Optimum Design of Multi-band Transformer with Multi-section for Two Arbitrary Complex Frequency-dependent Impedances

Transcription

1 Chinese Journal of Electronics Vol.21, No.1, Jan Optimum Design of Multi-band Transformer with Multi-section for Two Arbitrary Complex Frequency-dependent Impedances CHEN Ming (Institute of Microwave and Optics, Xi an University of Posts and Telecommunications, Xi an , China) Abstract N-frequency Transmission line s (TLTs) with N-section for two arbitrary complex frequency-dependent impedances is optimally designed based on an ideal transmission-line model. Optimum design equations of the proposed transmission line are derived and solved by using Particle swarm algorithm (PSO) and Levenberg-Marquardt algorithm(lma). Several numerical examples are given to verify the validity of the design. Key words Complex frequency-dependent impedances, Multi-band, Transmission line, Particle swarm algorithm, Levenberg-Marquardt algorithm. I. Introduction Multi-band impedances is neceery in modern communication systems due to its smaller size and less complexity in realizing maximu power transfer requirement [1]. In 2002, Chow et al. discovered a two-section that can match at a frequency and its first harmonic [2]. In 2003, Monzon developed a two-section construct that can match at two arbitrary frequencies for two different real impedances [3]. Then, Yongle Wu, et al. solved generalized dual-frequency for two arbitrary complex frequency-dependent impedances [4]. Almost at the same time, tri-band [5] and multi-band [6,7] s for real impedances were also investigated. Considering for general tri-and multi-band microwave circuits, including active circuits and systems, complex impedances varying with frequencies are very common, such as Low noise amplifiers (LNAs), Power amplifiers (PAs), mixers and microstrip antennas, therefore, design of a multi-band for two arbitrary complex frequency-dependent impedances is very imperative in modern communication systems. Though analytical methods is a highly effective in design of two-band for two arbitrary complex frequencydependent impedances [4], it is too complex to be difficult to achieve if bands to be matched exceed two, and feasible methods should be explored. In this paper, multi-band multisection Transmission line (TLTs) for two arbitrary complex frequency-dependent impedances is proposed and investigated. Based on an ideal transmission-line model, design equations of the proposed TLTs will be given in Section II. The validity of the the derived equations are proven by some numerical analytical examples in Section III. Finally, this paper presents conclusions in Section IV. II. Design Method 1. Optimum design equations A mult-band multi-section lossless transmission lines between two complex impedances (Z s = R s +jx s and Z L = R L + jx L) is illustrated in Fig.1(a), which is equiverlent to a two port network (see Fig.1(b)). Fig. 1. (a) Multi-section ; (b) Schematic diagram of a equivelent two port network for multi-section where [ ] A B ABCD = C D [ ] [ ] cos βl1 jz 1 sin βl 1 cos βl2 jz 2 sin βl 2 = j j sin βl 1 cos βl 1 sin βl 2 cos βl 2 Z 1 Z 2 Manuscript Received Oct. 2010; Accepted July This work is supported by the National Natural Science Foundation of China (No ).

