NUS/ECE EE4101 Antenna Arrays 1 Introduction Antenna arrays are becoming increasingly important in wireless communications. Advantages of using antenna arrays: 1. 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. Hon Tat Hui 1 Antenna Arrays NUS/ECE EE4101 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. Hon Tat Hui 2 Antenna Arrays NUS/ECE EE4101 2 Two Element Arrays z Far field observation point r1 Dipole 2 I 2 Ie j d Dipole 1 r I1 I r1 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). Hon Tat Hui 3 Antenna Arrays NUS/ECE EE4101 E1 = far-zone electric field produced by antenna 1 = E2 = far-zone electric field produced by antenna 2 = aˆ E1 aˆ E2 kI1d e jkr kd e jkr sin θ j cos θ I1 E1 j 4 r 2 4 r kI 2 d e jkr kd e jkr sin θ j cos θ I 2 E2 j 4 r1 2 4 r1 1 1 Use the following far-field approximations: 0 1 1 r1 r e jkr1 e jk r d cos Hon Tat Hui 4 Antenna Arrays NUS/ECE EE4101 The total E field is: E E1 E2 aˆ kd e jkr jkd cos aˆ j θ I I e cos 1 2 4 r kI1d e jkr j jkd cos aˆ j cos θ 1 e e 4 r kId e jkr aˆ j cos θ AF 4 r where Hon Tat Hui 5 Antenna Arrays NUS/ECE EE4101 AF Array Factor 1 e j e jkd cos j 1 e e jkd cos 2e j 1 kd cos 2 1 cos kd cos 2 The magnitude of the total E field is: 1 I1 I 2e jkd cos I1 jkr kId e E aˆ j cos θ AF 4 r radiation pattern of a single Hertzian dipole AF Hon Tat Hui 6 Antenna Arrays NUS/ECE EE4101 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: 1 1 1 AFn AF 2cos kd cos 2 Hon Tat Hui 7 where is a constant to make the maximum value of |AFn| equal to one. Antenna Arrays NUS/ECE EE4101 Examples of array patterns using pattern multiplication: Array pattern of a two-element array of Hertzian dipoles ( = 0°, and d = /4) 1 1 1 1 AFn 2cos kd cos 2cos cos 2 2 2 Hon Tat Hui 8 Antenna Arrays NUS/ECE EE4101 Array pattern of a two-element array of Hertzian dipoles ( = -90°, and d = /4) 1 1 1 1 AFn 2cos kd cos 2cos cos 2 2 2 2 Hon Tat Hui 9 Antenna Arrays NUS/ECE EE4101 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 Hon Tat Hui = |F()| Array factor |A()| = Array pattern (normalized) 10 Antenna Arrays NUS/ECE EE4101 3 N-Element Uniform Liner Arrays (ULAs) Dipoles are parallel to the z direction Far field observation point y r Dipole 1 rN-1 x Dipole N d An N-element uniform antenna array with an element separation d Hon Tat Hui 11 Antenna Arrays NUS/ECE EE4101 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 1 e j ( kd cos ) e j 2( kd cos ) e j ( N 1)( kd cos ) sin N N 1 j N 2 e 2 e j ( n1) n 1 sin 2 where kd cos and 0 , 2 Hon Tat Hui 12 Antenna Arrays NUS/ECE EE4101 The normalized array factor is: sin N 1 2 AFn sin 2 where is a constant to make the largest value of |AFn| equal to one. Note that is not necessarily equal to N. The relation between |AFn|, , d, and is shown graphically on next page. Note that |AFn| 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 |AFn| expression. Hon Tat Hui 13 Antenna Arrays NUS/ECE EE4101 |AFn( )| = kd cos + kd kdcos The relation between AFn,, d, and Hon Tat Hui 14 Antenna Arrays NUS/ECE EE4101 Properties of the normalized array factor AFn: 1. |AFn| is a periodic function of , with a period of 2. This is because |AFn( + 2)| = |AFn()|. 2. As cos() = cos(-), |AFn| is symmetric about the line of the array, i.e., = 0 & . Hence it is enough to know |AFn| for 0 . 