IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 9, SEPTEMBER 2012
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Design of Multiway Power Divider by Using Stepped-Impedance Transformers Yansheng Xu and Renato G. Bosisio, Fellow, IEEE
Abstract—In this paper, the design of multiway power dividers by interconnecting power dividers with fewer output ports is studied by transforming them into multisection stepped-impedance transformers. By using this approach, it is easy to design multiway power dividers with required equal ripples of input reflection ( ) within a wide passband. The interconnecting lines can be used as additional matching sections to improve the input matching. General properties of this type of multiway power dividers can be easily obtained. Design of multiway power divider with required input reflection level can be readily performed by using the published data and tables of stepped-impedance transformer in the literature. Both even- and odd-mode analyses are performed to obtain the optimal isolation resistor values. A prototype of a wideband four-way power divider with a rhombic architecture is designed, fabricated, and measured. Simulation and measurement results are in good agreement and validate the proposed approach. Index Terms—Broadband, Chebyshev polynomials, passive components, power combiner, power divider, power splitter, transmission lines, Wilkinson power divider.
this limitation, in most cases the optimal design cannot be obtained and the designed ripple values of the input reflection cannot be assigned in advance. In this paper, we transfer this problem into the design of stepped-impedance transformers and follow an optimization method based on the well-known Chebyshev polynomials presented in [2]–[6]. By using this approach, the general solutions for design of the multiway power divider can be obtained. The performances of the new power dividers are improved in comparison to the previous results in [9] and [10]. For simplicity, we first give a detailed study and calculation of the design of a four-way power divider by using evenand odd-mode analyses in Sections II– IV. An extension to the power dividers with more outputs (eight-way, 16-way, 32-way, and 64-way) is then made in Sections V and IV. Finally, a prototype of the wideband four-way power divider with a rhombic architecture is designed, fabricated, and measured. Simulation and measurement results are in good agreement and validate the proposed approach.
I. INTRODUCTION
P
OWER dividers are important components in microwave technology. A multiway power divider is a key component in phase-array antennas, power amplifiers, and six-port circuits. Wilkinson-type power dividers [1] are generally adopted, but it is planar only for two-way power division. Therefore, for an -way power divider, where is equal or larger than 3, it is generally realized by interconnecting two-way power dividers. In some cases, a multiway power divider is composed of interconnection of three-way or more-way power dividers to reduce the design complexity and difficulty. Although power dividers have been studied by many authors [2]–[13], the interconnection of power dividers with fewer ways of division into a multiway power divider has not been investigated in detail until recently. In [9] and [10], many calculations were made for the interconnection of two-way power dividers to achieve multiway power divider. However, the two-way power dividers used in the interconnections are restricted to the traditional design (see [10, Figs. 1 and 4]); they are all the same in different stages. Due to Manuscript received January 05, 2012; revised May 31, 2012; accepted June 06, 2012. Date of publication August 02, 2012; date of current version August 28, 2012. This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC). The authors are with the École Polytechnique de Montréal, Poly-Grames Research Centre, and the Centre de Recherche en Électronique Radiofréquence (CREER), Montreal, QC, Canada H3T 1J4 (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2012.2207911
II. DESIGN OF A FOUR-WAY POWER DIVIDER BY USING EVEN-MODE ANALYSIS A four-way power divider can be realized by interconnecting three two-way power dividers using the two-stage structure shown in Fig. 1. In Fig. 1, port 1 is the input port and ports 2–5 are the four output ports. The two-way one-section power divider of the first stage is composed of transmission lines , length , and also resistor with characteristic impedance . In turn, the two Wilkinson power dividers of stage 2 are composed of transmission lines with characteristic impedances . Two transmission lines , length , and also resistors and length are introduced with characteristic impedance to connect the power dividers of stages 1 and 2 [9], [10]. Following the generally accepted even-mode analysis [2]–[6], the even-mode equivalent circuit of the power divider in Fig. 1(a) can be expressed as in Fig. 1(b). Resistors and are removed in the even-mode equivalent circuit since they have no effect on the even-mode input and output signals. Following [2, Fig. 3] and due to the two outputs of the junction being connected in parallel, the impedances and resistors at the left-hand side of the junction are multiplied by a factor of 2 for each junction, as shown in Fig. 1(b). From Fig. 1(b), it is clear that we can use the design approach and design data of the stepped-impedance transformer as provided in the literature [2]–[6]. By this way, the design of this four-way power divider ) can be with needed reflection level of the input port ( readily obtained from the solution of the even-mode circuit shown in Fig. 1(b). The design procedure is as follows.
