Field Of Wireless Techniques Pervasively Grows

Published Date: 02 Nov 2017

As the field of wireless techniques pervasively grows in space-based platforms, radar,

wireless communications and personal electronics, the frequency of operation for the

electronic devices keeps increasing. As the frequency increases, the signal carrying large

bandwidth information becomes more and more directional. To reliably transmit/receive

the information in/from certain direction, a beam-forming device that produces true-time

delay, wideband, wide-angle, steerable beams is desirable. The narrow beam produces

good isolation between adjacent radiation elements; hence multiple beams are possible to

be simultaneously obtained by reusing the antenna structure. In doing so, multiple

functionalities can be incorporated in a single electronic system. This device,

implementable using a printed circuit board (PCB), is attractive in terms of low-profile

and mechanical properties such as light weight and strong resiliency.

A phased array is the essential device that utilizes the beam-forming network to radiate

energy into free space. Since the 1950s, it has been widely adopted in many radar and

satellite systems to perform electronically-controlled beam scanning. For decades, array

systems have been restricted for military applications due to high cost and complexity. In

recent years, low cost high performance array and its supporting devices have been

realized using printed circuit technology, thus array-based commercial applications such

as wireless point-to-point communications and auto-collision avoidance radar have

emerged. Besides, since the Federal Communications Commission (FCC) issued new

bands for commercial ultra-wideband (UWB) [1] and Extremely High Frequency (EHF)

applications [2], low-profile high performance arrays have been under investigation. The

low-cost high-performance beam-forming networks would facilitate new application

development.

The research performed in this dissertation aims at designing microwave lens-based

beam-forming networks and optimizing their performance. Traditional design theory will

be improved in terms of minimized phase errors and scanning capabilities. Microwave

lens for PCB implementation is emphasized. Accurate simulation and analysis method

2

based on electromagnetics full-wave simulation will be explored. Further improved

method based on hybrid ray tracing technique is developed to accelerate the simulation.

The latter is suitable for lens optimization or handling extremely large structures.

Besides, a novel technique that enables single lens to perform 360-degree scanning is

proposed. A couple of microwave lenses have been designed, fabricated and tested, and

they are covered throughout these topics. Their simulation and measurement data are

primarily used to support the proposals in this dissertation.

It is necessary to review some basic concepts before presenting the specific design

subjects, as the aforementioned designs are abided by the laws of electromagnetic

theories. The fundamentals of the antenna and array theory serve as one of the key

components to form the objective of the beam-forming network. Thus, in the remainder

of the session (1.1-1.3), we first review the basic knowledge on waves, antenna and array

that have extensively been used during the lens design and analysis process. Besides,

understanding the BFNs and previous researchers’ works have provided invaluable basis

for this dissertation to propose new formulations, simulation method and optimization

strategies. The basis of BFNs and some existing models are introduced in section 1.3.

Furthermore, because several existing EM simulation toolkits have been involved to

conduct performance evaluation and concept validation prior to the implementation, the

basic EM simulation methods are reviewed in section 1.4. Section 1.5 enlists the

contributions of this dissertation and the presentation of outlines for the rest of the

chapters.

1.1 Review of Basic Electromagnetics and Waves Concepts

Electronic devices operating at high frequencies usually have physical size comparable to

the wavelength, thus classical circuit theory hardly applies. It is the Maxwell equations

that provide the fundamental theories for many engineers to perform predictive design

and pursue solid explanation. In this section, we first review the Maxwell equations and

their time-domain and frequency-domain representations. Then focus is put on behaviors

of electrical magnetic fields and their relationship to energy. Furthermore, the waves in

real medium will be addressed with emphasis on their amplitude and phase characteristics

3

during propagation. Finally, it is worthwhile reviewing a few circuit model parameters

that intrinsically link to the behavior of the electronic designs. Note concepts covered in

this section primarily serve for understanding the microwave lens designs in this

dissertation rather than providing comprehensive electromagnetics basis. A detailed

derivation of the equations presented in this chapter can be found in Appendix A.

