Design of a compact microstrip balanced‐to‐balanced filtering power divider with real impedance‐transformation functionality

National Natural Science Foundation of China, Grant/Award Numbers: 61771247, 61571468; State Key Laboratory of Millimetre Waves, Grant/Award Number: K201921; International Science and Technology Cooperation Program of Jiangsu Province, Grant/Award Number: BZ2018027; University of Macau, Grant/Award Number: MYRG2017‐00007‐FST Abstract A microstrip balanced‐to‐balanced filtering power divider (FPD) design with impedance transformation via a coupled three‐line balun structure is proposed. The even/odd mode‐analysis method is used to construct a three‐port balun bandpass filter (BPF) with stub‐loaded resonators and a three‐line output structure after analysis and design to prescribe the filtering response of a proposed balanced‐to‐balanced FPD. To realize good isolation between the differential output ports, a complex impedance‐isolation network is loaded by means of coupled lines in the input‐feeding section. Theoretical formulas characterized by the coupling factors are provided with a specified source and load impedance ratio under the condition of perfect impedance matching at the centre frequency. According to the calculated S‐parameters, design rules are summarized to determine the overall circuit size according to desired bandwidth and return loss. For validation, two balanced‐to‐balanced FPDs centred at 2.4 GHz with 50‐to‐50 Ω matching and 50‐to‐100 Ω transformation are designed, fabricated and tested. Good agreement between the simulated and tested results of the designed balanced‐to‐balanced impedance‐transforming FPDs indicates not only high common‐mode rejection but also good port‐to‐port isolation and good matching at all ports.


| INTRODUCTION
For increased signal immunity to environmental noise, electromagnetic interference and crosstalk in wireless communication systems, most passive circuits such as bandpass filters (BPFs) [1,2], diplexers [3,4], couplers [5,6] and power dividers (PDs) [7][8][9][10][11][12][13][14][15][16] are designed with balanced input/output ports. In some balanced radio frequency communication systems, the trans-receiver modules usually require some of these components to be cascaded together-for example, the balanced BPF and PD function in the feeding networks of differential antenna arrays and the signal transmission blocks between antennas and low-noise amplifiers [10]. In addition, when designing antennas, impedance transformers are frequently integrated to achieve good impedance matching, as shown in Figure 1a. An effective way to achieve compact circuit design is by integrating these functions into one component as shown in Figure 1b, which leads to the design of a balanced-to-balanced impedance-transforming filtering power divider (FPD).
A balanced-to-balanced PD with bandpass filtering response was initially developed based on the Gysel PD in [17] using a ring structure and an extended coupled-resonator model design. Then, workers using a similar approach recently reported developing a compact balanced-to-balanced FPD comprising three λ g /2 open-circuited resonators and a short-stub-loaded resonator [18]. Both of the aforementioned designs have achieved narrow-band second-order Chebyshev bandpass response with no transmission zero (TZ) near the lower and upper cutoff frequencies of the desired passband. Additionally, by exploiting coupled-line structures and loaded stubs, planar balanced-tobalanced FPDs with wideband responses were designed in [19].

| BASIC THEORY OF SIX-PORT BALANCED-TO-BALANCED FILTERING POWER DIVIDER
As shown in Figure 1b, assuming that the designed circuit is symmetric with respect to PP 0 plane, even/odd mode-analysis method can be applied to derive the mixed-mode S-parameters of the six-port balanced-to-balanced circuit. Firstly, according to [25], the standard six-port S-parameters defined as [S std ] can be expressed by the two three-port S-matrices of the even/odd mode bisections [S e

