antennas & rf devices lab. lab seminar characteristic modes

Hanyang University

1/30 Antennas & RF Devices Lab.

Antennas & RF Devices Lab.

LAB Seminar

Characteristic Modes

Hong Youngtaek

Hanyang University

2/30

Contents

CHAPTER 5

CHARACTERISTIC MODE THEORY FOR N-PORT NETWORKS

• 5.1 Backgrounds

• 5.2 Characteristic Mode Formulations For N-Port Networks

• 5.3 Reactively Controlled Antenna Array Designs Using Characteristic Mode

5.3.1 Problem Formulation

5.3.2 Design and Optimization Procedure

5.3.3 Design Examples

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5.1 BACKGROUNDS

The characteristic mode (CM) theories for perfect electric conductors, for objects in

multilayered medium, and for dielectric objects are discussed in Chapters 2, 3, and 4,

respectively.

The characteristic currents themselves provide valuable information for the design of these

feeding structures.

These CM theories are more suitable to electromagnetic objects in electrical small and medium

sizes.

Characteristic modes can be also defined for electrically large multiport antenna systems

The CM theory for N port networks is extensively applied in the analysis and design of multiport

antenna systems , antenna arrays , and N-port loaded scatterers .

The goals in these studies were to obtain designated radiation pattern/scattering pattern over

desired frequencies or to achieve desired impedance properties at the ports. Both of the far-

field radiation performance and the impedance property are essentially determined by the

radiating currents over the antennas or scatterers.

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5.1 BACKGROUNDS

Therefore, techniques based on the characteristic currents on the ports were developed to

shape the current distributions over the antennas or scatterers.

In particular, the following design goals were achieved by shaping the current distributions

using various CM-based techniques:

• Realized mode decoupling elements designs at multiple input multiple output antenna ports to

decouple the mutual couplings [2, 3];

• Achieved broadband impedance bandwidth by loading the ports with non-Foster elements [4];

• Achieved various far-field radiation performances (e.g., power pattern, gain, and sidelobe level

(SLL)) by loading the ports with lumped reactive elements [5, 7, 8, 10];

• Synthesized power patterns by controlling the port currents using the far-field orthogonality

property [ 6];

• Achieved ultra-wideband antenna arrays by controlling the port excitations [9].

Fundamental theory of the characteristic modes for N-port networks in Section 5.2.

Beam scanning circular array, Yagi-Uda antennas and tightly coupled ultra-wideband antenna

aπay

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5.2 CHARACTERISTIC MODE FORMULATIONS FOR N-PORT NETWORKS

The CM theory for N-port networks was first developed by Harrington and Mautz in 1973

The CMs for N-port networks are computed from the N-port network’s impedance matrix.

Considering an N-port network as shown in Figure 5.1, an impedance matrix is defined to relate

the voltages and currents at each port.

From Figure 5.1, the total voltage and current at

the nth terminal plane are given by the following:

Therefore, the impedance matrix that relates the

total port voltages and currents can be written

as follows

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From Equation (5 .3), we can find that

Equation (5 .5) states that Zij is obtained by driving port j with the current I, keeping

all the other ports to be open-circuit( In = 0, n ≠ j), and calculating or measuring the

open-circuit voltage Vi at the port i.

Zij is the mutual impedance between ports i and j when all the other ports are open-circuit.

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The impedance matrix [Z] is generally a complex matrix for an arbitrary N-port network.

If the system does not contain any active devices or non-reciprocal material, [Z] is a

symmetrical matrix.

The characteristic modes for an N-port network are defined in a similar way as those for PEC

bodies and can be solved from the following generalized eigenvalue equation:

Jn is the nth characteristic current at the ports, and λn is the corresponding eigenvalue of the

nth characteristic mode.

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For convenience, the characteristic currents are normalized such that each of them delivers unit

power:

where denotes the transpose of the vectors. The orthogonality in the normalized

characteristic currents are given by:

( )T

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The characteristic currents form a set of basis to express the radiating currents on the N-port

antenna system, that is, the modal solution of the characteristic modes for N-port network:

where Voc is the open-circuit voltage vector that consists of the voltage for each port.

