Non-uniform transmission line ultra-wideband Wilkinson power divider.
Ultra-wideband frequency spectrum finds several key applications in many modern and emerging technologies, such as tactical and strategic communications , through-the-wall imaging , medical treatments [3, 4], and high data rate transmission [5,6]. Consequently, the need for microwave components that support such a range of frequencies is of utmost importance. Among various RF devices, the Wilkinson power divider (WPD), invented by E. Wilkinson , is a passive component that gained much interest in the literature due to its capability to achieve isolation between the output ports while maintaining a matched condition at all ports. Those significant characteristics qualified its adoption in countless front-end microwave subsystems. In , a reduced-size UWB power divider was proposed by implementing the transmission lines of a two-stage WPD using bridged T-coils. However, the accompanied complexity in the design and fabrication is a major drawback in such an approach. Bialkowski and Abbosh proposed a compact UWB out-of-phase uniplanar power divider formed by a slotline and a microstrip line T-junction along with wideband microstrip to slotline transitions . A miniaturized three-way power divider with ultra-wideband behavior was presented in  by utilizing broadside coupling via multilayer microstrip/slot transitions of elliptical shape. Tapered line transformers, which exhibit almost constant input impedance over a wide range of frequencies, were incorporated in the design of an UWB divider . Nevertheless, the resulting circuitry area was comparatively large. Different kinds of stubs such as open stubs , delta stubs , and radial stubs  were introduced as an approach in designing modified WPDs with extended bandwidth. As such, extra transmission lines were integrated in the proposed techniques.
In this paper, compact UWB WPD based on non-uniform transmission lines is proposed. The soul of the design approach depends mainly on the even/odd mode circuits, which results in the replacement of the uniform microstrip arms of the conventional power divider by variable-width impedance profiles. Furthermore, three isolation resistors are optimized and uniformly mounted between both arms to maintain an acceptable isolation and output ports' matching over the entire band.
The article organization is as follows: Section 2 discusses the design procedure of the proposed UWB NTLs power divider. Simulation and measurement results of a designed example are presented in Section 3. Finally, conclusions and remarks are given in Section 4.
2. ULTRA-WIDEBAND WPD DESIGN
A schematic diagram of the proposed device is shown in Fig. 1, which presents the modified UWB equal-split WPD. Each uniform impedance branch in the conventional divider is replaced by a single NTL transformer to achieve UWB operation.
Figure 2 illustrates the corresponding even and odd mode circuits. In Section 2.1 (even mode analysis), the design of the NTLs is presented; whereas in Section 2.2 (odd mode analysis), the values of the isolation resistors are derived to meet acceptable output ports' isolation and matching.
2.1. Even Mode Analysis
The even mode equivalent circuit is shown in Fig. 2(a), which presents a typical NTL of length d with a varying characteristic impedance Z(z) and propagation constant [beta](z) that matches a source impedance [Z.sub.s] to a load impedance [Z.sub.l]. In our case, [Z.sub.s] = 2[Z.sub.0] and [Z.sub.l] = [Z.sub.0]. The isolation resistors [R.sub.i]/2 (i = 1, 2, 3), due to the symmetric excitation at the two output ports, are terminated with an open circuit. The NTL is designed by enforcing the magnitude of the reflection coefficient [absolute value of [GAMMA]] to be zero (or very small) over the frequency range 3.1 GHz-10.6 GHz. The magnitude of the reflection coefficient at the input port can be expressed in terms of [Z.sup.e.sub.in] shown in Fig. 2(a), where [Z.sup.e.sub.in] can be calculated after obtaining the ABCD parameters of the NTL. Those ABCD parameters are found by subdividing such a transformer into K uniform electrically short segments each with length of [DELTA]z. It is worth to point out here that the number of the uniform short sections K is chosen to be 50. The ABCD matrix of the whole NTL transformer is obtained by multiplying the ABCD parameters of each section as follows :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)
where the ABCD parameters of the ith segment are :
[A.sub.i] = [D.sub.i] = cos([DELTA][theta]), (2a)
[B.sub.i] = [Z.sup.2]((i - 0.5)[DELTA]z)[C.sub.i] = jZ((i - 0.5)[DELTA]z) sin([DELTA][theta]), i = 1, 2, ..., K, (2b)
[DELTA][theta] = [2[pi]/[lambda]] [DELTA]z = [2[pi]/c] f [square root of [[epsilon].sub.eff]] [DELTA]z. (2c)
The effective dielectric constant, [[epsilon].sub.eff], of each section is calculated using the well-known microstrip line formulas given in . Then, the normalized non-uniform profile of the characteristic impedance Z(z), written in terms of a truncated Fourier series, is considered :
ln(Z(z)/[Z.sub.c]) = [c.sub.0] + [N.summation over (n=1)][[a.sub.n] cos (2[pi]nz/d) + [b.sub.n] sin (2[pi]nz/d)]. (3)
where [Z.sub.c], which equals [([Z.sub.s][Z.sub.l]).sup.0.5], is the characteristic impedance of the conventional WPD arm. Thus, an optimum designed NTL has its reflection coefficient magnitude over the frequency range (3.1 GHz-10.6 GHz), with an increment of [DELTA]f, as close as possible to zero. Therefore, the optimum values of the Fourier coefficients can be obtained through minimizing the following error function :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5a)
[[GAMMA].sub.in]([f.sub.j]) = [Z.sup.e.sub.in]([f.sub.j]) - [Z.sub.s]/[Z.sup.e.sub.in]([f.sub.j]) + [Z.sub.s], (5b)
[Z.sup.e.sub.in]([f.sub.j]) = A([f.sub.j])[Z.sub.l] + B([f.sub.j])/C([f.sub.j])[Z.sub.l] + D([f.sub.j])]. (5c)
Moreover, the error function in (4) should be restricted by some constraints such as easy fabrication and physical matching, as follows :
[[bar.Z].sub.min] [less than or equal to] [bar.Z](z) [less than or equal to] [[bar.Z].sub.max]. (6a)
[bar.Z](0) = [bar.Z](d) = 1. (6b)
To solve the above bound-constrained non-linear minimization problem, a trust-region-reflective algorithm  is adopted, which has strong convergence properties.