1. Maxwell Equations

The time-domain Maxwell equations in differential form are shown in (1-1)-(1-5), and

they are written in the MKS (meter, kilogram, second) unit system. This set of equations

consists of the classical Faraday law, Ampere law and Gauss law. Faraday law states that

the time varying magnetic flux produces electrical field, while the Ampere law originally

described that time varying electrical current produces rotational magnetic fields.

Maxwell proved that for these equations to be consistent there must be a displacement

current term D / t present, hence we presently call the ampere law as Maxwell-

Ampere law. A current source produces fields, and the Gauss law describes the

relationship between field and its enclosed source quantity. Because there is no magnetic

charge found in nature, the right hand of equation (1-4) is zero. Equation (1-5) states that

moving electrical charges produce electrical current.

t



   



B

E (Faraday law) (1-1)

t c i



    



H J J

D (Maxwell-Ampere law) (1-2)

v  D  q (Gauss law) (1-3)

 B  0 (Gauss law – magnetic form) (1-4)

v

t







q

J (Current continuity equation) (1-5)

where,

E : electric field intensity (volts/meter)

D : electric flux density (coulombs/meter2)

H : magnetic field intensity (amperes/meter)

B : magnetic flux density (webers/meter2)

Jc : electric conducting current density (amperes/meter2)

4

Ji : electric impressed current density (amperes/meter2)

qv : electric charge density (coulombs/meter2)

  : curl vector operator that describes the rotation of a vector field

 : divergence operator that measures the magnitude of a vector field’s source

The electric and magnetic flux densities are defined regardless of the material properties;

so that the source produces these quantities can be easily found by Gaussian integration

without mentioning the materials. The material properties appear in the constitutive

relations with the electrical and magnetic fields, as shown in (1-6)-(1-8).

D=E (1-6)

B=H (1-7)

c JE (1-8)

where,  is called permittivity of the medium,  is the permeability, and  refers to the

conductivity. According to [3], above relationships hold true if E , H and their time

derivatives are not very large.

It is noted that the vector quantities in the Maxwell equations are functions of both time

and space. If we express the field in a form of  (r,t)



E E , where r



represents the spatial

vector. It is found that the vector quantities in the Maxwell equations can be expressed in

a single frequency harmonic form shown in (1-9). If we substitute this as well as the

constitutive relationships into the Maxwell equations, the frequency-domain expression

of (1-10)-(1-11) is yielded. The frequency-domain representation of waves is adopted

through out this dissertation.

[ ()]

( )

( , ) | ( )|cos[ ( )]

Re{| ( ) | }

Re{| ( ) | }

Re{ ( ) }

e

e

e

j t r

j r j t

j t

r t E r t r

E r e

E r e e

E r e

 

 



 



 











  







E

(1-9)

i

E j H

H j E E J



 

   

     

(1-10)

(1-11)

5

2. Electrical, Magnetic Fields and Energy

The Maxwell equations (1-10)-(1-11) provide the governing laws for electrical, magnetic

fields and current behaviors in medium with properties of  ,  and . Any of the

quantity within the equations might be considered as unknowns depending on different

real life applications. Generally, for beam-forming network design, a typical problem

dealt with is that given impressed current J i excitations within a known geometry to

solve for the field performance. Similarly, when a beam former feeds array, conducting

current on the antenna would produce travelling fields in free space. The latter is a typical

radiation problem, which will be exclusively discussed in the section 1.2. In this section,

we focus on deriving the solution of planar waves from Maxwell functions and illustrate

their relationships with energy storage and propagations.

Assuming fields exist in a source-free region, then they must satisfy the Maxwell

equations of (1-10) and (1-11) providing J i  0 . Substitute one to the other, and it gives

electric vector wave equation (1-12) and magnetic vector wave equation (1-13).

   E( 2 j)E  0 (1-12)

   H( 2 j)H  0 (1-13)

Applying the vector identity of (1-14), equations (1-12) and (1-13) can be written as (1-

15) and (1-16).