Similar to the conventional CM theory for PEC objects, the modal solution in Equation (5.13) has

two important components to determine the modal expansion coefficient.

The denominator depends on the eigenvalue λn which is determined by the N-port antenna

system’s configuration and materials. The numerator is determined by an inner product of the

characteristic currents with the open-circuit voltage at each port.

As can be seen, the modal significance can be defined to show the intrinsic radiation capability

of each mode. It is given by:

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Considering the contributions from the modal significance and the couplings between the

characteristic currents and the voltage sources, the modal solution in Equation (5.13) can be

reduced to:

where an is a complex modal expansion coefficient for the nth mode

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5.3 REACTIVELY CONTROLLED ANTENNA ARRAY DESIGNS USING

CHARACTERISTIC MODES

Reactively controlled antenna array (RCAA) is attractive in wireless communication systems

due to its many distinct advantages including simple feed port configuration, high reliability,

small power consumption, and low cost .

RCAA is an N-port radiating system consisting of one element connected to an RF port and a

number of surrounding parasitic elements with reactive loadings that can be realized by

reversely biased varactor diodes.

Several circular antenna arrays are successfully synthesized according to main beam direction

and maximum SLL.

These antenna array designs prove that the characteristic modes for the N port network give

important decisions to the optimal loading designs for antenna arrays.

These examples also demonstrate the effectiveness of the proposed synthesis method based

on characteristic modes and differential evolution(DE) algorithm.

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Figure 5.2, There are six passive half-wavelength dipole antenna elements surrounding

an active element. The active dipole antenna is located at the center of the circular array.

The Thevenin equivalence of a general N-port reactively controlled antenna array is shown in

Figure 5.3. As can be seen, the terminal impedance characteristics of the antenna array are

represented by an impedance matrix [ZA], and the terminal impedance characteristics of the

loading/excitation network is described by a diagonal impedance matrix [ZL].

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Specifically, by applying a unit voltage source at each port when keeping all the other ports

open-circuit, the current at each port can be solved using the MoM.

Because unit voltage source is applied to each port, the voltage matrix is an identity matrix.

Evidently, if a loading/excitation network is connected to the multiport antenna system, the port

current should be computed from a new matrix equation

The goal of the reactively controlled antenna array design is to find a set of reactance loadings

such that the beam can steer to a designated direction and the maximum SLL can be

suppressed below a given level.

Solving the MoM matrix equation is quite time-consuming. The conventional method thus may

fail if antenna arrays with large number of elements are considered.

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In order to avoid the solution of the MoM matrix equation in each step of the cost function

evaluation, a new synthesis method based on the CM theory and DE algorithm is developed.

where an are the coefficients to be determined for designated radiation performance of the

RCAA. Similarly, the radiation field E radiated by J can be also written as the superposition of

the N characteristic field En that is produced by the characteristic currents J :

The goals of the array synthesis problem are to steer the maximum gain direction to a desired

direction φ。, maximize the front-to-back (FIB) ratio of the radiation pattern, and suppress the

maximum SLL by tuning the modal expansion coefficients an.

After getting the optimal set of modal expansion coefficients, the reactive loadings (diagonal

elements in the matrix [ZL]) can be determined directly from an.

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Recently, DE algorithm has been widely

applied to solve many electromagnetic

problems . It has been proven that the DE

algorithm has excellent search ability and

faster convergence rate over most of the

optimization algorithms such as the GA

and the PSO methods.

Figure 5.4 shows the flowchart of the DE

algorithm. Upon the solution space, cost

function (or objective function) and other

fundamental parameters are defined, and

the DE algorithm starts with the

initialization of the evolutionary

population.

