Novel design of a dual-frequency power divider using genetic algorithms

Transcription

1 Novel design of a dual-frequency power divider usg genetic algorithms D. KAMPITAKI, A. HATIGAIDAS, A. PAPASTERGIOU, P. LAARIDIS,. AHARIS Department of Electronics, Alexander Technological Educational Institute of Thessaloniki, Sdos, 574 Thessaloniki, Macedonia, GREECE Abstract: A novel design of a dual-frequency unequal power divider is presented. The design is optimized usg a genetic algorithm based method order to satisfy several requirements at two given frequencies. The method considers realistic conditions, sce it assumes lossy transmission les and unmatched real or frequency-dependent complex loads. The method can be applied a wide range of applications and responds successfully at any two given frequencies. Key-Words: Power dividers, Power splitters, Impedance matchg, Antenna feedg, Genetic algorithms 1. Introduction Power dividers have been widely studied, especially mobile dustry applications where the dividers are used to feed antenna arrays [1-4]. There have been studies for various power-split ratios, but mostly for sgle-frequency applications. The purpose of usg a divider is maly to split the power accordg to a desired split ratio. In addition, the divider has to provide impedance matchg the maximum possible bandwidth and to obta the highest possible power efficiency. For sgle-frequency applications these requirements are quite easily satisfied. However, dual-frequency applications the structure of the divider gets complicated because the requirements mentioned above have to be satisfied at once at two frequencies [5-12]. characteristic impedance were extracted from specifications of conventional-type radiofrequency (RF) cables [13]. In addition, the method assumes that the termal loads are not matched to the ma transmission le that feeds the power divider. Fig.1: The basic structure of a divider. The simplest divider topology is two branches termatg at two respective termal loads (Fig.1). In cases of feedg arrays composed of three or more elements, several power dividers are combed various topologies (Fig.2) order to feed all the array elements. The divider resultg from the proposed design method is made of lossy transmission les with typical values of characteristic impedance. The values of the attenuation constant and the values of the Fig.2: A combation of seven dividers for the feedg of eight elements. Considerg realistic conditions, this method does not take advantage of the periodic behavior of the put

2 impedance, which is observed only along lossless transmission les. The periodic behavior is the fact that the put impedance repeats along a lossless le every λ/2 along the le (λ is the wavelength). There are many papers like [7] that take advantage of that periodic behavior, but the consideration of lossless transmission les is just an ideal condition and far away from reality. In order to achieve all of the requirements at both frequencies, an optimization procedure is performed usg a genetic algorithm [14-17]. The objective of the algorithm is to maximize a specific function called fitness function. The fitness function depends on several physical parameters, which are the power-split ratios, the impedance-matchg bandwidths and the power efficiencies at the two respective frequencies for which the divider is designed. When these physical parameters reach their desired values, the fitness function fds its global maximum value and the algorithm termates successfully. lengths D 2a, D 2b, D 2c, D 2d and correspondg characteristic impedances o(2a), o(2b), o(2c), o(2d). Two termal loads, 1T and 2T, are connected at the respective ends of the two branches. The loads may be real or complex and are generally not matched to the ma le that feeds the divider. The factors used to determe the impedance-matchg bandwidth is usually the Standg Wave Ratio (SWR) and the Return Loss (RL). In fact, the impedance-matchg bandwidth is the frequency range where the SWR is below 2:1 or where the return loss is below 9.54dB (approximately B). The factor used our work is the return loss. The procedure of calculatg the return loss usg transmission le theory [18,19] is shown below. 2. Formulation As mentioned above, a simple power divider is composed of two branches of transmission le that feed two correspondg termal loads. Each branch is defed by its length and the type of transmission le used, resultg four parameters available to optimize the structure. Four parameters are not enough to solve the complex problem described above. An crease the number of parameters is necessary. Thus, the method considers that the branches of the divider are composed of more than one section of transmission le. After several trials, it was found that the appropriate number of transmission le sections at each branch is at least four. Consequently, the divider consists of eight sections, which contribute sixteen optimization parameters total. The optimization procedure makes use of conventional-type RF transmission les with typical values of characteristic impedance [13]. Additional electromagnetic characteristics of the transmission les are taken to account such as the phase velocity v p and the frequency dependence of the attenuation coefficient α=α(f). So, the complex propagation coefficient can be calculated for every type of transmission le and every frequency value by the followg expression: γ=α+ jβ=α ( f) + j2π f vp (1) The proposed structure of the divider is shown Fig.3. The branch 1 consists of four sections with lengths D 1a, D 1b, D, D and correspondg characteristic impedances o(1a), o(1b), o(), o(), while the branch 2 consists of four sections with Fig.3: The proposed structure of the dual-frequency divider. Given the value of 1T, the put impedance at the position 1A is calculated by the followg expression: = 1a o(1a ) 1T + o(1a ) tanh( γ1a D 1a ) + tanh( γ D ) o(1a) 1T 1a 1a (2) 1a is considered as termal load of the next section. Thus, the put impedance at the position 1B is given by