3. The maximum values of |AFn| occur when (see Supplementary Notes): 1 (kd cos ) m , m 0,1,2, 2 2 1 ( 2m ) max main beam directions cos 2 d Hon Tat Hui 15 Antenna Arrays NUS/ECE EE4101 Note that there may be more than one angles max corresponding to the same value of m because cos-1(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: 2nd grating lobe Hon Tat Hui 1st grating lobe Main lobe 16 1st grating lobe 2nd grating lobe Antenna Arrays NUS/ECE EE4101 2nd grating lobe 1st grating lobe 1st grating lobe Main lobe 2nd grating lobe |AFn( )| (1) 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 . Hon Tat Hui Visible region kd =kdcos 17 General conditions to avoid grating lobes with ϵ [0,2] and d ϵ [0.5,]: 1.For 0 < , the requirement is: kd + 2 2. For < 2, the requirement is: kd - 0 Antenna Arrays NUS/ECE EE4101 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 |AFn| cannot be determined by the formula in no. 3. 6. The main beam directions max are not related to N. They are functions of and d only. 7. The nulls of |AFn| occur when: n n 1,2,3, , N 2 n N ,2 N ,3 N , 2n 1 null null directions cos ( ) N 2 d Hon Tat Hui 18 Antenna Arrays NUS/ECE EE4101 Note that there may be more than one angles null corresponding to a single value of n because cos-1(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 = 1) to the first maximum (m = 0). This means a narrower main beam and an increase in the directivity or gain of the array. 10.The angle for the main beam direction (m = 0) can be controlled by varying or d. Hon Tat Hui 19 Antenna Arrays NUS/ECE EE4101 Example 1 A uniform linear array consists of 10 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 = 10 main beam dirction max Hon Tat Hui 2 m 2 60 cos 2 d 1 m2 cos 60 0.5 m2 45 360 315, 4 20 when m 1 Antenna Arrays NUS/ECE EE4101 Other values of corresponding to other values of m are outside the range of 0 2 and are not included. sin 5 cos 1 2 AFn 1 sin cos 2 2 10 Hon Tat Hui 21 Antenna Arrays NUS/ECE EE4101 4 Mutual Coupling in Transmitting Antenna Arrays 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. Hon Tat Hui 22 Antenna Arrays NUS/ECE EE4101 Consider two transmitting antennas as shown on next page. They are separated by a distance of d and the excitation voltage sources, Vs1 and Vs2, 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. Hon Tat Hui 23 Antenna Arrays NUS/ECE EE4101 Dipoles are parallel to the z direction Far field observation point, r y Dipole 1 x d Dipole 2 Two coupled dipoles Hon Tat Hui 24 Antenna Arrays NUS/ECE EE4101 Antenna 2 Antenna 1 Vs1 Terminal current I1 Zg1 Excitation voltage source Coupled voltage I1 a 1 b1 I2 a2 Vs2 V12 Antenna Self-impedance 25 V21 Z22 Zg2 Z11 Zg1 Hon Tat Hui a2 b2 d Vs1 Source internal Impedance Vs2 Zg2 a1 b1 I2 b2 Antenna Arrays NUS/ECE EE4101 Z12 mutual impedance with antenna 2 excited V12 I2 coupled voltage across antenna 1's open-circuit terminal excitation current at antenna 2's shorted terminal Voc12 I 2 I1 0,Vs1 0 Voc12 a I2 b Hon Tat Hui 26 c d Antenna Arrays NUS/ECE EE4101 Z 21 mutual impedance with antenna 1 excited V21 I1 coupled voltage across antenna 2's open-circuit terminal excitation current at antenna 1's shorted terminal Voc 21 I1 I 2 0,Vs 2 0 I1 Hon Tat Hui a b Voc21 c d 27 Note that for passive antennas, Z12 = Z21 Antenna Arrays NUS/ECE EE4101 Using the mutual impedance, the coupled voltages V12 and V21 can be expressed as follows: V12 Z12 I 2 V21 Z21I1 I1 and I2 are the actual terminal currents at the antennas when there is mutual coupling effect. From the antenna equivalent circuits, Vs1 V12 I1 Z g 1 Z11 Hon Tat Hui Vs 2 V21 I2 Z g 2 Z 22 28 Antenna Arrays NUS/ECE EE4101 Is1 and Is2 are the terminal currents at the antennas when there is no mutual coupling effect. Vs1 I s1 Z g1 Z11 Vs 2 I s2 Z g 2 Z11 Our aim is to express I1 and I2 in terms of Is1 and Is2. Vs 2 V21 I2 Z g 2 Z 22 Vs1 V12 I1 Z g 1 Z11 I1Z 21 I s2 Z g 2 Z 22 I 2 Z12 I s1 Z g1 Z11 Hon Tat Hui 29 Antenna Arrays NUS/ECE EE4101 From these two relations, we can find: I1 Z12 I s1 I s2 Z11 Z g1 Z12 Z 21 1 Z11 Z g1 Z 22 Z g 2 , I2 Z 21 I s2 I s1 Z 22 Z g 2 Z12 Z 21 1 Z11 Z g1 Z 22 Z g 2 That is: 1 I1 I s1 Z12 I s 2 D Hon Tat Hui 1 I s1 I 2 I s 2 Z 21 D 30 Antenna Arrays NUS/ECE where EE4101 Z12 Z 21 D 1 Z11 Z g1 Z 22 Z g 2 Z12 Z12 Z11 Z g1 Z 21 Z 21 Z 22 Z g 2 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 10 and 6). 