] and [S o ] as
Then, based on the defined balanced ports of the six-port network shown in Figure 1b Next, with the definitions of balanced output ports in the three-port even/odd-mode bisected circuits, mixed-mode Sparameters can also be applied to these two circuits as Finally, by substituting [S e ] and [S o ] calculated by (4) into (1) and then simplifying (2) with the updated (1), the mixed-mode S-parameters of a symmetrical six-port balanced- It can be found from (6a) that the DM filtering performance of the balanced-to-balanced FPD is determined only by the odd-mode bisection, while DM isolation and port matching depend on both even-and odd-mode counterparts. Further, by comparing the four groups of S-parameters, one can conclude that for high performance of the balanced FPD, the odd-mode bisected circuit should perform well as a balun BPF. For the even-mode counterpart, total reflection should occur at the single-ended port, while good matching is required at the balanced port. In addition, to achieve high CM suppression and low cross-mode conversion, none of the CM and DM signals can be transmitted to the single-ended output port in this case. Accordingly, a balun BPF with the desired filtering response can first be designed and analysed to determine the odd-mode bisection of a balanced-tobalanced FPD without an isolation network. Then, the differential port-matching property of the even-mode bisection (S e ddBB ) should be analysed with a lossy isolation network to achieve a high level of isolation between the differential output ports.

| DESIGN OF BALUN BANDPASS FILTER BASED ON THREE-LINE BALUN STRUCTURE
According to the analysis in Section 2, the DM response of a balanced-to-balanced FPD can be decided solely by a balun BPF, that is, a three-port single-to-balanced BPF. Thus, in this section, a compact balun BPF with an odd-mode bisection boundary condition is proposed and analysed as shown in Figure 2. As can be seen herein, the balun BPF can be divided into three sections: input coupled-line, loaded stubs and threeline balun. Thereinto, the section of three-line balun primarily plays the role of obtaining a pair of equal amplitude and outof-phase signals, while the loaded stubs are mainly used to improve frequency selectivity and can be adjusted according to the requirements. Compared with the design proposed in [27], the occupied circuit area of the proposed structure is reduced by sharing resonators in two out-of-phase output paths. In the following, a detailed analysis is provided to explain the working principles and summarize the design rules.

| Coupled three-line balun structure
In the first step, a balun structure comprising a coupled threeline where two open-circuited λ g /2 lines are deployed to be symmetrically coupled with one λ g /4 line is analysed. Herein, λ g is the guided wavelength at centre frequency f 0 , and Z kmn (k = a, b; mn = ee, oo, oe) stands for the characteristic impedances under the three fundamental modes of even-even, odd-odd and odd-even in the coupled three-line. Meanwhile, if the voltage ratios of side lines to the centre line under the even-even and odd-odd modes are respectively denoted as R V1 and R V2 , there should be Z aee /Z bee = Z aoo /Z boo = −R V1 R V2 /2 [28]. According to the electric field distribution, if an unbalanced signal is input into note b, a pair of balanced signals can be received at Ports 2 and 2'. That is to say, when Ports 2 and 2' are excited by a pair of DM signals, a superposed signal can be obtained at Port 1. On the other hand, if the signal incoming into Ports 2 and 2' is a CM pair, cancellation of the two coupled signals will occur on the central line, thus achieving high CM suppression at the centre frequency. To verify the effectiveness of the proposed balun, theoretical S-parameters for this three-port network are derived. Firstly, assume that the propagation constants of the three fundamental modes are equal. Then, according to [28], the three-port Z-matrix of the proposed balun can be derived as where [T A ] = [T D ] = diag(1, 1, cosθ 1 ), [T B ] = diag(0, 0, jZ 1 sinθ 1 ), [T C ] = diag(0, 0, j/Z 1 sinθ 1 ) and [Z load ] is indicated near the end of this section. Then, the S-matrix of the threeport network can be calculated as [29]  ). To realize S 21 = −S 2'1 at the centre frequency, a necessary condition of Z 1 = P−S can be derived from (9). Thus, in the following analysis, this condition should be satisfied to ensure equal amplitude and out-of-phase at the centre frequency.