3 1b = o(1b) + tanh( γ D ) 1a o(1b) 1b 1b + tanh( γ D ) o(1b) 1a 1b 1b (3) Both the put impedances of branch 1 at the respective positions 1C and 1D are calculated the same way: = = o() o() 1b + o() tanh( γ D ) + tanh( γ D ) o() 1b + o() tanh( γ D ) + tanh( γ D ) o() (4) (5) Similarly, the put impedances of branch 2 at the positions 2A, 2B, 2C, and 2D are calculated by the followg expressions: 2a 2b 2c 2d = = = = o(2a) o(2b) o(2c) o(2d) 2T+ o(2a) tanh( γ2a D 2a) + tanh( γ D ) o(2a) 2T 2a 2a 2a+ o(2b) tanh( γ2b D 2b) + tanh( γ D ) o(2b) 2a 2b 2b 2b+ o(2c) tanh( γ2c D 2c) + tanh( γ D ) o(2c) 2b 2c 2c 2c+ o(2d) tanh( γ2d D 2d) + tanh( γ D ) o(2d) 2c 2d 2d (6) (7) (8) (9) The position 1D is identical with 2D. Therefore, the put impedance of the divider is the parallel combation of and 2d : 1 1 = + 2d 1 (1) The ma le that feeds the divider is considered to have a characteristic impedance of 5 Ohm. Then, the return loss at the put of the divider is calculated decibels as follows: = 5 RL 2log [db] + 5 (11) The power-split ratio P 1T /P 2T is defed as the ratio between the average values of power consumed respectively by the two loads. All the power quantities are considered normalized with respect to the average put power of the divider. Therefore, the average put power P of the divider is set to 1 Watt. Given the value of, P can be expressed terms of the complex put voltage V of the divider: 1 2 P = V Real1 2 (12) where V is the amplitude of V and * is the complex conjugate value of. Given that P =1Watt, the above equation yields: V = 2 Real 1 e jϕ (13) where φ is the phase of V. Regardg V as reference voltage, we may assume that φ=. So, equation (13) is simplified as follows: V = 2 Real 1 (14) Once the voltage at the put of the divider is determed, it is easy to calculate the voltage at any position along the branches of the divider usg transmission le theory, [18,19]. Thus, the voltage at the positions 1C, 1B, 1A, and 1T is respectively calculated as follows: o() V1C = V cosh( γ D ) sh( γ D o() V1B = V1C cosh( γ D ) sh( γ D o(1b) V1A = V1B cosh( γ1b D 1b ) sh( γ1b D 1b 1b o(1a) V1T = V1A cosh( γ1a D 1a ) sh( γ1a D 1a 1a (15) (16) (17) (18) The voltage at the positions 2C, 2B, 2A, and 2T is respectively calculated by: o(2d) V2C = V cosh( γ2d D 2d) sh( γ2d D 2d) 2d o(2c) V2B = V2C cosh( γ2c D 2c) sh( γ2c D 2c) 2c o(2b) V2A = V2B cosh( γ2b D 2b) sh( γ2b D 2b) 2b o(2a) V2T = V2A cosh( γ2a D 2a) sh( γ2a D 2a) 2a (19) (2) (21) (22) The average values of power consumed respectively by 1T and 2T are calculated accordg to the followg expressions: 1 2 P1T = V1T Real 1 1T 2 (23)