2 Optimum Design of Multi-band Transformer with Multi-section for Two Arbitrary Complex Frequency-dependent [ ] cos βln jz 2 sin βl n j sin βl n cos βl n Z n where the wavenumber β =2π/λ 1. The scattering parameter S 11 (i.e., the S in in Fig.1(a)) of the two port network with arbitrary source and load impedances is [8] : S 11 = AZL + B CZ SZ L DZS (2) AZ L + B + CZ SZ L + DZ S where S 11 is the function of the frequency f, the characteristic impedances Z 1, Z 2,,Z n and the corresponding electric lengths θ 1,θ 2,,θ n (or the corresponding physical lengths l 1,l 2,,l n). For perfect matching at frequencies f 1,f 2,,f n, simultaneously, let S 11(f i) i=1,2, n = 0, i.e., whose real parts Re{S 11(f i) i=1,2, n} = 0 and imaginary parts Im{S 11(f i) i=1,2, n} = 0, simultaneously. For convenience, they can be rewritten as following equations group: F 1(Z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Re{S 11(f 1)} =0 F 2(Z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Re{S 11(f 2)} =0 F n(z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Re{S 11(f n)} =0 F n+1(z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Im{S 11(f 1)} =0 F n+2(z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Im{S 11(f 2)} =0 F 2n(Z 1,Z 2,,Z n,θ 1,θ 2,,θ n) Im{S 11(f n)} =0 (3) Apparently, there are 2n variables within 2n equations. The goal of the design is to find the characteristic impedances Z 1,Z 2,,Z n, and the corresponding electric lengths θ 1,θ 2,,θ n, or the corresponding physical lengths l 1,l 2,,l n. In order to assure desired parameter values are significant for compact microwave implementation, constraint conditions should be given as (1) Z 1 > 0,Z 2 > 0,,Z n > 0 (4a) 0 <θ i,ii=1,2, n <π/2 (4b) Eq.(3) with Eqs.(4a) and(4b) can be converted into following function optimum problem MinF = 2n i=1 F 2 i (Z 1,Z 2,,Z n,θ 1,θ 2,,θ n) (5) best and globle best position, the velocity matrix is updated according to following equation [9] V i mn =wv i 1 mn + c 1 rand 1 (P i mn X i 1 mn ) + c 2 rand 2 (G i n X i 1 mn ) (6) where the superscripts i and i 1 refer to the time index of the current and the previous iterations, rand 1 and rand 2 are uniformly distributed random numbers in the interval [0,1]. The parameters c 1 and c 2, called cognition and social acceleration, specify the relative weight of the personal best position versus the global best position. The position matrix is updated according to following equation X i = X i 1 + V i (7) The Algorithm flow are followings: (1) Determine spacifications, i.e., the matching frequencies f 1,f 2,,f n, the number n of the sections of, the source equivalent impedances Z si(f i) = R si(f i)+jx si(f i) i=1,2,,n, and the load impedances Z Li(f i)= R Li(f i)+jx Li(f i) i=1,2,,n. (2) Initialize the population - positions and velocities. (3) Use the current position vector of each particle to evaluate the fitness of the individual particle (pbest) using Eq.(5) with Eqs.(4a) and(4b). (4) Compare the particle s fitness value with the particle s pbest. If the current value is better (here better numerically means smaller) than pbest, then replace pbest by the current value, and the particle s position vector by the current position vector. (5) Compare the current fitness value with the global previous best. If the current value is better than gbest, then set gbest and G to the current value and position, respectively (6) Update the particle s velocity and position according to Eqs.(6) and (7). (7) Repeat starting from step (3) until a stopping criterion is met: a good objective value or a maximum number of iterations. And then output G. (8) Let G in step 7 as initial guess of LMA, and further solve Eq.(5) with Eqs.(4a) and(4b) sothatthemoreaccu- rate solution (resulting solution) is obtained by LMA until a maximum number of iteration of LMA is met. A Matlab programming code was developed for the solution process. with Eqs.(4a) and(4b). 2. Solution of the proposed optimum design equation by PSO and LMA To solve the Eq.(5) with Eqs.