    A  ( A) 2A (1-14)

2E(2 j)E 0 (1-15)

2H(2 j)H 0 (1-16)

In convention,  2  j is defined as the wave number quantity, k 2 . Thus (1-15)

and (1-16) have arrived at a form with linear operators on the electrical and magnetic

fields. These results of the rectangular components of the fields satisfy the Helmholtz

equations of (1-17).

2 k2 0 (1-17)

Let us take the depolarized component of x E for example, and assume it is independent of

x and y. Thus x E satisfies (1-18).

6

2

2

2x 0

x

d E k E

dz

  (1-18)

Details on solving this equation and the wave equations with forcing term J i , are

attached in Appendix A. Here we list the result in (1-19) without proof.

0

jkz

x E  E e (1-19)

The associated magnetic field is found by substituting (1-19) into (1-10).

0

0

jkz jkz

y

H Ee Ee

 

  

  (1-20)

where the ratio between the E field and H field is defined as the wave impedance. Now if

we apply relationship (1-9), the instantaneous fields are found as

0 ( , ) Re{ j t} cos( )

x x r t  E e  E t kz



E (1-21)

( , ) Re{ j t} 0 cos( )

y y

r t H e  E t kz



  



H (1-22)

These are waves traveling along +z direction, as shown in Figure 1- 1, the magnetic field

is in phase to the electrical field, both normal to the direction of propagation. The power

density of the fields can be found by calculating the Poynting vector, S. In frequency

domain, S for the fields in (1-21) and (1-22) is shown in (1-23).

2

S=E H* z E0



   (1-23)

Figure 1- 1. Linearly Polarized Uniform Traveling Wave along z Direction

H

E

x

y

z

7

Result in (1-23) indicates that real power flows along z direction with power density

value of 2

E0 / . In many electronic designs such as transmission line and antenna, it is

always desirable to have real S. However, in reality, when the electrical or magnetic

fields encounter discontinuities, in order to satisfy the boundary conditions, reflections

may occur. The reflected fields may combine with the original field forming an

imaginary portion of Poynting vector S. The imaginary power density represents the

reactive powers stored, which eventually generate heat in the devices. The following

paragraph illustrates how the other form of fields called standing waves can be formed.

Suppose the field of (1-19) travels in a bounded medium, due to discontinuity, it gets

total reflection and forms a 180o phase shift from the original incident wave. The total

field at the region is the superposition of both fields as below:

0 0

0 0

0 0

0

[cos( ) sin( )] [cos( ) sin( )]

[cos( ) sin( )] [ cos( ) sin( )]

2 sin( )

total

x x x

jkz jkz j

E E E

E e E e

E kz j kz E kz j kz

E kz j kz E kz j kz

j E kz



 

 

 

 

 

     

    

 

(1-24)

0 0

0 0

0 0

0

[cos( ) sin( )] [cos( ) sin( )]

[cos( ) sin( )] [cos( ) sin( )]

2 cos( )

total

y y y

jkz jkz j

H H H

Ee Ee

E kz j kz E kz j kz

E kz j kz E kz j kz

E kz



 

 

 

 



 

 

 

 

     

   



(1-25)

These co-existing waves are out of phase to each other, the phases in the time domain do

not depends on z, hence they are no longer traveling waves. Consequently, the Poynting

vector in (1-26) is an imaginary number, as shown in (1-26). The energy goes back and

forth between the E and H fields with respect to time. These more or less represent their

standing wave behaviors.

8

2

S=E H* zj2E0 sin(2kz)



  (1-26)

We emphasized the fields’ behavior and their relationships by reactive and real powers,

because these concepts are frequently met in many electronic devices design process.

Take the microwave tapered horn in Figure 1- 2 as an example, which is typical geometry

of the radiation port in microwave lens design, because of the tapering, fields’ reflection

happens within the taper. As what we have seen above, part of the energy is stored due to

the standing waves within the tapers. Note the tapering geometry, it plays essential rule to

minimize the amount of stored powers. Understanding these basic concepts help

construct improved designs. It is also worth pointing out that there is another form of

evanescent waves that can steal power from the electronic devices. We shall review that

in section 1.2 when discussing the near field of an antenna.

