Antenna Arrays 1 Introduction

Transcription

1 Introduction Antenna arrays are becoming increasingly important in wireless communications. Advantages of using antenna arrays:. They can provide the capability of a steerable beam (radiation direction change) as in smart antennas. 2. They can provide a high gain (array gain) by using simple antenna elements. 3. They provide a diversity gain in multipath signal reception. 4. They enable array signal processing.

2 An important characteristic of an array is the change of its radiation pattern in response to different excitations of its antenna elements. Unlike a single antenna whose radiation pattern is fixed, an antenna array s radiation pattern, called the array pattern, can be changed upon exciting its elements with different currents (both current magnitudes and current phases). This gives us a freedom to choose (or design) a certain desired array pattern from an array, without changing its physical dimensions. Furthermore, by manipulating the received signals from the individual antenna elements in different ways, we can achieve many signal processing functions such as spatial filtering, interference suppression, gain enhancement, target tracking, etc. 2

3 2 Two Element Arrays Dipole 2 d Dipole z I Ie 2 I I r j r Far field observation point r r d cos, 0 x Two Hertzian dipoles of length dl separated by a distance d and excited by currents with an equal amplitude I but a phase difference [0 ~ 2). 3

4 E = far-zone electric field produced by antenna = E 2 = far-zone electric field produced by antenna 2 = ˆ E a ˆ E 2 a ki de kde E j θ j θ I 4 r 2 4 r jkr jkr sin cos jkr jkr ki d e kd e E j sin θ j cosθ I r 2 4 r Use the following far-field approximations: 0 r r e jkr e 4 jk rd cos

5 The total E field is: E E aˆ aˆ E 2 aˆ kde j θ I I e 4 r jkr jkd cos cos 2 ki de jkdcos 4 r jkr j j cosθ e e jkr kide aˆ j 4 r cosθ AF where 5

6 AF Array Factor j e e jkdcos j kdcos cos 2 j jkd e e 2e cos kdcos 2 The magnitude of the total E field is: jkd cos I I2e I E jkr kide aˆ j cosθ AF 4 r radiation pattern of a single Hertzian dipole AF 6

7 Hence we see the total far-field radiation pattern E of the array (array pattern) consists of the original radiation pattern of a single Hertzian dipole multiplying with the magnitude of the array factor AF. This is a general property of antenna arrays and is called the principle of pattern multiplication. When we plot the array pattern, we usually use the normalized array factor which is: AFn AF 2cos kd cos 2 where is a constant to make the maximum value of AF n equal to one. 7

8 Examples of array patterns using pattern multiplication: Array pattern of a two-element array of Hertzian dipoles ( = 0, and d = /4) AFn 2cos kd cos 2cos cos

9 Array pattern of a two-element array of Hertzian dipoles ( = -90, and d = /4) AFn 2cos kd cos 2cos cos

10 In many practical arrays, the element radiation pattern is usually chosen to be non-directional, for example the -plane pattern of a Hertzian dipole or a half-wave dipole. Then in this case, the array radiation pattern will be totally determined by the array factor AF alone, as shown in the example below: = Element pattern F() Array factor A() = Array pattern (normalized) 0

11 3 N-Element Uniform Liner Arrays (ULAs) Dipoles are parallel to the z direction Dipole y d r x Dipole N Far field observation point r N- An N-element uniform antenna array with an element separation d

12 The principle of pattern multiplication can be extended to N-element arrays with identical antenna elements and equal inter-element separation (ULAs). If the excitation currents have the same amplitude but the phase difference between adjacent elements is (the progressive phase difference), the array factor for this array is: AF e e e j( kdcos ) j2( kdcos ) j( N)( kdcos ) sin N N ( ) 2 j N j n 2 e e n sin 2 where kd cos and 0, 2 2

13 The normalized array factor is: AF n sin N 2 sin 2 3 where is a constant to make the largest value of AF n equal to one. Note that is not necessarily equal to N. The relation between AF n,, d, and is shown graphically on next page. Note that AF n is a period function of, which is in turn a function of. The angle is in the real space and its range is 0 to 2. However, is not in the real space and its range can be greater than or smaller than 0 to 2, leading to the problem of grating lobes or not achieving the maximum values of the AF n expression.