(4a) and(4b), PSO [9] and LMA [10 12] are used. For the current problem, the design is to find the the characteristic impedances and lengths of N section TLTs matching at N arbitrary frequencies for two arbitrary complex frequency-dependent impedances Z s and Z L, under the conditions of minimizing the reflection coefficient. In order to apply the PSO with LM algorithms to the current design problem, let M is the number of particles in the swarm for an N-dimensional problem, X denotes position matrix, V the velocity matrix, P the personal matrix, G the globle best position. To move each particle closer to both its personal III. Numerical Examples In this section, two complex frequency-dependent impedances are considered, and three kinds of s (dual-band, and tri-band) with 6 types is presented to verify the proposed design. The real and imaginary parts of source and load equivalent impedances are given in Fig.2. Simulation of the frequency characteristics of designed is based on Eq.(5) with Eqs.(4a) and(4b). For the current problem, parameters of the PSO algorithm is chosen as follows: M = 50, c 1 = c 2 = 2, w = 0.9, V max = 2.5. The stopping criterion for PSO is either fitness value < 10 4 calculated from Eq.(5), or a maximum number of iterations = 10,000. And the stopping criterion for LMA is either value

3 162 Chinese Journal of Electronics 2012 < 10 4 calculated directly from Eq.(5), or a maximum number of iterations = 1000,000. In the beginning of the PSO algorithm, the impedance values and electric lengths are randomly initialized within the intervals [Z S,Z L], and (0,π/2), respectively. Fig. 2. Source and load impedances varying with the operating frequency 1. Numerical examples for dual-band two-section The goal is to match these two impedances at two different frequencies. Here, three situations are taken into account as follows: First, given Z S2 = j at f 2 =2GHz Z L2 = j at f 2 =4GHz model, obtained from Eq.(5) with Fig.2, obtained from Eq.(5) with Fig.2, are presented in type 3 of Table 1, and the corresponding dependence of reflection coefficient in db on frequency is presented in Fig.3. From the final matching responses in Fig.3, it can be observed clearly that the two complex frequency-dependent impedances are matched at two different frequencies, simultaneously. Note that the match bandwidth at each frequency depends on the values of matched impedances and the frequency ratio. From this figure, the bandwidth in which the reflection coefficient is lower than 20 db is 181MHz for f 1 = 1GHz and the corresponding matched impedances Z S1 and Z L1, and 208 MHz for f 2 = 2GHz and the corresponding matched impedances Z S2 and Z L2, and the bandwidth in which the reflection coefficient is lower than 20 db is 160 MHz for f 1 = 0.9GHz and the corresponding matched impedances Z S1 and Z L1, and 182 MHz for f 2 = 1.8GHz and the corresponding matched impedances Z S2 and Z L2, as well as the bandwidth in which the reflection coefficient is lower than 20 db is 140 MHz for f 1 = 1GHz and the corresponding matched impedances Z S1 and Z L1, and 206 MHz for f 2 = 4GHz and the corresponding matched impedances Z S2 and Z L2. Table 1. Design parameters of the dual-band Type Z 1 (Ω) Z 2 (Ω) l 1 /λ 1 l 2 /λ Z L2 = j at f 2 =2GHz model, obtained from Eq.(5) with Fig.2, are presented in type 1 of Table 1, and the corresponding dependence of reflection coefficient in db on frequency is presented in Fig.3. Second, given Z S1 = j at f 1 =0.9GHz Z S2 = j at f 2 =1.8GHz Z L1 = j at f 1 =0.9GHz Z L2 = j at f 2 =1.8GHz model, obtained from Eq.(5) with Fig.2, are presented in type 2 of Table 1, and the corresponding dependence of reflection coefficient in db on frequency is presented in Fig.3. Third, given Z S2 = j at f 2 =4GHz Fig. 3. Simulated results of db S 11 versus frequency To prove the validity of the proposed design, the TLTs were designed on an FR4 substrate (ε r = 4.55) of 1.6 mm thickness and simulated by the commercial software ADS. The EM simulation results are shown in Fig.4. From this figure, although the matched frequency f 1 and f 2 deviate respctively from the their original value due to discontinuity of microstrip line width, i.e., for type 1 f 1 from 1 GHz into GHz, and f 2 from 2 GHz into GHz, for type 2 f 1 from 0.9 GHz into GHz, and f 2 from 1.8 GHz into GHz, and for type 3 f 1 from 1 GHz into GHz, and f 2 from 4 GHz into GHz, the bandwidth in which the reflection coefficient is lower than 20 db is 188 MHz for f 1 =1GHzandthecorresponding matched impedances Z S1 and Z L1, and 209 MHz