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Fig. 2. Reflection at the input port in Fig. 1.
Fig. 1. (a) Schematic diagram of the studied four-way power dividers by interconnecting three two-way power dividers. (b) Even-mode circuit of the power divider of Fig. 1(a).
1) Fix the number of total sections. Attention should be paid to the interconnecting lines counted as individual sections. From the literature [2]–[6], [9], [10], the length of the interconnecting lines and also the lines and should be of a quarter-wavelength at the center frequency of the power divider for optimum performances and compact design. If longer interconnecting lines are needed, multiple quarter-wavelength lines with different characteristic impedances should be used. In Fig. 1(b), only one single quarter-wavelength interconnecting line is shown. However, more quarter-wavelength sections of interconnecting lines can be included as necessary and they will be denoted as and . 2) From the number of sections , the impedance ratio of input and output ports (in the case of the four-way power divider ) and the ripple value (maximum allowed value) of input port reflection ( ), the even-mode circuit shown in Fig. 1(b) can be fixed. We can then obtain the impedance values of different sections and in Fig. 1(b) according to the data and tables in the literature [2]–[6]. Attention should be paid to that, across each junction of the two-way power divider, a ratio of 2 is multiplied to the impedances and resistors, as shown in Fig. 1(b). In the case of a three-way power divider, this ratio should be 3 and so forth. 3) Transfer the impedance values obtained in 2) back to the case of power dividers, namely, across each junction of the two-way power divider the impedance values should be divided by 2 [please compare the values of impedances in Fig. 1(a) and (b)].
of the four-way power divider shown
4) Make simulation of the circuits shown in Fig. 1(a) and (b) to validate the design. The simulated data of the circuits in Fig. 1(a) and (b) should be the same. Simulations of the above four-way power dividers are performed by using commercial software (ADS).1 The simulated results of the four-way power dividers obtained by the proposed approach and in [9] and [10] are provided in the following. A comparison of the performances of the input reflection of four-way power dividers obtained by using the traditional approach [9], [10] and the proposed approach is shown in Fig. 2. It is noted that, in the traditional approach, the impedances and are all equal to 1.4142 and the input reflection is higher or the bandwidth is narrower than that obtained using the approach proposed in this paper. It should be pointed out that the possible ripple values of in the results of [9] and [10] are quite limited, e.g., in the case of no interconnecting lines [ in Fig. 1(a)] no ripple is present, in the case of , the ripple of is fixed to 15 dB. To the contrary, by putting different sets of impedance values according to the design procedure of [2]–[6], different ripple values can be obtained for all these architectures (namely, , , or with more sections of connecting lines). In Fig. 2, the ripples of in the designs using our approaches are set to be 20 dB and the values of the impedances are as follows. In the case of one quarter-wavelength interconnecting line, , and . In the case of no interconnecting line, , . Of course, other levels of ripple value of can easily be obtained by using our approaches and the design data in [2]–[6]. The lengths of and are one-quarter of a wavelength at the center frequency. 1Trade
name of Agilent Technologies Inc., Santa Clara, CA.
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TABLE I EXCITATIONS OF THE EVEN- AND ODD-MODE ANALYSIS
Fig. 3. Equivalent circuits of the four-way power divider of Fig. 1(a). (a) Equivalent circuit of the odd-even mode. (b) Equivalent circuit of the odd mode.