Figure 1- 2. Microwave Tapered Horn Example

3. Waves in Real Medium

In this dissertation, the microwave lens is designed on printed circuit board that has

certain dielectric material filled layer. High dielectric constant material leads to size

reduction, but the loss tangent of the medium tends to grow too. In this section, we

discuss the waves’ amplitude and phase variations in medium filled with lossy dielectric.

The general expression of the planar wave is (1-27). Assume it travels in a medium with

 value of (1-28), and  =0.

0 E(r) E e jkr (1-27)

Input Port

Output Port

Discontinuities

9

 r jrtan (1-28)

where tan is the loss tangent of the medium. From the definition of wave number in (1-

16), it is found

( tan) (1 tan)

r r r 2

k j j 

          (1-29)

Assume k r ,and substitute (1-29) into (1-27), this gives

tan /2

0 E(r) E er r ejr r          (1-30)

This is still planar wave traveling in the r direction, but with amplitude decaying ratio of

e r r tan /2      due to the loss tangent. The phase variation follows the relation of the real

portion of  . In Figure 1- 3, the amplitude decaying trend at different frequencies for

Rogers 3006 is illustrated. As the frequency increases, more attenuation occurs. Figure 1-

4 shows the phase variation along r direction for different materials. The solid reference

line in Figure 1- 4 represents the phase constant planes in the free space. The difference

between the two curves demonstrates that waves travelling in higher r

 medium can

receive the same phase variation but travel much shorter distance. This enables the size of

the printed lenses to be decreased by a factor of r

 from the ones in free space.

Figure 1- 3. Field Amplitude versus Traveling Distance in Rogers 3006 ( r

 =6.15,

tan =0.002) at Frequency of 6, 10 and 20 GHz

10

Figure 1- 4. Phase Front versus Traveling Distance for Different 

So far we have dealt with the planar wave representation in the form of (1-27). Beside of

the loss tangent of the material, the amplitude of the waves may decay in certain order of

|r|. Two typical cases are the spherical and cylindrical waves, as shown in (1-31) and (1-

32), which form circular phase fronts in the propagation domain.

( ) 0

| |

E r E e j k r

r

   (1-31)

( ) 0

| |

E r E e j k r

r

   (1-32)

4. Some Parameters for Circuit Analysis

The electric and magnetic fields contain energy, and the real power of the Poynting

vector leads to its propagation. This energy can be guided and travel along constrained

structures. When the dimension of the structure is much smaller than the wavelength,

circuit elements are usually derived from the Maxwell equations to facilitate the design.

Voltage V, occurs between two parts of the circuit elements and it is defined by line

integration of the E field, as shown in (1-33). Current I, standards for the amount of

Coulombs passed per second, relates to the conducting current density by (1-34), where

A is the cross-sectional area of the conductor. Resistance R, measures the opposition to

the passage of the current, and it determines the amount of current through the object for

11

a given potential difference of V, hence I=V/R. Capacitance C, defined by (1-35) depicts

the ability of storing charges (or electrical fields), where Q is the amount of charges. An

inductor L, which stores currents (or magnetic fields due to these currents), is defined by

the ratio between the magnetic flux over the current producing these fields, (1-36).

l

VEdl (1-33)

I  JcA (1-34)

C Q

V

 (1-35)

L

I



 (1-36)

where

s

   Bd A (1-37)

Now let us consider a transmission line with characteristic impedance of Z0 connecting to

load Zin, as shown in Figure 1- 5. For instant, in this dissertation, typically transmission

line is designed at Z0=50Ω. The load ZL might represent any port that attaches to the

transmission lines, e.g. horn antenna, terminal load, etc. At the discontinuity point,

voltage and current exist in both directions, here we expressed them as

VL=V++V- (1-38A)

IL=I++I- (1-38B)

Figure 1- 5. A Transmission Line with Impedance Z0 Terminated by Load ZL

It is not hard to find that equation (1-39) holds after applying the Ohm’s law. By

substituting (1-38) in (1-39), the reflection coefficient between the backward and forward

voltages is calculated as (1-40). This is an important design parameter, as it tells how

much field (and power) is reflected from the load component of ZL. We will shortly see

that ZL normally is a function of frequency, given constant transmission line Z0, and  is

Z Z0 ZL VL

IL

V+

12

used to characterize the bandwidth of the microwave devices. Besides, the real and

imaginary parts of the total system impedance Z can also be used to evaluate the power

efficiency.