| Analysis of impedance transformation
After the principle of a three-line balun structure is clarified, an input coupled-line with a short-circuited end can be included to achieve an impedance-transforming structure where the loaded stubs are not considered at first. Herein, two input impedances, Z inb and Z inc , which are seen from planes P b and P c , respectively, are derived to find the matching conditions with the uncertain port impedances. According to the Z-matrix of three-line balun expressed in (7) and transmission line theory, input impedance Z inb can be derived as where Z bij stands for the (i, j) element of [Z balun ] in (7).
Similarly, we can obtain the expression of Z inc as where Z cij represents the (i, j) element of coupled-line Z-matrix Then, to find the constraint condition of impedance matching at the centre frequency for the case without stubs, the equation of Z inc = Z inb * is solved with θ = θ c = 90°at f 0 . Afterwards, the following relationship can be found as (13) with Z 1 = P−S: This means that for a prescribed impedance-transforming ratio, R IT = Z 02 /Z 01 , the required input coupling strength k c can be calculated by the mode impedance parameters of the coupled three-line structure according to (13). Furthermore, higher impedance transformation can be achieved by increasing the coupling strength of the input coupled-line while keeping the parameters of the output structure unchanged.
Next, to enlarge the bandwidth and improve selectivity, loaded stubs with input admittance of Y ins are considered in the impedance-transforming network, and the input impedance seen from plane P c to differential Port B is derived as shown in (14): To keep perfect impedance matching unchanged at the centre frequency f 0 , Y ins should be equal to zero at f 0 according to Z insb = Z inc * = Z inb . Certainly, the loaded stubs can be constructed by different combinational forms with open-and short-circuited stubs. Herein, to achieve a three-pole filtering response with a compact structure, open-and short-circuited stubs are simultaneously adopted as a design example. Then, according to Y ins = j(tanθ 2 −cotθ 3 )/Z 2 , the electrical lengths of the two loaded-stubs should satisfy following condition: Further, with a given value of θ 2 , another two transmission poles can be found according to the property of conjugate match, Z insb = Z inc * , and the bandwidth can be enlarged by decreasing θ 2 . In addition, according to Y ins = ∞, a TZ can be obtained with which can be used to determine the value of θ 2 .

| Analysis of theoretical response
Based on foregoing analysis, the theoretical standard S-parameters of entire three-port network shown in Figure 2 is derived from the Z-matrix as where [Z o ] is Z-matrix of the three-port network deduced by the same method as (7). Then, with given impedance parameters and electrical lengths, theoretical responses can be predicted and plotted.
To verify Equations (13) and (15), a typical example of a balun BPF is displayed with theoretical response presented by mixed-mode S-parameters, as shown in Figure 3. In this example, the impedance parameters of the coupled three-line, that is, P, Q, R and S shown in (8), are extracted from an equalwidth three-line structure according to the Z-matrix of the three-line coupler in [28], and k c is calculated by (13). Initially, the responses of single-pole balun without loaded stubs are validated with three impedance-transforming ratios as shown in Figure 3a, where Z 02 :Z 01 respectively equals 1:2, 1:1 and 2:1. In these cases, the required values of k c are respectively 0.3, 0.42 and 0.59, and the dimensions of coupled three-lines are the same. As can be seen, perfect impedance matchings at the centre frequency are achieved under all three cases, and corresponding CM signals are also fully suppressed. Meanwhile, by increasing the coupling strength of input coupled-line, bandwidth is enlarged.
Then, the theoretical S-parameters of balun BPF considering the two loaded stubs are investigated as depicted in Figure 3b. Electrical length θ 3 is primarily determined by (15), and the value of θ 2 is obtained with (16). In Figure 3b, the navy lines with triangle symbols represent the initial results with θ 2 = 81.7°and θ 3 = 8.3°. It can be seen that perfect impedance matching has been achieved at the centre frequency, while the ripples in the passband are not equal due to the difference between the two loaded stubs. Thus, to achieve an equal-ripple response, a slight adjustment between θ 2 and θ 3 is required and is achieved by eliminating the restriction of (15) as shown in Figure 3b. As expected, equal ripple in the passband of the response has been easily achieved in this way.
Further, to provide guidance for both theoretical design and physical implementation, a set of examples are analysed based on the mapping relationship between physical sizes and characteristic parameters. Herein, the coupled lines are selected as the equal-width type, and the chosen substrate is Rogers RO4003C with a relative permittivity of ε r = 3.55, thickness of h = 0.508 mm and loss tangent of tanδ = 0.0027. The conductor is assumed as pure copper with a thickness of 35 μm. Then, four design graphs are extracted as shown in Figure 4, where Figure 4a,b are obtained from the Z-matrix of the three-line coupler at the half centre frequency [28], and Figure 4d is calculated from the extracted values shown in Figure 4a,b. Figure 4c is plotted with the data extracted from the transmission line calculator. Then, according to (13), dimensions of the coupled three-line can be determined from the input coupled-line based on Figure 4c,d with the same impedance value in the vertical axis.
Next, to investigate the realizable bandwidth of the balun BPF, five examples with different input coupled-line dimensions are studied with 20 dB return loss as shown in Table 1. As indicated, under the condition of (13), the bandwidths are mainly affected by the coupling gap of input coupled-line while the line width just has a little effect. Moreover, the wider the coupling gap, the smaller the bandwidth. With these design rules, a triple-mode balun BPF can be easily implemented with desired specifications.