It is interesting to note that the design in [9] and [10] of a four-way power divider by interconnecting a three one-section two-way power divider with 90 interconnecting lines ( and ) appears with equal ripples and , which is in agreement with the optimum design criteria of [2]–[6]. Therefore, the bandwidth of the input reflection of this case is wide and for the other cases in [9] and [10] (e.g., or ), the bandwidth of are very narrow. The reflection of the output ports and and the isolation between the output ports and can only be obtained from the even- and odd-mode analysis, and they will be shown in Section III. III. DESIGN OF A FOUR-WAY POWER DIVIDER USING EVEN- AND ODD-MODE ANALYSIS In this section, the even- and odd-mode analysis is studied for the calculation of the matching of the output ports and the isolation between them. Due to the complexity of this problem, an analysis of a four-way power divider will be performed first. Some important conclusions and valuable design rules can be obtained from the analysis of this simple case and a multiway power divider can be analyzed following the same approach. From the even- and odd-mode analysis, the equivalent circuits of the different cases are shown in Fig. 3 and the excitations of these cases are given in Table I. It is noted that the equivalent circuit of the even excitation is shown in Fig. 1(b) and the reflection of port 2 of this case is set to be equal to even. The -parameters of the output ports are calculated by the following equations: even even and
even
(1)
It is important to note that due to the symmetrical property of the two-way power divider in the second stage of Fig. 1(a), in the odd-mode analysis, point “A” shown in this figure [and also in Fig. 3(b)] is connected to ground. Consequently, the equivalent circuit of even–odd and odd–odd modes take the form as shown in Fig. 3(b), and it is separated from the part of the circuit on the left-hand side of point A. In our design procedure, the value of even is fixed by the calculation of the even-mode equivalent circuit shown in the last section and it is equal to even in the lossless case and is very near to ( ) even for the low-loss case. The optimal value of the reflection and isolation values are then obtained by minimizing and . From Fig. 3, it is noted that there is only one resistor in each equivalent circuit and we should select the values of and to obtain and , respectively, at the center frequency . At the center frequency , the electrical length of and are all equal to 90 and the value of can readily be found: . The values of for the cases in the last section using our proposed approach are: in the case of one quarter-wavelength interconnecting line, , and in the case of no interconnecting line, . The simulated curves of reflection and isolation are shown in Fig. 4. In this figure, the values of resistors and used in the cases of the traditional approach [10] are all fixed to . It is determined that the values of and calculated by our approach are also near to , and we can simply put them equal to in our design to obtain similar performances of reflection and isolation of the output ports. From the property of symmetry, the other reflection and isolation values of the output ports: and can also easily be found by the above approach. It is interesting to note that since for most cases the optimal value of even is equal to zero at the center frequency and we select and to make the values of and also equal to zero at this frequency, the values of and will also vanish at the center frequency . When the value of even is not equal to zero at the center frequency, the absolute values of should be equal to a quarter of the absolute value of even (or 12 dB lower) at the center frequency since, at this frequency, and are all equal to zero. For the design of multiway power dividers, the selection of the isolation resistors can be made by using the same approach described above. When the multiway power divider is designed by interconnecting two-way power dividers with more sections (larger than one), the design of the isolation resistors can be performed with reference to the approach in [15]. In these cases, more zero points of the reflection and
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in our even-mode design since even also has contributions to the values of . 4) The values of are determined by the superposition of the reflection coefficients of even-, even–odd-, odd-, modes shown in Figs. 1(b) and 3. Therefore, it is important to reduce the amplitude of these reflections to diminish the residual values of and obtain better output reflection and isolation. IV. DESIGN OF A FOUR-WAY POWER DIVIDER WITH MORE SECTIONS OF INTERCONNECTING LINES
