I. INTRODUCTION
During the last years, due to requirement of dual-band and multi-band wireless communication systems, the researches on dual-band and multi-band components become very popular [Reference Zhang, Hu and Zhang1–Reference Gence, Baktur and Jost4]. One of these components are dual-band N-way power dividers because they need to be connected to dual-band or multi-band antennas [Reference Wong5, Reference Anguera, Andújar, Huynh, Orlenius, Picher and Puente6]. In [Reference Shamaileh, Qaroot and Dib7], based on the bagly polygon technique, a 4-way power divider was presented. Also, in [Reference Kim, Jeon and Jeong8, Reference Kim, Jeon and Jeong9], multi-conductor lines were applied in order to design a 3-way and 4-way power dividers, respectively. But, the discussed dividers in [Reference Shamaileh, Qaroot and Dib7–Reference Kim, Jeon and Jeong9] work only in a single frequency band.
By the way, composite right and left handed transmission lines (CRLH-TLs) and extended CRLH-TLs (E-CRLH-TLs) are one of the main developments in the field of metamaterials. The behavior of these materials; i.e. ε < 0 and μ < 0, has been used to design various multi-band component [Reference Rennings, Otto, Mosig, Caloz and Wolff10–Reference Durán-Sindreu, Sisó, Bonache and Martín15]. Recently, the CRLH-TLs were used to design a dual-band 4-way Wilkinson power divider in a parallel configuration. However, the performance of the divider reported in [Reference Choi and Itoh16] is not good enough in terms of circuit size, and isolation because there is no isolation resistor between the output ports in this divider. By using series feed topology rather than parallel feed topology, the binary-tree transmission lines can be converted into a single feed line and consequently reducing significantly the overall dimension of the structure [Reference Antoniades17]. In fact, series power dividers (SPDs) are usually used over parallel dividers in applications where power needs to be equally divided to a large number of loading elements and where the physical area of the feed network is limited. In [Reference Antoniades17] (Chapter 4), zero-degree metamaterial phase-shifting was used to design a 4-way SPD, but the operation of this divider is limited around a single frequency. Also, two drawbacks of the presented divider are their low isolation and their low output return losses. Therefore, these SPDs would not function well as a power combiner and they are not recommended to use as a power combiner.
In this paper, based on the CRLH-TLs, a dual-band 3-way SPD was designed and fabricated in order to work at f 1 = 0.915 GHz (from 0.902 to 0.928 GHz) and f 1 = 2.440 GHz (from 2.400 to 2.483 GHz). It is useful to note that the first frequency band is in the ultra-high frequency band and the second one is in the industrial scientific and medical band. The proposed divider provides equal and in-phase signals for three output ports. Also, isolation resistors are implemented among output ports to improve the isolation and output return loss characteristics of the designed divider. It should be emphasized that, in contrast to [Reference Antoniades17, Reference Bemani and Nikmehr18], the proposed SPD in this paper can be used as a power combiner.
The organization of this article is as follows: Section II presents design methodology leading to dual-band CRLH-TLs and non-radiating CRLH-TL (NR-CRLH-TLs). Section III introduces a dual-band 3-way SPD and guideline for isolation and output return loss improvement. The simulated and the experimental results are discussed in Section IV. Finally, the conclusions are made in Section V.
II. THEORY: DUAL-BAND CRLH-TLS AND NR-CRLH-TLS
A) Analysis of CRLH-TLs
As we know, the non-linear phase response of CRLH-TLs can be engineered in order to produce desired phase shifts of ϕ 1 and ϕ 2 at two arbitrary frequencies of f 1 and f 2, and design dual-band CRLH-TLs [Reference Lin, Vincentis, Caloz and Itoh19]. Therefore, Under balance condition, the required parameters for dual-band CRLH-TLs are given by [Reference Bemani and Nikmehr3]
$$d = \displaystyle{{\phi_1\, f_1 - \phi_2\, f_2} \over {2\pi n\sqrt {LC} (\,f_2^2 - f_1^2 )}},$$
$$C_0 = \displaystyle{{n(\,f_2^2 - f_1^2 )} \over {2\pi Z_0 (\phi_1\, f_2 - \phi_2\, f_1 )f_1 f_2}}, $$
$$L_0 = \displaystyle{{nZ_0 (\,f_2^2 - f_1^2 )} \over {2\pi\, (\phi_1\, f_2 - \phi_2\, f_1 )f_1 f_2}}, $$
where d, Z 0, L, and C are the length, the characteristic impedance, the intrinsic inductance and the intrinsic capacitance per unit length of the right handed transmission line, respectively.