1 jkd cos Vector E =AF I1 I 2e magnitude, not I1 absolute value Hon Tat Hui 31 Antenna Arrays NUS/ECE EE4101 1 E I1 I 2e jkd cos I1 1 I s1 e jkd cos I s1 Z12 I s 2 I s 2 Z 21 I1D 1 I s1 I s 2e jkd cos Z12 I s 2 I s1e jkd cos I1D with Z12 Z 21 I s1 Is2 j jkd cos j jkd cos j 1 e e with Z12 e e e I1D I s1 I s1 j kd cos j kd cos j Z12e 1 e 1 e I1D original pattern Hon Tat Hui additional pattern 32 Antenna Arrays NUS/ECE EE4101 We see that the array pattern now consists of two parts: the original array pattern plus an additional pattern: Z12 e j 1 e j kd cos which has a reverse current phase - and a modified amplitude with a multiplication of a complex number Z’12ej. Note that all parameters in the above formula can be calculated except I1 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. Hon Tat Hui 33 Antenna Arrays NUS/ECE EE4101 Absolute value Example 2 Find the normalized array pattern |En| on the horizontal plane (=/2) of a two-monopole array with the following parameters with mutual coupling taken into account: I s1 1, I s 2 e j , I s1 I s 2 1, 150 d 4, 4 Z12 Z 21 21.8 - j 21.9 Ω Z11 Z 22 47.3 j 22.3 Ω Z g1 Z g 2 50 Ω d I s1 Hon Tat Hui 34 I s2 Antenna Arrays NUS/ECE EE4101 Solution I s1 1, Is2 e I s1 I s 2 1 I s1 0, j As the required array pattern |En| is on the horizontal plane, it is equal to the normalized array factor |AFn|. I s 2 150 2.62 rad 2 kd 4 2 Z12 Z 21 Z12 0.16 j 0.26 Z11 Z g 1 Z 22 Z g 2 Z12 Z 21 D 1 1.042 j 0.09 Z11 Z g1 Z 22 Z g 2 Hon Tat Hui 35 Antenna Arrays NUS/ECE EE4101 I s1 1 e j kd cos Z12 e j 1 e j kd cos E AF I1D 0.95 j 0.08 1 e j 2.62e j 2 cos I1 0.16 j 0.26 e j 2.62 1 e j 2.62e j 2 cos 0.94 j 0.37 j 2 cos 1 1.14 j 0.40 e I1 The pattern of f 1 1.14 j 0.40 e j 2 cos next page. Hon Tat Hui 36 is shown on Antenna Arrays NUS/ECE EE4101 f 1 1.14 j 0.40 e j 2 cos Hon Tat Hui 37 Antenna Arrays NUS/ECE EE4101 Normalization The pattern of f attains the maximum value when = 180°. When = 180°, E 180 0.94 j 0.37 j 2 cos 1 1.14 j 0.40 e 180 I1 1.83 I1 Hence we normalize |E| by this factor (1.83/|I1|) to get: Hon Tat Hui 38 Antenna Arrays NUS/ECE EE4101 0.94 j 0.37 1 1.14 j 0.40 e j 2 cos I1 En 1.83 I1 0.52 1 1.14 j 0.40 e j 2 cos The polar plot of |En| is shown on next page. Hon Tat Hui 39 Antenna Arrays NUS/ECE EE4101 En 0.52 1 1.14 j 0.40 e Hon Tat Hui 40 j 2 cos Antenna Arrays NUS/ECE EE4101 The case when there is no mutual coupling is shown below for comparison. 1 En no mutual coupling effect 1 e j e jkd cos where is a constant to make the largest value of |AFn| equal to one ( = 1.73). Hon Tat Hui 41 Antenna Arrays NUS/ECE EE4101 References: 1. C. A. Balanis, Antenna Theory, Analysis and Design, John Wiley & Sons, Inc., New Jersey, 2005. 2. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York, 1998. 3. David K. Cheng, Field and Wave Electromagnetic, AddisonWesley Pub. Co., New York, 1989. 4. John D. Kraus, Antennas, McGraw-Hill, New York, 1988. 5. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc., New Jersey, 2007. 6. Joseph A. Edminister, Schaum’s Outline of Theory and Problems of Electromagnetics, McGraw-Hill, Singapore, 1993. 7. Yung-kuo Lim (Editor), Problems and solutions on electromagnetism, World Scientific, Singapore, 1993. Hon Tat Hui 42 Antenna Arrays