Fig. 4. Performances of the four-way power divider of Fig. 1(a). (a) for different cases studied. different cases studied. (b)
for
It is interesting to point out that the interconnecting lines can also be used as matching sections. To this end, the length of these interconnecting line sections should be all equal to one quarter-wavelength at the center frequency. Simulation results on the cases with two and three sections of interconnecting lines of the four-way power divider are shown in Fig. 5(a) and (b). In Fig. 5(a), two sections of interconnecting lines are used, and in Fig. 5(b), three sections are adopted. The impedance values of the case of Fig. 5(a) are as follows. For the case of two interconnecting lines, , , , and . For the case of three sections of interconnecting lines, , , , , and . The isolation resistors are calculated following the approach presented in Section III. The value of the isolation resistor of the second stage is equal to for both cases shown in Fig. 5(a) and (b). The value of the isolation resistor of the first stage is for the case with two sections of interconnecting lines and for the case with three sections of interconnecting lines. It is clear from Figs. 2–5(b) that there are more ripples of the input reflection , and consequently, the bandwidth of becomes much wider. At the same time, the behavior of and are not changed much in comparison to the previous cases shown in Fig. 4. V. DESIGN OF AN EIGHT-WAY POWER DIVIDER
isolation will be obtained within the bandwidth of operation since more isolation resistors exist in each stage. From the above analyses, the following is concluded. 1) The isolation resistor of the last stage has much more effects on the values of the reflection and isolation than other isolation resistors in the previous stages. The effects of the isolation resistor of the first stage are the least. 2) When there is only one section and one isolation resistor in the last stage of the power divider, the optimal value of this resistor is always . The isolation resistors of other stages can be calculated separately stage by stage. 3) If it is preferred to have the best performances of at the center frequency, we should make even equal to zero at this frequency
When the four output ports of the four-way power divider in Fig. 1(a) are connected to two-way dividers, respectively, the total number of the output ports is equal to eight and an eight-way divider is obtained. The simulation results of design of such eight-way power dividers are shown in Fig. 6. Four different cases are studied, they are: no interconnecting line and with one section of a quarter-wavelength interconnecting line between different stages (namely, line with impedance between stages 1 and 2 and line with impedance between stages 2 and 3) for the proposed approach and the traditional approach described in [9] and [10] respectively. Fig. 6(a) shows the performances of input reflection of these four cases. Their behavior is basically the same as before. Among them, the case with one section of interconnecting line of the proposed approach has the widest bandwidth and its bandwidth is even wider than the case of four-way power divider shown in
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Fig. 5. Performances of the four-way power divider of Fig. 1(a) with two and three sections of interconnecting lines by using the proposed approach. (a) Case with two sections of interconnecting lines. (b) Case with three sections of interconnecting lines.
Fig. 2. However, the no interconnecting line case of the traditional approach [9], [10] is also an optimized case (in the sense of ), and hence, its performance is also quite good. In Fig. 6(b), and (c), the performance of and are shown. As before, they are almost the same for the four cases considered here. The characteristic impedances of different sections of the designs by the proposed approach are as follows. For the case of no interconnecting lines, , , and , For the case of one section of interconnecting line, , , , , and . It should be noted that, in this case, we have the impedance ratio (three stages of interconnection of two-way power dividers). All matching transmission-line impedances of the different stages ( and ) for the approach of [9], [10] are equal to 1.4142 and the values of and and the
Fig. 6. Performances of the eight-way power divider by interconnecting of two-way power dividers using different approaches. (a) Reflection at the input of different cases studied. (b) for different cases studied. (c) port for different cases studied.
impedances of the interconnecting lines and obtained by using our approach are all different from each other, as shown above. The impedances of the interconnecting sections for the traditional approach of [9] and [10] are all equal to .