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DE알고리즘은 1995년 R. Storn과 K. Price이 발표한

진화이론을 바탕으로 만들어진,

통계적(확률적으로~)

직접탐색(원하는 해를 직접 찾는~)

전역최적화(모든 수 영역에서 찾는~)

알고리즘입니다.

DE알고리즘은 목적함수를 최적화 시키는데 있어 적은

수의 변수로 조작이 가능하다는 장점을 가지고 있습니다.

먼저~ DE알고리즘이 과연 어떤 알고리즘인지 Flow

Chart부터 보겠습니다

간단히 설명하면,

윗쪽 행렬은 기존세대, 그 기존세대들의 각 개체를

진화과정에 투입하여 아래의 새로운 세대를 만들어내고,

이 과정을 계속해서 반복하면서, 결국 목적함수의 해를

찾아내는 알고리즘이죠.

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1. 최초 세대(모집단) 생성

DE알고리즘은 알고리즘의 연산이 시작되면 가장 먼저 사용자가 지정한 초기 탐색범위 내에서 아래와 같은 최초 세대(모집단)을 만들어냅니다.

P1 = [X1,1, X2,1, X3,1, ... , XNP,1 ]

Xi,1 = [ x1,i,1, x2,i,1, .... ,xD,i,1 ]

xj,i,1 = randj (0,1) × ( xjU - xjL ) + xjL

식으로는 위와 같이 표현할 수 있죠.

여기서 xjU와 xjL은 각각 j번째 파라미터의

초기탐색경계의 상한경계와 하한경계를 의미하고 모집단

개체들의 파라미터는 그 사이에서 랜덤값으로 선택합니다.

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2. 돌연변이

돌연변이 과정은 모집단에 포함된 3개의 개체를 이용해서

새로운 파라미터로 이루어진 하나의 돌연변이 개체를

생성하는 과정으로,

그 방법을 Flow Chart로 그려보면 아래와 같습니다.

그리고 식으로 표현하면 v = Xr3 + F × ( Xr1 - Xr2 )

이렇게 되죠.

위 식에서 v는 돌연변이개체, F는 가중인자를 의미하며

가중인자는 0 < F < 1인 값으로 알고리즘의 사용자가

설정하게 되어 있습니다.

또한 Xr1, Xr2, Xr3는 모집단 G에 포함된 개체로 각각 다른

개체이며,

미리 선택되어 있는 Target개체, 즉 진화과정에 투입된

개체를 제외한 다른 개체들 중에서 선택하도록 되어

있습니다.

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3. 교배

교배과정은 Trial개체를 생성하는 과정입니다.

Trial개체는 돌연변이 과정에서 생성된 돌연변이개체와

Target개체를 사용해서 만들어내는데,

이 Trial개체가 진화과정에 투입된 Target개체와 다음

세대의 구성개체가 되기 위해 경쟁을 하는 개체이며,

Trial개체를 생성하는 방법은 아래 식과 같습니다.

if randi (0,1) ≤ CR u(i) = v(i)

otherwise u(i) = Xtarget(i)

여기서 CR은 돌연변이 개체와 Target개체의 교배수준을

결정하는 상수로, 이는 알고리즘의 사용자가 미리

설정하게 되어 있습니다.

또한 u는 교배과정을 통해서 생성되는 Trial개체를

의미하며, u(i)는 Trial개체의 i번째 파라미터를 의미합니다.

0~1사이의 값들 중 무작위로 한 값을 선택하여,

그 값이 CR보다 더 작은 값을 가지고 있으면 돌연변이

개체의 파라미터를 Trial개체의 파라미터로 설정하고,

CR보다 더 큰 값을 가지고 있다면 Target개체의

파라미터를 Trial개체의 파라미터로 설정합니다.

CR값은 [0.0 1.0]의 범위에서 설정하며,

한 번 설정된 값은 알고리즘이 최적화를 완료할 때 까지

같은 값을 계속 사용합니다..

