A High-Frequency/Power Ratio Wilkinson Power Divider Based on Identical/Non-Identical Multi-T-Sections With Short-Circuited Stubs

In this article, a systematic procedure is presented for the design of a dual-band unequal-split Wilkinson power divider (WPD) with high frequency and power division ratios. The design methodology is based on replacing the impractical high-impedance transmission line in the conventional divider with cascaded T-section structures with short-circuited stubs. The proposed methodology is simple and adopts distributed elements without reactive components to achieve a large power division ratio. The use of short-circuited stubs allows for circuit miniaturization in comparison to counterpart open-stub techniques. Furthermore, a high frequency ratio is obtained through optimizing the electrical and physical parameters of the T-sections based on a transmission lines theory framework as demonstrated in a simulated and measured prototype. To validate the proposed procedure, two WPD prototypes operating at 1/2.7 GHz and 1/4.4 GHz with a 1:10 power division ratio are simulated and measured. The theoretical response, supported by electromagnetic simulations and measurements, justifies the design concept.

above-mentioned limitations [34]. We showed that replacing the high impedance branch in the conventional design with cascaded-T-sections of open-stubs allows for a high power division at the operating frequencies. The work presented herein expands that demonstrated in [34] and introduces the following new contributions: 1) The utilization of short-circuited stubs facilitates a miniaturized physical area, which in turn results in reduced losses (e.g., radiation losses) as compared to open-stubs.
2) The optimization-driven framework which is based on transmission lines theory optimizes the electrical and physical parameters of the T-sections for higher frequency ratios. In what follows, in Section II-A, the design equations of dual-band identical multi-T-sections with short-circuitedstubs are derived. Next, an optimization-driven framework is elaborated in Section II-B to develop non-identical multi-Tsections with higher frequency ratios. Steps for designing a dual-band unequal-split WPD are discussed in Section II-C. The resulting analytical response is given in Section III. Electromagnetic (EM) simulations and measurements are given in Section IV, followed by conclusions in Section V.

II. THEORY AND DESIGN EQUATIONS
In this section, the design procedure of the proposed high frequency and power division ratios dual-band WPD is elaborated. The major difference between the proposed methodology and the conventional divider design is in the use of identical multi-T-sections that facilitate a dual-band operation, and at the same time, eliminate the need for impractical high impedance transmission lines in high power division scenarios. Moreover, based on transmission lines theory, an optimization routine for non-identical structures is established to increase frequency ratio. In Sections II-A-II-C, the design equations, optimization process, and divider implementation are presented in greater detail. Fig. 1 shows a transmission line with electrical length θ h and high characteristic impedance Z h , subdivided into segments each with a length ofθ h < θ h and the proposed dual-band cascaded T-section network. Each T-section consists of two identical transmission lines Z u of length θ u and one shortcircuited stub Z s of length θ s . The mathematical formulation of these T-sections begins with matching the ABCD parameters of a single T-section with those in the conventional high-impedance transmission line as follows: By utilizing trigonometric identities on the parameters A and B at both sides, the following expressions are obtained:

A. HIGH-IMPEDANCE CHARACTERIZATION USING MULTI-T-SECTIONS
An expression of the T-section impedance, Z u , is obtained by substituting (2a) in (2b) as follows: Finally, an expression of the stub impedance, Z s , is derived by substituting (3) in (2a): It is noteworthy to point out that losses, which result in complex ABCD parameters, have a negligible impact on the T-section characteristics, especially when low-loss substrates are used. Therefore, a lossless transmission line model is used. According to (3) and (4), the proposed T-section maintains a dual-band operation under the following conditions: where θ u(f 1 ,f 2 ) and θ s(f 1 ,f 2 ) are the transmission line and short-circuited stub lengths, respectively, at the operating frequencies f 1 and f 2 (f 2 > f 1 ); whereas n and m are positive integers. The frequency ratio, R, is defined as Thus, a design equation with reference to f 1 can be derived from (5b) as: In this article, n = 1 and the positive sign in (6) are considered to realize a compact power divider design. In addition, θ s = θ u is set to obtain a physically realizable Z s ; whereas θ s = 2θ u in the case of multi-T-sections with open stubs [34]. Hence, circuit miniaturization is achieved. Fig. 2 shows Z u and Z s with different values ofθ h as a function of R.
Here, a 0.813-mm-thick Rogers RO4003C substrate with a relative permittivity of 3.55 is considered, which allows for a maximum microstrip impedance of Z max = 140 . The impedance Z h is chosen as 295 for a reason that will be discussed in the later sections.