4 1 2 P2T = V2T Real1 2T 2 (24) Fally, the power-split ratio is derived by: rp = P1T P2T (25) An additional factor useful for practical applications is the power efficiency factor, which is defed as the percentage of put power consumed by the two loads. Given that P =1Watt, the power efficiency factor is derived by the followg expression: ( ) e = P + P 1% (26) p 1T 2T Due to optimization for dual-frequency operation, the impedance-matchg bandwidth, the power-split ratio and the power efficiency factor must be calculated for both frequencies of operation. The optimization is performed usg a genetic algorithm, [14-17]. For the structure described above, the genetic algorithm uses 16 put parameters, which are the lengths and the respective transmission le types of the 8 sections that compose the divider. The objective of the algorithm is to maximize a suitably defed fitness function based on the requirements described above. In fact, the fitness function is a lear combation of the differences between the above physical parameters (impedance-matchg bandwidth, power-split ratio and power efficiency factor) and their respective desired values. The requirements are satisfied when each physical parameter comes up to its desired value. In that case, the fitness function fds its global maximum value and the algorithm termates successfully. 3. Results Initially, the proposed technique was applied at the mobile communication frequencies of and. The termal loads are equal each other and matched to the ma le that feeds the divider ( 1T = 2T = 5 Ohm). Three cases are studied this example concerng different values of power-split ratio: (a) r p =1 at both frequencies, (b) r p =1 at and r p =1.5 at, (c) r p =1.5 at and r p =1 at. The results of the three cases are summarized Table 1, while the frequency response of the divider is presented Fig.4. Specifically, the tables of all the examples show the power-split ratios and the power efficiency factors as well as the transmission le types and the lengths of the sections that come up from the optimization procedure. It is obvious that the resultg divider is very broadband. Even at frequencies between and, the return loss is less than B. Table 1: Structure characteristics of the optimized divider that resonates at & and feeds two matched termal loads 1T = 2T = 5Ω. EXAMPLE 1 Case a Case b Case c Resonant frequencies Required r p Resulted r p Resulted e p 96.1% 94.4% 93.7% 92.5% 93.1% 91.% RG-58C 7/8 Al. Jacket RG-174 Foam D 1a Length 127mm 4mm 216mm RG-9A RG-17A Belden-9913 (51Ω) (52Ω) D 1b Length 66mm 134mm 28mm RG-174 Belden-9913 RG-58A D Length 167mm 27mm 18mm 7/8 Al. Jacket 7/8 Al. Jacket RG-59 Foam Foam Foam D Length 9mm 87mm 171mm 3/4 Al. Jacket RG-174 1/2 Al. Jacket Foam Foam D 2a Length 317mm 411mm 316mm RG-8A RG-58 Foam RG-8A (52Ω) (52Ω) D 2b Length 47mm 232mm 324mm Belden /8 Al. Jacket 3/4 Al. Jacket Foam Foam D 2c Length 229mm 1mm 19mm 7/8 Al. Jacket 7/8 Al. Jacket RG-62A Foam Foam (93Ω) D 2d Length 44mm 43mm 114mm Example 1 Case a Case b Case c Frequency [] Fig.4: Frequency response of the optimized divider that resonates at & and feeds two matched termal loads 1T = 2T = 5Ω. In the second example the technique was applied at and considerg two unequal resistive loads 1T = 6 Ohm and 2T = 72 Ohm. The same cases are studied this example. The results are given Table 2, while Fig.5 exhibits an excellent response. Despite the fact that the loads are not matched to the ma feedg le, the optimization procedure is capable of fdg a very broadband