14 AF n ( ) = kd cos + kd kdcos The relation between AF n,, d, and 4

15 Properties of the normalized array factor AF n :. AF n is a periodic function of, with a period of 2. This is because AF n ( + 2) = AF n (). 2. As cos() = cos(-), AF n is symmetric about the line of the array, i.e., = 0 &. Hence it is enough to know AF n for The maximum values of AF n occur when (see Supplementary Notes): ( cos ), 0,,2, 2 2 kd m m max main beam directions cos ( 2 m) 2 d 5

16 Note that there may be more than one angles max corresponding to the same value of m because cos - (x) is a multi-value function. If there are more than one maximum angles max, the second and the subsequent maximum angles give rise to the phenomenon of grating lobes. The condition for grating lobes to occur is that d (disregarding the value of ) as shown below: 2 nd grating lobe st grating lobe Main lobe st grating lobe 2 nd grating lobe 6

17 2 nd grating lobe st grating lobe Main lobe AF n ( ) st grating lobe 2 nd grating lobe () When d 0.5, no grating lobes can be formed for whatever value of.(2) When d, grating lobe(s) is (are) formed for whatever value of. (3) When 0.5 <d<, formation of grating lobes depends on. Visible region kd =kdcos General conditions to avoid grating lobes with ϵ [0,2] and d ϵ [0.5,]:.For 0 <, the requirement is: kd For < 2, the requirement is: kd - 0 7

18 8 4. There are other angles corresponding to the maximum values for the minor lobes (minor beams) but these angles cannot be found from the formula in no. 3 above. 5. When and d are fixed, it is possible that can never be equal to 2m. In that case, the maximum values of AF n cannot be determined by the formula in no The main beam directions max are not related to N. They are functions of and d only. 7. The nulls of AF n occur when: n n, 2,3,, 2 N n N,2 N,3 N, 2n null null directions cos ( ) 2 d N

19 Note that there may be more than one angles null corresponding to a single value of n because cos - (x) is a multi-value function. 8. The null directions null are dependent on N. 9. The larger the number N, the closer is the first null (n = ) to the first maximum (m = 0). This means a narrower main beam and an increase in the directivity or gain of the array. 0.The angle for the main beam direction (m = 0) can be controlled by varying or d. 9

20 20 Example A uniform linear array consists of 0 half-wave dipoles with an inter-element separation d = /4 and equal current amplitude. Find the excitation current phase difference such that the main beam direction is at 60 ( max = 60 ). Solutions d = /4, max = 60, N = 0 main beam dirction max 60cos m2 2 d 2 m2cos600.5 m , when m 4

21 Other values of corresponding to other values of m are outside the range of 0 2 and are not included. sin 5 cos 2 AFn sin cos

22 4 Mutual Coupling in Transmitting What we studies before about antenna arrays has assumed that the antenna elements operate independently. In reality, antennas placed in close proximity to each other interact strongly. This interaction is called mutual coupling effect and it will distort the array characteristics, such as the array pattern, from those predicted based on the pattern multiplication principle. We need to consider the mutual coupling effect in order to apply the pattern multiplication principle. We study an example of a two-element dipole array. We characterize the mutual coupling effect using the mutual impedance. 22

23 23 Consider two transmitting antennas as shown on next page. They are separated by a distance of d and the excitation voltage sources, V s and V s2, have a phase difference of but an equal magnitude. Hence if there is no mutual coupling effect, the excitation currents also differ by a phase difference of and have an equal magnitude. When the mutual coupling effect is taken into account, the two coupled antennas can be modelled as two equivalent circuits. Now because of the mutual coupling effect, there is another excitation source (the controlled voltage source) in the equivalent circuit. This controlled voltage source is to model the coupled voltage from the other antenna.

24 Dipoles are parallel to the z direction y Far field observation point, r Dipole d Dipole 2 x Two coupled dipoles 24

25 Antenna Antenna 2 Terminal current V s Z g I I 2 a b a 2 d V s2 Z g2 b 2 Excitation voltage a V s V 2 Coupled voltage source I I 2 a 2 V s2 V 2 Z g Z Z g2 Z 22 Source internal Impedance b Antenna Self-impedance b 2 25

26 Z 2 mutual impedance with antenna 2 excited V2 I 2 coupled voltage across antenna 's open-circuit terminal excitation current at antenna 2's shorted terminal V oc2 I 2 I 0, V 0 s V oc2 a b I 2 c d 26