B. NON-IDENTICAL MULTI-T-SECTION OPTIMIZATION
As shown in Figs. 1 and 2, a high impedance of electrical length θ h = 90 • can be replaced with a dual-band identical single T-section with practical impedances (Z u,s ≤ 140 ). Furthermore, a higher R is obtained by subdividing Z h into segments each with a length ofθ h < θ h . However, for θ h = 45 • , the maximum value of R is 3.2. In other words, to design a dual-band identical multi-T-section with R > 3.2, θ h = 30 • must be considered (R max = 4.1). To increase R for a givenθ h , the transmission line parameters of the cascaded T-sections can be non-identical as Fig. 3 suggests. The cascaded ABCD parameters of the T-sections, [AB; CD] T , can be obtained by multiplying the ABCD matrices of all segments as follows: where [AB; CD] ij is the ABCD matrix of j th impedance in the i th section (i, j = 1, 2) and [AB; CD] si is the ABCD matrix of the short-circuited stub in the i th section. The ABCD parameters are computed at f 1,2 and are given as: where c is the speed of light. The effective dielectric constant, eff , of each section is found using the microstrip line formulas in [35]. In (8)-(9), Z uij and Z si are the impedances of the transmission lines and short stubs, respectively, and θ uij , d uij and θ si , d si are the electrical and the physical lengths of the transmission lines and short stubs, respectively, for the cascaded non-identical T-sections. The input impedance, Z in , of the cascaded T-sections terminated by a load impedance, Z L , is expressed in terms of the total ABCD matrix in (7) as follows [35]:

FIGURE 5. Schematic of an unequal-split WPD (black) and the T-sections that replace its branches (blue). R = 4.4.
Once Z in is determined, the reflection coefficient, , at f 1,2 can be calculated as follows: where Z s is the source impedance. Then, an error function at each frequency is defined as: Subsequently, the error vector resulting from applying (12) to both frequencies f 1,2 is used to formulate and minimize the following objective function: The parameters vector to be optimized in (13) consists of 12 elements: six physical lengths [d uij(i,j=1,2) 2) ]. The optimized vector must be within reasonable fabrication tolerances and meet matching conditions. Therefore, the following physical constraints are set: The constraints presented in (14) confine the transmission lines within minimum and maximum lengths such that miniaturization is maintained; and ensure impedance values within milling tolerance. The sequential quadratic programming algorithm is used to minimize (13) subject to (14) due to its performance in constrained optimization problems. Once this procedure is completed, corresponding widths and lengths of the transmission lines and short-circuited stubs are found from the impedances and electrical lengths based on the well-known formulas reported in [35]. Algorithm 1 presents a pseudo-code of the design steps for a non-identical multi-T-section with a high frequency ratio.

C. DESIGN OF UNEQUAL SPLIT WPDS WITH HIGH R
10 . Two 1: k 2 WPDs, where k 2 = 10, are designed to verify the design procedure. The first WPD is designed to operate at 1 and 2.7 GHz (θ u = θ s = 48.64 • ) taking into account the substrate mentioned earlier. Here, the two high-impedance branches Z 1 and Z 2 in the conventional WPD are subdivided into two 45 • segments and replaced with identical T-sections; whereas the low impedance branches, Z 3,4 , are replaced  with their equivalent dual-band single T-section structure. As the maximum frequency ratio with identical T-sections consideringθ h = 45 • is 3.2, the optimization routine is not necessary. The resulting impedances and electrical lengths are given in Table 1. The isolation resistor R is is given as Z 0 (k 2 + 1)/k = 174 [35].   with non-identical T-sections as R for Z 1 exceeds the maximum value with identical T-sections (i.e., R max = 3.2). The impedances and electrical lengths of the identical T-sections for this case are given in Table 2; whereas the optimized non-identical T-section parameters are reported in Table 3. As shown in Table 3 all optimized parameters are within the constraints described in (14a) and (14b). It is paramount to point out that there is no unique solution for the optimized parameters, and each optimization results in different sets of impedances and lengths. However, the optimal analytical response adjoined with a compact size is considered. The flowchart in Fig. 6 summarizes the design steps of a dual-band unequal-split WPD with high frequency ratio.
Meanwhile, it is imperative to discuss the key factors that limit the performance of the proposed WPDs. To this end, the effect of power division ratio, k 2 on maximum attainable frequency ratio, R max is studied. Based on Figs. 4 and 5, Z 1 has the maximum impedance. By utilizing (3)-(4), one can conclude that for a givenθ h , R max for Z 1 is the minimum for all impedance branches, Z (c=1,...4) . Hence, Z 1 governs R max for multi-T-section configurations. Therefore, a study of power split versus frequency ratio for Z 1 would suffice. For any given k 2 , Z 1 can be expressed as: where Z 0 is the characteristic impedance. Next, by adopting (3)-(4) considering Z 1 = Z h , the values of Z u,s are obtained, which in turn provide R max for the corresponding k 2 . Fig. 7 demonstrates R max for different values of k 2 . It is observed that R max reduces with the increase of k 2 for any givenθ h < θ h . The lower/upper limits on R max for a given k 2 can also be derived from the plot.