B) Analysis of NR-CRLH-TLs
As discussed in [Reference Antoniades and Eleftheriades20], any structure that support fast waves (the waves with superluminal phase velocity (V ϕ > V Light = c)) are prone to radiate into free space. Therefore, in microwave applications, in order to design non-radiating transmission line, each unit cell of the line must be designed such that the phase velocity of the line is less than that of light. This property can be achieved when the propagation constant of the line is more than that of free space, i.e. (β BL > K 0), because the phase velocity is defined as (V ϕ = ω/β).
On the other hand, as we know, for 0° CRLH-TLS, the propagation constant of the line is equal to zero. Therefore, the transmission line will be superluminal and will tend to radiate into free space [Reference Bemani and Nikmehr3]. In order to solve this problem the new topology which is known as NR-CRLH-TL has been proposed in [Reference Antoniades17]. The proposed transmission line has a phase velocity that is slower than the speed of the light and so will not prone to radiate. NR-CRLH-TLS is consisted of a conventional CRLH-TL and a conventional transmission line (C-TL) (as shown in Fig. 1). CRLH-TL must be designed such that the unit cell operates as non-radiating transmission line while the C-TL is inherently non-radiating. The phase response of NR-CRLH-TL is given by [Reference Antoniades17]
$$\eqalign{\Phi _{{\rm NR - CRLH - TL}} = & \phi _{{\rm CRLH - TL}} + \phi _{{\rm C - TL}} \cr = & n\left( { - \omega \sqrt {LC} d + \displaystyle{1 \over {\omega \sqrt {L_0 C_0}}}} \right) - \omega \sqrt {LC} d_{{\rm C - TL}}.} $$
Fig. 1. Topology of n-stage NR-CRLH-TL [Reference Antoniades17].
Similar to CRLH-TLs the non-linear phase response of NR-CRLH-TLs can be controlled in order to design dual-band NR-CRLH-TLs. Therefore, the required parameters for dual-band NR-CRLH-TLs are given by [Reference Bemani and Nikmehr18]
$$d = \displaystyle{{(\phi_1\, f_1 - \phi_2\, f_2 )f_1 + \phi _{{\rm C - TL}} (\,f_2^2 - f_1^2 )} \over {2\pi n\sqrt {LC} f_1 (\,f_2^2 - f_1^2 )}},$$
$$L_0 = \displaystyle{{nZ_0 (\,f_2^2 - f_1^2 )} \over {2\pi\, (\phi_1\, f_2 - \phi_2\, f_1 )f_1\, f_2}}, $$
$$C_0 = \displaystyle{{n(f_2^2 - f_1^2 )} \over {2\pi Z_0 (\phi _1\, f_2 - \phi _2 f_1 )f_1\, f_2}}, $$
where ϕ C−TL is the phase shift of conventional transmission line and can be written as
$$\phi _{{\rm C - TL}} = \phi _{{\rm C - TL}} \left\vert {_{f_1} = \left( {\displaystyle{{f_1} \over {f_2}}} \right)\phi _{{\rm C - TL}}} \right\vert_{f_2}. $$
III. DUAL-BAND 3-WAY SPD
Fig. 2 displays the topology of the proposed dual-band 3-way SPD. As it can be seen in this figure, this divider consists of three types of CRLH-TLs as follows
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(I) Three dual-band +90° and −90° CRLH-TLs with characteristic impedance of Z 0 = 86.6 Ω to provide a dual-band impedance transformation from the 50 Ω test equipment impedance to the load impedance of 150 Ω.
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(II) Three dual-band 0° and 180° NR-CRLH-TLs with characteristic impedance of Z 0 = 65.0 Ω between the output ports in order to feed three 150 Ω loads. Also, as discussed in [Reference Antoniades17] (Appendix B) that in order to achieve the maximum bandwidth for series power divider we need to use 0° (or 180°) transmission lines between output ports. Therefore, these loads are in parallel situation and resulting in a circuit with an input which is matched to 50 Ω. Also, two dual-band 0° and 180° NR-CRLH-TLs are used at the end of the second and fourth ports in order to provide in-phase signals at the output ports.
Fig. 2. Schematic diagram of the proposed dual-band 3-way SPD.
Based on the design equations in Section II, the various dual-band CRLH-TLs and NR-CRLH-TLs in the structure of the proposed equal 3-way SPD were designed. Table 1 summarizes the values of the loading elements, the length and width of the RH-TLs and other characteristics of these lines.