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For the traditional approach [9], [10], the values of the isolation resistors are all set to be equal to . For our proposed approach, the isolation resistors are calculated according to the procedure presented in Section III. The calculated values of the isolation resistor of the last stage (stage 3), , for both cases (no interconnecting line and one interconnecting line) are all equal to . The calculated values of the isolation resistor of the first stage are equal to 1.8739 for the no interconnecting line case and for the one interconnecting line case. As the isolation resistor in the middle stage , the calculated values are: 2.1345 for the no interconnecting line case and 2.0159 for the one interconnecting line case. VI. DESIGN OF MULTIWAY POWER DIVIDERS A multiway divider can be obtained by interconnecting twoway, three-way, or multiway dividers. The number of sections of the power divider and connecting line in different stages can also be different. However, the principle of design remains the same as before. Generally speaking, the number of sections of the composed multiway power divider can be calculated as
and
Fig. 7. Equivalent Chebyshev transformer for the 16-way power divider with one section of interconnecting line.
from the values of the Chebyshev transformer . In this procedure, attention should be paid to the values of in [14, Table X]. The values of are selected according to the required bandwidth of the input reflection of the design and are represented by in [14, Table X]. Calculated design data of 16-, 32-, and 64-way power dividers by using the proposed method and their comparison with the results obtained from the traditional method are given in the following. The frequency responses of input reflection of different design approaches are plotted in the same drawing for comparison. A. Design of 16-Way Power Divider
for all stages
(2)
where is the total number of sections of the resultant evenmode circuits, is the number of sections of the power divider of the th stage, is the number of sections of the interconnecting lines of the th stage, is the overall impedance ratio of the even-mode circuit, and is the number of outputs of the power dividers of the th stage. The design procedures of 16-, 32-, and 64-way power dividers are provided in this section as examples of multiway power design. When more-ways power dividers are designed, there will be a problem to get the section impedance values since, in the tables available of the literature, the impedance data are listed only when the number of sections is equal to or less than to , depending on the value of impedance ratio. To overcome this problem, the approach of [14] can be used. According to this approach, the values of section impedances for Chebyshev transformers are obtained from the known solutions of maximally flat transformers. At first, the values of ’s (the step voltage standing-wave ratios (VSWRs) of the maximally flat transformer) are obtained from [14, Table VII] and the values of are calculated. After that, the values of the Chebyshev transformer are calculated by multiplying them with the values of in [14, Table X]. Namely, (3) where stands for the th values of the Chebyshev transformer and stands for the th values of the maximally flat transformer with the same numbers of sections and impedance ratio. At last, the impedances of different sections of the Chebyshev transformer can be obtained
For 16-way power dividers obtained by interconnecting two-way power dividers, four stages are needed. The impedance ratio is equal to 16. For the one interconnecting line version, the calculated impedances of different sections of a Chebyshev transformer with impedance ratio and section number shown in Fig. 7 are , , , , , , and . The impedances of different stages of the designed 16-way power divider calculated from data of Fig. 7 [please refer to the schematic diagram of Fig. 1(a)] are , , , , , , and . The performance of the input reflection of the designed 16-way power divider is plotted in Fig. 8. For no interconnecting line version, there will be four sections only and the calculated impedances of different sections of a Chebyshev transformer with impedance ratio and section number are , , , and . The impedances of different stages of the designed 16-way power divider calculated from these data are , , , and . The performance of the input reflection of the designed 16-way power divider is plotted in Fig. 8. The frequency responses of the 16-way power dividers obtained by using the traditional approach [10] for both no interconnecting line and one interconnecting line cases are plotted on the same figure for comparison. From Fig. 8, it is clear that by using our proposed approach, the obtained bandwidth of the case of one interconnecting line is much wider ( to for 19.5 dB of ) than the traditional approach
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Fig. 9. Performances of the input reflection
Fig. 8. Performances of the input reflection
of the 16-way power dividers.