0 0

L

L

V V V

Z Z Z

 

  (1-39)

0

0

L

L

Z Z

Z Z



 



(1-40)

A beam-forming network typically has multiple inputs and outputs. Figure 1- 6 illustrates

the typical problem we will deal with in the following chapters. The N ports network

behavior is described by the scatter coefficient matrix of (1-41). Sij represents the

coupling between port i and j when all other ports are terminated. We shall keep in mind

that the matrix shown below only captures the network behavior at single frequency.

Should the frequency responds are required, SNP matrix at multiple frequencies have to

be calculated. The SNP file is a standard touchstone format that is used to describe NScatter-

Parameters.

Figure 1- 6. Typical N Port Network Structure

(1-41)

...

11 21 31 1

22 32 2

33 3

N

N

N

NN

S S S S

S S S

SNP S S

S

  

  

  

 

 

 

. . .

Port N

Port 1

Port 2

Port 3

Port 4

Port 5

13

1.2 Review of Antenna and Array Fundamentals

Typical microwave lens design involves topics on radiation component, wave

propagation in lossy medium, transmission line and array. Thus it is necessary to review

some of the fundamentals on antenna and array theories. The purpose of the following

paragraphs is to describe how antenna and array work. Two examples are used, one

elemental dipole and one regular dipole. The former is used to describe how the near field

and far field of the antenna behave, as well as the representation of pattern and gain. The

latter is adopted to illustrate how the Fourier theory relates to the far field of a real

antenna, and how it governs the behavior of an array. We will cover both topics to the

extent that helps understand the microwave lens design in this dissertation.

1. Antenna

The excitation of a real antenna is impressed electric current, as shown in equation (1-

11). Sometimes it might be easy to consider the current as magnetic current, because a

looped electric current can be regarded as magnetic dipole, and vise versa. We stay with

the electric current representation in this dissertation.

Let us assume a current J i(r0), where r0 implies the current distribution coordinate.

Now the fields at observation coordinate of r become function of J i . Typical method for

solving equation (1-10)-(1-11) is the potential theory. Because the divergence of H field

is zero, from vector identity, H can be expressed asH A, where A is called the

magnetic potential vector. If we substitute it back into (1-10), equation (1-42) is yielded,

due to which the electrical scalar potential can also be defined based on   E jA.

Reorganizing equations (1-10)-(1-11) and using potentials and vector identity, equation

(1-43) can be found.

 (EjA) 0 (1-42)

(  A) 2A(j ) (2 j)AJi (1-43)

Lorentz Gauge assumes that ( A)(j ) , after applying this condition. (1-

43) becomes an inhomogeneous equation of (1-44).

2 2

 Ak AJi (1-44)

14

A can be solved from equation (1-44), the result is shown in (1-45). Note the solution of

A , it is in the observation coordinate, while the integration is over source coordinate.

0

0

| |

0

0

0

( ) 1 ()

4 | |

jk r r

i

r

Ar J r e dv

 r r

 



  (1-45)

After obtaining the magnetic potential, the electrical and magnetic fields can be found

from the detailed derivation in Appendix A. The results are given in (1-46)-(1-47).