Table 1. Loading element values and pertinent characteristics of the two types of the dual-band line in Fig. 3 [Reference Bemani and Nikmehr18].
A) Isolation study of 3-way SPD
As discussed in [Reference Pozar21], the N-way divider or combiner can be matched at all ports, with good isolation between output ports by using isolation resistors between all ports. Therefore, as shown in Fig. 3, three isolation resistors; i.e. r 23, r 24, and r 34 are considered between output ports of the proposed 3-way SPD to improve isolation characteristics of this SPD. Since the NR-CRLH-TLs are of 0° and 180° at the two resonance frequencies, the joint of the three loads in Fig. 3, i.e. n 2, n 3, and n 4, can be described as one node. The same simplification can be applied to the vertical NR-CRLH-TLs in Fig. 3. Thus, the proposed equal 3-way SPD can be modeled as a 4-port network circuit. Furthermore, in order to calculate the values of isolation resistors, an input signal is applied at one of the output ports (For example port 4) while port 1 is well matched. The above scenario is shown in Fig. 3.
Fig. 3. The equivalent circuit of the proposed 3-way SPD (V 1, V 2, … and V 4 are the voltages of the nodes).
Because of the symmetry in this figure, i.e. three equal output lines, we can assume that r 23 = r 24 = r 34 = r. On the other hand, as is well known, the ideal isolation is obtained when the voltages of other output ports are zero. Regarding the Fig. 3, the transmission line voltages and currents giving rise to the following expressions [Reference Cheng22]
$$V_1 = V_2 \cos \theta + ji_{2o} Z_0 \sin \theta,$$
$$i_{2i} = i_{2o} \cos \theta + jV_{2o} \sin \theta /Z_0,$$
$$V_1 = V_3 \cos \theta + ji_{3o} Z_0 \sin \theta,$$
$$i_{3i} = i_{3o} \cos \theta + jV_{3o} \sin \theta /Z_0,$$
$$V_1 = V_4 \cos \theta + ji_{4o} Z_0 \sin \theta,$$
$$i_{4i} = i_{4o} \cos \theta + jV_{4o} \sin \theta /Z_0.$$
Also for the ports voltages we can write
$$V_1 = R_0 i_1 = 50i_1,$$
$$V_2 = R_0 i_2 = 50i_2,$$
$$V_3 = R_0 i_3 = 50i_3,$$
$$V_4 = V_s.$$
And for the isolation resistors the Kirchhoff's current law (KCL) equations are given by
$$V_2 - V_3 = r_{23} i_{23},$$
$$V_2 - V_4 = r_{24} i_{24},$$
$$V_3 - V_4 = r_{34} i_{34}.$$
Finally, for five nodes, the KCL equations can be written as
$$i_1 + i_{2i} + i_{3i} + i_{4i} = 0,$$
$$i_{2o} = i_2 + i_{23} + i_{24},$$
$$i_{23} + i_{3o} = i_3 + i_{34},$$
$$i_{24} + i_{34} + i_{4o} = i_4.$$
On the other hand, as is well known, the ideal isolation is obtained when the voltages of other output ports are zero. Therefore, a simple inspection of equations (9)–(12) with respect to the fact that V 2 = V 3 = 0 reveals that
$$r = \left. {\displaystyle{{Z_{{\rm CRLH}}^{\rm 2}} \over {50}}} \right\vert_{Z_{{\rm CRLH = 86}{\rm. 6}}} = 150\,\Omega. $$
IV. SIMULATION AND EXPERIMENTAL RESULTS
The proposed 3-way SPD in this paper was designed and simulated using Agilent-ADS simulator. Also, in order to fully account parasitic and electromagnetic effects within the structure, full wave simulations were carried out in the high frequency structures simulator (HFSS) which is a full wave simulator based on the finite element method. Finally, the designed SPD was fabricated as shown in Fig. 4. It should be mentioned that in this paper, standard size of 0402 were used for the loading elements.
Fig. 4. (a) Fabricated prototype of the proposed dual-band 3-way SPD, (b) backside of the isolation network.
The presented results in this section are compared among the circuit, HFSS, and measured results. Fig. 5 shows the measured and the simulated magnitude of S 11 for the proposed 3-way SPD. It should be mentioned that the small difference between measured and simulated results can be attributed to the small tolerance in substrate permittivity (FR4, ε r = 4.4±0.1) and non-ideal surface-mount technology (SMT) component with value variation of ±5%. As it can be seen, for the measured results the dual-band divider exhibits a measured |−10| dB |S 11| low frequency bandwidth of 0.36 GHz, from 0.85 to 1.21 GHz and a high frequency bandwidth of 0.41 GHz, from 2.36 to 2.77 GHz. In addition, the measured magnitude of S 22, S 33, and S 44 for the proposed 3-way SPD are shown in Fig. 6. This figure exhibits the significant improvement of the output return losses after using the isolation resistors.