( to for 15.2 dB of ). As for the case of no interconnecting line, the two approaches obtain similar bandwidth ( – ) for the level of 19.1 dB of our approach in comparison to 14.6 dB of the traditional approach [10]. By carefully checking the impedance data, it is found out that, in this case, the design by the traditional approach is near to optimum, and hence, the improvement of our approach is not so significant as in the case of one interconnecting line. However, the obtained level of the traditional approach is fixed to 4.6 dB in contrast to, that by using our approach, any level can be achieved as required.
B. Design of 32-Way Power Divider For 32-way power dividers obtained by interconnecting two-way power dividers, five stages are needed for the no interconnecting line version. The impedance ratio is equal to 32. For this case, the calculated impedances of different sections of a Chebyshev transformer with impedance ratio and section number are , , , , and . The impedances of different stages of the designed 32-way power divider calculated from these data are , , , , and . The performance of the input reflection of the above designed 32-way power divider is plotted in Fig. 9. The curves of the same type of power divider obtained by using the traditional approach of [10] are also plotted in the same figure for comparison. From Fig. 9, it is clear that the design of our approach has much wider bandwidth ( to for 17-dB level) than the results from the traditional approach ( – for 15.2-dB level).
of the 32-way power dividers.
C. Design of 64-Way Power Divider For 64-way power dividers obtained by interconnecting two-way power dividers, six stages are needed for the no interconnecting line version. The impedance ratio is equal to 64. For this case, the calculated impedances of different sections of a Chebyshev transformer with impedance ratio and section number are , , , , , and . The impedances of different stages of the designed 64-way power divider calculated from these data are , , , , , and . The performance of the input reflection of the designed 64-way power divider is plotted in Fig. 10. The curves of the same type of power divider obtained by using the traditional approach of [10] are also plotted in the same figure for comparison. From Fig. 10, it is clear that the design of our approach has much wider bandwidth (from to for 20-dB level) than the results from the traditional approach ( – for 14.9-dB level). If lower level of input reflection is required, interconnecting lines can be added to obtain more sections of impedance transformation. The calculated impedances of a Chebyshev transformer with impedance ratio 64 and section number and for the 64-way power divider are as follows. For (from impedance to ), , , , , , , and , For , , , , , , , , and . By using the approach included in the previous sections, the various impedances values of the different line sections of the designed 64-way power divider can readily be obtained. The data obtained are as follows. For , , , , , , , and ,
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Fig. 10. Performances of the input reflection viders.
of the 64-way power di-
Fig. 12. Layout of the designed prototype four-way power divider.
power dividers contain many sections to obtain ultra-wide bandwidth of isolation between the neighbor output ports. On the contrary, very low impedance value will occur only where many sections of matching lines appear at the input port before (or near) the input junction. However, such special cases are very rare to occur and can easily be overcome and avoided. VII. PROTOTYPE DESIGN, FABRICATION, AND MEASURED RESULTS
Fig. 11. Simulated results of eight sections.