2.2. Odd Mode Analysis
The odd-mode analysis is carried out to obtain the resistors' values needed to achieve the optimum output ports isolation and matching conditions. Fig. 2(b) shows the equivalent odd-mode circuit of the proposed divider , where the asymmetric excitation of the output ports results in terminating each [R.sub.i]/2 resistor with a short circuit.
Once the optimum values of the Fourier coefficients are determined by following the procedure described in Section 2.1, the NTL transformer will be subdivided into 3 sections, and the ABCD matrix for each section is calculated employing (1) and (2). Then, the total ABCD matrix of the whole network shown in Fig. 2(b) can be calculated as follows :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
It should be pointed out here that the isolation resistors are distributed uniformly (a resistor every d/3 distance) along the NTL transformer. Finally, and as illustrated in Fig. 2(b), the following equation can be written:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
Setting [V.sub.2] in (8) to zero, and solving for [V.sub.1]/[I.sub.1], one obtains:
[V.sub.1]/[I.sub.1] = B/D = [Z.sup.o.sub.in]. (9)
For perfect output port matching, the following condition should be satisfied:
[[GAMMA].sub.out] ([f.sub.j]) = [Z.sup.o.sub.in]([f.sub.j]) - [Z.sub.0]/[Z.sup.o.sub.in]([f.sub.j]) + [Z.sub.0]. (10)
where [f.sub.j](j = 1,2, ..., m) denotes the frequencies at which (10) is calculated. In the context of this article, [DELTA]f is set to 0.5 GHz. So, for a perfect output ports matching over the UWB range, the following error should be minimized:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11b)
This optimization problem is solved keeping in mind that [R.sub.1], [R.sub.2], and [R.sub.3] are the optimization variables to be determined.
3. SIMULATION AND EXPERIMENTAL RESULTS
In this section, based on the design procedure provided in Section 2, a design example of an UWB power divider is presented taking into account a characteristic impedance of 50 Q and a Rogers RO4003C substrate with a relative permittivity of 3.55, a thickness of 0.813 mm, and a loss tangent of 0.0027. The length of each NTL arm of the proposed WPD is set to 10 mm. The optimized Fourier coefficients of the designed nonuniform UWB power divider arm are given in Table 1; whereas Fig. 3 shows the corresponding variable impedance profile.
A photograph of the fabricated divider is shown in Fig. 4. It is worth mentioning that the optimum values of the isolation resistors [R.sub.1], [R.sub.2], and [R.sub.3] are found to be 150 Q, 301 Q, and 133 Q, respectively.
Figure 5 illustrates the resulting simulated and measured scattering parameters. The simulation results were obtained using HFSS , which is a finite element method-based full-wave EM simulator, while the experimental ones were obtained using an HP8510 VNA.
As shown in Fig. 5, the simulation and measurement results of the input and output ports match parameters [S.sub.11] and [S.sub.22] (which is equal to [S.sub.33]), respectively, as well as the isolation parameter [S.sub.23] are below -10dB over the 3.1-10.6GHz frequency band. Moreover, the simulated transmission parameter [S.sub.21] is in the range of -3.2 dB to -3.8 dB, whereas the measured values of the same parameter ranges between -3.2 dB to -4.2 dB over the frequency range of interest. Such results are in proximity to the theoretical value of -3dB. The small discrepancies between the simulations and measurements are thought to be due to the fabrication process and measurement errors (e.g., not taking the modeling of the SMA connectors into account). Fig. 6 shows the measured amplitude and phase imbalance between the two output ports of the proposed equal-split in-phase UWB divider.
