Antennas EEE6223 Past Paper PDF 2023-2024

EEE6223 Antennas 2023-2024 Antenna Arrays In order to achieve a higher gain, a number of single antenna elements such as dipoles can be used at the transmitter, or receiver, to form an array of dipoles. This will enhance...

EEE6223 Antennas 2023-2024 Antenna Arrays In order to achieve a higher gain, a number of single antenna elements such as dipoles can be used at the transmitter, or receiver, to form an array of dipoles. This will enhance the radiation or reception in a particular direction, depending on the relative phasing and positioning of the individual antennas. The radiation pattern in a particular plane is then given by the product of the individual element pattern and the array factor in that plane. The diagram below has been considered earlier and it will be used as a starting point to explain the array theory. The combined radiated field from two z-directed dipoles has been given in (64) as 𝑒 −𝑗𝛽𝑟1 𝑒 −𝑗𝛽𝑟2 𝐸𝜃 = 𝑗η𝛽∆𝑧 sinθ (𝐼1 + 𝐼2 ) 4𝜋𝑟1 4𝜋𝑟2 in which r1=r-(d/2)cosθ and r2=r+(d/2)cosθ. z r1 θ r θ r2 d y θ ϕ x 1 EEE6223 Antennas 2023-2024 φ Assuming, the first and second antennas are excited by complex currents of 𝐼o 𝑒 𝑗 2 φ and 𝐼o 𝑒 −𝑗 2 , respectively. The radiated field can be expressed as 𝑒 −𝑗𝛽𝑟 1 (84) 𝐸𝜃 = [𝑗η𝛽𝐼0 ∆𝑧 sinθ ] × [2cos ( (𝛽𝑑cos𝜃 + 𝜑))] 4𝜋𝑟 2 The first term on the RHS of (84) represents the radiation pattern of a single z- directed infinitesimal dipole given by (43). On the other hand, the 2nd RHS term denotes the array factor, which depends on the antennas’ number and orientations, separation distance d as well as the excitation phase φ. Equation (84) can be generalised as 𝐸𝑎𝑟𝑟𝑎𝑦 = 𝐸𝑠𝑖𝑛𝑔𝑙𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 × 𝐴𝑟𝑟𝑎𝑦 𝐹𝑎𝑐𝑡𝑜𝑟 (85) z r θ d y ϕ x However, if the dipole centres are located along the y-axis, then (85) is still applicable but the array factor changes to 1 𝐴𝐹 = 2cos ( (𝛽𝑑 sin𝜃sin𝜙 + 𝜑)) (86a) 2 Similarly, when the dipole centres located along the x-axis, the array factor changes to 2 EEE6223 Antennas 2023-2024 1 𝐴𝐹 = 2cos ( (𝛽𝑑sin𝜃 cos 𝜙 + 𝜑)) (86b) 2 Example Plot the normalised radiation patterns of two z-directed infinitesimal dipoles when d=0.5λ and the array is located along the z-direction assuming φ=0. How the pattern changes if the dipoles are replaced by point source elements. Solution Using (84), the normalised radiation patten of two dipoles can be expressed as 1 𝐹(𝜃,𝜙) = |sinθcos ( (𝛽𝑑cos𝜃 + 𝜑))| 2 The required radiation patterns are 1 Dipoles 𝐹(𝜃,𝜙) = |sinθcos ( (𝛽𝑑cos𝜃))| 2 1 Point sources 𝐹(𝜃,𝜙) = |cos ( (𝛽𝑑cos𝜃))| 2 The normalised far field patterns are shown below, where it can be noted that a narrower beam has bean achieved when dipole elements are utilised. 3 EEE6223 Antennas 2023-2024 Array Factor of N Elements Assuming the array consists of N elements with a separation distance, d, between adjacent elements. In addition, each element is driven using a complex current Ioejnφ, where z rN 0≤n≤N-1. This mean the antennas θ are fed using equal current amplitudes with a linear phase shift of φ between elements. This r2 type of antenna is therefore called a phased array. However, if φ=0 θ r1 then we have a uniform broadside array instead. d r θ The array factor for such an N element array can be derived by d θ considering an array of isotropic y elements that are located along the z-axis as shown in the diagram to the right. 𝐴𝐹 = 1 + 𝑒 𝑗𝛾 + 𝑒 𝑗2𝛾 + ⋯ + 𝑒 𝑗(𝑁−1)𝛾 (87) where 𝛾 = (𝛽𝑑 cos 𝜃 + 𝜑) (88) Equation (87) can be expressed as, 𝑁 𝐴𝐹 = ∑ 𝑒 𝑗(𝑛−1)𝛾 (89) 𝑛=1 After a few simplifications, a closed form normalised AF can be derived as 𝑁 1 sin ( 2 𝛾) (𝐴𝐹)𝑛 = [ ] (90) 𝑁 sin (1 𝛾) 2 4 EEE6223 Antennas 2023-2024 This represents the most general format of the array factor and can be used directly in (85) to calculate the radiation pattern of any liner array provided the far field pattern of a single element is known. The AF has a maximum value of N, and hence the normalised (AF)n has a maximum of 1 that can be achieved when γ=0. Furthermore, (89) can be used to calculate the far field pattern when the array is arranged along the x-axis, by modifying (88) to 𝛾 = (𝛽𝑑sin𝜃 cos 𝜙 + 𝜑) (91a) Similarly, when the array is along the y direction, then 𝛾 = (𝛽𝑑sin𝜃 sin 𝜙 + 𝜑) (91b) It should be noted that the main beam direction is determined by two factors; the radiation pattern of the single element and the array factor, which in turn depends on the excitation and the separation between adjacent elements, d. Therefore, the radiating elements must be chosen carefully depending on the required main beam direction. In addition, generally a single main beam is required instead of multiple beams, that are also