돌연변이과정에서 사용된 F와 교배과정에서 사용된

CR은 모두 [0.0 1.0]의 범위 내에서 사용자가 설정하는

상수로,

알고리즘이 해를 찾아가는 속도에 많은 영향을 주는

인자입니다.

이 상수들에 대해 F = 0.8~0.9, CR = 0.5~0.9의 값들을

권장하고 있지만,

가장 적합한 값은 몇 번의 시행착오를 통해서

얻어내야합니다.

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4. 선택

선택은 Target개체와 교배를 통해서 생성된 Trial개체 중,

다음 세대에 더 적합한 개체 개체를 선택하는 과정입니다.

적합성에 대한 판단기준은??

두 개체를 구성하는 파라미터를 각각 목적함수에 대입하여,

두 개체 중 Cost Value가 더 낮은 개체를 다음 세대의

개체로 합니다.

한 세대를 이루는 모든 개체를 위와 같은 진화과정에

투입하여

그 결과물로 다음 세대를 만들게 되면

이 다음 세대는 이전 세대보다 더 좋은 Cost Value값을

가진 세대가 되고,

이는 목적함수의 해에 더 근접한 세대로 진화했다는 것을

의미하게 됩니다.

이 과정을...

계속해서 반복합니다...

세대를 만들고..진화과정에 투입해서 또 다음 세대를

만들고..또 다음 세대..또또 다음 세대..

원하는 목적함수의 해가 찾아질때까지...

계속합니다...

http://blog.naver.com/i_aries/220330974243

http://egloos.zum.com/incredible/v/5265158

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To apply the DE algorithm to find the optimal modal expansion coefficients an such that the

main beam direction, the FIB ratio of the radiation pattern, and the SLL meet the specified value,

construction of a good cost function is of great importance.

The following cost function is defined for the reactively controlled beam steering antenna array

synthesis problem:

The term is included in the cost function to enforce the real part of the impedance loading

to be zero to allow purely reactive loadings.

The voltages implemented at the port of the all the elements can be expressed as:

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The impedance matrix [ZL] for the loading/excitation network is a diagonal matrix and is given

by:

The port currents [Iport] can be computed directly using the modal solution in Equation (5.21).

Because the impedance matrix [ZA] keeps unchanged and the voltage excitation [V port] in

Equation (5 .28) is known before optimization,

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Because we only need to constrain the main beam direction, backward radiations, and the value

of the loaded impedances in this design, the weighting factor w3 in the cost function of Equation

(5.27) is set to zero.

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Figure 5.5 shows the steered radiation patterns achieved by tuning the reactance loadings in the

seven-element circular aπay.

The backward radiations are suppressed to a

very low level in all of the three beams

Table 5.1

As can be seen, only purely reactance loadings

are necessary to steer the beam in different

directions.

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As the second design case, radiation pattern with low SSL is expected from a three-ring

concentric circular antenna aπay (CCAA). Figure 5.6 shows the configuration of the CCAA.

Because we only need to constrain the main

beam direction, the maximum SLL, and the

loaded impedance values, the weighting factor

w2 is set to zero in the cost function of Equation

(5.27).

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Two CCAA is designed with the CM-based approach. The radiuses of the two CCAAs are the

same, (r1,r2,r3) = (0.55λ,0 61λ,0 75λ) , and the number of antenna elements on the three rings for

the two CCAA are ( N1 ,N2,N3) = ( 4,6,8)and (8,10,12 ), respectively.

Figure 5.7 shows the normalized radiation patterns obtained using the optimal loaded reactance

values given in Table 5.2.

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As can be observed from Figure 5 .7, the obtained maximum SSL for the 19 (=4+6+8+ 1) and 31

(=8+ 10+ 12+ 1) element cases are - 22.17 and - 23.54dB,

It is evident that the SLL of the CCAAs is suppressed to a very low level by using the reactive

loadings. As compared to conventional low-sidelobe antenna array designs, the present

approach avoids the costly and complicated feeding networks.

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Thank you for your attention

antennas & rf devices lab. lab seminar characteristic modes

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