The measured phase imbalance is less than [+ or -] 10[degrees] over the entire design frequency range. Furthermore, the obtained amplitude imbalance is around [+ or -] 0.1 dB over the 3.1 GHz-10.6 GHz band proving the excellent degree of symmetry of the implemented structure. Fig. 7 depicts the simulated and measured group delay responses of the designed WPD.
Both results are almost flat over the UWB range and less than 0.2 ns with a mismatch thought to be due to the inhomogeneous substrate material used in this project (i.e., a horizontal dielectric constant different than a vertical dielectric constant). Table 2 shows a comparison between our proposed design and other UWB Wilkinson power dividers presented in  and [11-14] taking into consideration size, electrical performance, design approach, and fabrication technology.
In this paper, a general design of an ultra-wideband WPD incorporating Fourier-based impedance-varying profiles is presented. The design of the ultra-wideband NTLs was obtained from the even mode analysis of the WPD, whereas three isolation resistors were calculated through the odd mode circuit. For verification purposes, an equal-split UWB power divider was designed, simulated, and measured. The good agreement between simulation and measurement results over the 3.1-10.6 GHz frequency range proves the validity of the design procedure. The differences between simulated and experimental results could be due to the fabrication process, the effect of the connectors, and measurement errors.
Received 8 August 2013, Accepted 11 September 2013, Scheduled September 12 2013
This research was supported in part by NSF EARS Award 1247946. The authors thank Dr. M. Alam, Chair of the EECS Department and Dr. N. Naganathan, Dean of the College of Engineering for their continued support during the course of this project.
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Khair Shamaileh (1), *, Mohammad Almalkawi (1), Vijay Devabhaktuni (1), Nihad Dib (2), Bassem Henin (3), and Amin Abbosh (3)
(1) Electrical Engineering & Computer Science Department, University of Toledo, Toledo, OH 43606, USA
(2) Electrical Engineering Department, Jordan University of Science and Technology, Irbid 22110, Jordan
(3) School of ITEE, University of Queensland, St Lucia, QLD 4072, Australia
* Corresponding author: Khair Shamaileh (firstname.lastname@example.org).
Table 1. Fourier coefficients for the variable impedance profile in the UWB WPD design. [c.sub.0] [a.sub.1] [a.sub.2] [a.sub.3] [a.sub.4] [a.sub.5] 0.0191 0.0090 0.1335 0.0177 0.2015 0.1120 [b.sub.1] [b.sub.2] [b.sub.3] [b.sub.4] [b.sub.5] -0.4754 0.0733 -0.0701 0.1181 -0.1387 Table 2. Comparison between this work and different previous proposed designs. Arm S-Parameters over Design approach/ length 3.1-10.6 GHz (dB) Fabrication technology (mm) This work 10 [S.sub.11] [-28,-14] --Fourier-based [S.sub.22] [-22,-13] transmission lines [S.sub.23] [-30,-13] --Single-layer [S.sub.21] [-3.8,-3.2] substrate --No stubs are incorporated Ref.  20 [S.sub.11] [-19,-10] --Slotline and two [S.sub.22] [-12,-10] arms of a microstrip [S.sub.23] [-10, -7] t-junction [S.sub.21] [-4.1,-3.8] --Built on two-layer substrate --No stubs are incorporated Ref.  29.65 [S.sub.11] [-35,-19] --Tapered lines * [S.sub.22] [-40, -6] transformer [S.sub.23] [-48, -7] --Single-layer [S.sub.21] [-3.3,-3.1] substrate --No stubs are incorporated Ref.  24 [S.sub.11] [-25,-10] --Stub-loaded two- [S.sub.22] [-40,-12] section microstrip [S.sub.23] [-25,-10] --Single-layer [S.sub.21] [-4.2,-3.1] substrate --stubs incorporated in the design Ref.  6.5 [S.sub.11] [-45,-14] --Microstrip loaded [S.sub.22] [-55,-14] with stubs [S.sub.23] [-45,-10] --Single-layer [S.sub.21] [-4.1,-3.2] substrate --Delta stubs incorporated Ref.  7 [S.sub.11] [-50,-18] --Microstrip loaded [S.sub.22] [-20,-12] with stubs [S.sub.23] [-42,-12] --Single-layer [S.sub.21] [-3.5,-3.1] substrate --Butterfly radial stubs incorporated * The three isolation resistors case is taken in this comparison.
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|Author:||Shamaileh, Khair; Almalkawi, Mohammad; Devabhaktuni, Vijay; Dib, Nihad; Henin, Bassem; Abbosh, Amin|
|Publication:||Progress In Electromagnetics Research C|
|Date:||Nov 1, 2013|
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