27 Z 2 mutual impedance with antenna excited V2 I coupled voltage across antenna 2's open-circuit terminal excitation current at antenna 's shorted terminal V oc2 I I 0, V 0 2 s 2 I a b V c oc2 Note that for d passive antennas, Z 2 = Z 2 27

28 Using the mutual impedance, the coupled voltages V 2 and V 2 can be expressed as follows: V Z I V Z I I and I 2 are the actual terminal currents at the antennas when there is mutual coupling effect. From the antenna equivalent circuits, I V V V V I s 2 s2 2 2 ZgZ Zg2 Z22 28

29 I s and I s2 are the terminal currents at the antennas when there is no mutual coupling effect. I V s s2 s s2 ZgZ Zg2 Z I V Our aim is to express I and I 2 in terms of I s and I s2. I V Z I s 2 g s V Z IZ 2 2 Z Z g I 2 V Z I s2 2 g 2 22 s2 V Z Z IZ 2 Z g

30 I NUS/ECE From these two relations, we can find: Z Z I I I I 2 2 s s2 s2 s Z Zg Z22 Zg2, I2 Z2Z 2 Z2Z 2 Z Z Z Z Z Z Z Z g 22 g2 g 22 g2 That is: I I Z I I I Z I D D s 2 s2 2 s2 2 s 30

31 where D 2 2 Z Z Z Z Z Z Z Z g 22 g2 Z2 Z g Z2 Z 22 g 2 Z Z Now if we want to find the array pattern E on the horizontal plane (=/2) with mutual coupling effect, then E is just equal to the array factor (see pages 0 and 6). Vector magnitude, not absolute value E =AF I I I e 2 jkd cos

32 E I 2 jkd cos IsZ 2Is2Is2 Z 2Ise ID jkd cos ID jkd cos jkd cos I I e Z I I e with Z Z s s2 2 s2 s 2 2 I cos cos I e e Z e e with e ID Is jkdc os j jkd cos e Z 2e e ID s j jkd j jkd s2 j 2 Is I I e original pattern additional pattern 32

33 We see that the array pattern now consists of two parts: the original array pattern plus an additional pattern: j jkdcos 2 e Z e which has a reverse current phase - and a modified amplitude with a multiplication of a complex number Z 2 e j. Note that all parameters in the above formula can be calculated except I which will be removed after normalization. Normalization of the above formula can only be done when its maximum value is known, for example by numerical calculation. 33

34 Example 2 Absolute value Find the normalized array pattern E n on the horizontal plane (=/2) of a two-monopole array with the following parameters with mutual coupling taken into account: j I, I e, I I, 50 s s2 s s2 d 4, 4 Z2 Z j 2. 9 Ω Z Z j22.3 Ω Z Z 50 Ω g g2 d I s I s2 34

35 Solution I, I e I s s2 I s s2 j I 0, I rad kd s s Z Z Z 2. j. Z Zg Z22 Zg2 As the required array pattern E n is on the horizontal plane, it is equal to the normalized array factor AF n. D Z Z Z Z Z Z g 22 g2 35 j

36 E I cos j jkd cos e Z e e s jkd AF 2 ID 0.95 j0.08 e I j j2.62 j2.62 j j e e e 0.94 j j 0.40 I e j 2cos 2cos e j 2cos The pattern of next page. j 2cos f.4 j0.40 e is shown on 36

37 j 2cos f.4 j0.40 e 37

38 Normalization The pattern of f attains the maximum value when = 80. When = 80, E j j 0.40 I.83 I e j 2cos 80 Hence we normalize E by this factor (.83/ I ) to get: 38

39 E n 0.94 j j 0.40 I.83 I j0.40 e j 2cos e j 2cos The polar plot of E n is shown on next page. 39

40 E j 2cos n j e 40

41 The case when there is no mutual coupling is shown below for comparison. E n no mutual coupling effect j jkd e e where is a constant to make the largest value of AF n equal to one ( =.73). cos 4

42 References:. C. A. Balanis, Antenna Theory, Analysis and Design, John Wiley & Sons, Inc., New Jersey, W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York, David K. Cheng, Field and Wave Electromagnetic, Addison- Wesley Pub. Co., New York, John D. Kraus, Antennas, McGraw-Hill, New York, Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc., New Jersey, Joseph A. Edminister, Schaum s Outline of Theory and Problems of Electromagnetics, McGraw-Hill, Singapore, Yung-kuo Lim (Editor), Problems and solutions on electromagnetism, World Scientific, Singapore,