| Implementation
For demonstration, three design examples with impedancetransforming ratios (Z 02 :Z 01 ) of 1:1, 1:2 and 2:1 have been designed and simulated based on the rules shown in Figure 4 and Table 1. As the specifications, all three circuits are centred at 2.4 GHz with 20 dB return loss and 11% of fractional bandwidth (FBW). Herein, design parameters of the balun BPF with an impedance ratio of 1:1 can be obtained from Table 1  Abbreviation: FBW, fractional bandwidth. Figure 4b, the dimensions of the coupled three-line can be found. Finally, by analysing the theoretical response with Equations (15) and (16), all design parameters are obtained as shown in Table 2; the theoretical mixed-mode S-parameters of the three design examples are displayed in Figure 5a.  Figure 6 and Table 3. Simulated mixed-mode Sparameters are plotted in Figure 5b. As can be seen, the simulated FBW of 20 dB return loss at Port 1 is 12% and 11.9% with a centre frequency of 2.41 and 2.43 GHz, respectively corresponding to Z 02 :Z 01 = 1:1 (Z 02 = 50 Ω) and 2:1 (Z 02 = 100 Ω). The insertion losses of the differential-to-single signal conversion channel (S sd1B ) are 0.72 and 0.78 dB at centre frequency with common-to-single signal conversion of less than 30 and 31 dB over the entire operation band, respectively.

| Configuration of proposed six-port component
According to the basic theory of balanced-to-balanced FPD displayed in Section 2, the proposed three-port balun BPF can be served as the odd-mode bisection of a six-port balanced-to-balanced FPD to be constructed, and good balanced performance of the balun BPF can contribute to high CM suppression of the FPD. Then, to realize good isolation between two pairs of differential output ports, a grounded complex impedance network with Z iso = R + jX 1 can be loaded on the symmetrical plane by means of a coupled-line section as shown in Figure 7a. Herein, the adopted coupled-line should mainly affect response under even-mode excitation, while its influence on the odd-mode counterpart should be negligible. Thus, the length of additional coupled-line is approximately selected as the 10th wavelength, and the corresponding coupled section of the input line must be adjusted slightly. By appropriately adjusting the dimensions of the odd-mode bisected circuit shown in Figure 7b, the modified part can be determined as S i = S ci = 0.2 mm, W i = 0.4 mm and L i = 6 mm. In addition, it can be seen that between Ports 1 and 1 0 , there is a grounded lumped inductance loaded at the point on the symmetrical plane so as to improve the suppression of TA B L E 2 Design parameters of balun BPF with impedance transformation (Z 01 = 50 Ω, Z 2 = 85 Ω, Z 1 = P−S)   -487 cross-mode conversion (S cd12 ) and the CM rejection when a pair of CM signals is input into Port A. Detailed analysis is provided in the following section.

