utilizes a slot of width 3mm,

SF = 0.91, and SFt = 1.25. This design achieves a primary resonance frequency of 810MHz and a

secondary resonance of 1.15GHz. The impedances at these frequencies are 50.3+j1.6 and 50.1+j2.8,

respectively. The radiation patterns are broad, although the pattern at the secondary resonance is less

well behaved than the pattern at primary resonance.

Figure 33: Performance Characteristics of Second Dual Band PIFA Design


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Dimension No Slot Control Large Slot Small Slot Shift Slot

780 MHz

Single

Band PIFA

770

MHz/1.32

GHz PIFA

770

MHz/1.68

GHz PIFA

810

MHz/1.15

GHz PIFA

Build Your

Own Dual

Band PIFA

A (mm) 65.32 65.32 65.32 65.32 65.32 134.38 127.62 127.62 127.62 116.13

B (mm) 34.17 34.17 34.17 34.17 34.17 70.30 66.76 66.76 66.76 60.75

C (mm) N/A 11.16 11.16 11.16 12.15 N/A 20.25 15.19 24.30 30.37

D (mm) 14.52 14.52 14.52 14.52 14.52 29.87 28.36 28.36 28.36 25.81E (mm) 1.78 1.78 1.78 1.78 1.78 0.86 0.91 0.91 0.91 1.00

F (mm) 2.81 2.81 2.81 2.81 2.81 5.79 5.49 5.49 5.49 5.00

G (mm) N/A 1 3 0.1 1 N/A 3 3 3 3

H (mm) 3.04 3.04 3.04 3.04 3.04 6.25 5.94 5.94 5.94 5.41

I (mm) N/A 14.52 14.52 14.52 14.52 N/A 28.36 28.36 28.36 25.81

SF 1

SFt 1

Build your own dual

band PIFA

Table 5: PIFA Dimensions, Including a PIFA Dimension Calculator

Figure 34: Dual Band PIFA Dimension Definition


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IX. ConclusionThis paper examined several topics utilizing simulation techniques in antenna theory. The first

was that of tapered and uniform linear arrays. A simulation was devised that enabled study of the

properties of the uniform linear antenna array and tapered array in both sum and difference pattern

configurations. Differences between the approximate theoretical and simulation were discovered and

explained as a failing in the approximate theory due to approximations. The closed form solution was

used to demonstrate this.

A method of design a tapered array by simulation via a linear taper was presented and

performance data for this configuration was presented. The linear taper was applied to study a

difference pattern and steerable array patterns.

Next, the use of HFSS to analyze a PIFA was demonstrated. HFSS was used to study changes to

the PIFA that enable dual band performance. Several PIFA designs were implemented that provide

performance sufficient for operation in the 4GLTE band 14 frequency range. Several design guidelines

were noted and demonstrated.


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X. References[1] K. Melda, ECE 584 Antenna Theory and Design, Spring 2013: Take Home Midterm Project.

Unpublished work. Tucson AZ, March 2013

[2] C. Balanis, Antenna Theory: Analysis and Design. 3rd ed. John Wiley and Sons, Inc. Hoboken, NJ

2005

[3] K. Virga, Y. Rahmat-Samii, Low-Profile Enhanced-Bandwidth PIFA Antennas for Wireless

Communications Packaging in IEEE Transactions on Microwave Theory and Techniques, Vol. 45, No. 10,

Oct 1997


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XI. Appendix A: MATLAB Simulation of A Linear Antenna Array

%-------------------------------------------------------------------------% %Housekeeping- clean up previously used variables and the command screen%-------------------------------------------------------------------------% clear;clc;

%-------------------------------------------------------------------------% %Initialize Variables and Constants%-------------------------------------------------------------------------% %Basic variableslambda = 1;k = 2*pi / lambda;radians = pi/180;degrees = 1;

%Basic System/Design Parametersd = 1/2 * lambda;M = input('Please enter number of elements-\n>');NoiseFloor = 1e-6;

Excite_Mode = input('\nPlease select mode of excitation- \n1 = Uniform\n2 = Difference Pattern\n3 = Tapered Sum\n4 = Tapered Difference\n>', 's');

switch Excite_Modecase'1',

%Uniformly excited amplitude sum patternExcitation = ones(1,M);%No taper, no phase shift between sides of

arraycase'2',

%Uniformly excited amplitude difference patternExcitation = [ones(1,M/2),-ones(1,M/2)];%No taper, pi phase between

sidescase'3',

%A linear taper between 1 at center of array and 0.1 at edge of%array is used to synthesize a tapered sum patterna = 0.5;%input('Please enter taper parameter-\n>');Excitation = ([linspace(a,1,M/2),linspace(1,a,M/2)]);

case'4'%A linear taper between 1 at center of array and 0.1 at edge of%array is used to synthesize a tapered difference patterna = 0.5;%input('Please enter taper parameter-\n>');Excitation = ([linspace(a,1,M/2),-linspace(1,a,M/2)]);

otherwise,disp('Error: invalid excitation mode selected- Aborting simulation.')return

end

SteeringAngle = input('\nPlease select steering angle (in degrees)-\n>');

%Initialize working variablesTheta = (0:.01:180) * degrees;Beta = -k * d * cos(SteeringAngle*radians);


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Psi = k*d*cos(Theta*radians) + Beta;Efield = 0;

%-------------------------------------------------------------------------% %Calculate the electric field due to superposition of antenna elements%-------------------------------------------------------------------------%

for element = (1:1:M)Efield = Efield + Excitation(element)*exp(1i.*(element-1).*Psi);end

%-------------------------------------------------------------------------% %Normalize the electric field due to the M different antenna elements%-------------------------------------------------------------------------% Efield = Efield / M;

%-------------------------------------------------------------------------% %Calculate the Amplitude due to the electric field%-------------------------------------------------------------------------% Amplitude = abs(Efield);Amplitude = 20 * log10(Amplitude + NoiseFloor);

%-------------------------------------------------------------------------% %Normalize the total Amplitude to 0dB as max%-------------------------------------------------------------------------% Amplitude = Amplitude - max(Amplitude);

%-------------------------------------------------------------------------% %Plot angular amplitude distribution%-------------------------------------------------------------------------% figure(1);plot(Theta, Amplitude);

%-------------------------------------------------------------------------% %Find 3dB Points%-------------------------------------------------------------------------% [Y,Imax]=max(Amplitude+3);[Y,Izero]=min(abs(Amplitude+3));FWHM = 2*abs(Theta(Imax)-Theta(Izero));disp(' ');disp([' The FWHM is ', num2str(FWHM,5),'.']);disp(' ');

%-------------------------------------------------------------------------% %Find Local Maxima%Tabulate Results%-------------------------------------------------------------------------% [pks,locs] = findpeaks(Amplitude);ThetaMaxima = Theta(locs)';disp(' Theta Amplitude')disp([Theta(locs).' Amplitude(locs).'])

%-------------------------------------------------------------------------% %Calculate Directivity%-------------------------------------------------------------------------% P_Rad = trapz(Theta*radians,sin(Theta*radians).*10.^(Amplitude/10))*2*pi;


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U_Max = max(10.^(Amplitude/10));Directivity = 4*pi*U_Max/P_Rad;disp(' ');disp([' The directivity is ', num2str(Directivity,5), '.']);

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Antenna Theory and Design Project