4 Optimum Design of Multi-band Transformer with Multi-section for Two Arbitrary Complex Frequency-dependent for f 2 = 2 GHz and the corresponding matched impedances Z S2 and Z L2, and the bandwidth in which the reflection coefficient is lower than 20 db is 166 MHz for f 1 =0.9 GHz and the corresponding matched impedances Z S1 and Z L1, and 183 MHz for f 2 =1.8 GHz and the corresponding matched impedances Z S2 and Z L2, as well as the bandwidth in which the reflection coefficient is lower than 20 db is 138 MHz for and Z L1, and 197 MHz for f 2 = 4 GHz and the corresponding matched impedances Z S2 and Z L2. Therefore, Figs.4 and 5 showed a very good agreement between the predicted results using standard transmission line theory and full-wave simulation results. Z S3 = j at f 3 =2.4GHz Z L1 = j at f 1 =0.9GHz Z L2 = j at f 2 =1.8GHz Z L3 = j at f 3 =2.4GHz simulation results of the TLT from Eq.(5) with Fig.2, are presented in type 5 of Table 2, and the corresponding dependence of reflection coefficient in db on frequency is presented in Fig.5. Third, given Z S2 = j at f 2 =2.4GHz Z S3 = j at f 3 =3.9GHz Z L2 = j at f 2 =2.4GHz Z L3 = j at f 3 =3.9GHz Fig. 4. EM simulation results 2. Numerical examples for tri-band three-section The goal is to match these three impedances at three different frequencies. Here, three situations are taken into account as follows: First, given Z S2 = j at f 2 =2GHz Z S3 = j at f 3 =3GHz Z L2 = j at f 2 =2GHz Z L3 = j at f 3 =3GHz simulation results of the TLT from Eq.(5) with Fig.2 are presented in type 4 of Table 2, and the corresponding dependence of reflection coefficient in db on frequency is presented in Fig.5. Table 2. Design parameters of the three-band Type Z 1 (Ω) Z 2 (Ω) Z 3 (Ω) l 1 /λ 1 l 2 /λ 1 l 3 /λ Second, given Z S1 = j at f 1 =0.9GHz Z S2 = j at f 2 =1.8GHz simulation results of the TLT from Eq.(5) with Fig.2 are presented in type 6 of Table 2. The corresponding dependence of reflection coefficient in db on frequency is presented in Fig.5. Fig.5 showed that the two complex frequency-dependent impedances were matched at three different frequencies, simultaneously. Note that the match bandwidth at each frequency depends on the values of matched impedances and the frequency ratio. From this figure, the bandwidth in which the reflection coefficient is lower than 20 db is 164 MHz for and Z L1, 257 MHz for f 2 = 2 GHz and the corresponding matched impedances Z S2 and Z L2, and 203 MHz for f 3 =3 GHz and the corresponding matched impedances Z S3 and Z L3, and the bandwidth in which the reflection coefficient is lower than 20 db is 139 MHz for f 1 =0.9GHz and the corresponding matched impedances Z S1 and Z L1, 249 MHz for f 2 =1.8 GHz and the corresponding matched impedances Z S2 and Z L2, and 243 MHz for f 3 =2.4 GHz and the corresponding matched impedances Z S3 and Z L3, as well as the bandwidth in which the reflection coefficient is lower than 20 db is 164 MHz for and Z L1, 243 MHz for f 2 =2.4 GHz and the corresponding matched impedances Z S2 and Z L2, and 234 MHz for f 3 =3.9 GHz and the corresponding matched impedances Z S3 and Z L3. To prove the validity of the proposed design, the TLTs were designed designed on an FR4 substrate (ε r =4.55) of 1.6 mm thickness and simulated by the commercial software ADS. The EM simulation results were shown in Fig.5. From Fig.5, although the matched frequency f 1,f 2 and f 3 deviate respctively from the their original value due to discontinuity of microstrip line width, i.e., for type 1 f 1 from 1 GHz into GHz, f 2 from 2 GHz into GHz, and f 3 from 3 GHz into GHz, for type 2f 1 from 0.9 GHz into GHz, f 2 from 1.8 GHz into 1.81 GHz, and f 3 from 2.4 GHz into GHz, and for type 3f 1 from 1 GHz into GHz, f 2 from 2.4 GHz into GHz, and f 3 from 3.9 GHz into 3.87 GHz, the bandwidth in which the reflection coefficient is lower