For
,
of the 64-way power divider with seven and
, ,
,
, ,
, ,
and . The simulated values of the designed 64-way power divider (section number ) are shown in Fig. 11. By comparing Fig. 11 with the data of the 64-way power divider in [10], it is clear that the proposed approach can provide much better performances. It should be pointed out that the minimum number of sections of a 64-way power is . Here we have and , and it means that there are one or two additional sections of the interconnecting lines. In the above calculation, it is assumed that the interconnecting line is set at the last (the sixth) stage for and both the first and sixth stages for . From the above data, it is clear that the line impedances required in the design of practical multiway power dividers are mostly between 0.8 to and can be realized without difficulty. It can be shown that very high impedance values are needed only in the cases where many sections of interconnecting lines are used near the output ports or the output stage
To validate the proposed approach, a prototype of a four-way power divider is designed and fabricated. The working frequency range of this prototype is 0.5–1.5 GHz. The substrate is made from RT Doroid 5880 with a dielectric constant of 2.2 and thickness 0.787 mm (31 mil). The width of the input and output microstrip lines is equal to 2.38 mm and the widths of the microstrip lines of the first and second sections are 1.58 and 1.18 mm, respectively. The length of the arms of the first and second sections is 55.8 mm. To reduce the size of the power divider, no interconnecting line is included and the total section number is . To facilitate the connection of the first and the second sections, a rhombic configuration is adopted. The drawing of the layout is shown in Fig. 12. Altogether three rhombi are used: one for the first section and the other two for the second. The angle between the two arms of the rhombi is selected to be 60 , namely, it is a 30 angle between the arm of the first (input) power divider and the -axis of Fig. 12. The two rhombi of the second section of the four-way power divider are rotated by 45 relative to the -axis (see Fig. 12). These angle values (30 and 45 ) can be varied a little as necessary. The isolation resistors and have a square shape and are made by a material of 100 per square. A photograph of the fabricated prototype is shown in Fig. 13. The fabricated prototype was measured on an Anritsu automatic network analyzer. The measured performances of the prototype and their comparisons with the simulated data using ADS are shown in Fig. 14. The prototype four-way power divider is with 15-dB ripple. Good agreements between the simulation and measured data are achieved, only some discrepancies are observed
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Fig. 15. Measured -parameters of the designed four-way power divider with ripple 15 dB.
Fig. 13. Photograph of the designed four-way power divider with 15 dB.
ripple
when the -parameters are below 25 dB. It is clear that a simple, compact, and wideband (it covers from 0.5 to 1.5 GHz, namely, 3:1 frequency ratio) is obtained. The dissipation loss of the prototype measured is around 0.1–0.2 dB. To test the asymmetry of fabrication, measured and are plotted in Fig. 15. From Fig. 15, it is clear that excellent fabrication symmetry is achieved. The discrepancies of the power division and are less than 0.35 dB from 0.5 to 1.5 GHz. The minimum value of and is 6.37 dB from 0.5 to 1.4 GHz. The total area of this prototype is 100 100 mm. The dimensions of this designed prototype can be scaled for designs to cover other frequency bands. VIII. DISCUSSION In contrary to the results of [9] and [10], the bandwidth of of our approach becomes wider as the number of division increases. This is easy to understand. The parameters of the individual power dividers in the proposed approach are different for power dividers in other stages, and hence, the optimal performances can be obtained. For the approach in [9] and [10], all two-way power dividers are the same in different stages, and hence, the nonoptimal factors in different stages accumulate and the resultant performances become worse and worse as the number of stages increase. At the same time, the problem of significant performance deterioration by interconnecting even-number-section two-way dividers as the number of output ports increases [10] is also removed by using the proposed approach since the results obtained are all optimal. IX. CONCLUSION
Fig. 14. Simulated and measured results of the designed four-way power diripple 15 dB. (a) Results of parameter and . (b) Revider with and . sults of
A study of improved multiway power dividers by interconnecting power dividers with fewer way of division has been presented. The proposed approach of the design has many advantages in comparison to the previous approach in [9] and [10]: flexible improved performances of the designed power dividers (lower or wider bandwidth), the design performance of can be predicated in advance, the interconnecting lines are used as matching sections, the bandwidth of increases with the number of connecting stages, and so on. Even-