Fig. 5. Measured and simulated magnitude of S 11 for the proposed dual-band 3-way SPD.
Fig. 6. Measured magnitude of (a) S 22, (b) S 33, and (c) S 44 for the proposed dual-band 3-way SPD.
Also, Fig. 7 displays transmission characteristics to each of the output ports. It is obvious that the proposed dual-band divider provides equal and in-phase signals at output ports in the two passband. For demonstration, Fig. 8 shows the measured amplitude and phase balances of the proposed 3-way SPD which confirms that within the desired frequency bands, i.e. 0.902–0.928 GHz and 2.400–2.483 GHz, the measured amplitude and phase difference variations between output ports do not fluctuate beyond ±0.5 dB and ±10°, respectively, from the values at 0.915 and 2.440 GHz. In addition, Fig. 9 depicts, the measured isolation characteristics between output ports of the designed 3-way SPD with and without isolation resistors. This figure confirms the significant improvement for this parameter after employing the isolation resistors. For detailed performance of this dual-band 3-way SPD, a summary is given in Table 2. It can be concluded that, due to excellent return-loss at the output ports and high isolation, the proposed SPD in this paper would function well as a power combiner and can be recommended to use as a power combiner. It should be noted that the extra losses in output ports can be attributed to non-ideal SMT component and the loss of the FR4 substrate. Also, more losses at 2.44 GHz than in 0.915 GHz is from high dielectric loss in upper frequencies.
Fig. 7. Measured transmission characteristics to each of the output ports for the proposed dual-band 3-way SPD; (a) magnitude of S 21, S 31, and S 41, (b) insertion phase of S 21, S 31, and S 41.
Fig. 8. (a) Amplitude and (b) phase balances of the proposed 3-way SPD.
Fig. 9. Measured isolation between output ports of the proposed dual-band 3-way SPD; (a) with and (b) without isolation resistors.
Table 2. Dimensions and measured performances of the 3-way SPD.
V. CONCLUSION
A dual-band non-radiating 3-way equal in-phase series power divider has been proposed in this paper. The proposed divider has been designed and implemented to work at 0.915 and 2.440 GHz. The designed divider exhibits good input return loss as well as equal power split to output ports in dual bands. Also, due to its excellent return-loss at the output ports and its high isolation, this type of power divider would function well as a power combiner. In addition, the dual-band nature of this divider, its compact size, low cost, ability to set the output ports arbitrary apart and easy integration with planar devices therefore renders it ideal for feeding many analog circuits, such as multipliers, mixers, amplifiers, and planar antenna arrays. The design concept of this paper can be easily extended to any number of output ports with arbitrary frequency and power dividing ratio.
ACKNOWLEDGEMENTS
This paper is published as part of a research project supported by the University of Tabriz Research Affairs Office. The authors are grateful to the University of Tabriz for financial supports.
Mohammad Bemani received his B.S. degree in Electrical Engineering from the University of KN Toosi University of Technology, Tehran, Iran, in 2007, the M.S. and the Ph.D. degree from the University of Tabriz, Tabriz, Iran, in 2009 and 2013, respectively, in Electrical Engineering. From 2010 to 2013, he was a Teaching Assistant at the University of Tabriz, where he contributed to the design and teaching of undergraduate courses relating to electromagnetics. During the same period, he was a Research Assistant at the University of Tabriz, where he was involved in the development of microwave devices and antennas based on negative-refractive-index transmission-line (NRI-TL), metamaterials, and CRLH structures. His research interests include multi-band and UWB components, reconfigurable antenna, active circuits, dielectric resonator antennas, fractal antennas, composite right and left-handed structures, and negative-refractive-index transmission lines. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Tabriz, Tabriz, Iran.
Saeid Nikmehr Received his B.S. degree in Electrical Engineering from KN Toosi University of Technology, Tehran, Iran, and the M.S. and Ph.D. degrees, both from Syracuse University, Syracuse, NY, in 1988 and 1992, respectively. He is currently an Associate Professor in the Department of Electrical and Computer Engineering of University of Tabriz, Iran. His research interests involve microstrip antennas, microwave components design, and computational electromagnetics.