5 164 Chinese Journal of Electronics 2012 than 20 db is 159 MHz for f 1 = 1 GHz and the corresponding matched impedances Z S1 and Z L1, 256 MHz for f 2 =2 GHz and the corresponding matched impedances Z S2 and Z L2, and 199 MHz for f 3 = 3 GHz and the corresponding matched impedances Z S3 and Z L3, and the bandwidth in which the reflection coefficient is lower than 20 db is 126 MHz for f 1 =0.9 GHz and the corresponding matched impedances Z S1 and Z L1, 260 MHz for f 2 =1.8 GHz and the corresponding matched impedances Z S2 and Z L2, and 271 MHz for f 3 =2.4 GHzand the corresponding matched impedances Z S3 and Z L3, aswell as the bandwidth in which the reflection coefficient is lower than 20 db is 166 MHz for f 1 = 1 GHz and the corresponding matched impedances Z S1 and Z L1, 243 MHz for f 2 =2.4 GHz and the corresponding matched impedances Z S2 and Z L2, and 232 MHz for f 3 =3.9 GHz and the corresponding matched impedances Z S3 and Z L3. Therefore, as can be seen by comparing Figs.5 and 6, a very good agreement is obtained between the predicted results using standard transmission line theory and full-wave simulation results. Fig. 5. Simulated results of db S 11 versus frequency Fig. 6. EM simulation results In order to improve the accuracy of the design parameters of the proposed TLTs, full account of discontinuity of microwave line width is necessary, and it can be obtained by full-wave simulation with optimization design. On this issue will be discussed in subsequent articles. IV. Conclusion The design of multi-band multi-section TLTs for two arbitrary complex frequency-dependent impedances by using PSO and LMA were developed. Numerical examples of two-, threesection TLTs for two arbitrary complex frequency-dependent impedances were presented to verify the proposed structure and the design method. This can be used widely in dual-band or multi-band RF/microwave circuits and systems as an internal matching structure. Of course, further research is to solve the problem of the matched frequency offset due to the discontinuity of microstrip line width, by using optimization design of parameters based on electromagnetic simulation. References [1] W. Webb, The Complete Wireless Communications Professional: A Guide for Engineers and Managers. Boston, MA: Artech House, [2] Y.L. Chow and K.L. Wan, A of one-third wavelength in two sections-for a frequency and its first harmonic, IEEE Microw. Wireless Compon. Lett., Vol.12, No.1, pp.22 23, Jan [3] C. Monzon, A small dual-frequency in two sections, IEEE Trans. Microw. Theory Tech., Vol.51, No.4, pp , Apr [4] Y. Wu, Y.A. Liu, S.L. Li, C.P. Yu and X. Liu, A generalized dual-frequency for two arbitrary complex frequency-dependent impedances, IEEE Microw. Wireless Compon. Lett., Vol.19, No.12, pp , Dec [5] M. Chongcheawchamnan, S. Patisang, S. Srisathit, R. Phromloungsri, and S. Bunnjaweht, Analysis and design of a threesection transmission-line, IEEE Trans. Microw. Theory Tech., Vol.53, No.7, pp , July [6] M. Chen, Novel design method of a multi-section transmissionline using genetic algorithm techniques, 11th International Conference on Electrical Machines and Systems, ICEMS 2008, Wuhan, China, pp.17 20, Oct [7] Majid Khodier, Nihad Dib, Jehad Ababneh, Design of multiband multi-section transmission line using particle swarm optimization, Electr. Eng., Vol.90, pp , [8] D.A. Frickey, Conversion between S, Z, Y, h, ABCD, and T parameters which are valid for complex source and load impedances, IEEE Trans. Microwave Theory Tech., Vol.42, pp , [9] Kennedy J., Eberhart R.C. (1995), Particle swarm optimization. in Proceedings IEEE Int. Conf. Neural Networks, Vol.IV, Perth, Australia, pp , [10] Kenneth Levenberg, A method for the solution of certain nonlinear problems in least squares, The Quarterly of Applied Mathematics, No.2, pp , [11] D.W. Marquardt, An Algorithm for the Least-Squares Estimation of Nonlinear Parameters, SIAM Journal of Applied Mathematics, Vol.11, No.2, pp , June [12] H.D. Mittelmann, The Least Squares Problem. [web page], CHEN Ming received the B.S. degree in radio electronics from the Wuhan University, Wuhan, China, in 1982, the Ph.D. degree in electronic science and technology from Xi an Jiaotong University, Xi an, China in He is currently the Director of the Research Center of Microwave & Optics and a Professor with the Department of Electronic Information Engineering, Xi an University of Posts & Telecommunications, Xi an, China. His research and teaching interests include RF/microwave circuits, microwave photonics, and integrated optics. ( chenming5628@sina.com)