E j A 1 ( A)

j



 

     



(1-46)

H A (1-47)

It is observable that given any current distributions, the electrical and magnetic fields at

observation point r can be solved by two steps. 1) solve the potential by integration of the

current over its supporting structure; 2) solve the fields by relationships given in (1-46)-

(1-47). This seems straight forward, however, in reality, to sense the exact current

distribution on the supporting structure itself can be extremely hard. A typical way of

solving the current distribution Ji(r0) is assuming it has a format governed by linear

combinations of basis functions weighted by unknown coefficients. By substituting it into

(1-45), the fields can be solved. After applying boundary conditions, equations can be

yielded to solve these unknowns. We will discuss the concepts of solving J i(r0)and the

fields using numerical simulations in section 1.4.

To address more antenna basics, we use an elemental dipole that has been explicitly

solved in Appendix A as an example. The infinitesimal dipole lies along the z axis, as

shown in Figure 1-7. Assume the current on it is 0 0 0 0 JzJ(x)(y)(z ). According to

Appendix A, A(r) , E(r)and H(r)are solved as shown in (1-48)-(1-50).

Figure 1- 7. Elemental Dipole along z Axis

x y

z

r





r0 z

15



0 0 0 ( ) cos sin

4 4 4

A r zJejkr rJ ejkr J e jkr

r r r

  

  

  

   (1-48)



0

( ) sin ( 1)

4

H r J e jkr jk

r r

 





  (1-49)

0 0 2

2 2

( ) 2 cos ( 1) sin ( 1)

4 4

E r r j J ejkr jk jJ ejkr k jk

k r r r k r r r

 

  

 

 

      (1-50)

From (1-49)-(1-50), a couple of things are observed. First, the fields produced by

infinitesimal dipole along z direction travel along radial direction. Second, it produces E

field polarized waves along r and  , and H field polarized waves along . Third, both E

and H have components attenuating in the order(s) of1/ r . High order fields decay much

faster than that of the low order fields. In the far field, the fields can be approximated by

the lowest order term, resulting in equations (1-51) and (1-52). It is noticed that in the far

field E=ηH, and the cross product of the unit vector between E and H gives the wave

propagation direction.



0 ( ) sin

4

jkr

f

H r jkJ e

r

 





 (1-51)



0 ( ) sin

4

jkr

f

E r jkJ e

r

  





 (1-52)

High order fields exist at the near field of the elemental dipole, and this applies to most of

the real antennas. Different from the standing wave mentioned in section 1.1, the high

order fields behave like evanescent waves that die down shortly after increasing r. If we

look at the Poynting vector of the fields in (1-53), there are both reactive power, stored

along  and r , and real power that only travels along r direction. Figure 1- 8 shows the

radiated power density versus  in a linear scale. This type of pattern information is very

important to evaluate the beam width of the port design in microwave lens. If a radiation

element is designed to send signal into certain direction, the beam pattern is critical

information to calculate the amount of power delivered into a given receiving aperture.

* 02 2 02 2 3

2 3 2 3 2

sin 2 1( ) sin 1( 2 )

(4 ) (4 )

S E H J j jk rJ j k k

k r r r k r r r

   



 

        (1-53)

16

2. Array

Identical antennas operating close to each other are considered as an array. In doing so,

electronic steerable beams can be produced by simply varying the phase of the antennas.

This section covers the basis of array concepts by reviewing its relationship to spatial

Fourier transforms. The amplitude and phase tapering will be discussed by examples of

uniform spacing linear dipole array.

Figure 1- 8. Elemental Dipole Linear Scale Power Density Pattern versus 

In practical design, as far as radiation concerns, the far field is more of an interest. Thus

we can apply the far field approximation of rr0 before calculating the Green’s function

integration. As it is shown in the Appendix A, after applying Taylor expansion of

|rr0|, the Green’s function term in equation (1-45) becomes (1-54).