5 structure. The proposed method was also applied at the frequencies of and 14. The purpose of this example is to show that the method is generic and can be applied for any two frequencies, which are either multiples of one another or not. Table 2: Structure characteristics of the optimized divider that resonates at & and feeds two unequal resistive loads 1T = 6Ω and 2T = 72Ω. EXAMPLE 2 Case a Case b Case c Resonant frequencies Required r p Resulted r p Resulted e p 96.8% 95.5% 97.9% 97.1% 97.8% 97.% RG-174 RG-58B RG-58 Foam D 1a Length 69mm 47mm 6mm RG-58 Foam RG-9B RG-174 D 1b Length 47mm 55mm 62mm RG-11A RG-8A RG-58A (52Ω) D Length 133mm 61mm 42mm RG-62A 7/8 Al. Jacket 1/2 Al. Jacket (93Ω) Foam Foam D Length 87mm 88mm 98mm 1/2 Al. Jacket 1/2 Al. Jacket RG-59 Foam Foam Foam D 2a Length 118mm 194mm 11mm RG-58 Foam 7/8 Al. Jacket 7/8 Al. Jacket Foam Foam D 2b Length 128mm 15mm 49mm RG-59 Foam RG-58B RG-58 Foam D 2c Length 214mm 117mm 77mm RG-62A RG-62A 7/8 Al. Jacket (93Ω) (93Ω) Foam D 2d Length 46mm 52mm 46mm Two cases are studied this example: (a) r p =1 at both frequencies and (b) r p =1.5 at and r p =1 at 14. The loads are resistive with unequal values 1T = 6 Ohm and 2T = 72 Ohm. The results are given Table 3, while Fig.6 shows a wideband response. Table 3: Structure characteristics of the optimized divider that resonates at & 14 and feeds two unequal resistive loads 1T = 6Ω and 2T = 72Ω. EXAMPLE 3 Case a Case b Resonant frequencies Required r p Resulted r p Resulted e p 98.2% 97.7% 97.2% 96.% RG-58A RG-58 Foam D 1a Length 7mm 45mm Belden-9913 Belden-9913 D 1b Length 86mm 43mm 3/4 Al. Jacket RG-58B Foam D Length 73mm 47mm 3/4 Al. Jacket 7/8 Al. Jacket Foam Foam D Length 5mm 42mm RG-59 Foam RG-59A (73Ω) D 2a Length 198mm 423mm RG-62A 3/4 Al. Jacket (93Ω) Foam D 2b Length 48mm 24mm 1/2 Al. Jacket RG-58B Foam D 2c Length 111mm 78mm RG-59 Foam 7/8 Al. Jacket Foam D 2d Length 46mm 61mm Example 3 Case a Case b -1 Example 2 Case a Case b Case c Frequency [] Fig.5: Frequency response of the optimized divider that resonates at & and feeds two unequal resistive loads 1T = 6Ω and 2T = 72Ω Frequency [] Fig.6: Frequency response of the optimized divider that resonates at & 14 and feeds two unequal resistive loads 1T = 6Ω and 2T = 72Ω. The last example concerns the optimization of the divider considerg complex termal loads. In

6 practice, the complex loads are usually frequency dependent and thus they are more destructive to the matchg condition than the resistive loads. In order to show the ability of the method to work even for frequency dependent loads, two identical wire dipoles are assumed to be the termal loads of the divider. Table 4: Structure characteristics of the optimized divider that resonates at & and feeds two identical dipoles that have the freq. response of Fig.7. EXAMPLE 4 Resonant frequencies Required r p Resulted r p Resulted e p 97.4% 96.8% RG-58 Foam D 1a Length 83mm RG-8 Foam D 1b Length 13mm RG-8A (52Ω) D Length 42mm RG-59 Foam D Length 42mm RG-59 Foam D 2a Length 76mm RG-59 Foam D 2b Length 138mm RG-58 Foam D 2c Length 135mm RG-62A (93Ω) D 2d Length 9mm The two dipoles can be considered as elements of a base station antenna array used for dual-frequency operation at the mobile communication frequencies of and. The put impedance of the dipoles varies with frequency and is calculated by applyg the Method of Moments, [2-22]. The resultg values of the put impedance are generally complex, as given Fig.7, and they are used the optimization procedure as values of the termal loads. The case studied the example concerns r p =1 at and r p =1.5 at. Input Impedance [Ohm] Example 4 2 Real Part Imagary Part Frequency [] Fig.7: Input impedance of a wire dipole vs. frequency Example Frequency [] Fig.8: Frequency response of the optimized divider that resonates at & and feeds two identical dipoles that have the frequency response of Fig.7. Despite the frequency dependent loads, the optimization procedure is capable of fdg a structure that satisfies the design requirements. The results are given Table 4, while the frequency response of the divider is fairly good as shown Fig Conclusions Despite the itial requirement of a divider that resonates two frequency bands, our technique results very broadband structures. Therefore, the robustness of the technique exceeds our hopes. But this is not the only advantage. The proposed technique has the ability to satisfy many requirements at two frequencies simultaneously. The technique can be effectively used practical applications because it optimizes the divider considerg commercial RF cables and unmatched termal loads, which may be complex and frequency dependent. The optimization can be applied at any two frequencies. In addition, the proposed structure of the divider is compact and simple. Therefore, the divider can be easily implemented practice and can achieve realistic goals. References: [1] R. Steele, Mobile Radio Communications, Pentech Press Ltd, London 1992 [2] W.C.Y. Lee, Mobile Cellular Communication Systems, McGraw-Hill, New York 199 [3] W.C.Y. Lee, Mobile Communication Engeerg, McGraw-Hill, New York 1982 [4] K. Fujimoto, J.R. James, Mobile Antenna Systems Handbook, Artech House Inc, Boston 1994

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