| Analysis of even-mode equivalent circuit
As shown in Figure 7b, because the distributed parameters have been determined with an odd-mode circuit, the only undetermined parameters are the additional lumped elements. Obviously, the input coupled-line of the even-mode bisected circuit at Port 1 acts as a bandstop filter structure with no isolation network. However, owing to the coupling effect of the three-line coupled structure, the stopband frequency is shifted, which may degrade the performance in cross-mode suppression (S cdAB ). Thus, additional inductance (jX 2 ) is introduced; the analysis is provided in Figure 8. Compared with case (b), an additional parameter is introduced in case (c) to change the location of the original stopband TZ generated by the input coupled-line section. It is further found that the isolation impedance, Z iso , has little effect on the frequency location of the stopband. Thus, the initial inductance value of X 2 can be firstly determined by tuning the structure of case (c) without loading 2Z iso . Based on the dimensions of the oddmode structure, the initial inductance value can be roughly chosen as L X2 = 10 nH. To achieve high isolation between the differential output ports, the value of Z iso = R + jX 1 should be properly selected.
According to (6a), perfect isolation exists only in the case of S ddBBo = S ddBBe , but such a case is difficult to realize over the entire passband. Thus, an approximation is required to improve the isolation. Herein, the impedancematching property of differential Port B for the even-mode circuit is considered because the corresponding port of the odd-mode circuit has already been well matched in impedance. Assume that the cross-mode is fully suppressed, which means that the input DM signal at Port B is either reflected TA B L E 3 Dimensions (mm) of two balun BPFs (see also Figure 6 for a graphical representation)  For the case where differential Port B is well matched and Port 1 is fully reflected, the input impedance (Z iniso ) seen from the isolation network should be conjugate-matched with Z iso for a lossless and reciprocal network. Thus, the initial value of Z iso can be obtained as its real and imaginary parts, as shown in Figure 9, by virtue of the input impedance Z iniso extracted by connecting Ports 2 and 2 0 to Z 02 . Based on the condition of conjugate matching, R and X 1 can be calculated by setting 2R = Z iniso_Re and 2X = −Z iniso_Im . Herein, the impedance matching at centre frequency is considered. Accordingly, the initial resistance and reactance are chosen as R = 23.5 Ω and X 1 = 67.5 Ω for Z 02 = 50 Ω, while R = 20.5 Ω and X 1 = 69 Ω for Z 02 = 100 Ω, respectively. Because the values of reactance X 1 are both positive, an inductive element is adopted to meet the requirements. Further, the inductance values can be obtained by L X1 = X 1 /(2πf 0 ) = 4.48 and 4.58 nH, respectively.

| Implementation of balanced-tobalanced filtering power divider with and without impedance transformation
Till now, all the initial dimensions and lumped parameters of the six-port device shown in Figure Figure 10. Their frequency responses are measured by an Agilent N5244A four-port network analyser. For testing purposes, 50 Ω feeding lines are cascaded to all measured ports of the FPDs. For the circuit with load impedances of Z 02 = 100 Ω, the well-known through-reflect-line (TRL) calibration technique is applied to remove the parasitic effects of SMA connectors and additional 50 Ω feeding lines at the six ports.
Herein, four calibration kits are fabricated according to the measured frequency band as shown in Figure 11, where Lines 1 and 2 are respectively used for the frequency bands 0.6-4.2 and 2.4-7 GHz. Furthermore, for convenience of the four-port measurement, three sets of balanced-to-balanced impedancetransforming FPDs are fabricated with one pair of the differential ports connected to matching loads as shown in Figure 10b.
Finally, both the simulated and tested results of the devices with Z 02 = 50 Ω and 100 Ω are displayed in Figures 12 and 13, respectively. As can be observed in Figures 12a and 13a in terms of DM filtering performance, the measured centre frequencies are respectively located at 2.36 GHz with 3 dB FBW of 18.6% and 2.41 GHz with 3 dB FBW of 17.4%. The minimum insertion losses are 2.7 and 2.6 dB with corresponding 17.3 dB and 16.6 fdB return losses. Measured DM output port isolation levels achieve up to 24.9 and 26.5 dB over an ultrawide frequency band. For the CM performance shown in Figures 12b  and 13b, the realized rejection levels attain 48.9 and 46.3 dB for S ccBA and 42 and 34 dB for S ccBC , respectively. In addition, high rejection levels of the cross-mode conversion (S dc ) are obtained as shown in Figures 12c and 13c. As shown in (6c), the two cross-mode conversion matrices ([S dc ] and [S cd ]) are the transpose. Thus, DM to CM conversion suppression (S cd ) for the two structures can also be read from Figures 12c and 13c. The maximum phase and amplitude differences between S ddBA and S ddCA indicated in Figures 12d and 13d are 6°and 0.3 dB for Z 02 = 50 Ω and 5°and 0.3 dB for Z 02 = 100 Ω. F I G U R E 9 Investigation on the input impedance seen from reference point p (a) Real part, (b) imaginary part performance but also impedance-transformation characteristics and a compact size.