known as grating lobes and likely to appear for a larger separation distance, d, between elements. On the other hand, a smaller separation may result in a stronger mutual coupling between the elements as will be discussed later. Therefore, and depending on the individual radiating element, d needs to be chosen carefully. Broadside Arrays In many applications, it is desired to design an array with a broadside radiation pattern, i.e. the main beam direction is perpendicular to the array axis. For the example considered earlier of elements placed along the z-axis, the broadside beam is achieved at θ=90°. From (87), it can be noted that the maximum AF can be achieved when γ=0. Therefore, when the maximum is needed to be at θ=90°, (88) reduces to 𝛾 = (𝛽𝑑 cos 𝜃 + 𝜑)|θ=90° = 𝜑 = 0 (92) Therefore, the array elements must be excited using uniform amplitude anda progressive phase shift of 𝜑 = 0 to satisfy the broadside radiation with main 5 EEE6223 Antennas 2023-2024 beam orthogonal to the array axis. Similarly, if the array elements are placed in the xy plane, the broadside beam is achieved at θ=0° with a phase shift of φ=0°. The directivity of a large broadside array, with N elements, can be calculated as 𝑑 (93) 𝐷 = 2𝑁 ( ) 𝜆 Example Plot the array factor pattern for isotropic elements broadside array located along the x-axis when d=0.25λ and (i) N=3 and (ii) N=5. What is the directivity of these arrays? Solution The broadside beam will be located at θ=0°. From (91a) 𝛾 = (𝛽𝑑sin𝜃 cos 𝜙 + 𝜑)|𝜃=0 = 0, i.e. the required phase shift is φ=0. From (90) N=3 N=5 3 5 sin ( (𝛽𝑑sin𝜃 cos 𝜙)) sin ( (𝛽𝑑sin𝜃 cos 𝜙)) 1 2 1 2 (𝐴𝐹)𝑛 = (𝐴𝐹)𝑛 = 3 1 5 1 sin ( (𝛽𝑑sin𝜃 cos 𝜙)) sin ( (𝛽𝑑sin𝜃 cos 𝜙)) [ 2 ] [ 2 ] D=1.76 dBi D=4 dBi The radiation demonstrates a main beam direction that is orthogonal to the array axis. In addition, increasing the number of elements provided a considerably more directive pattern with a narrow beamwidth. Ordinary End-fire Arrays In contrast to broadside arrays, end-fire arrays are defined as those with the main beam direction along the array axis. For example, if the array elements are located 6 EEE6223 Antennas 2023-2024 along the z-axis, the main beam is achieved at θ=0°. Therefore, when the maximum is needed to be at θ=0°, (88) reduces to 𝛾 = (𝛽𝑑 cos 𝜃 + 𝜑)|θ=0° = 0 (94) From which it can be noted that the required phase shift between adjacent elements is φ=-βd. Similarly, if the array is located along the x-, or y-, axis then the respective main beam directions will be at ϕ=0° and ϕ=90°. The directivity of a large end-fire array, with N elements, can be calculated as 𝑑 (95) 𝐷 = 4𝑁 ( ) 𝜆 Example Plot the array factor pattern for isotropic elements end-fire array located along the x-axis when d=0.25λ and (i) N=3 and (ii) N=5. Solution The broadside beam will be achieved at θ=90°. From (91a) 𝛾 = (𝛽𝑑sin𝜃 cos 𝜙 + 𝜑)|(𝜃=90°,𝜙=0) = 0, i.e. the required phase shift is φ=-βd. Since d=0.25λ, then φ=-0.5π. The radiation pattern can be obtained by solving equation (90) using 𝛾 = (𝛽𝑑sin𝜃 cos 𝜙 − 0.5𝜋)). The radiation demonstrates a main beam direction that is located along the array axis. In addition, increasing the number of elements provided a considerably more directive pattern with a narrow beamwidth. 7 EEE6223 Antennas 2023-2024 A popular end-fire array example is the Yagi-Uda array, which consists of a fed resonant antenna that is backed by a slightly larger antenna known as reflector. In addition, a number of slightly short antennas are placed at the front of the fed element to form what is known as directors as demonstrated in the following diagram. d ~ Maximum gain Reflector Directors Classic Yagi-Uda arrays are constructed using dipoles. However, loop and microstrip antennas have also been used widely as elements of this kind of arrays as well as other antenna types. The gain of a Yagi-Uda array is mainly determined by the number of directors as well as the separations between elements. For example, an end-fire gain of ~11dBi can be achieved when six elements are used with a separation distance of 0.31λ between adjacent directors. Phased Arrays In addition to broadside and end-fire radiations, an array can be designed to radiate with a main beam in any particular direction that is defined as θ m. This can be achieved by the solution of 𝛾|𝜃=θ𝑚 = 0. This provides the required excitation phase of each element for z, x, and y directed arrays, respectively, 𝜑𝑧 = −𝛽𝑑 cos 𝜃𝑚 (96a) 𝜑𝑥 = −𝛽𝑑 sin 𝜃𝑚 cos 𝜙 (96b) 𝜑𝑦 = −𝛽𝑑 sin 𝜃𝑚 sin 𝜙 (96c) 8 EEE6223 Antennas 2023-2024 i.e. there should be a progressive phase shift between the currents at the centres of adjacent elements in a phased array as given by (96). Example Plot the array