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and odd-mode analyses have been performed and the optimal values of isolation resistors have been obtained. Design principles and examples are provided and simulated. Good performances are obtained. A prototype wideband four-way power divider with a rhombic architecture is designed, simulated, fabricated, and measured. Simulation results are in good agreement with the measurement. Wideband performances covering a 3:1 frequency range have been obtained by connecting three single-section two-way power dividers into a four-way power divider. ACKNOWLEDGMENT The authors are grateful to the staff of the Poly-Grames Research Centre, Montreal, QC, Canada, and the Centre de Recherche en Électronique Radiofréquence (CREER), Montreal, QC, Canada, for technical support. REFERENCES [1] E. J. Wilkinson, “An -way hybrid power divider,” IEEE Trans. Microw. Theory Tech., vol. MTT-8, no. 1, pp. 116–118, Jan. 1960. [2] S. B. Cohn, “A class of broadband three-port TEM mode hybrids,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 2, pp. 110–115, Feb. 1968. [3] H. Y. Yee, F.-C. Chang, and N. F. Audeh, “ -way TEM-mode broadband power dividers,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 10, pp. 682–688, Oct. 1970. [4] L. Young, “Tables for cascaded homogeneous quarter-wave transformers,” IRE Trans. Microw. Theory Tech., vol. MTT-7, no. 3, pp. 233–237, Apr. 1959. [5] L. Young, “Tables for cascaded homogeneous quarter-wave transformers (correction),” IRE Trans. Microw. Theory Tech., vol. MTT-8, no. 3, pp. 243–244, Mar. 1960. [6] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures. New York: McGraw-Hill, 1980. [7] S. W. Lee, C. S. Kim, K. S. Choi, J. S. Park, and D. Ahn, “A general design formula of multi-section power divider based on singly terminated filter design theory,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, vol. 2, pp. 1297–1300. [8] A. M. Abbosh, “A compact UWB three-way power divider,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 8, pp. 598–600, Aug. 2007. [9] J. Zhou and K. Morris, “Effects of interconnecting transmission lines on four-way Wilkinson power divider,” Microw. Opt. Technol. Lett., vol. 51, no. 12, pp. 2850–2852, 2009.
[10] J. Zhou, K. A. Morris, and M. J. Lancaster, “General design of multi-way multi-section power dividers by interconnecting two-way dividers,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 10, pp. 2208–2215, Oct. 2007. [11] I. Sakagami and T. Wuren, “Compact multi-way power dividers for dual-band, wideband and easy fabrication,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 489–492. [12] A. R. Barnes, M. T. Moore, M. B. Allenson, and R. G. Davis, “A compact 6 to 18 GHz power amplifier module with 10 W output power,” in IEEE MTT-S Int. Microw. Symp. Dig., 1999, pp. 959–962. [13] M. D. Abouzahra and K. C. Gupta, “Multi-way unequal power divider circuits using sector-shaped planar components,” in IEEE MTT-S Int. Microw. Symp. Dig., 1989, pp. 321–324. [14] L. Young, “Stepped-impedance transformers and filter protypes,” IRE Trans. Microw. Theory Tech., vol. MTT-10, no. 5, pp. 339–359, Sep. 1962. [15] S. B. Cohn, “A class of broadband three-port TEM-mode hybrids,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 2, pp. 110–116, Feb. 1968. Yansheng Xu graduated from Tsing Hua University, Beijing, China, in 1952. He received the Candidate of Technical Science degree from the Institute of Radio Physics and Electronics, Academy of Science, Moscow, Russia, in 1961. He then joined the Beijing Institute of Radio Measurements, where he was involved with radio communications and radar techniques. He then joined the Poly-Grames Research Centre, Department of Electrical Engineering, École Polytechnique de Montréal, Montréal, QC, Canada. His main research interests include microwave/millimeter-wave circuits and systems, microwave communications, microwave measurements, and microwave ferrite devices.
Renato G. Bosisio (M’79–F’95–LF’00) received the B.Sc. degree in mathematics and physics from McGill University, Montréal, QC, Canada, in 1951, and the M.Sc.A. degree in electrical engineering from the University of Florida, Gainesville, in 1963. In 1965, he became an Associate Professor with the École Polytechnique de Montréal, Montréal, QC, Canada, Head of the Electromagnetic and Microwave Group in 1971, Head and founder of the Advanced Microwave Research Group in 1990, and Emeritus Professor in 1995. He has authored or coauthored over 400 refereed papers. He holds 12 patents. His research interests involve microwave/millimeter-wave circuits and systems related to wireless localarea networks, automotive guidance systems, and point-to-point and satellite communication links.