| DESIGN EXTENSIONS
As mentioned earlier, the loaded stubs in the balun BPF can be replaced with other constructions for desired filtering performance. Herein, to improve the selectivity of the lower passband, an extended design of the balanced-to-balanced FPD with two TZs is proposed by loading an open-circuited stepped-impedance resonator (SIR), as shown in Figure 14. Herein, the input admittance of the loaded stubs can be expressed as (18) by assuming θ 2 = θ 3 = θ = 90°at f 0 : Obviously, Y ins = 0 when θ = 90°. Thus, according to the analysis in Section 3.2, the impedance matching at the centre frequency (Z insb = Z inc * = Z inb ) can be kept unchanged after the SIR stub is loaded. Thereby, the four design graphs shown in Figure 4 are still applicable in this extended design. In addition, according to the input impedance Z ins = 1/Y ins = 0, the locations of two TZs can be found at As can be seen, the two TZs are respectively located on either side of the centre frequency and can be tuned by controlling the impedance ratio of the loaded SIR. Therefore, after the dimensions of the input and output coupled lines are selected according to Figure 4, the bandwidth and return loss of the theoretical responses can both be tuned by changing the impedance ratio Z 2 /Z 3 .
Herein as an example, the theoretical filtering response of the balun BPF with two TZs is firstly designed with an FBW of 13.5% at a 20 dB return loss. In the design, the port impedances are set as 50 Ω, and the two impedance values Z 2 and Z 3 are respectively 11.7 and 127 Ω. The other parameters are the same as those indicated in Figure 3. With the initial size, the     Figure 15 depicts a comparison of the theory and simulated results, which shows a good agreement. Based on the design procedures, the proposed balanced-tobalanced FPD shown in Figure 14 is then realized and simulated, with the results shown in Figure 16 Figure 16a, the designed FPD achieved 13.8% FBW at a 20 dB return loss. The simulated isolation level between two pairs of balanced ports is higher than 24 dB with 18 dB output port matching. For CM suppression, the simulated results reached a level of 37 dB in the passband, as shown in Figure 16(b). Finally, the results of mode conversion as well as the phase and amplitude differences are depicted in Figures 16(c) and 16(d), respectively, and also show good performance by the extended design.

| CONCLUSIONS
This paper presents a balanced-to-balanced impedancetransforming FPD design based on a microstrip coupled three-line balun structure. The even/odd mode-analysis method is applied to synthesize the design procedure. By proposing a simple balun BPF with an impedance transformation analysis, the odd-mode equivalent circuit of a balanced-to-balanced FPD is formed and determined. Then, with the combination of two symmetrical balun BPFs via an isolation network, the proposed balanced-to-balanced FPD is constructed. For guidance, design graphs with determinations of both the physical sizes and the isolation network are displayed. Finally, two examples are implemented with fabrication and tested results, and one extended design is also discussed. Good results indicate that the proposed balanced FPDs are promising for application in some miniaturized balanced communication systems.