III. ANALYTICAL RESULTS
In this section, the analytical results for the two WPD examples are presented. Matching transformers are included to match the output ports to the 50 standard connectors. The analytical S-parameters of the divider with R = 2.7 are shown in Figs Hence, the proposed framework demonstrates an excellent dual-band performance of unequal split WPDs with high frequency and power division ratios.
Next, a sensitivity analysis is performed to evaluate the manufacturing tolerances in the proposed design. Worstcase tolerances (i.e., ±10%) in the physical parameters for WPD1 (e.g., widths and lengths of the transmission lines and short-circuited stubs) are studied. Figs. 9 and 10 show the effect of the tolerances in widths (W ui(i=1,..4) , W si(i=1,..4) ,   ui(i=1,..4) , d si(i=1,..4) , Table 1) on the S-parameters, respectively. As illustrated in Fig. 9, the proposed design handles tolerances encountered in widths. However, such tolerances in length introduce a shift in the operating frequencies, as shown in Fig. 10. Therefore, the proposed design is more sensitive for the variation in the lengths of the transmission lines and short-circuited stubs in the multi-T-sections.
The proposed methodology differs from other previous efforts in the following aspects: 1) The resulting design is planar and is built on a single-layer. 2) Circuit miniaturization is achieved by employing short stubs with θ s = θ u . On the other hand, θ s = 2θ u in the case where open stubs are utilized [34]. 3) Unlike [33], a distributed structure is adopted without reactive components. As such, characteristic distortions are avoided at high frequencies [27]. 4) The optimization-driven framework for the non-identical T-sections facilitates higher frequency ratios. Based on what was presented so far, it is not possible to achieve R > 4 without utilizing reactive components [33]. Our technique, however, achieves high R with only distributed transmission lines.

IV. SIMULATIONS AND MEASUREMENTS
In this section, EM simulations and measurements are presented and discussed for the aforementioned dualband 1:10 WPD examples. The prototypes are fabricated with a standard milling machine. The full-wave simulator ANSYS HFSS is used to simulate the two designs. Measurements are performed with a Rhode & Schwartz ZNB20 network analyzer. An off-wafer calibration was performed prior to the measurements to eliminate hardware error. To this end, a 3.5-mm short-open-load-through kit was employed to shift the measurement reference plane to the end of the probe tips. Fig. 11 shows photographs of the fabricated prototypes; whereas simulations and experimental results are shown in Figs. 12 and 13. Figs. 12(a)-b) show that the input/output ports matching for the WPD example with R = 2.7 are below −30 dB at the design frequencies. Fig. 12(c) shows that the simulated and measured values of S 21 and S 31 at f 1 = 1 GHz and f 2 = 2.7 GHz are −10.8 dB and −0.85 dB, respectively (−10.41 dB and −0.41 dB are the theoretical values for a 1:10 power division). Finally, Fig. 12(d) shows that the isolation between the output ports is better than −30 dB at the design frequencies. Fig. 13 shows the simulated and measured S-parameters for the WPD example with R = 4.4. Input/output ports matching below −30 dB are achieved at f 1 = 1 GHz and f 2 = 4.4 GHz; whereas the isolation between output ports is better than −20 dB. Finally, the transmission parameters, S 21 and S 31 , are found to be −11.5 dB and −1.83 dB, respectively, at the design frequencies. It is noteworthy to point out that the simulations and measurements demonstrated  in Figs. 8, 12, and 13 represent a dual-frequency (i.e., selective) operation. In other words, the desired electrical performance (e.g., matching, transmission) is met at the design frequencies. Future investigations can be devoted towards achieving a similar response over a broadband frequency range. In both WPD examples, the agreement between simulations and measurements are acceptable at the design frequencies. The measurements for WPD2 are slightly different from simulations, which can be attributed to fabrication errors. Fig. 14 depicts the measured phase difference between the output ports. As can be seen, the measured phase difference around the design frequencies in both examples is in the range of ±10 degrees.
A comparison between the proposed dual-band WPDs with other state-of-the-art designs is given in Table 4. The adopted technique, frequency ratio, power division ratio, S-parameters and circuitry area are set as benchmarks. It is observed that the proposed method facilitates higher power and frequency division ratios, k 2 and R, respectively, as compared to the state-of-the-art techniques. While it is possible to achieve a frequency ratio > 4 as reported in [33], the use of reactive components imposes distortions and parasitic effects  at higher frequencies. The proposed method also facilitates compact designs in contrast to [27], [33], [34] due to adopting short-circuited stubs. Therefore, the proposed method entails the design and realization of dual-frequency power dividers with high power/frequency ratios and intertwines miniaturized circuitry, ease in fabrication and inherent good performance.

V. CONCLUSION
A design methodology of dual-band WPDs with high frequency and power split ratios are proposed. First, a theoretical investigation to derive analytical formulas of cascaded identical T-sections with short-circuited stubs to replace impractical transmission lines in the conventional power divider is performed. Then, with the use of transmission line modeling, an optimization framework is proposed to increase the frequency ratio with non-identical multi-Tsections. To demonstrate the proposed method, two 1:10 WPDs with frequency ratios of 2.7 and 4.4 are designed, simulated, and measured taking into account rigorous mathematical analysis, optimization, and full-wave simulations. The two prototypes are planar and do not require reactive components. Furthermore, the use of short-circuited stubs together with an optimization framework facilitates  higher power/frequency ratios, while occupying minimal physical area. Future efforts can be devoted toward designing multi frequency/power ratios with wide/controllable fractional bandwidths and exploring different optimization techniques to maximize the frequency ratio for a given power ratio in non-identical multi-T-section configurations.