2 2 3

0 0 0

0 0 0 2

| | [ cos s2in ( )]

2 2 3

0 0 0 0

0 0 2

| | cos sin ( )

2

ejk r r ejk r r r r rr

r r r r r r

r r



 



 

   

 

   

 (1-54)

where 0 0  r r . When 0 rr, the denominator of (1-54) approaches r , and the high

order term in the phase can be dropped out too. (1-54) becomes

17

2 2

0 0 0

0 0

| | cos sin

2

0

( )

| |

jk r r jkr jkr jkr

r e e e

r r r



    



 (1-55)

Assume Dmax is the maximum dimension of the antenna. It has been proved in the

literature that as long as 2

max r(D ) / , the 2

0 r /rphase term can be ignored. Hence:

0

0 0 0

| |

cos

4 | 0 | 4 4

jk r r jkr jkr

e e ejkr e ej k r

r r r r



  

   

 



 (1-56)

Substitute the above result into equations (1-45)-(1-47), the H and E far fields are solved

as shown in (1-57)-(1-58).

 0

0

0 0 [ ( )]

4

jkr

jk r

f

r

H jke k Jr e dv

 r



     (1-57)

  0

0

0 0 0 [( ) ( )]

4

jkr

jk r

f

r

E jk e kJ k Jr e dv

r







      (1-58)

These equations apply to all types of real antennas. Interestingly, the integration parts of

both equations have a format of Fourier transforms between k and r0 domains. Let us

take (1-58) as an example, assume the effective current is named as J(r0), pattern is

denoted by f (,). The field pattern term (the integral portion) is actually the Fourier

transform of the current along the directions that are normal to propagation directionk , as

shown depicted by (1-59). The inverse transform is (1-60).

 0

0

0 0 ( , ) ( ) ( ) jk r

r

f   J k J r e  dv

    (1-59)

 0

0 3

( ) 1 ( )

(2 )

J r J k e j k rdk



 

    (1-60)

Suppose another identical antenna is located at r1r0, as illustrated by Figure 1- 9, we

now want to estimate the total radiation fields at observation point r . The solution is

approached in this way: first solve the field for the position translated antenna, and then

use superposition to estimate the total fields. Note this ignores the mutual couplings

between the two elements, however; in practical design, it has led to a reasonable

estimation.

18

Figure 1- 9. Shifted Current Source in Free Space

The current source at r1r0can be written as





1 1 0

1 1 1 0

( )

1 0 3 1 1

3 1 1

( ) 1 ( )

(2 )

1 [ ( ) ]

(2 )

j k r r

jk r jk r

J r r J k e dk

J k e e dk





  

 

   



 







(1-61)

The forward transform gives

 



1 1 0 0

0

1 1 0 0

0

( )

1 0 3 1 1 0

( )

3 1 10

( ) 1 { ( ) }

(2 )

1 ( ){ }

(2 )

jk r r jk r

r

jk r r jk r

r

J r r J k e dk e dv

J k e e dkdv





   

 

   



 



 

 

(1-62)

Apply the relationship (1-63) in (1-62), we get (1-64).

( 0) 3

ej k rrdk(2 )  (rr0 ) (1-63)

  0  1

0

3

1 0 3 1 1 0 0 1

( ) 1 ( )(2) ( ) ( )

(2 )

jk r jk r

r

J r r J k   r r e dv J k e



  

        (1-64)

Thus the pattern of a shifted antenna is a product between its original pattern and a phase

shift term.

1

1 0 0 ( , ) ( , ) jk r

r r r f   f   e 

  (1-65)

The total pattern for the two elements is

x

y

z

r0

r1

r

k

k '

19

( , ) ( , )(1 jk r1)

Tf   f  e  (1-66)

It is easy to extend the concepts to N identical antennas with relative position of ri r0 .

The total pattern is equal to the elemental pattern times the phase term due to the array

factor, as shown below

1

( , ) ( , )(1 i)

N

j k r

N

i

f   f  e 



  (1-67)

In reality, the array elements are not necessarily to be fed by the same amplitude and

phase as each other. As shown below, we develop the pattern expression for linear,

uniform spacing array, which is a typical feeding scheme for the microwave lens design.

Suppose each antenna of the linear array is fed by amplitude i a and phase i

 , as indicated

in Figure 1- 10, the current distribution in (1-59) will be weighted by factor of jk i

i a e  .

This constant factor will eventually be translated into the array factor of equation (1-67).

This leads to the particular result shown in (1-68).

Figure 1- 10. Uniform S