factor pattern for an isotropic elements phased array that is located along the x-axis when d=0.5λ, N=5 and the main beam is steered to θm=45°. Solution With the aid of (96b), the required progressive phased shift to steer the man beam to 45° is 𝜑𝑥 = 𝜋 − , and 𝛾 = 𝜋(sin 𝜃 − sin 𝜃𝑚 ), √2 where ϕ=0 has been assumed since the steered main beam will in located that plane. Therefore (90) can be expressed as 5 1 sin ( 𝛾) (𝐴𝐹)𝑛 = 2 5 1 sin ( (𝛾)) [ 2 ] It should be noted that practical phased arrays consist of actual antenna elements. As an example, if the isotropic antennas used in this example are replaced by dipoles then (84) needs to be utilised with the radiation pattern of a single dipole element given by (49). In this analysis, uniform amplitudes have been assumed and the required phase shifts can be achieved either using phase shifters in the feed network as illustrated in the following diagram. Alternatively, the phased shifters can be replaced transmission line sections with optimised lengths to ensure that each element is excited with the correct phase. However, the actual element’s current amplitude and phase may be different from the designed values due to the mutual coupling as will be explained next. 9 EEE6223 Antennas 2023-2024 φ1 φ2 φ3 φ4 Source Mutual Coupling It has been mentioned earlier that objects in proximity with the antenna could alter the performance due to the electromagnetic coupling between these objects and the antenna. This has been discussed for antennas in the vicinity of PEC and PMC surfaces. Since several radiating elements need to be utilised in order to form an array, electromagnetic coupling between various array elements represents a major challenge in the design of array antennas. This is known as mutual coupling and it may alter the excitation amplitude and/or phase of the array elements. This deteriorates the far field pattern and impacts the impedance matching bandwidth. Furthermore, coupling between various feed network lines results in further performance deteriorations. In the case of microstrip arrays, the existence of surface waves contributes considerably to the electromagnetic coupling across the array elements as well as the feed network. It should be noted that the presented array factor analysis assumed isotropic radiators without mutual coupling between the elements. However, this is not the case with practical antennas. Therefore, discrepancies are to be expect between a measured array’s radiation pattern and the array factor calculations. Despite this limitation, array factor calculations are essential in the design of arrays as they 10 EEE6223 Antennas 2023-2024 provide the needed physical insight into array radiation. In addition, array factor calculations are fast, simple and require no additional computing resources other than few lines of code. In order to understand the mutual coupling phenomena, an array of two parallel dipoles has been analysed, I1 where a current of I1 flows in the first dipole that triggers the radiation to free ~ I2 space. If a second, parasitic, dipole is placed at a distance d from the driven d antenna, then a current I2 will be induced due to the illumination by the radiated field. As a result, the parasitic antenna radiates its own fields that interfere with the current and radiation of the driven dipole. The input impedance of the first antenna can be expressed as 𝐼2 𝑍𝑖𝑛,1 = 𝑍11 + 𝑍 𝐼1 21 where Z11 is the self impedance of the driven dipole at the absence of the parasitic counterpart and Z21 represents the mutual impedance between the two antennas, which depends on the antenna shape, size and the separation distance between the two antennas. A larger separation distances provide a considerably lower mutual coupling. However, a separation distance of more than 0.5λ results in a larger array size as well as grating lobes in the pattern. Alternatively, the mutual coupling can be reduced using a number of techniques such as the incorporation of a deformed ground plane, or electromagnetic band gap, EBG, surfaces, for example. Those approaches reduce the mutual coupling over a certain frequency bandwidth. In practice, the mutual coupling is assessed using the scattering parameter S21, i.e. transmission coefficient, which can be easily measured to 11 EEE6223 Antennas 2023-2024 assess the mutual coupling for a particular design. Typical, S21≤-20dB represents an acceptable level of coupling that can be assumed to have a negligible impact on the performance. If the distance d is sufficiently larger so that the antennas are in the far field of each other, then S21 measurements can be followed to measure the relative gain of the antenna. 12